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Forschungsinstitut zur Zukunft der ArbeitInstitute for the Study of Labor
Exponent of Cross-sectional Dependence:Estimation and Inference
IZA DP No. 6318
January 2012
Natalia BaileyGeorge KapetaniosM. Hashem Pesaran
Exponent of Cross-sectional Dependence:
Estimation and Inference
Natalia Bailey University of Cambridge
George Kapetanios
Queen Mary, University of London
M. Hashem Pesaran University of Cambridge,
University of Southern California and IZA
Discussion Paper No. 6318 January 2012
IZA
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IZA Discussion Paper No. 6318 January 2012
ABSTRACT
Exponent of Cross-sectional Dependence: Estimation and Inference*
An important issue in the analysis of cross-sectional dependence which has received renewed interest in the past few years is the need for a better understanding of the extent and nature of such cross dependencies. In this paper we focus on measures of cross-sectional dependence and how such measures are related to the behaviour of the aggregates defined as cross-sectional averages. We endeavour to determine the rate at which the cross-sectional weighted average of a set of variables appropriately demeaned, tends to zero. One parameterisation sets this to be O(N 2α-2), for 1/2 < α ≤ 1. Given the fashion in which it arises, we refer to as the exponent of cross-sectional dependence. We derive an estimator of from the estimated variance of the cross-sectional average of the variables under consideration. We propose bias corrected estimators, derive their asymptotic properties and consider a number of extensions. We include a detailed Monte Carlo study supporting the theoretical results. Finally, we undertake an empirical investigation of using the S&P 500 data-set, and a large number of macroeconomic variables across and within countries. JEL Classification: C21, C32 Keywords: cross correlations, cross-sectional dependence, cross-sectional averages,
weak and strong factor models, Capital Asset Pricing Model Corresponding author: M. Hashem Pesaran Trinity College University of Cambridge CB2 1TQ Cambridge United Kingdom E-mail: [email protected]
* We are grateful to Jean-Marie Dufour and Oliver Linton for helpful comments and discussions. Natalia Bailey and Hashem Pesaran acknowledge financial support under ESRC Grant ES/I031626/1.
1 Introduction
Over the past decade there has been a resurgence of interest in the analysis of cross-sectional depen-
dence applied to households, firms, markets, regional and national economies. 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 alterna-
tive network formations or market inter-connections. Many of these topics are the subject of ongoing
research. In this paper we focus on measures of cross-sectional dependence and how such measures
are related to the behaviour of cross-sectional averages or aggregates.
Perhaps, the simplest and most concise way to motivate the need for determining the extent of
cross-sectional dependence is to view the matter from a simple statistical viewpoint. Let xit denote
a double array of random variables indexed by i = 1, 2, ..., N and t = 1, 2, ..., T, over space and time,
respectively. Then, a variety of analyses focus on weighted averages of xit over i. Examples include
the construction of portfolios, when xit are asset returns, or aggregate macroeconomic variables, when
xit are firm or consumer level data, such as firm sales, or individual consumption. Weighted averages
take the form xwt =∑N
i=1wNixit, where the weights wNi are granular in the sense that wNi = O(
1N
).
For simplicity, we set wNi = 1N and xwt = xt. Then, it is of considerable interest to determine the
behaviour of xt and, in particular, the rate at which xt, when appropriately standardised, tends to
zero. In the case of asset returns this determines the extent to which risk, associated with investing
in particular portfolios of assets, is diversifiable. In the case of firm sales this is of interest in relation
to the effect of idiosyncratic, firm level, shocks onto aggregate macroeconomic variables such as GDP.
In the case where xit are cross-sectionally independent, using standard law of large numbers, one
obtains the result that xt = Op(N−1/2
). However, in the more general and realistic case where xit
are cross-sectionally correlated we have1
V ar (xt) =1
N2
N∑i=1
σ2i,x
+1
N2
N∑i=1
N∑j=1,i 6=j
σij,x = N−1σ2xN + τN , (1)
where σ2i,x
= V ar(xit) and σij,x = Cov(xit, xjt), assuming, for simplicity, stationarity, over time, for xit.
The above result suggests two important conditions under which standard law of large numbers may
fail to apply to the cross-sectional averages. One relates to the absence of finite variances for individual
xit,2 while the other to the rate at which the remainder term, τN = 1
N2
∑Ni=1
∑Nj=1,i 6=j σij,x, grows with
N . Determining the extent of cross-sectional dependence, relates directly to the second way, in which
standard law of large numbers may not be applicable, and lead to phenomena of considerable interest
as discussed above. It is, therefore, of interest to investigate the rate at which τN declines with N .
We may parametrise this by letting τN = O(N2α−2
), with α measuring the degree of cross-sectional
dependence. We note that V ar (xt) cannot decline at a rate lower than N−1 sinceσ2xNN = O
(N−1
).
Consider now the estimator of τN , based on sample covariances, σij,x, namely τN = 1N2
∑Ni=1
∑Nj=1,i 6=j σij,x.
It is easily seen that τN = σ2x + Op
(N−1
), where σ2
x = 1T
∑Tt=1
(xt − 1
T
∑Tt=1 xt
)2. Therefore, the
rate at which V ar (xt) tends to zero with N , is governed by the rate at which σ2x tends to zero. But
1(1) can be equivalently stated in terms of correlations, but since correlations are more complex to handle we chooseto carry out the analysis in terms of covariances.
2This way has been recently explored, in the case of firm sales data, by Gabaix (2011), who noted that the failureof such data to have finite variances can lead to idiosyncratic firm shocks determining, to a considerable extent, GDPgrowth volatility.
2
this rate can not be faster than N−1, and hence the range of interest for α must lie in the range
−1 < 2α − 2 ≤ 0, or 1/2 < α ≤ 1, This paper focusses on the problem of identification, estimation
and inference regarding α, which we refer to as the exponent of cross-sectional dependence, bearing
in mind that α is defined by τN = O(N2α−2
).
However, it is important to note that different problems may require the determination of the
exponent of different summaries of the covariance matrix. For example, a summary focused upon by
Chudik, Pesaran, and Tosetti (2011) is the column sum norm of the covariance matrix. The exponent
of that is given by the parametrisation maxi∑N
j=1 |σij,x| = O (Nα). It is clear that this exponent need
not be the same as the one relating to τN . Therefore, while the exponent of cross-sectional dependence
is of great interest for the phenomena outlined above, one needs to be clear about the motivation for
using this measure of cross-sectional dependence.
Also, other measures of cross-sectional dependence can be considered. An important example is
the measure based on the upper bound for V ar (xt), which is N−1λmax (ΣN ), where ΣN = E (xtx′t),
xt = (x1t, x2t, ..., xNt)′ and λmax (ΣN ) denotes the maximum eigenvalue of ΣN . λmax (ΣN ) is an
object of considerable interest in the statistical literature on large data sets. However, work in the
area (see, e.g., Yin, Bai, and Krishnaiah (1988), Bai and Silverstein (1998), Hachem, Loubaton, and
Najim (2005a) and Hachem, Loubaton, and Najim (2005b)) suggests that as a statistical measure
of cross-sectional dependence λmax (ΣN ) could be difficult to analyse especially for temporally and
cross-sectionally dependent data. This is partly due to the fact that estimates of λmax (ΣN ) based on
sample estimates of ΣN , could be very poor when N is large relative to T , which is the type of data
sets often encountered in macroeconomics and finance. Therefore, we do not pursue its analysis in
this paper, although we acknowledge the need for further investigations in this area, comparing the
performance of estimates of cross-sectional dependence based on α and on N−1λmax (ΣN ).
The above measures of cross-sectional dependence are related to the degree of pervasiveness of
factors in unobserved factor models often used in the literature to model cross-sectional dependence.3
Consider the following canonical factor model
xit = ai + β′if t + uit,
where f t is the m× 1 vector of unobserved factors (m being fixed), and βi = (βi1, βi2, ..., βim)′ is the
associated vector of factor loadings, and Cov (uit, ujt) = ωij . The extent of cross-sectional dependence
in xit crucially depends on the nature of factor loadings. The degree of cross-sectional dependence will
be strong if βi is bounded away from 0 and the average value of βi is different from zero. In such a case
supiN−1∑N
j=1 |σij,x| = O (1) and τN = N−2∑N
i=1
∑Nj=1,i 6=j σij,x = O (1), for all i, which yields α = 1.
However, other configurations of factor loadings can also be entertained, that yield values of α in the
range (1/2, 1]. Since both ft and βi are unobserved, taking a strong stand on a particular value of α
might not be justified empirically. Accordingly, Chudik, Pesaran, and Tosetti (2011), Kapetanios and
Marcellino (2010) and Onatski (2011) have considered an extension of the above factor model which
allows for a wider spectrum of cross-sectional dependence behaviours by specifying that the factor
loadings βi may be functions of N and decline in magnitude as N → ∞ allowing for the possibility
of factors having a weaker effect than is the case for standard factor models.4 One such formulation
assumes that factor loadings decline with N , and βi = O(Nα−1), for any α < 1. This specification
has the same order for N−2∑N
i=1
∑Nj=1 β
′iβj =
(N−1
∑Ni=1 βi
)′ (N−1
∑Nj=1 βj
)= O(N2α−2), so long
3Factor models have a long pedigree both as a conceptual device for summarising multivariate data sets as well as anempirical framework with sound theoretical underpinnings both in finance and economics. Recent econometric researchon factor models include Bai and Ng (2002), Bai (2003), Stock and Watson (2002), and Pesaran (2006).
4While Chudik, Pesaran, and Tosetti (2011) and Kapetanios and Marcellino (2010) have considered the case where1/2 < α < 1, Onatski (2011) has focused on the case where a ≤ 1/2.
3
as N−2∑N
i=1
∑Nj=1,i 6=j ωij is of smaller order of magnitude than τN . The latter condition is satisfied
under an approximate factor model, so long as 2(α− 1) > −1, or if α > 1/2.
Although mathematically convenient, the assumption that all factor loadings vary with N (almost
uniformly) is rather restrictive in many economic applications. Therefore, we will not consider it in
detail, but only briefly as an alternative formulation. In this paper we consider a baseline formulation
where we allow βi to be fixed in N , but assume that only Nα of the N factor loadings are individually
important, in the sense that they are bounded away from zero in absolute value.5 More specifically,
we consider
βik = vik for i = 1, 2, ..., [Nαk ] , (2)
βik = ckρi−[Nαk ]+1k , for i = [Nαk ] + 1, [Nαk ] + 2, ..., N ,
for k = 1, 2, ...,m, where [Nαk ] is the integer part of Nαk , 1/2 < αk ≤ 1, |ρk| < 1, ck is a finite
constant, and vik ∼ iid(µvk , σ2vk
), with µvk 6= 0 and σ2vk
> 0. In effect, the factor loadings are
grouped into two categories, a strong category with effects that are bounded away from zero, and a
weak category with transitory effects that tend to zero exponentially. The focus of our analysis is on
α = maxk(αk), which is the cross-sectional exponent as defined above. As we shall see, since we are
interested in the behaviour of cross-sectional averages, our proposed estimator of α will be invariant
to the ordering of the factor loadings within each group. Also the exponential decay assumed for the
second group of factor loadings can be relaxed and replaced by the absolute summability condition,∑Ni=[Nαk ]+1 |βik| <∞, which is again invariant to the ordering of the units with weak dependence on
factors. It is important to recognize that while this formulation is the one we focus on, alternative
formulations can give the same exponent. While some details of our inferential theory does depend on
the nature of the specific formulation adopted, it is clear that very similar inferential procedure with
similar asymptotic and small sample properties can be easily developed for alternative formulations.
In cases where the common factors are observed the strength of the factors as measured by α can
be estimated directly in terms of the number of statistically significant βi coefficients. Denoting the
number of such statistically significant estimates of the loadings associated with the ith factor by Mi,
the estimates of αi can then be obtained as ln(Mi)/ ln(N). We shall consider such a ”direct” estimate
of α in our empirical application to Capital Asset Pricing Model.
Following the theoretical line of reasoning advanced above, in this paper we propose the use of the
variance of the cross-sectional average of the observed data, xt, to estimate and carry out inference on
α. Focusing, for simplicity, on a single factor representation, we show that6
V ar(xt) = κ2[N2α−2
]+N−1cN +O(Nα−2),
where κ2 = σ2fµ
2v, σ
2f is the variance of the factor process, µv is the mean of the factor loadings, and cN
is a bias term that is analysed in detail in the main body of the paper. Using this relationship, we can
provide a feasible estimator for α and derive inferential theory for it. The property of our proposed
estimator depends on the choice of κ. For an arbitrary but bounded value of κ, our estimator is
consistent but its rate of convergence at 1/ ln(N) is rather slow. However, given the identified nature
of σ2f in factor models it seems more sensible to consider estimating α for a given value of σ2
f . In the
factor literature the factor loadings are typically identified by setting σ2f = 1, assuming that the factors
are strong, and the idiosyncratic components are weakly cross correlated. However, since the aim here
is to estimate α, it is more sensible to fix κ2. The rationale behind this alternative normalization, and
its empirical implications will be discussed in detail. Further, we propose a second order bias corrected
5Note that α in our baseline formulation is not directly comparable to the α of the almost uniform loadings formulation.6We consider a general multi-factor representation, in detail, in Section 3.2.
4
estimator that addresses the term cN , and derive the asymptotic distribution of both estimators for a
given value of κ2. We consider extensions that relate to the presence of multiple factors, potentially
with multiple distinct exponents of cross-sectional dependence, temporal dependence in ft or uit, and
weak cross-sectional dependence in uit. It is worth noting that our baseline estimator is equivalent to
one obtained by setting up a regression framework whereby the logarithm of the partial sum process
of estimated factor loadings is regressed upon the logarithm of the cross-sectional dimension of the
partial sum.
To illustrate the properties of the proposed estimators of α and their asymptotic distributions,
we carry out a detailed Monte Carlo study that considers a battery of robustness checks. Finally, we
provide a number of empirical applications investigating the degree of inter-linkages in real and financial
variables in the global economy, the extent to which macroeconomic variables are interconnected across
and within countries, with special reference to the US and UK economies in the second case, and
present recursive estimates of α applied to excess returns on securities included in the Standard &
Poor 500 index. In these applications to ensure that our estimates of α are invariant to the scales of
measurement of the observations, xit, we base our analysis on the standardized variates, (xit− xi)/si,where xi and si are sample mean and standard deviation of xit, respectively, over the sample under
consideration.
The rest of the paper is organised as follows: Section 2 provides a formal characterisation of α in
the context of a single factor model, and discusses potential estimation strategies. This section also
presents the rudiments of the analysis of the variance of the cross-sectional average and motivates the
baseline estimator and bias corrected versions of it. Sections 3-3.3 present the theoretical results of
the paper. Section 3 provides the full inferential theory and discusses feasible estimation, including
estimators for the variance of the estimator of cross-sectional dependence. Section 3.2 presents an
extension of the theoretical analysis to a multi-factor setting and briefly touches upon an alternative
specification of factor loadings. Section 3.3 considers the normalization σ2fµ
2v = κ2, and Section 4
discusses the conditions under which estimation of α and κ2 can be carried out jointly. Section 5
presents a detailed Monte Carlo study. The empirical applications are discussed in Section 6. Finally,
Section 7 concludes. Proofs of all theoretical results are relegated to the Appendix.
2 Preliminaries and Motivations
As noted in the Introduction, to characterize the degree of cross-sectional dependence in xit we use a
possibly weak factor model. We begin with the following single factor specification
xit = ai + βift + uit for i = 1, 2, ..., N ; t = 1, 2, ..., T, (3)
where ft is an unobserved factor, βi are the associated factor loadings and ai are bounded constants
such that supi |ai| < K <∞. We make the following assumptions.
Assumption 1 The factor loadings are given by
βi = vi for i = 1, 2, ..., [Nα] , (4)
βi = cρi−[Nα]+1, for i = [Nα] + 1, [Nα] + 2, ..., N ,
where 1/2 < α ≤ 1, [Nα] is the integer part of Nα, |ρ| < 1, and {vi}[Nα]
i=1 is an identically, independently
distributed (IID) sequence of random variables with mean µv 6= 0, and variance σ2v <∞.
5
Assumption 2 The factor, ft, follows a linear stationary process given by
ft =∞∑j=0
ψf,jνf,t−j, (5)
where νft is an IID sequence of random variables with mean zero and finite variance and uniformly
finite ϕ-th moments for some ϕ > 4. We assume that
∞∑j=0
jζ |ψfj | <∞,
such that {ζ(ϕ− 2)}/{2(ϕ− 1)} ≥ 1/2. ft is distributed independently of the idiosyncratic errors, uit′,
and the factor loadings, βi, for all i, t and t′.
Assumption 3 For each i, uit follows a linear stationary process given by
uit =∞∑j=0
ψij
( ∞∑s=−∞
ξjsνj,t−s
), (6)
where νit, i = ...,−1, 0, ..., t = 0, ..., is a double sequence of IID random variables with mean zero and
uniformly finite variances, σ2νi and uniformly finite ϕ-th moments for some ϕ > 4. We assume that
supi
∞∑j=0
jζ |ψij | <∞, (7)
and
supi
∞∑s=−∞
|s|ζ |ξis| <∞ (8)
such that {ζ(ϕ− 2)}/{2(ϕ− 1)} ≥ 1/2.
It is worth briefly commenting on these assumptions. Assumption 1 has been motivated in Section
1, and implies that N−1∑N
i=1 β2i = Op
(Nα−1
), which is more general than the standard assumption in
the factor literature that requires N−1∑N
i=1 β2i to have a strictly positive limit (see, e.g., Assumption
B of Bai and Ng (2002)). It is easy to see that the standard assumption is satisfied only if α =
1. Assumptions 2 and 3 are mostly straightforward specifications of the factor and error processes
assuming a linear structure with sufficient restrictions to enable the use of central limit theorems. The
only noteworthy part of these assumptions relates to the cross-sectional dependence of the error terms.
Here, cross-sectional dependence is structured in a flexible linear way so as to mirror, to the extent
possible, the conditions assumed for stationary (weak) time series dependence. From Assumption 3 it
follows that E(u2it) = σ2
i = σ2νi
(∑∞s=−∞ ξ
2is
) (∑∞s=0 ψ
2is
)<∞.
A generalization of the factor model and the related Assumptions, 1 and 2, will be considered in
Section 3.2. In the rest of this Section, we motivate our proposed estimator for α.
We write (3) as
xt = a + β ft + ut,
where xt = (x1t, x2t, ..., xNt)′, a = (a1, a2, ..., aN )′, β= (β1, β2, ..., βN )′ and ut = (u1t, u2t, ..., uNt)
′. We
also note that under the above assumptions, Σβ = E(ββ′) − E(β)E(β′), with λmax(Σβ) < K < ∞;
6
Σu = E(utu′t), with λmax(Σu) < K < ∞, E(ft) = 0, E(f2
t ) = σ2f > 0, and ft and β are distributed
independently. Hence
xt − E(xt) = βft + ut,
Cov(xt) = E[(βft + ut) (βft + ut)
′]= E(ββ′)E(f2
t ) + E(utu
′t
)=[Σβ + E(β)E(β′)
]σ2f + Σu.
Consider now the cross-sectional averages of the observables defined by xt = τ ′Nxt/N , where τNis an N × 1 vector of ones. Therefore
V ar(xt) = N−2τ ′NCov(xt)τN = N−2τ ′N[σ2fΣβ + Σu
]τN+σ2
f
[τ ′NE(β)
N
]2
. (9)
But under (4), it follows that
N∑i=1
βi = [Nα]
1
[Nα]
[Nα]∑i=1
vi +cρ
[Nα]
(1− ρ(N−[Nα])
1− ρ
) =(vN +O
(N−α
))[Nα]
where vN = 1[Nα]
∑[Nα]i=1 vi is Op (1) and for simplicity, we set vi = 0, for i > [Nα]. Note that the
specification of βi for i > [Nα] need not be of the form given in (4). Any sequence of loadings, for
which∑N
i=[Nα]+1 βi = Op (1) is acceptable. Hence
N−1τ ′NE(β) = µv[Nα−1
]+O(N−1).
Also
N−2τ ′NΣβτN=N−2τ ′1NΣβ(1)τ 1N ≤[Nα−2
]λmax (Σβ) ,
where τ 1N is an [Nα]× 1 vector of ones and Σβ(1) is the upper [Nα]× [Nα] sub-matrix of Σβ. Using
the above results in (9) we now have
V ar(xt) ≤[Nα−2
]σ2fλmax (Σβ) +N−1cN + κ2
[N2α−2
].
whereτ ′NΣuτN
N= cN < K <∞. (10)
But, by assumption λmax (Σβ) < K <∞, and hence under 1 ≥ α > 1/2 we have
V ar(xt) = κ2[N2α−2
]+N−1cN +O(Nα−2). (11)
Depending on how σ2f is normalized, different estimators of α can be envisaged. Since, by assumption,
µv 6= 0, a natural normalization would be σ2f = 1/µ2
v or κ2 = 1. Then, a simple manipulation of (11)
yields
2(α− 1) ln(N) ≈ ln(σ2x) + ln
(1− N−1cN
σ2x
)≈ ln(σ2
x)− N−1cNσ2x
,
or
α ≈ 1 +1
2
ln(σ2x)
ln(N)− cN
2 [N ln(N)]σ2x
. (12)
7
Note that the third term on the RHS of (12) is of smaller order of magnitude than the other two
terms. In cases where α ≤ 1/2, the second term in the RHS of (11), that arises from the contribution
of the idiosyncratic components, will be at least as important as the contribution of a weak factor,
and, in consequence, α cannot be identified. However, for values of α > 1/2, α can be identified from
(12) using a consistent estimator of V ar(xt) = σ2x, given by
σ2x =
1
T
T∑t=1
(xt − x)2 , (13)
where x = T−1∑T
t=1 xt. A simple consistent estimator of α is given by
α = 1 +1
2
ln(σ2x)
ln(N). (14)
Further, in the case of exact factor models where Σu is a diagonal matrix, the third term in (12) can
be estimated by
cN = N−1N∑j=1
σ2j = σ2
N ,
where σ2j is the jth diagonal term of Σu and σ2
j is its estimator. This suggests the following modified
estimator of α
α = 1 +1
2
ln(σ2x)
ln(N)−
σ2N
2 [N ln(N)] σ2x
. (15)
Note that while cN , as an estimator for cN , is motivated by appealing to an exact factor model, it is
also valid for mild deviations from this model as discussed in the next Section. The above estimators
of α form the basis of the formal analysis that is carried out in Section 3.
3 Theoretical Derivations
In this section we provide a formal analysis of our proposed estimators.
3.1 Main Results
Our first set of theoretical results characterise the asymptotic behaviour of α. Introducing the ad-
ditional notations, σ2N = N−1
∑Ni=1 σ
2i , s
2f = 1
T
∑Tt=1
(ft − 1
T
∑Tt=1 ft
)2, µf = E(ft), and σ2
f =
E(ft − µf )2, we have:
Theorem 1 Let Assumptions 1-3 hold, m = 1 and α > 1/2. Then,
√min(Nα∗ , T )
(2 ln(N) (α− α∗)−
σ2N
N2α−1v2Ns
2f
)→d N (0, ω)
where
α∗ ≡ α∗N = α+ln(κ2)
2 ln (N), (16)
ω = limN,T→∞
[min(Nα, T )
TVf2 +
min(Nα, T )
Nα
4σ2v
µ2v
], (17)
Vf2 = V ar
(f2t
)+ 2
∞∑i=1
Cov(f2t , f
2t−i
),
and ft = (ft − µf )/σf .
8
This theorem shows that α, as an estimator of α, is subject to two sources of bias. The first relates
to the term ln(κ2)/ ln (N) in α∗, which arises due to the way identification of the factor loadings
depends on scaling of the factors as defined by σ2f . For example, this term vanishes if σ2
f is normalised
to 1/µ2v (assuming µv is non-zero). Under this normalization κ = 1, and α∗ = α. We will discuss
the implications of this restriction, and potential ways to circumvent it under certain conditions, in
Section 3.3.7 Clearly, κ2 is an important determinant of cross-sectional dependence in small samples,
and while its value is irrelevant for the probability limit of α, as shown in Theorem 1, one may wish
to focus on α∗ as a parameter of interest in small samples. The following corollary provides the
asymptotic properties of α when κ2 = 1.
Corollary 1 Suppose Assumptions 1 to 3 hold, m = 1 and α > 1/2. Under the normalisation
restriction κ2 = 1,
√min(Nα, T )
(2 ln(N) (α− α)−
σ2N
N2α−1v2Ns
2f
)→d N (0, ω) .
The second source of bias is the termσ2N
N2α−1v2Ns
2f
which is unobserved. A first order accurate
estimator of this term is given byσ2N
Nσ2x
where
σ2N =
1
N
N∑i=1
σ2i , σ
2i =
1
T
T∑t=1
u2it,
uit = xit − δixt, xt = xt−xσx
, and δi denotes the OLS estimator of the regression coefficient of xit on xt.
This suggests the following bias corrected estimator
α = α−σ2N
2 ln(N)Nσ2x
. (18)
The following theorem presents the asymptotic properties of α.
Theorem 2 Let Assumptions 1-3 hold, m = 1 and α > 1/2. Then, as long as either T 1/2
N4α−2 → 0 or
α > 4/7, √min(Nα∗ , T )2 ln(N) (α− α∗)→d N (0, ω) ,
where α∗ and ω are defined in (16) and (17), respectively.
As noted above this estimator is only first order accurate since Theorem 2 only holds under the spec-
ified assumptions concerning α and the assumed relative rate of growth of N and T . Fortunately, a sec-
ond order bias correction is available. This amounts to estimatingσ2N
N2α−1v2Ns
2f
byσ2N
ln(N)Nσ2x
(1 +
σ2N
Nσ2x
).
Accordingly, we define a second bias corrected estimator
α = α−σ2N
2 ln(N)Nσ2x
(1 +
σ2N
Nσ2x
), (19)
and give its asymptotic distribution in the following theorem.
7Note that as shown in (51) the rate of convergence of α to α without this restriction is a somewhat slow ln(N).
9
Theorem 3 Suppose Assumptions 1 to 3 hold, m = 1 and α > 1/2. Then,√min(Nα∗ , T )2 ln(N) (α− α∗)→d N (0, ω) ,
where α∗ and ω are defined in (16) and (17), respectively.
We consider α, even though α provides a more comprehensive bias correction because small sample
evidence suggests that α may outperform α for values of α in the middle of the admissible range (1/2, 1].
Obviously, equivalent Corollaries to Corollary 1 hold for α and α. The final result of this subsection
relates to providing a consistent estimator for ω. Let vi denote the OLS estimator of the regression
coefficient of xit on xt. Let{δ
(s)i
}Ni=1
and{v
(s)i
}Ni=1
denote the sequences of δi and vi, sorted according
to their absolute values in a descending order, and define the following estimators of σ2v/µ
2v
(σ2v
µ2v
)=
N2α−2N α∑i=1
(v
(s)i −
1N α
N α∑i=1v
(s)i
)2
N α − 1, for any finite value of κ, (20)
and
(σ2v
µ2v
)=
N α∑i=1
(δ
(s)i −
1N α
N α∑i=1δ
(s)i
)2
N α − 1, when κ = 1. (21)
Theorem 4 Suppose Assumptions 1 to 3 hold and m = 1. Let
ω =min(N α, T )
TVf2 +
4 min(N α, T )
N α
(σ2v
µ2v
),
where α = α, α, α, and(σ2vµ2v
)is given by (20) or (21) depending on the value of κ,
Vf2 =
1
T
T∑t=1
(st −
1
T
T∑t=1
st
)2
+l∑
j=1
1
T
T∑t=j+1
(st−j −
1
T
T∑t=1
st
)(st −
1
T
T∑t=1
st
) , (22)
st = (xt − x)2, x = T−1∑T
t=1 xt, and l→∞. Then,
ω − ω = op(1).
where ω is defined in (17), so long as l→∞, l = o (T ) and l = o(Nα−1/2T 1/2
).
3.2 Extensions
Consider now the following multiple factor extension of our basic setup:
xit = β′if t + uit =
m∑j=1
βijfjt + uit, i = 1, 2, ..., N,
where f t = (f1t, f2t, ..., fmt)′ is an m×1 vector of factors and βi is the associated vector of factor load-
ings (m is fixed). We make the following assumptions that generalise straightforwardly Assumptions
1 and 2.
10
Assumption 4 We assume that loadings are given by
βij = vij for i = 1, 2, ..., [Nαj ] , (23)
βij = cjρi−[Nαj ]+1
j , for i = [Nαj ] + 1, [Nαj ] + 2, ..., N ,
where 1/2 < αj ≤ 1, |ρj | < 1, and {vij}Nαj
i=1 is an IID sequence of random variables with mean µvj 6= 0,
and variance σ2vj <∞, for all j = 1, 2, ...,m.
Assumption 5 The m× 1 vector of factors, f t, follows a linear stationary process given by
f t =
∞∑j=0
ψfjνf,t−j, (24)
where νft is a sequence of IID random variables with mean zero and a finite variance matrix, Σνf ,
and uniformly finite ϕ-th moments for some ϕ > 4. The matrix coefficients, ψfj, satisfy the absolute
summability condition∞∑j=0
jζ∥∥ψf,j∥∥ <∞,
such that {ζ(ϕ−2)}/{2(ϕ−1)} ≥ 1/2. f t is distributed independently of the idiosyncratic errors, uit′ ,
for all i, t and t′.
Without loss of generality we assume that α = α1 ≥ αj , j = 2, ...,m and that factors are orthogonal.
We have
βjN = N−1N∑i=1
βji =Nj
N
(∑Nji=1 vjiNj
)+cjρjN
1− ρ(N−Nj)j
1− ρj
= Nαj−1vj +N−1Kρj (25)
and
V ar(βjN ) = N−2τ ′Σβjτ = O(Naj−2).
Note that
xt − E(xt) = β1Nf1t + β2Nf2t + ...βmNfmt + ut,
and
V ar(xt) = E(β1Nf1t + β2Nf2t + ...βmNfmt + ut
)2=
m∑j=1
E(β21N )σ2
jf + E(u2t ) =
m∑j=1
[E(β1N )
]2σ2jf +
m∑j=1
V ar(β1N )σ2jf + E(u2
t ).
Let βN =(β1N , ..., βmN
)′and vN = (v1N , ..., vmN )′. Then,
βN = Nα−1DN vN +N−1Kρ. (26)
where DN is anm×m diagonal matrix with diagonal elements given byNαj−α1 andKρ = (Kρ1 , ...,Kρm)′.
It is important to stress that (23) can be generalised. In particular, the results below hold so long as∑Ni=Nj+1 βji <∞, in which case, Kρj = Kj =
∑Ni=Nj+1 βji. Further, let
dT = v′NS−1/2ff fT − µ′vΣ
−1/2ff µf (27)
where Sff = Sff [sijf ] = 1T
∑Tt=1
(f t − fT
) (f t − fT
)′, Σff = Σff [σij,f ] = E
[(f t − µf
) (f t − µf
)′],
σif = σii,f , µf = E (f t), fT = 1T
∑Tt=1 f t, and µv = (µ1v, ..., µmv)
′ = E (vi). Then we have the fol-
lowing theorem which distinguishes between a number of different cases depending on the relative
values of the different exponents, αj , for j = 1, 2, ..,m.
11
Theorem 5 Suppose Assumptions 4 to 5 hold, α = α1 = α2 = ... = αm, and α > 1/2. Then,
√min(Nα∗ , T )
(2 ln(N) (α− α∗)−
σ2N
N2α−1v′NDNSffDN vN
)→d N (0, ωm) (28)
where
ωm = limN,T→∞
min (Nα, T )V ar(d2T ),
dT is defined by (27),
α∗ ≡ α∗N = α+ln(κ2)
2 ln (N),
and κ2 =∑m
i=1 µ2ivσ
2if .
Continue to assuming that α = α1 = α2 = ... = αm, and suppose that either α > 4/7 or T 1/2
N4α−2 → 0,
then √min(Nα∗ , T )2 ln(N) (α− α∗)→d N (0, ωm) , (29)
and √min(Nα∗ , T )2 ln(N) (α− α∗)→d N (0, ωm) . (30)
Further, if either
α1 = α > α2 + 1/4, (31)
or if
α2 < 3α/4, T b = N, b >1
4(α− α2), (32)
and α2 ≥ α3 ≥ ... ≥ αm, (28), (29) and (30) hold with ω replacing ωm, where ω is defined in (17) and
α∗ is now defined by (16).
Finally, if α > α2 ≥ α3... ≥ αm but neither (29) nor (30) hold, then (28), (29) and (30) hold with
ω replacing ωm, and
α∗ ≡ α∗N = α+ln(∑m
i=1N2(α1−αi)µ2
ivσ2if
)2 ln (N)
.
Up to now we have analysed estimators of the exponent of cross-sectional dependence assuming that
factor loadings take the form given in Assumption 1. We briefly examine an alternative formulation
for the loadings. As we noted in the Introduction this alternative specification is mathematically
convenient, yet economically restrictive. More specifically consider the following formulation of the
factor loadings:
Assumption 6 Suppose that the factor loadings are given by
βik = Nα−1vik, 1/2 < α ≤ 1 (33)
where {vik}Ni=1 is an i.i.d. sequence of random variables with mean µvk 6= 0, and variance σ2vk<∞.
It is easy to see that even in the case of this alternative specification of the loadings we have
τN =1
N2
N∑i=1
N∑j=1,i 6=j
σij,x = O(N2α),
and the following Corollary follows easily from the proofs of Theorems 1-3, and Lemma 6.
12
Corollary 2 Let Assumptions 6 and 2-3 hold, m = 1 and α > 1/2. Let α, α and α be defined as in
(14), (18) and (19) respectively. Then,
√min(N,T )
(2 ln(N) (α− α∗)−
σ2N
N2α−1v2Ns
2f
)→d N (0, ω) ,
where α∗ and ω are defined in (16) and (17), respectively. If µ2vσ
2f = 1,then,
√min(N,T )
(2 ln(N) (α− α)−
σ2N
N2α−1v2Ns
2f
)→d N (0, ω) .
As long as either α > 5/8 or T 1/2
N4α−2 → 0,√min(N,T )2 ln(N) (α− α∗)→d N (0, ω)
and √min(N,T )2 ln(N) (α− α∗)→d N (0, ω) .
Remark 1 It is of interest to consider circumstances where Assumption 6 fails but the above result
still holds. In particular, let
βi = Nα−1vi, 1/2 < α ≤ 1 (34)
where vi = vNi = vi + cNi and {vi}Ni=1 is an i.i.d. sequence of random variables with mean µv 6= 0,
and variance σ2v < ∞. Lemma 14 provides general conditions for this Assumption, under which, our
theoretical results hold. In this remark we explore a leading case of departure from Assumption 6 that
is covered by Lemma 14. Without loss of generality, we order units, such that cNi = N1−αηi for
i = 1, 2, ...,M where {ηi}Ni=1 is an i.i.d. sequence of random variables with mean µη 6= 0, and variance
σ2η < ∞. This implies that M units have loadings that are bounded away from zero. Then, using
Lemma 14, it is easy to see that the theorems relating to the asymptotic distribution of the estimators
continue to hold as long as M = o(Nα−1/2
).
3.3 Is the normalization restriction κ2 = 1 justified?
Since the cross-sectional exponent, α, is identified in terms of the factor loadings, and factor loadings
are in turn identified only up to a non-singular rotation matrix, it is clear that some restriction on
κ2 = µ2vσ
2f is needed before a sufficiently accurate estimator of α can be obtained. Although, as we
shall see below, it might be possible to jointly estimate α and κ under our preferred formulation of
the factor loadings, this is not the case in general. For example, joint estimation of κ and α does not
seem possible if the factor loadings are generated according to (33). It is, therefore, interesting to
investigate the extent to which the imposition of the restriction κ2 = 1 restrict the data generating
process of xit. A useful way to do this is to consider the extent to which setting κ2 = 1 imposes
restrictions on the second moment structure of xit. The restriction κ2 = 1 has no implications for the
second moments of the individual series, xi = (xi1, xi2, ..., xiT )′, and can only affect the cross-sectional
average of the second moments. We have investigated these effects in a not-for-publication appendix
that is available upon request, and conclude that they are reasonably modest.
13
4 Joint estimation of α and κ
In this section we show that it is in fact possible to jointly estimate α and κ, if the factor loadings
follow the set up given under Assumption 1. To see this, without loss of generality, suppose that βi are
ordered such that |β1| ≥ |β2| ≥ ....∣∣β[Nα]
∣∣, with βi = 0 for i = [Nα] + 1, ..., N . Let βn = (β1, β2, ...βn)′,
βn = 1n
∑ni=1 βi, and note that
E(βn) = n−1τ ′nE(βn) = µv, if n ≤ [Nα]
= n−1 [Nα]µv, if n > [Nα] .
Similarly, let Σβ,n = Cov(βn) for n ≤ [Nα] and bear in mind that
Σβ,N =
(Σβ,[Nα]×[Nα] 0
0 0(N−[Nα])×(N−[Nα])
)and note that
V ar(βn) = n−2τ ′nCov(βn)τ ′n ≤ n−1λmax(Σβ,n) = O(n−1), if n ≤ [Nα]
= n−2τ ′nCov(βn)τ ′n ≤[Nα]
n2λmax(Σβ,[Nα]×[Nα]) = O
([Nα]
n2
), if n > [Nα] .
Now let xnt be the cross-sectional average of xit over i = 1, 2, ..., n < N , where without loss of generality
we assume that xit is ordered by |βi|. Then, using the above results
V ar(xnt) = σ2fµ
2v+n
−1cn +O(n−1), if n ≤ [Nα] ,
= n−2[N2α
]σ2fµ
2v + n−1cn +O
([Nα]
n2
), if n > [Nα] .
where cn = n−1τ ′nΣu,nτn, E(unu′n), with un = (u1, u2, ..., un)′. This result can be written more
compactly as
V ar(xnt)− n−1cn = I([Nα]− n)κ2 + I([Nα]− n)O(n−1)
+ [1− I([Nα]− n)]
[n−2
[N2α
]κ2 +O
([Nα]
n2
)],
where I(A) is an indicator function that takes the value of 1 if A > 0 and zero otherwise. After some
simplifications we have
V ar(xnt)− n−1cn ={n−2
[N2α
]+ I([Nα]− n)
[1− n−2
[N2α
]]}κ2+
I([Nα]− n)
[O(n−1)−O
([Nα]
n2
)]+O
([Nα]
n2
)It is easily seen that for n = N , we obtain our earlier result (note that Nα ≤ N since α ≤ 1), namely
V ar(xNt)−N−1cN = κ2[N2α−2
]+O
([Nα−2
]).
But other values of n that are sufficiently large can also be used to provide information on κ and α.
First we need a consistent estimator of V ar(xnt) − n−1cn, when n is sufficiently large. To this end,
consider the OLS estimator of the slope coefficient in the regression of xit − xi on xNt − x, given by
vi =∑T
t=1 (xit − xi) (xNt − x) /∑T
t=1 (xNt − x)2, where x = N−1N∑i=1xi. Order the observations, xit,
as x(i)t, so that x(1)t is associated with v(1), x(2)t is associated with v(2), and so on, where v(1), v(2), ...
14
are the values of v1, v2, ..., vN ordered in a descending manner. Consider the following estimator of
V ar(xnt),
σ2xn =
∑Tt=1(xnt − xnT )2
T,
where xnt = n−1∑n
i=1 x(i)t, and xnT = T−1∑T
t=1 xnt. Similarly, estimate cn by cn = n−1∑n
j=1 σ2(j),
where σ2(j) is the estimator of σ2
(j), the standard error of the idiosyncratic component of x(j)t for
t = 1, 2, ..., T . Thus
cn =1
nT
n∑j=1
T∑t=1
u2(j)t,
where u(j)t = x(j)t − x(j) − v(s)j (xNt − xNT ).
Based on σ2xn , we obtain the following estimating equation
σ2xn − n
−1cn ={n−2
[N2α
]+ I([Nα]− n)
[1− n−2
[N2α
]]}κ2+
I([Nα]− n)
[O(n−1)−O
([Nα]
n2
)]+O
([Nα]
n2
)+ ξn,
where ξn is the estimation error. It is clear that only the estimates that are based on n > [Nα] are
informative for α. This is because for values of n < [Nα] we have σ2xn − n
−1cn = κ2 +O(n−1) + ξn.
The above results suggest that joint estimation of α and κ can be based on the minimization of
the following quadratic form in terms of α and κ:
Q(α, κ2) =∑
n≤[Nα]
(σ2xn − n
−1cn − κ2)2
+∑
n>[Nα]
(σ2xn − n
−1cn − n−2[N2α
]κ2)2, (35)
The first order condition for κ2 is∑n≤[Nα]
(σ2xn − n
−1cn − κ2(α))
+[N2α
] ∑n>[Nα]
n−2(σ2xn − n
−1cn − n−2[N2α
]κ2(α)
)= 0,
which yields
κ2(α) =
∑n≤[Nα]
(σ2xn − n
−1cn)
+[N2α
]∑Nn>[Nα] n
−2(σ2xn − n
−1cn)
[Nα] + [N4α]∑N
n>[Nα] n−4
.
Using this result we have
Q(α) = Q(α, κ2(α)) =∑
n≤[Nα]
(σ2xn − n
−1cn − κ2(α))2
+∑
n>[Nα]
(σ2xn − n
−1cn − n−2[N2α
]κ2(α)
)2.
The above expressions can be simplified. Let
Q1 =
[Nα]∑n=1
(σ2xn − n
−1cn)2, q1 =
[Nα]∑n=1
(σ2xn − n
−1cn)
Q2 =
N∑n=[Nα]+1
(σ2xn − n
−1cn)2, q2 =
N∑n=[Nα]+1
n−2(σ2xn − n
−1cn)
N(α) = [Nα] +[N4α
] N∑n=[Nα]+1
n−4
15
which yields
κ2(α) =q1 +
[N2α
]q2
N(α). (36)
Then,
Q(α) = Q1 +Q2 + κ4(α) [Nα] +[N4α
]κ4(α)
N∑n=[Nα]+1
n−4 − 2κ2(α)q1 − 2κ2(α)[N2α
]q2
= Q1 +Q2 + κ4(α)N(α)− 2κ2(α)[q1 +
[N2α
]q2
]= Q1 +Q2 +
[q1 +
[N2α
]q2
N(α)
]2
N(α)− 2[q1 +
[N2α
]q2
] [q1 +[N2α
]q2
N(α)
]
= Q1 +Q2 −[q1 +
[N2α
]q2
]N(α)
2
.
Further, note that Q1 +Q2 =∑N
n=1
(σ2xn − n
−1cn)2
which does not depend on α. Then,[q1+[N2α]q2]
N(α)
2
can be maximised straightforwardly by grid search, which gives an estimator of α (denoted by α).
κ2 can then be estimated using (36). The consistency of α is established in Appendix IV, although
we have not been able to establish the rate at which this estimator is consistent. But judging by the
results obtained in the structural break literature we conjecture that the rate at which α converges to
α is ln(N), and to obtain a better convergence rate some a priori restriction on κ might be needed.
5 Monte Carlo Study
We investigate the small sample properties of the proposed estimators of α, through a detailed Monte
Carlo simulation study. We consider the following two factor model
xit = β1if1t + β2if2t + uit, (37)
for i = 1, 2, ..., N , and t = 1, 2, ..., T. The factors are generated as
fjt = ρjfj,t−1 +√
1− ρ2jζjt, j = 1, 2, for t = −49,−48, ..., 0, 1, ..., T,
with fj,−50 = 0, for j = 1, 2. The shocks are generated as
uit = φiui,t−1 +√
1− φ2i εit, for i = 1, 2, ..., N and t = −49,−48, ..., 0, 1, ..., T, with ui,−50 = 0,
ζjt ∼ IIDN(0, 1), εit ∼ IIDN(0, σ2i ), where σ2
i ∼ IID χ2, i = 1, 2, ..., N.
Therefore, by construction σ2fj
= 1, for j = 1, 2. In the first instance we set the loadings of the second
factor equal to zero, β2i = 0, and focus on the properties of β1i. We consider the following generating
mechanism:
β1i = v1i, for i = 1, 2, ...,M (N)
β1i = ρi−Ml , for i = M + 1,M + 2, ..., N
where v1i ∼ IIDU(µv1 − 0.5, µv1 + 0.5), M = [Na] and ρl = 0.5. The above parametrization ensures
that µ2v1σ2f1
= 1, as discussed in the development of the theory. Initially, where m = 1, we consider
the following experiments.
16
Experiment A A basic design, where the factor, f1t, is serially uncorrelated and the errors, uit,
are serially uncorrelated and cross-sectionally independent, namely when
ρ1 = 0, φi = 0, i = 1, 2, ..., N
and εit ∼ IIDN(0, σ2i ), for all i and t.
Experiment B This design is as in Experiment A, but allows for temporal dependence in the
factor, so that
ρ1 = 0.5, φi = 0, i = 1, 2, ..., N
and εit ∼ IIDN(0, σ2i ), for all i and t.
Experiment C This design is as in Experiment A, where we continue to set ρ1 = 0, but al-
low the idiosyncratic errors, uit, to be cross-sectionally dependent according to a first order spatial
autoregressive model. Let ut = (u1t, u2t, ..., uNt)′, and set ut as
ut = Qεt, εt = σεηt; ηt ∼ IIDN(0, IN ),
where Q = (IN − θS)−1, and
S =
0 1 0 . . . 0
1/2 0 1/2 . . . 0...
. . .. . .
. . .
0 0 . . . 0 1/20 0 . . . 1 0
,
We set θ = 0.2, and set σ2ε = N/Tr(QQ′) which ensures that N−1
∑Ni=1 var(uit) = 1.
Experiment D Next, we take into account the second factor as well and generate its loadings
as
β2i = v2i, for i = 1, 2, ...,M2 (N)
β2i = ρi−M2l , for i = M2 + 1,M2 + 1, ..., N,
where v2i ∼ IIDU(µv2 − 0.5, µv2 + 0.5), M2 = Na2 and ρl = 0.5. We examine the case where α2 = a
and set σf1 = σf2 = 1 and µv1 = µv2 =√
0.5. The rest of the parameters are se as in Experiment A,
namely
α2 = a, ρj = 0, φi = 0, for i = 1, 2, ..., N, j = 1, 2
and uit ∼ IIDN(0, 1), for all i and t.
For all experiments we consider values of α = 0.70, 0.75, ..., 0.90, 0.95, 1.00, N = 100, 200, 500, 1000
and T = 100, 200, 500. All experiments are based on 2000 replications. For each replication, the values
of α, ρj , ρl, φi and S are given as set out above. These parameters are fixed across all replications.
The values of vji, j = 1, 2 are drawn randomly (N of them) for each replication.
In the case where the leading factor (f1t) is serially uncorrelated, the statistic for making inference
about α (when κ = 1) is given by (see theorems 1, 3 and 4 )(1
TVf21
+4
N α
σ2v
µ2v
)−1/2
2 ln(N) (α− α)→d N(0, 1),
17
for α = α or α. Note that when the leading factor is serially uncorrelated then Vf21
= E(f41t)/σ
4f1− 1,
where E(f41t)/σ
4f is consistently estimated by
E(f41t)/σ
4f1
=
∑Tt=1 (xt − x)4
T,
where xt =(N−1
∑Ni=1 xit
)/σx. Also σ2
v/µ2v , the estimator of σ2
v/µ2v, is given by
σ2v
µ2v
=
N α∑i=1
(δ
(s)i −
1N α
N α∑i=1δ
(s)i
)2
N α − 1, with α = α or α,
where{δ
(s)i
}denotes the sequence of δi sorted according to their absolute values in a descending
order, and δi is the OLS estimator of the regression coefficient of xit on xt = (xt − x)/σx. Also see
Theorem 4 and the discussions that proceed it. The above expressions apply irrespective of whether
the model contains one or two factors.
Size of the tests is computed under H0 : α = α0, using a two-sided test where α0 takes values in the
range [0.70, 1.00], as indicated above. Power is computed under the alternatives Ha : αa = α0 + 0.05
(power+), and Ha : αa = α0 − 0.05 (power-). All results are scaled by 100.
Finally, when ρj 6= 0, the case of serially correlated factors, we use a corrected variance estimator
of ft. The relevant formula for the test statistic is given by[1
T
[Vf2(q)
]+
4
N α
σ2v
µ2v
]−1/2
2 ln(N) (α− α)→d N(0, 1), (38)
for α = α or α. Vf2(q) is computed by first estimating an AR(q) process for zt = zt − z, where
zt = (xt − x)2 , xt =(
1N
∑Ni=1 xit
)/σx, x = T−1
∑Tt=1 xt and z = T−1
∑Tt=1 zt ,and then V
f2(q) =
σ2z/(1− γ1 − γ2 − ...− γq)2, where σz is the regression standard error and γi is the ith estimated AR
coefficient fitted to zt. The lag order is set to q = T 1/3, and σ2v/µ
2v is computed as before. Note that
this correction is not the standard Newey-West one but based on AR approximations. We found that
this correction has better finite samples properties and hence we use this in both the Monte Carlo
study and the empirical applications of Section 6.
5.1 Results
The results for experiment A are summarized in Table A1 giving the bias, Root Mean Square Error
(RMSE), size and power when α is used as the estimator of α. Table A2 presents the same set
of summary statistics for experiment A when the further bias-corrected estimator, α, is considered.
We only report results for values of α over the range [0.70, 1.0]. Recall that α is identified only if
α > 1/2, and for asymptotically valid inference on α it is further required that α > 4/7, unless
T 1/2/N (4α−2) → 0, as N and T →∞. (See Theorem 2).
Both estimators perform reasonably well with α having a slightly smaller bias for values of α in
the range [0.70− 0.85], but overall there is little to choose between the two estimators, and therefore
in what follows we focus on α as the estimator of α. As predicted by the theory, the bias and RMSE
of α decline with both N and T , and tend to be smaller for larger values of α. A similar pattern can
also be seen in size and power of the tests based on α. There is evidence of some size distortion when
α is below 0.75, but it tends towards the nominal 5% level as α is increased. The size distortion is also
18
reduced as N and T are increased. The power of the test also rises in α, N and T , and approaches
unity quite rapidly. However, the power function seems to be asymmetric with the power tending to be
higher for alternatives below the null (denoted by Power-) as compared to the alternatives above the
null (denoted by Power+). This asymmetry is particularly marked for low values of α and disappears
as α is increased.
The results for Experiment B where the factor is allowed to be serially correlated are summarized
in Table B. As compared to the baseline case, we see some deterioration in the results, particularly
for relatively small values of N and T . The RMSEs are slightly higher, the size distortions slightly
larger, and the power slightly lower. But these differences tend to vanish as N and T are increased.
The effects of allowing for weak cross-sectional dependence in the idiosyncratic errors, uit, on
estimation of α are summarized in Table C for Experiment C. Considering the moderate nature of the
spatial dependence introduced into the errors (with the spatial parameter, θ, set to 0.2), the results
are not that different from the ones reported in Table A2, for the baseline experiments. However, one
would expect greater distortions as θ is increased, although the effects of introducing weak dependence
in the idiosyncratic errors are likely to be less pronounced if higher values of α are considered. For
values of α near the borderline value of 1/2, it will become particularly difficult to distinguish between
factor and spatial dependent structures.
Finally, the results of Experiment D where one additional factor is included in the baseline case
are summarized in Table D. As can be seen, the results are hardly affected by the addition of the new
factor to the data generating process. Consistent with the one-factor case of Experiment A, both the
bias and RMSE of α fall gradually as N and α are increased, while tests of the null hypothesis based
on α, are correctly sized for α > 0.7 in this case as well. Similar observations can also be made with
respect to the power.
The above Monte Carlo experiments, although limited in scope, clearly illustrate the potential
utility of the estimation and inferential procedure proposed in the paper for the analysis of cross-
sectional dependence. The results are broadly in agreement with the theory and are reasonably robust
to departures from the basic model. Although, the results tend to deteriorate somewhat when we
consider serially correlated factors or weak cross-sectional dependence in the idiosyncratic errors, the
estimated values of α tend to retain a high degree of accuracy even for moderate sample sizes. It is
also worth bearing in mind that in most empirical applications the interest will be on estimates of α
that are close to unity, as it is for these values that a factor structure makes sense as compared to
spatial or other network models of cross-sectional dependence. It is, therefore, helpful that the quality
of the small sample results tend to improve when values of α close to unity are considered.
6 Empirical Applications
In this section we provide estimates of the exponent of cross-sectional dependence, α, for a number
of panel data sets used extensively in economics and finance. Specifically, we consider three types
of data sets: quarterly cross-country data used in global modelling, large quarterly data sets used in
empirical factor model literature, and monthly stock returns on the constituents of Standard and Poor
500 index.
6.1 Cross-country dependence of macro-variables
Table 1 presents estimates of α for real output growth, inflation, rate of change of real equity prices,
and interest rates (short and long) computed over 33 countries (when available). The data is from
Cesa-Bianchi, Pesaran, Rebucci, and Xu (2012), and covers the period 1979Q2-2009Q4, which updates
19
the earlier GVAR (global vector autoregressive) data sets used in Pesaran, Schuermann, and Weiner
(2004), and Dees, di Mauro, Pesaran, and Smith (2007).8 We provide both bias corrected estimates,
α and α, computed using available cross-country time series, yit, over the full sample period. The
observations were standardized as xit = (yit− yi)/si, where yi is the sample mean of each time series,
and si is the corresponding standard deviations. Table 1 also reports the 95% confidence bands, the
cross section dimension (N) and the time series dimension (T ) for each of the variables. Although,
there are 33 countries in the GVAR data set, not all variables are available for all the 33 countries.
For example, the short term (3 months) interest rate data is not available for Saudi Arabia, and real
equity prices and long term interest rate data (10 year government bond) are available only for some
of the countries.
We first note that the two different estimates of α provided in Table 1 are very close, which are
in line with the Monte Carlo results reported in Tables A1 and A2. Focusing on α, we observe that
the point estimates range between 0.754 for cross dependence of GDP growth rates, to 0.968 for long
term interest rates. The exponent of cross-sectional dependence for short term interest rates and real
equity prices at 0.907 and 0.881 are also quite high, indicating that financial variables are more strongly
correlated as compared to the real variables. The reported confidence bands all lie above 0.5, but none
cover unity either apart from the case of long term interest rates (marginally), suggesting that whilst
a factor structure might be a good approximation for modelling global dependencies, the value of
α = 1 typically assumed in the empirical factor literature might be exaggerating the importance of the
common factors for modelling cross-sectional dependence at the expense of other forms of dependencies
that originate from trade or financial inter-linkages that are more local or regional rather than global
in nature.
Table 1: Exponent of cross-country dependence of macro-variablesN T α∗0.025 α α∗0.975 α∗0.025 α α∗0.975
Real GDP growth, q/q 33 122 0.691 0.754 0.818 0.689 0.752 0.816Inflation, q/q 33 123 0.778 0.851 0.924 0.777 0.850 0.924
Real equity price change, q/q 26 122 0.797 0.881 0.966 0.796 0.881 0.966Short-term interest rates 32 123 0.831 0.907 0.983 0.831 0.907 0.983Long-term interest rates 18 123 0.864 0.968 1.072 0.864 0.968 1.072
*95% level confidence bands
6.2 Within-country dependence of macro-variables
An important strand in the empirical factor literature, promoted through the work of Forni, Hallin,
Lippi, and Reichlin (2000), Forni and Lippi (2001) and Stock and Watson (2002), uses factor models to
forecast a few key macro variables such as output growth, inflation or unemployment rate with a large
number of macro-variables, that could exceed the number of available time periods. It is typically
assumed that the macro variables satisfy a strong factor model with α = 1. We estimated α using the
quarterly data sets used in Eklund, Kapetanios, and Price (2010). For the US the data set comprises
95 variables over the period 1960Q2 to 2008Q3. For the UK the data set covers 94 variables spanning
from 1977Q1 to 2008Q2. The estimates of α, computed from the standardized data sets as explained
in the previous subsection, together with their 95% confidence bands are summarized in Table 2.
For both data sets the estimates, α and α, are identical up to two decimal places. For the US data
set the point estimate (α) of α, at 0.739, is slightly larger than the estimate obtained for the UK at
8This version of GVAR data set can be downloaded fromhttp://www-cfap.jbs.cam.ac.uk/research/gvartoolbox/download.html
20
Table 2: Exponent of within-country dependence of macro-variablesUS UK
1960Q2-2008Q3 1977Q1-2008Q2N=95, T=194 N=94, T=126
α∗0.025 α α∗0.975 α∗0.025 α α∗0.975
0.689 0.739 0.788 0.636 0.715 0.793α∗0.025 α α∗0.975 α∗0.025 α α∗0.975
0.689 0.738 0.788 0.635 0.713 0.792*95% level confidence bands
0.715. The 95% confidence bands for both data sets are well above the threshold value of 0.50, but are
well short of 1.0 at the upper end of the band. Once again there is some evidence of a common factor
dependence, but the evidence is not as strong as it is assumed in the literature.
6.3 Cross-sectional exponent of stock returns
One of the important considerations in the analysis of financial markets is the extent to which asset
returns are interconnected. This is encapsulated in the capital asset pricing model (CAPM) of Sharpe
(1964) and Lintner (1965), and the arbitrage pricing theory (APT) of Ross (1976). Both theories have
factor representations with at least one strong common factor and an idiosyncratic component that
could be weakly correlated (see, for example, Chamberlain (1983)). The strength of the factors in
these asset pricing models is measured by the exponent of the cross-sectional dependence, α. When
α = 1, as it is typically assumed in the literature, all individual stock returns are significantly affected
by the factor(s), but there is no reason to believe that this will be the case for all assets and at all
times. The disconnect between some asset returns and the market factor(s) could occur particularly at
times of stock market booms and busts where some asset returns could be driven by non-fundamentals.
Therefore, it would be of interest to investigate possible time variations in the exponent α for stock
returns. Note that under our methodology the market factor associated with the CAPM specification
is implied by the data rather than imposed by use of a specific market portfolio composition which
can be limiting, as explained in Roll (1977).
We base our empirical analysis on monthly excess returns of the securities included in the Standard
& Poor 500 (S&P 500) index of large cap U.S. equities market, and estimate α recursively using rolling
samples of size 120 months (10 years) and 60 months ( 5 years). Due to the way the composition
of S&P 500 changes over time, we compiled returns on all 500 securities at the end of each month
over the period from September 1989 to September 2011, and included in the rolling samples only
those securities that had a sufficiently long history in the month under consideration. On average we
ended up with 439 securities at the end of each month for the rolling samples of size 10 years, and
476 securities when we used a rolling sample of size 5 years. The one-month US treasury bill rate
was chosen as the risk free rate (rft), and excess returns computed as rit = rit − rft, where rit is the
monthly return on the ith security in the sample inclusive of dividend payments (if any).9 Recursive
estimates of α were then computed using the standardized observations xit = (rit − ri)/si, where riis the sample mean of the excess returns over the selected rolling sample, and si is the corresponding
standard deviations.
The recursive estimates of α based on 10 years and 5 years rolling windows are given in Figure 1.
We also computed rolling standard errors for the estimates, αt, using the serial correlation correction
discussed in Section 5. Based on these standard errors, the 95% confidence bands of the recursive
9For further details of data sources and definitions see Pesaran and Yamagata (2012).
21
estimates were on average ±0.02 around the point estimates for both rolling sample sizes considered.
These bands are not shown in Figure 1, since the bands are relatively narrow and we aim to highlight
the time variations in the estimates of α.
Figure 1: αt associated with S&P 500 securities’ excess returns - 5-yr and 10-yr rolling samples
0.83
0.84
0.85
0.86
0.87
0.88
0.89
0.9
0.91
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0.94
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89
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0
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93
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-93
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94
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99
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-01
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-02
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04
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-06
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-07
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08
Nov
-08
Sep-
09
Jul-1
0
May
-11
5-year estimates of α (ᾶ) 10-year estimates of α (ᾶ)
The figure covers 23 years of monthly recursive estimates of α, and yet fall in a relatively narrow
range of 0.854 − 0.917 in the case of the 10 years rolling samples, and in a slightly wider range of
0.838 − 0.932 in the case of 5 years rolling samples. These estimates clearly show a high degree of
inter-linkages across individual securities, although the null hypothesis that α = 1 is clearly rejected.
More importantly, there are clear trends in the estimates of α. The estimates based on the 10 years
rolling samples fall from a high of 0.92 in 1990 to a low of 0.85 just before the burst of the dot-com
bubble in 1999-2000. The estimates of α then stabilize around 0.86 over the period 2000− 2008, then
fall quite dramatically towards the end of 2008 at the time of the market crash, before starting to
rise again to its present level of 0.90 in September 2011. The factors behind these fluctuations are
complex and reflect the relative importance of micro and macro fundamentals prevailing in financial
markets. A standard factor model does not seem able to fully account for the changing nature of the
dependencies in securities market over the 1989-2011 period. A similar result is also obtained when
α is estimated using 5-year rolling samples, although as to be expected the estimates are variable.
Indeed, the upward movements in the estimates based on 5-year windows are more pronounced both
in the latest crisis as well as in the period of 1997-2000 that saw smaller crises caused by the Asian
economic turmoil, LTCM and the bursting of the dot-com bubble.
The patterns observed in the above estimates of α are in line with changes in the degree of
correlations in equity markets. It is generally believed that correlations of returns in equity markets
rise at times of financial crises, and it would be of interest to see how our estimates of α relate to return
correlations. To this end in Figure 2 we compare the estimates of α to average pair-wise correlation
coefficients of excess returns (ρN ) on securities included in S&P 500 index, using 10-year and 5-year
rolling windows.10 As the plots in these figures show, our estimates of α closely follow the rolling
10Denote the correlation of excess returns on i and j securities by ρij , the pair-wise average correlation of the marketis then computed as ρN = (1/N(N − 1))
∑N−1i=1
∑Nj=i+1 ρij , where N is the number of securities under consideration.
22
estimates of ρN .
Figure 2: Average pair-wise correlations of excess returns for securities in the S&P 500 index and theassociated αt estimate computed using 10-year and 5-year rolling samples
0.15
0.20
0.25
0.30
0.35
0.40
0.85
0.87
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-89
Au
g-9
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91
Ju
n-9
2
May-9
3
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r-9
4
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95
Feb
-96
Jan-9
7
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v-9
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-00
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g-0
1
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02
Ju
n-0
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r-0
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06
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-07
Jan-0
8
Dec-0
8
No
v-0
9
Oct-
10
Sep
-11
10-year estimates of α (ᾶ) based on excess returns (lhs)
10-year average pairwise correlations of excess returns (rhs)
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.83
0.85
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-89
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l-9
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v-9
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Ju
l-9
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v-9
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-99
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v-0
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-04
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-09
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-11
5-year estimates of α (ᾶ) based on excess returns (lhs)
5-year average pairwise correlations of excess returns (rhs)
Further, it would be of interest to see how our estimates of α compare with estimates obtained
using excess returns on market portfolio as a measure of the unobserved factor. This approach starts
with capital asset pricing model (CAPM) and assumes that the single factor in CAPM regressions can
be approximated by a stock market index. Under these assumptions, as noted in the Introduction,
a direct estimate of α is given by αd = ln(M)/ ln(N), where M denotes the estimated number of
non-zero betas, and N is the total number of securities under consideration.11 M can be consistently
estimated (as N and T →∞) by the number of t-tests of βi = 0 in the CAPM regressions
rit − rft = ai + βi (rmt − rft) + uit, for i = 1, 2, ..., N, (39)
that end up in rejection of the null hypothesis at a chosen significance level, where rmt is a broadly
defined stock market index. In our application we choose the value-weighted return on all NYSE,
AMEX, and NASDAQ stocks to measure rmt,12 and select 1% as the significance level of the tests.
Such estimates of α obtained recursively using 10-year and 5-year rolling windows are shown in the
two plots in Figure 3. For ease of comparison, these plots also include our (indirect) estimates of α
based on the same data sets (except for the marker return, rmt, which is not used). The two sets of
estimates co-move over most of the period and tend to become closer in the aftermath of the dot-com
bubble and during the recent financial crisis. The correlation coefficient of the two sets of estimates is
0.923 for the ones based on 10-year rolling samples, and 0.739 for the ones based on the 5-year rolling
samples. The two sets of estimates, however, differ in scale, with the direct estimates being closer to
unity. The scale of the direct estimates clearly depends on the measure of market return, the level of
significance chosen, and the assumption that the model contains only one single factor with α > 1/2,
Almost identical estimates are also obtained if we use returns instead of excess returns.11Recall that M = [Nα], where M is the true number of non-zero betas.12The return data on market index was obtained from Ken French’s data library.
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html
23
and in consequence is subject to a high degree of uncertainty.13 Nevertheless, it is reassuring that the
direct and indirect estimates of α in this application tend to move together closely.
Figure 3: Direct (αd) and indirect (α) estimates of cross-sectional exponent of the market factor (usingexcess returns on S&P 500 securities) based on 10-year and 5-year rolling samples
0.83
0.85
0.87
0.89
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Sep-8
9Ju
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Dec-0
9S
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1
10-year indirect α estimates (ᾶ ) 10-year direct α estimates (1% significance level)
0.83
0.85
0.87
0.89
0.91
0.93
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-89
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-96
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-97
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-07
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-08
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-09
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-11
5-year indirect α estimates (ᾶ) 5-year direct α estimates (1% significance level)
There is also a further consideration when comparing the estimates of α and αd. Under CAPM the
errors, uit, in (39) are assumed to be cross-sectionally weakly correlated, namely that the cross-sectional
exponent of the errors, say αu, must be ≤ 1/2. But this need not be the case in reality. Although we
do not observe uit, under CAPM the OLS residuals from regressions of rit − rft on rmt − rft, denoted
by uit, provide an accurate estimate of uit up to Op(T−1/2), and can be used to compute consistent
estimates of αu.14 The bias-adjusted estimates of αu, denoted by αu, and computed using standardized
residuals over 5-year and 10-year rolling samples, are displayed in Figure 4. Interestingly enough, these
estimates, although much smaller than those estimated using excess returns, nevertheless tend to be
larger than the threshold value of 1/2, suggesting the presence factors other than the market factor
influencing individual security returns. The influence of residual factor(s) is rather weak initially
(around 0.60), but starts to rise in the years leading to the dot-com bubble and reaches the pick of
0.80 in the middle of 2000 and stays at around that level for the period up to 2006− 2008 (depending
on whether the 10-year or 5-year rolling samples are used), then begins to fall significantly after the
start of the recent financial crisis, and currently stands at around 0.63. Although, special care must be
exercised when interpreting these estimates (both because αu is estimated using residuals and the fact
that α tends to be biased upward particularly when α < 0.75), nevertheless their patterns over time
are indicative of some departures from CAPM during the period 1999 − 2006. Also, it is interesting
that the rolling estimates of αu tend to move in opposite directions to the estimates of α computed
over the same rolling samples. Weakening of the market factor tends to coincide with strengthening of
the residual factor(s), thus suggesting that correlations across returns could remain high even during
13The distribution theory of the direct estimator of α is complicated by the cross dependence of the errors in theunderlying CAPM regressions and its consideration is outside the scope of the present paper.
14A formal proof and analysis when α is estimated from regression residuals is beyond the scope of the present paper.
24
periods where the cross-sectional exponent of the dominant factor is relatively low, once the presence
of multiple factors with exponents exceeding 0.5 is acknowledged.
Figure 4: Estimates of cross-sectional exponent of residuals (αu) from CAPM regressions using 5-yearand 10-year rolling samples
0.55
0.57
0.59
0.61
0.63
0.65
0.67
0.69
0.71
0.73
0.75
0.77
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Jul-9
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-91
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-92
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-93
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-98
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-11
5-year estimates of α associated with residuals of market factor10-year estimates of α associated with residuals of market factor
7 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
cross-sectional dependence especially when it is prevalent enough to materially affect the outcome of
the analysis.
In this paper we propose a relatively simple method of measuring the extent of inter-connections in
large panel data sets in terms of a single parameter that we refer to as the exponent of cross-sectional
dependence. We find that this exponent can accommodate a wide range of cross-sectional dependence
manifestations while retaining its simple and tractable form. We propose consistent estimators of
the cross-sectional exponent and derive their asymptotic distribution under plausible conditions. The
inference problem is complex, as it involves handling a variety of bias terms and, from an econometric
point of view, has noteworthy characteristics such as nonstandard rates of convergence. We provide a
feasible and relatively straightforward estimation and inference implementation strategy.
A detailed Monte Carlo study suggests that the estimated measure has desirable small sample
properties. We apply our measure to three widely analysed classes of data sets. In all cases, we find
that the results of the empirical analysis accord with prior intuition. For example, in the case of cross
country applications we obtain larger estimates for the cross-sectional exponent of equity returns as
compared to those estimated for cross country output growths and inflation. For individual securities
25
in S&P 500 index, the estimates of cross-sectional exponents are systematically high but not equal to
unity, a widely maintained assumption in the theoretical multi-factor literature.
We conclude by pointing out some of the implications of our analysis for large N factor models of
the type analysed by Bai and Ng (2002), Bai (2003), and Stock and Watson (2002). This literature
assumes that all factors have the same cross-section exponent of α = 1, which, as our empirical
applications suggest, may be too restrictive, and it is important that implications of this assumption’s
failure are investigated. Chudik, Pesaran, and Tosetti (2011), Kapetanios and Marcellino (2010) and
Onatski (2011) discuss some of these implications, namely that when 1/2 < α < 1, factor estimates are
consistent but their rates of convergence are different (slower) as compared to the case where α = 1,
and in particular their asymptotic distributions may need to be modified. Methods used to determine
the number of factors in large data sets, discussed in, e.g., Bai and Ng (2002), Onatski (2009) and
Kapetanios (2010), are invalid and will select the wrong number of factors, even asymptotically.15
Finally, the use of estimated factors in regressions for forecasting or other modelling purposes might
not be justified under the conditions discussed in Bai and Ng (2006).
References
Bai, J. (2003): “Inferential Theory for Factor Models of Large Dimensions,” Econometrica, 71(1), 135–171.
Bai, J., and S. Ng (2002): “Determining the Number of Factors in Approximate Factor Models,” Econometrica, 70(1),191–221.
(2006): “Confidence Intervals for Diffusion Index Forecasts and Inference for Factor-Augmented Regressions,”Econometrica, 74(4), 1133–1150.
Bai, Z. D., and J. W. Silverstein (1998): “No Eigenvalues Outside the Support of the Limiting Spectral Distributionof Large Dimensional Sample Covariance Matrices,” Annals of Probability, 26(1), 316–345.
Cesa-Bianchi, A., M. H. Pesaran, A. Rebucci, and T. Xu (2012): “China’s Emergence in the World Economyand Business Cycles in Latin America,” Forthcoming in Economia, Journal of the Latin American and CaribbeanEconomic Association.
Chamberlain, G. (1983): “Funds, Factors and Diversification in Arbitrage Pricing Theory,” Econometrica, 51(5),1305–1323.
Chudik, A., M. H. Pesaran, and E. Tosetti (2011): “Weak and Strong Cross Section Dependence and Estimationof Large Panels,” Econometrics Journal, 14(1), C45–C90.
Davidson, J. (1994): Stochastic Limit Theory. Oxford University Press.
Dees, S., F. di Mauro, M. H. Pesaran, and L. V. Smith (2007): “Exploring the International Linkages of the EuroArea: a Global VAR Analysis,” Journal of Applied Econometrics, 22(1), 1–38.
Eklund, J., G. Kapetanios, and S. Price (2010): “Forecasting in the Presence of Recent Structural Change,”Working Paper 406, Bank of England.
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15It is interesting to note that another contribution to this literature (Onatski (2010)) does not assume strong factorsand, therefore, the suggested method will be valid in our framework. Also, Kapetanios and Marcellino (2010) suggestmodifications to the methods of Bai and Ng (2002) that enable their use in the presence of weak factors.
26
(2005b): “The Empirical Eigenvalue Distribution of a Gram Matrix with a Given Variance Profile,” Annales del’Institut Henri Poincare, 42(6), 649–670.
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Onatski, A. (2009): “Testing Hypotheses about the Number of Factors in Large Factor Models,” Econometrica, 77(5),1447–1479.
(2010): “Determining the Number of Factors Form Empirical Distribution of Eigenvalues,” Review of Economicsand Statistics, 92(4), 1004–1016.
(2011): “Asymptotics of the Principal Components Estimator of Large Factor Models with Weakly InfluentialFactors,” Working paper, University of Cambridge.
Pesaran, M. H. (2006): “Estimation and Inference in Large Heterogeneous panels with a Multifactor Error Structure,”Econometrica, 74(4), 967–1012.
Pesaran, M. H., T. Schuermann, and S. Weiner (2004): “Modeling Regional Interdependencies Using a GlobalError-Correcting Macroeconometric Model,” Journal of Business and Economic Statistics, 22(2), 129–162.
Pesaran, M. H., and T. Yamagata (2012): “Testing CAPM with a Large Number of Assets,” Forthcoming WorkingPaper, University of Cambridge.
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Ross, S. A. (1976): “The Arbitrage Theory of Capital Asset Pricing,” Journal of Economic Theory, 13, 341–360.
Sharpe, W. (1964): “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk,” Journal ofFinance, 19(3), 425–442.
Stock, J. H., and M. W. Watson (2002): “Macroeconomic Forecasting Using Diffusion Indexes,” Journal of Businessand Economic Statistics, 20(1), 147–162.
Stout, W. F. (1974): Almost Sure Convergence. Academic Press.
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Appendix I: Statement of Lemmas
Lemma 1 Under Assumptions 2 and 3, ft, {uit}∞t=1, and {uit}∞i=1 are Lr-bounded, L2-NED processes of size −ζ, forsome r > 2. This result holds uniformly over i, in the case of {uit}∞t=1, and over t, in the case of {uit}∞i=1.
Lemma 2 Under Assumptions 2 and 3,
1√T
T∑t=1
(ft − f
) (√Nut
)→d N(0, σ2
∞),
where
σ2∞ = lim
N→∞
∑Ni=1
(σ2i +
∑∞j=1,j 6=i σij
)N
, (40)
σ2i = E(u2
it), and σij = E(uituit−j).
27
Lemma 3 Under Assumption 3,
1√2T
T∑t=1
[(√Nut
)2
− E(√
Nut)2]→d N(0, V ),
where
V = limN→∞
(V ar
((√Nut
)2)
+
∞∑j=1
Cov
((√Nut
)2
,(√
Nut−j)2))
.
Lemma 4 Under Assumptions 1-3, σ2N − σ
2∞ = Op
(T−1
)+Op
((NT )−1/2
).
Lemma 5 Under Assumptions 1-2,√
min(Nα, T )(ln(s2
f v2N )− ln(σ2
fµ2v))→d N (0, ω) .
Lemma 6 Under Assumption 6 and Assumptions 2-3,√
min(N,T )(ln(s2
f v2N )− ln(σ2
fµ2v))→d N (0, ω) .
Lemma 7 Under Assumptions 1-3, and as long as α > 4/7,√min(Nα, T )
(σ2N
N2α−1v2Ns
2f
−σ2N
Nσ2x
)= Op
(√min(Nα, T )N2−4α
).
Lemma 8 Under Assumptions 1-2, and as long as α > 1/2,√min(Nα, T ) ln(N)
(σ2N
N2α−1v2Ns
2f
−σ2N
Nσ2x
(1 +
σ2N
Nσ2x
))= op (1) .
Lemma 9 Let{v
(s)i
}Ni=1
and{δ
(s)i
}Ni=1
be the reorderings of {vi}Ni=1 and{δi}Ni=1
where∣∣∣v(s)i
∣∣∣ ≥ ∣∣∣v(s)i+1
∣∣∣, ∀i and ∣∣∣δ(s)i
∣∣∣ ≥∣∣∣δ(s)i+1
∣∣∣, ∀i. Under Assumptions 1-3, and assuming that limT,N→∞ T−1Nα <∞, we have
N2α−2Nα∑i=1
(v
(s)i − 1
Nα
Nα∑i=1
v(s)i
)2
Nα − 1− σ2
v
µ2v
= op (1) . (41)
If σ2fµ
2v = 1, then
Nα∑i=1
(δ
(s)i − 1
Nα
Nα∑i=1
δ(s)i
)2
Nα − 1− σ2
v
µ2v
= op (1) .
Lemma 10 Under Assumptions 1-2, we have Vf2 − Vf2 = op(1), as long as l→∞, l = o (T ) and l = o
(Nα−1/2T 1/2
).
Lemma 11 Under Assumptions 1-2, and assuming αj = α, for all j = 1, ...,m,√min(Nα, T )
(ln(v′NSff vN )− ln(µ′vΣffµv)
)→d N (0, ωm) ,
where µv = E (vi), Σff = E((f t − µf
)′ (f t − µf
)),
ωm = limN,T→∞
min (Nα, T )E
(((v′N fT − µ
′vµf
)2 − E ((v′N fT − µ′vµf)2))2),
µf = E (f t) and fT = 1T
∑Tt=1 f t.
Lemma 12 Under Assumptions 1-2, and assuming α > α2 > ... > αm,√min(Nα, T )
(ln(v′NSff vN )− ln(µ′vΣffµv)
)→d N (0, ω) .
Lemma 13 Under either Assumption 1 or 6 and Assumptions 2-3, and α > α2 ≥ α3 ≥ ... ≥ αm,√min (Nα, T ) ln (N) ln(µ′vDNΣffDNµv)− ln
(µ2
1vσ21f
)= o (1) ,
if either α2 − α < −0.25 or, if T b = N , α2 < 3α/4 and
b >1
4 (α− α2). (42)
Lemma 14 Let βi = Nα−1vi, 1/2 < α ≤ 1, where vi = vNi = vi + cNi and {vi}Ni=1 is an i.i.d. sequence of random
variables with mean µv 6= 0, and variance σ2v <∞. Let cN = 1
N
∑Ni=1 cNi. Under Assumptions 2-3, (a) α, α and α are
consistent estimators of α, if cN = op (Nc) for all c > 0, (b) Corollary 2 holds, if√NcN = op (1).
28
Appendix II: Proofs of Theorems
Proof of Theorem 1
We start by noting that
σ2x =
1
T
T∑t=1
(xt −
1
T
T∑t=1
xt
)2
=1
T
T∑t=1
x2t − x2,
where xt = βNft + ut, and x = T−1∑Tt=1 xt = βN f + u. Further, let
Kρ = KNρ =
N∑j=M+1
βi = ρ
(1− ρ(N−M)
1− ρ
); M = [Nα]. (43)
Then we have
σ2x = β2
Ns2f + 2βN
[1
T
T∑t=1
(ft − f
)ut
]+
[1
T
T∑t=1
u2t − u2
],
s2f =
1
T
T∑t=1
(ft − f
)2 →p σ2f > 0, as T →∞.
Therefore
ln(σ2x
)= ln(β2
Ns2f ) + ln
1 +2βN
[1T
∑Tt=1
(ft − f
)ut]
+[
1T
∑Tt=1 u
2t − u2
]β2Ns
2f
.
But under Assumption 1, βN = (M/N)vN +N−1Kρ, where vN = M−1∑Mi=1 vi, we have
ln(β2Ns
2f ) = ln
(Nα−1vNsf +
KρsfN
)2
= 2(α− 1) ln(N) + ln(s2f v
2N ) + 2 ln
(1 +
Kρ
NαvN
)= 2(α− 1) ln(N) + ln(s2
f v2N ) +Op(N
−α).
Hence, recalling from (14) that α = 1+ ln(σ2x)/2 ln (N),we have
2 ln(N) (α− α)− ln(s2f v
2N ) = ln
1 +2βN
[1T
∑Tt=1
(ft − f
)ut]
+[
1T
∑Tt=1 u
2t − u2
]β2Ns
2f
+Op(N−α). (44)
However,
ln
1 +2βN
[1T
∑Tt=1
(ft − f
)ut]
+[
1T
∑Tt=1 u
2t − u2
]β2Ns
2f
=2βN
[1T
∑Tt=1
(ft − f
)ut]
+[
1T
∑Tt=1 u
2t − u2
]β2Ns
2f
+ op (aTN ) ,
(45)where
aTN =2βN
[1T
∑Tt=1
(ft − f
)ut]
+[
1T
∑Tt=1 u
2t − u2
]β2Ns
2f
.
Consider the first term of the RHS of (45). We have,
2βN[
1T
∑Tt=1
(ft − f
)ut]
β2Ns
2f
=
2√TN
[1
σf√T
∑Tt=1
(ft − f
) (√Nut
)](sf βN
)(sf/σf )
.
We note that sf/σf = 1 +Op(T−1/2). But, by Lemma 2 (as N and T →∞)
1
σf√T
T∑t=1
(ft − f
) (√Nut
)→p N(0, σ2
∞), (46)
where σ2∞ is as in (40).
We need to determine the probability order of 1/βN . We note that
1
βN− 1
Nα−1vN=
1
Nα−1vN +KρN
− 1
Nα−1vN=
−KρN−1
N2α−2v2N +KρNα−2vN
= −N1−2αKρv−1N
(vN +KρN
−α)−1= Op
(N1−2α) ,
29
and hence
2βN[
1T
∑Tt=1
(ft − f
)ut]
β2Ns
2f
=
2√TN
[1
σf√T
∑Tt=1
(ft − f
) (√Nut
)]sf βN (sf/σf )
(47)
= Op(T−1/2N1/2−α
)+Op
(T−1/2N1/2−2α
).
Consider now the second term on the RHS of (45). We have
1
β2N
− 1
N2α−2v2N
=−Nα−2KρvN −N−2K2
ρ
N4α−4v4N +N3α−4Kρv3
N +N2α−4K2ρ v
2N
=
−(N2−3αKρv
3N +N2−4αK2
ρ v2N
) (v2N +N−αKρvN +N−2αK2
ρ
)= Op
(N2−3α) .
Note that since, by Lemma 1 and Theorems 17.5 and 19.11 of (Davidson, 1994, Theorem 15.18),√NTu = Op(1), then,
since s2f/σ
2f = 1 +Op(T
−1/2) and 0 < σ2f <∞,
u2(Nα−1vN +
KρN
)2
s2f
=
(√NTu
)2
NT(Nα−1vN +
KρN
)2
s2f
= Op(T−1N1−2α) . (48)
Similarly,
1T
∑Tt=1 u
2t(
Nα−1vN +KρN
)2
s2f
=
1
N√T
{1√T
∑Tt=1
[(√Nut)
2 − σ2N
]+√T σ2
N
}(Nα−1vN +
KρN
)2
s2f
=
σ2N
N√T
{1√T
∑Tt=1
[(√NutσN
)2
− 1
]+√T
}(Nα−1vN +
KρN
)2
s2f
=
σ2N
N√T
1√T
∑Tt=1
[(√NutσN
)2
− 1
](Nα−1vN +
KρN
)2
s2f
+σ2N
N(Nα−1vN +
KρN
)2
s2f
.
Note thatσ2N
N(Nα−1vN +
KρN
)2
s2f
− σ2N
N2α−1v2Ns
2f
= Op(N1−3α). (49)
But, by Lemma 3,
1√2T
T∑t=1
[(√NutσN
)2
− 1
]→d N(0, 1),
andσ2N
N√T
(1√T
∑Tt=1
[(√NutσN
)2
− 1
])(Nα−1vN +
KρN
)2
s2f
= Op(T−1/2N1−2α) +Op(T
−1/2N1−3α). (50)
Therefore, collecting all results derived above, and keeping the highest order terms of the RHS of (44), (47), (48), (49)and (50), we have
2 ln(N) (α− α)− ln(s2f v
2N )− σ2
N
N2α−1v2Ns
2f
= Op(
max(T−1/2N1/2−α, T−1N1−2α, T−1/2N1−2α, N1−3α, N−α
)).
Since α > 1/2, in the first instance this implies that
α− α = Op
(1
ln(N)
), (51)
which establishes the consistency of α as an estimate of α as N and T →∞, in any order.Consider now the derivation of the asymptotic distribution of α. We have
ln(N) (α− α)− σ2N
N2α−1v2Ns
2f
= ln(s2f v
2N ) +
2√TN
[1
σf√T
∑Tt=1
(ft − f
) (√Nut
)]sfNα−1vN (sf/σf )
+
(√NTu
)2
NT(Nα−1vN +
KρN
)2
s2f
+
σ2N
N√T
1√T
∑Tt=1
[(√NutσN
)2
− 1
](Nα−1vN +
KρN
)2
s2f
+Op(N−α).
30
We first examine ln(s2f v
2N ). By Lemma 5 we have
√min(Nα, T )
(ln(s2
f v2N )− ln(σ2
fµ2v))→d N (0, ω) .
Further, since α > 1/2,
√min(Nα, T )
2√TN
[1
σf√T
∑Tt=1
(ft − f
) (√Nut
)]sfNα−1vN (sf/σf )
= Op(√
min(Nα, T )T−1/2N1/2−α)
= op(1).
Similarly, √min(Nα, T )
(√
NTu)2
NT(Nα−1vN +
KρN
)2
s2f
= Op(√
min(Nα, T )T−1N1−2α)
= op(1),
and
√min(Nα, T )
σ2N
N√T
1√T
∑Tt=1
[(√NutσN
)2
− 1
](Nα−1vN +
KρN
)2
s2f
= Op(√
min(Nα, T )T−1/2N1−2α)
= op(1).
Thus, √min(Nα, T )
(ln(N) (α− α∗N )− σ2
N
N2α−1v2Ns
2f
)→d N (0, ω) ,
where α∗N= α+ ln (σ2fµ
2v)/ ln (N).
Proof of Theorem 2
We need to show that as long as either α > 4/7 or T 1/2/N4α−2 → 0,√
min(Nα, T ) ln(N)
(σ2N
N2α−1v2Ns2f−
σ2N
Nσ2x
)= op (1) .
The result follows immediately by Lemma 7.
Proof of Theorem 3
The result follows immediately by Lemma 8.
Proof of Theorem 4
The proof follows immediately from Lemmas 9 and 10.
Proof of Theorem 5
Under the general factor model, we have
σ2x = β
′
NSff βN + 2β′N
[1
T
T∑t=1
(f t − f
)ut
]+
[1
T
T∑t=1
u2t − u2
],
where
Sff =1
T
T∑t=1
(f t − f
) (f t − f
)′ →p Σff > 0, as T →∞.
We have that
β′NSff βN = N2α−2v′NDNSffDN vN + 2Nα−2v′NDNSffKρ +N−2K′ρSffKρ = N2α−2v′NDNSffDN vN +O
(Nα−2) .
Soln(β′NSff βN
)= ln
(N2α−2v′NDNSffDN vN + 2Nα−2v′NDNSffKρ +N−2K′ρSffKρ
)=
2(α− 1) ln (N) + ln(v′NDNSffDN vN
)+ ln
(1 +
2N−αv′NDNSffKρ +N−2αK′ρSffKρ
v′NDNSffDN vN
).
Then,
31
ln(σ2x
)= ln(β
′
NSff βN ) + ln
1 +2β′N
[1T
∑Tt=1
(f t − f
)ut]
+[
1T
∑Tt=1 u
2t − u2
]β′
NSff βN
+Op(N−α
),
ln(σ2x
)= 2(α− 1) ln(N) + ln(v′NDNSffDN vN )
+ ln
1 +2β′N
[1T
∑Tt=1
(f t − f
)ut]
+[
1T
∑Tt=1 u
2t − u2
]β′
NSff βN
+Op(N−α
).
So
2 ln(N) (α− α)− ln(v′NDNSffDN vN ) = ln
1 +2β′N
[1T
∑Tt=1
(f t − f
)ut]
β′
NSff βN+
[1T
∑Tt=1 u
2t − u2
]β′
NSff βN
+Op(N−α
),
or
2 ln(N) (α− α)− ln(v′NDNSffDN vN ) =2β′N
[1T
∑Tt=1
(f t − f
)ut]
β′
NSff βN+
[1T
∑Tt=1 u
2t − u2
]β′
NSff βN+Op
(N−α
)+ op(BN,T ),
where
BN,T =2β′N
[1T
∑Tt=1
(f t − f
)ut]
β′
NSff βN+
[1T
∑Tt=1 u
2t − u2
]β′
NSff βN.
Similarly to (47),
2β′N
[1T
∑Tt=1
(f t − f
)ut]
β′
NSff βN= Op
(T−1/2N1/2−α
)+Op
(T−1/2N1/2−2α
).
Also [1T
∑Tt=1 u
2t − u2
]β′
NSff βN=
σ2N
1√T
∑Tt=1
[(√NutσN
)2
− 1
]√TN2α−1v′NDNSffDN vN
+σ2N
N2α−1v′NDNSffDN vN+Op(N
1−2αT−1),
where
σ2N
1√T
∑Tt=1
[(√NutσN
)2
− 1
]√TN2α−1v′NDNSffDN vN
= Op(T−1/2N1−2α
).
So,
2 ln(N) (α− α)−ln(v′NDNSffDN vN )− σ2N
N2α−1v′NDNSffDN vN= Op
(max
(T−1/2N1/2−α, T−1N1−2α, T−1/2N1−2α, N1−3α, N−α
)).
Using the above derivations gives straightforward extensions of Lemmas 7 and 8. Using these, we get
√min(Nα, T )
(σ2N
N2α−1v′NDNSffDN vN−
σ2N
Nσ2x
)= Op
(√min(Nα, T )N2−4α
)and √
min(Nα, T ) ln(N)
(σ2N
N2α−1v′NDNSffDN vN−
σ2N
Nσ2x
(1 +
σ2N
Nσ2x
))= op (1) ,
which together with Lemmas 11, 12 and 13 prove the theorem.
Appendix III: Proofs of Lemmas
Proof of Lemma 1
By the Marcinkiewicz–Zygmund inequality (see, e.g., (Stout, 1974, Theorem 3.3.6)),
supiE(|uit|r) = sup
iE
({∞∑l=0
(ψil
∞∑s=−∞
ξisvst−l
)}r)≤ c
(supi
(∞∑l=0
|ψil|2)
supi
(∞∑
s=−∞
|ξis|2))r/2(
supi,t
E(|vit|r)),
32
so uit is Lr-bounded if supi supt E(|vt|r) < ∞ which holds by Assumption 2. Moreover, writing ‖·‖r for the Lr-norm,we have, by Minkowski’s inequality,
supi
∥∥∥uit − E(uit|Fvit,|m|)∥∥∥
2= sup
i
∥∥∥∥∥∥∞∑
j=m+1
ψij
∑|s|≥m
ξisνst
∥∥∥∥∥∥2
≤ supi,t‖vit‖2
(supi
∞∑j=m+1
|ψij |
)supi
∑|s|≥m
|ξis|
,
(52)
for any integer m > 0 where Fνit,|m| is the σfield generated by {vis; i, s ≤ t−m} ∪ {vis; i, s ≥ t+m}. But, Assumption 2
implies that supi limm→∞mζ∑∞
j=m+1 |ψij | = O (1) and supi limm→∞mζ(∑
|s|≥m |ξis|)
= O (1) . Consequently {uit}∞t=1
and {uit}∞i=1 and Lr-bounded, L2-NED processes of size −ζ, uniformly over i and t. Similarly, we can show that ft is
an Lr-bounded (r ≥ 2) L2-NED processes of size −ζ.
Proof of Lemma 2
We have 1√T
∑Tt=1
(ft − f
) (√Nut
)= 1√
T
∑Tt=1 zt, where zt =
(ft − f
) (√Nut
). We have that zt is a station-
ary process such that E (zt) = 0. We note that by Lemma 1 and Theorem 24.6 of Davidson (1994), we have that
E
((√Nut
)2)
= 1N
∑Ni=1 σ
2i <∞. Further, by Theorem 17.8 of Davidson (1994), we have that sums of L2-bounded,
L2-NED triangular arrays of size −ζ are L2-bounded, L2-NED triangular arrays of size −ζ as well, implying, givenLemma 1, that
√Nut is an L2-bounded, L2-NED triangular arrays of size −ζ. Further, by the Marcinkiewicz–Zygmund
inequality,
E(∣∣∣√Nut∣∣∣r) = E
(∣∣∣∣∣ 1√N
N∑i=1
∞∑l=0
(ψil
∞∑s=−∞
ξisvst−l
)∣∣∣∣∣r)≤ c
(1
N
N∑i=1
(∞∑l=0
|ψil|2)(
∞∑s=−∞
|ξis|2))r/2
supi,t
E(|vit|r) ≤
(53)
c
(supi
(∞∑l=0
|ψil|2)
supi
(∞∑
s=−∞
|ξis|2))r/2(
supi,t
E(|vit|r))<∞.
As a result,√Nut is a Lr-bounded, L2-NED triangular arrays of size −ζ.
Finally, since {√Nut} and {ft} are Lr-bounded (r ≥ 2) L2-NED processes of size −ζ on a φ-mixing process of size
−η (η > 1), then, by Example 17.17 of Davidson (1994), {zt} is L2-NED of size −{ζ(ϕ − 2)}/{2(ϕ − 1)} ≤ −1/2 on a
φ-mixing process of size −η. Since νit and νft are i.i.d. processes they are also φ-mixing processes of any size. In view
of Theorem 17.5(ii) of Davidson (1994), this in turn implies that {zt} is an L2-mixingale of size −1/2, if 2η > ζ, which
automatically holds by the i.i.d. property of νit and νft. This, implies the result of the Lemma by Theorem 24.6 of
(Davidson, 1994, Theorem 15.18).
Proof of Lemma 3
By Lemma 2,√Nut is a Lr-bounded, L2-NED triangular arrays of size −ζ. By Example 17.17 of Davidson (1994), and
(53),(√
Nut)2
is Lr-NED of size −{ζ(ϕ− 2)}/{2(ϕ− 1)} ≤ −1/2, r > 4. Then, by Theorem 24.6 of (Davidson, 1994,
Theorem 15.18), the result follows.
Proof of Lemma 4
We need to show that σ2N − σ2
∞ = Op(T−1
)+ Op
((NT )−1/2
). We have that σ2
N = 1NT
∑Ni=1
∑Tt=1 u
2it, where
uit = xit − δixt. Then, 1NT
∑Ni=1
∑Tt=1 u
2it = 1
NT
∑Ni=1
∑Tt=1 u
2it + 1
NT
∑Ni=1
∑Tt=1
(u2it − u2
it
), where uit = xit −
δixt. Following similar lines to those of the proof of Lemma 3 we have that 1NT
∑Ni=1
∑Tt=1 u
2it →p σ2
∞. Further,1NT
∑Ni=1
∑Tt=1
(u2it − σ2
∞)
= O(
(NT )−1/2)
. Next, we examine 1NT
∑Ni=1
∑Tt=1
(u2it − u2
it
). It is sufficient to consider
1NT
∑Ni=1
∑Tt=1 uit (uit − uit). Noting that xt
N1−αxt= Op (1), and denoting equality in order of probability by =p, we
have
1
NT
N∑i=1
T∑t=1
uit (uit − uit) =1
NT
N∑i=1
(δi − δi
) T∑t=1
xtuit =p
1
NT
(1
1T
∑Tt=1 (N1−αxt)
2
)N∑i=1
( 1√T
T∑t=1
(N1−αxt
)uit
)2
− E
( 1√T
T∑t=1
(N1−αxt
)uit
)2+
33
+1
NT
(1
1T
∑Tt=1 (N1−αxt)
2
)N∑i=1
E( 1√
T
T∑t=1
(N1−αxt
)uit
)2 .
But E
((1√T
∑Tt=1
(N1−αxt
)uit)2)<∞ and 1
T
∑Tt=1
(N1−αxt
)2= Op (1), which implies that
1
NT
(1
1T
∑Tt=1 (N1−αxt)
2
)N∑i=1
E( 1√
T
T∑t=1
(N1−αxt
)uit
)2 = Op
(1
T
).
Further,(
1√T
∑Tt=1
(N1−αxt
)uit)2
− E((
1√T
∑Tt=1
(N1−αxt
)uit)2)
is a NED process over i, which implies that
1
NT
(1
1T
∑Tt=1 (N1−αxt)
2
)N∑i=1
( 1√T
T∑s=
(N1−αxt
)uit
)2
− E
( 1√T
T∑s=
(N1−αxt
)uit
)2 = Op
(1
T√N
),
proving the required result.
Proof of Lemma 5
We have that
ln(s2f v
2N )− ln(σ2
fv2∞) = ln
(s2f v
2N
σ2fv
2∞
)= ln
(s2f
σ2f
)+ ln
(v2N
v2∞
)=
(s2f − σ2
f
σ2f
)+
(v2N − µ2
v
µ2v
)+
Op((s2f − σ2
f
)2)+Op
((v2N − µ2
v
)2).
But, under Assumption 2,
√T
(s2f − σ2
f
σ2f
)=
1√T
T∑t=1
{[(ft − f)/σf
]2 − 1}→d N
(0, V
f2
),
where f = 1T
∑Tt=1 ft, and
Vf2 = E
(([(ft − µf )/σf ]2 − 1
)2)+
∞∑i=1
Cov((
[(ft − µf )/σf ]2 − 1) (
[(ft−i − µf )/σf ]2 − 1)).
Further, recalling that vN = 1Nα
∑Nα
i=1 vi,√Nα
(v2N−µ
2v
µ2v
)=√Nα
(vN−µvµv
)(vN+µvµv
). But vN+µv
µv→p 2, and
√Nα
(vN − µvµv
)→d N
(0,σ2v
µ2v
). (54)
Further, E
[(s2f−σ
2f
σ2f
)(v2N−µ
2v
µ2v
)]= 0, implying that
√min(Nα, T )
(ln(s2
f v2N )− ln(σ2
fv2∞))→d N (0, ω) , where ω =
limN,T→∞
[min(Nα,T )
TVf2 + min(Nα,T )
Nα4σ2v
µ2v
].
Proof of Lemma 6
The proof follows easily along the same lines as that of Lemma 5. In the present case under Assumption 6 we have
vN = N−1∑Ni=1 vi, and thus
√N(v2N−µ
2v
µ2v
)=√N(vN−µvµv
)(vN+µvµv
), and vN+µv
µv→p 2. Therefore,
√N(vN−µvµv
)→d
N(
0,σ2vµ2v
).
Proof of Lemma 7
We need to show that √min(Nα, T )
(σ2N
N2α−1v2Ns
2f
−σ2N
Nσ2x
)= op (1) . (55)
We haveσ2N
N2α−1v2Ns
2f
−σ2N
Nσ2x
=σ2N
N2α−1v2Ns
2f
−σ2N
N2α−1v2Ns
2f
+σ2N
N2α−1v2Ns
2f
−σ2N
Nσ2x
.
34
Butσ2N
N2α−1v2Ns
2f
−σ2N
N2α−1v2Ns
2f
= Op(T−1/2N1−2α
),
which is negligible as a bias. Next,σ2N
N2α−1v2Ns
2f
−σ2N
Nσ2x
= σ2N
( σ2N
σ2N
)(1
N2α−1v2Ns
2f
− 1
Nσ2x
)= σ2
N
( σ2N
σ2N
)(1
N2α−1v2Ns
2f
)N2α−1 (N2−2ασ2
x − v2Ns
2f
)( 1
Nσ2x
).
But by the proof of Theorem 1, we have
v2Ns
2f −N2−2ασ2
x = Op(T−1/2N1/2−α
)+Op
(T−1N1−2α)+Op(N
1−3α) +Op(N−α) +Op
(N1−2α) .
So
σ2N
( σ2N
σ2N
)(1
N2α−1v2Ns
2f
)N2α−1 (N2−2ασ2
x − v2Ns
2f
)( 1
Nσ2x
)=
Op(T−1/2N1/2−αN1−2α
)+Op
(T−1N1−2αN1−2α)+Op(N
1−3αN1−2α) +Op(N−αN1−2α) +Op
(N1−2αN1−2α) =
Op(T−1/2N3/2−3α
)+Op
(T−1N2−4α)+Op(N
2−5α) +Op(N1−3α) +Op
(N2−4α) .
Therefore, as long as α > 4/7, (55) holds, proving the Lemma.
Proof of Lemma 8
We have that
σ2N
( σ2N
σ2N
)(1
N2α−1v2Ns
2f
)N2α−1
(N1−2α
(1
T
T∑t=1
(√Nut)
2 − σ2N
))(1
Nσ2x
)= Op
(N2−4αT−1/2
),
which is negligible as long as α > 1/2. To see the above result note the following. We have
1
T
T∑t=1
( 1√N
N∑i=1
uit
)2− σ2
N =
1
T
T∑t=1
( 1√N
N∑t=1
uit
)2− σ2
∞
+
(σ2∞ −
1
NT
N∑i=1
T∑t=1
u2it
)−
(1
NT
N∑i=1
T∑t=1
u2it −
1
NT
N∑i=1
T∑t=1
u2it
).
But it is straightforward to show that 1T
∑Tt=1
((1√N
∑Ni=1 uit
)2)− σ2
∞ = Op(T−1/2) and σ2
∞ − 1NT
∑Ni=1
∑Tt=1 u
2it =
Op(T−1/2). Finally by Lemma 4, 1
NT
∑Ni=1
∑Tt=1 u
2it− 1
NT
∑Ni=1
∑Tt=1 u
2it = op(T
−1/2). So, 1T
∑Tt=1
((1√N
∑Nt=1 uit
)2)−σ2
N = Op(T−1/2).
Proof of Lemma 9
The first step in the proof is to show that the number of cross-sectional units that are misclassified, i.e., that are includedin the variance calculation when their loading is not a function of any vi, is op (Nα). The first thing to note is that weabstract from the possibility that any vi = 0. By the fact that Pr (vi = 0) = 0, it follows that the number of units thathave vi = 0 is op (Nα). Without loss of generality, we further assume that units whose loadings do not depend on anyvi have zero loadings. The probability that op (Nα) units are misclassified is bounded from above by
C1N−α+c Pr
((∪[Nα]i=1 {|vi − vi| > ε}
)∪(∪Ni=[Nα]+1 {|vi| > ε}
)),
for some C1 > 0, some ε > 0, and any c > 0, where Pr((∪[Nα]i=1 {|vi − vi| > ε}
)∪(∪Ni=[Nα]+1 {|vi| > ε}
))is an upper
bound for the probability that one unit is misclassified. But
Pr((∪[Nα]i=1 {|vi − vi| > ε}
)∪(∪Ni=[Nα]+1 {|vi| > ε}
))≤
[Nα]∑i=1
Pr (|vi − vi| > ε) +
N∑i=[Nα]+1
Pr (|vi| > ε) .
But, by Markov’s inequality Pr (|vi − vi| > ε) ≤ E(|vi−vi|2)ε2
= O(T−1
), i = 1, ..., [Nα] and Pr (|vi| > ε) ≤ E(|vi|2)
ε2=
O(T−1
), i = [Nα] + 1, ..., N. Therefore, the probability that op (Nα) units are misclassified is o
(N1−α+cT−1
)for all
35
c > 0. Since α > 1/2, this probability goes to zero as long as limT,N→∞ T−1Nα <∞, proving the first step in the proof.
Next, we show the first part of the Lemma assuming that we observe which units have non zero loadings. Recall that,assuming that units whose loadings do not depend on any vi have zero loadings, xit = vi
vN
(N1−αxt
)+ uit. We analyse
vi by defining it to be the estimated regression coefficient of the regression of xit on N1−αxt rather than xit on xt. Thisis clearly the case once the normalisation N2α−2 is taken into account in (41). Let v
(1)i = vi
µvand vNi = vi
vN. We need to
show that 1Nα−1
Nα∑i=1
(vi − 1
Nα
∑Nα
i=1 vi)2
− σ2vµ2v
= op (1) . We have
1
Nα − 1
Nα∑i=1
(vi −
1
Nα
Nα∑i=1
vi
)2
− σ2v
µ2v
=1
Nα − 1
Nα∑i=1
(vi −
1
Nα
Nα∑i=1
vi
)2
− 1
Nα − 1
Nα∑i=1
(vNi −
1
Nα
Nα∑i=1
vNi
)2
+
1
Nα − 1
Nα∑i=1
(vNi −
1
Nα
Nα∑i=1
vNi
)2
− 1
Nα − 1
Nα∑i=1
(v
(1)i −
1
Nα
Nα∑i=1
v(1)i
)2
+1
Nα − 1
Nα∑i=1
(v
(1)i −
1
Nα
Nα∑i=1
v(1)i
)2
− σ2v
µ2v
.
But by the law of large numbers for i.i.d. random variables with finite variance
1
Nα − 1
Nα∑i=1
(v
(1)i −
1
Nα
Nα∑i=1
v(1)i
)2
− σ2v
µ2v
= Op(N−1/2
).
It is sufficient to show that
1
Nα − 1
Nα∑i=1
(vi −
1
Nα
Nα∑i=1
vi
)2
− 1
Nα − 1
Nα∑i=1
(vNi −
1
Nα
Nα∑i=1
vNi
)2
= op (1) (56)
and1
Nα − 1
Nα∑i=1
(vNi −
1
Nα
Nα∑i=1
vNi
)2
− 1
Nα − 1
Nα∑i=1
(v
(1)i −
1
Nα
Nα∑i=1
v(1)i
)2
= op (1) . (57)
For (56), it is sufficient that 1Nα−1
Nα∑i=1
(vi − vNi) = op (1) . Recall that xit = vivN
(N1−αxt
)+ uit. So
1
Nα − 1
Nα∑i=1
(vi − vNi) =
1Nα−1
Nα∑i=1
(∑Tt=1 xtuit
)∑Tt=1 x
2t
=1
(Nα − 1) vN(∑T
t=1 f2t
) Nα∑i=1
T∑t=1
ftuit.
But∑Nα
i=1
∑Tt=1 ftuit = Op
((NαT )1/2
)and
∑Tt=1 f
2t = Op (T ) . So
Nα∑i=1
(∑Tt=1 xtuit
)Nα − 1
(∑Tt=1 x
2t
) = Op(T−1N−α (NαT )1/2
)= Op
(T−1/2N−α/2
)= op (1) .
For (57), it is sufficient that 1Nα−1
Nα∑i=1
(vNi − v(1)
i
)= op (1) . We have,
1
Nα − 1
Nα∑i=1
(vNi − v(1)
i
)=
(1vN− 1
µv
) Nα∑i=1
vi
Nα − 1.
But 1Nα−1
Nα∑i=1
vi = Op (1) . Also 1vN− 1µv
= 1vNµv
(vN − µv) = Op(N−α/2
). So, overall 1
N−1
Nα∑i=1
(vi − 1
N
∑Ni=1 vi
)2
− σ2vµ2v
=
op (1) , proving the result.
Proof of Lemma 10
We need to show Vf2 − Vf2 = op(1). We have V
f2 − Vf2 = Vf2 − Vf2 + V
f2 − Vf2 where
Vf2 =
1
T
T∑t=1
(qt −
1
T
T∑t=1
qt
)2
+
l∑j=1
(1
T
T∑t=j+1
(qt−j −
1
T
T∑t=1
qt
)(qt −
1
T
T∑t=1
qt
)),
36
and qt =(ft−fsf
)2
. But, by Theorem 25.3 of Davidson (1994) and Assumption 3, we have that Vf2 − Vf2 = op(1), as
long as l→∞ and l = o(T ). Then, it is sufficient to examine
1
T
T∑t=1
(ftsf− xt
)=
1
T
T∑t=1
(ftsf− xtσx
)=
1
sfT
T∑t=1
(ft −
xtNα−1vN
)=
1
sfT
T∑t=1
(ft −
vNNα−1ft + 1
N
∑Ni=1 uit
Nα−1vN
)=N−α
sfT
T∑t=1
N∑i=1
uit =N−α
σfT
T∑t=1
N∑i=1
uit + op
(N−α
σfT
T∑t=1
N∑i=1
uit
).
But,∑Tt=1
∑Ni=1 uit = Op
((NT )1/2
). So, 1
T
∑Tt=1
(ftsf− xt
)= Op
(N1/2−αT−1/2
). Thus, V
f2−Vf2 = Op(lN1/2−αT−1/2
),
proving the Lemma.
Proof of Lemma 11
Without loss of generality we consider the case of two factors. The result extends straightforwardly to m factors. Wefurther assume, for simplicity, that factors are independent from each other. Then,
ln(v2
1Ns21f + 2v1N v2Ns12f + v2
2Ns22f
)− ln
(σ2
1fµ21v + σ2
2fµ22v
)= ln
(v2
1Ns21f + 2v1N v2Ns12f + v2
2Ns22f
σ21fµ
21v + σ2
2fµ22v
).
Then,
ln
(v2
1Ns21f + 2v1N v2Ns12f + v2
2Ns22f
σ21fµ
21v + σ2
2fµ22v
)=v2
1Ns21f + 2v1N v2Ns12f + v2
2Ns22f
σ21fµ
21v + σ2
2fµ22v
− 1 = (58)(v2
1Ns21f − σ2
1fµ21v
)+(v2
2Ns22f − σ2
2fµ22v
)+ 2v1N v2Ns12f
σ21fµ
21v + σ2
2fµ22v
=(v2
1Ns21f − v2
1Nσ21f + v2
1Nσ21f − σ2
1fµ21v
)+(v2
2Ns22f − v2
2Nσ22f + v2
2Nσ22f − σ2
2fµ22v
)+ 2µ1vµ2vs12f
σ21fµ
21v + σ2
2fµ22v
=
µ21v
(s2
1f − σ21f
)σ2
1fµ21v + σ2
2fµ22v
+σ2
1f
(v2
1N − µ21v
)σ2
1fµ21v + σ2
2fµ22v
+µ2
2v
(s2
2f − σ22f
)σ2
1fµ21v + σ2
2fµ22v
+σ2
2f
(v2
2N − µ22v
)σ2
1fµ21v + σ2
2fµ22v
+2µ1vµ2vs12f
σ21fµ
21v + σ2
2fµ22v
.
Note that
v1N v2Ns12f = v1N v2Ns12f − v1Nµ2vs12f + v1Nµ2vs12f − v1Nµ2vσ12f + v1Nµ2vσ12f − 2µ1vµ2vσ12f =
s12f v1N (v2N − µ2v) + v1Nµ2v (s12f − σ12f ) + σ12fµ2v (v1N − 2µ1v) =
(s12f − σ12f ) v1N (v2N − µ2v) + σ12f v1N (v2N − µ2v) + v1Nµ2v (s12f − σ12f ) + σ12fµ2v (v1N − 2µ1v) .
But(s12f − σ12f ) v1N (v2N − µ2v) = op(T
−1/2),
and σ12f = 0, and so
(s12f − σ12f ) v1N (v2N − µ2v) + σ12f v1N (v2N − µ2v) + v1Nµ2v (s12f − σ12f ) + σ12fµ2v (v1N − 2µ1v)
= v1Nµ2vs12f =
(v1N
µ1v
)µ1vµ2vs12f + op
(T−1/2
).
Then,µ2iv
(s2if − σ2
if
)σ2
1fµ21v + σ2
2fµ22v
=µ2ivσ
2if
σ21fµ
21v + σ2
2fµ22v
(s2if − σ2
if
)σ2if
, i = 1, 2,
σ2if
(v2iN − µ2
iv
)σ2
1fµ21v + σ2
2fµ22v
=µ2ivσ
2if
σ21fµ
21v + σ2
2fµ22v
(v2iN − µ2
iv
)µ2iv
, i = 1, 2,
or if σ21fµ
21v + σ2
2fµ22v = 1,
µ2iv
(s2if − σ2
if
)= µ2
ivσ2if
(s2if − σ2
if
)σ2if
, i = 1, 2,
σ2if
(v2iN − µ2
iv
)= µ2
ivσ2if
(v2iN − µ2
iv
)µ2iv
, i = 1, 2.
Assuming loadings of factors and factors are independent of each other and across factors, gives
µ2ivσ
2if
(√T
(s2if − σ2
if
)σ2if
)= µ2
ivσ2if
(1√T
T∑t=1
{[(fit − fi)/σif
]2 − 1})→d N
(0,(µ2ivσ
2if
)2µ
(4)i
), i = 1, 2,
37
µ2ivσ
2if
(√Nα
(v2iN − µ2
iv
µ2iv
))= µ2
ivσ2if
(√Nα
(viN − µiv
µiv
)(viN + µiv
µiv
))→d N
(0, 2σ2
ivµ2ivσ
2if
), i = 1, 2.
Further,
µ1vµ2v
√Ts12f = µ1vµ2v
σ1fσ2f√T
T∑t=1
(f1t − f1
σ1f
)(f2t − f2
σ2f
)→d N
(0, µ2
1vσ21fµ
22vσ
22f
).
Further, by factor independence
E
((1√T
T∑t=1
{[(fit − fi)/σif
]2 − 1})( 1√
T
T∑t=1
(f1t − f1
σ1f
)(f2t − f2
σ2f
)))= 0, i = 1, 2.
So √min (Nα, T )
T
(µ2
1vσ21f
(√T
(s2
1f − σ21f
)σ2
1f
)+ µ2
2vσ22f
(√T
(s2
2f − σ22f
)σ2
2f
)+ 2µ1vµ2v
√Ts12f
)+
√min (Nα, T )
Nα
(µ2
1vσ21f
(√Nα
(v2
1N − µ21v
µ21v
))+ µ2
2vσ22f
(√Nα
(v2
2N − µ22v
µ22v
)))→d
N
(0, min(Nα,T )
T
((µ2
1vσ21f
)2µ
(4)1 +
(µ2
2vσ22f
)2µ
(4)2 + 2µ2
1vσ21fµ
22vσ
22f
)+ min(Nα,T )
Nα
(2σ2
1vµ21vσ
21f + 2σ2
2vµ22vσ
22f
)).
Proof of Lemma 12
Again, without loss of generality we look at the case of two factors. The result again extends straightforwardly. Wefurther assume, for simplicity, that factors are independent from each other. Then,
ln(v2
1Ns21f + 2Nα2−αv1N v2Ns12f +N2(α2−α)v2
2Ns22f
)− ln
(σ2
1fµ21v +N2(α2−α)σ2
2fµ22v
)=
ln
(v2
1Ns21f + 2Nα2−αv1N v2Ns12f +N2(α2−α)v2
2Ns22f
σ21fµ
21v +N2(α2−α)σ2
2fµ22v
).
Then, similarly to the proof of Lemma 11
ln
(v2
1Ns21f + 2Nα2−αv1N v2Ns12f +N2(α2−α)v2
2Ns22f
σ21fµ
21v +N2(α2−α)σ2
2fµ22v
)=
µ21v
(s2
1f − σ21f
)σ2
1fµ21v +N2(α2−α)σ2
2fµ22v
+σ2
1f
(v2
1N − µ21v
)σ2
1fµ21v +N2(α2−α)σ2
2fµ22v
+
N2(α2−α)µ22v
(s2
2f − σ22f
)σ2
1fµ21v +N2(α2−α)σ2
2fµ22v
+N2(α2−α)σ2
2f
(v2
2N − µ22v
)σ2
1fµ21v +N2(α2−α)σ2
2fµ22v
+2Nα2−αµ1vµ2vs12f
σ21fµ
21v +N2(α2−α)σ2
2fµ22v
.
Then,µ2
1v
(s2
1f − σ21f
)σ2
1fµ21v +N2(α2−α)σ2
2fµ22v
=µ2
1vσ21f
σ21fµ
21v +N2(α2−α)σ2
2fµ22v
(s2
1f − σ21f
)σ2
1f
,
σ21f
(v2
1N − µ21v
)σ2
1fµ21v +N2(α2−α)σ2
2fµ22v
=µ2
1vσ21f
σ21fµ
21v +N2(α2−α)σ2
2fµ22v
(v2
1N − µ21v
)µ2
1v
,
N2(α2−α)µ22v
(s2
2f − σ22f
)σ2
1fµ21v +N2(α2−α)σ2
2fµ22v
=µ2
2vσ22f
σ21fµ
21v +N2(α2−α)σ2
2fµ22v
N2(α2−α)(s2
2f − σ22f
)σ2
2f
, (59)
N2(α2−α)σ22f
(v2
2N − µ22v
)σ2
1fµ21v +N2(α2−α)σ2
2fµ22v
=µ2
2vσ22f
σ21fµ
21v +N2(α2−α)σ2
2fµ22v
N2(α2−α)(v2
2N − µ22v
)µ2
2v
. (60)
But, then it is obvious that the Lemma holds since (59) and (60) are op (1) , when multiplied by min(√
T ,√Nα)
respectively, as well as min(√
T ,√Nα)Nα2−αµ1vµ2vs12f .
38
Proof of Lemma 13
We analyse the population counterpart of ln (v′NDNSffDN vN ) assuming for simplicity that Σff is diagonal and α >α2 ≥ α3 ≥ ... ≥ αm. We have
ln(µ′vDNΣffDNµv) = ln
(µ2
1vσ21f +N2(α2−α)
m∑j=2
N2(αj−α2)µ2jvσ
2jf
).
Then,
ln(µ′vDNΣffDNµv)−ln(µ2
1vσ21f
)= ln
1 +N2(α2−α)∑m
j=2
(N2(αj−α2)µ2
jvσ2jf
)µ2
1vσ21f
=
∑mj=2
(N2(αj−α2)µ2
jvσ2jf
)µ2
1vσ21f
N2(α2−α).
So √min(Nα, T ) ln (N)
(ln(v′NDNSffDN vN )− ln(µ′vDNΣffDNµv)
)=
√min(Nα, T ) ln (N)
(ln(v′NDNSffDN vN )− ln(µ2
1vσ21f ))−√
min(Nα, T ) ln (N)N2(α2−α)
∑mj=2
(N2(αj−α2)µ2
jvσ2jf
)µ2
1vσ21f
.
We need
N2(α2−α)
∑mj=2
(N2(αj−α2)µ2
jvσ2jf
)µ2
1vσ21f
= o(
min (Nα, T )−1/2 ln (N)−1).
This holds if√
min(Na, T )N2(α2−α) = o(1). If T < Nα then a sufficient condition for the above to hold is α2−α < −0.25.
Otherwise, the sufficient condition is α2 < 3α/4. But, this condition is implied by α2 − α < −0.25 as long as α ≤ 1.
An alternative condition that relates to the relative rate of growth of N and T is that α2 < 3α/4 and T b = N and
1/(4b) + α2 − α < 0 or b > 14(α−α2)
Proof of Lemma 14
We note that the first part of the Lemma holds if
ln(s2f v
2N )
ln(N)= op (1) . (61)
We have
ln(s2f v
2N ) = ln
(s2f
)+ 2 ln (vN ) = ln
(s2f
)+ 2 ln
(1
N
N∑i=1
vi + cN
).
So (61) holds if 1N
∑Ni=1 vi + cN = op (Nc) for all c > 0, which holds if cN = op (Nc) for all c > 0, proving
the first part of the Lemma. For the second part of the Lemma we reconsider (54). We have√N (vN − µv) =
√N(
1N
∑Ni=1 vi + cN − µv
). But,
√N(
1N
∑Ni=1 vi − µv
)→d N
(0, σ2
v
). Therefore,
√NcN = op (1) is sufficient for
the second part of the Lemma to hold.
Appendix IV: Proof of consistency of joint estimation of α and κ
Without loss of generality, we consider a simplified version of the estimation problem. For simplicity, we assume vi = 0for n > [Nα]. By the first part of the proof of Lemma 9, we can disregard the matter of misclassification, when orderingthe estimated factor loadings. We start by noting that
σ2xn =
1
T
T∑t=1
(xnt −
1
T
T∑t=1
xnt
)2
=1
T
T∑t=1
x2nt − x2
n,
where
xnt = βnft + unt, xn =1
T
T∑t=1
xnt = βnf + un.
We have that
39
σ2xn = β2
ns2f + 2βn
[1
T
T∑t=1
(ft − f
)unt
]+
[1
T
T∑t=1
u2nt − u2
n
],
or
σ2xn =
κ2 +(v2ns
2f − κ2
)+ 2βn
[1T
∑Tt=1
(ft − f
)unt]
+[
1T
∑Tt=1 u
2nt − u2
n
], if n ≤ [Nα]
[Nα]nκ2 + [Nα]
n
(v2
[Nα]s2f − κ2
)+ 2βn
[1T
∑Tt=1
(ft − f
)unt]
+[
1T
∑Tt=1 u
2nt − u2
n
], if n > [Nα]
,
since
βn =1
n
n∑i=1
vi =
{vn, if n ≤ [Nα]
[Nα]nv[Nα], if n > [Nα]
,
since vi = 0 for n > [Nα]. Let
ξn =
(v2ns
2f − κ2
)+ 2βn
[1T
∑Tt=1
(ft − f
)unt]
+[
1T
∑Tt=1 u
2nt − u2
n
], if n ≤ [Nα]
[Nα]n
(v2
[Nα]s2f − κ2
)+ 2βn
[1T
∑Tt=1
(ft − f
)unt]
+[
1T
∑Tt=1 u
2nt − u2
n
], if n > [Nα]
.
Then,
σ2xn =
{κ2 + ξn, if n ≤ [Nα]
[Nα]nκ2 + ξn, if n > [Nα]
. (62)
But 1T
∑Tt=1
(ft − f
)unt = Op
(n−1/2T−1/2
), uniformly over n; 1
T
∑Tt=1 u
2nt = Op
(n−1
), uniformly over n; u2
n =
Op(n−1
), uniformly over n; v2
ns2f−κ2 = Op
(min
(n−1/2, T−1/2
)), uniformly over n.; and v2
[Nα]s2f−κ2 = Op
(min
(N−α/2, T−1/2
)).
Therefore,ξn = op (1) , uniformly over n. (63)
We estimate ( 62) by NLLS. Define
ξn =
{σ2xn − κ2, if n ≤ [Nα]
σ2xn −
[Nα]nκ2, if n > [Nα]
,
where κ2and α are the NLLS estimators of κ2 and α, and
dn = ξn − ξn =
{κ2 − κ2, if n ≤ [Nα]
[Nα]nκ2 − [Nα]
nκ2, if n > [Nα]
.
So,
1
N
N∑n=1
ξ2n =
1
N
N∑n=1
ξ2n +
1
N
N∑n=1
d2n +
1
N
N∑n=1
ξndn.
By the definition of the NLLS estimator
limn,T→∞
Pr
(1
N
N∑n=1
ξ2n ≤
1
N
N∑n=1
ξ2n
)= 1. (64)
If eitherp limn,T→∞
κ2 − κ2 6= 0 (65)
orp limn,T→∞
α− α 6= 0, (66)
then 1N
∑Nn=1 d
2n = Op (1). Further, by (63),
1
N
N∑n=1
ξndn = op (1) .
Therefore, if either p limn→∞ κ2 − κ2 6= 0 or p limn→∞ α− α 6= 0, then
limn,T→∞
Pr
(1
N
N∑n=1
ξ2n −
1
N
N∑n=1
ξ2n − C > 0
)> ε,
for some C > 0 and ε > 0. But this contradicts (64). Therefore, neither (65) nor (66) can hold, proving consistency.
40
Tab
leA
1:
Bia
s,R
MS
E,
size
and
pow
er(×
100)
for
theα
esti
mat
eof
the
cros
s-se
ctio
nal
exp
onen
t-
case
ofa
seri
ally
un
corr
elate
dfa
ctor
an
dcr
oss
-sec
tion
all
yin
dep
end
ent
idio
syn
crati
cer
rors
(β2i
=0,f 1t∼IIDN
(0,1
),uit∼IIDN
(0,σ
2 i),v 1i∼IIDU
(0.5,1.5
),N
=100,2
00,5
00,1
000
an
dT
=100,2
00,5
00)
α0.7
00.7
50.8
00.8
50.9
00.9
51.0
0α
0.7
00.7
50.8
00.8
50.9
00.9
51.0
0
N\T
100
N\T
100
100
Bia
s0.9
90.6
30.4
50.3
50.2
50.1
6-0
.07
100
Siz
e11.1
08.2
56.4
06.1
05.6
55.7
57.0
5R
MSE
2.4
82.2
12.0
41.9
21.8
31.7
71.7
2P
ow
er+
52.7
561.3
066.5
070.4
575.0
078.3
584.5
0P
ow
er-
84.7
083.6
084.2
085.4
085.6
586.3
584.5
0200
Bia
s0.5
10.3
10.1
90.1
00.0
60.0
2-0
.09
200
Siz
e9.1
57.5
56.1
05.7
05.0
05.3
06.4
5R
MSE
1.9
01.7
31.6
21.5
41.4
81.4
51.4
3P
ow
er+
76.1
082.3
086.4
090.0
092.1
093.6
593.2
5P
ow
er-
91.5
092.2
093.1
593.1
593.8
594.1
093.2
5500
Bia
s0.1
40.0
4-0
.00
-0.0
3-0
.05
-0.0
7-0
.11
500
Siz
e9.6
08.0
57.6
56.9
56.8
56.8
07.1
0R
MSE
1.4
41.3
51.3
01.2
51.2
31.2
21.2
1P
ow
er+
94.5
096.7
597.6
598.4
598.9
099.4
098.4
0P
ow
er-
96.2
597.2
097.6
597.9
598.2
598.3
598.4
01000
Bia
s0.1
00.0
40.0
10.0
0-0
.01
-0.0
1-0
.03
1000
Siz
e7.8
56.4
55.7
05.6
55.2
55.3
55.5
5R
MSE
1.2
01.1
31.0
91.0
61.0
41.0
31.0
3P
ow
er+
98.9
599.4
099.5
599.7
599.7
099.7
599.8
5P
ow
er-
99.2
099.3
599.5
599.6
599.8
599.8
599.8
5
200
200
100
Bia
s1.0
20.6
70.5
00.3
70.2
70.1
8-0
.08
100
Siz
e9.0
06.7
55.1
54.4
54.2
53.9
05.5
5R
MSE
2.0
21.7
81.6
21.5
11.4
11.3
31.2
8P
ow
er+
65.1
576.0
082.8
086.9
090.0
093.4
598.0
0P
ow
er-
95.3
595.7
096.1
596.9
097.6
098.2
098.0
0200
Bia
s0.5
80.3
70.2
60.1
90.1
30.0
9-0
.02
200
Siz
e8.8
56.8
55.7
55.1
05.0
55.1
55.9
5R
MSE
1.5
21.3
61.2
51.1
71.1
11.0
71.0
4P
ow
er+
89.5
593.7
096.2
098.2
098.8
599.6
599.8
5P
ow
er-
99.0
099.4
099.5
099.6
599.8
599.8
599.8
5500
Bia
s0.1
80.0
80.0
40.0
2-0
.00
-0.0
1-0
.05
500
Siz
e7.0
56.0
55.8
55.7
55.8
55.7
06.2
5R
MSE
1.0
70.9
90.9
40.9
00.8
70.8
50.8
4P
ow
er+
99.6
099.9
5100.0
099.9
5100.0
0100.0
0100.0
0P
ow
er-
99.9
099.9
0100.0
0100.0
0100.0
0100.0
0100.0
01000
Bia
s0.0
90.0
3-0
.00
-0.0
2-0
.03
-0.0
4-0
.06
1000
Siz
e7.3
06.6
55.6
05.6
55.5
05.5
56.0
5R
MSE
0.9
00.8
40.8
00.7
80.7
60.7
60.7
5P
ow
er+
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0P
ow
er-
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0
500
500
100
Bia
s1.0
80.7
20.5
20.4
00.3
10.2
3-0
.03
100
Siz
e9.8
06.0
54.7
04.1
53.3
53.3
05.2
5R
MSE
1.7
71.4
81.3
11.1
91.0
91.0
10.9
3P
ow
er+
78.0
087.9
593.7
096.6
598.6
099.5
099.9
5P
ow
er-
99.3
599.4
099.6
599.6
599.8
099.9
599.9
5200
Bia
s0.5
40.3
50.2
30.1
60.1
20.0
8-0
.03
200
Siz
e8.0
55.8
04.6
04.6
54.6
04.6
05.7
0R
MSE
1.2
11.0
60.9
60.8
80.8
20.7
70.7
3P
ow
er+
98.3
599.5
099.7
599.9
5100.0
0100.0
0100.0
0P
ow
er-
99.9
5100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0500
Bia
s0.2
50.1
60.1
00.0
70.0
50.0
4-0
.00
500
Siz
e6.6
05.2
54.9
04.4
54.0
54.2
54.8
5R
MSE
0.8
00.7
20.6
60.6
10.5
80.5
60.5
5P
ow
er+
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0P
ow
er-
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
01000
Bia
s0.1
20.0
60.0
40.0
20.0
10.0
0-0
.02
1000
Siz
e6.8
05.6
55.6
55.0
05.3
05.3
05.4
5R
MSE
0.6
40.5
80.5
40.5
10.4
90.4
80.4
8P
ow
er+
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0P
ow
er-
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0
41
Tab
leA
2:
Bia
s,R
MS
E,
size
and
pow
er(×
100)
for
theα
esti
mat
eof
the
cros
s-se
ctio
nal
exp
onen
t-
case
ofa
seri
ally
un
corr
elate
dfa
ctor
an
dcr
oss
-sec
tion
all
yin
dep
end
ent
idio
syn
crati
cer
rors
(β2i
=0,f 1t∼IIDN
(0,1
),uit∼IIDN
(0,σ
2 i),v 1i∼IIDU
(0.5,1.5
),N
=100,2
00,5
00,1
000
an
dT
=100,2
00,5
00)
α0.7
00.7
50.8
00.8
50.9
00.9
51.0
0α
0.7
00.7
50.8
00.8
50.9
00.9
51.0
0
N\T
100
N\T
100
100
Bia
s0.3
80.3
30.3
20.2
90.2
20.1
5-0
.08
100
Siz
e11.9
09.5
07.0
06.2
05.7
55.8
07.1
0R
MS
E2.5
12.2
42.0
51.9
31.8
41.7
71.7
3P
ow
er+
61.6
565.5
568.2
071.0
575.4
078.5
084.5
0P
ow
er-
75.6
078.3
581.2
584.2
585.3
586.1
584.5
0200
Bia
s0.1
60.1
70.1
30.0
80.0
50.0
2-0
.09
200
Siz
e10.5
07.7
06.5
05.9
05.0
55.3
06.4
5R
MS
E1.9
51.7
51.6
31.5
41.4
91.4
51.4
3P
ow
er+
80.2
083.6
087.1
090.1
592.1
093.6
593.2
5P
ow
er-
87.4
590.4
592.4
593.0
593.6
594.1
093.2
5500
Bia
s-0
.03
-0.0
2-0
.02
-0.0
3-0
.05
-0.0
7-0
.11
500
Siz
e10.6
08.4
57.7
56.8
56.9
06.8
07.1
0R
MS
E1.4
91.3
71.3
01.2
61.2
41.2
21.2
1P
ow
er+
95.3
096.8
097.7
098.4
598.9
099.4
098.4
0P
ow
er-
94.4
096.6
597.6
097.9
098.2
598.3
598.4
01000
Bia
s0.0
10.0
10.0
1-0
.00
-0.0
1-0
.01
-0.0
31000
Siz
e8.0
56.6
55.8
05.6
55.2
55.3
55.5
5R
MS
E1.2
31.1
31.0
91.0
61.0
41.0
31.0
3P
ow
er+
99.1
099.4
099.6
099.7
599.7
099.7
599.8
5P
ow
er-
98.8
599.3
099.5
599.6
599.8
599.8
599.8
5
200
200
100
Bia
s0.4
20.3
80.3
60.3
10.2
40.1
7-0
.09
100
Siz
e8.9
06.9
05.5
54.5
54.4
03.9
05.5
5R
MS
E1.9
51.7
61.6
21.5
11.4
11.3
31.2
8P
ow
er+
73.9
079.8
084.2
087.5
590.4
093.5
097.9
5P
ow
er-
89.0
592.8
594.9
096.5
597.4
598.1
097.9
5200
Bia
s0.2
30.2
30.2
00.1
70.1
30.0
9-0
.02
200
Siz
e10.0
07.6
06.3
05.4
05.1
05.1
05.9
5R
MS
E1.5
11.3
61.2
61.1
71.1
21.0
71.0
4P
ow
er+
92.3
594.6
596.3
598.2
098.8
599.6
599.8
5P
ow
er-
97.0
098.5
599.4
099.6
599.8
599.8
599.8
5500
Bia
s0.0
20.0
20.0
20.0
1-0
.00
-0.0
2-0
.05
500
Siz
e8.4
56.4
55.8
05.8
05.8
05.7
06.2
5R
MS
E1.0
91.0
00.9
40.9
00.8
70.8
50.8
4P
ow
er+
99.6
5100.0
0100.0
099.9
5100.0
0100.0
0100.0
0P
ow
er-
99.7
099.9
0100.0
0100.0
0100.0
0100.0
0100.0
01000
Bia
s0.0
00.0
0-0
.01
-0.0
2-0
.03
-0.0
4-0
.06
1000
Siz
e8.1
56.7
05.6
55.7
05.5
05.6
06.0
5R
MS
E0.9
10.8
50.8
00.7
80.7
60.7
60.7
5P
ow
er+
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0P
ow
er-
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0
500
500
100
Bia
s0.4
90.4
40.3
90.3
40.2
90.2
2-0
.03
100
Siz
e7.5
55.3
54.7
54.2
53.5
03.3
55.3
0R
MS
E1.6
01.4
11.2
81.1
91.0
91.0
10.9
3P
ow
er+
86.0
591.0
094.8
097.0
098.6
599.5
099.9
5P
ow
er-
97.3
598.7
099.5
099.6
599.8
099.9
599.9
5200
Bia
s0.1
90.2
10.1
80.1
40.1
10.0
8-0
.03
200
Siz
e7.8
55.9
54.9
04.6
54.6
04.6
05.7
0R
MS
E1.1
71.0
40.9
50.8
80.8
20.7
70.7
3P
ow
er+
98.8
099.6
099.8
099.9
5100.0
0100.0
0100.0
0P
ow
er-
99.7
599.9
0100.0
0100.0
0100.0
0100.0
0100.0
0500
Bia
s0.0
90.1
00.0
80.0
70.0
50.0
4-0
.00
500
Siz
e7.0
05.0
54.9
04.4
04.0
54.2
54.8
5R
MS
E0.8
00.7
20.6
60.6
10.5
80.5
60.5
5P
ow
er+
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0P
ow
er-
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
01000
Bia
s0.0
30.0
40.0
30.0
20.0
10.0
0-0
.02
1000
Siz
e7.2
05.9
55.6
55.1
05.3
05.3
05.4
5R
MS
E0.6
40.5
80.5
40.5
10.4
90.4
80.4
8P
ow
er+
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0P
ow
er-
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0
42
Tab
leB
:B
ias,
RM
SE
,si
zean
dp
ower
(×10
0)fo
rth
eα
esti
mat
eof
the
cros
s-se
ctio
nal
exp
onen
t-
case
ofa
seri
ally
corr
elate
dfa
ctor
an
dcr
oss
-sec
tion
ally
ind
epen
den
tid
iosy
ncr
ati
cer
rors
(β2i
=0,ρ1
=0.
5,ζ 1t∼IIDN
(0,1
),uit∼IIDN
(0,σ
2 i),v 1i∼IIDU
(0.5,1.5
),N
=100,2
00,5
00,1
000
an
dT
=100,2
00,5
00)
α0.7
00.7
50.8
00.8
50.9
00.9
51.0
0α
0.7
00.7
50.8
00.8
50.9
00.9
51.0
0
N\T
100
N\T
100
100
Bia
s0.6
60.2
80.0
9-0
.04
-0.1
5-0
.23
-0.4
8100
Siz
e11.9
09.7
59.2
58.3
58.9
010.2
512.1
5R
MS
E2.5
62.3
92.2
82.2
12.1
62.1
22.1
4P
ow
er+
55.1
560.7
564.9
567.2
069.7
072.2
567.1
5P
ow
er-
76.2
071.8
070.7
071.0
570.2
070.9
567.1
5200
Bia
s0.2
50.0
6-0
.05
-0.1
1-0
.16
-0.2
0-0
.31
200
Siz
e11.7
010.6
510.3
510.9
011.1
011.1
011.8
0R
MSE
2.1
11.9
91.9
11.8
61.8
21.7
91.7
8P
ow
er+
72.7
576.5
079.3
081.0
082.8
083.9
081.7
5P
ow
er-
84.1
583.4
083.5
583.2
583.4
583.4
081.7
5500
Bia
s-0
.07
-0.1
7-0
.23
-0.2
6-0
.28
-0.2
9-0
.34
500
Siz
e12.3
011.8
011.7
011.6
511.8
011.9
012.4
0R
MSE
1.6
61.5
91.5
61.5
31.5
11.5
01.5
0P
ow
er+
89.4
090.8
091.8
592.1
092.5
592.9
591.5
5P
ow
er-
91.1
590.6
090.9
090.8
091.2
591.5
591.5
51000
Bia
s-0
.14
-0.1
9-0
.21
-0.2
2-0
.22
-0.2
3-0
.25
1000
Siz
e13.6
512.6
012.7
012.6
512.8
512.6
013.3
5R
MSE
1.4
81.4
31.3
91.3
81.3
61.3
51.3
5P
ow
er+
95.3
095.5
096.2
596.3
596.6
096.9
596.3
0P
ow
er-
95.3
595.8
095.9
596.0
596.1
096.0
596.3
0
200
200
100
Bia
s0.9
10.5
30.3
40.2
20.1
20.0
2-0
.23
100
Siz
e10.7
58.2
07.2
06.1
05.5
06.4
07.8
0R
MSE
2.1
71.9
51.8
21.7
11.6
41.5
91.5
6P
ow
er+
63.1
070.5
575.9
080.8
084.2
587.0
089.9
0P
ow
er-
92.1
090.1
590.3
590.8
591.5
091.1
089.9
0200
Bia
s0.3
70.1
70.0
5-0
.03
-0.0
6-0
.10
-0.2
0200
Siz
e8.9
07.7
07.0
56.7
58.2
08.2
58.7
5R
MSE
1.6
01.4
91.4
01.3
51.3
21.2
91.2
7P
ow
er+
86.7
091.0
593.0
594.0
595.1
596.2
597.9
5P
ow
er-
96.7
596.7
596.9
597.7
097.8
098.2
097.9
5500
Bia
s0.1
20.0
1-0
.04
-0.0
6-0
.09
-0.1
0-0
.15
500
Siz
e9.1
08.4
58.4
08.6
08.5
08.7
09.1
5R
MSE
1.2
31.1
71.1
31.1
11.1
01.0
91.0
9P
ow
er+
98.0
598.7
599.0
099.3
599.2
599.3
599.5
5P
ow
er-
99.6
599.7
099.7
599.6
599.7
099.5
599.5
51000
Bia
s0.0
4-0
.02
-0.0
5-0
.07
-0.0
8-0
.09
-0.1
11000
Siz
e8.9
57.6
07.7
07.5
57.9
07.7
08.2
5R
MSE
1.0
30.9
80.9
60.9
50.9
40.9
30.9
3P
ow
er+
99.4
599.5
599.8
099.8
099.8
099.8
099.9
5P
ow
er-
99.9
0100.0
0100.0
099.9
599.9
599.9
599.9
5
500
500
100
Bia
s1.0
30.6
70.4
90.3
70.2
60.1
8-0
.07
100
Siz
e9.8
07.2
55.9
05.0
04.5
54.7
56.9
0R
MSE
1.8
51.6
01.4
31.3
31.2
41.1
61.1
0P
ow
er+
74.3
084.6
089.6
593.2
595.3
597.3
099.5
0P
ow
er-
98.2
097.9
098.8
599.0
599.3
099.6
599.5
0200
Bia
s0.5
40.3
30.2
00.1
40.0
80.0
4-0
.07
200
Siz
e7.8
06.7
55.7
05.4
55.6
55.4
07.0
0R
MSE
1.2
71.1
31.0
40.9
90.9
40.9
10.8
8P
ow
er+
96.5
098.3
099.1
599.7
5100.0
099.9
599.9
5P
ow
er-
99.8
099.9
599.9
599.9
0100.0
0100.0
099.9
5500
Bia
s0.1
90.0
90.0
3-0
.01
-0.0
3-0
.04
-0.0
8500
Siz
e7.7
06.3
56.5
55.8
06.2
56.4
57.4
0R
MSE
0.9
00.8
20.7
80.7
40.7
20.7
10.7
0P
ow
er+
99.9
5100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0P
ow
er-
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
01000
Bia
s0.1
00.0
40.0
1-0
.01
-0.0
2-0
.03
-0.0
51000
Siz
e7.5
56.7
56.0
06.5
06.3
56.7
56.9
5R
MSE
0.7
40.6
90.6
60.6
40.6
30.6
10.6
1P
ow
er+
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0P
ow
er-
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0
43
Tab
leC
:B
ias,
RM
SE
,si
zean
dp
ower
(×10
0)fo
rth
eα
esti
mat
eof
the
cros
s-se
ctio
nal
exp
onen
t-
case
ofa
seri
ally
un
corr
elate
dfa
ctor
an
dcr
oss
-sec
tion
all
yd
epen
den
tid
iosy
ncr
ati
cer
rors
(β2i
=0,f 1t∼IIDN
(0,1
),uit
-θ
=0.
2,v 1i∼IIDU
(0.5,1.5
),N
=100,2
00,5
00,1
000
an
dT
=100,2
00,5
00)
α0.7
00.7
50.8
00.8
50.9
00.9
51.0
0α
0.7
00.7
50.8
00.8
50.9
00.9
51.0
0
N\T
100
N\T
100
100
Bia
s1.4
91.0
30.7
30.5
30.3
70.2
4-0
.04
100
Siz
e10.4
07.1
06.0
05.2
05.4
55.2
56.9
5R
MSE
2.5
52.2
32.0
21.8
91.7
91.7
41.6
9P
ow
er+
44.1
054.5
062.6
569.5
073.7
577.5
585.6
0P
ow
er-
91.4
089.9
088.6
088.7
088.0
587.7
585.6
0200
Bia
s0.7
50.4
60.2
50.1
20.0
4-0
.01
-0.1
4200
Siz
e8.0
55.9
55.5
05.2
54.9
55.3
56.0
5R
MSE
1.8
21.6
41.5
21.4
51.4
01.3
81.3
7P
ow
er+
74.3
581.1
586.7
090.6
593.3
594.3
094.8
0P
ow
er-
94.5
094.6
094.7
594.6
595.3
595.5
094.8
0500
Bia
s0.4
00.2
00.1
00.0
4-0
.00
-0.0
3-0
.07
500
Siz
e6.1
55.7
05.8
55.4
56.5
06.1
56.4
5R
MSE
1.3
71.2
61.2
11.1
81.1
71.1
51.1
4P
ow
er+
94.5
097.4
598.4
098.8
099.2
099.4
098.8
5P
ow
er-
98.7
098.7
098.9
599.0
598.9
098.9
098.8
51000
Bia
s0.2
40.1
10.0
4-0
.01
-0.0
3-0
.04
-0.0
71000
Siz
e7.1
06.7
56.3
06.2
06.3
56.4
06.7
0R
MSE
1.1
91.1
41.1
01.0
81.0
81.0
71.0
6P
ow
er+
98.9
099.3
099.5
599.7
099.7
599.8
099.6
0P
ow
er-
99.4
099.6
599.6
099.5
599.5
599.5
599.6
0
200
200
100
Bia
s1.4
20.9
50.6
60.4
70.3
30.2
0-0
.09
100
Siz
e9.3
56.0
04.4
54.2
53.9
54.1
05.9
5R
MSE
2.1
91.8
41.6
41.5
01.3
91.3
21.2
7P
ow
er+
56.6
570.0
580.1
586.3
091.1
594.1
597.4
0P
ow
er-
97.8
096.9
596.8
597.3
598.1
098.0
097.4
0200
Bia
s0.8
70.5
60.3
60.2
30.1
40.0
8-0
.04
200
Siz
e7.3
05.3
05.1
05.4
05.1
55.1
55.9
5R
MSE
1.5
81.3
81.2
61.1
81.1
21.0
81.0
5P
ow
er+
85.7
092.1
596.4
597.9
598.8
599.5
099.6
5P
ow
er-
99.7
099.6
599.5
599.6
599.6
099.6
599.6
5500
Bia
s0.4
50.2
60.1
60.0
90.0
50.0
2-0
.03
500
Siz
e7.6
55.8
05.2
05.4
05.4
05.1
56.1
0R
MSE
1.1
31.0
00.9
30.8
90.8
70.8
50.8
4P
ow
er+
99.0
599.8
599.9
5100.0
0100.0
0100.0
0100.0
0P
ow
er-
99.9
5100.0
0100.0
0100.0
0100.0
0100.0
0100.0
01000
Bia
s0.2
90.1
50.0
80.0
30.0
1-0
.00
-0.0
21000
Siz
e6.8
06.3
06.2
56.0
06.1
56.2
06.4
5R
MSE
0.9
10.8
40.8
00.7
80.7
70.7
70.7
6P
ow
er+
99.9
5100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0P
ow
er-
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0
500
500
100
Bia
s1.5
91.0
90.8
00.6
00.4
40.3
20.0
4100
Siz
e12.0
57.0
04.9
04.0
53.5
03.8
55.1
5R
MSE
2.0
81.6
71.4
21.2
51.1
31.0
40.9
4P
ow
er+
62.6
579.7
089.5
094.8
097.8
599.2
0100.0
0P
ow
er-
99.9
599.9
099.9
099.9
5100.0
0100.0
0100.0
0200
Bia
s0.9
60.6
60.4
40.3
00.2
20.1
50.0
3200
Siz
e9.0
05.4
55.1
04.2
04.5
53.9
04.9
5R
MSE
1.4
01.1
40.9
80.8
70.8
00.7
40.7
0P
ow
er+
95.8
598.5
099.6
599.9
5100.0
0100.0
0100.0
0P
ow
er-
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0500
Bia
s0.4
60.2
70.1
60.1
00.0
50.0
2-0
.02
500
Siz
e6.8
05.1
55.3
04.9
05.1
04.4
55.0
0R
MSE
0.8
80.7
50.6
80.6
30.6
00.5
70.5
5P
ow
er+
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0P
ow
er-
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
01000
Bia
s0.3
10.1
70.0
90.0
50.0
20.0
1-0
.01
1000
Siz
e6.3
55.4
54.9
04.3
04.1
04.5
04.9
0R
MSE
0.6
80.5
90.5
40.5
10.4
90.4
80.4
6P
ow
er+
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0P
ow
er-
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0
44
Tab
leD
:B
ias,
RM
SE
,si
zean
dp
ower
(×10
0)fo
rth
eα
esti
mat
eof
the
cros
s-se
ctio
nal
exp
onen
t-
case
oftw
ose
rial
lyu
nco
rrel
ate
dfa
ctors
and
cross
-sec
tion
all
yin
dep
end
ent
idio
syn
crati
cer
rors
(α2
=α
,f jt
anduit∼IIDN
(0,1
),v ji∼IIDU
(0.5,1.5
),j
=1,
2,
N=
100,2
00,5
00,1
000
an
dT
=100,2
00,5
00)
α0.7
00.7
50.8
00.8
50.9
00.9
51.0
0α
0.7
00.7
50.8
00.8
50.9
00.9
51.0
0
N\T
100
N\T
100
100
Bia
s1.0
60.8
90.7
20.4
30.3
10.2
2-0
.18
100
Siz
e6.9
06.4
05.7
54.6
05.2
05.4
07.0
0R
MSE
2.3
32.1
52.0
01.8
51.7
71.7
11.6
8P
ow
er+
51.9
557.2
563.4
570.8
575.9
079.2
084.0
0P
ow
er-
88.0
088.7
589.0
087.5
088.2
588.3
084.0
0200
Bia
s0.6
70.4
10.3
00.2
10.1
70.0
9-0
.09
200
Siz
e6.8
05.6
05.3
04.8
05.1
05.0
05.8
0R
MSE
1.8
01.6
51.5
61.5
01.4
61.4
21.4
0P
ow
er+
75.0
081.3
085.4
088.1
591.0
592.4
594.6
5P
ow
er-
94.7
094.4
094.7
594.6
095.1
595.0
594.6
5500
Bia
s0.2
30.1
50.0
60.0
4-0
.01
-0.0
4-0
.11
500
Siz
e7.3
06.5
56.7
06.6
56.5
56.7
07.1
0R
MSE
1.3
81.3
11.2
61.2
21.2
01.1
91.1
8P
ow
er+
95.1
596.6
097.9
098.3
098.9
099.1
098.3
5P
ow
er-
98.2
098.2
598.0
598.3
598.2
598.3
098.3
51000
Bia
s0.1
20.0
5-0
.01
-0.0
2-0
.05
-0.0
5-0
.09
1000
Siz
e6.7
56.7
06.4
56.2
06.3
06.3
06.6
0R
MSE
1.1
61.1
11.0
81.0
61.0
61.0
51.0
5P
ow
er+
99.1
599.4
599.7
599.8
599.8
599.9
099.4
5P
ow
er-
99.4
099.4
599.4
099.5
599.6
099.6
599.4
5
200
200
100
Bia
s1.1
70.9
80.8
30.5
50.4
10.3
2-0
.08
100
Siz
e7.0
05.8
54.4
54.3
04.4
54.4
05.5
0R
MSE
2.0
21.8
31.6
71.5
01.4
01.3
31.2
5P
ow
er+
63.8
070.9
577.1
585.1
089.0
093.4
097.9
0P
ow
er-
97.1
598.1
098.3
598.0
098.4
098.5
597.9
0200
Bia
s0.7
40.4
60.3
40.2
50.2
00.1
3-0
.05
200
Siz
e7.0
05.7
04.6
54.8
54.8
05.2
06.1
0R
MSE
1.5
11.3
31.2
21.1
51.0
91.0
61.0
3P
ow
er+
87.0
593.6
596.7
098.2
598.9
099.4
099.8
0P
ow
er-
99.5
099.4
099.5
599.8
599.8
599.8
599.8
0500
Bia
s0.2
50.1
80.1
00.0
80.0
30.0
1-0
.06
500
Siz
e5.3
05.1
55.1
04.9
55.0
05.8
05.7
5R
MSE
1.0
40.9
70.9
20.8
80.8
60.8
60.8
4P
ow
er+
99.6
099.8
5100.0
0100.0
0100.0
0100.0
0100.0
0P
ow
er-
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
01000
Bia
s0.1
70.1
20.0
60.0
40.0
10.0
1-0
.03
1000
Siz
e6.0
05.7
55.4
06.2
05.7
56.1
06.2
5R
MSE
0.8
70.8
20.7
90.7
70.7
60.7
50.7
5P
ow
er+
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0P
ow
er-
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0
500
500
100
Bia
s1.2
01.0
20.8
60.5
70.4
40.3
7-0
.02
100
Siz
e9.4
07.5
56.5
05.2
04.6
04.5
56.3
5R
MSE
1.8
41.6
41.4
51.2
51.1
21.0
40.9
2P
ow
er+
74.0
083.0
088.9
594.0
597.1
098.9
0100.0
0P
ow
er-
99.3
599.5
599.9
599.9
5100.0
0100.0
0100.0
0200
Bia
s0.7
80.5
10.3
90.2
90.2
50.1
8-0
.01
200
Siz
e8.1
05.4
04.7
04.3
04.3
04.3
05.3
0R
MSE
1.3
21.1
00.9
70.8
80.8
10.7
60.7
1P
ow
er+
95.7
098.8
599.6
599.9
0100.0
0100.0
0100.0
0P
ow
er-
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0500
Bia
s0.3
10.2
40.1
40.1
20.0
70.0
4-0
.02
500
Siz
e5.3
04.2
54.5
54.6
05.0
04.9
55.8
5R
MSE
0.8
10.7
20.6
60.6
20.5
90.5
70.5
6P
ow
er+
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0P
ow
er-
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
01000
Bia
s0.2
00.1
40.0
90.0
70.0
40.0
3-0
.00
1000
Siz
e4.9
54.9
55.1
04.9
55.0
05.1
55.2
0R
MSE
0.6
40.5
80.5
40.5
20.5
10.5
00.4
9P
ow
er+
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0P
ow
er-
100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0100.0
0
45
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