Exponent of Cross-sectional Dependence for Residuals∗

Natalia BaileyMonash University

George KapetaniosKing’s College London

M. Hashem PesaranUniversity of Southern California, and Trinity College, Cambridge

April 4, 2019

Abstract

In this paper, we focus on estimating the degree of cross-sectional dependence in theerror terms of a classical panel data regression model. For this purpose we propose anestimator of the exponent of cross-sectional dependence denoted by α, which is based onthe number of non-zero pair-wise cross correlations of these errors. We prove that ourestimator, α, is consistent and derive the rate at which α approaches its true value. Wealso propose a resampling procedure for the construction of confidence bounds around theestimator of α. We evaluate the finite sample properties of the proposed estimator by useof a Monte Carlo simulation study. The numerical results are encouraging and supportiveof the theoretical findings. Finally, we undertake an empirical investigation of α for theerrors of the CAPM model and its Fama-French extensions using 10-year rolling samplesfrom S&P 500 securities over the period Sept 1989 - May 2018.

Keywords: Pair-wise correlations, Cross-sectional dependence, Cross-sectional aver-ages, Weak and strong factor models, CAPM and Fama-French Factors.

JEL Codes: C21, C32

∗This paper is written in honour of the 125th anniversary of PC Mahalanobis, for his important contributionsto statistics and econometrics. We would like to acknowledge helpful comments from Alex Chudik. NataliaBailey and Hashem Pesaran acknowledge financial support under ESRC Grant ES/I031626/1.

1 Introduction

Interest in the analysis of cross-sectional dependence applied to households, firms, markets,

regional and national economies has become prominent over the past decade, especially so in

the aftermath of the latest financial crisis given its effects on the global economy. Researchers

in many fields have turned to network theory, spatial and factor models to obtain a better

understanding of the extent and nature of such cross dependencies. There are many issues

to be considered: how to test for the presence of cross-sectional dependence, how to measure

the degree of cross-sectional dependence, how to model cross-sectional dependence, and how

to carry out counterfactual exercises under alternative network formations or market inter-

connections. Many of these topics are the subject of ongoing research. In this paper we focus

on measuring cross-sectional dependence.

Bailey, Kapetanios and Pesaran (2016, BKP hereafter) give a thorough account of the ratio-

nale and motivation behind the need for determining the extent of cross-sectional dependence,

be it in finance, micro or macroeconomics. They focus on the asymptotic behaviour of the

variance of the cross section average of the observations on a double array of random variables,

say xit, indexed by i = 1, 2, . . . , N and t = 1, 2, . . . , T, over space and time. In particular, they

analyse the rate at which this variance tends to zero and show that it depends on the degree or

exponent of cross-sectional dependence which they denote by α. They explore a factor model

setting as a vehicle for characterising strong and semi-strong covariance structures as defined in

Chudik et al. (2011). They relate these to the degree of pervasiveness of factors in unobserved

factor models often used in the literature to model cross-sectional dependence.

In this paper we build on BKP and extend the analysis in two respects. First, we consider

a more generic setting which does not require a common factor representation and holds more

generally for both moderate to sizable cross-sectional dependence. We achieve this by directly

considering the significance of individual pair-wise correlations, and do not concern ourselves

with the factors that might underlie these pair-wise correlations. Second, we consider estimating

the exponent of cross-sectional dependence, α, of the residuals obtained from a panel data

regression model.

We propose a new estimator of α based on the number of statistically significant pair-wise

correlations of the residuals from the panel regression under consideration. To establish the

statistical significance of the correlation coefficients we adopt the thresholding multiple testing

(MT) estimator proposed by Bailey et al. (2018), BPS. Other thresholding estimators can also

be used. See, for example, Bickel and Levina (2008) or Karoui (2008) and Cai and Liu (2011) or

Fan et al. (2013). The MT testing procedure advanced by BPS has the advantage that it directly

considers the statistical significance of the correlation coefficient which is invariant to scales.

Other thresholding procedures focus on the sample covariances and resort to cross validation to

identify the threshold. Bickel and Levina (2008) use universal thresholding, namely comparing

all the sample covariances to the same threshold value, whilst Cai and Liu (2011) propose an

‘adaptive’ thresholding procedure that allows for differing thresholds across the different pairs

of sample covariances. Other contributions to this literature include the work of Huang et al.

(2006), Rothman et al. (2009), Cai and Zhou (2011) and Cai and Zhou (2012), Wang and

1

Zhou (2010), and Fan et al. (2011).1 All these contributions apply the thresholding procedure

to sample covariances and do not apply to the residuals from a panel regression model that

concerns us in this paper. It is also important to bear in mind that when estimating α we do

not assume that the underlying error covariance matrix is sparse, as is assumed in the literature

on regularization of the sample covariance. Our objective is to estimate the degree of sparsity

of the covariance matrix rather than assume sparsity for the purpose of consistent estimation

of the covariance matrix or its inverse. What matters for estimation of α is to ensure that all

non-zero entries of the correlation matrix are correctly identified.

We establish consistency of our estimator under the assumptions of exogeneity of regressors

and symmetry of the error distribution. We also explain how the derivations can be extended

to the case when weakly exogenous variables are present, as for example in a dynamic panel

data setting. The proposed estimator is simple to compute and is shown to perform well in

small samples, for a variety of correlation matrices, irrespective of whether the cross correlations

are generated from a multi-factor structure or specified by a given correlation matrix with a

specified degree of sparsity. This is especially the case as compared to basing the estimation of

α on the largest eigenvalue of the correlation matrix, which performs particularly poorly. The

rate of convergence of our preferred estimator is complex and depends on an interplay of the

cross-sectional and time dimensions, N and T . The Monte Carlo results also show that the error

in estimating α is smaller for values of α close to unity, which is likely to be of greater interest

in practice. The problem of making inference about the value of α raises additional technical

difficulties and will not be addressed in this paper. In practice, bootstrap techniques can be

used to obtain confidence bounds around our proposed estimator. We provide some Monte

Carlo results in support of estimating the empirical distribution of the proposed estimator of

α, using cross-sectional resampling as suggested in Kapetanios (2008). Finally, we provide an

empirical application investigating the degree of inter-linkages between financial variables using

the Standard & Poor’s 500 index. We present 10-year rolling estimates of α applied to excess

returns on securities included in the S&P 500 data set as well as α estimates applied to the

residuals obtained from the CAPM and its Fama-French extensions used extensively in the

finance literature.

The rest of the paper is organised as follows: Section 2 discusses alternative characterisa-

tions of α, the exponent of cross-sectional dependence, and the conditions under which these

measures are equivalent as N → ∞. Section 3 sets up the panel data model and discusses its

underlying assumptions. Section 4 proposes the estimator of α in terms of the number of statis-

tically significant non-zero pair-wise correlations of the residuals. Section 5 presents the main

theoretical results of the paper for a static panel data model with strictly exogenous regressors.

Extensions to dynamic panels or panels with weakly exogenous regressors are discussed in the

sub-section 5.2 while inference of α by use of bootstrap procedures is discussed in sub-section

5.3. Section 6 presents a detailed Monte Carlo simulation study. The empirical application

is discussed in Section 7. Finally, Section 8 concludes. Proofs of all theoretical results are

provided in the Appendix.

1Shrinkage procedures have also been proposed in the literature for regularization of covariance matrices.See, for example, Stein (1956), Ledoit and Wolf (2003) and Ledoit and Wolf (2004). However, the shrinkageprocedure does not set any elements of the covariance matrix to zero, and is not suitable for estimation of αwhich builds on the support recovery properties of the estimated covariance matrix.

2

2 Degrees of cross-sectional dependence: alternative mea-

sures

Our analysis focuses on the covariance matrix of εt = (ε1t, ε2t, . . . , εNt)′, where εt is the N × 1

vector of errors from a panel data regression model. Let ΣN = E (εtε′t) = (σij), and denote

its largest eigenvalue by λmax (ΣN) > 0. The errors εit are said to be strongly cross-sectionally

correlated, if λmax (ΣN) = (N), where denotes exact order of magnitude, and they are said

to be weakly cross-sectionally correlated, if λmax (ΣN) is bounded in N . All intermediate cases

can be parameterized in terms of the exponent αλ, such that

λmax (ΣN) = (Nαλ). (1)

The weak and strong cross dependence cases then relate to αλ = 0 and αλ = 1, respectively. It

is important to emphasise that the exponent, αλ, is an asymptotic concept, in the sense that

αλ can be identified only as N →∞, as the definition in (1) makes clear.

Suppose now that the cross dependence of εit is characterized by the following approximate

multiple-factor error process

εit = β′ift + uit, (2)

where ft is the m × 1 vector of unobserved common factors with zero means, and βi =

(βi1, βi2, . . . , βim)′ is the associated m × 1 vector of factor loadings, and uit is the idiosyn-

cratic component assumed to have mean zero and the covariance matrix V =E(utu′t), where

ut = (u1t, u2t, . . . , uNt)′. Then

ΣN = E (εtε′t) = BB′ + V,

where B = (β1,β2, . . . ,βN)′, and without loss of generality we have set E (ftf′t) = Im. To

identify the factor component from the idiosyncratic component we assume that λmax (V) =

O(1), but allow the factor loadings to satisfy the condition

B′B =N∑i=1

βiβ′i = (Nαβ), αβ > 0, (3)

where αβ measures the degree to which the factors are pervasive, in the sense that they have

non-zero effects on the individual errors, εit. In what follows we refer to αβ as the exponent of

factor loadings. In the standard approximate factor models it is assumed that αβ = 1, whilst

in practice, where the possibility of weak factors can not be ruled out, αβ could be a parameter

of interest to be estimated.

To see how αβ and αλ are related note that

λmax (ΣN) ≤ ‖BB′ + V‖1 ≤ ‖B‖1 ‖B‖∞ + ‖V‖1 ,

where ‖B‖1 and ‖B‖∞ are column and row norms of B, respectively. To ensure that V ar(εit)

is bounded we must have ‖B‖∞ < K. Also to ensure that λmax (V) = O(1), we must have

‖V‖1 < K. Therefore, the rate at which λmax (ΣN) can rise with N is controlled by

‖B‖1 = max1≤j≤N

N∑i=1

|βij| . (4)

3

Setting∑N

i=1 |βij| = (Nαβj ), for j = 1, 2, . . . ,m, then ‖B‖1 = (Nαβ), where αβ = maxj(αβj).

Then

λmax (ΣN) = (Nαβ) +O(1).

To distinguish the effects of the factor component from those of the idiosyncractic component

we must have αβ > 0. Comparing this result with (1) establishes that αλ = αβ > 0, as N →∞.

The above analysis suggests two alternative ways of estimating αλ. A direct procedure

would be to base the estimate of αλ on λmax (ΣN) and set

λmax (ΣN) = O (Nαλ) = κNαλ ,

where κ is a constant independent of N . Then

αλ =ln [λmax (ΣN)]

ln (N)− ln (κ)

ln (N). (5)

In order to identify αλ, as N →∞, we set κ = 1, so that (5) becomes

αλ =ln [λmax (ΣN)]

ln (N). (6)

In this form, the value of αλ is susceptible to the scaling of elements in εt. For this reason

we focus our attention rather on the corresponding correlation matrix RN = (ρij) given by

RN = D−1/2N ΣND

−1/2N ,

where

DN = diag(σii, i = 1, 2, . . . , N). (7)

Hence, (6) finally becomes

αλ =ln [λmax (RN)]

ln (N), (8)

and αλ has fixed bounds at zero and unity, as N →∞.

Developing a theory based on the maximum eigenvalue of the correlation matrix RN can

be challenging. To avoid some of the technical problems involved in estimating λmax (RN),

and noting that V ar(εt) ≤ N−1λmax (ΣN) , BKP propose basing the estimation of αλ on

V ar(εt), where εt = N−1∑N

i=1 εit. In the case where εit has a factor representation, BKP

show that V ar(εt) = O[max

(N2(αβ−1), N−1

)], which reduces to Std(εt) = O

(N(αβ−1)

), if

2 (αβ − 1) > −1, or if αβ > 1/2. This means that at least N1/2 of the factor loadings must

have non-zero values for ΣN to differ sufficiently from a diagonal ΣN .

In this paper we consider an alternative estimation strategy that does not require εit to have

a factor representation. Since

λmax (RN) ≤ ‖RN‖1 = max1≤j≤N

N∑i=1

|ρij| ,

we focus directly on estimation of ρij and distinguish between values of ρij that are close to

zero and those that are significantly different from zero, and measure the exponent of cross-

sectional dependence in terms of the number of significant (non-zero) cross-correlation coef-

ficients. Specifically, we define α such that MN = N2α where M is the number of non-zero

4

elements of RN which can be written equivalently as MN = τ ′N∆NτN , where τN is an N × 1

vector of ones and ∆N = (δij) is an N × N matrix of population correlation indicators with

typical elements given by

δij = I (ρij 6= 0) , i, j = 1, 2, . . . , N,

in which I(A) is equal to unity if A is true and zero otherwise. Note that by construction

δii = 1. Hence,

α =ln (MN)

ln(N2)=

ln (τ ′N∆NτN)

lnN2. (9)

Cross-sectional independence refers to the case when RN = IN , and α = 1/2, while the case

of cross-sectional strong dependence corresponds to all pair-wise correlation coefficients being

non-zero such that α = 1. Note that by construction 1/2 ≤ α ≤ 1, with α = 1/2 arising when

∆N = IN , and α = 1 if ρij 6= 0 for all i and j.

Other exponents of cross-sectional dependence can be defined by focussing only on the off-

diagonal elements of RN and consider the following exponent of cross-sectional dependence:2

α =ln [τ ′N (∆N − IN) τN ]

lnN (N − 1),

assuming that ∆N 6= IN . Unlike α the above measure is not defined if RN = IN . The two

measures coincide, namely α = α = 1, if ρij 6= 0 for all i and j, as N →∞. In cases where εithave a multi-factor error representation given by (2), the largest exponent of the factor loadings

is given by αβ > 0. Assuming, for simplicity that V is diagonal, it then readily follows that

τ ′N (∆N − IN) τN = N2αβ − Nαβ , where (N2αβ −Nαβ) = Nαβ (Nαβ − 1) is the total number

of off-diagonal non-zero pair-wise cross correlations of the errors. In such a multi-factor error

set up we have

α =ln (τ ′N∆NτN)

lnN2=

ln (N2αβ −Nαβ +N)

lnN2,

and

α =ln [τ ′N (∆N − IN) τN ]

lnN (N − 1)=

ln (N2αβ −Nαβ)

lnN (N − 1).

Recall that we must have αβ > 1/2 for factors to be distinguishable from the idiosyncratic

components. It is then easily seen that limN→∞ α = limN→∞ α = αβ. However, the two

measures could differ if N is not sufficiently large. In finite samples α can be written in terms

of α by first solving the quadratic equation

N2αβ + (N −Nαβ) = N2α, (10)

2One can also consider only the distinct off-diagonal elements of RN and define α as

α =ln[

12τ′N (∆N − IN ) τN

]ln[

12N (N − 1)

] =ln [τ ′N (∆N − IN ) τN ]− ln(2)

ln [N (N − 1)]− ln(2)

=α− ln(2)

ln[N(N−1)]

1− ln(2)ln[N(N−1)]

→ α, as N →∞.

5

for αβ, namely

αβ =ln(

1 +√

1− 4 (N −N2α))− ln 2

lnN. (11)

Since [Nαβ ] can only take positive or zero values the second root of (10) is clearly redundant.

In what follows, we focus on α since it is defined even if RN = IN , and α is suitably scaled to

lie in the range (1/2, 1] .

3 Panel data model

Consider the panel data regression model

yit = γ′ixit + εit, for i = 1, 2, . . . , N ; t = 1, 2, . . . , T, (12)

where xit is a k× 1 vector of observed regressors, γi is the associated vector of coefficients, and

εit are the model’s errors. We are interested in estimating the exponent of the cross-sectional

dependence of the errors, εit, defined by (9). First, we obtain residuals eit computed as

eit = yit − x′itγi = yit − x′it(X′iXi)

−1X′iyi, (13)

where Xi is the T × k matrix of observations on the regressors for the ith unit, and yi is the

T × 1 vector of observations on the dependent variable of the ith unit. We assume that the

regressors are strictly exogenous.

We define the standardized errors, ξit, and the associated standardized residuals, zit, as

ξit =εit

(T−1ε′iMiεi)1/2, (14)

zit =eit

(T−1e′iMiei)1/2

=eit

(T−1e′iei)1/2, (15)

where εi = (εi1, εi2, . . . , εiT )′, ei = (ei1, ei2, . . . , eiT )′, and Mi = IT− Xi(X′iXi)

−1X′i. Note that

(since ei = Miεi)

zit =εit + x′it(γi − γi)(T−1ε′iMiεi)

1/2= ξit − a′itξi, (16)

where

ait = Xi (X′iXi)

−1xit,

and ξi = (ξi1, ξi2, . . . , ξiT )′.

Further, in what follows we assume that the error terms are symmetrically distributed.

Assumption 1 Conditional on Xi, the errors of the panel data model, (12), (a) εit are sym-

metrically distributed with zero means and variances 0 < c < σ2i < K <∞, (b) εit are serially

independent, (c) εit and εjt are distributed independently if E (εitεjt) = 0, for all i 6= j.

Under the above assumption, and using (14) it readily follows that

E (ξi |Xi ) = 0, (17)

6

and

E(ξiξ′j |Xi,Xj ) = ρijIT . (18)

Our main analysis will condition on the observed regressors. Remark 2 will discuss an

unconditional version of our results. For the observed regressors, we make the following as-

sumption:

Assumption 2 The k×1 vector of regressors xit in (12) is bounded: supi,t ‖xit‖ <∞. Further,

for some T0, we have

infT>T0,i

λmin

(X′iXi

T

)> 0.

Under Assumption 1 it readily follows that

E (a′itξi |Xi ) = x′it(X′iXi)

−1X′iE (ξi |Xi ) = 0,

where x′it is the tth row of Xi = (xi1,xi2, . . . ,xiT )′. Hence,

E (zit |Xi ) = 0, (19)

which in turn implies that zit is a martingale difference process with respect to the non-

decreasing information set, Ωit = (xi1,xi2, . . . ,xit). By construction ΩiT ≡ Xi. Indeed, since

Ωit is a subset of ΩiT , then by the chain rule of conditional expectations we have

E (zit |Ωit ) = E [E (zit |ΩiT ) |Ωit ] .

But in view of (19), E (zit |ΩiT ) = 0, and it also follows that E (zit |Ωit ) = 0, for all i and t.

We also require the following assumption that sets a lower bound condition on the non-zero

values of the pair-wise correlations.

Assumption 3 Let ρmin = infi,j (|ρij| |ρij 6= 0). Then,

limT→∞

ln(T )√Tρmin

= 0. (20)

Remark 1 This assumption is needed for successful recovery of non-zero pair-wise correlations,

and is weaker than requiring ρmin > 0, as it allows ρmin to tend to zero with N or T or both, so

long as its rate of decline is slower than ln(T )/√T .

4 Consistent estimation of α

Consider the sample estimate of the pair-wise correlation coefficients of the residuals from units

i and j,

ρij =T−1e′iej

(T−1e′iei)1/2 (T−1e′jej)1/2 , (21)

7

where ei = (ei1, ei2, . . . , eiT )′, eit is defined by (13), and by construction the sample mean of eitis exactly zero. We can re-write (21) equivalently as

ρij = T−1T∑t=1

zitzjt, (22)

where zit is defined by (15). In order to identify whether the pair-wise correlation coefficients

ρij are significantly different from zero we follow Bailey et al. (2018) and apply the multiple

testing estimator associated with ρij. This is defined by

ρij = ρijI

(|ρij| >

cp(n, δ)√T

), (23)

where

cp(n, δ) = Φ−1(

1− p/2

nδ

), n =

1

2N(N − 1), (24)

n is the number of tests carried out, p is the nominal size of the individual test, which can

be set to 1%, 5% or 10%, Φ−1(.) is the inverse of the standard normal distribution function,

and δ is a tuning parameter to be set a priori. This thresholding method is based on the

notion that for each unit (i, j) pairs we carry out a total of 12N(N − 1) individual tests of the

null hypothesis that ρij = 0 where j 6= i, i, j = 1, 2, . . . , N . Such tests can result in spurious

outcomes especially when N is larger than T . The critical value function, cp(n, δ), is therefore

adjusted using parameter δ to take account of the effects of the multiple testing procedure

for the estimation of α. It is important to bear in mind that the multiple testing problem

encountered here differs from the standard one studied in the literature by Bonferroni (1935),

Holm (1979) and others. Our focus here is on identifying the range of values for δ such that

α can be consistently estimated, rather than controlling the overall size of the multiple tests

being carried out.

Accordingly, we propose to estimate α by

α =ln(τ ′∆τ

)2 lnN

, (25)

where ∆ = (δij), with

δij =

ρij, if |ρij| > cp(n,δ)√

T, for i 6= j

1, for i = j0, otherwise

.

5 Theoretical Derivations

5.1 Main Results

To establish that α converges to α, in addition to Assumptions 1, 2 and 3, we also require the

following additional technical sub-exponential assumption:

8

Assumption 4 There exist sufficiently large positive constants C0, C1, s > 0, such that

supi,t Pr (|εit| > α) ≤ C0 exp (−C1αs) , for all α > 0. (26)

This assumption is used to allow a relatively simple bounding of an infinite sum of proba-

bilities, needed for the proof of Lemmas 2 and 3 in the Appendix. It can be relaxed to allow

for fatter tails, at the expense of smaller allowable values for N .

The rate of convergence of α to α is given in Theorem 1 below:

Theorem 1 Consider the panel data regression model (12) and suppose that Assumptions 1-4

hold. Let α and δ be defined by (25), (23), and (24). Then, conditional on the observed xit, as

N, T →∞, and if, for some d > 0, N = O(T d) = o (exp(T )) ,

2 (lnN) (α− α) = O(N2(1−α−κδ))+O

(N2(1−α) exp

(−C0T

C1))

+O(N−α

)+O

(N1−2α) =

(27)

O(T 2d(1−α−κδ))+O

(exp

[2d(1− α) ln(T )− C0T

C1])

+O(T−dα

)+O

(T d(1−2α)

)for any 0 < κ < 1, and some C0, C1 > 0.

As long as δ is set large enough (δ > 1−α), the first term on the RHS of (27) can be made

sufficiently small.

Remark 2 If we do not wish to condition on the observed xit, one could obtain the result

of Theorem 1, unconditionally, if it is assumed that the regressors satisfy the following sub-

exponential condition for some s > 0,

supi,t Pr (‖xit‖ > α) ≤ C0 exp (−C1αs) , for all α > 0, (28)

and if for some T0,(

X′iXi

T

)−1exists for all T > T0. Under these conditions on xit (which

replace Assumption 2), we can then use Lemma A6 of Chudik et al. (2018) to establish suitable

probability bounds on(

ξ′iXi

T

)(X′iXi

T

)−1 (X′iξiT

), and to show that, for some C0, C1 > 0, and

0 < π < 1,

supi,j

Pr(|∑T

t=1zitzjt| >√Tcp (n, δ)

)≤ sup

i,jPr(|∑T

t=1ξitξjt| > (1− π)√Tcp (n, δ)

)+exp

(−C0T

C1).

5.2 Extension to panels with weakly exogenous regressors

In the case of panels with lagged dependent variables, the use of OLS residuals for estimation of

α could still be justifiable so long as T is sufficiently large, such that the time series bias in the

estimated residuals is not too large. This is supported by the Monte Carlo evidence provided

for dynamic panels below.

However, the mathematical proofs provided above will not be applicable to the OLS residuals

if the panel regression model, (12), contains weakly exogenous regressors, such as lagged values

of yit. An alternative approach which avoids some of the technical issues associated with the use

9

of OLS residuals would be to base the estimation of α on the recursive residuals. Specifically,

one could consider the recursive residuals defined by

eit = yit − γ′i,t−1xit,

where

γi,t−1 =

(t−1∑τ=1

xiτx′iτ

)−1( t−1∑τ=1

xiτyiτ

).

Then the pair-wise correlations based on these recursive residuals are given by

ρij = (T − h)−1T∑t=h

zitzjt,

where

zit =eitσit,

and

σ2it =

1

t

t∑τ=1

e2iτ .

Here h is the size of the training period, which needs to be set by the researcher. It is then

easily seen that under cross-sectional independence, zitzjt is a martingale process with respect

to Ωιi,t−1, where Ωιi,t−1 = (yiτ , xiτ ; for τ = t−1, t−2, . . . , 1). This and other related results then

allow us to apply the mathematical analysis of the previous sections to the recursive residuals,

after suitable adjustments. The main open question is what critical value to use when checking

the significance of ρij, and hence the threshold value in the determination of the indicators δijdefined above. This issue will not be pursued in this paper.

5.3 Confidence intervals for α

Quantifying the uncertainty surrounding the proposed estimator of α is clearly of interest.

Given the complexity of developing asymptotic inferential theory, a fruitful avenue is to use

bootstrap procedures. In the case of panel datasets a number of important data features

matter. One possibility is to consider a parametric bootstrap where residuals are resampled.

Again there are a number of ways to construct such a bootstrap method. The first is to

resample, with replacement, from the rows of the residual matrix with the rows referring to

time periods. This procedure applies if the residuals are serially uncorrelated, although block

resampling methods can be considered to deal with the serial correlation. It is important to

resample whole rows as otherwise the nature of the cross sectional dependence of the original

sample, on which α depends, is not inherited by the bootstrap sample. Initial experimentation

suggests that coverage rates for such bootstrap methods are very low. An alternative is to

use some estimation method for large dimensional covariance matrices, such as thresholding,

to estimate the covariance matrix of the residuals and then implement a wild bootstrap. This

approach has been considered, for example, by Goncalves and Perron (2018). However, its

desirable properties depend on the true covariance being sparse and certainly sparser that the

10

structures we consider in this paper. Another alternative is to resample from columns (cross

section units) of the residual matrix, namely to resample across units keeping all the residuals

(observations) on a given together. This procedure is robust to the serial correlation problem.

This resampling procedure, initially proposed by Kapetanios (2008), will be used to obtain

bounds on the estimator of α in the Monte Carlo section below.

6 Monte Carlo Simulations

We investigate the small sample properties of our proposed estimator of α, defined by (25),

using a number of different simulation designs, allowing for dynamics as well as non-Gaussian

errors. We consider the following relatively general dynamic panel data model

yit = ai + ϑiyi,t−1 + γixit + εit, for i = 1, 2, . . . , N ; t = 2, 3, . . . , T, (29)

with exogenous, but serially correlated, regressors:

xit = ρixxi,t−1 +(1− ρ2ix

)1/2νit, for i = 1, 2, . . . , N ; t = −50,−49, . . . , 0, . . . , T,

where ρix ∼ IIDU (0, 0.95), and νit ∼ IIDN (0, 1) , with xi,−50 = νi,−50 for i = 1, 2, . . . , N. The

first 50 observations are disregarded for all units to minimize the effects of the initial values on

the observations used in the estimation.

We consider the following cases: (i) a static panel data model, where ϑi = 0, ai ∼IIDN (1, 1), and γi ∼ IIDN (1, 1) , for i = 1, 2, . . . , N ; and (ii) a dynamic panel data model

with exogenous regressors, where ϑi ∼ IIDU (0, 0.95), γi ∼ IIDN (1, 1) , and ai ∼ IIDN (1, 1),

for i = 1, 2, . . . , N .3 Our estimator is robust to possible correlations between the fixed effects,

αi, and the regressors, xit.

We consider two different designs for generating the errors, εit, both with the same exponent

of cross-sectional dependence, α :

Design 1 We draw N × 1 vector bN = (b1, b2, . . . , bN)′ as Uniform (0.7, 0.9) for the first

Nb (≤ N) elements, where Nb = [Nαβ ] and set the remaining elements to zero. Then, we

construct the correlation matrix RN given by

RN = IN + bNb′N −B2N , (30)

where BN = Diag (bN) is an N ×N diagonal matrix with its ith diagonal element given

by bi. The degree of sparseness of RN is determined by the choice of αβ. If αβ = 0 then

RN = IN and α = 1/2, while if αβ = 1 then all elements of RN will be non-zero and we

have α = 1. For all intermediate values of αβ, RN will have a total of [Nαβ (Nαβ − 1) +N ]

non-zero elements. The exact relationship between α and αβ is given by (11). Further,

we generate the variances of εit as σii ∼ IID 0.5 [1 + 0.5χ2(2)] , for i = 1, 2, . . . , N, and

set DN = Diag(σii, i = 1, 2, . . . , N). We now generate εit so that its correlation matrix is

3We also consider (iii) a dynamic panel data model with no exogenous regressors, where ϑi ∼ IIDU (0, 0.95)and γi = 0, for i = 1, 2, . . . , N . Simulation results for case (iii) are similar to those for cases (i) and (ii) and areavailable in the online supplement, Tables S1a-S1d.

11

equal to RN . To this end we first obtain matrix PN as the Cholesky factor of RN , and

then set WN = D1/2N PN = (wij), where wij is the (i, j) element of WN . We then generate

εit as

εit =N∑j=1

wijujt, i = 1, 2, . . . , N ; t = 1, 2, . . . , T, (31)

where ujt are IID draws from Gaussian or non-Gaussian distributions, to be specified

below.4

Design 2 The second design closely follows the set up in BKP and employs the two-factor

specification given by

εit = βi1f1t + βi2f2t + uit, for i = 1, 2, . . . , N ; t = 1, 2, . . . , T, (32)

where fjt ∼ IIDN(0, 1), j = 1, 2 and uit ∼ IIDN(0, 1) for i = 1, 2, . . . , N . With regard

to the factor loadings, we generate them as follows:

βi1 = vi1, for i = 1, 2, . . . , [Nαβ1 ] (33)

βi1 = 0, for i = [Nαβ1 ] + 1, [Nαβ1 ] + 2, . . . , N

βi2 = vi2, for i = 1, 2, . . . , [Nαβ2 ] ,

βi2 = 0, for i = [Nαβ2 ] + 1, [Nαβ2 ] + 1, . . . , N,

where βi2 are then randomised across i to achieve independence from βi1. The loadings

are generated as vij ∼ IIDU(µvj − 0.2, µvj + 0.2), for j = 1, 2. We examine the case

where αβ2 < αβ1 = αβ and consider values of αβ and αβ2 such that αβ2 =2αβ3

. We set

µv2 = 0.71 and µv1 =

√µ2v −N

2(αβ2−αβ)µ2v2

such that µ2v1

+ µ2v2

= µ2v = 0.75 - see BKP

for further details. Both µv1 and µv2 are chosen such that µvj 6= 0, j = 1, 2, without µ′vjs

being too distant from zero either. As before, the exact relationship between α and αβ is

given by (11).

In both designs, we examine two cases for the innovations ujt: (i) Gaussian, where ujt ∼IIDN(0, 1) for j = 1, 2, . . . , N ; (ii) non-Gaussian, where ujt follows a multivariate t-distribution

with v degrees of freedom. This is achieved by generating ujt as

ujt =

(v− 2

χ2v,t

)1/2

νjt, for j = 1, 2, . . . , N,

where νjt ∼ IIDN(0, 1) and χ2v,t is a chi-squared random variate with v = 8 degrees of freedom.

For the estimation of α, in each replication r = 1, 2, . . . , R, we first compute the OLS

residuals

e(r)it = yit − a(r)i − ϑ

(r)i yi,t−1 − γ(r)i xit, i = 1, 2, . . . , N ; t = 2, 3, . . . , T,

where a(r)i , ϑ

(r)i , and γ

(r)i are the OLS estimators of the regressions of yit on an intercept, yi,t−1

and xit, computed using the observations t = 2, 3, . . . , T, for each i. In the case of the static

4Note that εt = WNut, and V ar(εt) = WNW′N = D

1/2N PNP′ND

1/2N = D

1/2N RND

1/2N , as required.

12

regressions where ϑi = 0, the residuals are computed from regressions of yit on an intercept and

xit using the observations t = 1, 2, . . . , T . The sample covariance matrix, Σ(r)

N = (σ(r)ij ), is then

computed as

σ(r)ij = T−1

T∑t=2

e(r)it e

(r)jt , for i, j = 1, 2, . . . , N,

and diagonal elements σ(r)ii collected in D

(r)

N = Diag(σ(r)ii , i = 1, 2, . . . , N). The corresponding

sample correlation matrix is then given by

R(r)

N = D(r)−1/2N Σ

(r)

N D(r)−1/2N .

We evaluate the bias and root mean squared error (RMSE) of the exponent of cross-sectional

dependence α computed as in (25) with the critical value, cp(n, δ), given by (24). For p and

δ we consider the values p = 0.05, 0.10 and δ = 1/2, 1/3.5 Further, we compute the bias-

corrected version of the exponent of cross-sectional dependence estimator developed in BKP,

α, and compare its performance with that of α.6 However, it is important to bear in mind

that BKP provide theoretical justification for their estimator only in the case of demeaned

observations, namely xit − xi, and do not consider residuals from panel regressions as we do in

this paper. As a by-product, this paper also provides Monte Carlo evidence on the properties

of the estimator, α, when applied to residuals from panel regressions.

Finally, for the construction of confidence intervals for α, we propose the following bootstrap

procedure:

Bootstrap In each replication, r = 1, 2, . . . , R, we collect residuals in matrix E(r) =(e(r)it

)T×N

and proceed to resample (with replacement) from the columns a total number of B times.

More precisely, in each bootstrap b = 1, 2, . . . , B, the N cross section units are reshuf-

fled with replacement to generate a new matrix of residuals E(r),(b) =(e(r),(b)(i)t

)T×N

,

where (i) denotes the resampled index of the N units. For each b, we then compute

the correlation matrix R(r),(b)N corresponding to residuals E(r),(b) and estimate α, which

we denote by α(r),(b). The bootstrap estimates of α are collected in the vector α(r),B =(α(r),(1), α(r),(2), . . . , α(r),(B)

)′from where we obtain the estimates of α that correspond to

the 0.05 and 0.95 percentiles, and which we denote by α(r),B0.05 and α

(r),B0.95 , respectively. Fi-

nally, we evaluate the frequency at which α(r),B0.05 < α(r) < α

(r),B0.95 across the R replications

of each MC experiment.

5The value of δ = 1/4 was also considered. Results are in line with those for δ = 1/3 and are available inthe online supplement, Tables S2a-S2d.

6Recall that α corresponds to the most robust bias-adjusted estimator of the exponent of cross-sectionaldependence considered in Bailey et al. (2016) and allows for both serial correlation in the factors and weakcross-sectional dependence in the error terms. It is given by

α = α(µ2v

)= 1 +

1

2

ln(σ2x)

ln(N)−

ln(µ2v

)2 ln (N)

− cN2 [N ln(N)] σ2

x

, (34)

where σ2x, µ

2v and cN are consistent estimators of σ2

x, µ2v, and cN - see BKP for further details. We use four

principal components when estimating cN .

13

For all experiments we consider the values of α = 0.55, 0.60, . . . , 0.90, 0.95, 1.00, and the

sample sizes N, T = 100, 200, 500, and carry out all experiments with R = 2, 000 replications

and B = 500 bootstraps. The values of ai, ϑi and γi are drawn randomly in each replication.

6.1 Small sample results

First, we consider the small sample performance of our proposed estimator, α, and investigate

its robustness to different choices of p and δ, that govern the critical value, cp(n, δ), used in

estimating it. The results for Gaussian errors are provided in Tables A1a to A2b, and the

results for non-Gaussian errors are reported in Tables A3a to A4b. Each table reports bias and

RMSE of α computed using the residuals from either static or dynamic panel data regressions.

Tables A1a and A1b give the results for static and dynamic panels, respectively, when the

cross-sectional dependence in the errors are generated according to Design 1, whilst the same

results for Design 2 are summarized in Tables A2a and A2b. Similarly, the results in Tables

A3a and A3b give bias and RMSE of α for static and dynamic panels when the errors are

non-Gaussian and the cross correlations are generated according to Design 1, whilst the same

results for Design 2 are provided in Tables A4a and A4b. In each of these tables, the left panels

give bias and RMSE for p = 0.05, and the right panels for p = 0.10, whilst the top panels give

the results for δ = 1/2, and the bottom panels for δ = 1/3. Specifically, each Table gives four

sets of results for the combinations (p, δ), with p = 0.05, 0.10 and δ = 1/3, 1/2.

Comparing the left and right panels of the tables, it is clear that α is robust to the choice

of p, irrespective of the value of δ, and for all N and T combinations. Observing that n =

N (N − 1) /2 is quite large even for moderate values of N , the effective p-value of the underlying

individual tests is given by 2p/nδ, which is likely to be dominated by the choice of δ as compared

to p. Therefore, the test outcomes are more likely to be robust to the choice of p as compared

to δ.

Turning to the choice of δ, comparing the results reported in top and bottom panels of the

tables, we note that for all N and T combinations the choice of δ = 1/2 produces smaller bias

and RMSE as compared to δ = 1/3 for values of α close to 1/2 (α ≤ 0.75). The reverse is true

when considering values of α close to unity (α > 0.80). Again, this is consistent with the result

of Theorem 1 which requires δ to be larger than 1− α. Hence, for α → 1/2 setting δ = 1/2 is

more appropriate, while as α→ 1 values of δ below 1/2 are more appropriate. In cases where

there is no priori information regarding the range in which the true value of α might fall, the

simulation results suggest setting δ to its upper bound value of δ = 1/2.

Overall, irrespective of whether we consider static or dynamic panel regressions, with Gaus-

sian or non-Gaussian errors, the tabulated results show that the small sample performance of

α improves as the true exponent of cross-sectional dependence, α, rises from 0.55 towards 1.0,

uniformly over N and T combinations. This finding holds for both Designs, although α gen-

erally performs better when Design 1 is used to generate the error cross-sectional dependence.

Further, both bias and RMSE of α diminish as N rises for all values of T considered. These re-

sults are in line with our main theoretical findings as set out in Theorem 1. It is also interesting

to note that under Design 1, the bias and RMSE of α are particularly small for values of α in

the range of 0.9−1, even if we consider dynamic panels with non-Gaussian errors. For example,

14

for T = 100 and N = 500, p = 0.05, δ = 1/3, the bias and RMSE of estimating α = 0.95 by α,

in the case of dynamic panels with non-Gaussian errors are −0.00008 and 0.00067, respectively.

(see Table A3b).

Tables A2a and A2b summarize the results for static and dynamic panel data models,

respectively, when the error cross-sectional dependence is generated by Design 2 (the two-factor

structure). Compared with Tables A1a and A1b, both bias and RMSE are more sizeable across

the range of α when T = 100. However, as T increases, the performance of α improves for all

values of α, especially when α approaches unity, as to be expected. Perhaps, the signal-to-noise

ratio implied by (32) becomes somewhat distorted when the T dimension is short and adversely

affects the accuracy of the multiple testing procedure used to identify the non-zero elements of

the error correlation matrix, RN . To verify this conjecture, we repeated the same experiments

attaching a scaling parameter of ς =√

1/2 to uit in (32), in line with the simulation setup in

BKP. Performance of α is much improved in this case and comparable to those shown in Tables

A1a and A1b, even when T is small.7 Our conclusions regarding the robustness of α to the

choice of p and δ arrived at under Design 1 continue to hold for Design 2.

We now consider the small sample performance of α relative to that of α, the estimator of

α proposed in BKP. α is a biased-corrected estimator of α based on the standard deviation

of the cross-sectional average of the residuals. As noted earlier, the asymptotic properties of

α are established only for demeaned observations, but it is conjectured that these asymptotic

properties are likely to hold even if α is computed using residuals from panel regressions. For

comparison we consider bias and RMSE of α computed using p = 0.05 and δ = 1/2, and

note that similar results are obtained for other choices of p and δ. Table B1 compares the

resulting bias and RMSE of the two α estimators when applied to residuals obtained from a

static (top panel) and dynamic panel data models (bottom panel). These results refer to Design

1 with Gaussian errors. Both estimators perform well for all values of α, and irrespective of

whether the panel regressions are static or dynamic. This is particularly so for values of α >

0.8. In comparative terms, α outperforms α on average, for all values of α, and all N and

T combinations. The superior performance α is more pronounced when α ≤ 0.75 uniformly

over N and T . The results for Design 2 (where the cross-sectional dependence of the errors

are generated using a two-factor specification) are summarized in Table B2 which has the same

format as Table B1. In the case of these experiments, α (the estimator proposed by BKP)

performs better than α when T is small (T = 100), but the bias and RMSE of α becomes more

comparable to α as both N and T rise. As noted above, scaling uit by ς in (32) eliminates this

relative outperformance of α.8

Further, Table C displays bias and RMSE results for estimates of the exponent of cross-

sectional dependence, given by (8), and computed using the maximum eigenvalue of the corre-

lation matrices RN derived from the residuals from a static or dynamic panel data model with

Gaussian errors generated under Designs 1 or 2. It is clear that all eigenvalue based estimates

of α perform rather poorly even for large values of N and T , and even for values of α close to

1.

7These results are available in the online supplement - see Table S3.8Comparison of α and α when errors are non-Gaussian are provided in Tables S4a-S4b.of the online supple-

ment.

15

Turning to the problem of sampling uncertainty, we compute bootstrapped confidence inter-

vals for the estimate α in the case of Design 1 with Gaussian errors. The results are summarised

in Table D and give the average coverage rates (in percent) over the R simulations of α being

between the 5th and 95th percentiles of its empirical distribution. The related confidence in-

tervals are constructed by applying the resampling procedure with replacement to the residuals

obtained for the static and dynamic panel data models, with the results summarized in the

top and bottom panels of Table D, respectively. We set δ = 1/2 in the critical value function

cp(n, δ), and p = 0.05 (left panel) p = 0.10 (right panel), in each replication of the MC exper-

iments and in each bootstrap. For α = 0.60 coverage is low for small values of N , but rises

steadily towards unity as N increases and for all values of T considered. For 0.65 ≤ α ≤ 0.95

coverage stands universally at 100% but drops significantly when α = 1.00. Indeed, in this case

for small T coverage is relatively high and improves as N rises but drops toward zero when the

T dimension is increased for all values of N considered. This is to be expected since, as noted

earlier, the error in estimating α becomes negligible when α = 1.00.

Overall, we can conclude that using multiple testing for identifying non-zero elements of

RN when computing α is computationally attractive, has sound theoretical properties, with

comparable performance to the estimator of the exponent of cross-sectional dependence, α,

developed in BKP.

7 An Empirical Application: identifying the weak factor

component of CAPM

In their paper BKP investigate the extent to which excess returns on the Standard & Poor’s

500 (S&P 500) securities are interconnected through the market factor by computing rolling

estimates of α, the degree of cross-sectional dependence of S&P 500 securities. According to

asset pricing theories such as the capital asset pricing model (CAPM) of Sharpe (1964) and

Lintner (1965), and arbitrage pricing (APT) of Ross (1976), such estimates of α should be close

to unity at all times. This is because both CAPM and APT assume security returns have a

common factor representation with at least one strong common factor, with the idiosyncratic

component being weakly correlated - see also the approximate factor model due to Chamberlain

(1983). Such a factor structure implies that all individual stock returns are significantly affected

by the common factor(s) and in consequence they are all pair-wise correlated with varying

degrees.

The subsequent analysis in BKP reveals that a disconnect between some asset returns and

the market factor does occur particularly at times of stock market bubbles and crashes where

these asset returns could be driven by non-fundamentals. In this paper, we focus on the

exponent of cross-sectional dependence of the residuals obtained from different versions of the

CAPM model, and provide rolling estimates of the exponent of the cross-sectional dependence

of the errors from CAPM and related APT models. This is important since under CAPM, after

allowing for the market factor, the errors can not be cross-sectionally strongly correlated. It

is therefore of interest to see if this is in fact true at all times, or if there are episodes where

market factors are not sufficient to capture all the significant interdependencies that might exist

16

across the security returns.

We update the BKP analysis and consider monthly excess returns of the securities included

in the S&P 500 index over the period from September 1989 to May 2018. We obtain estimates

of α using rolling samples of 120 months (10 years) to capture possible time variations in the

degree of cross-sectional dependence.9 Since the composition of the S&P 500 index changes

over time, we compiled returns on all 500 securities at the end of each month and included

in our analysis only those securities that had at least 10 years of data in the month under

consideration. On average, we ended up with 442 securities at the end of each month for the

10-year rolling samples. The one-month US treasury bill rate was chosen as the risk-free rate

(rft), and excess returns were computed as rit = rit − rft, where rit is the monthly return on

the ith security in the sample inclusive of dividend payments (if any).10

First, following BKP, we estimated α for excess security returns to see the extent to which

securities in S&P 500 index are fully interconnected at all times. As noted above, under CAPM

we would expect estimates of α to be close to unity. To this end we used our proposed estimator,

α, defined by (25), and computed 10-year rolling estimates αt, for t = September 1989 to May

2018 - a total of 345 estimates - with p = 0.05. To check the robustness of the estimates to the

choice of δ we computed the rolling estimates for δ = 1/2 and 1/3.11 The resultant estimates

are shown in Figure 1.12 As expected, estimates of αt, for δ = 1/2, lie below those of αt, using

δ = 1/3, but the series track each other very closely. Also the quantitative differences between

the two estimates are not that large. Specifically, all the 345 rolling estimates αt (δ = 1/2) fall

in the interval 0.82 to 0.96, whilst the corresponding estimates αt (δ = 1/3) all lie in the range

0.86 − 0.97. These estimates show a high degree of inter-linkages across individual securities,

and are very close to unity at the start and at the end of the sample, with important departures

from unity in between. Considering αt (δ = 1/2), it recorded lows around 0.86 in 1998 before

recovering temporarily and falling further to 0.84. This episode coincides with the Russian and

the Long-Term Capital crises of 1998 which originated in bond markets but rapidly transmitted

through international equity markets. The α estimates remained low, falling to 0.82 − 0.83,

around the turn of the century which saw the burst of the Dotcom bubble and 9/11 terrorist

attacks in the US. During the less volatile period of 2003 − 2007, α rose slightly to about

0.85 before a new low of 0.82 was recorded in August 2008, around the time of the sub-prime

mortgage crisis in the US, and the ensuing global financial meltdown and economic recession.13

The estimates of α gradually increase to pre-1997 levels of 0.92 − 0.93 in 2011 and have since

remained in this range to the end of our sample, May 2018.

We now turn our attention to the exponent of cross-sectional correlations of the error terms

of the CAPM model, and two well known extensions using additional Fama and French factors.

9We also consider rolling samples of size 60 months (5 years). Results for this setting are shown in the onlinesupplement - Figures 3 and 4.

10For further details of data sources and definitions see Pesaran and Yamagata (2012).11For the remaining parameters in (25) we set p = 0.05 and n = Nt(Nt − 1)/2, where Nt is the number of

securities in a given 10-year rolling window (t = 1, 2, . . . , 345).12The same estimates including their 90% confidence intervals are shown in Figures 5 and 6 of the online

supplement, where in critical value cp(n, δ) we set δ = 1/2 and 1/3, respectively.13The measured increase in α estimates during 2003 − 2007 is partly attributed to the length of the rolling

windows being set to 10 years (120 months). Opting for 5-year rolling windows (60 months) produces morepronounced increments in α estimates which is expected - see Figure 3 in the online supplement.

17

Figure 1: 10-year rolling estimates of the exponent of cross-sectional correlation (αt) of S&P500 securities’ excess returns

0.75

0.80

0.85

0.90

0.95

1.00

Sep-

89

Sep-

91

Sep-

93

Sep-

95

Sep-

97

Sep-

99

Sep-

01

Sep-

03

Sep-

05

Sep-

07

Sep-

09

Sep-

11

Sep-

13

Sep-

15

Sep-

17

10-year rolling estimates of α ( with δ=1/2) based on excess returns10-year rolling estimates of α ( with δ=1/3) based on excess returns

αα

Specifically, the first regression is the usual CAPM one-factor representation given by

rit − rft = ai + βi (rmt − rft) + u1i,t, for i = 1, 2, . . . , N, (35)

where rmt is the market return computed as the value-weighed returns on all NYSE, AMEX,

and NASDAQ stocks. The second and third regressions assume the following extensions to (35)

proposed by Fama and French (2004):

rit − rft = ai + β1i (rmt − rft) + β2ismbt + u2i,t, for i = 1, 2, . . . , N (36)

and

rit − rft = ai + βi (rmt − rft) + β2ismbt + β3ihmlt + u3i,t, for i = 1, 2, . . . , N, (37)

where smbt stands for average return on the three small portfolios minus the average return

on the three big portfolios formed by size, while hmlt refers to the average return on securities

with high book value to market value ratio minus the average return of securities with low book

value to market value ratio.14

As noted previously, under CAPM we would expect the errors, u1i,t, to be cross-sectionally

weakly correlated, with αu1 to be close to 1/2. But this need not be the case in reality. In

fact the introduction of FF factors, smbt and hmlt, could be viewed as an attempt to ensure

cross-sectionally weakly correlated errors for the augmented CAPM model. It is therefore of

interest to consider the estimates of α for the errors, u1i,t, u2i,t, and u3i,t, and see if they are close

to 1/2 as required by the theory. To this end, we compute 10-year rolling estimates of α based

on the pair-wise correlations of the OLS residuals u1i,t, u2i,t and u3i,t in the panel regressions

14For further details of data sources and definitions see Pesaran and Yamagata (2012).

18

(35), (36) and (37), respectively. These estimates denoted by αujt, j = 1, 2, 3 for t = September

1989 to May 2018, are shown in Figure 2.15,16

Figure 2: 10-year rolling estimates of the exponent of cross-sectional correlation (αt) of residualsfrom CAPM and its two Fama-French extensions

0.60

0.62

0.64

0.66

0.68

0.70

0.72

0.74

0.76

0.78Se

p-89

Sep-

91

Sep-

93

Sep-

95

Sep-

97

Sep-

99

Sep-

01

Sep-

03

Sep-

05

Sep-

07

Sep-

09

Sep-

11

Sep-

13

Sep-

15

Sep-

17

10-year rolling estimates of α ( with δ=1/2) for the residuals of the CAPM model

10-year rolling estimates of α ( with δ=1/2) for the residuals of the CAPM model augmented by the SMB factor

10-year rolling estimates of α ( with δ=1/2) for the residuals of the CAPM model augmented by SMB and HML factors

α

α

α

Notes: CAPM model includes excess market returns, CAPM model augmented by SMBincludes excess market returns and small minus big (SMB) firm returns, and CAPM modelaugmented by SMB and HML includes excess market returns, small minus big (SMB) firmreturns and high minus low (HML) firm returns as regressors in (35), (36) and (37), respec-tively.

As expected, estimates of α based on the residuals are smaller compared to the estimates

obtained for the securities themselves (as depicted in Figure 1). It is also interesting that all

the three estimates αu1t, αu2t and αu3t are closely clustered over the two sub-periods September

1989 to September 1997, and February 2011 to May 2018, suggesting that the standard CAPM

model provides an adequate characterisation of the cross-sectional correlations of securities,

and the additional FF factors are not required in these sub-periods. It is also worth nothing

that, over these two sub-periods, estimates of α fall in the narrow range of 0.63-0.71 which are

sufficiently small and support CAPM as an adequate model for characterising cross-correlations

of S&P 500 security returns. In contrast, the estimates αu1t, αu2t and αu3t tend to diverge over

the period from October 1997 to January 2011, and more importantly they all start to rise

sharply, suggesting important departures from the basic CAPM model. Using only the market

15We set p = 0.05 and δ = 1/2 when estimating α for the residuals, since a priori we would expect the truevalue of α for the errors of CAPM models to be close to 1/2. See the discussion in Section 6.

16The same estimates including their 90% confidence intervals are shown in Figures 7, 8 and 9 of the onlinesupplement, respectively.

19

factor, as in (35), results in αu1t jumping to levels around 0.74 − 0.76. Adding smbt to (35)

reduces the α estimates of the resulting residuals to 0.69−0.73, suggesting that the size portfolio

does have some influence on individual security returns during this period. Adding the second

FF factor (as in (37)), further reduces the estimates of α to the range 0.66−0.68. These results

are also in line with the sharp drop in the estimates of α we reported for the excess returns

during the period 1998 − 2010, and provide further evidence in favour of the argument that

the presence of factors other than the market factor, namely smbt and hmlt, tend to become

relevant during periods of financial crises and turmoils.

8 Conclusions

Cross-sectional dependence and the extent to which it occurs in large multivariate data sets is

of great interest for a variety of economic, econometric and financial analyses. Such analyses

vary widely. Examples include the effects of idiosyncratic shocks on aggregate macroeconomic

variables, the extent to which financial risk can be diversified by investing in disparate assets

or asset classes and the performance of standard estimators such as principal components when

applied to data sets with unknown collinearity structures. A common characteristic of such

analyses is the need to quantify the degree of cross-sectional dependence, especially when it is

prevalent enough to materially affect the outcome of the analysis.

In this paper we generalize the work of Bailey et al. (2016) by proposing a method of

measuring the extent of inter-connections in the residuals of large panel data sets in terms of

a single parameter, α. We refer to this as the exponent of cross-sectional dependence of the

residuals. We show that this exponent can be used to characterize the degree of sparsity of

correlation matrices, or the prevalence of factors in multi-factor representations routinely used

in economic and financial analysis. We propose a simple consistent estimator of the cross-

sectional exponent and derive the rate at which it approaches its true value. We also propose

a resampling procedure for the construction of confidence bounds around the estimator of α.

A detailed Monte Carlo study suggests that the proposed estimator has desirable small

sample properties especially when α > 3/4. We apply our measure to the widely analysed

Standard & Poor’s 500 data set. We find that for individual securities in S&P 500 index, the

10-year rolling estimates of cross-sectional exponents are sufficiently close to unity over the two

sub-periods 1989− 1997 and 2011− 2018, but not during the intervening period 1998− 2010,

when markets have been subject to a number of financial turmoils, starting with the LTCM crisis

and the Dotcom bubble, and ending with the credit crunch of 2007− 2008. These results carry

over when we consider the cross-sectional dependence of errors from the CAPM model and its

multi-factor extensions using Fama-French factors. Estimates of α based on the residuals from

the CAPM model lend support to CAPM during the sub-periods 1989− 1997 and 2011− 2018,

but not when we consider the period 1998− 2010.

20

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22

Appendices

Appendix I: Statement of Lemmas

Lemma 1 Consider the panel data regression model (12) and suppose that Assumptions 1-4 hold. Then,

supi,j

Pr(|∑Tt=1zitzjt − E (zitzjt|Ωi,j,t) | >

√Tcp (n, δ)

)≤ sup

i,jPr(|∑Tt=1ξitξjt − E (ξisξjs|Ωi,j,t) | > (1− π)

√Tcp (n, δ)

)+ exp

(−C0T

C1),

for some C0, C1 > 0, where zit = eit

(T−1e′iMiei)1/2 .

Lemma 2 Let

W 0NT = N−2α

N∑i 6=j

I

(∣∣∣∣∣T−1T∑t=1

zitzjt

∣∣∣∣∣ > cp(n, δ)√T|ρij = 0

)where zit = eit

(T−1e′iMiei)1/2 . Under Assumptions 1-4,

E(W 0NT

)= O

(N2(1−α−κδ)

)+O

(N2(1−α) exp

(−C0T

C1)),

for any 0 < κ < 1, and some C0, C1 > 0, where δ is defined below (23).

Lemma 3 Let

W 1NT = N−2α

N∑i 6=j

I

(∣∣∣∣∣T−1T∑t=1

zitzjt

∣∣∣∣∣ ≤ cp(n, δ)√T|ρij 6= 0

)Under Assumptions 1-4, and as long as N = o (exp(T )), infi,j |E(ρij)| > 0,

E(W 1NT

)= O

(exp

(−C0T

C1)),

for some C0, C1 > 0, where δ is defined below (23) and zit = eit

(T−1e′iMiei)1/2 .

Appendix II: Proofs of Lemmas

Proof of Lemma 1

Recall ai,t = (ai,t1, . . . , ai,tT )′

= Xi(X′iXi)

−1xit. Then, using (15),

T∑t=1

zitzjt =

T∑t=1

(ξit +

T∑s=1

ai,tsξis

)(ξjt +

T∑s=1

aj,tsξjs

)=

T∑t=1

ξitξjt +

T∑t=1

ξit

(T∑s=1

aj,tsξjs

)+

T∑t=1

ξjt

(T∑s=1

ai,tsξis

)+

T∑t=1

(T∑s=1

ai,tsξis

)(T∑

s′=1

aj,ts′ξjs′

)

=

4∑i=1

Ai.

23

We focus on A4. A2 and A3 can be treated similarly. We have

T∑t=1

(T∑s=1

ai,tsξis

)(T∑

s′=1

aj,ts′ξjs′

)=

T∑s=1

T∑s′=1

ξisξjs′T∑t=1

ai,tsaj,ts′

T∑s=1

T∑s′=1

ξisξjs′ =

T∑s=1

ξisξjs +

T∑s=1

ξis

T∑s6=s′,s′=1

ξjs′

Further,

T∑s=1

T∑s′=1

ξisξjs′T∑t=1

ai,tsaj,ts′ =

T∑s=1

ξisξjs

(T∑t=1

ai,tsaj,ts′

)+

T∑s=1

ξis

T∑s 6=s′,s′=1

ξjs′

( T∑t=1

ai,tsaj,ts′

).

Define

aT,ij,ss′ = TT∑t=1

ai,tsaj,ts′ .

By Assumption 2, supT,i,j,s,s′ aT,ij,ss′ < K <∞. Further, define

ξ1,ijs = ξisξjsaT,ij,ss′

and

ξ2,ijs = ξis

1√T

T∑s6=s′,s′=1

ξjs′

aT,ij,ss′ .

Then,T∑s=1

ξisξjs

(T∑t=1

ai,tsaj,ts′

)+

T∑s=1

ξis

T∑s6=s′,s′=1

ξjs′

( T∑t=1

ai,tsaj,ts′

)=

1

T

T∑s=1

ξ1,ijs +1√T

T∑s=1

ξ2,ijs.

It can be easily seen that ξ1,ijs − E(ξ1,ijs|Ωi,j,t

)and ξ2,ijs − E

(ξ2,ijs|Ωi,j,t

)are martingale difference

series with finite variances, and that if ρij = 0 then E(ξ1,ijs|Ωi,j,t

)= E

(ξ2,ijs|Ωi,j,t

)= 0. Define cij,t =

E (zitzjt|Ωi,j,t). Then, using Lemma A3 of Chudik et al. (2018), it easily follows that

supi,j

Pr(|∑Tt=1zitzjt − cij,t| >

√Tcp (n, δ)

)≤ sup

i,jPr(|∑Tt=1ξitξjt − E (ξisξjs|Ωi,j,t) | > (1− π)

√Tcp (n, δ)

)+ exp

(−C0T

C1),

for some C0, C1 > 0.

Proof of Lemma 2

Note that W 0NT can be written as

W 0NT = N−2α

N∑i 6=j

I

(∣∣∣∣∣T−1T∑t=1

zitzjt

∣∣∣∣∣ > cp (n, δ)√T|ρij = 0

)

Since by assumption ξit and ξjt are distributed independently when ρij = 0, it also follows that

E (zitzjt |Ωi,j,t ) = E (ξitξjt |Ωi,j,t ) + E (ξitqjt |Ωi,j,t ) + E (ξjtqit |Ωi,j,t ) + E (qitqjt |Ωi,j,t )= 0,

24

and hence zitzjt,Ωi,j,t , is a zero mean martingale difference sequence, where Ωi,j,t = Ωi,t ∪ Ωj,t. Then, wenote that zitzjt is a normalised process since zit = eit

(T−1e′iei)1/2 . Then, noting Lemma 1, using Chudik et al.

(2018), and, in particular, their Lemmas A3 (which provides a martingale difference exponential probabilityinequality under (26)), A4 (which handles exponential probability tails for products of random variables) and

A9 (which handles the normalisation by(T−1e′iei

)1/2), we have for any 0 < π < 1, any bounded sequence,

dT > 0, and some C0, C1 > 0,

supi,j

Pr(|∑Tt=1zitzjt| >

√Tcp (n, δ) |ρij = 0

)≤ exp

[− (1− π)2cp (n, δ)

2

2(1 + dT )

]+ exp

(−C0T

C1). (38)

Note that (1−π)2

1+dT< 1, but can be made arbitrarily close to 1, and that, by Lemma A2 of Chudik et al. (2018),

exp[−bc2p (n, δ)

]= O

(n−2bδ

). Then, for any 0 < κ < 1, and some C0, C1 > 0,

supi.j

Pr(|∑Tt=1zitzjt| >

√Tcp (n, δ) |ρij = 0

)= O(n−κδ)+O

(exp

(−C0T

C1))

= O(N−2κδ)+O(exp

(−C0T

C1)).

Then,for some some C0, C1 > 0, any 0 < κ < 1, and if N = O(T d)

E(W 0NT ) = N−2α

N∑i 6=j

Pr

(∣∣∣∣∣T−1T∑t=1

zitzjt

∣∣∣∣∣ > cp (n, δ)√T|ρij = 0

)= O

(N2(1−α−κδ)

)+O

(N2(1−α) exp

(−C0T

C1))

=

O(T 2d(1−α−κδ)

)+O

(exp

[2d(1− α) ln(T )− C0T

C1]).

Proof of Lemma 3

We need to derive hN,T in

Pr

[∣∣∣∣∣T∑t=1

zitzjt

∣∣∣∣∣ ≤ √Tcp (n, δ) |ρij 6= 0

]= O (hN,T ) .

Let cij,t = E (zitzjt|Ωi,j,t), and note the inequality

Pr (|X +B| ≤ C) ≤ Pr (|X| > |B| − C) ,

where X is a random variable, B and C are constants, and |B| ≥ C > 0. Then, for some 0 < π < 1

Pr

[∣∣∣∣∣T∑t=1

(zitzjt − cij,t) + cij,t − E (cij,t) + E (cij,t)

∣∣∣∣∣ ≤ √Tcp (n, δ) |ρij 6= 0

]

≤ Pr

[∣∣∣∣∣T∑t=1

(zitzjt − cij,t) + cij,t − E (cij,t)

∣∣∣∣∣ >∣∣∣∣∣T∑t=1

E(cij,t)

∣∣∣∣∣−√Tcp (n, δ) |ρij 6= 0

]≤

≤ Pr

[∣∣∣∣∣T∑t=1

(zitzjt − cij,t)

∣∣∣∣∣ > (1− π)

[∣∣∣∣∣T∑t=1

E(cij,t)

∣∣∣∣∣−√Tcp (n, δ)

]|ρij 6= 0

]+ (39)

Pr

[∣∣∣∣∣T∑t=1

cij,t − E (cij,t)

∣∣∣∣∣ > π

[∣∣∣∣∣T∑t=1

E(cij,t)

∣∣∣∣∣−√Tcp (n, δ)

]|ρij 6= 0

](40)

But, by (28), (40) is bounded by exp(−C0T

C1)

for some C0, C1 > 0. We consider

Pr

[∣∣∣∣∣T∑t=1

(zitzjt − cij,t)

∣∣∣∣∣ > (1− π)

[∣∣∣∣∣T∑t=1

E(cij,t)

∣∣∣∣∣−√Tcp (n, δ)

]|ρij 6= 0

]

But, by (20) of Assumption 1, limN,T→∞

√Tcp(n,δ)∑Tt=1 E(cij,t)

= 0. Therefore, using again Lemma 1 and (38) of Lemma

2, we have

supij,

Pr

[∣∣∣∣∣T∑t=1

(zitzjt − cij,t)

∣∣∣∣∣ > (1− π)

[∣∣∣∣∣T∑t=1

E(cij,t)

∣∣∣∣∣−√Tcp (n, δ)

]|ρij 6= 0

]≤ exp (−CT ) ,

for some C > 0, proving the result.

25

Appendix III: Proof of Theorem 1

We prove that α converges to α under our assumptions of exogenous regressors and symmetrically distributederrors. We first note that

α =1

2

ln(

τ ′∆τN2α N

2α)

lnN=

1

2

ln(

τ ′∆τN2α

)lnN

+ α,

(lnN) (α− α) =1

2ln

(τ ′∆τ

N2α− 1 + 1

), (41)

and

2 (lnN) (α− α) =τ ′∆τ

N2α− 1 +O

[τ ′∆τN2α

− 1

]2 .

Further

N−2ατ ′∆τ = N−2αN∑i 6=j

I

(|ρij | >

cp(n, δ)√T

),

and

N−2ατ ′∆τ − 1 = N−2αN∑i6=j

I

(|ρij | >

cp(n, δ)√T|ρij = 0

)(42)

+

N−2αN∑i 6=j

I

(|ρij | >

cp(n, δ)√T|ρij 6= 0

)− 1

= W 0

NT +W 1NT +O

(N1−2α

).

We now have

N−2αN∑i 6=j

I

(|ρij | >

cp(n, δ)√T|ρij 6= 0

)− 1

= N−2α(N2α −Nα)− 1−N−2αN∑i 6=j

I

(|ρij | ≤

cp(N, δ)√T|ρij 6= 0

)

= −N−2αN∑i 6=j

I

(|ρij | ≤

cp(n, δ)√T|ρij 6= 0

)−N−α.

Hence,

N−2ατ ′∆τ − 1 = N−2αN∑i6=j

I

(|ρij | >

cp(n, δ)√T|ρij = 0

)−

N−2αN∑i 6=j

I

(|ρij | ≤

cp(n, δ)√T|ρij 6= 0

)−N−α,

which we write more compactly as

N−2ατ ′∆τ − 1 = W 0NT − W 1

NT +Op(N−α

),

where

W 0NT = N−2α

N∑i 6=j

I

(|ρij | >

cp(n, δ)√T|ρij = 0

),

W 1NT = N−2α

N∑i 6=j

I

(|ρij | ≤

cp(n, δ)√T|ρij 6= 0

).

26

It is worth noting that E∣∣W 0

NT

∣∣ = E(W 0NT

), and E

∣∣∣W 1NT

∣∣∣ = E(W 1NT

), and hence,

E∣∣∣N−2ατ ′∆τ − 1

∣∣∣ ≤ E (W 0NT

)+ E

(W 1NT

)+O

(N−α

)+O

(N1−2α

).

Lemmas 2-3 provide bounds for E(W 0NT

)and E

(W 1NT

)proving the result.

27

Tab

leA

1a:

Bia

san

dR

MS

E(×

100)

for

theα

esti

mat

eof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

oma

stati

cp

anel

dat

am

od

elC

ross

corr

elati

on

sare

gen

erate

du

sin

gD

esig

n1

wit

hG

au

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

c p(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

05c p

(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

10δ

=1/

2δ

=1/

2T

NB

ias

TN

Bia

s10

010

00.

204

-0.6

09-0

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-0.3

55-0

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01-0

.259

-0.2

79-0

.094

-0.0

1610

010

00.

634

-0.3

24-0

.405

-0.2

45-0

.218

0.04

2-0

.230

-0.2

61-0

.083

-0.0

0920

00.

066

-0.4

26-0

.156

-0.3

02-0

.254

-0.1

01-0

.165

-0.0

80-0

.058

-0.0

2520

00.

395

-0.2

19-0

.034

-0.2

25-0

.205

-0.0

70-0

.143

-0.0

65-0

.046

-0.0

1550

00.

090

-0.1

20-0

.162

-0.1

44-0

.149

-0.0

35-0

.042

-0.0

46-0

.050

-0.0

4150

00.

332

0.02

1-0

.079

-0.0

93-0

.115

-0.0

09-0

.021

-0.0

29-0

.034

-0.0

2520

010

00.

243

-0.5

84-0

.557

-0.3

28-0

.267

0.01

7-0

.240

-0.2

62-0

.078

0.00

020

010

00.

686

-0.2

91-0

.372

-0.2

11-0

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0.05

6-0

.218

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51-0

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0.00

020

00.

118

-0.3

80-0

.120

-0.2

71-0

.226

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74-0

.139

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57-0

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0.00

020

00.

470

-0.1

610.

003

-0.1

98-0

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52-0

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51-0

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0.00

050

00.

158

-0.0

64-0

.114

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01-0

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0.00

7-0

.001

-0.0

06-0

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0.00

050

00.

418

0.08

1-0

.035

-0.0

59-0

.084

0.01

90.

005

-0.0

04-0

.010

0.00

050

010

00.

277

-0.5

56-0

.549

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25-0

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0.01

9-0

.239

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62-0

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0.00

050

010

00.

718

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51-0

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10-0

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0.05

9-0

.216

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51-0

.073

0.00

020

00.

157

-0.3

67-0

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65-0

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73-0

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56-0

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0.00

020

00.

519

-0.1

390.

020

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88-0

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49-0

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50-0

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0.00

050

00.

189

-0.0

47-0

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-0.0

97-0

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0.00

90.

000

-0.0

06-0

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0.00

050

00.

470

0.10

8-0

.021

-0.0

53-0

.080

0.02

10.

006

-0.0

04-0

.010

0.00

0T

NR

MSE

TN

RM

SE

100

100

0.31

10.

633

0.59

60.

364

0.29

60.

051

0.26

40.

282

0.10

20.

041

100

100

0.71

40.

407

0.43

80.

269

0.23

30.

072

0.23

50.

263

0.08

70.

026

200

0.14

50.

438

0.17

40.

308

0.26

10.

124

0.17

10.

093

0.07

00.

053

200

0.43

60.

255

0.09

40.

233

0.21

10.

089

0.14

70.

072

0.05

30.

033

500

0.11

80.

137

0.17

70.

157

0.16

50.

073

0.07

80.

076

0.07

70.

079

500

0.34

50.

068

0.09

80.

104

0.12

40.

045

0.04

90.

050

0.05

20.

052

200

100

0.34

70.

611

0.57

10.

338

0.27

20.

042

0.24

20.

263

0.07

90.

000

200

100

0.76

90.

381

0.41

00.

243

0.21

70.

081

0.22

20.

252

0.07

50.

000

200

0.18

30.

391

0.13

30.

274

0.22

80.

076

0.14

00.

057

0.03

50.

000

200

0.50

90.

210

0.08

20.

207

0.18

80.

058

0.12

80.

052

0.03

30.

000

500

0.17

30.

075

0.11

60.

102

0.10

70.

010

0.00

40.

007

0.01

10.

000

500

0.42

90.

100

0.05

00.

063

0.08

50.

021

0.00

80.

006

0.01

10.

000

500

100

0.37

40.

584

0.56

30.

337

0.26

80.

043

0.24

00.

263

0.07

80.

000

500

100

0.79

80.

356

0.39

70.

245

0.21

10.

083

0.22

00.

252

0.07

50.

000

200

0.21

80.

379

0.12

60.

268

0.22

50.

075

0.13

80.

057

0.03

50.

000

200

0.56

00.

193

0.08

90.

199

0.18

40.

056

0.12

60.

051

0.03

30.

000

500

0.20

30.

063

0.10

80.

098

0.10

40.

011

0.00

40.

007

0.01

10.

000

500

0.48

00.

124

0.04

20.

057

0.08

20.

023

0.00

90.

005

0.01

00.

000

δ=

1/3

δ=

1/3

TN

Bia

sT

NB

ias

100

100

1.50

70.

288

-0.0

13-0

.003

-0.0

720.

127

-0.1

77-0

.232

-0.0

70-0

.005

100

100

3.07

11.

418

0.71

20.

452

0.21

20.

289

-0.0

78-0

.181

-0.0

50-0

.002

200

1.43

30.

445

0.35

40.

011

-0.0

660.

010

-0.0

97-0

.038

-0.0

32-0

.006

200

2.97

01.

475

0.97

40.

391

0.15

70.

136

-0.0

28-0

.004

-0.0

17-0

.003

500

1.53

30.

715

0.31

40.

123

0.00

80.

061

0.02

1-0

.001

-0.0

15-0

.009

500

3.03

71.

641

0.84

90.

421

0.17

00.

146

0.06

60.

021

-0.0

04-0

.005

200

100

1.58

50.

330

0.01

80.

030

-0.0

580.

141

-0.1

69-0

.226

-0.0

650.

000

200

100

3.15

91.

461

0.75

10.

491

0.21

80.

307

-0.0

75-0

.176

-0.0

460.

000

200

1.53

10.

529

0.40

30.

040

-0.0

460.

024

-0.0

87-0

.031

-0.0

260.

000

200

3.08

91.

572

1.03

00.

423

0.17

60.

148

-0.0

200.

001

-0.0

140.

000

500

1.68

40.

813

0.37

00.

161

0.03

20.

078

0.03

40.

009

-0.0

060.

000

500

3.21

71.

764

0.91

80.

464

0.19

60.

163

0.07

60.

028

0.00

10.

000

500

100

1.62

70.

386

0.04

10.

035

-0.0

440.

144

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67-0

.225

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640.

000

500

100

3.19

21.

517

0.77

80.

495

0.24

10.

310

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71-0

.176

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450.

000

200

1.61

30.

564

0.43

20.

054

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390.

028

-0.0

83-0

.030

-0.0

250.

000

200

3.17

51.

616

1.06

70.

441

0.18

50.

154

-0.0

160.

003

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130.

000

500

1.78

30.

867

0.39

90.

173

0.04

00.

082

0.03

70.

010

-0.0

050.

000

500

3.32

71.

827

0.95

70.

483

0.20

70.

168

0.08

00.

030

0.00

10.

000

TN

RM

SE

TN

RM

SE

100

100

1.57

10.

448

0.24

80.

170

0.14

30.

153

0.18

70.

236

0.07

50.

015

100

100

3.12

41.

492

0.79

00.

514

0.28

00.

316

0.12

30.

192

0.06

10.

008

200

1.46

10.

490

0.37

80.

098

0.09

30.

053

0.10

20.

046

0.03

60.

016

200

2.99

11.

498

0.99

30.

416

0.18

40.

153

0.05

50.

032

0.02

60.

009

500

1.54

10.

722

0.32

10.

133

0.03

70.

067

0.03

20.

019

0.02

30.

022

500

3.04

41.

646

0.85

40.

426

0.17

60.

150

0.07

00.

027

0.01

30.

013

200

100

1.65

20.

475

0.25

50.

180

0.13

70.

167

0.18

10.

230

0.06

80.

000

200

100

3.21

11.

531

0.82

60.

555

0.28

60.

335

0.12

20.

188

0.05

80.

000

200

1.55

90.

570

0.42

60.

109

0.08

20.

052

0.09

20.

037

0.02

80.

000

200

3.10

91.

598

1.04

80.

447

0.20

30.

163

0.05

30.

032

0.02

20.

000

500

1.69

20.

820

0.37

70.

167

0.04

40.

081

0.03

60.

013

0.00

70.

000

500

3.22

31.

769

0.92

30.

469

0.20

10.

166

0.07

90.

031

0.00

80.

000

500

100

1.69

30.

524

0.25

40.

191

0.13

80.

170

0.17

90.

229

0.06

80.

000

500

100

3.24

51.

589

0.85

40.

561

0.30

70.

338

0.12

10.

187

0.05

90.

000

200

1.64

10.

604

0.45

60.

116

0.08

00.

057

0.09

00.

037

0.02

80.

000

200

3.19

61.

640

1.08

50.

465

0.21

10.

171

0.05

30.

034

0.02

30.

000

500

1.79

00.

874

0.40

50.

180

0.05

10.

085

0.04

00.

014

0.00

70.

000

500

3.33

21.

832

0.96

30.

487

0.21

20.

171

0.08

30.

033

0.00

80.

000

Not

es:

Par

am

eter

sof

the

stat

icp

anel

dat

am

od

el,(2

9),

are

gen

erate

das:ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5),ϑi

=0

an

dγi∼IIDN

(1,1

),fo

ri

=1,

2,...,N

.G

auss

ian

erro

rsar

ege

ner

ated

asuit∼IIDN

(0,1

)in

(31).

Des

ign

1ass

um

esb i

∼U

(0.7,0.9

)fo

rth

efirs

tNb

(≤N

)el

emen

tsof

vect

or

bN

,i

=1,2,...,N

,in

the

con

stru

ctio

nof

the

corr

elat

ion

mat

rix

ofth

eer

rors

,RN

=I N

+bN

b′ N−

B2 N

giv

enby

(30),

wh

ere

BN

=D

iag

(bN

).c p

(n,δ

)co

rres

pon

ds

toth

ecr

itic

al

valu

eu

sed

inth

em

ult

iple

test

ing

pro

ced

ure

show

nin

(23).

Th

enu

mb

erof

rep

lica

tion

sis

set

toR

=2000.

28

Tab

leA

1b

:B

ias

an

dR

MS

E(×

100)

for

theα

esti

mat

eof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

oma

dyn

amic

pan

eld

ata

mod

elw

ith

exog

enou

sre

gres

sors

Cro

ssco

rrel

atio

ns

are

gen

erate

du

sin

gD

esig

n1

wit

hG

au

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

c p(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

05c p

(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

10δ

=1/

2δ

=1/

2T

NB

ias

TN

Bia

s10

010

00.

198

-0.6

21-0

.594

-0.3

63-0

.295

-0.0

07-0

.261

-0.2

84-0

.099

-0.0

2110

010

00.

625

-0.3

30-0

.412

-0.2

45-0

.221

0.03

9-0

.230

-0.2

64-0

.085

-0.0

1220

00.

064

-0.4

32-0

.161

-0.3

09-0

.262

-0.1

08-0

.173

-0.0

88-0

.070

-0.0

3320

00.

398

-0.2

18-0

.035

-0.2

29-0

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74-0

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69-0

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-0.0

1950

00.

084

-0.1

23-0

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58-0

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46-0

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59-0

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4950

00.

326

0.01

8-0

.084

-0.1

02-0

.120

-0.0

16-0

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37-0

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-0.0

3020

010

00.

255

-0.5

73-0

.561

-0.3

30-0

.266

0.01

8-0

.240

-0.2

63-0

.078

0.00

020

010

00.

687

-0.2

80-0

.376

-0.2

16-0

.200

0.05

7-0

.217

-0.2

52-0

.074

0.00

020

00.

123

-0.3

81-0

.118

-0.2

72-0

.225

-0.0

74-0

.139

-0.0

56-0

.035

0.00

020

00.

469

-0.1

600.

008

-0.1

98-0

.183

-0.0

51-0

.126

-0.0

50-0

.033

0.00

050

00.

156

-0.0

63-0

.113

-0.1

01-0

.106

0.00

7-0

.001

-0.0

07-0

.011

0.00

050

00.

417

0.08

2-0

.034

-0.0

60-0

.084

0.01

90.

005

-0.0

04-0

.010

0.00

050

010

00.

281

-0.5

50-0

.550

-0.3

22-0

.261

0.02

0-0

.240

-0.2

61-0

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0.00

050

010

00.

737

-0.2

46-0

.361

-0.2

05-0

.190

0.06

0-0

.216

-0.2

49-0

.073

0.00

020

00.

153

-0.3

66-0

.108

-0.2

66-0

.222

-0.0

72-0

.137

-0.0

56-0

.035

0.00

020

00.

514

-0.1

390.

022

-0.1

90-0

.179

-0.0

49-0

.125

-0.0

50-0

.033

0.00

050

00.

185

-0.0

49-0

.104

-0.0

97-0

.104

0.00

90.

000

-0.0

06-0

.011

0.00

050

00.

467

0.10

7-0

.020

-0.0

53-0

.080

0.02

10.

006

-0.0

04-0

.010

0.00

0T

NR

MSE

TN

RM

SE

100

100

0.30

20.

644

0.60

70.

375

0.30

50.

056

0.26

60.

289

0.10

80.

045

100

100

0.70

20.

411

0.44

40.

270

0.23

80.

070

0.23

50.

267

0.09

00.

027

200

0.15

30.

444

0.18

00.

316

0.27

20.

126

0.18

30.

106

0.09

40.

064

200

0.43

90.

255

0.09

60.

237

0.21

90.

088

0.15

40.

080

0.06

80.

041

500

0.11

70.

144

0.18

50.

179

0.17

40.

102

0.10

20.

102

0.10

70.

090

500

0.34

00.

073

0.10

20.

119

0.13

00.

066

0.06

70.

069

0.07

30.

060

200

100

0.35

40.

600

0.57

30.

339

0.27

20.

043

0.24

10.

264

0.07

90.

000

200

100

0.76

50.

374

0.41

30.

246

0.21

60.

082

0.22

10.

253

0.07

50.

000

200

0.19

00.

392

0.13

10.

274

0.22

60.

076

0.13

90.

057

0.03

50.

000

200

0.51

10.

208

0.08

50.

207

0.18

70.

057

0.12

70.

051

0.03

30.

000

500

0.16

90.

074

0.11

60.

102

0.10

70.

010

0.00

40.

007

0.01

10.

000

500

0.42

70.

100

0.04

90.

063

0.08

50.

021

0.00

80.

005

0.01

10.

000

500

100

0.37

50.

580

0.56

30.

332

0.26

70.

045

0.24

10.

262

0.07

90.

000

500

100

0.81

50.

360

0.40

20.

236

0.20

80.

084

0.22

00.

251

0.07

50.

000

200

0.21

30.

378

0.12

30.

270

0.22

40.

074

0.13

80.

057

0.03

50.

000

200

0.55

30.

195

0.08

90.

199

0.18

30.

056

0.12

70.

051

0.03

30.

000

500

0.19

80.

065

0.10

70.

098

0.10

40.

011

0.00

40.

007

0.01

10.

000

500

0.47

80.

123

0.04

20.

057

0.08

20.

023

0.00

90.

006

0.01

00.

000

δ=

1/3

δ=

1/3

TN

Bia

sT

NB

ias

100

100

1.50

90.

279

-0.0

26-0

.005

-0.0

740.

125

-0.1

77-0

.233

-0.0

71-0

.006

100

100

3.07

01.

397

0.70

80.

455

0.20

80.

290

-0.0

77-0

.181

-0.0

49-0

.003

200

1.43

10.

450

0.35

90.

009

-0.0

690.

009

-0.0

99-0

.040

-0.0

35-0

.008

200

2.97

41.

486

0.98

30.

387

0.15

60.

135

-0.0

29-0

.004

-0.0

19-0

.004

500

1.53

20.

716

0.31

10.

119

0.00

70.

059

0.01

8-0

.004

-0.0

19-0

.011

500

3.03

81.

642

0.84

80.

418

0.16

90.

145

0.06

50.

020

-0.0

07-0

.006

200

100

1.56

90.

341

0.01

70.

027

-0.0

540.

142

-0.1

68-0

.227

-0.0

650.

000

200

100

3.12

21.

471

0.75

00.

487

0.23

00.

307

-0.0

73-0

.178

-0.0

470.

000

200

1.55

20.

529

0.40

70.

042

-0.0

450.

026

-0.0

85-0

.030

-0.0

260.

000

200

3.10

61.

571

1.03

50.

427

0.17

70.

150

-0.0

170.

003

-0.0

140.

000

500

1.68

60.

819

0.37

10.

158

0.03

20.

078

0.03

40.

010

-0.0

060.

000

500

3.22

01.

765

0.91

90.

462

0.19

60.

163

0.07

60.

029

0.00

10.

000

500

100

1.63

40.

384

0.04

00.

040

-0.0

400.

146

-0.1

67-0

.223

-0.0

640.

000

500

100

3.19

61.

512

0.77

80.

502

0.24

10.

310

-0.0

72-0

.174

-0.0

450.

000

200

1.60

90.

563

0.43

20.

057

-0.0

380.

029

-0.0

83-0

.030

-0.0

250.

000

200

3.17

21.

611

1.06

80.

443

0.18

90.

152

-0.0

150.

003

-0.0

130.

000

500

1.78

00.

864

0.40

10.

174

0.04

10.

082

0.03

60.

010

-0.0

050.

000

500

3.32

11.

824

0.96

10.

483

0.20

80.

168

0.07

90.

030

0.00

20.

000

TN

RM

SE

TN

RM

SE

100

100

1.57

60.

441

0.24

20.

165

0.14

40.

150

0.18

70.

238

0.07

60.

015

100

100

3.12

31.

466

0.78

70.

514

0.27

40.

315

0.12

20.

192

0.06

20.

008

200

1.45

90.

493

0.38

30.

095

0.09

80.

050

0.10

50.

049

0.04

30.

020

200

2.99

61.

509

1.00

10.

411

0.18

50.

152

0.05

60.

034

0.02

90.

012

500

1.54

10.

724

0.31

90.

130

0.03

60.

069

0.03

70.

029

0.03

30.

026

500

3.04

41.

648

0.85

30.

423

0.17

50.

149

0.07

00.

029

0.01

80.

015

200

100

1.63

20.

486

0.24

70.

175

0.13

50.

169

0.18

00.

230

0.06

90.

000

200

100

3.17

41.

540

0.82

30.

548

0.29

40.

335

0.12

00.

188

0.05

90.

000

200

1.57

90.

568

0.43

00.

109

0.08

40.

053

0.09

10.

037

0.02

80.

000

200

3.12

71.

595

1.05

50.

451

0.20

50.

166

0.05

30.

034

0.02

30.

000

500

1.69

40.

826

0.37

70.

165

0.04

50.

081

0.03

70.

013

0.00

70.

000

500

3.22

61.

771

0.92

50.

467

0.20

10.

166

0.07

90.

032

0.00

70.

000

500

100

1.70

00.

529

0.26

20.

182

0.13

40.

171

0.17

90.

228

0.06

80.

000

500

100

3.24

61.

581

0.85

60.

564

0.30

40.

337

0.12

10.

187

0.05

80.

000

200

1.63

60.

602

0.45

40.

116

0.08

10.

056

0.09

00.

037

0.02

80.

000

200

3.19

31.

635

1.08

60.

466

0.21

60.

168

0.05

40.

034

0.02

30.

000

500

1.78

70.

871

0.40

70.

180

0.05

20.

085

0.03

90.

014

0.00

70.

000

500

3.32

71.

830

0.96

50.

487

0.21

40.

171

0.08

20.

033

0.00

80.

000

Not

es:

Par

amet

ers

ofth

ed

yn

amic

pan

eld

ata

mod

el,

(29),

are

gen

erate

das:ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5),ϑi∼U

(0,0.9

5)

an

dγi∼IIDN

(1,1

),fo

ri

=1,

2,...,N

.G

auss

ian

erro

rsar

ege

ner

ated

asuit∼IIDN

(0,1

)in

(31).

Des

ign

1ass

um

esb i

∼U

(0.7,0.9

)fo

rth

efi

rstNb

(≤N

)el

emen

tsof

vec

tor

bN

,i

=1,

2,...,N

,in

the

con

stru

ctio

nof

the

corr

elat

ion

matr

ixof

the

erro

rs,RN

=I N

+bN

b′ N−

B2 N

giv

enby

(30),

wher

eBN

=D

iag

(bN

).c p

(n,δ

)co

rres

pon

ds

toth

ecr

itic

alva

lue

use

din

the

mu

ltip

lete

stin

gpro

ced

ure

show

nin

(23).

Th

enu

mb

erof

rep

lica

tion

sis

set

toR

=2000.

29

Tab

leA

2a:

Bia

san

dR

MS

E(×

100)

for

theα

esti

mat

eof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

oma

stati

cp

anel

dat

am

od

elC

ross

corr

elati

on

sare

gen

erate

du

sin

gD

esig

n2

wit

hG

au

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

c p(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

05c p

(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

10δ

=1/

2δ

=1/

2T

NB

ias

TN

Bia

s10

010

0-0

.880

-2.3

78-2

.922

-3.0

26-3

.232

-3.1

31-3

.509

-3.6

48-3

.576

-3.5

9810

010

0-0

.092

-1.5

33-2

.041

-2.1

19-2

.328

-2.2

20-2

.609

-2.7

63-2

.699

-2.7

3820

0-1

.221

-2.4

80-2

.800

-3.2

67-3

.464

-3.4

34-3

.595

-3.5

91-3

.658

-3.7

1420

0-0

.542

-1.7

21-1

.988

-2.4

35-2

.620

-2.5

90-2

.750

-2.7

54-2

.824

-2.8

8350

0-1

.501

-2.6

58-3

.266

-3.5

81-3

.780

-3.7

79-3

.859

-3.9

34-4

.011

-4.0

5750

0-0

.910

-1.9

52-2

.492

-2.7

75-2

.953

-2.9

48-3

.026

-3.0

98-3

.175

-3.2

2220

010

00.

705

-0.0

90-0

.160

0.12

00.

009

0.20

5-0

.145

-0.2

99-0

.242

-0.3

0420

010

01.

207

0.29

40.

129

0.34

60.

192

0.36

2-0

.007

-0.1

77-0

.131

-0.2

0720

00.

375

-0.0

680.

134

-0.0

86-0

.146

-0.0

63-0

.191

-0.2

06-0

.269

-0.3

2620

00.

784

0.23

60.

360

0.09

40.

001

0.06

6-0

.071

-0.0

97-0

.166

-0.2

2850

00.

219

-0.0

24-0

.086

-0.1

58-0

.206

-0.1

53-0

.212

-0.2

67-0

.325

-0.3

6850

00.

541

0.20

90.

094

-0.0

12-0

.079

-0.0

38-0

.104

-0.1

63-0

.222

-0.2

6650

010

01.

043

0.29

50.

247

0.56

30.

433

0.62

50.

264

0.08

40.

114

-0.0

0150

010

01.

463

0.58

10.

426

0.67

20.

502

0.66

60.

289

0.09

80.

121

0.00

020

00.

683

0.31

30.

550

0.32

60.

254

0.32

90.

203

0.16

10.

087

-0.0

0120

01.

031

0.52

90.

674

0.40

10.

298

0.35

50.

218

0.16

90.

091

0.00

050

00.

530

0.35

50.

331

0.24

90.

205

0.24

60.

176

0.11

70.

056

-0.0

0150

00.

799

0.50

40.

412

0.29

40.

229

0.25

90.

184

0.12

10.

058

0.00

0T

NR

MSE

TN

RM

SE

100

100

1.25

22.

688

3.30

13.

469

3.69

93.

623

3.97

44.

094

4.02

94.

051

100

100

0.82

61.

877

2.41

22.

542

2.75

22.

671

3.02

33.

155

3.09

73.

137

200

1.45

52.

748

3.16

73.

655

3.86

83.

859

4.00

24.

009

4.07

04.

130

200

0.89

42.

002

2.35

12.

795

2.98

72.

976

3.11

23.

125

3.18

93.

251

500

1.66

22.

899

3.58

03.

938

4.15

34.

176

4.26

04.

336

4.41

14.

456

500

1.11

52.

198

2.79

13.

105

3.29

23.

306

3.38

63.

459

3.53

33.

580

200

100

0.80

30.

377

0.36

90.

346

0.29

80.

354

0.31

20.

405

0.35

90.

399

200

100

1.27

20.

451

0.31

40.

438

0.30

50.

426

0.21

70.

275

0.23

90.

281

200

0.45

70.

291

0.30

70.

274

0.29

60.

251

0.31

50.

318

0.36

70.

415

200

0.82

40.

336

0.42

10.

223

0.19

90.

198

0.20

60.

208

0.25

10.

299

500

0.30

30.

241

0.26

20.

299

0.32

80.

295

0.32

60.

366

0.41

80.

456

500

0.57

10.

283

0.21

40.

196

0.21

10.

197

0.21

60.

252

0.30

00.

337

500

100

1.06

80.

337

0.27

20.

571

0.44

20.

631

0.27

90.

116

0.13

60.

002

500

100

1.49

50.

621

0.45

10.

682

0.51

00.

672

0.30

30.

127

0.14

10.

001

200

0.69

70.

325

0.55

30.

329

0.25

70.

332

0.20

80.

166

0.09

30.

001

200

1.04

80.

542

0.67

80.

405

0.30

10.

357

0.22

30.

174

0.09

70.

001

500

0.53

40.

357

0.33

20.

250

0.20

60.

246

0.17

70.

118

0.05

70.

001

500

0.80

40.

507

0.41

40.

294

0.23

00.

260

0.18

40.

122

0.05

90.

001

δ=

1/3

δ=

1/3

TN

Bia

sT

NB

ias

100

100

1.12

1-0

.410

-1.0

23-1

.161

-1.4

37-1

.372

-1.7

89-1

.971

-1.9

26-1

.985

100

100

2.94

71.

111

0.20

8-0

.136

-0.5

73-0

.609

-1.0

93-1

.323

-1.3

12-1

.401

200

0.92

6-0

.377

-0.7

70-1

.305

-1.5

54-1

.571

-1.7

58-1

.790

-1.8

71-1

.940

200

2.75

01.

070

0.35

2-0

.393

-0.7

87-0

.898

-1.1

40-1

.210

-1.3

13-1

.396

500

0.90

5-0

.329

-1.0

08-1

.398

-1.6

34-1

.672

-1.7

75-1

.859

-1.9

39-1

.991

500

2.68

81.

017

0.01

4-0

.590

-0.9

50-1

.065

-1.2

10-1

.316

-1.4

07-1

.466

200

100

2.12

00.

962

0.59

00.

671

0.42

50.

538

0.13

4-0

.064

-0.0

38-0

.131

200

100

3.64

92.

070

1.33

81.

167

0.75

80.

763

0.29

30.

047

0.03

9-0

.080

200

1.90

00.

990

0.85

50.

438

0.24

60.

250

0.08

00.

027

-0.0

56-0

.130

200

3.43

62.

034

1.51

20.

863

0.52

00.

428

0.20

30.

113

0.00

9-0

.082

500

1.87

01.

040

0.62

20.

337

0.17

30.

156

0.05

9-0

.017

-0.0

85-0

.135

500

3.39

42.

001

1.20

00.

682

0.38

40.

290

0.15

00.

052

-0.0

29-0

.088

500

100

2.30

51.

160

0.79

70.

898

0.64

10.

748

0.33

80.

123

0.13

10.

000

500

100

3.78

52.

210

1.48

71.

322

0.90

50.

903

0.42

90.

171

0.15

00.

000

200

2.07

31.

186

1.05

70.

633

0.43

40.

431

0.26

10.

191

0.09

90.

000

200

3.57

12.

177

1.65

30.

999

0.65

00.

553

0.32

80.

224

0.11

20.

000

500

2.06

01.

231

0.81

50.

514

0.34

80.

322

0.21

60.

137

0.06

40.

000

500

3.55

62.

152

1.34

60.

811

0.51

00.

406

0.25

90.

157

0.07

10.

000

TN

RM

SE

TN

RM

SE

100

100

1.34

90.

991

1.45

81.

608

1.84

11.

798

2.15

32.

306

2.26

72.

325

100

100

3.02

91.

332

0.83

20.

857

1.04

91.

073

1.42

91.

611

1.60

31.

685

200

1.10

20.

867

1.21

01.

653

1.88

31.

906

2.06

12.

097

2.17

02.

241

200

2.79

81.

230

0.79

20.

864

1.13

21.

222

1.41

01.

477

1.56

61.

648

500

1.01

70.

766

1.32

51.

693

1.91

61.

966

2.06

32.

144

2.22

02.

272

500

2.71

11.

136

0.64

30.

938

1.22

41.

335

1.46

51.

562

1.64

71.

703

200

100

2.16

91.

026

0.64

90.

711

0.46

90.

568

0.21

60.

175

0.15

30.

187

200

100

3.68

82.

111

1.37

11.

190

0.78

00.

780

0.32

60.

140

0.12

00.

121

200

1.92

21.

014

0.87

30.

463

0.28

40.

281

0.15

50.

125

0.13

50.

181

200

3.45

32.

049

1.52

30.

874

0.53

40.

440

0.22

80.

145

0.08

90.

119

500

1.87

81.

049

0.63

30.

357

0.20

80.

192

0.12

40.

111

0.14

40.

180

500

3.39

92.

005

1.20

50.

688

0.39

40.

301

0.16

90.

093

0.08

80.

122

500

100

2.34

51.

200

0.82

40.

910

0.65

10.

755

0.35

00.

149

0.15

00.

001

500

100

3.82

32.

246

1.51

31.

336

0.91

70.

911

0.44

10.

192

0.16

80.

000

200

2.09

11.

200

1.06

30.

638

0.43

70.

434

0.26

60.

195

0.10

50.

000

200

3.58

62.

190

1.66

11.

005

0.65

50.

556

0.33

20.

228

0.11

70.

000

500

2.06

51.

234

0.81

70.

515

0.34

90.

322

0.21

70.

138

0.06

50.

000

500

3.56

02.

155

1.34

80.

813

0.51

10.

407

0.26

00.

158

0.07

30.

000

Not

es:

Par

am

eter

sof

the

stat

icp

anel

dat

am

od

el,(2

9),

are

gen

erate

das:ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5),ϑi

=0

an

dγi∼IIDN

(1,1

),fo

ri

=1,

2,...,N

.D

esig

n2

assu

mes

atw

o-fa

ctor

mod

elw

ith

[Nαβ]

and

[Nαβ2]

non

-zer

olo

ad

ings

for

the

firs

tan

dse

con

dfa

ctor,

resp

ecti

vely

.W

ese

t:αβ

2=

2αβ/3

,w

her

eαβ

rela

tes

toα

un

der

(11)

,f jt

anduit∼IIDN

(0,1

),v ij∼IIDU

(µvj−

0.2,µvj

+0.2

),j

=1,

2,µv

=0.

87,µv2

=0.

71,µv1

=√ µ

2 v−N

2(αβ2−αβ

)µ

2 v2),

in(3

2)

an

d

(33)

.T

he

nu

mb

erof

rep

lica

tion

sis

set

toR

=20

00.

30

Tab

leA

2b

:B

ias

an

dR

MS

E(×

100)

for

theα

esti

mat

eof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

oma

dyn

amic

pan

eld

ata

mod

elw

ith

exog

enou

sre

gres

sors

Cro

ssco

rrel

atio

ns

are

gen

erate

du

sin

gD

esig

n2

wit

hG

au

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

c p(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

05c p

(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

10δ

=1/

2δ

=1/

2T

NB

ias

TN

Bia

s10

010

0-1

.016

-2.5

71-3

.183

-3.3

19-3

.568

-3.4

79-3

.838

-3.9

98-3

.922

-3.9

5210

010

0-0

.216

-1.6

91-2

.254

-2.3

54-2

.602

-2.5

09-2

.879

-3.0

47-2

.978

-3.0

2420

0-1

.320

-2.6

42-3

.032

-3.5

58-3

.745

-3.7

63-3

.918

-3.9

25-4

.003

-4.0

5720

0-0

.627

-1.8

58-2

.175

-2.6

71-2

.846

-2.8

56-3

.011

-3.0

25-3

.104

-3.1

6350

0-1

.610

-2.8

49-3

.517

-3.8

90-4

.107

-4.1

24-4

.222

-4.3

04-4

.388

-4.4

3950

0-1

.007

-2.1

19-2

.705

-3.0

35-3

.227

-3.2

35-3

.330

-3.4

09-3

.490

-3.5

4320

010

00.

689

-0.1

09-0

.179

0.08

8-0

.030

0.16

6-0

.181

-0.3

29-0

.272

-0.3

3620

010

01.

194

0.28

00.

115

0.32

90.

162

0.33

2-0

.035

-0.1

99-0

.153

-0.2

3020

00.

357

-0.0

890.

115

-0.1

20-0

.177

-0.0

95-0

.227

-0.2

40-0

.301

-0.3

6020

00.

774

0.22

00.

345

0.06

8-0

.022

0.04

3-0

.099

-0.1

22-0

.189

-0.2

5350

00.

209

-0.0

33-0

.095

-0.1

70-0

.215

-0.1

65-0

.225

-0.2

81-0

.337

-0.3

8050

00.

538

0.20

40.

087

-0.0

22-0

.085

-0.0

47-0

.114

-0.1

73-0

.230

-0.2

7550

010

01.

040

0.29

40.

248

0.56

50.

436

0.62

70.

261

0.08

10.

115

-0.0

0150

010

01.

466

0.58

30.

430

0.67

60.

504

0.67

00.

286

0.09

50.

121

0.00

020

00.

685

0.31

30.

548

0.32

40.

253

0.33

00.

202

0.16

20.

086

-0.0

0120

01.

031

0.52

80.

672

0.39

90.

298

0.35

50.

217

0.17

00.

089

0.00

050

00.

527

0.35

30.

330

0.24

90.

204

0.24

60.

176

0.11

70.

055

-0.0

0150

00.

797

0.50

20.

412

0.29

40.

229

0.25

90.

183

0.12

20.

057

0.00

0T

NR

MSE

TN

RM

SE

100

100

1.36

02.

876

3.54

93.

762

4.03

13.

981

4.30

14.

455

4.37

84.

411

100

100

0.87

12.

032

2.62

12.

783

3.03

82.

976

3.29

63.

453

3.38

33.

430

200

1.53

62.

902

3.39

83.

954

4.17

04.

208

4.35

84.

367

4.44

54.

497

200

0.95

52.

133

2.53

63.

040

3.23

33.

258

3.40

43.

418

3.49

73.

553

500

1.76

93.

101

3.84

94.

273

4.51

24.

554

4.65

94.

740

4.82

24.

870

500

1.20

72.

375

3.02

43.

391

3.59

73.

625

3.72

53.

803

3.88

13.

932

200

100

0.79

40.

393

0.39

30.

369

0.32

60.

359

0.35

50.

440

0.39

80.

446

200

100

1.25

70.

440

0.31

80.

438

0.30

60.

415

0.24

20.

301

0.26

90.

317

200

0.44

80.

292

0.30

10.

306

0.33

00.

284

0.35

40.

355

0.40

10.

453

200

0.81

60.

323

0.41

20.

229

0.21

30.

207

0.23

00.

233

0.27

50.

327

500

0.29

50.

245

0.27

40.

306

0.32

90.

296

0.33

50.

376

0.42

20.

463

500

0.56

70.

281

0.21

70.

197

0.20

90.

194

0.22

10.

258

0.30

10.

342

500

100

1.06

50.

332

0.27

20.

573

0.44

30.

634

0.27

60.

118

0.13

50.

002

500

100

1.50

10.

622

0.45

70.

686

0.51

30.

676

0.30

00.

129

0.14

10.

001

200

0.69

80.

326

0.55

10.

328

0.25

70.

332

0.20

70.

166

0.09

20.

001

200

1.04

80.

540

0.67

60.

403

0.30

10.

357

0.22

10.

174

0.09

50.

001

500

0.53

10.

355

0.33

10.

250

0.20

50.

246

0.17

60.

118

0.05

70.

001

500

0.80

20.

505

0.41

30.

295

0.23

00.

260

0.18

40.

123

0.05

90.

001

δ=

1/3

δ=

1/3

TN

Bia

sT

NB

ias

100

100

1.04

5-0

.523

-1.1

86-1

.344

-1.6

52-1

.597

-1.9

97-2

.190

-2.1

41-2

.207

100

100

2.88

51.

029

0.09

1-0

.271

-0.7

32-0

.774

-1.2

47-1

.487

-1.4

74-1

.568

200

0.89

1-0

.465

-0.9

00-1

.471

-1.7

12-1

.762

-1.9

47-1

.984

-2.0

75-2

.143

200

2.71

71.

001

0.25

4-0

.516

-0.9

00-1

.042

-1.2

81-1

.359

-1.4

69-1

.550

500

0.84

0-0

.446

-1.1

57-1

.577

-1.8

22-1

.868

-1.9

84-2

.074

-2.1

57-2

.213

500

2.64

10.

932

-0.0

96-0

.725

-1.0

94-1

.216

-1.3

73-1

.485

-1.5

78-1

.640

200

100

2.11

20.

951

0.58

10.

663

0.40

60.

518

0.11

2-0

.078

-0.0

52-0

.148

200

100

3.66

22.

072

1.34

51.

171

0.74

90.

749

0.27

60.

039

0.03

0-0

.091

200

1.88

20.

979

0.84

10.

418

0.23

10.

234

0.06

10.

011

-0.0

70-0

.146

200

3.41

52.

020

1.49

60.

845

0.50

80.

416

0.18

90.

102

-0.0

01-0

.093

500

1.86

61.

037

0.61

70.

330

0.17

00.

151

0.05

3-0

.022

-0.0

88-0

.140

500

3.39

21.

999

1.19

80.

678

0.38

20.

286

0.14

60.

048

-0.0

31-0

.091

500

100

2.31

11.

162

0.79

90.

900

0.64

60.

752

0.33

50.

122

0.13

20.

000

500

100

3.79

82.

219

1.48

91.

327

0.91

30.

909

0.42

80.

170

0.15

20.

000

200

2.07

71.

187

1.05

70.

632

0.43

30.

431

0.26

00.

191

0.09

80.

000

200

3.57

32.

179

1.65

70.

999

0.64

90.

553

0.32

70.

224

0.11

10.

000

500

2.05

31.

226

0.81

10.

513

0.34

80.

322

0.21

60.

137

0.06

30.

000

500

3.54

92.

147

1.34

10.

810

0.50

90.

406

0.25

80.

157

0.07

10.

000

TN

RM

SE

TN

RM

SE

100

100

1.28

91.

085

1.61

31.

794

2.06

82.

041

2.36

62.

538

2.49

12.

557

100

100

2.97

31.

295

0.86

30.

961

1.20

61.

240

1.58

41.

785

1.77

31.

861

200

1.07

40.

929

1.33

51.

827

2.05

62.

113

2.27

62.

310

2.39

82.

463

200

2.76

61.

179

0.79

30.

977

1.25

51.

378

1.57

51.

640

1.74

31.

819

500

0.96

90.

873

1.48

81.

895

2.13

02.

189

2.30

32.

387

2.46

72.

521

500

2.66

71.

086

0.71

51.

086

1.39

01.

509

1.65

41.

756

1.84

31.

904

200

100

2.16

11.

015

0.64

50.

708

0.45

70.

555

0.21

60.

188

0.17

10.

213

200

100

3.70

02.

111

1.38

01.

196

0.77

40.

768

0.31

60.

140

0.12

70.

139

200

1.90

61.

006

0.86

10.

448

0.27

30.

272

0.15

30.

131

0.14

80.

198

200

3.43

22.

036

1.50

80.

858

0.52

20.

430

0.21

70.

140

0.09

30.

130

500

1.87

41.

046

0.62

90.

351

0.20

40.

186

0.12

10.

112

0.14

20.

183

500

3.39

72.

003

1.20

20.

684

0.39

10.

296

0.16

50.

092

0.08

50.

123

500

100

2.35

11.

200

0.82

50.

912

0.65

60.

759

0.34

80.

151

0.15

10.

001

500

100

3.83

52.

253

1.51

51.

342

0.92

40.

916

0.44

00.

194

0.16

90.

000

200

2.09

51.

201

1.06

40.

636

0.43

70.

434

0.26

40.

196

0.10

40.

000

200

3.58

92.

193

1.66

41.

004

0.65

40.

556

0.33

00.

228

0.11

60.

000

500

2.05

91.

229

0.81

30.

514

0.34

90.

322

0.21

60.

138

0.06

50.

000

500

3.55

32.

150

1.34

30.

811

0.51

00.

406

0.25

90.

158

0.07

20.

000

Not

es:

Par

amet

ers

ofth

ed

yn

amic

pan

eld

ata

mod

el,

(29),

are

gen

erate

das:ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5),ϑi∼U

(0,0.9

5)

an

dγi∼IIDN

(1,1

),fo

ri

=1,

2,...,N

.es

ign

2as

sum

esa

two-

fact

orm

od

elw

ith

[Nαβ]

an

d[N

αβ2]

non

-zer

olo

ad

ings

for

the

firs

tan

dse

con

dfa

ctor,

resp

ecti

vely

.W

ese

t:αβ

2=

2αβ/3,

wh

ereαβ

rela

tes

toα

un

der

(11)

,f jt

anduit∼IIDN

(0,1

),v ij∼IIDU

(µvj−

0.2,µvj

+0.

2),j

=1,

2,µv

=0.

87,µv2

=0.

71,µv1

=√ µ

2 v−N

2(αβ2−αβ

)µ

2 v2),

in

(32)

and

(33)

.T

he

num

ber

ofre

pli

cati

ons

isse

ttoR

=2000.

31

Tab

leA

3a:

Bia

san

dR

MS

E×

100)

for

theα

esti

mat

eof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

om

ast

atic

pan

eld

ata

mod

elC

ross

corr

elat

ion

sare

gen

erate

du

sin

gD

esig

n1

wit

hn

on

-Gau

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

c p(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

05c p

(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

10δ

=1/

2δ

=1/

2T

NB

ias

TN

Bia

s10

010

02.

272

0.83

70.

313

0.22

10.

042

0.17

6-0

.159

-0.2

43-0

.103

-0.0

4710

010

03.

653

1.84

70.

991

0.67

00.

318

0.35

0-0

.048

-0.1

79-0

.068

-0.0

3020

02.

184

0.89

90.

639

0.17

90.

027

0.02

0-0

.114

-0.0

76-0

.078

-0.0

6220

03.

487

1.79

91.

201

0.53

90.

252

0.15

6-0

.032

-0.0

26-0

.048

-0.0

4150

02.

138

1.15

30.

485

0.21

30.

030

0.03

5-0

.036

-0.0

67-0

.090

-0.0

8650

03.

317

1.92

90.

948

0.49

30.

199

0.13

70.

029

-0.0

23-0

.057

-0.0

5920

010

02.

680

1.15

00.

594

0.35

90.

156

0.28

6-0

.090

-0.1

88-0

.050

0.00

020

010

04.

140

2.23

51.

324

0.82

30.

446

0.46

10.

013

-0.1

35-0

.030

0.00

020

02.

604

1.30

90.

890

0.32

30.

134

0.11

1-0

.022

-0.0

04-0

.017

0.00

020

03.

993

2.28

81.

499

0.70

30.

364

0.23

60.

051

0.03

1-0

.005

0.00

050

02.

642

1.44

00.

792

0.41

80.

178

0.14

00.

069

0.02

70.

000

0.00

050

03.

934

2.28

61.

311

0.71

70.

345

0.22

40.

112

0.04

70.

007

0.00

050

010

02.

988

1.36

60.

677

0.43

10.

202

0.30

4-0

.079

-0.1

77-0

.047

0.00

050

010

04.

527

2.50

71.

450

0.92

50.

513

0.48

90.

028

-0.1

19-0

.025

0.00

020

03.

035

1.52

21.

012

0.39

90.

164

0.15

7-0

.019

0.00

1-0

.014

0.00

020

04.

505

2.56

51.

667

0.80

70.

408

0.29

70.

056

0.03

80.

000

0.00

050

03.

170

1.71

90.

910

0.44

90.

203

0.16

20.

079

0.02

90.

003

0.00

050

04.

577

2.64

01.

470

0.77

00.

384

0.25

50.

127

0.05

00.

011

0.00

0T

NR

MSE

TN

RM

SE

100

100

2.87

01.

712

1.03

20.

720

0.45

40.

330

0.28

70.

282

0.14

40.

098

100

100

4.18

32.

530

1.54

31.

081

0.64

10.

493

0.28

80.

243

0.11

20.

068

200

3.01

81.

642

1.16

10.

731

0.58

60.

219

0.21

40.

139

0.12

00.

112

200

4.19

32.

418

1.67

51.

017

0.72

60.

312

0.20

30.

114

0.08

60.

078

500

3.06

61.

984

1.03

10.

793

0.39

70.

250

0.15

30.

149

0.15

90.

145

500

4.12

72.

686

1.44

61.

000

0.51

90.

323

0.15

10.

120

0.11

40.

104

200

100

3.19

11.

765

1.19

00.

747

0.45

60.

459

0.21

10.

217

0.07

50.

003

200

100

4.57

22.

725

1.80

11.

136

0.70

00.

633

0.24

50.

195

0.07

60.

002

200

3.15

51.

943

1.34

90.

662

0.46

20.

240

0.20

00.

076

0.03

40.

002

200

4.46

52.

817

1.90

91.

000

0.64

90.

358

0.24

60.

103

0.04

00.

001

500

3.24

01.

965

1.26

50.

867

0.43

90.

230

0.13

90.

108

0.02

70.

001

500

4.45

72.

753

1.74

61.

129

0.59

60.

320

0.18

80.

128

0.03

40.

001

500

100

3.35

11.

900

1.01

50.

680

0.41

20.

445

0.17

60.

202

0.06

40.

000

500

100

4.83

22.

903

1.71

51.

134

0.68

20.

624

0.20

40.

174

0.06

40.

000

200

3.54

11.

946

1.27

80.

616

0.34

70.

294

0.13

80.

066

0.03

20.

000

200

4.91

72.

913

1.90

90.

996

0.56

00.

424

0.17

70.

093

0.03

70.

000

500

3.70

42.

155

1.29

80.

660

0.40

80.

239

0.12

80.

057

0.04

00.

000

500

5.01

83.

016

1.81

00.

964

0.56

70.

334

0.17

90.

079

0.04

70.

000

δ=

1/3

δ=

1/3

TN

Bia

sT

NB

ias

100

100

5.75

03.

447

2.10

51.

411

0.78

50.

634

0.12

7-0

.084

-0.0

24-0

.018

100

100

8.49

65.

631

3.68

12.

501

1.49

21.

069

0.39

40.

058

0.03

5-0

.010

200

6.20

23.

781

2.49

61.

376

0.77

70.

461

0.14

30.

069

-0.0

02-0

.021

200

8.93

65.

912

3.98

52.

381

1.41

40.

837

0.35

50.

177

0.04

3-0

.013

500

6.77

54.

358

2.48

81.

433

0.75

30.

444

0.20

30.

073

-0.0

03-0

.027

500

9.46

96.

422

3.91

32.

350

1.30

30.

749

0.36

70.

154

0.03

2-0

.017

200

100

6.30

43.

897

2.49

31.

587

0.92

90.

757

0.18

9-0

.046

0.00

40.

000

200

100

9.06

06.

128

4.12

42.

705

1.66

11.

210

0.46

20.

094

0.05

90.

000

200

6.79

54.

372

2.87

31.

591

0.90

90.

543

0.22

80.

116

0.02

70.

000

200

9.53

26.

541

4.40

62.

637

1.57

10.

929

0.45

00.

225

0.06

70.

000

500

7.52

44.

846

2.98

31.

731

0.93

10.

527

0.26

90.

119

0.03

30.

000

500

10.2

066.

939

4.46

82.

694

1.51

20.

841

0.43

40.

196

0.06

10.

000

500

100

6.74

74.

228

2.66

41.

727

1.02

90.

796

0.21

0-0

.022

0.01

20.

000

500

100

9.53

26.

485

4.33

52.

877

1.78

71.

264

0.49

20.

128

0.06

90.

000

200

7.39

04.

731

3.11

41.

747

0.98

00.

633

0.23

90.

128

0.03

30.

000

200

10.1

576.

940

4.69

52.

827

1.66

91.

043

0.46

70.

241

0.07

50.

000

500

8.30

45.

326

3.23

41.

844

1.00

70.

585

0.29

70.

127

0.03

90.

000

500

10.9

897.

458

4.76

42.

843

1.61

70.

920

0.47

20.

209

0.06

80.

000

TN

RM

SE

TN

RM

SE

100

100

6.19

23.

978

2.53

61.

756

1.05

40.

778

0.37

50.

220

0.09

90.

045

100

100

8.83

66.

029

4.01

82.

794

1.72

11.

212

0.58

80.

264

0.12

20.

029

200

6.72

24.

248

2.89

21.

766

1.15

60.

603

0.30

10.

152

0.06

60.

045

200

9.32

06.

270

4.31

32.

708

1.73

20.

974

0.48

80.

248

0.09

00.

030

500

7.32

14.

922

2.89

01.

828

1.02

80.

621

0.29

10.

154

0.06

50.

053

500

9.86

76.

859

4.25

02.

686

1.55

50.

922

0.45

50.

225

0.07

00.

035

200

100

6.65

24.

277

2.87

51.

848

1.14

90.

926

0.36

60.

189

0.08

80.

002

200

100

9.32

16.

410

4.42

52.

920

1.85

21.

369

0.60

70.

254

0.12

80.

001

200

7.14

34.

765

3.20

71.

839

1.14

60.

657

0.38

90.

181

0.06

40.

001

200

9.79

06.

839

4.67

72.

848

1.77

61.

038

0.59

20.

286

0.10

10.

001

500

7.87

75.

184

3.32

52.

062

1.15

60.

631

0.35

10.

196

0.06

00.

001

500

10.4

647.

200

4.74

92.

975

1.71

70.

946

0.51

60.

268

0.08

80.

001

500

100

6.98

84.

518

2.87

71.

903

1.17

60.

924

0.33

40.

163

0.08

00.

000

500

100

9.70

86.

698

4.50

33.

028

1.91

61.

383

0.59

30.

243

0.12

50.

000

200

7.67

64.

985

3.31

61.

905

1.10

50.

746

0.32

80.

175

0.06

30.

000

200

10.3

627.

130

4.86

12.

963

1.78

01.

146

0.54

30.

285

0.10

30.

000

500

8.58

15.

585

3.48

32.

007

1.15

50.

663

0.35

20.

158

0.07

20.

000

500

11.1

897.

655

4.96

22.

985

1.74

80.

996

0.52

70.

239

0.09

90.

000

Not

es:

Par

am

eter

sof

the

stat

icp

anel

dat

am

od

el,(2

9),

are

gen

erate

das:ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5),ϑi

=0

an

dγi∼IIDN

(1,1

),fo

ri

=1,

2,...,N

.

Non

-Gau

ssia

ner

rors

are

gen

erat

edasuit

=( v−

2χ2 v,t

) 1/2ν it,

fori

=1,2,...,N

,ν it∼IIDN

(0,1

)an

dχ

2 v,t

isa

chi-

squ

are

dra

nd

om

vari

ate

wit

hv=

8d

egre

esof

free

dom

,in

(31)

.D

esig

n1

assu

mesb i

∼U

(0.7,0.9

)fo

rth

efi

rstNb

(≤N

)el

emen

tsof

vect

or

bN

,i

=1,

2,...,N

,in

the

con

stru

ctio

nof

the

corr

elati

on

matr

ixof

the

erro

rs,RN

=I N

+bN

b′ N−

B2 N

give

nby

(30)

,w

her

eBN

=D

iag

(bN

).c p

(n,δ

)co

rres

pon

ds

toth

ecr

itic

al

valu

eu

sed

inth

em

ult

iple

test

ing

pro

ced

ure

show

nin

(23)

.T

he

nu

mb

erof

rep

lica

tion

sis

set

toR

=2000.

32

Tab

leA

3b

:B

ias

an

dR

MS

E(×

100)

for

theα

esti

mat

eof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

oma

dyn

amic

pan

eld

ata

mod

elw

ith

exog

enou

sre

gres

sors

Cro

ssco

rrel

atio

ns

are

gen

erate

du

sin

gD

esig

n1

wit

hn

on

-Gau

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

c p(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

05c p

(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

10δ

=1/

2δ

=1/

2T

NB

ias

TN

Bia

s10

010

02.

224

0.77

00.

293

0.19

00.

027

0.17

1-0

.178

-0.2

56-0

.110

-0.0

5410

010

03.

581

1.76

50.

966

0.62

00.

305

0.34

7-0

.067

-0.1

88-0

.073

-0.0

3520

02.

032

0.82

10.

589

0.14

0-0

.032

0.00

5-0

.128

-0.0

90-0

.096

-0.0

7720

03.

308

1.69

91.

143

0.49

50.

187

0.14

1-0

.042

-0.0

37-0

.059

-0.0

5150

02.

043

0.96

60.

483

0.17

10.

015

0.01

7-0

.051

-0.0

83-0

.105

-0.1

0550

03.

212

1.71

30.

954

0.44

60.

187

0.11

90.

015

-0.0

35-0

.067

-0.0

7220

010

02.

719

1.15

90.

531

0.33

80.

149

0.26

9-0

.085

-0.1

83-0

.051

0.00

020

010

04.

192

2.23

91.

248

0.79

60.

437

0.44

00.

017

-0.1

29-0

.032

0.00

020

02.

632

1.28

00.

894

0.33

40.

116

0.12

2-0

.025

-0.0

05-0

.016

0.00

020

04.

013

2.24

71.

499

0.71

30.

340

0.24

90.

046

0.02

9-0

.003

0.00

050

02.

622

1.42

00.

791

0.37

00.

164

0.13

60.

069

0.02

60.

001

0.00

050

03.

915

2.25

41.

300

0.65

90.

326

0.21

80.

112

0.04

50.

008

0.00

050

010

02.

974

1.35

00.

694

0.40

50.

203

0.30

3-0

.076

-0.1

79-0

.048

0.00

050

010

04.

489

2.48

71.

462

0.89

10.

511

0.48

60.

033

-0.1

23-0

.027

0.00

020

02.

913

1.47

41.

005

0.40

50.

170

0.14

6-0

.013

-0.0

02-0

.014

0.00

020

04.

383

2.50

21.

652

0.81

30.

415

0.28

40.

062

0.03

40.

000

0.00

050

03.

138

1.67

40.

918

0.47

80.

198

0.17

60.

078

0.02

90.

001

0.00

050

04.

541

2.59

21.

482

0.80

40.

377

0.27

30.

125

0.05

00.

008

0.00

0T

NR

MSE

TN

RM

SE

100

100

2.81

81.

527

0.98

10.

768

0.44

40.

457

0.25

70.

301

0.15

60.

110

100

100

4.11

62.

368

1.49

61.

097

0.63

20.

593

0.23

10.

253

0.12

20.

076

200

2.73

01.

507

1.09

60.

686

0.41

60.

262

0.21

30.

159

0.15

60.

136

200

3.91

72.

278

1.59

10.

959

0.53

20.

341

0.19

40.

133

0.11

40.

097

500

2.93

91.

547

1.05

50.

623

0.54

90.

314

0.16

50.

161

0.17

00.

175

500

3.99

72.

246

1.48

30.

848

0.65

20.

370

0.15

30.

121

0.12

10.

127

200

100

3.25

61.

787

1.04

20.

722

0.41

40.

404

0.33

10.

244

0.07

10.

002

200

100

4.64

52.

741

1.65

41.

111

0.65

60.

580

0.36

10.

231

0.07

10.

001

200

3.29

91.

992

1.49

00.

790

0.42

30.

291

0.20

50.

099

0.05

40.

006

200

4.57

22.

834

2.00

41.

101

0.59

90.

412

0.24

70.

123

0.06

20.

005

500

3.19

51.

991

1.38

60.

626

0.40

30.

229

0.15

50.

075

0.04

00.

002

500

4.41

42.

754

1.82

80.

908

0.55

90.

316

0.20

10.

097

0.04

70.

001

500

100

3.36

31.

821

1.09

50.

623

0.43

60.

403

0.18

90.

210

0.06

50.

000

500

100

4.81

32.

851

1.77

11.

073

0.69

60.

588

0.22

00.

185

0.06

40.

000

200

3.25

71.

957

1.35

60.

704

0.37

10.

286

0.23

70.

060

0.03

50.

000

200

4.67

22.

885

1.95

21.

060

0.58

00.

410

0.27

40.

085

0.04

10.

000

500

3.65

11.

969

1.28

70.

764

0.37

00.

331

0.14

30.

053

0.01

50.

000

500

4.97

12.

860

1.81

01.

064

0.53

60.

420

0.19

00.

075

0.02

10.

000

δ=

1/3

δ=

1/3

TN

Bia

sT

NB

ias

100

100

5.65

93.

338

2.05

81.

326

0.77

00.

632

0.10

9-0

.090

-0.0

28-0

.021

100

100

8.37

75.

505

3.63

32.

386

1.47

21.

067

0.37

30.

053

0.03

1-0

.012

200

6.00

83.

638

2.42

11.

324

0.69

00.

444

0.13

80.

058

-0.0

08-0

.028

200

8.74

35.

747

3.89

02.

309

1.30

80.

816

0.35

20.

165

0.04

0-0

.017

500

6.64

54.

087

2.49

91.

366

0.74

00.

422

0.18

50.

065

-0.0

08-0

.034

500

9.33

26.

129

3.92

22.

265

1.28

50.

719

0.34

40.

147

0.02

9-0

.021

200

100

6.34

93.

883

2.40

51.

544

0.92

10.

730

0.19

1-0

.037

0.00

20.

000

200

100

9.10

26.

090

4.03

22.

643

1.65

31.

178

0.46

00.

106

0.05

60.

000

200

6.81

54.

303

2.87

41.

600

0.87

90.

558

0.22

00.

112

0.02

80.

000

200

9.55

96.

462

4.40

72.

641

1.53

40.

944

0.43

80.

218

0.06

80.

000

500

7.51

04.

793

2.96

11.

652

0.90

00.

517

0.26

90.

117

0.03

40.

000

500

10.1

976.

880

4.44

22.

604

1.47

10.

828

0.43

20.

194

0.06

20.

000

500

100

6.70

44.

202

2.67

31.

683

1.02

10.

796

0.21

6-0

.027

0.00

90.

000

500

100

9.48

36.

464

4.33

02.

827

1.77

51.

265

0.49

70.

119

0.06

50.

000

200

7.27

14.

659

3.08

91.

752

0.99

30.

612

0.24

50.

122

0.03

30.

000

200

10.0

356.

857

4.66

12.

829

1.68

31.

016

0.47

20.

233

0.07

50.

000

500

8.25

95.

286

3.25

31.

885

0.99

30.

609

0.29

20.

128

0.03

50.

000

500

10.9

427.

418

4.78

42.

888

1.59

80.

948

0.46

60.

209

0.06

40.

000

TN

RM

SE

TN

RM

SE

100

100

6.10

43.

821

2.48

11.

718

1.04

10.

848

0.30

50.

223

0.10

60.

049

100

100

8.72

45.

874

3.96

82.

707

1.70

41.

259

0.52

40.

261

0.12

50.

032

200

6.46

64.

085

2.79

41.

699

0.95

40.

613

0.29

20.

165

0.08

40.

058

200

9.08

66.

094

4.20

12.

624

1.53

50.

974

0.48

20.

253

0.10

00.

039

500

7.17

74.

504

2.93

41.

700

1.10

50.

629

0.27

30.

141

0.06

70.

067

500

9.72

26.

462

4.28

72.

559

1.60

00.

909

0.42

80.

212

0.07

10.

046

200

100

6.71

24.

274

2.73

21.

811

1.11

10.

871

0.45

80.

236

0.08

50.

001

200

100

9.37

56.

387

4.28

82.

868

1.82

31.

315

0.66

90.

301

0.12

40.

001

200

7.21

24.

731

3.25

91.

907

1.09

00.

709

0.38

20.

192

0.08

50.

004

200

9.84

86.

783

4.70

72.

894

1.71

71.

083

0.57

80.

291

0.11

90.

004

500

7.84

75.

147

3.34

81.

876

1.11

20.

620

0.36

00.

171

0.07

20.

001

500

10.4

447.

151

4.75

02.

805

1.66

60.

930

0.52

30.

248

0.09

80.

001

500

100

6.95

64.

478

2.91

31.

839

1.17

40.

897

0.35

10.

177

0.07

80.

000

500

100

9.66

66.

666

4.51

52.

959

1.90

81.

363

0.60

50.

250

0.12

10.

000

200

7.48

34.

928

3.32

01.

944

1.12

80.

722

0.40

20.

165

0.06

70.

000

200

10.1

927.

055

4.84

52.

987

1.80

21.

116

0.59

80.

274

0.10

60.

000

500

8.53

15.

484

3.49

82.

093

1.13

00.

737

0.35

60.

155

0.04

70.

000

500

11.1

377.

573

4.98

03.

065

1.72

21.

064

0.52

70.

237

0.07

60.

000

Not

es:

Par

amet

ers

ofth

ed

yn

amic

pan

eld

ata

mod

el,

(29),

are

gen

erate

das:ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5),ϑi∼U

(0,0.9

5)

an

dγi∼IIDN

(1,1

),fo

r

i=

1,2,...,N

.N

on-G

auss

ian

erro

rsar

ege

ner

ated

asuit

=( v−

2χ2 v,t

) 1/2ν it,

fori

=1,2,...,N

,ν it∼IIDN

(0,1

)an

dχ

2 v,t

isa

chi-

squ

are

dra

nd

om

vari

ate

wit

h

v=

8d

egre

esof

free

dom

,in

(31)

.D

esig

n1

assu

mesb i

∼U

(0.7,0.9

)fo

rth

efi

rstNb

(≤N

)el

emen

tsof

vect

or

bN

,i

=1,2,...,N

,in

the

const

ruct

ion

of

the

corr

elat

ion

mat

rix

ofth

eer

rors

,RN

=I N

+bN

b′ N−

B2 N

giv

enby

(30),

wh

ere

BN

=D

iag

(bN

).c p

(n,δ

)co

rres

pon

ds

toth

ecr

itic

al

valu

eu

sed

inth

em

ult

iple

test

ing

pro

ced

ure

show

nin

(23)

.c p

(n,δ

)co

rres

pon

ds

toth

ecr

itic

al

valu

eu

sed

inth

em

ult

iple

test

ing

pro

ced

ure

show

nin

(23).

Th

enu

mb

erof

rep

lica

tion

sis

set

toR

=20

00.

33

Tab

leA

4a:

Bia

san

dR

MS

E(×

100)

for

theα

esti

mat

eof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

oma

stati

cp

anel

dat

am

od

elC

ross

corr

elati

on

sare

gen

erate

du

sin

gD

esig

n2

wit

hn

on

-Gau

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

c p(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

05c p

(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

10δ

=1/

2δ

=1/

2T

NB

ias

TN

Bia

s10

010

01.

116

-0.9

49-2

.057

-2.5

27-3

.024

-3.0

83-3

.551

-3.7

45-3

.713

-3.7

5410

010

02.

735

0.49

7-0

.797

-1.4

07-2

.004

-2.1

34-2

.645

-2.8

79-2

.869

-2.9

3120

00.

933

-1.0

04-1

.930

-2.7

75-3

.210

-3.3

57-3

.631

-3.6

93-3

.777

-3.8

4420

02.

503

0.35

6-0

.758

-1.7

38-2

.272

-2.4

85-2

.792

-2.8

79-2

.979

-3.0

5550

00.

898

-1.0

48-2

.261

-3.0

08-3

.456

-3.6

15-3

.782

-3.9

01-3

.998

-4.0

6350

02.

404

0.23

8-1

.163

-2.0

48-2

.582

-2.7

95-2

.993

-3.1

26-3

.229

-3.2

9620

010

02.

764

1.26

50.

642

0.53

50.

221

0.27

0-0

.159

-0.3

46-0

.318

-0.3

9420

010

04.

186

2.32

21.

376

1.03

90.

575

0.53

00.

042

-0.1

87-0

.184

-0.2

8120

02.

655

1.33

20.

886

0.30

70.

020

-0.0

36-0

.226

-0.2

83-0

.361

-0.4

3020

04.

035

2.31

11.

535

0.75

10.

324

0.18

5-0

.053

-0.1

41-0

.235

-0.3

1550

02.

623

1.35

90.

641

0.19

0-0

.076

-0.1

38-0

.258

-0.3

42-0

.412

-0.4

6050

03.

925

2.23

61.

209

0.55

80.

175

0.04

3-0

.112

-0.2

15-0

.294

-0.3

4650

010

03.

337

1.82

41.

180

1.09

60.

741

0.79

40.

349

0.12

10.

125

-0.0

0250

010

04.

715

2.80

61.

821

1.48

10.

974

0.92

60.

424

0.15

90.

140

-0.0

0120

03.

163

1.85

11.

417

0.82

20.

525

0.46

70.

270

0.18

70.

094

-0.0

0220

04.

521

2.76

41.

968

1.15

50.

717

0.56

90.

324

0.21

30.

104

-0.0

0150

03.

186

1.89

61.

174

0.69

70.

431

0.35

30.

223

0.13

50.

060

-0.0

0250

04.

510

2.73

91.

667

0.97

10.

576

0.42

60.

258

0.15

10.

065

-0.0

01T

NR

MSE

TN

RM

SE

100

100

2.16

51.

928

2.63

13.

035

3.49

93.

576

3.99

94.

181

4.16

24.

206

100

100

3.41

31.

803

1.73

32.

023

2.48

32.

605

3.05

33.

266

3.27

13.

336

200

2.17

91.

962

2.49

93.

204

3.62

43.

796

4.06

64.

140

4.22

74.

292

200

3.32

51.

811

1.67

52.

227

2.67

52.

893

3.18

43.

281

3.38

43.

460

500

2.48

92.

230

2.88

93.

493

3.90

54.

068

4.23

74.

357

4.45

24.

516

500

3.48

22.

058

2.07

82.

589

3.02

03.

217

3.40

93.

540

3.64

23.

709

200

100

3.21

31.

794

1.13

10.

851

0.54

10.

478

0.38

30.

473

0.45

10.

508

200

100

4.56

02.

717

1.71

61.

255

0.75

90.

637

0.28

90.

313

0.30

70.

373

200

3.21

61.

863

1.23

10.

650

0.38

80.

317

0.37

20.

413

0.47

20.

532

200

4.51

42.

751

1.82

80.

996

0.52

10.

334

0.24

00.

272

0.33

40.

399

500

3.27

91.

932

1.10

20.

598

0.39

40.

343

0.40

00.

467

0.52

90.

571

500

4.49

22.

735

1.59

70.

857

0.43

40.

274

0.26

50.

330

0.39

40.

440

500

100

3.56

92.

057

1.36

01.

190

0.80

60.

821

0.37

70.

156

0.14

40.

010

500

100

4.92

63.

012

1.98

01.

573

1.03

90.

955

0.45

20.

190

0.15

80.

009

200

3.46

62.

107

1.55

60.

915

0.57

80.

486

0.28

00.

193

0.10

00.

004

200

4.79

23.

000

2.11

41.

256

0.77

70.

594

0.33

70.

219

0.10

90.

003

500

3.60

52.

250

1.43

20.

871

0.53

00.

390

0.23

80.

140

0.06

20.

004

500

4.86

73.

055

1.91

11.

144

0.68

00.

469

0.27

70.

157

0.06

80.

003

δ=

1/3

δ=

1/3

TN

Bia

sT

NB

ias

100

100

5.02

92.

438

0.80

5-0

.099

-0.8

95-1

.169

-1.7

67-2

.070

-2.1

01-2

.197

100

100

7.90

24.

856

2.73

61.

406

0.29

7-0

.215

-0.9

57-1

.367

-1.4

67-1

.611

200

5.51

72.

858

1.22

4-0

.128

-0.9

37-1

.333

-1.7

41-1

.900

-2.0

40-2

.139

200

8.40

25.

243

3.06

81.

291

0.15

0-0

.479

-1.0

25-1

.275

-1.4

66-1

.596

500

6.30

73.

401

1.32

0-0

.089

-0.9

74-1

.409

-1.7

29-1

.926

-2.0

60-2

.141

500

9.14

75.

730

3.11

21.

241

0.01

7-0

.644

-1.0

95-1

.363

-1.5

34-1

.633

200

100

6.27

03.

901

2.47

91.

776

1.07

00.

858

0.27

6-0

.022

-0.0

62-0

.190

200

100

8.91

15.

987

3.99

12.

793

1.74

61.

286

0.55

80.

157

0.04

9-0

.125

200

6.77

44.

318

2.87

21.

642

0.90

20.

562

0.20

60.

044

-0.0

90-0

.194

200

9.45

96.

401

4.32

62.

628

1.53

40.

950

0.44

60.

192

0.00

5-0

.131

500

7.48

64.

767

2.86

91.

599

0.81

90.

447

0.16

1-0

.009

-0.1

22-0

.192

500

10.1

316.

809

4.29

82.

520

1.37

80.

770

0.35

10.

108

-0.0

43-0

.133

500

100

6.77

24.

331

2.85

52.

129

1.37

81.

155

0.55

50.

225

0.16

4-0

.001

500

100

9.41

26.

412

4.34

53.

101

2.00

21.

517

0.76

50.

332

0.20

40.

000

200

7.25

24.

723

3.22

21.

942

1.17

90.

822

0.45

70.

274

0.12

60.

000

200

9.92

66.

788

4.64

22.

883

1.75

31.

145

0.62

80.

355

0.15

50.

000

500

8.09

85.

256

3.26

01.

907

1.08

90.

687

0.38

50.

206

0.08

40.

000

500

10.7

307.

297

4.68

02.

806

1.60

70.

960

0.51

90.

265

0.10

40.

000

TN

RM

SE

TN

RM

SE

100

100

5.48

93.

043

1.69

01.

275

1.49

31.

667

2.14

02.

409

2.45

42.

551

100

100

8.22

45.

216

3.11

91.

827

1.03

40.

949

1.33

51.

668

1.77

51.

915

200

6.02

53.

453

1.91

91.

205

1.43

91.

730

2.08

52.

242

2.38

12.

480

200

8.75

85.

623

3.43

31.

738

0.95

20.

995

1.35

91.

581

1.76

31.

890

500

6.88

64.

072

2.17

81.

368

1.53

01.

809

2.08

22.

266

2.39

62.

478

500

9.55

16.

175

3.59

41.

836

1.05

41.

131

1.43

31.

665

1.82

71.

926

200

100

6.57

14.

206

2.73

61.

945

1.19

80.

928

0.37

50.

198

0.19

20.

263

200

100

9.14

16.

221

4.19

32.

934

1.85

11.

344

0.61

40.

229

0.14

50.

182

200

7.13

44.

656

3.12

11.

838

1.03

50.

635

0.27

90.

164

0.18

30.

257

200

9.72

86.

666

4.53

92.

801

1.65

31.

015

0.48

90.

229

0.11

50.

180

500

7.86

75.

131

3.17

31.

828

0.97

40.

530

0.23

40.

151

0.20

30.

258

500

10.4

127.

093

4.55

22.

724

1.51

80.

844

0.39

20.

157

0.12

50.

186

500

100

6.95

24.

506

2.99

62.

219

1.44

21.

186

0.58

30.

252

0.18

10.

008

500

100

9.55

06.

551

4.46

23.

182

2.06

21.

551

0.79

30.

357

0.22

00.

007

200

7.45

84.

915

3.36

12.

045

1.24

50.

853

0.47

30.

282

0.13

20.

002

200

10.0

816.

940

4.76

42.

980

1.82

01.

181

0.64

80.

365

0.16

00.

001

500

8.33

35.

484

3.45

72.

060

1.19

20.

739

0.40

90.

215

0.08

70.

001

500

10.9

027.

473

4.84

32.

941

1.70

41.

014

0.54

70.

276

0.10

70.

001

Not

es:

Par

am

eter

sof

the

stat

icp

anel

dat

am

od

el,(2

9),

are

gen

erate

das:ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5),ϑi

=0

an

dγi∼IIDN

(1,1

),fo

ri

=1,

2,...,N

.D

esig

n2

assu

mes

atw

o-fa

ctor

mod

elw

ith

[Nαβ]an

d[N

αβ2]n

on

-zer

olo

ad

ings

for

the

firs

tan

dse

con

dfa

ctor,

resp

ecti

vel

y.W

ese

t:αβ

2=

2αβ/3

,w

her

eαβ

rela

tes

toα

un

der

(11)

andf jt∼IIDN

(0,1

).N

on-G

auss

ian

erro

rsare

gen

erate

das:uit

=( v−

2χ2 v,t

) 1/2ν it,

fori

=1,

2,...,N

,ν it∼IIDN

(0,1

)an

dχ

2 v,t

isa

chi-

squ

are

d

ran

dom

vari

ate

wit

hv=

8d

egre

esof

free

dom

,in

(31)

.v ij∼IIDU

(µvj−

0.2,µvj

+0.

2),j

=1,

2,µv

=0.

87,µv2

=0.

71,µv1

=√ µ

2 v−N

2(αβ2−αβ

)µ

2 v2),

in(3

2)

and

(33)

.T

he

nu

mb

erof

rep

lica

tion

sis

set

toR

=20

00.

34

Tab

leA

4b

:B

ias

an

dR

MS

E(×

100)

for

theα

esti

mat

eof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

oma

dyn

amic

pan

eld

ata

mod

elw

ith

exog

enou

sre

gres

sors

Cro

ssco

rrel

atio

ns

are

gen

erate

du

sin

gD

esig

n2

wit

hn

on

-Gau

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

c p(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

05c p

(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

10δ

=1/

2δ

=1/

2T

NB

ias

TN

Bia

s10

010

01.

016

-1.1

03-2

.241

-2.7

71-3

.270

-3.3

57-3

.838

-4.0

45-4

.008

-4.0

3610

010

02.

662

0.36

5-0

.939

-1.5

97-2

.202

-2.3

58-2

.883

-3.1

24-3

.111

-3.1

6220

00.

813

-1.1

92-2

.165

-3.0

69-3

.534

-3.7

08-3

.969

-4.0

29-4

.118

-4.1

8420

02.

380

0.19

1-0

.959

-1.9

89-2

.541

-2.7

74-3

.072

-3.1

59-3

.261

-3.3

3750

00.

669

-1.2

97-2

.534

-3.2

94-3

.752

-3.9

26-4

.100

-4.2

16-4

.312

-4.3

8050

02.

163

-0.0

04-1

.413

-2.2

97-2

.832

-3.0

55-3

.256

-3.3

87-3

.488

-3.5

6020

010

02.

749

1.24

90.

619

0.50

60.

180

0.23

5-0

.194

-0.3

87-0

.359

-0.4

3120

010

04.

150

2.29

41.

352

1.01

30.

542

0.50

10.

014

-0.2

17-0

.216

-0.3

0720

02.

578

1.26

70.

834

0.25

6-0

.018

-0.0

70-0

.268

-0.3

16-0

.397

-0.4

6420

03.

950

2.24

11.

482

0.70

10.

292

0.15

7-0

.086

-0.1

66-0

.263

-0.3

4050

02.

568

1.31

10.

610

0.16

1-0

.098

-0.1

61-0

.281

-0.3

66-0

.436

-0.4

8550

03.

869

2.18

81.

178

0.53

10.

155

0.02

5-0

.130

-0.2

33-0

.312

-0.3

6550

010

03.

419

1.88

81.

222

1.11

30.

749

0.79

20.

349

0.12

50.

124

-0.0

0250

010

04.

812

2.88

61.

868

1.50

50.

989

0.92

70.

426

0.16

40.

139

-0.0

0120

03.

202

1.88

21.

439

0.83

90.

535

0.47

10.

272

0.19

10.

092

-0.0

0220

04.

557

2.79

61.

992

1.17

30.

727

0.57

50.

326

0.21

60.

102

-0.0

0150

03.

166

1.87

91.

158

0.68

50.

422

0.34

80.

221

0.13

30.

059

-0.0

0250

04.

489

2.72

21.

651

0.95

90.

567

0.42

00.

256

0.14

90.

065

-0.0

01T

NR

MSE

TN

RM

SE

100

100

2.11

72.

014

2.78

23.

266

3.74

93.

863

4.30

64.

507

4.48

84.

515

100

100

3.35

81.

765

1.79

72.

176

2.67

62.

840

3.30

73.

537

3.54

03.

591

200

2.21

92.

155

2.74

43.

521

3.96

94.

155

4.40

64.

474

4.55

94.

626

200

3.28

51.

877

1.86

12.

502

2.96

73.

189

3.46

93.

562

3.66

13.

737

500

2.42

42.

367

3.10

43.

732

4.15

14.

331

4.50

64.

622

4.71

74.

785

500

3.32

12.

061

2.23

22.

785

3.22

33.

432

3.62

83.

757

3.85

83.

930

200

100

3.19

11.

751

1.07

00.

792

0.49

20.

453

0.40

70.

521

0.50

20.

555

200

100

4.53

32.

687

1.67

51.

210

0.70

70.

603

0.28

70.

347

0.34

70.

408

200

3.13

51.

816

1.21

00.

658

0.43

10.

357

0.42

40.

456

0.52

40.

579

200

4.42

12.

684

1.78

60.

970

0.52

60.

343

0.27

50.

304

0.37

40.

436

500

3.27

01.

965

1.18

10.

708

0.48

60.

394

0.42

50.

483

0.54

50.

591

500

4.45

82.

730

1.63

20.

920

0.51

20.

328

0.29

10.

341

0.40

60.

455

500

100

3.67

42.

127

1.38

61.

183

0.79

10.

808

0.36

60.

154

0.14

30.

006

500

100

5.05

13.

105

2.02

71.

583

1.03

70.

947

0.44

40.

189

0.15

70.

004

200

3.61

82.

254

1.66

11.

001

0.63

10.

509

0.29

10.

198

0.09

90.

006

200

4.91

43.

122

2.20

71.

335

0.82

80.

619

0.34

90.

225

0.10

90.

005

500

3.50

22.

134

1.31

90.

778

0.46

80.

362

0.22

60.

135

0.06

10.

003

500

4.79

22.

967

1.81

91.

063

0.62

10.

439

0.26

30.

151

0.06

70.

002

δ=

1/3

δ=

1/3

TN

Bia

sT

NB

ias

100

100

4.96

72.

337

0.69

1-0

.247

-1.0

47-1

.341

-1.9

60-2

.264

-2.2

93-2

.378

100

100

7.86

84.

785

2.65

51.

301

0.18

6-0

.344

-1.1

12-1

.516

-1.6

17-1

.751

200

5.38

62.

703

1.05

9-0

.320

-1.1

44-1

.548

-1.9

52-2

.111

-2.2

51-2

.349

200

8.27

25.

102

2.92

51.

133

-0.0

17-0

.647

-1.1

90-1

.440

-1.6

32-1

.760

500

6.06

03.

176

1.11

0-0

.281

-1.1

55-1

.589

-1.9

10-2

.103

-2.2

34-2

.319

500

8.90

55.

513

2.91

71.

072

-0.1

36-0

.790

-1.2

39-1

.502

-1.6

69-1

.771

200

100

6.21

63.

864

2.45

61.

755

1.04

60.

840

0.25

6-0

.043

-0.0

86-0

.208

200

100

8.87

95.

963

3.97

92.

784

1.73

21.

277

0.54

50.

142

0.03

1-0

.136

200

6.67

44.

238

2.81

51.

590

0.87

20.

538

0.18

20.

028

-0.1

08-0

.211

200

9.35

06.

310

4.26

22.

570

1.49

80.

925

0.42

50.

179

-0.0

09-0

.142

500

7.41

14.

705

2.82

81.

567

0.79

90.

433

0.15

0-0

.020

-0.1

32-0

.203

500

10.0

626.

749

4.25

52.

487

1.35

60.

756

0.34

20.

100

-0.0

50-0

.141

500

100

6.88

24.

427

2.91

92.

162

1.39

91.

163

0.55

90.

232

0.16

5-0

.001

500

100

9.53

26.

514

4.41

63.

144

2.02

91.

530

0.77

20.

340

0.20

50.

000

200

7.28

44.

756

3.25

01.

961

1.19

30.

830

0.46

00.

279

0.12

5-0

.001

200

9.94

16.

805

4.66

32.

897

1.76

51.

152

0.63

20.

360

0.15

40.

000

500

8.07

35.

235

3.24

31.

895

1.07

90.

680

0.38

10.

203

0.08

40.

000

500

10.7

057.

276

4.66

12.

793

1.59

80.

953

0.51

50.

262

0.10

30.

000

TN

RM

SE

TN

RM

SE

100

100

5.44

02.

965

1.63

81.

304

1.61

81.

840

2.34

92.

627

2.66

82.

753

100

100

8.20

05.

156

3.05

81.

759

1.04

11.

045

1.50

01.

839

1.94

42.

073

200

5.92

33.

366

1.88

51.

347

1.66

01.

950

2.30

12.

456

2.59

02.

688

200

8.64

35.

511

3.33

61.

690

1.03

61.

148

1.52

71.

750

1.92

72.

055

500

6.65

53.

889

2.06

51.

402

1.64

81.

947

2.22

52.

407

2.53

62.

622

500

9.31

65.

974

3.42

91.

730

1.06

21.

215

1.54

01.

773

1.93

32.

036

200

100

6.53

44.

180

2.71

41.

916

1.16

30.

902

0.35

20.

211

0.21

90.

287

200

100

9.12

16.

208

4.18

62.

924

1.83

41.

330

0.59

40.

222

0.15

30.

198

200

7.02

54.

571

3.06

31.

791

1.01

60.

622

0.27

60.

175

0.21

00.

281

200

9.61

36.

569

4.47

22.

744

1.62

20.

994

0.47

50.

225

0.13

20.

198

500

7.79

95.

083

3.15

51.

831

0.99

60.

554

0.25

50.

156

0.20

70.

266

500

10.3

467.

040

4.52

22.

710

1.52

10.

855

0.40

40.

160

0.12

60.

191

500

100

7.08

04.

614

3.06

52.

246

1.45

41.

188

0.57

90.

255

0.18

10.

003

500

100

9.68

56.

665

4.54

23.

226

2.08

51.

559

0.79

50.

361

0.22

00.

002

200

7.54

45.

006

3.43

92.

111

1.29

40.

881

0.48

80.

290

0.13

20.

003

200

10.1

316.

996

4.82

13.

029

1.86

01.

206

0.66

20.

373

0.16

10.

003

500

8.28

55.

432

3.40

02.

006

1.14

60.

710

0.39

40.

208

0.08

60.

001

500

10.8

637.

433

4.79

92.

899

1.66

70.

988

0.53

10.

268

0.10

50.

001

Not

es:

Par

amet

ers

ofth

ed

yn

amic

pan

eld

ata

model

,(2

9),

are

gen

erate

das:ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5),ϑi∼U

(0,0.9

5)

an

dγi∼IIDN

(1,1

),fo

ri

=1,

2,...,N

.D

esig

n2

assu

mes

atw

o-fa

ctor

mod

elw

ith

[Nαβ]

an

d[N

αβ2]

non

-zer

olo

ad

ings

for

the

firs

tan

dse

con

dfa

ctor,

resp

ecti

vely

.W

ese

t:

αβ

2=

2αβ/3,

wh

ereαβ

rela

tes

toα

un

der

(11)

andf jt∼IIDN

(0,1

).N

on

-Gau

ssia

ner

rors

are

gen

erate

das:

uit

=( v−

2χ2 v,t

) 1/2ν it,

fori

=1,2,...,N

,

ν it∼IIDN

(0,1

)an

dχ

2 v,t

isa

chi-

squ

ared

ran

dom

vari

ate

wit

hv=

8d

egre

esof

free

dom

,in

(31).

v ij∼IIDU

(µvj−

0.2,µvj

+0.

2),j

=1,

2,µv

=0.

87,

µv2

=0.

71,µv1

=√ µ

2 v−N

2(αβ2−αβ

)µ

2 v2),

in(3

2)an

d(3

3).

Th

enu

mb

erof

rep

lica

tion

sis

set

toR

=2000.

35

Tab

leB

1:C

omp

aris

onof

Bia

san

dR

MS

E(×

100)

for

theα

andα

esti

mat

esof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

oma

stat

ican

dd

yn

amic

pan

eld

ata

mod

elw

ith

exog

enou

sre

gres

sors

Cro

ssco

rrel

atio

ns

are

gen

erate

du

sin

gD

esig

n1

wit

hG

au

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Bia

sR

MSE

Sta

tic

model

:ϑi

=0

fori

=1,

2,...,N

TN

αT

Nα

100

100

0.20

4-0

.609

-0.5

84-0

.355

-0.2

89-0

.001

-0.2

59-0

.279

-0.0

94-0

.016

100

100

0.31

10.

633

0.59

60.

364

0.29

60.

051

0.26

40.

282

0.10

20.

041

200

0.06

6-0

.426

-0.1

56-0

.302

-0.2

54-0

.101

-0.1

65-0

.080

-0.0

58-0

.025

200

0.14

50.

438

0.17

40.

308

0.26

10.

124

0.17

10.

093

0.07

00.

053

500

0.09

0-0

.120

-0.1

62-0

.144

-0.1

49-0

.035

-0.0

42-0

.046

-0.0

50-0

.041

500

0.11

80.

137

0.17

70.

157

0.16

50.

073

0.07

80.

076

0.07

70.

079

200

100

0.24

3-0

.584

-0.5

57-0

.328

-0.2

670.

017

-0.2

40-0

.262

-0.0

780.

000

200

100

0.34

70.

611

0.57

10.

338

0.27

20.

042

0.24

20.

263

0.07

90.

000

200

0.11

8-0

.380

-0.1

20-0

.271

-0.2

26-0

.074

-0.1

39-0

.057

-0.0

350.

000

200

0.18

30.

391

0.13

30.

274

0.22

80.

076

0.14

00.

057

0.03

50.

000

500

0.15

8-0

.064

-0.1

14-0

.101

-0.1

060.

007

-0.0

01-0

.006

-0.0

110.

000

500

0.17

30.

075

0.11

60.

102

0.10

70.

010

0.00

40.

007

0.01

10.

000

500

100

0.27

7-0

.556

-0.5

49-0

.325

-0.2

610.

019

-0.2

39-0

.262

-0.0

780.

000

500

100

0.37

40.

584

0.56

30.

337

0.26

80.

043

0.24

00.

263

0.07

80.

000

200

0.15

7-0

.367

-0.1

10-0

.265

-0.2

23-0

.073

-0.1

38-0

.056

-0.0

350.

000

200

0.21

80.

379

0.12

60.

268

0.22

50.

075

0.13

80.

057

0.03

50.

000

500

0.18

9-0

.047

-0.1

05-0

.097

-0.1

040.

009

0.00

0-0

.006

-0.0

110.

000

500

0.20

30.

063

0.10

80.

098

0.10

40.

011

0.00

40.

007

0.01

10.

000

TN

αT

Nα

100

100

1.22

0-0

.126

-0.2

87-0

.090

-0.0

300.

216

-0.0

96-0

.115

0.05

10.

105

100

100

2.20

81.

607

1.43

51.

046

0.78

60.

610

0.43

80.

324

0.18

80.

105

200

0.46

3-0

.305

-0.0

46-0

.191

-0.0

910.

040

-0.0

220.

056

0.07

00.

093

200

1.39

21.

292

1.00

60.

826

0.58

90.

416

0.29

10.

206

0.13

60.

093

500

0.16

2-0

.182

-0.1

16-0

.010

-0.0

160.

091

0.08

00.

080

0.07

00.

080

500

1.05

70.

996

0.76

20.

575

0.38

60.

295

0.20

30.

142

0.09

70.

080

200

100

1.91

60.

155

-0.2

08-0

.091

-0.0

930.

147

-0.1

34-0

.168

-0.0

010.

050

200

100

2.57

61.

473

1.15

20.

842

0.60

50.

452

0.34

60.

275

0.13

90.

050

200

0.53

0-0

.375

-0.1

43-0

.221

-0.1

760.

003

-0.0

75-0

.002

0.02

20.

045

200

1.12

31.

036

0.76

70.

615

0.45

40.

298

0.22

20.

139

0.08

60.

045

500

0.14

4-0

.244

-0.1

88-0

.086

-0.0

810.

052

0.04

40.

036

0.03

10.

040

500

0.74

20.

759

0.57

50.

425

0.29

10.

202

0.14

10.

094

0.05

60.

040

500

100

6.30

12.

825

0.92

80.

325

0.04

50.

186

-0.1

33-0

.190

-0.0

300.

018

500

100

6.67

53.

383

1.62

20.

932

0.56

50.

415

0.29

10.

256

0.11

10.

018

200

1.49

50.

021

0.03

3-0

.221

-0.1

77-0

.037

-0.1

04-0

.027

-0.0

120.

017

200

1.86

90.

881

0.60

70.

477

0.34

30.

212

0.17

80.

098

0.05

70.

017

500

0.14

8-0

.269

-0.2

16-0

.118

-0.0

990.

022

0.01

70.

011

0.00

70.

015

500

0.54

70.

544

0.42

70.

280

0.20

50.

123

0.08

90.

055

0.03

10.

015

Dynam

icm

odel

wit

hex

ogen

ous

regr

esso

rs:ϑi∼U

(0,0.9

5)fo

ri

=1,

2,...,N

TN

αT

Nα

100

100

0.19

8-0

.621

-0.5

94-0

.363

-0.2

95-0

.007

-0.2

61-0

.284

-0.0

99-0

.021

100

100

0.30

20.

644

0.60

70.

375

0.30

50.

056

0.26

60.

289

0.10

80.

045

200

0.06

4-0

.432

-0.1

61-0

.309

-0.2

62-0

.108

-0.1

73-0

.088

-0.0

70-0

.033

200

0.15

30.

444

0.18

00.

316

0.27

20.

126

0.18

30.

106

0.09

40.

064

500

0.08

4-0

.123

-0.1

69-0

.158

-0.1

58-0

.046

-0.0

54-0

.059

-0.0

66-0

.049

500

0.11

70.

144

0.18

50.

179

0.17

40.

102

0.10

20.

102

0.10

70.

090

200

100

0.25

5-0

.573

-0.5

61-0

.330

-0.2

660.

018

-0.2

40-0

.263

-0.0

780.

000

200

100

0.35

40.

600

0.57

30.

339

0.27

20.

043

0.24

10.

264

0.07

90.

000

200

0.12

3-0

.381

-0.1

18-0

.272

-0.2

25-0

.074

-0.1

39-0

.056

-0.0

350.

000

200

0.19

00.

392

0.13

10.

274

0.22

60.

076

0.13

90.

057

0.03

50.

000

500

0.15

6-0

.063

-0.1

13-0

.101

-0.1

060.

007

-0.0

01-0

.007

-0.0

110.

000

500

0.16

90.

074

0.11

60.

102

0.10

70.

010

0.00

40.

007

0.01

10.

000

500

100

0.28

1-0

.550

-0.5

50-0

.322

-0.2

610.

020

-0.2

40-0

.261

-0.0

780.

000

500

100

0.37

50.

580

0.56

30.

332

0.26

70.

045

0.24

10.

262

0.07

90.

000

200

0.15

3-0

.366

-0.1

08-0

.266

-0.2

22-0

.072

-0.1

37-0

.056

-0.0

350.

000

200

0.21

30.

378

0.12

30.

270

0.22

40.

074

0.13

80.

057

0.03

50.

000

500

0.18

5-0

.049

-0.1

04-0

.097

-0.1

040.

009

0.00

0-0

.006

-0.0

110.

000

500

0.19

80.

065

0.10

70.

098

0.10

40.

011

0.00

40.

007

0.01

10.

000

TN

αT

Nα

100

100

1.21

9-0

.011

-0.2

92-0

.137

-0.0

330.

188

-0.0

85-0

.123

0.05

60.

106

100

100

2.16

01.

722

1.41

01.

091

0.80

90.

619

0.44

60.

317

0.19

20.

106

200

0.47

2-0

.353

-0.0

80-0

.172

-0.1

020.

043

-0.0

310.

057

0.06

90.

094

200

1.48

01.

319

1.04

40.

812

0.60

80.

421

0.30

70.

205

0.13

60.

094

500

0.15

2-0

.178

-0.1

28-0

.053

-0.0

310.

092

0.09

10.

082

0.07

30.

081

500

1.05

90.

986

0.81

00.

564

0.40

70.

294

0.21

00.

146

0.09

90.

081

200

100

1.93

80.

167

-0.1

58-0

.096

-0.0

830.

155

-0.1

19-0

.170

-0.0

030.

051

200

100

2.59

81.

444

1.15

20.

863

0.60

10.

476

0.34

60.

276

0.13

50.

051

200

0.54

7-0

.363

-0.1

01-0

.227

-0.1

49-0

.011

-0.0

810.

001

0.01

80.

046

200

1.16

61.

023

0.76

50.

612

0.43

90.

294

0.22

10.

139

0.08

50.

046

500

0.13

4-0

.238

-0.2

00-0

.100

-0.0

690.

048

0.04

20.

034

0.03

20.

040

500

0.74

10.

746

0.58

90.

403

0.29

60.

205

0.13

80.

093

0.05

80.

040

500

100

6.14

52.

677

0.95

20.

313

0.04

30.

184

-0.1

42-0

.187

-0.0

310.

018

500

100

6.49

43.

260

1.65

80.

900

0.55

40.

410

0.29

20.

256

0.11

10.

018

200

1.48

90.

038

0.02

4-0

.205

-0.1

91-0

.031

-0.1

04-0

.028

-0.0

100.

017

200

1.89

60.

881

0.60

60.

464

0.35

40.

204

0.17

80.

097

0.05

90.

017

500

0.16

5-0

.264

-0.2

12-0

.124

-0.1

030.

020

0.01

70.

009

0.00

50.

015

500

0.57

10.

561

0.40

40.

284

0.20

60.

130

0.08

60.

056

0.03

10.

015

Not

es:

Rem

ain

ing

par

amet

ers

ofth

ep

anel

dat

am

od

el,

(29),

are

gen

erate

das:ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5)

an

dγi∼IIDN

(1,1

),fo

ri

=1,

2,...,N

.G

auss

ian

erro

rsar

ege

ner

ated

asuit

∼IIDN

(0,1

)in

(31).

Des

ign

1ass

um

esb i

∼U

(0.7,0.9

)fo

rth

efi

rstNb

(≤N

)el

emen

tsof

vec

tor

bN

,i

=1,

2,...,N

,in

the

con

stru

ctio

nof

the

corr

elat

ion

mat

rix

ofth

eer

rors

,RN

=I N

+bN

b′ N−

B2 N

giv

enby

(30),

wh

ere

BN

=D

iag

(bN

).α

isco

mp

ute

du

sin

gc p

(n,δ

)w

ithn

=N

(N−

1)/2

,δ

=1/

2an

dp

=0.

05in

the

mu

ltip

lete

stin

gp

roce

du

resh

own

in(2

3).α

corr

esp

on

ds

toth

em

ost

rob

ust

esti

mato

rof

the

exp

on

ent

of

cros

s-se

ctio

nal

dep

end

ence

con

sid

ered

inB

aile

yet

al.

(2016)

an

dall

ows

for

both

seri

al

corr

elati

on

inth

efa

ctors

an

dw

eak

cross

-sec

tion

al

dep

end

ence

inth

eer

ror

term

s.W

eu

sefo

ur

pri

nci

pal

com

pon

ents

wh

enes

tim

ati

ngc N

inth

eex

pre

ssio

nfo

rα

.T

he

nu

mb

erof

rep

lica

tion

sis

set

toR

=2000.

36

Tab

leB

2:C

om

pari

son

ofB

ias

an

dR

MS

E(×

100

)fo

rth

eα

andα

ofth

ecr

oss-

sect

ion

alex

pon

ent

ofth

eer

rors

ofa

stat

ican

dd

yn

am

icp

anel

dat

am

od

elw

ith

exog

enou

sre

gres

sors

Cro

ssco

rrel

atio

ns

are

gen

erate

du

sin

gD

esig

n2

wit

hG

au

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Bia

sR

MSE

Sta

tic

model

:ϑi

=0,

fori

=1,

2,...,N

TN

αT

Nα

100

100

-0.8

80-2

.378

-2.9

22-3

.026

-3.2

32-3

.131

-3.5

09-3

.648

-3.5

76-3

.598

100

100

1.25

22.

688

3.30

13.

469

3.69

93.

623

3.97

44.

094

4.02

94.

051

200

-1.2

21-2

.480

-2.8

00-3

.267

-3.4

64-3

.434

-3.5

95-3

.591

-3.6

58-3

.714

200

1.45

52.

748

3.16

73.

655

3.86

83.

859

4.00

24.

009

4.07

04.

130

500

-1.5

01-2

.658

-3.2

66-3

.581

-3.7

80-3

.779

-3.8

59-3

.934

-4.0

11-4

.057

500

1.66

22.

899

3.58

03.

938

4.15

34.

176

4.26

04.

336

4.41

14.

456

200

100

0.70

5-0

.090

-0.1

600.

120

0.00

90.

205

-0.1

45-0

.299

-0.2

42-0

.304

200

100

0.80

30.

377

0.36

90.

346

0.29

80.

354

0.31

20.

405

0.35

90.

399

200

0.37

5-0

.068

0.13

4-0

.086

-0.1

46-0

.063

-0.1

91-0

.206

-0.2

69-0

.326

200

0.45

70.

291

0.30

70.

274

0.29

60.

251

0.31

50.

318

0.36

70.

415

500

0.21

9-0

.024

-0.0

86-0

.158

-0.2

06-0

.153

-0.2

12-0

.267

-0.3

25-0

.368

500

0.30

30.

241

0.26

20.

299

0.32

80.

295

0.32

60.

366

0.41

80.

456

500

100

1.04

30.

295

0.24

70.

563

0.43

30.

625

0.26

40.

084

0.11

4-0

.001

500

100

1.06

80.

337

0.27

20.

571

0.44

20.

631

0.27

90.

116

0.13

60.

002

200

0.68

30.

313

0.55

00.

326

0.25

40.

329

0.20

30.

161

0.08

7-0

.001

200

0.69

70.

325

0.55

30.

329

0.25

70.

332

0.20

80.

166

0.09

30.

001

500

0.53

00.

355

0.33

10.

249

0.20

50.

246

0.17

60.

117

0.05

6-0

.001

500

0.53

40.

357

0.33

20.

250

0.20

60.

246

0.17

70.

118

0.05

70.

001

TN

αT

Nα

100

100

1.17

20.

453

0.38

30.

727

0.53

10.

700

0.31

00.

144

0.19

80.

104

100

100

2.34

91.

859

1.54

31.

430

1.07

31.

002

0.61

40.

393

0.30

10.

104

200

0.25

00.

045

0.46

70.

320

0.27

20.

358

0.22

40.

207

0.15

20.

093

200

1.45

31.

314

1.14

40.

878

0.67

20.

590

0.41

20.

312

0.20

50.

093

500

-0.1

48-0

.003

0.18

30.

210

0.20

40.

269

0.21

30.

164

0.12

00.

080

500

1.06

60.

940

0.75

50.

583

0.45

70.

400

0.30

10.

218

0.14

40.

080

200

100

2.80

61.

587

1.10

81.

329

0.84

40.

894

0.42

80.

181

0.18

10.

051

200

100

3.43

82.

401

1.86

21.

879

1.27

71.

157

0.69

90.

414

0.28

80.

051

200

0.90

60.

459

0.68

70.

431

0.30

10.

357

0.21

80.

178

0.11

70.

046

200

1.47

81.

251

1.17

90.

869

0.61

20.

540

0.37

30.

262

0.16

40.

046

500

0.24

60.

109

0.21

90.

197

0.17

20.

229

0.17

40.

127

0.08

10.

040

500

0.78

10.

714

0.61

40.

468

0.36

30.

325

0.23

90.

166

0.10

00.

040

500

100

7.42

54.

897

3.57

23.

447

2.35

51.

976

1.18

40.

587

0.37

40.

019

500

100

7.72

35.

218

3.88

13.

685

2.63

32.

223

1.45

10.

825

0.52

20.

019

200

2.63

72.

154

2.10

21.

495

0.94

00.

763

0.49

00.

275

0.15

60.

018

200

2.95

62.

505

2.39

71.

811

1.23

10.

984

0.67

00.

383

0.22

30.

018

500

0.62

30.

529

0.65

20.

399

0.30

50.

281

0.18

50.

121

0.06

40.

016

500

0.94

80.

917

0.95

30.

625

0.47

90.

370

0.24

20.

156

0.08

50.

016

Dynam

icm

odel

wit

hex

ogen

ous

regr

esso

rs:ϑi∼U

(0,0.9

5)fo

ri

=1,

2,...,N

TN

αT

Nα

100

100

-1.0

16-2

.571

-3.1

83-3

.319

-3.5

68-3

.479

-3.8

38-3

.998

-3.9

22-3

.952

100

100

1.36

02.

876

3.54

93.

762

4.03

13.

981

4.30

14.

455

4.37

84.

411

200

-1.3

20-2

.642

-3.0

32-3

.558

-3.7

45-3

.763

-3.9

18-3

.925

-4.0

03-4

.057

200

1.53

62.

902

3.39

83.

954

4.17

04.

208

4.35

84.

367

4.44

54.

497

500

-1.6

10-2

.849

-3.5

17-3

.890

-4.1

07-4

.124

-4.2

22-4

.304

-4.3

88-4

.439

500

1.76

93.

101

3.84

94.

273

4.51

24.

554

4.65

94.

740

4.82

24.

870

200

100

0.68

9-0

.109

-0.1

790.

088

-0.0

300.

166

-0.1

81-0

.329

-0.2

72-0

.336

200

100

0.79

40.

393

0.39

30.

369

0.32

60.

359

0.35

50.

440

0.39

80.

446

200

0.35

7-0

.089

0.11

5-0

.120

-0.1

77-0

.095

-0.2

27-0

.240

-0.3

01-0

.360

200

0.44

80.

292

0.30

10.

306

0.33

00.

284

0.35

40.

355

0.40

10.

453

500

0.20

9-0

.033

-0.0

95-0

.170

-0.2

15-0

.165

-0.2

25-0

.281

-0.3

37-0

.380

500

0.29

50.

245

0.27

40.

306

0.32

90.

296

0.33

50.

376

0.42

20.

463

500

100

1.04

00.

294

0.24

80.

565

0.43

60.

627

0.26

10.

081

0.11

5-0

.001

500

100

1.06

50.

332

0.27

20.

573

0.44

30.

634

0.27

60.

118

0.13

50.

002

200

0.68

50.

313

0.54

80.

324

0.25

30.

330

0.20

20.

162

0.08

6-0

.001

200

0.69

80.

326

0.55

10.

328

0.25

70.

332

0.20

70.

166

0.09

20.

001

500

0.52

70.

353

0.33

00.

249

0.20

40.

246

0.17

60.

117

0.05

5-0

.001

500

0.53

10.

355

0.33

10.

250

0.20

50.

246

0.17

60.

118

0.05

70.

001

TN

αT

Nα

100

100

1.09

20.

325

0.31

20.

674

0.49

60.

664

0.29

10.

130

0.18

70.

105

100

100

2.27

41.

834

1.49

21.

393

1.04

70.

958

0.59

70.

386

0.28

90.

105

200

0.15

7-0

.008

0.43

20.

286

0.25

10.

341

0.22

20.

201

0.14

80.

094

200

1.47

01.

363

1.16

30.

888

0.68

70.

586

0.41

80.

308

0.20

10.

094

500

-0.1

60-0

.033

0.16

20.

193

0.18

80.

257

0.20

40.

157

0.11

60.

081

500

1.11

00.

978

0.77

20.

587

0.45

70.

399

0.29

80.

215

0.14

20.

081

200

100

2.79

11.

573

1.13

21.

355

0.83

20.

878

0.41

40.

168

0.18

70.

051

200

100

3.39

52.

377

1.89

01.

911

1.28

01.

161

0.70

80.

414

0.30

50.

051

200

0.88

10.

432

0.67

90.

416

0.29

00.

345

0.21

10.

175

0.11

40.

046

200

1.46

31.

254

1.19

70.

864

0.62

30.

542

0.37

20.

261

0.16

20.

046

500

0.26

10.

128

0.24

00.

206

0.18

10.

237

0.18

00.

133

0.08

10.

040

500

0.80

10.

733

0.63

00.

473

0.36

90.

332

0.24

40.

174

0.10

10.

040

500

100

7.38

84.

904

3.52

93.

455

2.33

41.

980

1.18

80.

579

0.37

70.

019

500

100

7.71

45.

231

3.84

93.

696

2.62

02.

222

1.44

80.

829

0.52

70.

019

200

2.60

32.

107

2.04

61.

437

0.88

30.

723

0.45

90.

266

0.14

30.

018

200

2.93

22.

460

2.34

31.

753

1.17

60.

939

0.64

20.

370

0.20

90.

018

500

0.63

60.

531

0.64

40.

397

0.29

20.

274

0.18

30.

121

0.06

30.

016

500

0.95

50.

904

0.93

50.

620

0.47

20.

363

0.24

00.

156

0.08

40.

016

Not

es:

Rem

ain

ing

par

amet

ers

ofth

ep

anel

dat

am

od

el,

(29),

are

gen

erate

das:ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5)

an

dγi∼IIDN

(1,1

),fo

ri

=1,

2,...,N

.D

esig

n2

assu

mes

atw

o-fa

ctor

mod

elw

ith

[Nαβ]

and

[Nαβ2]

non

-zer

olo

ad

ings

for

the

firs

tan

dse

con

dfa

ctor,

resp

ecti

vely

.W

ese

t:αβ

2=

2αβ/3

,w

her

eαβ

rela

tes

toα

un

der

(11)

,f jt

anduit∼IIDN

(0,1

),v ij∼IIDU

(µvj−

0.2,µvj

+0.2

),j

=1,

2,µv

=0.

87,µv2

=0.

71,µv1

=√ µ

2 v−N

2(αβ2−αβ

)µ

2 v2),

in(3

2)

an

d

(33)

.α

isco

mp

ute

du

sin

gc p

(n,δ

)w

ithn

=N

(N−

1)/2

,δ

=1/2

an

dp

=0.0

5in

the

mu

ltip

lete

stin

gp

roce

du

resh

own

in(2

3).α

corr

esp

on

ds

toth

em

ost

rob

ust

esti

mat

orof

the

exp

onen

tof

cros

s-se

ctio

nal

dep

end

ence

con

sid

ered

inB

ailey

etal.

(2016)

an

dall

ows

for

both

seri

al

corr

elati

on

inth

efa

ctors

an

dw

eak

cros

s-se

ctio

nal

dep

end

ence

inth

eer

ror

term

s.W

eu

sefo

ur

pri

nci

pal

com

pon

ents

wh

enes

tim

ati

ngc N

inth

eex

pre

ssio

nfo

rα

.T

he

num

ber

of

rep

lica

tion

sis

set

toR

=20

00.

37

Tab

leC

:B

ias

an

dR

MS

E(×

100)

ofα

esti

mate

sof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsu

sin

gth

em

axim

um

eige

nva

lue

ofth

eir

sam

ple

corr

elati

on

mat

rix

ina

stat

ican

dd

yn

amic

pan

eld

ata

mod

elw

ith

exog

enou

sre

gre

ssor

sC

ross

corr

elat

ion

sare

gen

erate

du

sin

gD

esig

ns

1an

d2

wit

hG

au

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Bia

sR

MSE

Des

ign

1:Sta

tic

model

-ϑi

=0

,fo

ri

=1,

2,...,N

TN

αes

tim

ate

usi

ng

eige

nva

lue

appro

ach

TN

αes

tim

ate

usi

ng

eige

nva

lue

appro

ach

100

100

-14.

080

-12.

332

-10.

881

-10.

088

-9.8

17-9

.406

-9.7

05-9

.782

-9.7

00-9

.706

100

100

14.1

3112

.391

10.9

4210

.152

9.88

99.

471

9.77

19.

845

9.76

39.

772

200

-11.

253

-9.6

74-8

.509

-8.3

38-8

.267

-8.1

89-8

.364

-8.3

52-8

.443

-8.5

3420

011

.291

9.72

38.

564

8.39

38.

323

8.24

58.

418

8.40

68.

496

8.58

850

0-7

.719

-6.6

71-6

.427

-6.4

69-6

.682

-6.7

50-6

.931

-7.0

69-7

.197

-7.2

8350

07.

751

6.71

06.

471

6.51

56.

728

6.79

66.

977

7.11

47.

239

7.32

820

010

0-1

5.98

7-1

3.47

9-1

1.63

4-1

0.52

0-1

0.07

6-9

.572

-9.8

06-9

.839

-9.6

50-9

.641

200

100

16.0

2413

.519

11.6

7410

.563

10.1

179.

614

9.84

39.

874

9.68

59.

676

200

-13.

357

-10.

916

-9.2

94-8

.883

-8.6

09-8

.368

-8.4

50-8

.371

-8.3

95-8

.375

200

13.3

8410

.946

9.32

78.

915

8.64

08.

400

8.47

98.

400

8.42

38.

402

500

-10.

018

-8.1

02-7

.384

-7.1

10-7

.032

-6.9

77-7

.013

-7.1

23-7

.159

-7.1

7750

010

.037

8.12

57.

408

7.13

57.

055

7.00

07.

038

7.14

87.

182

7.19

950

010

0-1

7.22

1-1

4.20

8-1

2.09

5-1

0.83

0-1

0.23

0-9

.670

-9.8

26-9

.780

-9.5

77-9

.513

500

100

17.2

4814

.236

12.1

2210

.858

10.2

569.

694

9.84

69.

800

9.59

69.

529

200

-14.

655

-11.

707

-9.7

62-9

.195

-8.7

99-8

.474

-8.5

01-8

.360

-8.3

37-8

.320

200

14.6

7511

.728

9.78

29.

213

8.81

68.

490

8.51

58.

373

8.35

08.

332

500

-11.

613

-9.0

02-7

.948

-7.4

76-7

.302

-7.0

85-7

.074

-7.0

95-7

.106

-7.1

1450

011

.626

9.01

57.

961

7.48

97.

314

7.09

67.

085

7.10

57.

115

7.12

4D

esig

n1:

Dynam

icm

odel

wit

hex

ogen

ous

regr

esso

rs-ϑi∼U

(0,0.9

5)fo

ri

=1,

2,...,N

TN

αes

tim

ate

usi

ng

eige

nva

lue

appro

ach

TN

αes

tim

ate

usi

ng

eige

nva

lue

appro

ach

100

100

-14.

273

-12.

513

-11.

143

-10.

300

-10.

022

-9.6

08-9

.905

-10.

049

-9.9

07-9

.935

100

100

14.3

2312

.572

11.2

1010

.368

10.0

949.

678

9.96

710

.112

9.97

19.

997

200

-11.

387

-9.7

44-8

.615

-8.5

08-8

.457

-8.3

29-8

.544

-8.5

36-8

.669

-8.7

2320

011

.425

9.79

28.

669

8.56

18.

512

8.38

48.

596

8.59

08.

725

8.77

650

0-7

.782

-6.7

13-6

.555

-6.6

33-6

.792

-6.8

89-7

.059

-7.2

31-7

.380

-7.4

0350

07.

813

6.75

36.

599

6.68

06.

836

6.93

47.

108

7.27

77.

426

7.44

620

010

0-1

6.10

9-1

3.52

4-1

1.71

7-1

0.65

7-1

0.16

9-9

.691

-9.8

70-9

.912

-9.7

64-9

.682

200

100

16.1

4413

.564

11.7

6010

.701

10.2

109.

733

9.90

79.

949

9.79

89.

718

200

-13.

395

-10.

991

-9.3

55-8

.997

-8.6

83-8

.451

-8.4

96-8

.453

-8.4

70-8

.438

200

13.4

2011

.022

9.38

89.

028

8.71

28.

482

8.52

78.

481

8.50

08.

464

500

-10.

067

-8.1

54-7

.470

-7.1

69-7

.125

-7.0

15-7

.111

-7.1

89-7

.205

-7.2

6950

010

.087

8.17

67.

493

7.19

37.

149

7.03

97.

134

7.21

27.

227

7.29

250

010

0-1

7.23

7-1

4.21

7-1

2.11

6-1

0.86

2-1

0.27

6-9

.736

-9.8

76-9

.808

-9.6

32-9

.554

500

100

17.2

6614

.245

12.1

4310

.889

10.3

029.

759

9.89

69.

826

9.65

09.

570

200

-14.

701

-11.

724

-9.7

74-9

.242

-8.8

28-8

.512

-8.5

08-8

.399

-8.3

83-8

.344

200

14.7

2111

.744

9.79

49.

260

8.84

48.

528

8.52

28.

412

8.39

58.

357

500

-11.

622

-9.0

37-7

.975

-7.5

25-7

.308

-7.1

10-7

.118

-7.1

31-7

.143

-7.1

4550

011

.635

9.05

17.

989

7.53

77.

320

7.12

17.

129

7.14

07.

152

7.15

4D

esig

n2:

Sta

tic

model

-ϑi

=0

,fo

ri

=1,

2,...,N

TN

αes

tim

ate

usi

ng

eige

nva

lue

appro

ach

TN

αes

tim

ate

usi

ng

eige

nva

lue

appro

ach

100

100

-20.

441

-19.

904

-19.

231

-18.

859

-18.

828

-18.

720

-19.

151

-19.

367

-19.

377

-19.

458

100

100

20.4

9419

.971

19.3

0618

.939

18.9

1118

.804

19.2

3519

.451

19.4

6119

.542

200

-16.

689

-16.

128

-15.

570

-15.

791

-15.

961

-16.

051

-16.

357

-16.

489

-16.

672

-16.

813

200

16.7

3616

.185

15.6

3615

.859

16.0

3016

.122

16.4

2716

.560

16.7

4316

.885

500

-12.

052

-11.

948

-12.

233

-12.

612

-13.

038

-13.

298

-13.

613

-13.

879

-14.

103

-14.

260

500

12.0

8311

.990

12.2

8212

.664

13.0

9213

.355

13.6

7013

.938

14.1

6214

.320

200

100

-23.

448

-21.

789

-20.

444

-19.

630

-19.

337

-19.

024

-19.

310

-19.

402

-19.

314

-19.

324

200

100

23.4

8821

.833

20.4

9119

.678

19.3

8419

.072

19.3

5619

.447

19.3

5919

.369

200

-19.

823

-18.

132

-16.

909

-16.

718

-16.

586

-16.

469

-16.

617

-16.

628

-16.

714

-16.

780

200

19.8

5318

.167

16.9

4716

.756

16.6

2416

.507

16.6

5416

.665

16.7

5116

.817

500

-15.

427

-14.

141

-13.

740

-13.

655

-13.

743

-13.

750

-13.

888

-14.

025

-14.

149

-14.

240

500

15.4

5014

.168

13.7

7013

.686

13.7

7413

.781

13.9

1914

.056

14.1

8014

.271

500

100

-25.

468

-23.

006

-21.

249

-20.

175

-19.

708

-19.

258

-19.

452

-19.

476

-19.

322

-19.

289

500

100

25.4

9623

.034

21.2

7720

.201

19.7

3219

.281

19.4

7519

.498

19.3

4219

.309

200

-21.

987

-19.

443

-17.

732

-17.

272

-16.

946

-16.

678

-16.

734

-16.

667

-16.

696

-16.

715

200

22.0

0819

.465

17.7

5317

.292

16.9

6416

.695

16.7

5116

.683

16.7

1216

.731

500

-17.

787

-15.

539

-14.

644

-14.

250

-14.

124

-13.

980

-14.

009

-14.

065

-14.

127

-14.

177

500

17.8

0115

.553

14.6

5714

.263

14.1

3713

.993

14.0

2214

.078

14.1

4014

.189

Des

ign

2:D

ynam

icm

odel

wit

hex

ogen

ous

regr

esso

rs-ϑi∼U

(0,0.9

5)fo

ri

=1,

2,...,N

TN

αes

tim

ate

usi

ng

eige

nva

lue

appro

ach

TN

αes

tim

ate

usi

ng

eige

nva

lue

appro

ach

100

100

-20.

516

-20.

025

-19.

395

-19.

051

-19.

063

-18.

960

-19.

387

-19.

614

-19.

633

-19.

726

100

100

20.5

6820

.092

19.4

6919

.130

19.1

4419

.043

19.4

6919

.696

19.7

1519

.809

200

-16.

722

-16.

214

-15.

701

-15.

947

-16.

112

-16.

236

-16.

543

-16.

686

-16.

880

-17.

026

200

16.7

6516

.269

15.7

6516

.015

16.1

8216

.307

16.6

1616

.759

16.9

5517

.100

500

-12.

067

-12.

003

-12.

324

-12.

736

-13.

174

-13.

452

-13.

782

-14.

057

-14.

287

-14.

454

500

12.0

9912

.047

12.3

7512

.791

13.2

3113

.511

13.8

4214

.119

14.3

4914

.516

200

100

-23.

502

-21.

871

-20.

547

-19.

757

-19.

473

-19.

155

-19.

429

-19.

532

-19.

454

-19.

459

200

100

23.5

4221

.917

20.5

9619

.807

19.5

2219

.204

19.4

7719

.578

19.5

0019

.505

200

-19.

885

-18.

203

-16.

982

-16.

815

-16.

671

-16.

562

-16.

722

-16.

742

-16.

825

-16.

896

200

19.9

1518

.238

17.0

1916

.853

16.7

1016

.600

16.7

6116

.780

16.8

6316

.934

500

-15.

445

-14.

156

-13.

756

-13.

679

-13.

765

-13.

786

-13.

930

-14.

068

-14.

191

-14.

277

500

15.4

6614

.182

13.7

8513

.708

13.7

9513

.816

13.9

6014

.098

14.2

2114

.307

500

100

-25.

472

-23.

054

-21.

306

-20.

220

-19.

759

-19.

304

-19.

518

-19.

538

-19.

386

-19.

356

500

100

25.5

0123

.085

21.3

3620

.249

19.7

8619

.329

19.5

4119

.559

19.4

0719

.376

200

-21.

991

-19.

433

-17.

726

-17.

280

-16.

950

-16.

703

-16.

762

-16.

700

-16.

729

-16.

755

200

22.0

1219

.455

17.7

4817

.300

16.9

6816

.721

16.7

7916

.716

16.7

4516

.770

500

-17.

821

-15.

585

-14.

692

-14.

280

-14.

152

-14.

010

-14.

044

-14.

095

-14.

160

-14.

209

500

17.8

3515

.600

14.7

0714

.295

14.1

6614

.024

14.0

5714

.108

14.1

7314

.222

Not

es:

Rem

ain

ing

par

amet

ers

ofth

ep

anel

dat

am

od

el,

(29),

are

gen

erate

das:ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5)

an

dγi∼IIDN

(1,1

),fo

ri

=1,

2,...,N

.G

auss

ian

erro

rsar

ege

ner

ated

asuit∼IIDN

(0,1

)in

(31).

For

det

ail

son

Des

ign

s1

an

d2

refe

rto

footn

ote

sof

Tab

les

A1a

an

dA

2a.

Th

enu

mb

erof

rep

lica

tion

sis

set

toR

=20

00.

38

Tab

leD

:C

over

age

(×100

)fo

rth

eα

esti

mat

eof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

oma

stati

can

dd

yn

am

icp

anel

dat

am

od

elw

ith

exog

enou

sre

gres

sors

Cro

ssco

rrel

ati

on

sare

gen

erate

du

sin

gD

esig

n1

wit

hG

au

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

c p(n,δ

)w

ithn

=N

(N−

1)/2

,δ

=1/

2an

dp

=0.

05c p

(n,δ

)w

ithn

=N

(N−

1)/2

,δ

=1/

2an

dp

=0.

10Sta

tic

model

:ϑi

=0

fori

=1,

2,...,N

TN

TN

100

100

0.00

20.0

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

65.9

510

010

00.

0027

.35

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0053

.70

200

0.00

89.3

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

92.3

020

00.

0090

.15

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0085

.30

500

0.00

99.7

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

99.6

050

00.

0099

.70

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0098

.90

200

100

0.00

19.8

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.15

200

100

0.00

25.0

510

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.00

200

0.00

88.4

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.65

200

0.00

88.8

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.25

500

0.00

99.9

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

6.35

500

0.00

99.9

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

2.85

500

100

0.00

20.9

510

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.00

500

100

0.00

28.2

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.00

200

0.00

88.9

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.00

200

0.00

90.2

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.00

500

0.00

99.7

510

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.00

500

0.00

99.9

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.00

Dynam

icm

odel

wit

hex

ogen

ous

regr

esso

rs:ϑi∼U

(0,0.9

5)fo

ri

=1,

2,...,N

TN

TN

100

100

0.00

18.3

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

69.9

510

010

00.

0024

.35

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0058

.20

200

0.00

86.7

510

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

94.6

020

00.

0088

.20

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0089

.60

500

0.00

99.6

510

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

99.8

550

00.

0099

.70

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0099

.65

200

100

0.00

20.7

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.10

200

100

0.00

26.9

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.10

200

0.00

88.9

510

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.70

200

0.00

90.0

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.30

500

0.00

99.7

510

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

7.60

500

0.00

99.8

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

4.25

500

100

0.00

21.5

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.00

500

100

0.00

28.0

510

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.00

200

0.00

88.2

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.00

200

0.00

89.4

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.00

500

0.00

99.7

010

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.00

500

0.00

99.5

510

0.00

100.

0010

0.00

100.

0010

0.00

100.

0010

0.00

0.00

Not

es:

Rem

ain

ing

par

amet

ers

ofth

ep

anel

dat

am

od

el,

(29),

are

gen

erate

das:ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5)

an

dγi∼IIDN

(1,1

),fo

ri

=1,

2,...,N

.G

auss

ian

erro

rsar

ege

ner

ated

asuit∼IIDN

(0,1

)in

(31).

Des

ign

1ass

um

esb i

∼U

(0.7,0.9

)fo

rth

efirs

tNb

(≤N

)el

emen

tsof

vect

or

bN

,i

=1,2,...,N

,in

the

con

stru

ctio

nof

the

corr

elat

ion

mat

rix

ofth

eer

rors

,RN

=I N

+bN

b′ N−

B2 N

giv

enby

(30),

wh

ere

BN

=D

iag

(bN

).α

an

dit

sem

pir

ical

dis

trib

uti

on

are

com

pu

ted

usi

ngc p

(n,δ

)w

ithn

=N

(N−

1)/2,δ

=1/

2an

dp

=0.

05,

0.10

inth

em

ult

iple

test

ing

pro

ced

ure

show

nin

(23).

Cov

erage

refe

rsto

the

freq

uen

cyat

wh

ichα

(r),B

0.0

5<α

(r)<α

(r),B

0.9

5,

wh

erer

=1,

2,...,R

andb

=1,

2,...,B

.T

he

nu

mb

erof

rep

lica

tions

isse

ttoR

=2000

an

dth

enu

mb

erof

boots

trap

sis

set

toB

=50

0.

39

An Online Supplement for

Exponent of Cross-sectional Dependence for Residuals

Natalia BaileyDepartment of Econometrics and Business Statistics,

Monash University

George KapetaniosKing’s Business School,King’s College London

M. Hashem PesaranDepartment of Economics,

University of Southern California and Trinity College, Cambridge

April 4, 2019

This online supplement provides additional Monte Carlo and empirical results.

Appendix A

Additional Monte Carlo results

The Monte Carlo results provided in the tables below are based on the designs set out in Section6 of the paper.

1

Tab

leS

1a:

Bia

san

dR

MS

E(×

100)

for

theα

esti

mat

eof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

oma

dyn

amic

pan

eld

ata

mod

elw

ith

no

exog

enou

sre

gres

sors

Cro

ssco

rrel

atio

ns

are

gen

erate

du

sin

gD

esig

n1

wit

hG

au

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

c p(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

05c p

(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

10δ

=1/

2δ

=1/

2T

NB

ias

TN

Bia

s10

010

00.

211

-0.6

18-0

.595

-0.3

57-0

.287

-0.0

02-0

.259

-0.2

81-0

.094

-0.0

1710

010

00.

639

-0.3

27-0

.415

-0.2

46-0

.216

0.04

3-0

.230

-0.2

62-0

.083

-0.0

0920

00.

074

-0.4

24-0

.155

-0.3

02-0

.257

-0.1

02-0

.166

-0.0

81-0

.064

-0.0

2920

00.

406

-0.2

14-0

.030

-0.2

25-0

.207

-0.0

70-0

.144

-0.0

66-0

.050

-0.0

1750

00.

087

-0.1

21-0

.164

-0.1

49-0

.152

-0.0

36-0

.043

-0.0

53-0

.051

-0.0

4250

00.

331

0.02

0-0

.081

-0.0

96-0

.117

-0.0

10-0

.022

-0.0

34-0

.035

-0.0

2520

010

00.

267

-0.5

79-0

.553

-0.3

29-0

.270

0.01

7-0

.241

-0.2

63-0

.078

0.00

020

010

00.

709

-0.2

85-0

.366

-0.2

20-0

.204

0.05

7-0

.218

-0.2

51-0

.074

0.00

020

00.

124

-0.3

85-0

.117

-0.2

71-0

.226

-0.0

74-0

.139

-0.0

56-0

.035

0.00

020

00.

471

-0.1

630.

006

-0.1

97-0

.183

-0.0

51-0

.127

-0.0

50-0

.033

0.00

050

00.

153

-0.0

64-0

.112

-0.1

01-0

.106

0.00

7-0

.001

-0.0

07-0

.011

0.00

050

00.

416

0.08

1-0

.033

-0.0

59-0

.084

0.01

80.

005

-0.0

04-0

.010

0.00

050

010

00.

282

-0.5

55-0

.549

-0.3

23-0

.264

0.02

0-0

.239

-0.2

62-0

.078

0.00

050

010

00.

734

-0.2

45-0

.358

-0.2

08-0

.196

0.06

0-0

.216

-0.2

49-0

.073

0.00

020

00.

154

-0.3

63-0

.107

-0.2

66-0

.224

-0.0

72-0

.138

-0.0

56-0

.035

0.00

020

00.

520

-0.1

350.

023

-0.1

91-0

.181

-0.0

48-0

.125

-0.0

50-0

.032

0.00

050

00.

188

-0.0

48-0

.104

-0.0

97-0

.103

0.00

90.

000

-0.0

06-0

.011

0.00

050

00.

468

0.10

9-0

.020

-0.0

52-0

.079

0.02

10.

006

-0.0

04-0

.010

0.00

0T

NR

MSE

TN

RM

SE

100

100

0.32

30.

640

0.60

80.

367

0.29

60.

055

0.26

30.

284

0.10

10.

041

100

100

0.72

00.

403

0.44

70.

271

0.23

20.

074

0.23

50.

264

0.08

70.

025

200

0.15

60.

436

0.17

40.

308

0.26

40.

117

0.17

50.

096

0.08

80.

061

200

0.44

90.

251

0.09

40.

234

0.21

40.

083

0.14

90.

074

0.06

50.

038

500

0.11

70.

140

0.18

00.

165

0.16

80.

085

0.08

50.

099

0.08

40.

078

500

0.34

40.

070

0.10

00.

109

0.12

70.

054

0.05

50.

067

0.05

70.

051

200

100

0.35

90.

602

0.56

60.

339

0.27

50.

043

0.24

20.

263

0.07

90.

000

200

100

0.78

60.

379

0.40

70.

252

0.21

80.

082

0.22

20.

252

0.07

50.

000

200

0.18

70.

396

0.13

10.

274

0.22

70.

076

0.14

00.

057

0.03

50.

000

200

0.51

00.

209

0.08

80.

206

0.18

80.

058

0.12

80.

052

0.03

30.

000

500

0.16

80.

075

0.11

50.

102

0.10

60.

009

0.00

40.

007

0.01

10.

000

500

0.42

70.

098

0.04

90.

063

0.08

50.

021

0.00

80.

006

0.01

10.

000

500

100

0.37

90.

586

0.56

20.

333

0.27

00.

044

0.24

10.

262

0.07

80.

000

500

100

0.81

30.

360

0.39

90.

240

0.21

30.

084

0.22

10.

251

0.07

50.

000

200

0.21

00.

376

0.12

40.

269

0.22

50.

075

0.13

80.

057

0.03

50.

000

200

0.55

60.

192

0.09

20.

200

0.18

50.

056

0.12

60.

051

0.03

30.

000

500

0.20

20.

063

0.10

80.

098

0.10

40.

011

0.00

40.

007

0.01

10.

000

500

0.47

90.

124

0.04

20.

056

0.08

10.

023

0.00

90.

006

0.01

00.

000

δ=

1/3

δ=

1/3

TN

Bia

sT

NB

ias

100

100

1.51

50.

274

-0.0

30-0

.008

-0.0

680.

129

-0.1

77-0

.232

-0.0

69-0

.005

100

100

3.09

91.

393

0.69

90.

444

0.21

50.

296

-0.0

79-0

.180

-0.0

49-0

.002

200

1.45

30.

459

0.36

30.

014

-0.0

650.

010

-0.0

98-0

.039

-0.0

34-0

.007

200

3.00

51.

495

0.98

80.

398

0.15

70.

135

-0.0

30-0

.004

-0.0

19-0

.004

500

1.54

40.

723

0.31

50.

123

0.00

70.

061

0.02

1-0

.003

-0.0

15-0

.009

500

3.05

21.

650

0.85

30.

422

0.16

90.

147

0.06

60.

020

-0.0

04-0

.005

200

100

1.60

40.

335

0.02

40.

022

-0.0

610.

142

-0.1

69-0

.225

-0.0

650.

000

200

100

3.16

31.

461

0.75

20.

480

0.22

00.

306

-0.0

73-0

.176

-0.0

460.

000

200

1.55

20.

528

0.40

80.

042

-0.0

450.

024

-0.0

86-0

.031

-0.0

260.

000

200

3.10

31.

570

1.03

70.

425

0.18

00.

149

-0.0

190.

002

-0.0

140.

000

500

1.68

40.

816

0.37

30.

160

0.03

20.

077

0.03

50.

009

-0.0

060.

000

500

3.21

71.

765

0.92

30.

464

0.19

60.

162

0.07

70.

028

0.00

10.

000

500

100

1.63

30.

378

0.03

60.

033

-0.0

500.

146

-0.1

67-0

.224

-0.0

630.

000

500

100

3.21

11.

498

0.77

50.

491

0.23

40.

313

-0.0

71-0

.173

-0.0

440.

000

200

1.62

10.

574

0.43

30.

054

-0.0

410.

029

-0.0

84-0

.029

-0.0

250.

000

200

3.18

31.

620

1.06

80.

440

0.18

20.

155

-0.0

160.

004

-0.0

130.

000

500

1.77

50.

869

0.40

00.

175

0.04

20.

082

0.03

70.

010

-0.0

050.

000

500

3.32

21.

833

0.96

10.

485

0.21

00.

168

0.08

00.

030

0.00

20.

000

TN

RM

SE

TN

RM

SE

100

100

1.57

90.

437

0.23

90.

168

0.14

00.

156

0.18

80.

236

0.07

40.

014

100

100

3.14

81.

464

0.77

50.

509

0.28

20.

323

0.12

20.

191

0.06

10.

008

200

1.48

20.

503

0.38

70.

097

0.09

50.

049

0.10

40.

047

0.04

20.

019

200

3.02

71.

518

1.00

60.

422

0.18

60.

151

0.05

60.

033

0.02

80.

011

500

1.55

20.

730

0.32

20.

132

0.03

70.

068

0.03

40.

029

0.02

50.

022

500

3.05

81.

656

0.85

80.

427

0.17

50.

150

0.07

00.

029

0.01

40.

013

200

100

1.67

00.

487

0.25

50.

181

0.13

40.

168

0.18

10.

230

0.06

90.

000

200

100

3.21

41.

531

0.82

40.

543

0.28

30.

334

0.12

20.

188

0.05

90.

000

200

1.58

00.

568

0.43

10.

108

0.08

40.

053

0.09

20.

037

0.02

90.

000

200

3.12

31.

592

1.05

50.

449

0.20

80.

166

0.05

40.

032

0.02

40.

000

500

1.69

10.

822

0.37

90.

167

0.04

40.

080

0.03

80.

013

0.00

70.

000

500

3.22

21.

770

0.92

80.

470

0.20

10.

165

0.08

00.

031

0.00

80.

000

500

100

1.70

10.

520

0.25

30.

180

0.13

40.

171

0.18

00.

228

0.06

80.

000

500

100

3.26

11.

568

0.84

60.

553

0.29

80.

341

0.12

20.

186

0.05

90.

000

200

1.64

70.

612

0.45

80.

114

0.08

00.

057

0.09

00.

037

0.02

80.

000

200

3.20

31.

643

1.08

60.

464

0.20

80.

172

0.05

30.

036

0.02

30.

000

500

1.78

30.

876

0.40

70.

181

0.05

40.

085

0.04

00.

014

0.00

70.

000

500

3.32

81.

838

0.96

60.

490

0.21

60.

171

0.08

30.

033

0.00

80.

000

Not

es:

Par

amet

ers

ofth

ed

yn

amic

pan

eld

ata

mod

el,

(29),

are

gen

erate

das:

ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5),ϑi∼U

(0,0.9

5)

an

dγi

=0,

for

i=

1,2,...,N

.G

auss

ian

erro

rsar

ege

ner

ated

asuit∼IIDN

(0,1

)in

(31).

Des

ign

1ass

um

esb i

∼U

(0.7,0.9

)fo

rth

efi

rstNb

(≤N

)el

emen

tsof

vec

tor

bN

,i

=1,

2,...,N

,in

the

con

stru

ctio

nof

the

corr

elat

ion

matr

ixof

the

erro

rs,RN

=I N

+bN

b′ N−

B2 N

giv

enby

(30),

wh

ere

BN

=D

iag

(bN

).c p

(n,δ

)co

rres

pon

ds

toth

ecr

itic

alva

lue

use

din

the

mu

ltip

lete

stin

gpro

ced

ure

show

nin

(23).

Th

enu

mb

erof

rep

lica

tion

sis

set

toR

=2000.

2

Tab

leS

1b:

Bia

san

dR

MS

E(×

100)

for

theα

esti

mat

eof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

oma

dyn

amic

pan

eld

ata

mod

elw

ith

no

exog

enou

sre

gres

sors

Cro

ssco

rrel

ati

on

sare

gen

erate

du

sin

gD

esig

n2

wit

hG

au

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

c p(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

05c p

(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

10δ

=1/

2δ

=1/

2T

NB

ias

Bia

s10

010

0-0

.928

-2.4

37-3

.044

-3.1

37-3

.332

-3.2

47-3

.622

-3.7

52-3

.669

-3.6

9310

010

0-0

.125

-1.5

82-2

.137

-2.2

06-2

.403

-2.3

21-2

.710

-2.8

51-2

.777

-2.8

1620

0-1

.276

-2.5

35-2

.883

-3.3

50-3

.548

-3.5

38-3

.722

-3.7

28-3

.791

-3.8

3720

0-0

.595

-1.7

63-2

.057

-2.4

99-2

.683

-2.6

70-2

.854

-2.8

66-2

.932

-2.9

8350

0-1

.531

-2.7

37-3

.388

-3.7

23-3

.923

-3.9

37-4

.023

-4.0

94-4

.174

-4.2

3950

0-0

.941

-2.0

29-2

.601

-2.8

97-3

.078

-3.0

84-3

.167

-3.2

37-3

.316

-3.3

8020

010

00.

716

-0.0

78-0

.151

0.11

20.

008

0.20

1-0

.158

-0.3

07-0

.249

-0.3

1220

010

01.

211

0.29

80.

129

0.33

90.

188

0.35

6-0

.018

-0.1

83-0

.136

-0.2

1220

00.

368

-0.0

730.

136

-0.0

88-0

.145

-0.0

67-0

.198

-0.2

12-0

.273

-0.3

3120

00.

787

0.23

50.

362

0.09

30.

004

0.06

3-0

.077

-0.1

02-0

.168

-0.2

3150

00.

219

-0.0

26-0

.082

-0.1

54-0

.199

-0.1

48-0

.210

-0.2

65-0

.321

-0.3

6550

00.

546

0.20

90.

097

-0.0

09-0

.073

-0.0

33-0

.102

-0.1

61-0

.218

-0.2

6350

010

01.

044

0.29

80.

250

0.56

60.

432

0.62

90.

260

0.08

60.

113

0.00

050

010

01.

464

0.58

10.

431

0.67

60.

501

0.66

90.

284

0.10

00.

119

0.00

020

00.

674

0.31

00.

547

0.32

50.

254

0.32

90.

203

0.16

20.

085

-0.0

0120

01.

022

0.52

40.

672

0.39

90.

298

0.35

50.

218

0.17

10.

089

0.00

050

00.

529

0.35

50.

331

0.24

90.

205

0.24

70.

176

0.11

70.

056

-0.0

0150

00.

797

0.50

40.

412

0.29

40.

230

0.26

00.

184

0.12

10.

058

0.00

0T

NR

MSE

TN

RM

SE

100

100

1.30

22.

747

3.43

03.

585

3.79

23.

754

4.09

24.

217

4.14

04.

162

100

100

0.85

01.

930

2.51

82.

637

2.82

92.

789

3.13

03.

258

3.19

03.

230

200

1.49

42.

797

3.25

23.

742

3.96

23.

974

4.15

24.

164

4.22

14.

265

200

0.92

82.

042

2.42

32.

862

3.05

83.

064

3.23

63.

253

3.31

33.

363

500

1.70

43.

009

3.73

84.

119

4.34

04.

378

4.46

74.

532

4.61

34.

675

500

1.16

02.

306

2.93

63.

264

3.45

63.

484

3.56

73.

630

3.71

03.

772

200

100

0.81

00.

360

0.35

70.

337

0.30

80.

355

0.32

70.

409

0.36

50.

405

200

100

1.27

70.

449

0.30

50.

427

0.31

00.

425

0.22

20.

277

0.24

40.

286

200

0.45

90.

300

0.31

30.

276

0.29

50.

260

0.31

90.

325

0.37

20.

419

200

0.83

10.

341

0.42

70.

226

0.19

80.

203

0.20

70.

212

0.25

40.

301

500

0.30

10.

237

0.25

80.

288

0.31

00.

280

0.31

70.

357

0.40

40.

446

500

0.57

40.

280

0.21

40.

188

0.19

60.

184

0.20

80.

243

0.28

70.

329

500

100

1.06

90.

339

0.27

40.

575

0.44

00.

635

0.27

50.

120

0.13

40.

002

500

100

1.50

00.

623

0.45

70.

686

0.50

90.

675

0.29

80.

130

0.13

90.

001

200

0.68

70.

322

0.55

00.

328

0.25

70.

332

0.20

80.

167

0.09

20.

001

200

1.03

90.

538

0.67

60.

403

0.30

10.

357

0.22

20.

175

0.09

50.

001

500

0.53

40.

357

0.33

20.

250

0.20

60.

247

0.17

70.

118

0.05

70.

002

500

0.80

30.

507

0.41

40.

295

0.23

00.

261

0.18

40.

122

0.05

90.

001

δ=

1/3

δ=

1/3

TN

Bia

sT

NB

ias

100

100

1.11

5-0

.445

-1.0

89-1

.228

-1.4

93-1

.453

-1.8

71-2

.042

-1.9

89-2

.049

100

100

2.94

21.

087

0.16

1-0

.183

-0.6

08-0

.668

-1.1

53-1

.376

-1.3

63-1

.451

200

0.91

3-0

.396

-0.8

12-1

.348

-1.5

94-1

.626

-1.8

32-1

.870

-1.9

49-2

.012

200

2.74

71.

058

0.32

5-0

.425

-0.8

13-0

.936

-1.1

94-1

.272

-1.3

71-1

.451

500

0.89

1-0

.383

-1.0

88-1

.487

-1.7

25-1

.771

-1.8

77-1

.958

-2.0

42-2

.106

500

2.69

20.

986

-0.0

41-0

.655

-1.0

21-1

.144

-1.2

91-1

.395

-1.4

90-1

.559

200

100

2.12

30.

960

0.58

70.

663

0.42

50.

534

0.12

4-0

.068

-0.0

40-0

.134

200

100

3.67

12.

077

1.34

91.

164

0.76

20.

760

0.28

60.

045

0.03

8-0

.083

200

1.89

20.

989

0.85

50.

434

0.24

80.

248

0.07

60.

025

-0.0

57-0

.132

200

3.42

52.

029

1.50

80.

860

0.52

00.

426

0.20

00.

112

0.00

8-0

.083

500

1.87

01.

040

0.62

50.

340

0.17

70.

159

0.06

0-0

.015

-0.0

81-0

.133

500

3.39

62.

001

1.20

30.

685

0.38

70.

292

0.15

10.

052

-0.0

26-0

.086

500

100

2.30

51.

161

0.80

10.

900

0.64

10.

752

0.33

30.

126

0.12

90.

000

500

100

3.78

02.

216

1.48

71.

324

0.90

80.

908

0.42

50.

174

0.14

80.

000

200

2.06

51.

182

1.05

80.

633

0.43

40.

431

0.26

20.

192

0.09

80.

000

200

3.56

92.

180

1.65

71.

001

0.65

00.

553

0.32

90.

225

0.11

00.

000

500

2.06

21.

231

0.81

50.

514

0.34

90.

323

0.21

60.

137

0.06

40.

000

500

3.56

42.

157

1.34

90.

813

0.51

20.

408

0.25

90.

157

0.07

10.

000

TN

RM

SE

RM

SE

100

100

1.35

51.

026

1.52

71.

678

1.90

21.

893

2.23

82.

390

2.34

42.

402

100

100

3.02

71.

330

0.85

50.

895

1.08

61.

142

1.48

71.

674

1.66

61.

745

200

1.09

00.

883

1.25

31.

696

1.92

41.

967

2.15

12.

189

2.26

22.

323

200

2.79

41.

222

0.79

10.

889

1.15

81.

266

1.47

61.

548

1.63

61.

711

500

1.01

70.

848

1.43

91.

816

2.04

12.

099

2.19

62.

269

2.35

12.

414

500

2.71

81.

134

0.71

21.

035

1.32

91.

447

1.57

31.

663

1.75

41.

821

200

100

2.17

31.

024

0.64

50.

700

0.47

00.

566

0.21

40.

173

0.15

50.

189

200

100

3.71

12.

118

1.38

21.

187

0.78

50.

777

0.32

00.

135

0.12

20.

122

200

1.91

51.

016

0.87

50.

460

0.28

50.

281

0.15

10.

126

0.13

70.

181

200

3.44

22.

045

1.51

90.

872

0.53

30.

439

0.22

40.

144

0.09

00.

119

500

1.87

81.

048

0.63

70.

358

0.20

60.

190

0.11

90.

105

0.13

50.

175

500

3.40

12.

006

1.20

80.

690

0.39

50.

301

0.16

80.

090

0.08

00.

118

500

100

2.34

21.

199

0.82

60.

911

0.65

10.

758

0.34

60.

152

0.14

90.

001

500

100

3.81

52.

250

1.51

31.

339

0.91

90.

915

0.43

70.

196

0.16

60.

000

200

2.08

21.

196

1.06

40.

638

0.43

80.

434

0.26

50.

197

0.10

30.

000

200

3.58

42.

193

1.66

41.

007

0.65

50.

556

0.33

30.

230

0.11

60.

000

500

2.06

81.

234

0.81

70.

515

0.35

00.

323

0.21

70.

138

0.06

50.

000

500

3.56

82.

160

1.35

10.

815

0.51

30.

408

0.26

00.

158

0.07

20.

000

Not

es:

Par

amet

ers

ofth

ed

yn

amic

pan

eld

ata

mod

el,

(29),

are

gen

erate

das:

ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5),ϑi∼U

(0,0.9

5)

an

dγi

=0,

for

i=

1,2,...,N

.D

esig

n2

assu

mes

atw

o-fa

ctor

mod

elw

ith

[Nαβ]

an

d[N

αβ2]

non

-zer

olo

ad

ings

for

the

firs

tan

dse

con

dfa

ctor,

resp

ecti

vely

.W

ese

t:αβ

2=

2αβ/3,

wh

ereαβ

rela

tes

toα

un

der

(11)

,f jt

anduit∼IIDN

(0,1

),v ij∼IIDU

(µvj−

0.2,µvj

+0.

2),j

=1,

2,µv

=0.

87,µv2

=0.

71,µv1

=√ µ

2 v−N

2(αβ2−αβ

)µ

2 v2),

in

(32)

and

(33)

.T

he

num

ber

ofre

pli

cati

ons

isse

ttoR

=2000.

3

Tab

leS

1c:

Bia

san

dR

MS

E(×

100)

for

theα

esti

mat

eof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

om

ad

yn

am

icp

anel

dat

am

od

elw

ith

no

exog

enou

sre

gres

sors

Cro

ssco

rrel

ati

on

sare

gen

erate

du

sin

gD

esig

n1

wit

hn

on

-Gau

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

c p(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

05c p

(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

10δ

=1/

2δ

=1/

2T

NB

ias

TN

Bia

s10

010

02.

253

0.81

20.

371

0.20

10.

044

0.18

8-0

.164

-0.2

50-0

.103

-0.0

5010

010

03.

623

1.81

61.

052

0.64

20.

327

0.36

4-0

.053

-0.1

84-0

.067

-0.0

3220

02.

107

0.94

40.

664

0.16

2-0

.008

0.03

0-0

.108

-0.0

80-0

.085

-0.0

6620

03.

396

1.84

01.

242

0.52

30.

214

0.16

6-0

.021

-0.0

26-0

.052

-0.0

4450

02.

050

1.04

30.

488

0.20

40.

012

0.03

4-0

.040

-0.0

72-0

.096

-0.0

9250

03.

226

1.80

20.

952

0.48

30.

183

0.14

00.

026

-0.0

27-0

.061

-0.0

6420

010

02.

761

1.16

40.

575

0.38

50.

144

0.26

9-0

.096

-0.1

83-0

.049

0.00

020

010

04.

226

2.26

21.

302

0.85

40.

432

0.44

10.

005

-0.1

30-0

.028

0.00

020

02.

628

1.26

80.

853

0.33

70.

136

0.10

9-0

.028

-0.0

07-0

.016

0.00

020

04.

016

2.24

21.

454

0.71

80.

363

0.23

50.

042

0.02

7-0

.003

0.00

050

02.

659

1.46

80.

747

0.39

60.

170

0.15

90.

071

0.02

80.

000

0.00

050

03.

961

2.31

71.

259

0.69

20.

334

0.24

80.

114

0.04

80.

007

0.00

050

010

03.

035

1.32

10.

722

0.44

40.

218

0.29

4-0

.071

-0.1

74-0

.043

0.00

050

010

04.

588

2.46

21.

491

0.94

20.

531

0.47

50.

038

-0.1

17-0

.021

0.00

020

02.

930

1.52

91.

049

0.41

10.

180

0.14

6-0

.019

-0.0

03-0

.013

0.00

020

04.

403

2.56

71.

713

0.82

40.

427

0.28

60.

057

0.03

20.

001

0.00

050

03.

062

1.71

70.

888

0.49

20.

215

0.16

80.

076

0.02

80.

001

0.00

050

04.

461

2.64

81.

446

0.82

00.

396

0.26

10.

123

0.05

00.

008

0.00

0T

NR

MSE

TN

RM

SE

100

100

2.88

81.

621

1.23

60.

625

0.56

10.

382

0.27

40.

283

0.14

00.

100

100

100

4.18

32.

465

1.73

40.

993

0.73

50.

551

0.27

20.

235

0.10

70.

069

200

2.85

11.

818

1.22

10.

717

0.40

10.

401

0.28

30.

147

0.13

20.

117

200

4.04

12.

563

1.73

60.

981

0.54

10.

472

0.29

20.

125

0.09

30.

082

500

2.84

11.

783

1.05

00.

679

0.33

10.

282

0.19

20.

166

0.16

40.

163

500

3.92

22.

464

1.46

90.

917

0.44

90.

351

0.18

90.

131

0.11

80.

120

200

100

3.39

91.

838

1.15

60.

836

0.37

50.

398

0.20

60.

226

0.06

70.

001

200

100

4.76

22.

789

1.75

91.

229

0.62

30.

574

0.23

10.

207

0.06

80.

001

200

3.17

31.

903

1.24

90.

742

0.57

30.

215

0.24

50.

069

0.04

40.

001

200

4.48

72.

766

1.81

01.

066

0.73

30.

335

0.27

90.

094

0.05

10.

000

500

3.21

22.

004

1.10

10.

750

0.46

30.

339

0.17

70.

105

0.02

80.

003

500

4.45

32.

797

1.59

41.

028

0.60

90.

428

0.21

70.

127

0.03

50.

002

500

100

3.41

11.

715

1.18

60.

728

0.42

70.

380

0.21

80.

210

0.08

70.

000

500

100

4.90

42.

765

1.84

71.

171

0.70

20.

565

0.25

20.

187

0.09

20.

000

200

3.31

22.

098

1.43

20.

749

0.42

30.

264

0.10

80.

056

0.03

30.

000

200

4.71

73.

015

2.04

81.

102

0.62

90.

394

0.14

80.

079

0.04

00.

000

500

3.46

82.

134

1.18

20.

818

0.45

40.

249

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50.

052

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50.

000

500

4.80

73.

004

1.71

11.

117

0.61

60.

346

0.16

50.

075

0.02

20.

000

δ=

1/3

δ=

1/3

TN

Bia

sT

NB

ias

100

100

5.72

23.

405

2.16

01.

368

0.79

30.

652

0.12

0-0

.086

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23-0

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100

100

8.46

75.

599

3.74

42.

451

1.50

21.

085

0.38

40.

059

0.03

7-0

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200

6.09

93.

823

2.55

41.

360

0.72

60.

473

0.16

00.

072

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03-0

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200

8.83

15.

954

4.05

12.

362

1.35

40.

849

0.37

80.

182

0.04

4-0

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500

6.68

34.

203

2.49

41.

419

0.73

80.

455

0.20

00.

070

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03-0

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500

9.37

36.

256

3.92

12.

329

1.28

90.

763

0.36

30.

151

0.03

3-0

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200

100

6.36

13.

929

2.46

01.

621

0.91

60.

731

0.17

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0.00

80.

000

200

100

9.12

96.

167

4.08

22.

744

1.64

21.

179

0.44

40.

106

0.06

40.

000

200

6.81

84.

315

2.81

61.

609

0.90

60.

542

0.21

30.

111

0.02

80.

000

200

9.56

46.

471

4.34

22.

652

1.56

50.

926

0.43

10.

219

0.06

80.

000

500

7.56

94.

881

2.92

21.

698

0.91

10.

561

0.27

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121

0.03

30.

000

500

10.2

536.

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4.40

82.

656

1.48

80.

881

0.43

80.

198

0.06

10.

000

500

100

6.81

94.

181

2.70

01.

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1.04

50.

780

0.22

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0.01

70.

000

500

100

9.61

16.

456

4.36

72.

898

1.81

21.

239

0.50

70.

129

0.07

60.

000

200

7.29

54.

739

3.16

51.

761

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70.

616

0.24

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119

0.03

40.

000

200

10.0

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4.75

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1.70

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0.47

30.

229

0.07

60.

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500

8.17

45.

343

3.20

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906

1.02

10.

593

0.29

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127

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50.

000

500

10.8

617.

477

4.73

62.

913

1.63

00.

929

0.46

60.

208

0.06

40.

000

TN

RM

SE

TN

RM

SE

100

100

6.18

03.

909

2.68

91.

670

1.11

90.

835

0.35

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0.09

30.

045

100

100

8.81

65.

980

4.15

22.

713

1.76

61.

264

0.57

10.

247

0.11

80.

030

200

6.58

14.

355

2.96

21.

722

0.98

80.

713

0.39

50.

169

0.06

90.

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200

9.19

06.

355

4.38

82.

661

1.58

31.

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0.57

50.

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0.09

10.

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500

7.16

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694

2.91

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90.

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9.72

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200

100

6.78

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339

2.81

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0.34

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206

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001

200

100

9.43

96.

473

4.36

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994

1.78

91.

310

0.58

00.

273

0.12

80.

000

200

7.17

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700

3.10

71.

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1.19

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0.39

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170

0.07

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9.82

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760

4.58

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885

1.80

01.

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0.58

40.

275

0.11

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7.90

85.

236

3.20

01.

981

1.14

50.

732

0.36

80.

198

0.06

00.

001

500

10.5

047.

252

4.64

02.

901

1.69

61.

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0.52

90.

271

0.08

80.

001

500

100

7.06

94.

413

2.97

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0.38

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185

0.11

20.

000

500

100

9.79

76.

629

4.57

53.

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1.94

11.

330

0.63

10.

263

0.15

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000

200

7.52

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3.42

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1.17

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713

0.31

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158

0.06

60.

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200

10.2

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171

4.96

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1.84

11.

112

0.53

50.

266

0.10

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000

500

8.40

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3.41

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142

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0.33

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11.0

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110

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000

Not

es:

Par

amet

ers

ofth

ed

yn

amic

pan

eld

ata

mod

el,

(29),

are

gen

erate

das:

ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5),ϑi∼U

(0,0.9

5)

an

dγi

=0,

for

i=

1,2,...,N

.N

on-G

auss

ian

erro

rsar

ege

ner

ated

asuit

=( v−

2χ2 v,t

) 1/2ν it,

fori

=1,2,...,N

,ν it∼IIDN

(0,1

)an

dχ

2 v,t

isa

chi-

squ

are

dra

nd

om

vari

ate

wit

h

v=

8d

egre

esof

free

dom

,in

(31)

.D

esig

n1

assu

mesb i

∼U

(0.7,0.9

)fo

rth

efi

rstNb

(≤N

)el

emen

tsof

vect

or

bN

,i

=1,2,...,N

,in

the

const

ruct

ion

of

the

corr

elat

ion

mat

rix

ofth

eer

rors

,RN

=I N

+bN

b′ N−

B2 N

giv

enby

(30),

wh

ere

BN

=D

iag

(bN

).c p

(n,δ

)co

rres

pon

ds

toth

ecr

itic

al

valu

eu

sed

inth

em

ult

iple

test

ing

pro

ced

ure

show

nin

(23)

.T

he

nu

mb

erof

rep

lica

tion

sis

set

toR

=2000.

4

Tab

leS

1d:

Bia

san

dR

MS

E(×

100)

for

theα

esti

mat

eof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

oma

dyn

amic

pan

eld

ata

mod

elw

ith

no

exog

enou

sre

gres

sors

Cro

ssco

rrel

ati

on

sare

gen

erate

du

sin

gD

esig

n2

wit

hn

on

-Gau

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

c p(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

05c p

(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

10δ

=1/

2δ

=1/

2T

NB

ias

Bia

s10

010

01.

186

-0.8

98-2

.047

-2.5

70-3

.052

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25-3

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14-3

.780

-3.8

2010

010

02.

840

0.56

4-0

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30-2

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-2.1

63-2

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35-2

.926

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8920

00.

953

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04-1

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65-3

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93-3

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05-3

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7020

02.

530

0.36

8-0

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11-2

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95-2

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73-3

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6050

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712

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98-2

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40-3

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08-4

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5950

02.

201

0.07

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73-2

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89-3

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15-3

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7520

010

02.

816

1.27

90.

648

0.53

10.

198

0.24

1-0

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70-0

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2420

010

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235

2.34

91.

391

1.04

30.

563

0.50

80.

021

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05-0

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0420

02.

495

1.21

50.

809

0.25

5-0

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49-0

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91-0

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3420

03.

846

2.17

11.

441

0.68

60.

290

0.16

8-0

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49-0

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1750

02.

592

1.33

00.

625

0.17

4-0

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47-0

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50-0

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6950

03.

906

2.21

21.

194

0.54

40.

168

0.03

6-0

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-0.2

20-0

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-0.3

5250

010

03.

405

1.87

31.

221

1.12

50.

752

0.79

80.

355

0.12

40.

127

-0.0

0250

010

04.

805

2.87

21.

872

1.52

00.

993

0.93

30.

432

0.16

30.

142

-0.0

0120

03.

125

1.82

41.

398

0.81

30.

519

0.46

30.

266

0.18

60.

094

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0220

04.

487

2.73

61.

948

1.14

30.

708

0.56

50.

320

0.21

20.

104

-0.0

0150

03.

153

1.87

61.

159

0.68

60.

425

0.35

00.

222

0.13

40.

059

-0.0

0250

04.

471

2.71

31.

649

0.95

80.

569

0.42

20.

256

0.14

90.

065

-0.0

01T

NR

MSE

TN

RM

SE

100

100

2.40

92.

073

2.71

83.

131

3.55

63.

628

4.05

94.

266

4.24

34.

287

100

100

3.64

42.

010

1.86

02.

119

2.53

92.

647

3.10

73.

338

3.34

13.

408

200

2.28

72.

057

2.59

93.

322

3.75

33.

940

4.19

74.

254

4.35

84.

418

200

3.40

41.

901

1.77

12.

336

2.78

63.

011

3.29

63.

377

3.49

23.

564

500

2.16

92.

100

2.89

93.

554

3.97

24.

144

4.31

94.

442

4.53

84.

595

500

3.16

01.

817

2.02

12.

622

3.07

13.

278

3.47

63.

611

3.71

23.

773

200

100

3.28

11.

792

1.11

00.

820

0.50

90.

451

0.40

10.

507

0.48

70.

548

200

100

4.63

72.

751

1.72

81.

249

0.73

70.

609

0.28

90.

339

0.33

70.

406

200

2.86

71.

563

1.02

60.

508

0.34

90.

321

0.40

40.

434

0.49

30.

549

200

4.17

92.

465

1.62

30.

842

0.44

10.

309

0.26

20.

291

0.35

20.

413

500

3.18

41.

857

1.05

70.

575

0.37

80.

334

0.39

20.

459

0.52

00.

566

500

4.41

22.

662

1.54

80.

824

0.41

50.

263

0.25

70.

320

0.38

50.

434

500

100

3.76

42.

240

1.49

81.

264

0.84

00.

829

0.38

50.

159

0.14

60.

009

500

100

5.11

93.

185

2.11

81.

660

1.08

40.

970

0.46

50.

196

0.16

10.

008

200

3.40

22.

053

1.51

70.

892

0.56

10.

477

0.27

50.

192

0.10

00.

004

200

4.73

62.

948

2.07

51.

231

0.75

70.

584

0.33

00.

217

0.11

00.

003

500

3.51

82.

167

1.35

70.

815

0.50

00.

378

0.23

30.

138

0.06

10.

005

500

4.79

22.

983

1.84

41.

089

0.64

80.

454

0.27

00.

154

0.06

70.

004

δ=

1/3

δ=

1/3

TN

Bia

sT

NB

ias

100

100

5.16

12.

533

0.85

2-0

.095

-0.9

01-1

.189

-1.8

10-2

.115

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49-2

.246

100

100

8.03

44.

962

2.79

91.

420

0.29

9-0

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93-1

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05-1

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200

5.53

82.

858

1.19

2-0

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04-1

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20-1

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22-2

.218

200

8.40

85.

235

3.03

91.

248

0.09

9-0

.543

-1.0

85-1

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31-1

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500

6.09

33.

222

1.16

9-0

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67-1

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94-1

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17-2

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500

8.92

85.

545

2.95

81.

121

-0.0

73-0

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51-1

.415

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78-1

.672

200

100

6.31

13.

936

2.50

51.

789

1.06

80.

846

0.26

3-0

.033

-0.0

78-0

.206

200

100

8.96

66.

036

4.03

02.

820

1.75

51.

287

0.55

00.

151

0.03

7-0

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200

6.56

94.

157

2.76

01.

563

0.85

60.

535

0.18

70.

036

-0.0

95-0

.195

200

9.24

86.

224

4.19

52.

534

1.47

40.

914

0.42

40.

182

0.00

0-0

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500

7.48

14.

754

2.85

91.

586

0.81

20.

442

0.15

9-0

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-0.1

24-0

.195

500

10.1

376.

805

4.29

52.

513

1.37

20.

766

0.35

00.

107

-0.0

43-0

.135

500

100

6.84

84.

402

2.90

82.

167

1.39

71.

164

0.56

30.

229

0.16

7-0

.001

500

100

9.48

56.

474

4.39

23.

140

2.01

91.

528

0.77

50.

337

0.20

70.

000

200

7.22

54.

701

3.20

61.

933

1.17

00.

818

0.45

30.

274

0.12

60.

000

200

9.88

56.

757

4.61

72.

867

1.74

11.

140

0.62

50.

355

0.15

60.

000

500

8.05

15.

221

3.23

51.

890

1.07

90.

681

0.38

10.

204

0.08

4-0

.001

500

10.6

857.

261

4.65

42.

787

1.59

60.

953

0.51

50.

262

0.10

30.

000

TN

RM

SE

TN

RM

SE

100

100

5.71

13.

248

1.86

81.

395

1.56

91.

708

2.19

22.

470

2.51

32.

614

100

100

8.41

75.

394

3.26

51.

932

1.14

51.

011

1.38

71.

721

1.82

41.

967

200

6.07

13.

497

1.95

91.

290

1.53

41.

825

2.17

42.

317

2.46

42.

558

200

8.78

55.

643

3.44

11.

759

1.01

01.

066

1.43

11.

640

1.82

81.

950

500

6.60

23.

810

1.94

41.

247

1.52

61.

840

2.12

52.

313

2.44

12.

517

500

9.28

55.

935

3.37

61.

651

0.95

61.

127

1.46

31.

702

1.86

11.

955

200

100

6.63

84.

256

2.77

11.

960

1.19

50.

911

0.36

30.

211

0.21

40.

286

200

100

9.21

76.

285

4.24

42.

967

1.86

31.

342

0.60

50.

231

0.15

50.

197

200

6.83

14.

394

2.92

31.

685

0.94

10.

585

0.26

40.

173

0.19

80.

266

200

9.44

96.

415

4.34

12.

647

1.55

10.

954

0.45

90.

223

0.12

70.

186

500

7.82

25.

079

3.13

11.

793

0.95

30.

521

0.22

90.

139

0.19

30.

252

500

10.3

897.

060

4.52

22.

695

1.49

90.

836

0.39

00.

150

0.11

50.

179

500

100

7.10

04.

652

3.11

52.

299

1.48

91.

206

0.59

80.

260

0.18

50.

007

500

100

9.67

56.

665

4.56

03.

259

2.10

71.

574

0.81

20.

366

0.22

40.

006

200

7.41

94.

878

3.33

12.

026

1.22

70.

844

0.46

70.

281

0.13

20.

001

200

10.0

326.

900

4.73

02.

956

1.80

01.

171

0.64

20.

363

0.16

10.

001

500

8.26

75.

427

3.40

52.

015

1.16

00.

721

0.40

00.

211

0.08

60.

002

500

10.8

457.

423

4.79

82.

902

1.67

50.

996

0.53

70.

271

0.10

70.

002

Not

es:

Par

amet

ers

ofth

ed

yn

amic

pan

eld

ata

mod

el,

(29),

are

gen

erate

das:

ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5),ϑi∼U

(0,0.9

5)

an

dγi

=0,

for

i=

1,2,...,N

.D

esig

n2

assu

mes

atw

o-fa

ctor

mod

elw

ith

[Nαβ]

an

d[N

αβ2]

non

-zer

olo

ad

ings

for

the

firs

tan

dse

con

dfa

ctor,

resp

ecti

vely

.W

ese

t:αβ

2=

2αβ/3,

wh

ereαβ

rela

tes

toα

un

der

(11)

andf jt∼IIDN

(0,1

).N

on

-Gau

ssia

ner

rors

are

gen

erate

das:

uit

=( v−

2χ2 v,t

) 1/2ν it,

fori

=1,2,...,N

,ν it∼IIDN

(0,1

)

andχ

2 v,t

isa

chi-

squ

ared

ran

dom

vari

ate

wit

hv=

8d

egre

esof

free

dom

,in

(31).

v ij∼

IIDU

(µvj−

0.2,µvj

+0.2

),j

=1,2

,µv

=0.8

7,µv2

=0.7

1,

µv1

=√ µ

2 v−N

2(αβ2−αβ

)µ

2 v2),

in(3

2)an

d(3

3).

Th

enu

mb

erof

rep

lica

tion

sis

set

toR

=2000.

5

Tab

leS

2a:

Bia

san

dR

MS

E(×

100)

for

theα

esti

mat

eof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

oma

stati

cp

anel

dat

am

od

elC

ross

corr

elati

on

sare

gen

erate

du

sin

gD

esig

n1

wit

hG

au

ssia

nan

dn

on

-Gau

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

c p(n,δ

)w

ithn

=N

(N−

1)/2

,δ

=1/

4an

dp

=0.

05c p

(n,δ

)w

ithn

=N

(N−

1)/2

,δ

=1/

4an

dp

=0.

10G

auss

ian

erro

rsG

auss

ian

erro

rsT

NB

ias

Bia

s10

010

03.

117

1.45

10.

736

0.46

60.

220

0.29

4-0

.075

-0.1

79-0

.049

-0.0

0210

010

05.

785

3.47

02.

112

1.35

60.

779

0.62

70.

124

-0.0

76-0

.010

-0.0

0120

03.

367

1.75

01.

145

0.49

60.

219

0.17

1-0

.009

0.00

6-0

.013

-0.0

0320

06.

148

3.75

72.

441

1.32

10.

716

0.45

30.

145

0.08

20.

016

-0.0

0150

03.

913

2.21

41.

191

0.61

30.

276

0.20

20.

095

0.03

50.

001

-0.0

0450

06.

774

4.22

12.

470

1.36

50.

698

0.42

50.

208

0.08

80.

021

-0.0

0220

010

03.

205

1.49

50.

774

0.50

50.

227

0.31

2-0

.072

-0.1

75-0

.045

0.00

020

010

05.

846

3.48

62.

139

1.39

80.

788

0.64

40.

123

-0.0

75-0

.006

0.00

020

03.

491

1.85

11.

203

0.52

80.

239

0.18

3-0

.001

0.01

0-0

.010

0.00

020

06.

231

3.84

52.

499

1.35

20.

734

0.46

50.

153

0.08

50.

018

0.00

050

04.

100

2.33

91.

264

0.66

10.

304

0.21

80.

104

0.04

10.

005

0.00

050

06.

942

4.33

82.

539

1.41

60.

728

0.44

30.

217

0.09

30.

024

0.00

050

010

03.

238

1.55

10.

802

0.50

90.

250

0.31

5-0

.068

-0.1

74-0

.044

0.00

050

010

05.

863

3.54

62.

162

1.39

10.

806

0.64

40.

125

-0.0

75-0

.006

0.00

020

03.

566

1.89

31.

238

0.54

80.

247

0.18

90.

003

0.01

2-0

.009

0.00

020

06.

299

3.88

42.

533

1.36

80.

741

0.47

10.

157

0.08

60.

018

0.00

050

04.

208

2.40

61.

307

0.68

10.

315

0.22

50.

108

0.04

30.

006

0.00

050

07.

028

4.40

62.

587

1.43

90.

742

0.45

00.

222

0.09

50.

024

0.00

0T

NR

MSE

RM

SE

100

100

3.16

91.

525

0.81

20.

527

0.28

70.

322

0.12

10.

190

0.06

10.

008

100

100

5.82

23.

515

2.16

01.

398

0.82

30.

657

0.19

00.

125

0.05

60.

004

200

3.38

61.

771

1.16

30.

518

0.24

30.

187

0.05

10.

035

0.02

50.

008

200

6.16

03.

772

2.45

61.

338

0.73

30.

466

0.16

40.

097

0.03

60.

004

500

3.91

82.

220

1.19

60.

619

0.28

10.

205

0.09

80.

039

0.01

20.

011

500

6.77

84.

224

2.47

31.

369

0.70

30.

429

0.21

10.

091

0.02

60.

006

200

100

3.25

81.

564

0.84

80.

568

0.29

30.

340

0.12

10.

187

0.05

80.

000

200

100

5.88

23.

530

2.18

51.

442

0.83

10.

675

0.19

10.

124

0.05

60.

000

200

3.51

01.

875

1.22

10.

550

0.26

20.

198

0.05

30.

036

0.02

10.

000

200

6.24

43.

861

2.51

41.

368

0.75

10.

478

0.17

30.

100

0.03

50.

000

500

4.10

52.

344

1.26

90.

665

0.30

80.

222

0.10

70.

044

0.01

00.

000

500

6.94

54.

341

2.54

21.

420

0.73

20.

446

0.22

00.

096

0.02

70.

000

500

100

3.29

11.

622

0.87

60.

575

0.31

40.

344

0.12

00.

186

0.05

90.

000

500

100

5.90

23.

594

2.21

01.

438

0.85

10.

675

0.19

30.

125

0.05

90.

000

200

3.58

51.

916

1.25

50.

570

0.27

00.

206

0.05

50.

039

0.02

20.

000

200

6.31

23.

899

2.54

71.

385

0.75

70.

486

0.17

70.

103

0.03

60.

000

500

4.21

22.

411

1.31

20.

685

0.32

00.

228

0.11

20.

046

0.01

10.

000

500

7.03

14.

410

2.59

11.

443

0.74

60.

454

0.22

60.

098

0.02

80.

000

Non

-Gau

ssia

ner

rors

Non

-Gau

ssia

ner

rors

TN

Bia

sT

NB

ias

100

100

8.56

85.

689

3.72

42.

532

1.51

21.

081

0.40

20.

062

0.03

6-0

.010

100

100

12.0

708.

641

5.99

34.

153

2.60

81.

779

0.83

10.

291

0.12

7-0

.006

200

9.53

06.

389

4.33

32.

622

1.56

90.

929

0.40

80.

203

0.05

4-0

.012

200

12.9

989.

290

6.52

44.

199

2.62

11.

572

0.77

70.

390

0.12

7-0

.007

500

10.7

247.

428

4.64

92.

841

1.60

70.

920

0.45

90.

198

0.05

0-0

.014

500

14.0

8410

.250

6.83

64.

378

2.59

51.

492

0.77

00.

348

0.10

7-0

.008

200

100

9.13

06.

188

4.16

82.

735

1.68

01.

223

0.47

00.

098

0.06

10.

000

200

100

12.5

839.

125

6.43

74.

381

2.80

11.

935

0.90

90.

331

0.15

20.

000

200

10.1

207.

023

4.75

92.

884

1.73

11.

024

0.50

50.

252

0.07

70.

000

200

13.5

329.

905

6.96

84.

496

2.80

71.

679

0.89

00.

444

0.15

00.

000

500

11.4

397.

948

5.22

03.

202

1.82

91.

017

0.52

80.

239

0.07

70.

000

500

14.7

0210

.742

7.41

34.

766

2.84

81.

606

0.84

70.

390

0.13

20.

000

500

100

9.60

36.

544

4.37

92.

909

1.80

91.

277

0.50

00.

132

0.07

10.

000

500

100

13.0

549.

478

6.67

64.

564

2.95

92.

007

0.94

80.

375

0.16

60.

000

200

10.7

397.

425

5.05

33.

082

1.83

31.

144

0.52

30.

269

0.08

60.

000

200

14.1

0810

.304

7.27

94.

715

2.93

41.

827

0.91

40.

468

0.16

10.

000

500

12.2

068.

470

5.52

93.

365

1.94

71.

106

0.57

00.

255

0.08

50.

000

500

15.4

0411

.243

7.73

44.

956

2.99

31.

719

0.90

20.

412

0.14

20.

000

TN

RM

SE

RM

SE

100

100

8.90

56.

084

4.05

82.

823

1.74

01.

224

0.59

50.

266

0.12

30.

029

100

100

12.3

058.

914

6.23

84.

387

2.79

91.

917

0.99

10.

436

0.19

70.

018

200

9.88

96.

726

4.64

72.

937

1.87

51.

066

0.53

70.

273

0.09

80.

027

200

13.2

409.

528

6.76

44.

449

2.86

71.

701

0.89

10.

457

0.16

60.

018

500

11.0

677.

813

4.95

53.

152

1.84

91.

089

0.54

80.

268

0.08

20.

029

500

14.3

1310

.519

7.06

64.

624

2.80

51.

649

0.86

00.

414

0.13

40.

018

200

100

9.39

06.

467

4.46

62.

949

1.87

11.

381

0.61

40.

257

0.12

90.

001

200

100

12.7

609.

319

6.65

84.

552

2.96

12.

076

1.03

30.

447

0.21

00.

001

200

10.3

617.

303

5.01

73.

088

1.93

01.

132

0.64

40.

313

0.11

10.

001

200

13.6

9410

.100

7.16

24.

658

2.97

01.

778

1.01

00.

503

0.18

30.

000

500

11.6

628.

179

5.47

43.

460

2.02

51.

121

0.61

00.

309

0.10

30.

001

500

14.8

5210

.903

7.60

24.

968

3.01

51.

705

0.92

60.

454

0.15

80.

000

500

100

9.77

86.

756

4.54

73.

059

1.93

71.

396

0.60

00.

246

0.12

70.

000

500

100

13.1

749.

623

6.80

04.

685

3.06

82.

114

1.03

50.

458

0.21

50.

000

200

10.9

317.

603

5.21

23.

213

1.94

11.

245

0.59

80.

313

0.11

30.

000

200

14.2

3510

.427

7.39

94.

820

3.02

61.

917

0.98

00.

510

0.18

80.

000

500

12.3

788.

643

5.70

63.

498

2.07

01.

180

0.62

50.

286

0.11

40.

000

500

15.5

1911

.362

7.86

25.

062

3.09

61.

787

0.95

50.

443

0.16

90.

000

Not

es:

Par

am

eter

sof

the

stat

icp

anel

dat

am

od

el,(2

9),

are

gen

erate

das:ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5),ϑi

=0

an

dγi∼IIDN

(1,1

),fo

ri

=1,

2,...,N

.

Gau

ssia

ner

rors

are

gen

erat

edasuit∼IIDN

(0,1

)in

(31).

Non

-Gau

ssia

ner

rors

are

gen

erate

dasuit

=( v−

2χ2 v,t

) 1/2ν it,

fori

=1,

2,...,N

,ν it∼IIDN

(0,1

)an

d

χ2 v,t

isa

chi-

squ

ared

ran

dom

vari

ate

wit

hv=

8d

egre

esof

free

dom

,in

(31).

Des

ign

1ass

um

esb i

∼U

(0.7,0.9

)fo

rth

efi

rstNb

(≤N

)el

emen

tsof

vect

or

bN

,i

=1,

2,...,N

,in

the

con

stru

ctio

nof

the

corr

elat

ion

matr

ixof

the

erro

rs,RN

=I N

+bN

b′ N−

B2 N

giv

enby

(30),

wh

ere

BN

=D

iag

(bN

).c p

(n,δ

)co

rres

pon

ds

toth

ecr

itic

alva

lue

use

din

the

mu

ltip

lete

stin

gpro

ced

ure

show

nin

(23).

Th

enu

mb

erof

rep

lica

tion

sis

set

toR

=2000.

6

Tab

leS

2b

:B

ias

and

RM

SE

(×100)

for

theα

esti

mat

eof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

om

ast

atic

pan

eld

ata

mod

elC

ross

corr

elati

on

sare

gen

erate

du

sin

gD

esig

n2

wit

hG

au

ssia

nan

dn

on

-Gau

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

c p(n,δ

)w

ithn

=N

(N−

1)/2

,δ

=1/

4an

dp

=0.

05c p

(n,δ

)w

ithn

=N

(N−

1)/2

,δ

=1/

4an

dp

=0.

10G

auss

ian

erro

rsG

auss

ian

erro

rsT

NB

ias

TN

Bia

s10

010

03.

000

1.15

40.

241

-0.1

09-0

.553

-0.5

91-1

.078

-1.3

09-1

.300

-1.3

8910

010

05.

848

3.44

71.

981

1.21

20.

465

0.21

1-0

.407

-0.7

35-0

.792

-0.9

3120

03.

189

1.40

80.

603

-0.2

00-0

.634

-0.7

71-1

.028

-1.1

09-1

.218

-1.3

0520

06.

118

3.68

92.

255

1.02

00.

278

-0.0

67-0

.450

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15-0

.773

-0.8

9050

03.

662

1.73

50.

529

-0.2

12-0

.655

-0.8

22-0

.996

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17-1

.215

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7950

06.

695

4.02

02.

156

0.92

10.

159

-0.2

06-0

.493

-0.6

77-0

.808

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9020

010

03.

693

2.10

21.

360

1.18

20.

768

0.76

90.

297

0.04

90.

041

-0.0

7920

010

06.

255

4.03

22.

697

2.05

61.

346

1.13

20.

530

0.19

10.

120

-0.0

4620

03.

826

2.30

51.

683

0.97

30.

589

0.47

10.

231

0.13

10.

020

-0.0

7520

06.

517

4.25

02.

952

1.80

11.

105

0.78

10.

420

0.24

10.

083

-0.0

4450

04.

262

2.57

11.

549

0.88

90.

506

0.36

10.

194

0.08

0-0

.009

-0.0

7350

07.

065

4.53

92.

811

1.65

40.

951

0.61

20.

337

0.16

30.

040

-0.0

4450

010

03.

829

2.24

11.

509

1.33

50.

914

0.90

80.

432

0.17

20.

150

0.00

050

010

06.

320

4.10

52.

786

2.15

21.

435

1.21

60.

615

0.26

90.

188

0.00

020

03.

954

2.44

01.

816

1.10

00.

711

0.58

70.

347

0.23

30.

116

0.00

020

06.

589

4.33

03.

033

1.88

01.

183

0.85

60.

495

0.30

60.

143

0.00

050

04.

410

2.70

71.

679

1.00

20.

614

0.46

10.

287

0.17

00.

076

0.00

050

07.

169

4.63

42.

902

1.73

11.

025

0.67

90.

398

0.22

10.

094

0.00

0T

NR

MSE

TN

RM

SE

100

100

3.08

01.

365

0.83

50.

847

1.03

31.

059

1.41

41.

596

1.59

01.

672

100

100

5.88

63.

501

2.07

61.

363

0.78

90.

675

0.78

60.

996

1.04

01.

158

200

3.22

71.

519

0.89

90.

752

0.99

71.

100

1.29

41.

370

1.46

31.

548

200

6.13

33.

716

2.30

51.

144

0.61

90.

572

0.73

00.

855

0.97

81.

086

500

3.67

51.

789

0.77

10.

676

0.94

81.

092

1.24

01.

348

1.43

91.

499

500

6.70

04.

031

2.18

81.

022

0.51

40.

564

0.73

60.

881

0.99

31.

068

200

100

3.73

22.

143

1.39

31.

205

0.79

00.

785

0.32

90.

140

0.12

00.

120

200

100

6.28

54.

061

2.72

12.

075

1.36

21.

144

0.54

80.

226

0.15

30.

074

200

3.84

12.

319

1.69

30.

983

0.60

10.

481

0.25

10.

157

0.08

60.

110

200

6.52

84.

260

2.96

11.

808

1.11

20.

787

0.42

90.

251

0.10

30.

069

500

4.26

62.

575

1.55

20.

893

0.51

10.

368

0.20

50.

105

0.07

20.

102

500

7.06

84.

541

2.81

31.

656

0.95

40.

615

0.34

00.

170

0.06

30.

065

500

100

3.86

72.

277

1.53

51.

349

0.92

50.

916

0.44

40.

194

0.16

80.

000

500

100

6.35

04.

134

2.80

92.

168

1.44

81.

225

0.62

70.

286

0.20

40.

000

200

3.96

82.

452

1.82

41.

106

0.71

60.

590

0.35

10.

237

0.12

10.

000

200

6.59

94.

339

3.04

11.

886

1.18

80.

860

0.50

00.

310

0.14

80.

000

500

4.41

42.

710

1.68

11.

003

0.61

50.

462

0.28

70.

171

0.07

70.

000

500

7.17

24.

636

2.90

41.

732

1.02

60.

680

0.39

90.

222

0.09

50.

000

Non

-Gau

ssia

ner

rors

Non

-Gau

ssia

ner

rors

TN

Bia

sT

NB

ias

100

100

7.97

34.

917

2.78

51.

443

0.32

6-0

.192

-0.9

39-1

.352

-1.4

54-1

.599

100

100

11.5

657.

997

5.25

93.

336

1.75

60.

878

-0.1

01-0

.684

-0.9

01-1

.123

200

9.01

55.

756

3.46

81.

595

0.37

6-0

.308

-0.8

89-1

.162

-1.3

66-1

.504

200

12.5

738.

804

5.87

93.

441

1.73

70.

678

-0.1

43-0

.580

-0.8

80-1

.073

500

10.4

406.

818

3.96

21.

870

0.47

1-0

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37-1

.146

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38-1

.448

500

13.8

669.

790

6.35

73.

666

1.75

20.

582

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86-0

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07-1

.056

200

100

8.97

96.

043

4.03

12.

821

1.76

51.

298

0.56

50.

161

0.05

2-0

.123

200

100

12.3

548.

843

6.16

54.

314

2.78

51.

939

0.97

00.

395

0.17

2-0

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200

10.0

366.

865

4.66

02.

858

1.68

31.

041

0.50

10.

224

0.02

3-0

.121

200

13.3

959.

656

6.76

14.

363

2.67

61.

648

0.86

00.

421

0.12

4-0

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500

11.3

527.

797

5.02

23.

004

1.67

60.

941

0.44

70.

162

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11-0

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500

14.5

9610

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7.14

64.

495

2.62

51.

491

0.74

90.

320

0.06

9-0

.074

500

100

9.48

36.

468

4.38

63.

128

2.01

91.

528

0.77

10.

335

0.20

50.

000

500

100

12.8

259.

246

6.49

44.

590

3.00

12.

122

1.12

40.

517

0.27

30.

000

200

10.4

967.

246

4.97

03.

107

1.89

41.

226

0.67

20.

376

0.16

20.

000

200

13.8

1010

.006

7.04

54.

582

2.85

51.

790

0.98

30.

525

0.21

60.

000

500

11.9

338.

278

5.40

03.

284

1.89

31.

115

0.59

70.

299

0.11

50.

000

500

15.1

1410

.992

7.50

94.

766

2.82

61.

640

0.86

70.

420

0.15

60.

000

TN

RM

SE

TN

RM

SE

100

100

8.29

25.

272

3.16

21.

856

1.03

90.

938

1.31

81.

652

1.76

11.

901

100

100

11.7

798.

225

5.48

13.

535

1.97

01.

143

0.69

90.

982

1.17

31.

375

200

9.34

76.

109

3.80

01.

975

0.99

50.

889

1.22

81.

463

1.65

51.

789

200

12.7

949.

037

6.09

73.

649

1.94

90.

962

0.65

30.

882

1.13

21.

310

500

10.7

867.

197

4.36

12.

315

1.12

80.

915

1.18

41.

437

1.61

51.

723

500

14.0

9510

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6.62

03.

929

2.02

30.

944

0.67

60.

918

1.14

61.

288

200

100

9.20

86.

276

4.23

22.

961

1.87

01.

356

0.62

10.

232

0.14

60.

181

200

100

12.5

159.

010

6.31

74.

430

2.87

51.

993

1.01

40.

430

0.20

50.

121

200

10.2

897.

114

4.86

53.

027

1.80

01.

105

0.54

10.

255

0.11

10.

168

200

13.5

699.

836

6.92

34.

504

2.78

11.

712

0.89

60.

439

0.14

70.

112

500

11.5

958.

048

5.25

53.

196

1.81

11.

014

0.48

50.

193

0.10

30.

161

500

14.7

6010

.721

7.32

34.

653

2.74

61.

563

0.78

70.

338

0.09

90.

111

500

100

9.62

16.

606

4.50

33.

210

2.08

01.

561

0.79

90.

360

0.22

10.

007

500

100

12.9

229.

346

6.58

64.

660

3.05

62.

155

1.15

10.

539

0.28

70.

006

200

10.6

427.

390

5.08

83.

202

1.96

11.

262

0.69

20.

385

0.16

80.

001

200

13.9

1010

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7.13

94.

664

2.91

61.

828

1.00

50.

536

0.22

20.

001

500

12.0

828.

434

5.54

83.

410

1.98

61.

169

0.62

50.

311

0.11

90.

001

500

15.2

1311

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7.61

94.

867

2.90

61.

691

0.89

60.

434

0.16

10.

001

Not

es:

Par

am

eter

sof

the

stat

icp

anel

dat

am

od

el,(2

9),

are

gen

erate

das:ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5),ϑi

=0

an

dγi∼IIDN

(1,1

),fo

ri

=1,

2,...,N

.D

esig

n2

assu

mes

atw

o-fa

ctor

mod

elw

ith

[Nαβ]

and

[Nαβ2]

non

-zer

olo

ad

ings

for

the

firs

tan

dse

con

dfa

ctor,

resp

ecti

vely

.W

ese

t:αβ

2=

2αβ/3

,w

her

eαβ

rela

tes

toα

un

der

(11)

andf jt∼IIDN

(0,1

).G

auss

ian

erro

rsare

giv

enby:uit∼IIDN

(0,1

)an

dn

on

Gau

ssia

ner

rors

are

gen

erate

das:uit

=( v−

2χ2 v,t

) 1/2ν it,

fori

=1,

2,...,N

,ν it∼IIDN

(0,1

)an

dχ

2 v,t

isa

chi-

squ

are

dra

nd

om

vari

ate

wit

hv=

8d

egre

esof

free

dom

,in

(31).v ij∼IIDU

(µvj−

0.2,µvj

+0.2

),j

=1,

2,

µv

=0.

87,µv2

=0.

71,µv1

=√ µ

2 v−N

2(αβ2−αβ

)µ

2 v2),

in(3

2)

an

d(3

3).

Th

enu

mb

erof

rep

lica

tion

sis

set

toR

=2000.

7

Tab

leS

2c:

Bia

san

dR

MS

E(×

100

)fo

rth

eα

esti

mat

eof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

oma

dyn

amic

pan

eld

ata

mod

elw

ith

exog

enou

sre

gres

sors

Cro

ssco

rrel

atio

ns

are

gen

erate

du

sin

gD

esig

n1

wit

hG

au

ssia

nan

dn

on

-Gau

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

c p(n,δ

)w

ithn

=N

(N−

1)/2

,δ

=1/

4an

dp

=0.

05c p

(n,δ

)w

ithn

=N

(N−

1)/2

,δ

=1/

4an

dp

=0.

10G

auss

ian

erro

rsG

auss

ian

erro

rsT

NB

ias

TN

Bia

s10

010

03.

118

1.43

10.

731

0.46

90.

217

0.29

5-0

.074

-0.1

79-0

.048

-0.0

0310

010

05.

795

3.45

12.

105

1.35

70.

780

0.63

10.

123

-0.0

75-0

.008

-0.0

0120

03.

378

1.75

91.

154

0.49

40.

218

0.17

0-0

.010

0.00

5-0

.015

-0.0

0420

06.

150

3.77

02.

457

1.32

10.

717

0.45

30.

146

0.08

20.

014

-0.0

0250

03.

916

2.21

21.

190

0.61

10.

275

0.20

10.

093

0.03

4-0

.001

-0.0

0550

06.

783

4.21

82.

466

1.36

30.

697

0.42

40.

208

0.08

70.

020

-0.0

0220

010

03.

169

1.50

60.

773

0.50

20.

238

0.31

2-0

.069

-0.1

76-0

.046

0.00

020

010

05.

829

3.49

42.

147

1.38

80.

795

0.64

00.

125

-0.0

75-0

.008

0.00

020

03.

510

1.84

71.

206

0.53

30.

240

0.18

60.

002

0.01

2-0

.010

0.00

020

06.

257

3.84

52.

503

1.36

40.

736

0.46

80.

157

0.08

70.

018

0.00

050

04.

097

2.34

21.

265

0.65

80.

303

0.21

90.

104

0.04

20.

005

0.00

050

06.

937

4.34

22.

540

1.41

30.

727

0.44

40.

217

0.09

40.

024

0.00

050

010

03.

242

1.54

60.

801

0.51

60.

250

0.31

5-0

.069

-0.1

72-0

.045

0.00

050

010

05.

851

3.52

82.

170

1.39

80.

806

0.64

20.

125

-0.0

72-0

.007

0.00

020

03.

568

1.88

91.

240

0.54

90.

252

0.18

70.

005

0.01

2-0

.010

0.00

020

06.

308

3.88

52.

529

1.37

50.

747

0.46

80.

159

0.08

60.

017

0.00

050

04.

201

2.40

11.

309

0.68

10.

317

0.22

40.

107

0.04

30.

006

0.00

050

07.

024

4.39

92.

589

1.44

00.

745

0.44

90.

221

0.09

60.

025

0.00

0T

NR

MSE

TN

RM

SE

100

100

3.17

11.

500

0.80

90.

528

0.28

20.

321

0.12

10.

191

0.06

20.

008

100

100

5.83

33.

494

2.15

21.

399

0.82

50.

659

0.18

80.

124

0.05

80.

004

200

3.39

91.

781

1.17

10.

515

0.24

30.

186

0.05

20.

036

0.02

70.

011

200

6.16

33.

785

2.47

11.

337

0.73

30.

467

0.16

50.

098

0.03

30.

005

500

3.92

12.

217

1.19

50.

616

0.28

00.

205

0.09

80.

040

0.01

50.

012

500

6.78

74.

221

2.47

01.

367

0.70

10.

428

0.21

20.

091

0.02

50.

006

200

100

3.22

01.

574

0.84

50.

562

0.30

20.

340

0.11

90.

187

0.05

90.

000

200

100

5.86

53.

538

2.19

21.

429

0.83

90.

670

0.19

00.

122

0.05

60.

000

200

3.52

91.

869

1.22

40.

555

0.26

40.

201

0.05

50.

038

0.02

20.

000

200

6.27

03.

860

2.51

71.

380

0.75

30.

482

0.17

80.

103

0.03

50.

000

500

4.10

22.

347

1.27

00.

663

0.30

80.

222

0.10

70.

045

0.01

00.

000

500

6.94

04.

345

2.54

51.

417

0.73

10.

447

0.22

10.

097

0.02

70.

000

500

100

3.29

11.

615

0.87

70.

578

0.31

30.

343

0.12

00.

186

0.05

80.

000

500

100

5.88

43.

573

2.21

91.

444

0.85

00.

671

0.19

40.

126

0.05

70.

000

200

3.58

81.

911

1.25

70.

570

0.27

60.

202

0.05

60.

039

0.02

30.

000

200

6.32

13.

901

2.54

31.

391

0.76

40.

481

0.18

00.

103

0.03

50.

000

500

4.20

72.

406

1.31

30.

686

0.32

20.

228

0.11

10.

046

0.01

10.

000

500

7.02

74.

403

2.59

31.

444

0.74

90.

452

0.22

50.

099

0.02

80.

000

Non

-Gau

ssia

ner

rors

Non

-Gau

ssia

ner

rors

TN

Bia

sT

NB

ias

100

100

8.44

75.

562

3.67

72.

417

1.49

11.

079

0.38

10.

057

0.03

2-0

.012

100

100

11.9

738.

498

5.93

24.

013

2.58

51.

768

0.80

50.

286

0.12

4-0

.007

200

9.34

06.

220

4.23

12.

548

1.46

00.

907

0.40

50.

192

0.05

1-0

.015

200

12.8

099.

116

6.41

74.

116

2.49

61.

543

0.77

80.

375

0.12

6-0

.009

500

10.5

887.

135

4.65

72.

749

1.58

70.

886

0.43

30.

191

0.04

7-0

.017

500

13.9

509.

966

6.83

64.

270

2.56

91.

448

0.73

60.

340

0.10

5-0

.010

200

100

9.17

26.

148

4.07

62.

673

1.67

41.

191

0.46

80.

110

0.05

80.

000

200

100

12.6

369.

066

6.34

94.

291

2.79

51.

901

0.90

00.

343

0.14

60.

000

200

10.1

466.

941

4.76

12.

886

1.69

21.

040

0.49

30.

245

0.07

80.

000

200

13.5

549.

818

6.96

54.

486

2.76

11.

695

0.87

40.

435

0.15

10.

000

500

11.4

297.

889

5.19

23.

109

1.78

41.

002

0.52

60.

238

0.07

80.

000

500

14.6

9810

.680

7.38

14.

666

2.79

21.

586

0.84

30.

389

0.13

30.

000

500

100

9.55

16.

522

4.37

52.

858

1.79

71.

278

0.50

50.

124

0.06

70.

000

500

100

13.0

039.

468

6.68

14.

519

2.93

92.

009

0.95

00.

362

0.16

00.

000

200

10.6

267.

339

5.01

63.

082

1.84

91.

115

0.52

90.

261

0.08

60.

000

200

14.0

0210

.220

7.24

24.

716

2.95

21.

790

0.92

10.

457

0.16

10.

000

500

12.1

608.

434

5.54

93.

413

1.92

61.

135

0.56

40.

255

0.08

00.

000

500

15.3

5511

.209

7.75

75.

006

2.97

01.

751

0.89

40.

413

0.13

70.

000

TN

RM

SE

TN

RM

SE

100

100

8.79

35.

928

4.00

92.

737

1.72

31.

271

0.53

10.

263

0.12

60.

031

100

100

12.2

138.

756

6.18

24.

264

2.78

01.

936

0.93

20.

431

0.19

80.

019

200

9.66

16.

547

4.53

02.

851

1.68

01.

063

0.53

20.

277

0.10

70.

037

200

13.0

299.

348

6.64

64.

355

2.68

01.

687

0.89

10.

453

0.17

10.

024

500

10.9

257.

431

4.99

03.

023

1.88

11.

068

0.51

60.

255

0.08

30.

039

500

14.1

7610

.176

7.08

74.

492

2.81

11.

609

0.81

80.

402

0.13

40.

026

200

100

9.44

26.

443

4.33

02.

897

1.84

31.

328

0.67

60.

304

0.12

50.

001

200

100

12.8

229.

271

6.53

74.

469

2.94

12.

028

1.06

30.

486

0.20

60.

001

200

10.4

167.

244

5.04

73.

129

1.87

01.

177

0.62

90.

316

0.12

80.

003

200

13.7

3510

.029

7.17

34.

675

2.91

01.

817

0.99

10.

499

0.19

60.

003

500

11.6

438.

128

5.46

73.

298

1.97

01.

104

0.61

50.

292

0.11

40.

001

500

14.8

4210

.846

7.58

24.

820

2.95

21.

681

0.92

80.

442

0.16

70.

000

500

100

9.73

36.

723

4.55

92.

989

1.92

91.

377

0.61

20.

253

0.12

20.

000

500

100

13.1

259.

607

6.81

54.

624

3.04

92.

099

1.04

20.

455

0.20

90.

000

200

10.7

737.

524

5.19

13.

233

1.96

51.

212

0.65

00.

301

0.11

60.

000

200

14.1

0110

.346

7.37

14.

834

3.05

11.

875

1.01

80.

495

0.18

90.

000

500

12.3

288.

570

5.72

63.

576

2.04

41.

246

0.62

40.

284

0.09

30.

000

500

15.4

6811

.304

7.88

55.

133

3.07

01.

847

0.95

10.

442

0.15

00.

000

Not

es:

Par

amet

ers

ofth

ed

yn

amic

pan

eld

ata

model

,(2

9),

are

gen

erate

das:ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5),ϑi∼U

(0,0.9

5)

an

dγi∼IIDN

(1,1

),

fori

=1,2,...,N

.G

auss

ian

erro

rsar

ege

ner

ated

asuit∼IIDN

(0,1

)in

(31).

Non-G

au

ssia

ner

rors

are

gen

erate

dasuit

=( v−

2χ2 v,t

) 1/2ν it,

fori

=1,2,...,N

,

ν it∼IIDN

(0,1

)an

dχ

2 v,t

isa

chi-

squ

ared

ran

dom

vari

ate

wit

hv=

8d

egre

esof

free

dom

,in

(31).

Des

ign

1ass

um

esb i

∼U

(0.7,0.9

)fo

rth

efi

rstNb

(≤N

)el

emen

tsof

vect

orbN

,i

=1,

2,...,N

,in

the

con

stru

ctio

nof

the

corr

elati

on

matr

ixof

the

erro

rs,RN

=I N

+bN

b′ N−

B2 N

giv

enby

(30),

wh

ere

BN

=D

iag

(bN

).c p

(n,δ

)co

rres

pon

ds

toth

ecr

itic

alva

lue

use

din

the

mu

ltip

lete

stin

gp

roce

du

resh

own

in(2

3).

Th

enu

mb

erof

rep

lica

tion

sis

set

toR

=2000.

8

Tab

leS

2d

:B

ias

and

RM

SE

(×100)

for

theα

esti

mat

eof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

oma

dyn

amic

pan

eld

ata

mod

elw

ith

exog

enou

sre

gres

sors

Cro

ssco

rrel

atio

ns

are

gen

erate

du

sin

gD

esig

n2

wit

hG

au

ssia

nan

dn

on

-Gau

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

c p(n,δ

)w

ithn

=N

(N−

1)/2

,δ

=1/

4an

dp

=0.

05c p

(n,δ

)w

ithn

=N

(N−

1)/2

,δ

=1/

4an

dp

=0.

10G

auss

ian

erro

rsG

auss

ian

erro

rsT

NB

ias

TN

Bia

s10

010

02.

937

1.07

20.

123

-0.2

44-0

.710

-0.7

55-1

.231

-1.4

71-1

.460

-1.5

5510

010

05.

779

3.38

21.

900

1.11

30.

353

0.09

5-0

.515

-0.8

50-0

.907

-1.0

5020

03.

160

1.34

70.

512

-0.3

15-0

.740

-0.9

08-1

.162

-1.2

50-1

.366

-1.4

5220

06.

113

3.65

52.

200

0.94

70.

210

-0.1

63-0

.545

-0.7

16-0

.880

-0.9

9750

03.

620

1.66

20.

436

-0.3

31-0

.784

-0.9

57-1

.142

-1.2

68-1

.368

-1.4

3550

06.

666

3.97

22.

092

0.83

70.

066

-0.3

06-0

.603

-0.7

90-0

.922

-1.0

0820

010

03.

709

2.10

31.

368

1.18

50.

759

0.75

50.

280

0.04

20.

032

-0.0

9020

010

06.

276

4.04

32.

708

2.06

31.

341

1.12

40.

518

0.18

80.

115

-0.0

5220

03.

807

2.29

01.

669

0.95

60.

577

0.45

90.

217

0.12

00.

011

-0.0

8420

06.

494

4.22

92.

939

1.78

31.

094

0.77

10.

409

0.23

20.

077

-0.0

5050

04.

264

2.57

21.

549

0.88

60.

504

0.35

80.

190

0.07

7-0

.010

-0.0

7550

07.

057

4.53

32.

808

1.64

90.

948

0.60

90.

334

0.16

10.

039

-0.0

4650

010

03.

840

2.25

01.

510

1.34

00.

921

0.91

40.

431

0.17

20.

152

0.00

050

010

06.

358

4.13

52.

802

2.16

31.

447

1.22

60.

615

0.27

00.

191

0.00

020

03.

955

2.44

01.

818

1.09

90.

709

0.58

70.

345

0.23

30.

114

0.00

020

06.

603

4.33

33.

042

1.88

11.

183

0.85

60.

494

0.30

60.

142

0.00

050

04.

407

2.70

41.

675

1.00

10.

614

0.46

10.

287

0.17

00.

075

0.00

050

07.

166

4.63

02.

898

1.72

91.

024

0.67

90.

398

0.22

20.

094

0.00

0T

NR

MSE

TN

RM

SE

100

100

3.02

31.

327

0.86

30.

947

1.18

71.

223

1.56

71.

769

1.75

81.

847

100

100

5.82

03.

447

2.01

11.

305

0.77

70.

712

0.88

01.

119

1.16

21.

287

200

3.20

01.

471

0.87

40.

844

1.11

01.

247

1.45

01.

523

1.63

21.

711

200

6.12

93.

683

2.25

91.

100

0.62

90.

644

0.84

10.

964

1.10

31.

205

500

3.63

51.

730

0.75

70.

787

1.09

41.

248

1.41

21.

524

1.61

61.

681

500

6.67

13.

984

2.13

30.

974

0.55

20.

663

0.86

41.

015

1.13

01.

209

200

100

3.74

72.

142

1.40

31.

210

0.78

30.

774

0.32

00.

141

0.12

60.

138

200

100

6.30

54.

072

2.73

32.

082

1.35

81.

137

0.53

90.

222

0.15

20.

085

200

3.82

32.

305

1.67

90.

967

0.58

90.

471

0.24

00.

150

0.08

80.

120

200

6.50

54.

239

2.94

61.

791

1.10

10.

778

0.41

80.

243

0.10

00.

074

500

4.26

82.

575

1.55

20.

890

0.51

00.

364

0.20

20.

103

0.06

80.

103

500

7.05

94.

536

2.81

01.

651

0.95

10.

612

0.33

80.

168

0.06

00.

065

500

100

3.87

72.

283

1.53

51.

354

0.93

20.

921

0.44

30.

196

0.17

00.

000

500

100

6.38

64.

162

2.82

52.

178

1.46

01.

235

0.62

70.

289

0.20

60.

000

200

3.96

92.

453

1.82

61.

105

0.71

40.

590

0.34

90.

237

0.12

00.

000

200

6.61

34.

343

3.04

91.

887

1.18

80.

860

0.49

80.

310

0.14

70.

000

500

4.41

12.

707

1.67

71.

002

0.61

50.

462

0.28

70.

171

0.07

70.

000

500

7.16

84.

632

2.90

01.

731

1.02

50.

680

0.39

90.

223

0.09

50.

000

Non

-Gau

ssia

ner

rors

Non

-Gau

ssia

ner

rors

TN

Bia

sT

NB

ias

100

100

7.94

34.

850

2.70

71.

341

0.21

7-0

.321

-1.0

92-1

.499

-1.6

03-1

.738

100

100

11.5

427.

952

5.21

23.

270

1.68

50.

788

-0.2

11-0

.790

-1.0

15-1

.228

200

8.88

45.

619

3.33

01.

443

0.21

7-0

.469

-1.0

46-1

.320

-1.5

24-1

.660

200

12.4

498.

683

5.76

13.

318

1.60

90.

552

-0.2

65-0

.700

-0.9

98-1

.191

500

10.2

096.

610

3.77

71.

710

0.32

9-0

.446

-0.9

67-1

.272

-1.4

59-1

.571

500

13.6

519.

595

6.18

43.

520

1.62

80.

472

-0.2

88-0

.733

-0.9

95-1

.147

200

100

8.94

66.

018

4.02

02.

811

1.75

01.

289

0.55

30.

147

0.03

3-0

.135

200

100

12.3

288.

823

6.15

44.

304

2.76

91.

932

0.96

10.

386

0.15

7-0

.085

200

9.92

36.

769

4.59

22.

799

1.64

61.

015

0.47

90.

211

0.01

0-0

.131

200

13.2

829.

557

6.68

54.

302

2.63

51.

622

0.84

00.

410

0.11

4-0

.084

500

11.2

797.

734

4.97

52.

969

1.65

20.

926

0.43

70.

154

-0.0

18-0

.120

500

14.5

2810

.481

7.09

74.

456

2.59

91.

475

0.74

00.

314

0.06

5-0

.079

500

100

9.60

16.

570

4.45

73.

173

2.04

71.

540

0.77

80.

343

0.20

60.

000

500

100

12.9

469.

356

6.57

94.

646

3.03

42.

138

1.13

30.

527

0.27

50.

000

200

10.5

117.

264

4.99

13.

122

1.90

71.

233

0.67

60.

381

0.16

20.

000

200

13.8

2010

.018

7.06

04.

596

2.86

41.

799

0.98

60.

531

0.21

70.

000

500

11.9

108.

257

5.38

23.

272

1.88

41.

108

0.59

30.

296

0.11

50.

000

500

15.0

9410

.975

7.49

44.

755

2.81

61.

632

0.86

20.

417

0.15

60.

000

TN

RM

SE

TN

RM

SE

100

100

8.27

35.

217

3.10

41.

788

1.04

41.

032

1.48

21.

822

1.92

82.

059

100

100

11.7

638.

187

5.44

23.

474

1.91

71.

104

0.78

01.

106

1.30

11.

495

200

9.23

05.

995

3.70

11.

910

1.03

71.

023

1.38

71.

625

1.81

21.

946

200

12.6

768.

927

5.99

33.

555

1.86

90.

927

0.74

11.

002

1.24

91.

432

500

10.5

597.

000

4.19

42.

191

1.07

70.

965

1.27

31.

531

1.70

91.

821

500

13.8

819.

852

6.45

33.

794

1.91

70.

878

0.70

60.

981

1.21

21.

358

200

100

9.18

76.

261

4.22

62.

951

1.85

21.

342

0.60

20.

225

0.15

20.

196

200

100

12.4

958.

996

6.31

04.

422

2.85

81.

983

0.99

90.

421

0.19

60.

130

200

10.1

707.

014

4.79

42.

967

1.76

71.

083

0.52

60.

249

0.12

40.

184

200

13.4

509.

732

6.84

24.

440

2.74

11.

686

0.88

00.

432

0.14

60.

124

500

11.5

267.

991

5.21

73.

175

1.80

71.

020

0.49

30.

195

0.10

10.

166

500

14.6

9410

.662

7.27

84.

621

2.73

01.

559

0.78

90.

337

0.09

50.

114

500

100

9.75

36.

720

4.58

33.

254

2.10

41.

569

0.80

10.

364

0.22

10.

002

500

100

13.0

539.

466

6.67

84.

719

3.08

82.

170

1.15

80.

547

0.29

00.

002

200

10.6

897.

444

5.14

33.

251

2.00

01.

287

0.70

60.

395

0.16

90.

003

200

13.9

4010

.145

7.17

74.

700

2.94

61.

851

1.01

80.

546

0.22

40.

002

500

12.0

478.

398

5.51

03.

374

1.95

41.

145

0.61

00.

303

0.11

70.

001

500

15.1

8711

.076

7.59

24.

842

2.88

11.

670

0.88

20.

426

0.15

90.

000

Not

es:

Par

amet

ers

ofth

ed

yn

amic

pan

eld

ata

model

,(2

9),

are

gen

erate

das:ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5),ϑi∼U

(0,0.9

5)

an

dγi∼IIDN

(1,1

),fo

ri

=1,

2,...,N

.D

esig

n2

assu

mes

atw

o-fa

ctor

mod

elw

ith

[Nαβ]

an

d[N

αβ2]

non

-zer

olo

ad

ings

for

the

firs

tan

dse

con

dfa

ctor,

resp

ecti

vely

.W

ese

t:αβ

2=

2αβ/3,

wh

ereαβ

rela

tes

toα

un

der

(11)

andf jt∼

IIDN

(0,1

).G

au

ssia

ner

rors

are

giv

enby:uit

∼IIDN

(0,1

)an

dn

on

Gau

ssia

ner

rors

are

gen

erat

edas

:uit

=( v−

2χ2 v,t

) 1/2ν it,

fori

=1,

2,...,N

,ν it∼IIDN

(0,1

)an

dχ

2 v,t

isa

chi-

squ

are

dra

nd

om

vari

ate

wit

hv=

8d

egre

esof

free

dom

,in

(31).v ij

∼IIDU

(µvj−

0.2,µvj

+0.2

),j

=1,

2,µv

=0.

87,µv2

=0.

71,µv1

=√ µ

2 v−N

2(αβ2−αβ

)µ

2 v2),

in(3

2)

an

d(3

3).

Th

enu

mb

erof

rep

lica

tion

sis

set

toR

=2000.

9

Tab

leS

3:B

ias

an

dR

MS

E(×

100)

for

theα

esti

mat

eof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

oma

stati

cp

anel

dat

am

od

el

Cro

ssco

rrel

ati

on

sare

gen

erate

du

sin

gD

esig

n2

wit

hG

au

ssia

ner

rors

scale

dbyς

=√ 1/

2α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00c p

(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

05c p

(n,δ

)w

ithn

=N

(N−

1)/2

andp

=0.

10δ

=1/

2δ

=1/

2T

NB

ias

TN

Bia

s10

010

00.

812

0.02

7-0

.037

0.23

40.

107

0.29

5-0

.080

-0.2

44-0

.199

-0.2

7510

010

01.

253

0.35

90.

203

0.42

30.

256

0.42

40.

037

-0.1

37-0

.100

-0.1

8420

00.

446

0.04

60.

268

0.03

9-0

.039

0.03

2-0

.112

-0.1

38-0

.213

-0.2

7920

00.

802

0.29

80.

449

0.17

90.

076

0.13

3-0

.015

-0.0

49-0

.125

-0.1

9350

00.

303

0.11

40.

070

-0.0

03-0

.061

-0.0

23-0

.096

-0.1

64-0

.234

-0.2

8850

00.

575

0.29

70.

205

0.10

10.

029

0.06

0-0

.016

-0.0

84-0

.152

-0.2

0420

010

01.

013

0.27

80.

233

0.55

00.

416

0.61

60.

252

0.07

30.

105

-0.0

0320

010

01.

426

0.55

80.

406

0.66

00.

487

0.65

90.

280

0.09

10.

114

-0.0

0220

00.

649

0.28

90.

530

0.31

10.

241

0.31

90.

192

0.15

30.

077

-0.0

0420

00.

982

0.49

60.

650

0.38

40.

286

0.34

60.

210

0.16

30.

083

-0.0

0250

00.

497

0.33

50.

317

0.23

90.

195

0.23

60.

168

0.10

90.

048

-0.0

0450

00.

748

0.47

40.

393

0.28

10.

219

0.25

10.

177

0.11

50.

052

-0.0

0350

010

01.

046

0.30

10.

254

0.57

40.

444

0.63

60.

275

0.09

20.

120

0.00

050

010

01.

465

0.58

70.

432

0.68

20.

509

0.67

40.

296

0.10

30.

124

0.00

020

00.

688

0.32

10.

558

0.33

50.

262

0.33

80.

213

0.16

80.

092

0.00

020

01.

035

0.53

50.

680

0.40

80.

304

0.36

20.

225

0.17

50.

094

0.00

050

00.

534

0.36

10.

339

0.25

60.

213

0.25

30.

183

0.12

30.

060

0.00

050

00.

801

0.50

80.

418

0.29

90.

235

0.26

50.

189

0.12

60.

061

0.00

0T

NR

MSE

TN

RM

SE

100

100

0.87

50.

315

0.30

20.

388

0.31

90.

417

0.32

50.

390

0.37

20.

428

100

100

1.30

40.

470

0.33

10.

495

0.35

30.

486

0.25

40.

275

0.26

10.

309

200

0.49

60.

235

0.35

80.

238

0.24

30.

242

0.27

00.

292

0.34

30.

405

200

0.83

20.

362

0.48

90.

259

0.20

10.

228

0.19

00.

204

0.24

10.

295

500

0.34

10.

210

0.20

20.

197

0.20

90.

214

0.24

00.

286

0.34

60.

397

500

0.59

10.

329

0.25

10.

179

0.15

20.

171

0.16

70.

196

0.24

70.

293

200

100

1.03

80.

325

0.26

30.

561

0.42

70.

623

0.26

90.

115

0.12

70.

013

200

100

1.46

10.

604

0.43

70.

672

0.49

80.

666

0.29

60.

127

0.13

50.

009

200

0.66

30.

303

0.53

40.

316

0.24

60.

323

0.19

80.

158

0.08

50.

012

200

0.99

90.

510

0.65

50.

390

0.29

10.

350

0.21

50.

168

0.09

00.

008

500

0.50

10.

338

0.31

80.

240

0.19

60.

237

0.16

90.

110

0.05

10.

010

500

0.75

30.

477

0.39

50.

283

0.22

10.

252

0.17

80.

116

0.05

40.

007

500

100

1.07

10.

343

0.27

90.

582

0.45

10.

642

0.28

90.

123

0.14

10.

000

500

100

1.49

80.

630

0.46

00.

693

0.51

80.

680

0.31

00.

133

0.14

50.

000

200

0.70

20.

333

0.56

10.

339

0.26

60.

341

0.21

70.

173

0.09

80.

000

200

1.05

30.

550

0.68

50.

413

0.30

80.

365

0.23

00.

179

0.10

00.

000

500

0.53

80.

363

0.34

00.

257

0.21

30.

254

0.18

40.

124

0.06

10.

000

500

0.80

70.

511

0.41

90.

300

0.23

60.

266

0.19

00.

127

0.06

20.

000

δ=

1/3

δ=

1/3

TN

Bia

sT

NB

ias

100

100

2.10

40.

973

0.61

70.

712

0.46

10.

578

0.15

9-0

.038

-0.0

16-0

.116

100

100

3.60

32.

048

1.33

81.

180

0.77

60.

785

0.30

50.

063

0.05

4-0

.071

200

1.82

60.

977

0.88

00.

472

0.28

00.

284

0.11

00.

055

-0.0

32-0

.109

200

3.33

81.

987

1.50

20.

869

0.53

00.

445

0.21

90.

130

0.02

3-0

.069

500

1.78

41.

026

0.65

00.

381

0.22

40.

205

0.10

40.

025

-0.0

45-0

.101

500

3.25

51.

936

1.18

60.

692

0.40

80.

317

0.17

80.

079

-0.0

03-0

.065

200

100

2.26

11.

131

0.77

40.

886

0.62

80.

743

0.33

10.

119

0.12

6-0

.001

200

100

3.73

52.

174

1.45

71.

308

0.89

70.

902

0.42

50.

169

0.14

6-0

.001

200

2.01

01.

144

1.03

00.

617

0.42

30.

424

0.25

50.

187

0.09

4-0

.001

200

3.50

42.

131

1.62

70.

983

0.64

00.

547

0.32

30.

221

0.10

8-0

.001

500

1.96

91.

175

0.78

20.

495

0.33

60.

314

0.21

10.

133

0.06

1-0

.001

500

3.45

02.

084

1.30

40.

787

0.49

50.

398

0.25

40.

153

0.06

9-0

.001

500

100

2.30

41.

161

0.79

80.

903

0.64

40.

752

0.34

10.

126

0.13

30.

000

500

100

3.78

02.

208

1.48

91.

325

0.90

60.

905

0.43

10.

173

0.15

10.

000

200

2.07

31.

190

1.06

00.

637

0.43

70.

436

0.26

50.

194

0.10

10.

000

200

3.57

52.

182

1.65

71.

004

0.65

30.

556

0.33

10.

227

0.11

30.

000

500

2.06

21.

233

0.81

70.

516

0.35

10.

325

0.21

90.

139

0.06

60.

000

500

3.55

52.

152

1.34

80.

813

0.51

20.

408

0.26

10.

159

0.07

20.

000

TN

RM

SE

TN

RM

SE

100

100

2.14

81.

029

0.67

10.

750

0.50

50.

610

0.25

50.

186

0.18

00.

212

100

100

3.64

32.

090

1.37

31.

206

0.80

10.

805

0.34

90.

161

0.14

50.

143

200

1.84

81.

002

0.89

70.

496

0.31

20.

314

0.17

30.

145

0.14

20.

182

200

3.35

52.

004

1.51

40.

882

0.54

50.

459

0.24

40.

165

0.10

40.

123

500

1.79

11.

033

0.65

70.

391

0.24

00.

225

0.14

10.

105

0.12

10.

158

500

3.26

01.

940

1.19

00.

697

0.41

40.

325

0.19

10.

108

0.08

00.

108

200

100

2.30

01.

173

0.80

50.

901

0.64

10.

752

0.34

70.

152

0.14

60.

006

200

100

3.77

42.

214

1.48

81.

327

0.91

30.

913

0.44

10.

198

0.16

60.

004

200

2.02

91.

159

1.03

80.

624

0.42

90.

428

0.26

10.

192

0.10

10.

004

200

3.52

12.

147

1.63

70.

992

0.64

70.

551

0.32

90.

226

0.11

40.

003

500

1.97

51.

179

0.78

40.

497

0.33

70.

315

0.21

20.

134

0.06

20.

003

500

3.45

42.

088

1.30

70.

790

0.49

70.

399

0.25

50.

154

0.07

00.

002

500

100

2.34

61.

205

0.83

00.

919

0.65

60.

759

0.35

60.

154

0.15

30.

000

500

100

3.81

92.

250

1.52

11.

344

0.92

20.

916

0.44

70.

199

0.17

00.

000

200

2.09

31.

206

1.06

90.

644

0.44

30.

439

0.27

00.

199

0.10

70.

000

200

3.59

22.

197

1.66

71.

013

0.66

10.

561

0.33

70.

231

0.11

90.

000

500

2.06

71.

237

0.82

00.

518

0.35

30.

326

0.22

00.

140

0.06

70.

000

500

3.55

92.

156

1.35

10.

815

0.51

30.

409

0.26

20.

160

0.07

40.

000

Not

es:

Par

am

eter

sof

the

stat

icp

anel

dat

am

od

el,(2

9),

are

gen

erate

das:ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5),ϑi

=0

an

dγi∼IIDN

(1,1

),fo

ri

=1,

2,...,N

.D

esig

n2

assu

mes

atw

o-fa

ctor

mod

elw

ith

[Nαβ]an

d[N

αβ2]n

on

-zer

olo

ad

ings

for

the

firs

tan

dse

con

dfa

ctor,

resp

ecti

vel

y.W

ese

t:αβ

2=

2αβ/3

,w

her

eαβ

rela

tes

toα

un

der

(11)

,f jt

anduit∼IIDN

(0,1

)sc

aled

byς

=√ 1

/2,v ij∼IIDU

(µvj−

0.2,µvj

+0.2

),j

=1,

2,µv

=0.

87,µv2

=0.

71,µv1

=√ µ

2 v−N

2(αβ2−αβ

)µ

2 v2),

in(3

2)an

d(3

3).

Th

enu

mb

erof

rep

lica

tion

sis

set

toR

=2000.

10

Tab

leS

4a:

Com

par

ison

of

Bia

san

dR

MS

E(×

100)

for

theα

andα

esti

mat

esof

the

cros

s-se

ctio

nal

exp

onen

tof

the

erro

rsfr

oma

stat

ican

dd

yn

amic

pan

eld

ata

mod

elw

ith

exog

enou

sre

gres

sors

Cro

ssco

rrel

ati

on

sare

gen

erate

du

sin

gD

esig

n1

wit

hn

on

-Gau

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Bia

sR

MSE

Sta

tic

model

:ϑi

=0,

fori

=1,

2,...,N

TN

αT

Nα

100

100

2.27

20.

837

0.31

30.

221

0.04

20.

176

-0.1

59-0

.243

-0.1

03-0

.047

100

100

2.87

01.

712

1.03

20.

720

0.45

40.

330

0.28

70.

282

0.14

40.

098

200

2.18

40.

899

0.63

90.

179

0.02

70.

020

-0.1

14-0

.076

-0.0

78-0

.062

200

3.01

81.

642

1.16

10.

731

0.58

60.

219

0.21

40.

139

0.12

00.

112

500

2.13

81.

153

0.48

50.

213

0.03

00.

035

-0.0

36-0

.067

-0.0

90-0

.086

500

3.06

61.

984

1.03

10.

793

0.39

70.

250

0.15

30.

149

0.15

90.

145

200

100

2.68

01.

150

0.59

40.

359

0.15

60.

286

-0.0

90-0

.188

-0.0

500.

000

200

100

3.19

11.

765

1.19

00.

747

0.45

60.

459

0.21

10.

217

0.07

50.

003

200

2.60

41.

309

0.89

00.

323

0.13

40.

111

-0.0

22-0

.004

-0.0

170.

000

200

3.15

51.

943

1.34

90.

662

0.46

20.

240

0.20

00.

076

0.03

40.

002

500

2.64

21.

440

0.79

20.

418

0.17

80.

140

0.06

90.

027

0.00

00.

000

500

3.24

01.

965

1.26

50.

867

0.43

90.

230

0.13

90.

108

0.02

70.

001

500

100

2.98

81.

366

0.67

70.

431

0.20

20.

304

-0.0

79-0

.177

-0.0

470.

000

500

100

3.35

11.

900

1.01

50.

680

0.41

20.

445

0.17

60.

202

0.06

40.

000

200

3.03

51.

522

1.01

20.

399

0.16

40.

157

-0.0

190.

001

-0.0

140.

000

200

3.54

11.

946

1.27

80.

616

0.34

70.

294

0.13

80.

066

0.03

20.

000

500

3.17

01.

719

0.91

00.

449

0.20

30.

162

0.07

90.

029

0.00

30.

000

500

3.70

42.

155

1.29

80.

660

0.40

80.

239

0.12

80.

057

0.04

00.

000

TN

αT

Nα

100

100

2.18

10.

661

0.21

70.

273

0.17

80.

399

0.05

8-0

.009

0.09

80.

104

100

100

3.49

72.

601

2.03

61.

541

1.16

60.

901

0.63

90.

451

0.27

20.

104

200

1.14

50.

159

0.30

60.

023

0.10

00.

174

0.06

80.

105

0.10

80.

093

200

2.49

91.

945

1.62

41.

164

0.94

60.

615

0.41

90.

286

0.19

00.

093

500

0.48

50.

174

0.03

80.

073

0.09

20.

156

0.14

30.

116

0.09

40.

080

500

1.86

21.

757

1.11

70.

943

0.64

80.

468

0.31

10.

223

0.13

40.

080

200

100

3.29

31.

318

0.61

50.

402

0.22

10.

372

0.01

6-0

.075

0.05

00.

050

200

100

4.21

72.

571

2.02

21.

392

0.98

80.

825

0.51

40.

342

0.21

60.

050

200

1.44

60.

283

0.27

40.

061

0.01

40.

117

0.01

00.

066

0.05

70.

045

200

2.31

81.

734

1.22

00.

906

0.65

50.

474

0.38

10.

233

0.13

90.

045

500

0.60

10.

104

0.07

10.

075

0.04

20.

121

0.09

70.

074

0.05

20.

040

500

1.52

81.

103

1.04

00.

757

0.46

40.

330

0.24

10.

210

0.09

40.

040

500

100

8.33

04.

834

2.30

01.

191

0.59

40.

484

0.05

1-0

.078

0.03

30.

018

500

100

8.77

55.

465

2.96

01.

806

1.12

70.

815

0.44

70.

300

0.19

40.

018

200

3.32

61.

257

0.73

30.

201

0.06

60.

137

-0.0

040.

040

0.02

70.

017

200

3.90

51.

988

1.37

20.

775

0.52

90.

408

0.25

40.

179

0.10

40.

017

500

0.97

90.

238

0.11

10.

061

0.02

60.

097

0.06

70.

043

0.02

90.

015

500

1.52

50.

917

0.81

00.

485

0.41

00.

250

0.15

80.

104

0.08

50.

015

Dynam

icm

odel

wit

hex

ogen

ous

regr

esso

rs:ϑi∼U

(0,0.9

5)fo

ri

=1,

2,...,N

TN

αT

Nα

100

100

2.22

40.

770

0.29

30.

190

0.02

70.

171

-0.1

78-0

.256

-0.1

10-0

.054

100

100

2.81

81.

527

0.98

10.

768

0.44

40.

457

0.25

70.

301

0.15

60.

110

200

2.03

20.

821

0.58

90.

140

-0.0

320.

005

-0.1

28-0

.090

-0.0

96-0

.077

200

2.73

01.

507

1.09

60.

686

0.41

60.

262

0.21

30.

159

0.15

60.

136

500

2.04

30.

966

0.48

30.

171

0.01

50.

017

-0.0

51-0

.083

-0.1

05-0

.105

500

2.93

91.

547

1.05

50.

623

0.54

90.

314

0.16

50.

161

0.17

00.

175

200

100

2.71

91.

159

0.53

10.

338

0.14

90.

269

-0.0

85-0

.183

-0.0

510.

000

200

100

3.25

61.

787

1.04

20.

722

0.41

40.

404

0.33

10.

244

0.07

10.

002

200

2.63

21.

280

0.89

40.

334

0.11

60.

122

-0.0

25-0

.005

-0.0

160.

000

200

3.29

91.

992

1.49

00.

790

0.42

30.

291

0.20

50.

099

0.05

40.

006

500

2.62

21.

420

0.79

10.

370

0.16

40.

136

0.06

90.

026

0.00

10.

000

500

3.19

51.

991

1.38

60.

626

0.40

30.

229

0.15

50.

075

0.04

00.

002

500

100

2.97

41.

350

0.69

40.

405

0.20

30.

303

-0.0

76-0

.179

-0.0

480.

000

500

100

3.36

31.

821

1.09

50.

623

0.43

60.

403

0.18

90.

210

0.06

50.

000

200

2.91

31.

474

1.00

50.

405

0.17

00.

146

-0.0

13-0

.002

-0.0

140.

000

200

3.25

71.

957

1.35

60.

704

0.37

10.

286

0.23

70.

060

0.03

50.

000

500

3.13

81.

674

0.91

80.

478

0.19

80.

176

0.07

80.

029

0.00

10.

000

500

3.65

11.

969

1.28

70.

764

0.37

00.

331

0.14

30.

053

0.01

50.

000

TN

αT

Nα

100

100

2.10

40.

586

0.25

50.

282

0.17

30.

365

0.06

3-0

.034

0.10

50.

105

100

100

3.36

82.

626

1.98

61.

530

1.17

60.

983

0.61

00.

434

0.27

30.

105

200

1.09

60.

120

0.25

50.

033

0.09

40.

165

0.06

30.

123

0.10

90.

093

200

2.41

51.

888

1.55

11.

145

0.84

70.

638

0.42

90.

309

0.20

40.

093

500

0.46

10.

050

0.07

00.

138

0.12

00.

163

0.13

70.

118

0.09

30.

081

500

1.87

81.

414

1.23

50.

914

0.68

80.

487

0.30

20.

218

0.13

40.

081

200

100

3.37

11.

310

0.53

40.

356

0.21

40.

347

0.01

6-0

.068

0.05

50.

051

200

100

4.32

22.

617

1.82

21.

382

0.95

50.

765

0.57

50.

388

0.22

60.

051

200

1.45

10.

234

0.33

60.

056

0.01

10.

121

0.02

50.

067

0.05

90.

046

200

2.45

91.

627

1.45

90.

955

0.69

40.

499

0.37

20.

252

0.16

20.

046

500

0.60

30.

097

0.04

90.

061

0.04

40.

125

0.09

60.

071

0.05

70.

040

500

1.72

41.

229

1.12

60.

660

0.46

60.

318

0.27

20.

155

0.10

70.

040

500

100

8.19

44.

656

2.33

51.

190

0.55

70.

499

0.05

1-0

.072

0.03

50.

018

500

100

8.62

45.

271

3.00

31.

796

1.08

80.

819

0.45

70.

335

0.18

30.

018

200

3.27

51.

278

0.70

60.

195

0.06

90.

117

0.00

60.

036

0.02

80.

017

200

3.85

22.

096

1.35

20.

771

0.55

20.

407

0.33

20.

173

0.11

60.

017

500

0.97

60.

245

0.12

20.

076

0.03

10.

113

0.06

90.

045

0.02

50.

015

500

1.63

00.

927

0.82

70.

572

0.38

00.

368

0.20

40.

104

0.05

80.

015

Not

es:

Rem

ain

ing

par

amet

ers

ofth

ep

anel

dat

am

od

el,

(29),

are

gen

erate

das:ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5)

an

dγi∼IIDN

(1,1

),fo

ri

=1,

2,...,N

.

Non

-Gau

ssia

ner

rors

are

gen

erat

edasuit

=( v−

2χ2 v,t

) 1/2ν it,

fori

=1,2,...,N

,ν it∼IIDN

(0,1

)an

dχ

2 v,t

isa

chi-

squ

are

dra

nd

om

vari

ate

wit

hv=

8d

egre

esof

free

dom

,in

(31)

.D

esig

n1

assu

mesb i

∼U

(0.7,0.9

)fo

rth

efi

rstNb

(≤N

)el

emen

tsof

vect

or

bN

,i

=1,

2,...,N

,in

the

con

stru

ctio

nof

the

corr

elati

on

matr

ixof

the

erro

rs,RN

=I N

+bN

b′ N−

B2 N

give

nby

(30)

,w

her

eBN

=D

iag

(bN

).α

isco

mp

ute

du

sin

gc p

(n,δ

)w

ithn

=N

(N−

1)/

2,δ

=1/

2an

dp

=0.

05

inth

em

ult

iple

test

ing

pro

ced

ure

show

nin

(23)

.α

corr

esp

ond

sto

the

most

rob

ust

esti

mato

rof

the

exp

on

ent

of

cros

s-se

ctio

nal

dep

end

ence

con

sid

ered

inB

ail

eyet

al.

(201

6)an

dal

low

sfo

rb

oth

seri

alco

rrel

atio

nin

the

fact

ors

an

dw

eak

cross

-sec

tion

al

dep

end

ence

inth

eer

ror

term

s.W

euse

fou

rp

rin

cip

al

com

pon

ents

wh

enes

tim

atin

gc N

inth

eex

pre

ssio

nfo

rα

.T

he

nu

mb

erof

rep

lica

tion

sis

set

toR

=2000.

11

Tab

leS

4b

:C

omp

aris

onof

Bia

san

dR

MS

E(×

100)

for

theα

andα

ofth

ecr

oss-

sect

ion

alex

pon

ent

of

the

erro

rsof

ast

ati

can

dd

yn

amic

pan

eld

ata

mod

elw

ith

exog

enou

sre

gres

sors

Cro

ssco

rrel

ati

on

sare

gen

erate

du

sin

gD

esig

n2

wit

hn

on

-Gau

ssia

ner

rors

α0.

550.

600.

650.

700.

750.

800.

850.

900.

951.

00α

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Bia

sB

ias

Sta

tic

model

:ϑi

=0,

fori

=1,

2,...,N

TN

αα

100

100

1.11

6-0

.949

-2.0

57-2

.527

-3.0

24-3

.083

-3.5

51-3

.745

-3.7

13-3

.754

100

100

2.16

51.

928

2.63

13.

035

3.49

93.

576

3.99

94.

181

4.16

24.

206

200

0.93

3-1

.004

-1.9

30-2

.775

-3.2

10-3

.357

-3.6

31-3

.693

-3.7

77-3

.844

200

2.17

91.

962

2.49

93.

204

3.62

43.

796

4.06

64.

140

4.22

74.

292

500

0.89

8-1

.048

-2.2

61-3

.008

-3.4

56-3

.615

-3.7

82-3

.901

-3.9

98-4

.063

500

2.48

92.

230

2.88

93.

493

3.90

54.

068

4.23

74.

357

4.45

24.

516

200

100

2.76

41.

265

0.64

20.

535

0.22

10.

270

-0.1

59-0

.346

-0.3

18-0

.394

200

100

3.21

31.

794

1.13

10.

851

0.54

10.

478

0.38

30.

473

0.45

10.

508

200

2.65

51.

332

0.88

60.

307

0.02

0-0

.036

-0.2

26-0

.283

-0.3

61-0

.430

200

3.21

61.

863

1.23

10.

650

0.38

80.

317

0.37

20.

413

0.47

20.

532

500

2.62

31.

359

0.64

10.

190

-0.0

76-0

.138

-0.2

58-0

.342

-0.4

12-0

.460

500

3.27

91.

932

1.10

20.

598

0.39

40.

343

0.40

00.

467

0.52

90.

571

500

100

3.33

71.

824

1.18

01.

096

0.74

10.

794

0.34

90.

121

0.12

5-0

.002

500

100

3.56

92.

057

1.36

01.

190

0.80

60.

821

0.37

70.

156

0.14

40.

010

200

3.16

31.

851

1.41

70.

822

0.52

50.

467

0.27

00.

187

0.09

4-0

.002

200

3.46

62.

107

1.55

60.

915

0.57

80.

486

0.28

00.

193

0.10

00.

004

500

3.18

61.

896

1.17

40.

697

0.43

10.

353

0.22

30.

135

0.06

0-0

.002

500

3.60

52.

250

1.43

20.

871

0.53

00.

390

0.23

80.

140

0.06

20.

004

TN

αT

Nα

100

100

2.00

70.

792

0.56

40.

800

0.58

00.

729

0.32

70.

146

0.19

90.

104

100

100

3.30

32.

266

1.72

41.

516

1.11

41.

016

0.62

50.

395

0.30

10.

104

200

0.64

50.

087

0.43

70.

268

0.24

10.

330

0.21

40.

199

0.14

90.

092

200

2.03

41.

611

1.25

90.

928

0.70

00.

580

0.41

70.

308

0.20

20.

092

500

0.00

6-0

.107

0.05

70.

124

0.15

10.

237

0.19

50.

154

0.11

00.

080

500

1.64

91.

307

0.94

60.

667

0.48

00.

400

0.29

80.

216

0.13

70.

080

200

100

3.80

42.

191

1.41

61.

521

0.96

30.

939

0.45

90.

196

0.19

70.

051

200

100

4.58

93.

065

2.15

52.

028

1.40

01.

209

0.73

10.

422

0.30

50.

051

200

1.49

30.

719

0.81

60.

501

0.34

20.

382

0.23

20.

190

0.12

10.

046

200

2.26

01.

570

1.33

50.

929

0.65

10.

560

0.38

60.

271

0.16

90.

046

500

0.44

70.

118

0.21

40.

199

0.18

30.

238

0.17

80.

129

0.08

10.

040

500

1.25

80.

876

0.67

80.

493

0.37

50.

337

0.24

30.

170

0.10

10.

040

500

100

9.13

95.

921

3.98

13.

658

2.47

82.

109

1.24

50.

613

0.39

30.

019

500

100

9.49

66.

288

4.31

23.

896

2.76

22.

355

1.49

30.

836

0.53

50.

019

200

3.78

62.

622

2.23

01.

557

0.96

30.

765

0.48

70.

280

0.15

30.

018

200

4.22

43.

014

2.53

51.

877

1.24

80.

976

0.67

20.

394

0.22

10.

018

500

0.99

30.

664

0.72

00.

436

0.31

50.

290

0.18

50.

122

0.06

60.

016

500

1.41

21.

062

1.01

90.

667

0.49

20.

382

0.24

20.

158

0.08

60.

016

Dynam

icm

odel

wit

hex

ogen

ous

regr

esso

rs:ϑi∼U

(0,0.9

5)fo

ri

=1,

2,...,N

TN

αT

Nα

100

100

1.01

6-1

.103

-2.2

41-2

.771

-3.2

70-3

.357

-3.8

38-4

.045

-4.0

08-4

.036

100

100

2.11

72.

014

2.78

23.

266

3.74

93.

863

4.30

64.

507

4.48

84.

515

200

0.81

3-1

.192

-2.1

65-3

.069

-3.5

34-3

.708

-3.9

69-4

.029

-4.1

18-4

.184

200

2.21

92.

155

2.74

43.

521

3.96

94.

155

4.40

64.

474

4.55

94.

626

500

0.66

9-1

.297

-2.5

34-3

.294

-3.7

52-3

.926

-4.1

00-4

.216

-4.3

12-4

.380

500

2.42

42.

367

3.10

43.

732

4.15

14.

331

4.50

64.

622

4.71

74.

785

200

100

2.74

91.

249

0.61

90.

506

0.18

00.

235

-0.1

94-0

.387

-0.3

59-0

.431

200

100

3.19

11.

751

1.07

00.

792

0.49

20.

453

0.40

70.

521

0.50

20.

555

200

2.57

81.

267

0.83

40.

256

-0.0

18-0

.070

-0.2

68-0

.316

-0.3

97-0

.464

200

3.13

51.

816

1.21

00.

658

0.43

10.

357

0.42

40.

456

0.52

40.

579

500

2.56

81.

311

0.61

00.

161

-0.0

98-0

.161

-0.2

81-0

.366

-0.4

36-0

.485

500

3.27

01.

965

1.18

10.

708

0.48

60.

394

0.42

50.

483

0.54

50.

591

500

100

3.41

91.

888

1.22

21.

113

0.74

90.

792

0.34

90.

125

0.12

4-0

.002

500

100

3.67

42.

127

1.38

61.

183

0.79

10.

808

0.36

60.

154

0.14

30.

006

200

3.20

21.

882

1.43

90.

839

0.53

50.

471

0.27

20.

191

0.09

2-0

.002

200

3.61

82.

254

1.66

11.

001

0.63

10.

509

0.29

10.

198

0.09

90.

006

500

3.16

61.

879

1.15

80.

685

0.42

20.

348

0.22

10.

133

0.05

9-0

.002

500

3.50

22.

134

1.31

90.

778

0.46

80.

362

0.22

60.

135

0.06

10.

003

TN

αT

Nα

100

100

1.85

40.

704

0.48

10.

745

0.54

20.

707

0.30

50.

139

0.18

60.

105

100

100

3.17

72.

262

1.72

51.

472

1.08

91.

006

0.61

40.

388

0.28

70.

105

200

0.60

90.

031

0.38

50.

237

0.21

50.

316

0.20

20.

192

0.14

40.

093

200

2.04

21.

618

1.24

40.

914

0.69

00.

574

0.41

20.

303

0.19

90.

093

500

0.01

0-0

.106

0.06

00.

124

0.15

10.

242

0.19

70.

156

0.11

20.

080

500

1.52

21.

155

0.86

20.

636

0.47

60.

402

0.29

70.

215

0.13

90.

080

200

100

3.81

62.

133

1.37

11.

412

0.90

10.

891

0.42

40.

181

0.18

20.

051

200

100

4.63

63.

039

2.15

21.

945

1.34

41.

159

0.69

70.

422

0.29

40.

051

200

1.46

40.

701

0.81

10.

496

0.35

20.

379

0.22

80.

185

0.11

80.

046

200

2.27

31.

579

1.35

10.

952

0.68

20.

575

0.38

60.

274

0.16

90.

046

500

0.44

10.

101

0.20

40.

190

0.17

80.

237

0.17

80.

128

0.08

10.

040

500

1.40

41.

045

0.81

30.

565

0.41

00.

343

0.24

50.

169

0.10

20.

040

500

100

9.05

95.

866

3.98

03.

556

2.41

71.

994

1.20

20.

591

0.37

00.

019

500

100

9.45

06.

268

4.33

33.

831

2.71

92.

248

1.46

50.

843

0.51

50.

019

200

3.74

72.

521

2.19

31.

494

0.92

50.

732

0.47

80.

272

0.14

80.

018

200

4.20

12.

926

2.50

21.

816

1.23

30.

955

0.66

90.

387

0.21

80.

018

500

1.03

00.

646

0.70

50.

433

0.32

10.

286

0.19

10.

125

0.06

60.

016

500

1.45

61.

044

1.00

60.

661

0.49

60.

375

0.25

00.

161

0.08

60.

016

Not

es:

Rem

ain

ing

par

amet

ers

ofth

ep

anel

dat

am

od

el,

(29),

are

gen

erate

das:ai∼IIDN

(1,1

),ρix

∼U

(0,0.9

5)

an

dγi∼IIDN

(1,1

),fo

ri

=1,

2,...,N

.D

esig

n2

assu

mes

atw

o-fa

ctor

mod

elw

ith

[Nαβ]

and

[Nαβ2]

non

-zer

olo

ad

ings

for

the

firs

tan

dse

con

dfa

ctor,

resp

ecti

vely

.W

ese

t:αβ

2=

2αβ/3

,w

her

eαβ

rela

tes

toα

un

der

(11)

andf jt∼IIDN

(0,1

).G

auss

ian

erro

rsare

giv

enby:uit∼IIDN

(0,1

)an

dn

on

Gau

ssia

ner

rors

are

gen

erate

das:uit

=( v−

2χ2 v,t

) 1/2ν it,

fori

=1,

2,...,N

,ν it∼IIDN

(0,1

)an

dχ

2 v,t

isa

chi-

squ

are

dra

nd

om

vari

ate

wit

hv=

8d

egre

esof

free

dom

,in

(31).v ij∼IIDU

(µvj−

0.2,µvj

+0.2

),j

=1,

2,

µv

=0.

87,µv2

=0.

71,µv1

=√ µ

2 v−N

2(αβ2−αβ

)µ

2 v2),

in(3

2)

an

d(3

3).

Th

enu

mb

erof

rep

lica

tion

sis

set

toR

=2000.

12

Appendix B

Additional Empirical results

The figures below provide the estimates displayed in Figures 1 and 2 but using the shorter5-year rolling samples.

Figure 3: 5-year rolling estimates of the exponent of cross-sectional correlation (αt) of S&P 500securities’ excess returns

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Sep-

89

Sep-

91

Sep-

93

Sep-

95

Sep-

97

Sep-

99

Sep-

01

Sep-

03

Sep-

05

Sep-

07

Sep-

09

Sep-

11

Sep-

13

Sep-

15

Sep-

17

5-year rolling estimates of α ( with δ=1/2) based on excess returns5-year rolling estimates of α ( with δ=1/3) based on excess returns

αα

Figure 4: 5-year rolling estimates of the exponent of cross-sectional correlation (αt) of residualsfrom CAPM and its two Fama-French extensions

0.55

0.60

0.65

0.70

0.75

Sep-

89

Sep-

91

Sep-

93

Sep-

95

Sep-

97

Sep-

99

Sep-

01

Sep-

03

Sep-

05

Sep-

07

Sep-

09

Sep-

11

Sep-

13

Sep-

15

Sep-

17

5-year rolling estimates of α ( with δ=1/2) for the residuals of the CAPM model

5-year rolling estimates of α ( with δ=1/2) for the residuals of the CAPM model augmented by the SMB factor

5-year rolling estimates of α ( with δ=1/2) for the residuals of the CAPM model augmented by SMB and HML factors

α

α

α

Notes: CAPM model includes excess market returns, CAPM model augmented by SMB includesexcess market returns and small minus big (SMB) firm returns, and CAPM model augmented bySMB and HML includes excess market returns, small minus big (SMB) firm returns and high minuslow (HML) firm returns as regressors in (35), (36) and (37), respectively.

13

The figures below provide the estimates displayed in Figures 1 and 2 including their 95%confidence bands.

Figure 5: 10-year rolling estimates of the exponent of cross-sectional correlation (αt) of S&P500 securities’ excess returns, with 90% confidence intervals

0.80

0.82

0.84

0.86

0.88

0.90

0.92

0.94

0.96

0.98Se

p-89

Sep-

91

Sep-

93

Sep-

95

Sep-

97

Sep-

99

Sep-

01

Sep-

03

Sep-

05

Sep-

07

Sep-

09

Sep-

11

Sep-

13

Sep-

15

Sep-

17

10-year rolling estimates of α ( with δ=1/2) based on excess returns

10-year rolling estimates of 0.05 percentile of α's empirical distribution

10-year rolling estimates of 0.95 percentile of α's empirical distribution

α

Notes: In the critical value, cp(n, δ) we set p = 0.05 and δ = 1/2.

Figure 6: 10-year rolling estimates of the exponent of cross-sectional correlation (αt) of S&P500 securities’ excess returns, with 90% confidence intervals

0.80

0.82

0.84

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

Sep-

89

Sep-

91

Sep-

93

Sep-

95

Sep-

97

Sep-

99

Sep-

01

Sep-

03

Sep-

05

Sep-

07

Sep-

09

Sep-

11

Sep-

13

Sep-

15

Sep-

17

10-year rolling estimates of α ( with δ=1/3) based on excess returns

10-year rolling estimates of 0.05 percentile of α's empirical distribution

10-year rolling estimates of 0.95 percentile of α's empirical distribution

α

Notes: In the critical value, cp(n, δ) we set p = 0.05 and δ = 1/3.

14

Figure 7: 10-year rolling estimates of the exponent of cross-sectional correlation (αt) of residualsfrom CAPM model, with 90% confidence intervals

0.60

0.62

0.64

0.66

0.68

0.70

0.72

0.74

0.76

0.78

Sep-

89

Sep-

91

Sep-

93

Sep-

95

Sep-

97

Sep-

99

Sep-

01

Sep-

03

Sep-

05

Sep-

07

Sep-

09

Sep-

11

Sep-

13

Sep-

15

Sep-

17

10-year rolling estimates of α ( with δ=1/2) for the residuals of the CAPM model

10-year rolling estimates of 0.05 percentile of α's empirical distribution

10-year rolling estimates of 0.95 percentile of α's empirical distribution

α

Notes: CAPM model includes excess market returns as regressor in (35). In the criticalvalue, cp(n, δ) we set p = 0.05 and δ = 1/2. Confidence intervals are computed using theboostrap procedure described in Section 6.

Figure 8: 10-year rolling estimates of the exponent of cross-sectional correlation (αt) of residualsfrom CAPM model augmented by SMB, with 90% confidence intervals

0.60

0.62

0.64

0.66

0.68

0.70

0.72

0.74

0.76

0.78

Sep-

89

Sep-

91

Sep-

93

Sep-

95

Sep-

97

Sep-

99

Sep-

01

Sep-

03

Sep-

05

Sep-

07

Sep-

09

Sep-

11

Sep-

13

Sep-

15

Sep-

17

10-year rolling estimates of α ( with δ=1/2) for the residuals of the CAPM model augmented by the SMB factor

10-year rolling estimates of 0.05 percentile of α's empirical distribution

10-year rolling estimates of 0.95 percentile of α's empirical distribution

α

Notes: CAPM model augmented by SMB includes excess market returns and small minusbig (SMB) firm returns as regressors in (36). In the critical value, cp(n, δ) we set p = 0.05and δ = 1/2. Confidence intervals are computed using the boostrap procedure described inSection 6.

15

Figure 9: 10-year rolling estimates of the exponent of cross-sectional correlation (αt) of residualsfrom CAPM model augmented by SMB and HML, with 90% confidence intervals

0.60

0.62

0.64

0.66

0.68

0.70

0.72

0.74

0.76

0.78

Sep-

89

Sep-

91

Sep-

93

Sep-

95

Sep-

97

Sep-

99

Sep-

01

Sep-

03

Sep-

05

Sep-

07

Sep-

09

Sep-

11

Sep-

13

Sep-

15

Sep-

1710-year rolling estimates of α ( with δ=1/2) for the residuals of the CAPM model augmented by SMB and HML factors10-year rolling estimates of 0.05 percentile of α's empirical distribution

10-year rolling estimates of 0.95 percentile of α's empirical distribution

α

Notes: CAPM model augmented by SMB and HML includes excess market returns, smallminus big (SMB) firm returns and high minus low (HML) firm returns as regressors in (37).In the critical value, cp(n, δ) we set p = 0.05 and δ = 1/2. Confidence intervals are computedusing the boostrap procedure described in Section 6.

16