Testing serial correlation in semiparametric panel data models

*Corresponding author. E-mail: [email protected]

Journal of Econometrics 87 (1998) 207—237

Testing serial correlation in semiparametric paneldata models

Q. Li!,*, C. Hsiao"

! Department of Economics, University of Guelph, Guelph, Ontario, Canada N1G 2W1" Department of Economics, University of Southern California, Los Angeles, CA 90089, USA

Received 1 July 1996; received in revised form 1 November 1997

Abstract

We propose three test statistics for testing serial correlation in a semiparametricpartially linear panel data model that could allow lagged dependent variables as explana-tory variables. The first is for testing zero first-order serial correlation, the second fortesting higher-order serial correlations and the third testing for individual effects. The teststatistics are shown to have asymptotic normal or chi-square distributions under the nullhypothesis of a martingale difference error process. We conduct some Monte Carloexperiments to examine the finite sample performances of the proposed tests. We alsodiscuss the generalization to testing serial correlation in a nonparametric framework.( 1998 Elsevier Science S.A. All rights reserved.

JEL classification: C12; C14; C52

Keywords: Partially linear model; Dynamic panel data model; Testing serial correlation;Individual effects; Semiparametric estimation

1. Introduction and motivation

This paper proposes three methods to test serial correlation or the presence ofthe random individual effects in a panel data context when models are onlypartially specified. Testing for serial correlation has long been a standardpractice in applied econometric analysis because if the residuals are serially

0304-4076/98/$— see front matter ( 1998 Elsevier Science S.A. All rights reserved.PII S 0 3 0 4 - 4 0 7 6 ( 9 8 ) 0 0 0 1 3 - X

correlated, not only the least squares estimator is inefficient, it can be inconsist-ent if the regressors contain lagged dependent variables. Moreover, strong serialcorrelation is often an indication of omitting important explanatory variables orfunctional form misspecification. Testing autocorrelation or random individualeffects is important because the choice of an appropriate estimation procedurefor a given panel data model crucially depends on the error structure assumedby the model. Panel data studies often simply assume the presence of randomindividual effects without considering any formal test. For example, a variety of(parametric and nonparametric) GMM estimators have been proposed fordynamic models with random effects or fixed effects. But the estimation of themodels could be considerably simplified if the errors are not autocorrelated.

In a time series context with parametric specification, the conventionalmethod to test for the absence of first-order serial correlation is to first estimatethe model assuming no serial correlation, then use the estimated residual toconstruct a Durbin—Watson (1950) statistic when the regressors are exogenousor Durbin (1970) h-statistic when the regressors contain lagged dependentvariable. To test higher-order serial correlation, a commonly used test is the LMtests of Breusch (1978) and Godfrey (1978). Although the Q-test of Box andPierce (1970) is also a commonly used test for testing higher-order serialcorrelation, the Q-test is not valid when the regressors contain both exogenousand lagged dependent variables. To test for the absence of individual effects ina panel data model, a popular test is the Breusch—Pagan (1980) test. In thispaper, we will generalize the Durbin h-statistic, the Breusch (1978) and Godfrey(1978) LM statistics and the Breusch—Pagan (1980) test to the case ofa semiparametric partially linear panel data model. When a model is onlypartially specified, say the partially linear model considered by Engle et al.(1986), and Robinson (1988), we have to appeal to asymptotic theory to justifythe validity of the conventional testing procedure. This requires the number ofobservations to go to infinity and the serial correlation pattern of the explana-tory variables to satisfy some kind of mixing conditions for time series data.However, panel data typically contains a large number of individuals, it may notbe unreasonable to assume that observations are independently distributedacross individuals. Hence, for panels with large N, less restrictive conditions areneeded and some (standard) form of the law of large numbers and central limittheorems can be applied to observations across individuals. It not only providesthe theoretical basis of deriving the semiparametric tests for serial correlation,but also is empirically feasible.

In Section 2 we set up the basic model. Section 3 proposes a semiparametrictest for the absence of first-order serial correlation and discusses its relation tothe Durbin—Watson and the Durbin h statistics. Section 4 extends the results ofSection 3 to the case of testing higher-order serial correlation. Section 5 pro-vides a test for the absence of individual effects (e.g., Balestra and Nerlove, 1966;Hsiao, 1986). Monte Carlo studies of the size and power of these tests are

208 Q. Li, C. Hsiao / Journal of Econometrics 87 (1998) 207–237

1See Kniesner and Li (1994) for an empirical application of using semiparametric partial linearpanel data model with w

itcontains y

i,t~1. For estimation of a parametric dynamic panel data

models, see the pioneering study of Balestra and Nerlove (1966), and the more recent work byAnderson and Hsiao (1982), Ahn and Schmidt (1995), Arellano and Bover (1995), Keane and Runkle(1992), and Pesaran and Smith (1995).

2 If the data follows a stationary process, then ft( ) )"f( ) ). We do not require stationary assump-

tion in this paper.

provided in Section 6. Finally, Section 7 discusses the generalization to testingserial correlation in a nonparametric framework.

2. The model

We consider the following semiparametric partially linear model:

yit"z@

itc#h(w

it)#u

it(i"1,2,N; t"!¸#1,2,0, 1,2,¹) (1)

with uit

satisfying

E(uitDw

it, z

it)"0, (2)

where zit

is of dimension r]1 and wit

is of dimension q]1. We allow thepossibility that w

itand/or z

itcontain ¸ (¸*0) lagged values of y

it.1 The

functional form of h( ) ) is unknown to the researcher and the functional form ofp2(z

it,w

it)"E(u2

itDzit, w

it) is not specified. We are interested in testing whether the

error uitis serially uncorrelated. The case we consider is the panel data with large

N and small ¹. Hence all the asymptotics in this paper are for NPR fora fixed value of ¹.

In order to test whether uit

is serially uncorrelated, we need to get a consistentestimator of c, which in turn requires some nonparametric estimation tech-niques either to estimate h( ) ) or to eliminate it from the regression equation.Following Robinson (1988), we will use kernel estimation method to estimatesome unknown conditional expectations and then to eliminate h( ) ) beforeestimating c. We will use cross-sectional data (for a fixed value of t) to estimatethe density of w

it, f

t(w

it),2 and E(A

itDw

it). Denote by fK

itand AK

itthe kernel

estimators of ft(w

it) and E(A

itDw

it), respectively. Then

fKit"

1

Naq+jEi

Kit,jt

, (3)

and

AKit"

1

Naq+jEi

AjtK

it, jt/fKit, (4)

Q. Li, C. Hsiao / Journal of Econometrics 87 (1998) 207–237 209

where

Kit, jt

"KAwit!w

jta B

is the kernel function and a is the smoothing parameter. We use product kernelK(w

it)"<q

l/1k(w

it,l), where k is the univariate kernel and w

it,lis the lth compon-

ent of wit. In this paper A

itwill be y

it, z

it, u

it, h

it,h(w

it), m

it"E(z

itDw

it) and

vit,z

it!m

it. Also we use the short-hand notations +

jEi,+

ifor +N

j/1, jEiand

+Ni/1

. Similarly we will also use +tand +

sEtto denote +T

t/1and +T

s/1, sEt.

Applying the linear operator (Naq)~1+iK

it, jt/fKjt

to Eq. (1) and interchangingi with j, we get

yLit"zL @

itc#hK (w

it)#uL

it, (5)

where yLit, zL

it, hK (w

it) and uL

itare all defined as the right-hand-side of Eq. (4) with

Ajt

replaced by yjt, z

jt, h(w

jt) and u

jt, respectively. They are the estimated condi-

tional expectations of yit, z

it, h(w

it) and u

itgiven w"w

it. For example,

yLit"(Naq)~1+

jEiyjtK

it, jt/fKit

is a nonparametric kernel estimator of E(yitDw

it)

based on (yjt, w

jt)Nj/1

, and uLit"(Naq)~1+

jEiujtK

it, jt/fKit

is the infeasible kernelestimator of E(u

itDw

it)"0 based on (u

jt, w

jt)Nj/1

. Subtracting Eq. (5) from Eq. (1)leads to

yit!yL

it"(z

it!zL

it)@c#(h

it!hK

it)#u

it!uL

it. (6)

In Eq. (6), hit!hK

it"o

1(1) and uL

it"o

1(1). Hence, regressing y

it!yL

iton

zit!zL

itwill lead to consistent (in fact JN-consistent) estimator of c. The

random denominator fKit

in the kernel estimation will, when fKit

is too small, cause

technical difficulties in proving JN-consistency of an estimator of c based onEq. (6). In the nonparametric kernel estimation literature, two methods are oftenused to avoid this difficulty. One method is to use density-weighted approach toget rid of the random denominator (e.g., Powell et al., 1989) and the othermethod is to trim out small values of fK

it(e.g., Robinson, 1988). The advantage of

the former method is that no trimming parameter (a nuisance parameter) isneeded in the estimation. The disadvantage is that, the proofs usually are moretedious than those based on trimming methods. In this paper we choose to usethe density-weighted approach to avoid the issue of using random denominatorin the kernel estimation.

We propose to estimate c by

cL"S~1(z~zL )fK

S(z~zL )fK , (y~yL )fK

"c#S~1(z~zL )fK

S(z~zL )fK , *u~uL `(h~hK )+fK , (7)

where we use notations similar to Robinson (1988), for scalar or column-vectorsequences A

itand B

it, S

AfK , BfK"(1/n)+

i+

tA

itfKit

B@it

fKit

and SAfK

"SAfK , AfK

withn,N¹.


To derive the asymptotic distribution of cL , we first need some regularityassumptions. We use the definition of Robinson (1988) for the class of kernelfunctions K

l: RPR, for l*1 and the class of Gal, some a'0 and l'0. These

notations represent that, for a positive integer l, if g3Gal, then g is l orderdifferentiable, g and its derivatives (up to the order l) are all bounded by somefunctions that have finite ath moment. Let y

idenote the 1]¹ row vector

(yi1,2,y

iT), and z

iand w

iare similarly defined. Also define m

it"E(z

itDw

it) and

vit"z

it!m

it. We assume that

(A1) (i) (yi, z

i, w

i), i"1, 2,2,N, are independently and identically distributed

(i.i.d.) as (y, z, w). (ii) For all t, w1t

admits a density functionft( ) )3G=l~1

, h( ) )3G4l , and m( ) )3G4l , where l*2 is an integer. (iii)E(u

1tDz1t, w

1t)"0, both u

1tand v

1thave finite (4#d)th moments for

some d'0. E(u21tDz1t,w

1t)"p2(z

1t, w

1t) is continuous in w

1t.

(A2) As NPR, and namax(2q~4, q)PR, na2lP0; k3Kl, (A2)@ As NPR,namax(2q~4, q)PR, na4lP0; k3Kl.

Condition (A1)(ii) requires that ft( ) ), h( ) ) and m( ) ) all satisfy some moment

conditions and are differentiable up to order l. The condition ensures that allthese functions admit Taylor expansions up to order l. Together with k3Kl (alth order kernel), they guarantee that the bias in the kernel nonparametricestimation is of the order O(av) (e.g., E(hK (z))!h(z)"O(av)).

We will derive the limiting distributions of our test statistics (which will begiven in the next three sections) under the null hypothesis

H0: The error u

itfollows a martingale difference process.

The next lemma establishes the asymptotic distribution of cL .

¸emma 1. ºnder (A1), (A2)@ and H0, we have

(i) Jn(cL!c)"B~1(1/Jn)+i+

tvituitf2it#o

1(1)

$P N(0,R), where R"B~1AB~1,

A"(1/¹)+tE[v

1tv@1tu21t

f41t] and B"(1/¹)+

tE[v

1tv@1t

f21t].

(ii) RK "BK ~1AK BK ~1 is a consistent estimator for R, where AK "(1/n)+

i+

tuJ 2it(z

it!zL

it)(z

it!zL

it)@fK 4

it, BK "(1/n)+

i+

t(z

it!zL

it)(z

it!zL

it)@fK 2

itand uJ

it"y

it!yL

it!(z

it!zL

it)@cL .

Proof. The proof follows exactly the same steps as the proof of Theorem 1 of Li(1996), with the results of Lemma A.3(i)—(iv) of this paper replacing the Lemmas2—5 of Li (1996) in the proof.

Note that the conditions of Lemma 1 requires l'maxM(q/2)!1, q/4N. Hencea second-order kernel (l"2) can be used for q)5. Li and Stengos (1996)

considered the problem of JN-consistent estimation of c under stronger condi-tions than our (A2)@.


3. Testing zero first-order serial correlation

In this section, we suggest a test statistic for testing the absence of first-orderserial correlation. Under the null hypothesis of a martingale difference error, wehave o"E(u

itui,t~1

)"0 and of"E(u

itfitui,t~1

fi,t~1

)"0. In principle, a teststatistic for testing first-order serial correlation can be constructed based oneither of the sample analogue of o or o

f. However, since we consider

a semiparametric model and we estimate uit

using nonparametric kernelmethod, it is easier to construct a test statistic based on sample analogue ofof

rather than o because the former avoids the technical difficulty associatedwith the use of the random denominator in the kernel estimation. Therefore weshall use o

fas the basis to construct our test statistic.

The error uit

of the semiparametric model (1) is estimated by

uJit"y

it!yL

it!(z

it!zL

it)@cL"y

it!yL

it!(z

it!zL

it)@c!(z

it!zL

it)@(cL!c)

"uit!uL

it#(h

it!hK

it)!(z

it!zL

it)@(cL!c). (8)

From Eq. (8), it is easy to see that uJit"u

it#o

1(1) because (cL!c)"

O1(N~1@2), h

it!hK

it"o

1(1) and uL

it"o

1(1) (recall that uL

itis the kernel estimator

of E(uitDw

it)"0). Hence our test statistic for testing zero first-order serial correla-

tion is given by

In"JnS

uJ fK , uJ ~1fK ~1"

1

Jn+i

+t

uJituJi,t~1

fKit

fKi,t~1

. (9)

¹heorem 1. ºnder (A1), (A2) and H0,

(i) InP$ N(0, p2

0), where p2

0"(1/¹)+

tE[M(u

1,t~1f1,t~1

!v@1t

f1tB~1U)u

1tf1tN2]

with U"E[v1t

f1tu1,t~1

f1,t~1

].(ii) pL 2

0"(1/n)+

i+

tM[uJ

i,t~1fKi,t~1

!(zit!zL

it)@fK

itBK ~1UK ]uJ

itfKitN2 with UK "(1/n)+

i+

t(z

it!zL

it) fK

ituJi,t~1

fKi,t~1

, is a consistent estimator for p20. Hence

Sn,I

n/pL

0P$ N(0,1).

The proof of Theorem 1(i) and (ii) are given by Propositions A.8 and A.9 in theAppendix.

Note that if both wit

and zit

are strictly exogenous variables, then U"0. Theresult of Theorem 1 can be simplified. Moreover, condition (A2) can be replacedby (A2)@. Hence from Theorem 1, we immediately have the following corollary.

Corollary 1. ºnder (A1), (A2)@ and H0, if in addition, E(u

1tDw

1s, z

1s,

w1s{

, z1s{

)"0 for all t, s and s@. ¹hen

(i) In

$P N(0, p2

1), where p2

1"(1/¹)+

tE[(u

1,t~1f1,t~1

u1t

f1t)2].


(ii) pL 21"(1/n)+

i+

t[uJ

i,t~1fKi,t~1

uJit

fKit]2 is a consistent estimator for p2

1. Hence

Sn"I

n/pL

1$

P N(0,1).

Proof. The proof follows the same steps as in the proof of Theorem 1 with U"0since E(u

i,t~1Dw

it, w

i,t~1, z

it)"0. Moreover, note that when both w

itand z

itare

strictly exogenous variables, the result of Lemma A.6(a) in the appendix can besharpened to S

(h~hK )fK ,u~1fK ~1"O

1(a2l#N~1@2al@2). The proof of this result fol-

lows exactly the same steps as in the proof of Lemma A.3(a). This is the reasonwhy we can replace (A2) by a weaker condition (A2)@ in Corollary 1.

Remark 3.1. The conditions of Theorem 1 are different from that of Corollary 1.A stronger condition Na2lP0 in (A2) is used to prove Theorem 1. This isbecause in order to prove Theorem 1, we need to show that, among other things,S(h~hK )fK ,uL ~1fK ~1

"o1(N~1@2). From Lemma 6(a), we know that S

(h~hK )fK , u~1fK ~1"

O1(al#N~1@2al@2). For this to be o

1(N~1@2), we need the condition Na2lP0.

Note that (A2) implies that a second-order kernel can be used if q)3. To seethis, note that (A2) requires that l'max(q!2, q/2). When l"2 (a second-order kernel), this gives q(4 or q)3 because q is a positive integer.

Remark 3.2. The results of Theorem 1 can be easily understood by noting thatthe asymptotic null distribution of I

nis identical to the case that E(y

itDw

it) and

E(zitDw

it) are known. To see this, we get from Eq. (1),

yit!E(y

itDw

it)"(z

it!E(z

itDw

it))@c#u

it,v@

itc#u

it. (10)

Least squares estimator of c based on Eq. (10), say cJ , is obviously JN-consis-tent.

Jn(cJ!c)"C1

n+i

+t

vitv@itD

~1 1

Jn+

i+t

vituit

"[BI ]~11

Jn+i

+t

vituit#o

1(1), (11)

where BI "(1/¹)+tE[v

1tv@1t]. Then using uN

it"u

it!z@

it(cJ!c), we get

IIn,

1

Jn+i

+t

uNituNi,t~1

"

1

Jn+i

+t

Muitui,t~1

!(cJ!c)@vitui,t~1

!uitvi,t~1

(cJ!c)

#(cJ!c)@vitv@i,t~1

(cJ!c)N

,C1n!C

2n(cJ!c)!C

3n(cJ!c)#(cJ!c)@C

4n(cJ!c). (12)


It is easy to see that C3n"O

1(1) (because E(u

itDzi,t~1

,wi,t~1

)"0) and C4n"

O1(N1@2). Also (1/Jn)C

2n"(1/n)+

i+

tvitui,t~1

1P (1/¹)+

tE[v

1tu1,t~1

],UI by

a law of large number. Hence

IIn"

1

Jn+i

+t

Muitui,t~1

!Jn(cL!c)@UI #o1(1)#MO

1(N~1@2)#O

1(N~1@2)N

"

1

Jn+i

+t

uit[u

i,t~1!v@

itBI ~1UI ]#o

1(1)

$P N(0, pJ 2

0),

where the second equality follows from (Eq. (11)), pJ 20"(1/¹)+

tE

M[(u1,t~1

!v@1tBI ~1UI )u

1t]2N, which is the same as our Theorem 1 if one adds the

density weight factors.

Remark 3.3. When w and z are strictly exogenous, the Intest using pL

1(Corollary

1) is analogous to the Durbin—Watson (1950) test. When zit

contains lagged yit,

say yi,t~1

, the Intest using pL

0is analogous to the Durbin (1970) h-statistic. This

can be seen by noting that in (Eq. (12)), the only term in (1/Jn)C2n

that does notconverge to zero is the term (1/n)+

i+

tui,t~1

yi,t~1

. Therefore, the variance ofInwill have to be adjusted by the square of this term multiplied by n times the

variance of the coefficient estimator of yi,t~1

.

4. Testing higher-order serial correlation

We show in this section that the results of Section 3 can be easily generalizedto the case of testing higher-order serial correlation. Denoteos"E(u

itfitui,t~s

fi,t~s

) (s"1,2,2,p; p(¹). Obviously, os"0 under H

0. We

are interested in jointly testing: o1"o

2"2"o

p"0. Denotes

oLs"(1/n)+

i+

tuJituJi,t~s

fKit

fKi,t~s

(n"N(¹!p)) and oL "(oL1,2,oL

p)@. Let

¸n,p

"JnoL .

The next corollary gives the asymptotic distribution of ¸n,p

under H0.

Corollary 2. ºnder the same conditions as given in ¹heorem 1, we have

1. Jn¸n,p

$P N(0, X), where X is a p]p matrix with its (l, s)th element given by

Xls"(1/(¹!p)+T

t/.!9Ml,sNE[u2

1tf21t(u

1,t~lf1,t~l

!v@1t

f1tB~1U

l)(u

1,t~sf1,t~s

!

v@1t

f1tB~1U

s)], and U

s"+T

t/sE[v

1tf1tu1,t~s

f1,t~s

]/¹.


2. X can be consistently estimated by XK , where (XK )ls"(1/n)+

i+

tuJ 2it

fK 2it(uJ

i,t~lfKi,t~l

!(zit!zL

it)@ fK

itBK ~1UK

l)(uJ

i,t~sfKi,t~s

!(zit!zL

it)@ fK

itBK ~1UK

s)],

and UKs"n~1+

i+

t(z

it!zL

it) fK

ituJi,t~s

fKi,t~s

. Hence

»n,p

"n¸@n,p

(XK )~1¸n,p

P$ s2(p).

Proof. Similar to the proof of Theorem 1, one can show that

JnoLs"(1/Jn)+

i+

tuit

fit(u

i,t~sfi,t~s

!v@it

fitB~1U

s)#o

1(1). Hence

JnoLs

$P N(0, X

ss) by Lindberg—Levi central limit theorem (because ¹ is finite).

The result of Corollary 2(i) follows by a multivariate central limit theorem.Finally the proof that XK

ls"X

ls#o

1(1) follows the same arguments as

pL 20"p2

0#o

1(1).

Remark 4.1. Similar to the arguments of Remark 3.2, one can easily show thatCorollary 2 shows that the asymptotic distribution of the »

n,ptest is the same as

in the case that E(yitDw

it) and E(z

itDw

it) are known. Hence our »

n,ptest is similar to

the LM type tests of Breusch (1978) and Godfrey (1978). One main difference isthat we use residuals from a semiparametric model rather than from a paramet-ric model to construct our test because our model is only partially specified. Alsosince we do not derive our test from the maximum likelihood principle, we donot need the normality assumption for the error term u

it.

5. Testing for individual effects

In this section we show that the test statistic In

for testing zero first-orderserial correlation can be easily generalized to test for the absence of individualeffects. In the parametric regression case, the popular Breusch—Pagan (1980) testis often used for testing individual effects in a parametric regression model. The

Breusch—Pagan test is based on +i+

t+

s:tuituis/JN¹(¹!1)/2. Our test will be

based on the density weighted version of this quantity:

Jn"

+i+

t+

s:tuJituJis

fKit

fKis

JN¹(¹!1)/2.

Given the results of Theorem 1, it is easy to see that the following corollaryholds.

Corollary 3. ºnder the same conditions as in ¹heorem 1,

(i) Jn

$P N(0, p2

2), where p2

2"(2/¹(¹!1))+

t+

s:t+

s{:tE[u2

1tf21t(u

1sf1s!v@

1t

f1tB~1U

t~s)(u

1s{f1s{

!v@1t

f1tB~1U

t~s{)]


(ii) pL 22"(2/N¹(¹!1))+

i+

t+

s:t+

s{:tMuJ 2

itfK2it[uJ

isfKis!(z

it!zL

it)@ fK

itBK ~1UK

t~s]

[uJis{

fKis{!(z

it!zL

it)@ fK

itBK ~1UK

t~s{]N is a consistent estimator for p2

2. Hence

¹n"J

n/pL

2$

P N(0,1).

Proof. Similar to the proof of Theorem 1, one can show that

Jn"+

i

+t

+s:t

J2/N¹(¹!1)uit

fit(u

isfis!v@

itfitB~1U

t~s)#o

1(1).

By Lindberg central limit theorem, we have

Jn"

1

JNC+i giD

$P N(0,p2

2)

under H0, where

gi"+

t

+s:t

J2/¹(¹!1)uit

fit(u

isfis!v@

itfit

B~1Ut~s

)

has mean zero and variance

2

¹(¹!1)+t

+s:t

+s{:t

E[u2it

f2it(u

1sf1s!v@

1tf1tB~1U

t~s)

](u1s{

f1s{

!v@1t

f1tB~1U

t~s{)]"p2

2.

The estimator pL 22"p2

2#o

1(1) follows the same arguments as pL 2

0"p2

0#o

1(1).

Remark 5.1. A popular error structure specification for panel data model is theone-way error components model (e.g., Balestra and Nerlove, 1966; Hsiao,1986). If the individual effects exist, then the composite error term will exhibitconstant serial correlations at different time lags. One would expect the ¹

nand

the »n,p

tests to have higher power than the Sn

test in rejecting the null of noserial correlation. On the other hand, if the error follows a first-order autoreg-ressive or moving average process, then the higher-order serial correlationcoefficients are either zero or quickly converge to zero, one would expect theSntest to have a higher power in rejecting the null of no serial correlation than

the »n,p

and the ¹ntests.

Remark 5.2. If zit

or wit

contain lagged dependent variable, one would expectthat the I

n, the »

n,pand the ¹

ntests to give the correct size asymptotically under

the null hypothesis. However, under the alternative hypothesis, the estimated


serial correlation coefficient are likely to be biased toward zero. One wouldexpect this downward bias will reduce the power somewhat under the alterna-tive hypothesis, just like the Durbin h-test in the linear model case. This shouldbe particularly apparent in the one way error component model becauseconditioning on lagged dependent variable, the serial correlations for all orderswill be simultaneously reduced. To see this, consider a parametric specificationwith the error following an one-way error component structure.

yit"z@

itc#g(y

i,t~1, b)#k

i#l

it, (13)

where ki

is i.i.d. (0, p2k), lit

is i.i.d. (0, p2l ), and g is a known function withb a unknown (possibly vector) parameter. Assuming that z

itis independent of

both kiand l

isfor all t and s, then we can rewrite (Eq. (13)) as

yit"z@

itc#g(y

i,t~1, b)#h(y

i,t~1)#e

it, (14)

where h(yi,t~1

)"E(kiDzit, y

i,t~1)"E(k

iDy

i,t~1) and e

it"k

i!h(y

i,t~1)#l

it. The

functional form of h is not specified. Obviously E(eitDzit, y

i,t~1)"0. Hence

E(yitDzit, y

i,t~1)"z@

itc#g(y

i,t~1, b)#h(y

i, t~1). If we let w

it"y

i, t~1in (Eq. (1))

and define h(yi, t~1

)"g(yi,t~1

, b)#h(yi, t~1

). Then Eqs. (1) and (14) are notdistinguishable because the error e

itin (Eq. (14)) satisfies the same condition (2)

as uit. However, the serial correlation coefficient in (Eq. (13)) is equal to

p2k/(p2k#p2l ) and the serial correlation coefficient in (Eq. (14)) is equal top*2k /(p*2k #p2l ), where p*2k "var[k

i!h(y

i, t~1)](p2k.

Remark 5.3. Like the parametric case, if the ¹n

test fails to reject the nullhypothesis, it can be viewed as evidence against one way error componentsspecification of the error process. On the other hand, if the ¹

ntest rejects the null

hypothesis, it only rejects the hypothesis that uit

is a martingale differenceprocess. To further test whether the serial correlation is due to higher-orderserial correlation, or due to unobserved individual effects, is beyond the scope ofthe present paper. But this is certainly worthy of effort for further research.

6. Monte-Carlo experiments

In this section we report some simulation results to examine the finite sampleperformance of the test statistics S

n, »

n,pand ¹

n. We use the following data

generating processes (DGP):

DGP1: yit"b

1#z

itc1#x

itb2#u

it,z

itc1#h

1(x

it)#u

it,

DGP2: yit"b

1#y

i,t~1c2#x

itb3#u

it,y

i,t~1c2#h

2(x

it)#u

it,


DGP3: yit"b

1#z

itc3#y

i,t~1b4#u

it,z

itc3#h

3(y

i,t~1)#u

it.

DGP4: yit"b

1#y

i,t~1c4#x

itb5#x2

itb6#u

it,y

i,t~1c4#h

4(x

it)#u

it.

DGP5: yit"b

1#z

itc5#y

i,t~1b7#x

itb8#x

ityi,t~1

b9#u

it

,zitc5#h

5(x

it, y

i,t~1)#u

it.

We generate six different processes for the error uit:

(a) uit’s are i.i.d. N(0,1); (a white noise error),

(b) uit"0.3u

i,t~1#e

it; (an AR(1) error),

(c) uit"0.4e

i,t~1#e

it; (a MA(1) error),

(d) uit"0.3u

i,t~2#e

it; (an AR(2) error),

(e) uit"0.4e

i,t~2#e

it; (a MA(2) error),

(f) uit"k

i#l

it. (an one-way EC error).

In (b)—(e), eit’s are i.i.d. N(0, 1). In (f) k

i’s are i.i.d. N(0,1) and l

it’s are i.i.d. N(0,1),

also kiand l

jtare independent. Case (a) corresponds to the null hypothesis H

0.

Cases (b) to (f) correspond to the alternatives with E(uituis)O0 for some sOt.

The explanatory variables xit, z

it’s and the parameters are chosen as follows:

1. For DGP1, zit’s are i.i.d. N(0, 1), x

it"0.3x

i,t~1#g

itwith g

itbeing i.i.d.

N(0,1). (c1, b

1, b

2)"(1, 1, 1). Also x, z and u are all independent to each

other.2. For DGP2, x

itis the same AR(1) process as in (1) above. x and u are

independent of each other. (c2, b

1, b

3)"(0.5, 1, 1).

3. For DGP3, xit

is the same AR(1) process as in (1) above. x and u areindependent to each other. (c

3, b

1, b

4)"(1, 1, 0.5).

4. For DGP4, xit

is the same as in DGP3. (c4, b

1, b

5, b

6)"(0.5, 1, 1, 1).

5. For DGP5, xit

and zit

are independent i.i.d. N(0, 1) variates.(c

5, b

1, b

7, b

8, b

9)"(1, 1, 0.5, 1, 0.3).

We use a standard normal kernel function and the smoothing parameter ischosen via a

x,t"x

t,sdN~1@5 for DGP1, DGP2, DGP4 and DGP5; and

ay,t"0.5y

t~1,sdN~1@5 for DGP3 and DGP5, where x

t,sdand y

t~1,sdare the

sample standard deviations for MxitNNi/1

and Myi,t~1

NNi/1

respectively. The numberof replications are 2000 for size estimation and 1000 for power estimation. Wefix ¹"10. The results for S

nare given in Tables 1—3; the results for »

n,pare

given in Tables 4—6; and the results for ¹nare given in Tables 7—9.

First for the Sn

test, Table 1 reports the estimated size for the Sn

test. Theestimated sizes for DGP1, DGP3 and DGP5 are quite good and it underesti-mates the sizes for DGP2 and DGP4, perhaps because both cases use lagged


Tab

le1

S n-tes

t,ca

se(a

):I.I.D

.er

ror

(est

imat

edsize

)

DG

P1

DG

P2

DG

P3

DG

P4

DG

P5

N1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%

500.

009

0.05

60.

109

0.00

30.

021

0.05

20.

004

0.03

70.

085

0.00

50.

029

0.06

40.

006

0.04

80.

094

100

0.01

20.

054

0.10

60.

001

0.03

30.

068

0.00

50.

042

0.09

00.

005

0.03

00.

069

0.00

80.

058

0.10

4

Tab

le2

S n-tes

t,ca

se(b

):A

R(1

)er

ror

(est

imat

edpo

wer

)

DG

P1

DG

P2

DG

P3

DG

P4

DG

P5

N1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%

501.

001.

001.

000.

890

0.97

90.

993

0.96

20.

996

0.99

90.

963

0.99

41.

000.

818

0.95

40.

978

Tab

le3

S n-tes

t,ca

se(c

):M

A(1

)er

ror

(est

imat

edpow

er)

DG

P1

DG

P2

DG

P3

DG

P4

DG

P5

N1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%

501.

001.

001.

001.

001.

001.

001.

001.

001.

000.

999

0.99

91.

000.

966

0.99

40.

999


Tab

le4

»n,2-t

est,

case

(a):

I.I.D

.er

ror

(est

imat

edsize

)

DG

P1

DG

P2

DG

P3

DG

P4

DG

P5

N1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%

500.

009

0.05

50.

109

0.00

40.

023

0.05

70.

007

0.03

60.

075

0.00

70.

034

0.06

90.

006

0.03

70.

089

100

0.01

00.

051

0.10

70.

004

0.02

60.

058

0.00

50.

037

0.08

10.

008

0.03

70.

074

0.00

70.

039

0.09

1

Tab

le5

»n,2-t

est,

case

(d):

AR

(2)er

ror

(est

imat

edpo

wer

)

DG

P1

DG

P2

DG

P3

DG

P4

DG

P5

N1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%

501.

001.

001.

000.

953

0.98

40.

993

0.94

40.

988

0.99

40.

938

0.98

80.

997

0.69

90.

890

0.94

3

Tab

le6

»n,2-t

est,

case

(e):

MA

(2)er

ror

(est

imat

edpow

er)

DG

P1

DG

P2

DG

P3

DG

P4

DG

P5

N1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%

501.

001.

001.

000.

995

0.99

91.

000.

996

1.00

1.00

0.99

51.

001.

000.

858

0.96

80.

985


Tab

le7

¹n-t

est,

case

(a):

I.I.D

.er

ror

(est

imat

edsize

)

DG

P1

DG

P2

DG

P3

DG

P4

DG

P5

N1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%

500.

011

0.04

40.

097

0.00

20.

024

0.04

90.

015

0.05

50.

092

0.00

40.

035

0.07

90.

012

0.05

40.

110

100

0.00

90.

050

0.09

40.

004

0.03

40.

062

0.00

90.

054

0.10

60.

005

0.03

90.

081

0.01

20.

052

0.10

7

Tab

le8

¹n-t

est,

case

(b):

AR

(1)er

ror

(est

imat

edpo

wer

)

DG

P1

DG

P2

DG

P3

DG

P4

DG

P5

N1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%

501.

001.

001.

000.

054

0.16

10.

248

0.23

70.

418

0.51

20.

124

0.26

80.

380

0.26

60.

432

0.53

110

01.

001.

001.

000.

324

0.51

90.

638

0.53

70.

798

0.80

80.

280

0.47

90.

605

0.50

90.

690

0.77

620

01.

001.

001.

000.

669

0.83

90.

891

0.85

40.

943

0.97

10.

608

0.79

80.

870

0.83

80.

938

0.96

6

Tab

le9

¹n-t

est,

case

(f):

one-

way

EC

(est

imat

edpow

er)

DG

P1

DG

P2

DG

P3

DG

P4

DG

P5

N1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%

501.

001.

001.

000.

585

0.82

90.

913

0.99

51.

001.

000.

959

0.99

10.

997

1.00

1.00

1.00

100

1.00

1.00

1.00

0.98

90.

998

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00


dependent variables as explanatory variables in the parametric parts. However,as N increases, the estimated sizes increase toward their nominal sizes. Tables 2and 3 show that the S

ntest is quite powerful in detecting first-order serial

correlation for all the DGPs considered.Secondly for the »

n,ptest, from Table 4 we see that the estimated size for the

»n,p

test is similar to Table 1 of the Sntest. It has good estimated size for DGP1,

DGP3 and DGP5 and again it underestimates the size for DGP2 and DGP4.Tables 5 and 6 show that the »

n,ptest is quite powerful for detecting the AR(2)

and MA(2) serially correlated errors. Note that for the MA(2) error of case (e),E(u

itui,t~1

)"0, hence the Sn

test has only trivial power against such a MA(2)process.

Finally for the ¹ntest, Table 7 gives the estimated sizes for DGP1—DGP5. In

general, the size estimation is quite good except that for DGP2, it underesti-mates the size for DGP2. Tables 8 and 9 report estimated power for the ¹

ntest

against an AR(1) and an one-way error component error structure, respectively.The ¹

ntest is quite powerful in detecting one-way error structure, but as

expected, it is not very powerful against an AR(1) error process.The Monte-Carlo simulation results show that our proposed tests perform

reasonably well in finite samples for the DGPs we considered. However, theabove results are only for conditional homoskedastic normal errors. Also ourtest statistics depend on the smoothing parameter a. Below we will reportsimulation results for the S

ntest using slightly different smoothing parameters as

well as a nonnormal error and some conditional heteroskedastic error cases.First we increased and decreased the smoothing parameter a by 20% and

examine the sensitivity of the estimated sizes of our test statistics to differentvalues of a. In particular, we replace a

x,tby a`

x,t,1.2x

t,sdN~1@5 or a~

x,t,

0.8xt,sd

N~1@5, and replace ay,t

by a`y,t"0.6y

t~1, sdN~1@5 a~

y,t"0.4y

t~1,sdN~1@5.

The results for size estimation for different DGPs are given in Table 10, wherea` denotes a`

x,tfor DGP1, DGP2 and DGP4; a`

y,tfor DGP3 and (a`

x,t, a`

y,t) for

DGP5. a~ is similarly defined. The sample size is N"50 and ¹"10. FromTable 10, we observe that for most cases, the estimated sizes only change slightlywhen the smoothing parameters increased or decreased by 20% except forDGP5. For DGP5 with a"a`, the S

ntest overestimates the size.

Next, we changed the error term from a normal distribution to a t-distribu-tion with 5 degree of freedom. The estimated sizes for a t

5error are given in

Table 11. We see that the estimated sizes are quite close to their nominal levelsfor DGP1, DGP3, and DGP5, and the estimated sizes are below their nominallevels for DGP2 and DGP4. These results are quite similar to their correspond-ing normal disturbance cases.

Finally, we used a conditional heteroskedastic error: uit"p

iteit, where e

itis

i.i.d. N(0,1), and pit"J(1#x2

it#z2

it)/3 for DGP1, p

it"J(1#x2

it#y2

i,t~1)/10

for DGP2, pit"J(1#z2

it#y2

i,t~1)/10 for DGP3, p

it"J(1#x2

it#y2

i,t~1)/25


Tab

le10

S n-tes

tfo

rdi

ffere

ntsm

ooth

ing

par

amet

er(e

stim

ated

size

,N"

50)

DG

P1

DG

P2

DG

P3

DG

P4

DG

P5

a1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%

a`0.

013

0.05

90.

110

0.00

20.

019

0.04

90.

008

0.04

70.

092

0.00

30.

027

0.05

80.

015

0.06

80.

132

a~0.

009

0.04

60.

099

0.00

30.

024

0.05

40.

005

0.04

00.

075

0.00

40.

032

0.07

30.

004

0.03

90.

085

Tab

le11

S n-tes

tw

ith

t 5er

ror

term

(est

imat

edsize

)

DG

P1

DG

P2

DG

P3

DG

P4

DG

P5

N1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%

500.

004

0.04

60.

096

0.00

20.

018

0.05

20.

005

0.03

70.

082

0.00

30.

022

0.05

60.

007

0.04

70.

102

Tab

le12

S n-tes

tw

ith

cond

itio

nalhe

ter.

erro

r(e

stim

ated

size

)

DG

P1

DG

P2

DG

P3

DG

P4

DG

P5

N1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%1%

5%10

%

500.

007

0.04

60.

093

0.00

30.

013

0.03

70.

008

0.04

30.

092

0.01

10.

049

0.10

10.

013

0.06

10.

119


3 If part of the variables of zit

is discrete, say zit"(z

1,it, z

2,it), z

1,itis discrete and z

2,itis continuous.

We can redefine z1,it

and (z2,it

,wit) as the new z

itand w

it, respectively.

for DGP4, and pit"J(1#x2

it#z

it#y2

i,t~1)/14 for DGP5. All the errors have

unconditional variances close to one. The results is given in Table 12. Theestimation results for DGP1, DGP2 and DGP3 are very similar to theircorresponding conditional homoskedastic error cases (compare with Table 1).However, for DGP4, the estimated sizes are much closer to their nominal valuesfor this conditional heteroskedastic case, and the S

ntest over estimates the sizes

for DGP5 in the conditional heteroskedastic error case.

7. The case of nonparametric regression model

In this section we show that our test statistics based on a semiparametricpartially linear model can be easily extended to the case of a nonparametricpanel data regression model of the form:

yit"g(z

it, w

it)#u

it(15)

where the functional form of g( ) ) is not specified. Without loss of generality, weassume that z

it3Rr (r*0, r"0 is allowed in this section) are discrete variables

and wit3Rq (q*1) are continuous variables.3 In this case, we estimate g(z

it, w

it)

by

gJit"(Naq)~1+

jEi

yjtK

it,jtPit, jt

/fKit,1

,

where Pit, jt

"<rl/1

I(zl,it!z

l,jt), z

l,itis the lth component of z

it, I( ) ) is the

indicator function with I(0)"1 and I(x)"0 for DxDO0, and

fKit,1

"(Naq)~1+jEi

Kit,jt

Pit,jt

. (16)

The nonparametric residual is given by

uJit,1

"yit!gJ

it"g

it#u

it!gL

it,1!uL

it,1, (17)

where gLit,1

and uLit,1

are defined by

gLit,1

"(Naq)~1+jEi

gjtK

it,jtP

it,jt/fKit,1

, (18)

and

uLit,1

"(Naq)~1+jEi

ujtK

it,jtP

it,jt/fKit,1

. (19)


4When part of the regressors are discrete and the rest are continuous, the convergence rate ofnonparametric kernel estimates is determined by that of the continuous variables, see the pioneerwork of Bierens (1983), and the more recent works of Delgado and Mora (1995) and Baltagi et al.(1996). Hence, the proof of Corollary 4 follows the same arguments as in the proof of Theorem1 where the nonparametric estimation are only associated with continuous variables.

The test statistic for testing the absence of first-order serial correlation ina nonparametric regression model becomes

In,1

"

1

Jn+i

+t

uJit,1

uJit~1,1

fKit,1

fKit~1,1

. (20)

To derive the asymptotic distribution of In,1

, we need some regularity condi-tions. Let (A1)@ be the same as (A1) with the joint-density function f

t,1(z, w)

replacing ft(w) and g(z, w) replacing h(w), also all the smoothness conditions in

(A1)@ are with respect to w for a fixed value of z, for example ft(w)3G=l~1

of (A1)becomes, for any fixed value of z, f

t,1(w, z)3G=l~1

in (A1)@. Then by the proof ofTheorem 1, we immediately have the following corollary.

Corollary 4. Under (A1)@,(A2) and H0, we have I

n,1/pL

0,1

$P N(0,1), where

pL 20,1

"n~1+i+

t[uJ

it,1fIit,1

uJit~1,1

fIit~1,1

]2 is a consistent estimator for p20,1

"

1/¹+tE[(u

1tft,1

(x1t)u

1t~1ft~1,1

(x1t~1

))2] where xit"(z

it, w

it).

Proof. The proof of Corollary 4 follows from a similar argument as that of theproof of Theorem 1. Hence, we only provide a sketch of the proof below.4 ThefK , gL , uL below are defined by Eqs. (16), (18) and (19), respectively.

In,1

"n1@2S*u~uL `(g~gL )+fK ,*u~1~uL ~1`(g~1~gL ~1)+fK ~1

"n1@2SufK ,u~1fK ~1

#n1@2MSufK ,uL ~1fK ~1

#SuL fK ,u~1fK ~1

#SuL fK ,uL ~1fK ~1

N

#n1@2MSufK ,(g~1~gL ~1)fK ~1

#SuL fK ,(g~1~gL ~1)fK ~1

#S(g~gL )fK ,(g~1~gL ~1)fK ~1

N

#n1@2MS(g~gL )fK ,u~1fK ~1

#S(g~gL )fK ,uL ~1fK ~1

N

,n1@2SufK ,u~1fK ~1

#(I)#(II)#(III)

"n1@2Suf, u~1f~1

#o1(1),

where (I)"o1(1), (II)"o

1(1) and (III)"o

1(1) follow from similar proofs of

Lemmas A.4, A.5 and A.6, respectively, and n1@2SufK ,u~1fK ~1

"n1@2Suf,u~1f~1

#o1(1)

follows from a similar proof as Lemma A.5(a). Now, n1@2Suf,u~1f~1

$P N(0, p2

0,1)

by Lindeberg—Levi central limit theorem and pL 20,1

"p20,1

#o1(1) follows the

same proof as pL 20"p2

0#o

1(1).


Remark 7.1. An interesting result of Corollary 4 is that p20,1

has asimple expression whether (z

it, w

it) is strictly exogenous or not. This is in

contrast to the results of Theorem 1 and Corollary 1 where the asymptoticvariance of the test statistic depends on whether (z

it, w

it) and u

i,t~1are correlated

or not.

Remark 7.2. When g( ) ) is a linear function, say g(zit, w

it)"z@

itc#w@

itb, Eq. (15)

reduces to a linear model and the appropriate tests for testing first-order serialcorrelation are the popular Durbin—Watson or Durbin-h test when (z

it, w

it)

contains yi,t~1

. However, Durbin—Watson or Durbin-h test is unable to dis-criminate between serial correlation and regression functional form misspecifi-cation. Rejection of no serial correlation by Durbin—Watson or Durbin-h testmay be due to regression functional form misspecification rather than serialcorrelation in the error. An important advantage of using a nonparametricfunctional form of Eq. (15) when testing for serial correlation is that it allows theresearchers to separate the issues of functional form misspecification fromgenuine autocorrelation in the errors. However, nonparametric estimation ofEq. (15) suffers from the well-known curse of dimensionality problem. In thisrespect, the semiparametric model (1) provides a compromise. It has a moreflexible functional form compared with a linear regression model, and the issueof the curse of dimensionality problem is not as serious as a completelynonparametric specification.

Similar generalizations also hold for testing higher-order serial corre-lation and for testing individual effects. Define ¸

np,1and J

n,1the same

way as ¸np

and Jnwith uJ

itand fK

itreplaced by uJ

it,1and fK

it,1, respectively. Then we

have

Corollary 5. ºnder (A1)@,(A2) and H0, we have

(a) n¸@n,p,1

(XK1)~1¸

n,p,1P$ s2(p), where the (l, s)th element of XK

1is given by

(XK1)ls"(1/n)+

i+

tuJit,1

fKit,1

uJit~l,1

fKit~l,1

uJit~s,1

fKit~s,1

;

(b) Jn,1

/pL2,1

P$ N(0,1), where pL 22,1

"(2/N¹(¹!1))+i+

t+

s:t+

s{:tuJ 2it,1

fK2it,1

uJis,1

fKis,1

uJis{,1

fKis{,1

.

Note that similar to the result of Corollary 4, both XK1

and pL 22

have simpleexpressions whether (z

it, w

it) is strictly exogenous or not.

8. Acknowledgements

We would like to thank two referees and an associate editor for helpfulcomments that greatly improved the paper. C. Hsiao’s research is supported inpart by Natural Science Foundation grants SBR 94-09540 and SBR 96-19330.


Q. Li’s work is supported by a grant from the Social Sciences and HumanitiesResearch Council of Canada and a grant from the Natural Sciences EngineeringResearch Council of Canada.

Appendix A. Proof of Theorem 1

We first present some lemmas. In all the lemmas given below, we assume theconditions in Theorem 1 hold. Also e"u or v and g"h or m. Hence g3G4l . Notethat both m and v are p]1 vectors. However, when we evaluate the order ofsome terms, we will often only consider the case of p"1 for m and v, because thep'1 vector case follows from the same proof as in the scalar case and theCauchy inequality.

When we evaluate the order of some terms, if there is no confusion, we will useN, (N!1) and (N!2) interchangeably. We will use E

it( ) ) and E

it,js( ) ) to denote

conditional expectations E( ) Dwit) and E( ) Dw

it,w

js). w

> t,(w

1t,2,w

Nt). Also for

a p]1 vector x, we will use D ) D to denote its Euclidean norm. The notationA&B means A"O(B). We will write f( ) ) for f

t( ) ) to simplify the notation.

¸emma A.1. (a) DE1t(g(w

1t)!g(w

2t))K((w

1t!w

2t)/a)D)D

g(w

1t)aq`l,

(b) DE1t(g(w

1t)!g(w

2t))g(w

1t)K((w

1t!w

2t)/a)D)D

g(w

1t)g(w

1t)aq`l,

(c) DE1t(a~qK((w

1t!w

2t)/a))!f(w

1t)D)D

f(w

1t)al,

(d ) DE[(g2t!g

1t)2K2

1t,2te21tK

1t,3tK

1t,4t] D)Ca2`3qE DH2

g(w

1t)e2

1t(w

1t) D"

O(a2`3q), where Dg( ) ) has finite fourth moment and D

f( ) ) has finite moment of any

order. e"u or v and e21t(w

1t)"E(e2

1tDw

1t).

Proof. See the proof of Lemma 5 of Robinson (1988) for (a). (b) follows directlyfrom (a). (c) is Lemma 4 of Robinson (1988). (d) follows by change-of-variable:wlt!w

1t"ag

l(l"2,3,4) and then using Dg(w

1t#ag

l)!g(w

1t)D)aH

g(w

1t)Dg

lD

(see (A2)).

Define StAfK ,BK fK

"(1/N)+iA

itBK @itfK 2it. St

AK fK ,BK fKand St

AfK ,BfKare similarly defined.

¸emma A.2.(a) St

(g~gL )fK"O

1(a2(Naq)~1#a2l), where g"h or m;

(b) St(g~gL )fK ,efK "O

1(N~1@2al#a(Naq@2)~1), where g"h or m and e"u or v;

(c) St(g~gL )fK ,eL fK "O

1(a(Naq@2)~1#N~1@2al), where g"h or m and e"u or v;

(d) Stu,vL

"O1(N~1a~q@2), St

uL ,v"O

1(N~1a~q@2), St

u,vL"O

1(N~1a~q@2).

Proof. (a) was proved in the proof of Lemma 1 of Li (1996). The proof of (d)follows exactly the same steps as in the proof of Lemma A.7 of this appendix.The proofs for (b) and (c) are very similar to the proofs of the Lemmas 3 and 4 ofLi (1996). Here we will only provide a proof for (b), the proof of (c) is similar to(b).


(b) First we evaluate E[St(g~gL )fK ,efK ].

E[St(g~gL )fK ,efK ]2"EGC

1

n+i

+t

(git!gL

it) fK

iteit

fKitD

]C1

n+i{

+t{

(gi{t{!gL

i{t{) fK

i{t{ei{t{

fKi{t{DH

"EGC1

nN2a2q+i

+t

+jEi

+lEi

(git!g

jt)K

it,jteitK

it,ltD]C

1

nN2a2q+i{

+t{

+j{Ei{

+l{Ei{

(git{!g

j{t{)K

it{,j{t{ei{t{

Ki{t{,l{t{DH

&EGC1

nN2a2q+i

+t

+jEi

+lEi

(git!g

jt)K

it,jteitK

it,ltD]C

1

N3a2q+i{

+j{Ei{

+l{Ei{

(gi{t!g

j{t)K

i{t,j{tei{tK

i{t,l{tDH"

1

nN5a4q+i

+t

+jEi

+lEi

+j{Ei

+l{Ei

E[(git!g

jt)

]Kit,jt

e2itK

it,lt(g

it!g

j{t)K

it,j{tK

it,l{t]

,RH0.

Case (i) i, j, l, j@,l@ are all different. In this case

RH0(*)"

1

N6a4qO

1(N5a4q`2l)"O(N~1a2l)

by Lemma A.1(a).Case (ii) i, j, l, j@, l@ take four different values. In this case, it has the same order

as j@"j, which is (N2a4q)~1DE[(g1t!g

2t)2K2

1t,2te21tK

1t,3tK

1t,4t]D"

O((N2a2~q)~1) by Lemma A.1(d). Hence RH0(**)"O((N2a2~q)~1).

Case (iii) i, j, l, j@, l@ take at most three different values. Using the samearguments as case (ii) above, it is easy to see that

RH0(***)

"

1

N3a4qO(a2~2q)"O((N3a2~2q)~1),

which has a smaller order than RH0(**)

. Note that the factor a2 comes from thefact that D(g

it!g

jt)(g

it!g

2j{)D)a2Dg

1g2H2

g(w

it)D by using change-of-variable:

wjt!w

it"ag

1and w

j{t!w

it"ag

2(since jOi and j@Oi).


We have shown that RH0"O(N~1a2l#a2(N2aq)~1). Hence St(g~gL )fK ,efK "

O1(N~1@2al#a(Naq@2)~1).From Lemma A.2, we immediately have the following lemma.

¸emma A.3.(a) S

g~gL"O

1(a2(Naq)~1#a2l), where g"h or m;

(b) S(g~gL )fK ,efK "O


(c) S(g~gL )fK ,eL fK "O


(d) SufK ,vL fK

"O1(N~1a~q@2b~1), S

uL fK ,vfK"O

1(N~1a~q@2), S

ufK ,vL fK"O

1(N~1a~q@2).

Proof. SAfK ,BfK

,(1/¹)+tStAfK ,BfK

has the same order as StAfK ,BfK

(for any 1)t)¹)because ¹ is finite. Lemma A.3 follows directly from Lemma A.2. Note that allthe terms in Lemma A.3 are of the order of o

1(N~1@2) by (A2).

¸emma A.4.(a) S

ufK ,uL ~1fK ~1"O

1(N~1a~q@2), (b) S

uL fK ,u~1fK ~1"O

1(N~1a~q@2), (c) S

uL fK ,uL ~1fK ~1"

O1(N~1a~q@2).

Proof. (a)

E[(SufK ,uL ~1 fK ~1

)2]"EGC1

n+i

+t

uit

fKituLi,t~1

fKi,t~1D

2

H"EGC

1

n+i

+t

uit

fKituLi,t~1

fKi,t~1D

]C1

n+i{

+t{

ui{t{

fKi{t{

uLi{,t{~1

fKi{,t{~1DH

&EC1

nN+i

+t

+i{

E[uit

fKituLi,t~1

fKi,t~1

ui{t

fKi{tuLi{,t~1

fKi{,t~1

]D"

1

n N5a4q+i

+t

+i{

+jEi

+lEi

+j{Ei{

+l{Ei{

]E[uitK

it,jtul,t~1

Ki,t~1§l,t~1

ui{tK

i{t,j{tul{,t~1

Ki{,t~1§l{,t~1

]

"

1

n N5a4q+i

+t

+jEi

+lEi

+j{Ei

+l{Ei

]E[u2itul,t~1

ul{,t~1

Kit,jt

Ki,t~1§l,t~1

Kit,j{t

Ki,t~1§l{,t~1

]

,RH1,


where in the fourth equality, we used the fact that if iOi@, E[uitK

it,jtul,t~1

Ki,t~1§l,t~1

ui{tK

it,j{tul{,t~1

Ki{,t~1§l{,t~1

]" E[Kit,jt

Ki,t~1§l,t~1

Kit,j{t

Ki{,t~1§l{,t~1

E(uitui{tul,t~1

ul{,t~1

Dw> t,w

> t~1)]"0. This is because u

itis independent of u

i{tand

E(uitDw

> t,w

> t~1,u

> ,t~1)"0. This kind of argument will be used several times in

this appendix.We consider RH1 for three different cases: (i) the summation indices i, j, l, j@, l@

are all different from each other; (ii) i, j, l, j@, l@ take four different values; and (iii)i, j, l, j@, l@ take at most three different values.

For case (i), it is easy to see that RH1(*)"0 because u

l,t~1is independent of all

the variables in RH1(*)

except wl,t~1

and E(ul,t~1

Dwl,t~1

)"0.For case (ii), using the same argument as we did in the case (i) above, it is easy

to see that RH1(**)O0 if and only if l"l@. In this case, RH1

(**)becomes

RH1(**)"C

1

n N5a4q+i

+t

+jEi

+lEi

+j{Ei

E[u2itu2l, t~1

Kit, jt

K2i, t~1§ l, t~1

Kit, j{t

]D"

1

nN5a4qO(N4a3q)"O((N2aq)~1).

Finally it is easy to see that RH1(***)

"(nN5a4q)~1O(N3a2q)"O(N~3a~2q).Summarizing the above, we showed that RH1"EM[S

ufK ,u~1fK ~1]2N"

O(N~2a~q). Hence SufK ,u~1fK ~1

"O1(N~1a~q@2)"o

1(N~1@2).

The proofs of (b) and (c) follow the same arguments as in the proof of (a) andare therefore omitted here. However, these proofs are available from the authorsupon request.

¸emma A.5.(a) S

ufK ,(h~1~hK ~1)fK ~1"O

1(a(Naq@2)~1#N~1@2al),

(b) SuL fK ,(h~1~hK ~1)fK ~1

"O1(a(Naq@2)~1#N~1@2al),

(c) S(h~hK )fK ,(h~1~hK ~1)fK ~1

"O1(a2(Naq)~1#a2l).

Proof. (a) follows the same proof as SufK ,(h~hK )fK "O

1(a(Naq@2)~1#N~1@2al) of

Lemma A.3(b). This is because E(uitDw

i,t~s)"0 for s*0. Hence replacing

E(uitDw

it)"0 in the proof of Lemma A.3(b) by E(u

itDw

i,t~1)"0 gives the proof of

Lemma A.5(a).(b) follows the same proof as Lemma A.3(c).(c) follows from Lemma A.3(a) and the Cauchy’s inequality.

¸emma A.6.

(a) S(h~hK )fK ,u~1fK ~1

"O1(al#N~1@2al@2), (b) S

(h~hK )fK ,uL ~1fK ~1"O

1(N~1@2al@2).


Proof. (a) We first show that E[S(h~hK )fK ,u~1fK ~1

]"O(al).

E[S(h~hK )fK , u~1fK ~1

]"1

nN2a2q+i

+t

+jEi

+lEi

]E[(hit!h

jt)K

it, jtui, t~1

Ki, t~1§ l, t~1

]

,RH2.

We consider RH2 for two different cases: (i) lOj and (ii) l"j. For case (i), wehave

RH2(*)"

1

¹a2q+t

E[(h1t!h

2t)K

1t,2tu1,t~1

K1,t~1§3,t~1

]

"

1

¹a2q+t

E[(h1t!h

2t)K

1t,2tg(w

1t, w

1, t~1)K

1, t~1§ 3, t~1]

)

al¹aq

+tE[DDh(w1t

)g(w1t,w

1,t~1)K

1,t~1§3,t~1D] (by Lemma A.1(b))

"O(al).

Obviously for case (ii) RH2(ii)"(1/Na2q)O

1(a2q)"O(N~1)"o(al). Hence

E[S(h~hK )fK ,u~1fK ~1

]"O(al).The second moment of S

(h~hK )fK ,u~1fK ~1is:

E[S(h~hK )fK ,u~1fK ~1

]2

"EGC1

nN2a2q+i

+t

+jEi

+lEi

E[(hit!h

jt)K

it, jtui, t~1

Ki, t~1§ l, t~1D

2

H"

1

n2N4a4q+i

+t

+jEi

+lEi

+i{

+t{

+j{Ei{

+l{Ei{

]E[(hit!h

jt)K

it, jtº

i, t~1K

i, t~1§ l, t~1][(h

i{t{!h

j{t{)

]Ki{t{,j{t{

ui{,t{~1

Ki{,t{~1§l{,t{~1

]

&

1

n N5a4q+i

+t

+jEi

+lEi

+i{

+j{Ei{

+l{Ei{

]EM[(hit!h

jt)K

it, jtº

i, t~1K

i, t~1§ l, t~1][(h

i{t!h

j{t)

]Ki{t, j{t

ui{, t~1

Ki{, t~1§ l{, t~1

]N

,RH3.


Case (i) i, j, l, i@, j@, l@ are all different from each other. In this case, the twoterms in the two square brackets are independent, we have RH3

(i)"O([E(S

(h~hK )fK ,u~1fK ~1]2)"O(a2l).

Case (ii) i, j, l, i@, j@, l@ take five different values. It is easy to see that in this caseRH3

ii"(1/N6a4q)O(N5a2l`4q)"O(N~1a2l).

Case (iii) i, j, l, i@, j@, l@ take at most four different values. In this case, RH3(iii)

"

(1/N6a4q)O(N4a3q)"O(N~2a~q).Thus we have shown that RH3"O(a2l#N~2a~q). Hence S

(h~hK )fK ,u~1fK ~1"

O1(al#N~1a~q@2).(b) The proof follows the same argument as in the RH3 part of (a). The reason

that (b) does not have a O(al) order term is that E[S(h~hK )fK ,u~1fK ~1

]"O(N~1)(rather than O(al)). To see this is indeed the true, we have

E[S(h~hK )fK ,uL ~1fK ~1

]"1

n N2a2q+i

+t

+jEi

+lEi

E[(hit!h

jt)K

it, jtul, t~1

Ki, t~1§ l, t~1

]

,RH4.

We consider RH4 for two different cases: (i) lOj and (ii) l"j. For case (i), wehave

RH4(i)"

1

¹a2q+t

E[(h1t!h

2t)K

1t, 2tu3, t~1

K1, t~1§ 3, t~1

]"0

because u3,t~1

is independent of all the variables except w3,t~1

andE(u

3,t~1Dw

3,t~1)"0.

Obviously RH4(ii)"(1/nN2a2q)+

i+

t+

jEiE[(h

1t!h

2t)K

1t, 2tu2, t~1

K1, t~1§ 2, t~1

]"(1/Na2q)O(a2q)"O(N~1). Hence E[S(h~hK )fK ,u~1fK ~1

]"O(N~1).The second moment of S

(h~hK )fK ,uL ~1fK"O(N~2a~q) by the similar proof as in the

proof of (a) RH3(ii)#RH3

(iii)"O(N~2a~q) above. Hence S

(h~hK )fK ,u~1fK ~1"

O1(N~1a~q@2).

¸emma A.7. (a) SufK ,u~1fK ~1

!SufK ,u~1f~1

"O1(N~1@2al#N~1a~q@2).

(b) SufK ,u~1f~1

!Suf,u~1f~1

"O1(N~1@2al#N~1a~q@2).

Proof. (a) SufK ,u~1fK ~1

!SufK ,u~1f~1

"(1/n)+i+

tuitfKitui, t~1

(fKi, t~1

!fi, t~1

)"(1/nN2aq)+

i+

t+

jEi+

lEiuitK

it,jtui, t~1

(a~qKi, t~1§ l, t~1

!fi, t~1

). It is easy to seethat S

ufK ,u~1fK ~1!S

ufK ,u~1f~1has mean zero because E(u

itDw

> t,w

> t~1,u

i,t~1)"0 (un-

der H0). Its second moment is

E[SufK ,u~1fK ~1

!SufK ,u~1f~1

]2

"

1

n2N4a2q+i

+t

+jEi

+lEi

+i{

+t{

+j{Ei{

+l{Ei{

]EM[uitK

it, jtui, t~1

(a~qKi, t~1§l, t~1

!fi, t~1

)]

][ui{t{

Ki{t{, j{t{

ui{, t{~1

(a~qKi{, t{~1§l{, t{~1

!fi{, t{~1

)]N


&

1

n2N4a2q+i

+jEi

+lEi

+i{

+j{Ei{

+l{Ei{

]EM[uitK

it, jtui, t~1

(a~qKi, t~1§l, t~1

!fi, t~1

)]

[ui{tK

i{t,j{tui{, t~1

(a~qKi{, t~1§l{, t~1

!fi{, t~1

))]N

"

1

n2N4a2q+i

+jEi

+lEi

+j{Ei

+l{Ei

EM[u2itK

it, jtui, t~1

(a~qKi, t~1§l, t~1

!fi, t~1

)]

][Kit, j{t

ui, t~1

(a~qKi, t~1§l{, t~1

!fi, t~1

))]N

,RH5,

because if i@Oi, uit

is independent of all the variables except (wit, w

i,t~1) and

E(uitDw

it, w

i,t~1)"0.

Case (i) if i, j, l, j@, l@ are all different from each other, RH5(i)"

(1/N6a2q)O1(N5a2l`2q)"O(N~1a2l) by Lemma A.1(c).

Case (ii) i, j, l, j@, l@ take four different values. (1) If lOl@ and both are differentfrom any other indices, we have RH5

(ii),(1)"(1/N6a2q)O

1(N4a2l`2q)"

O(N~2a2l) by Lemma A.1(c). (2) l"l@, RH5(ii),(2)

"(1/N6a2q)O

1(N4aq)"O(N~2a~q) by Lemma A.1(d). (3) lOl@ with l or l@ equal one other

index. RH5(ii),(3)

"(1/N6a2q)O(N4al`2q)"O(N~2al). Hence RH5(ii)"

O(N~2a~q).Case (iii) when i, j, l, j@, l@ take at most three different values. It is easy to see

that RH5(iii)

"(1/N6a2q)O(N3aq)"O(N~3a~q).Summarizing the above, we have shown that RH5"O(N~1a2l#N~2a~q).

Hence SufK ,u~1fK ~1

!SufK ,u~1f~1

"O1(N~1@2al#N~1a~q@2)"o

1(N~1@2).

(b) SufK ,u~1f~1

!Suf,u~1f~1

"(1/n)+i+

tuitui,t~1

fi,t~1

(fKit!f

it)"(1/nNaq)+

i+

t+

jEiuit

ui,t~1

fi,t~1

(a~qKit,jt

!fit). Obviously S

ufK ,u~1f~1!S

uf,u~1f~1has mean

zero and its second moment is

E[SufK ,u~1f~1

!Suf,u~1f~1

]2

"

1

n2N2a2q+i

+t

+jEi

+i{

+t{

+j{Ei{

EM[uitui, t~1

fi, t~1

(a~qKit, jt

!fit)]

][ui{t{

ui{,t{~1

fi{,t{~1

(a~qKi{t{,j{t{

!fi{t{

)]N

"

1

n2N2a2q+i

+t

+jEi

+t{

+j{Ei{

EM[u2itu2i, t~1

fi, t~1

(a~qKit, jt

!fit)]

][fi, t{~1

(a~qKit{,j{t{

!fit{)]N

&

1

nN3a2q+i

+t

+jEi

+j{Ei{

EM[u2itu2i, t~1

fi, t~1

(a~qKit, jt

!fit)]

][fi, t~1

(a~qKit, j{t

!fit)]N

,RH6.


Case (i) if i, j, l, j@, l@ are all different, RH6(i)"(1/N4a2q)O(N5a2q`2l)"

O(N~1a2l).Case (ii) i, j, l, j@, l@ take at most four different values, RH6

(ii)"(1/N4a2q)

O(N4aq)"O(N~2a~q).Hence RH6"O(N~1a2l#N~2a~q) and S

ufK ,u~1f~1!S

uf,u~1f~1"

O1(N~1@2al#N~1a~q@2)"o

1(N~1@2).

Proposition A.8. ºnder the same conditions as ¹heorem 1, then InP

$N(0, p2

0).

Proof. Using Eqs. (8) and (9), we have

In"n1@2S

*u~uL `(h~hK )~(z~zL )(cL~c)+fK ,*u~1~uL ~1`(h~1~hK ~1)~(z~1~zL ~1)(cL ~c)+fK ~1

"n1@2SufK ,u~1~uL ~1`(h~1~hK ~1)~(z~1~zL ~1)(cL ~c)+fK ~1

!n1@2SuL fK ,*u~1~uL ~1`(h~1~hK ~1)~(z~1~zL ~1)(cL ~c)+fK ~1

#n1@2S(h~hK )fK ,*u~1~uL ~1`(h~1~hK ~1)~(z~1~zL ~1)(cL~c)+fK ~1

!n1@2S(z~zL )(cL ~c)fK ,*u~1~uL ~1`(h~1~hK ~1)~(z~1~zL ~1)(cL ~c)+fK ~1

,A1n#A

2n#A

3n#A

4n.

We will first show that A1n"JnS

uf,u~1f~1#o

1(1), A

2n"o

1(1), A

3n"o

1(1),

and A4n"!n1@2S

vfK ,u~1fK ~1(cL!c)@#o

1(1)"!n1@2U(cL!c)@#o

1(1), where

U"(1/¹)+tE[v

1tf1tu1, t~1

f1, t~1

].

Proof of A1n"JnS

uf,u~1f~1#o

1(1): Define B

1nfrom A

1n"JnS

ufK ,u~1fK ~1#

B1n

. JnSufK ,u~1fK ~1

"JnSuf,u~1f~1

#o1(1) follows from n1@2DS

ufK ,u~1fK ~1!

Suf,u~1f~1

D)n1@2DSufK ,u~1fK ~1

!SufK ,u~1f~1

D#n1@2DSufK ,u~1f~1

!Suf,u~1f~1

D"o1(1) by

Lemma A.7.

B1n"n1@2S

ufK ,*~uL ~1`(h~1~hK ~1)~(z~1~zL ~1)(cL ~c)+fK ~1

"n1@2M!SufK ,uL ~1fK ~1

#SufK ,(h~1~hK ~1)fK ~1

!(cL!c)@[SufK ,v~1fK ~1

!SufK ,vL ~1fK ~1

#SufK ,(m~1~mK ~1)fK ~1

]N.

SufK ,uL ~1fK ~1

"o1(N~1@2) by Lemma A.4 (a) and S

ufK ,(h~1~hK ~1)fK ~1"o

1(N~1@2) by

Lemma A.5 (b). Also cL!c"O1(N~1@2). Thus we only need to show

SufK ,v~1fK ~1

, SufK ,vL ~1fK ~1

and SufK ,(m~1~mK ~1)fK ~1

are all o1(1).

By Cauchy’s inequality and the facts that SufK"O

1(1), S

vL ~1fK ~1"o

1(1) and

S(m~1~mK ~1)fK ~1

"o1(1), we get S

ufK ,vL ~1fK ~1and S

ufK ,(m~1~mK ~1)fK ~1are both o

1(1). Below we

show that SufK ,v~1fK ~1

"o1(1).

By the fact that fKit!f

it"o

1(1), it is obvious that DS

ufK ,v~1fK ~1!S

uf,v~1f~1D)

DSufK ,v~1fK ~1

!SufK ,v~1f~1

D#DSufK ,v~1f~1

!Suf,v~1f~1

D"o1(1). Hence it suffices to show

Suf,v~1f~1

"o1(1). S

uf,v~1f~1"(1/N)+N

i/1M¹+T

t/1uitf (w

it)[z

i,t~1!E(z

i,t~1Dw

i,t~1)]

f (wi,t~1

)N,(1/N)+Ni/1

giP

$E(g

1)"0 by a law of large number because g

i’s are

i.i.d. with


E(gi)"EMu

itf(w

it)[z

i,t~1!E(z

i,t~1Dw

i,t~1)] f (w

i,t~1)N"EM f (w

it)[z

i,t~1!

E(zi,t~1

Dwi,t~1

)] f (wi,t~1

) E (uitDw

it, w

i, t~1, z

it, z

i, t~1)N"0.

Thus we have shown that A1n"JnS

uf,u~1f~1#o

1(1).

Proof of A2n"o

1(1)

A2n"n1@2S

uL fK ,*u~1~uL ~1`(h~1~hK ~1)~(z~1~zL ~1)(cL~c)+fK ~1

"n1@2MSuL fK ,u~1fK ~1

!SuL fK ,uL ~1fK ~1

#SuL fK ,(h~1~hK ~1)fK ~1

!SuL fK ,z~1~zL ~1)fK ~1

(cL!c)N.

SuL fK ,u~1fK ~1

"o1(N~1@2) by Lemma A.4 (b), S

uL fK ,uL ~1fK ~1"o

1(N~1@2) by Lemma A.4

(c) and SuL fK ,(h~1~hK ~1)fK ~1

"o1(N~1@2) by Lemma A.5(b). Finally S

(uL fK ,z~1~zL ~1)fK ~1"

o1(1) by Cauchy inequality and the facts that S

uL fK"o

1(1) and

S(z~1~zL ~1)fK ~1

"O1(1). Hence S

uL fK ,z~1~zL ~1)fK ~1(cL!c)"o

1(N~1@2) because

cL!c"O1(N~1@2).

Proof of A3n"o

1(1)

A3n"n1@2S

(h~hK )fK ,*u~1~uL ~1`(h~1~hK ~1)~(z~1~zL ~1)(cL~c)+fK ~1

"n1@2MS(h~hK )fK ,u~1fK ~1

!S(h~hK )fK ,uL ~1fK ~1

#S(h~hK )fK ,(h~1~hK ~1)fK ~1

!S(h~hK )fK ,z~1~zL ~1)fK ~1

(cL!c)N.

S(h~hK )fK ,u~1fK ~1

"o1(N~1@2) by Lemma A.5(a), S

(h~hK )fK ,uL ~1fK ~1"o

1(N~1@2) by

Lemma A.6(a) and S(h~hK )fK ,(h~1~hK ~1)fK ~1

"o1(N~1@2) by Lemma A.5(c). Finally

S(h~hK )fK ,z~1~zL ~1)fK ~1

"o1(1) by Cauchy inequality and the facts that

S(z~1~zL ~1)fK ~1

"O1(1) and S

(h~hK )fK "o1(1). Hence S

(h~hK )fK ,z~1~zL ~1)fK ~1(cL!c)"

o1(N~1@2) because cL!c"O

1(N~1@2).

Proof of A4n"!n1@2U(cL!c)@#o

1(1)

A4n"!n1@2S

(z~zL )(cL ~c)fK ,*u~1~uL ~1`(h~1~hK ~1)~(z~1~zL ~1)(cL~c)+fK ~1

"!n1@2(cL!c)@MS(z~zL )fK ,u~1fK ~1

!S(z~zL )fK ,uL ~1fK ~1

#S(z~zL )fK ,(h~1~hK ~1)fK ~1

!S(z~zL )fK ,(z~1~zL ~1)fK ~1

(cL!c)N.

Using the fact that cL"O1(N~1@2), it sufficies to show that (i)

S(z~zL )fK ,u~1fK ~1

"U#o1(1), (ii) S

(z~zL )fK ,uL ~1fK ~1and (iii) S

(z~zL )fK ,(h~1~hK ~1)fK ~1are both

o1(1), and (iv) S

(z~zL )fK ,(z~1~zL ~1)fK ~1is O

1(1). (iv) is obviously true. Using Cauchy’s

inequality and the facts that SuL ~1fK ~1

"o1(1), S

(h~1~hK ~1)fK ~1"o

1(1) and

S(z~zL )fK

"O1(1), lead to (ii) and (iii). Finally for (i) using z

it!zL

it"

(mit!mK

it)#v

it!vL

it, it is easy to see that S

(z~zL )fK ,u~1fK ~1"S

(m~mK )fK ,u~1fK ~1#

SvfK ,u~1fK ~1

!SvL fK ,u~1fK ~1

"SvfK ,u~1fK ~1

#o1(1)"S

vf,u~1f~1#o

1(1), and that

E[Svf,u~1f~1

]"U, var[Svf,u~1f~1

]"o(1). Hence (i) also holds.


Now summarizing the above results, we have shown that In"JnS

uf,u~1f~1#

Jn(cL!c)@U#o1(1). Using Lemma 1(i), we have JnS

uf,u~1f~1!Jn(cL!c)@U"

(1/JN)+i(1/J¹)+

t[u

i,t~1fi,t~1

!v@itfitB~1U]u

itfit,(1/JN)+

igi P

$ N(0,p20) by

Levi—Lindeberg central limit theorem, because giis i.i.d. with zero mean and

finite variance E[g21]"(1/¹)+

tE[(u

1,t~1f1,t~1

!v@1tf1tB~1U)u

1tf1t]2"p2

0.

Proposition A.9. ºnder the same conditions as ¹heorem 1, then pL 20"

p20#o

1(1).

Proof. Using the similar arguments as in the proof of Proposition A.8, it is easyto show that BK "B#o

1(1), UK "U#o

1(1). Then it is straightforward to show

that pL 20"(1/n)+

i+

t[(u

i,t~1fi,t~1

!v@itfitB~1U)u

itfit]2#o

1(1),(1/N)+

iBi#o

1(1),

where Bi"(1/¹)+

t[(u

i,t~1fi,t~1

!v@itfitUB~1)u

itfit]2. (1/N)+

iB

i1

P E[B1]"p2

0by a law of large number.

References

Ahn, S.C., Schmidt, P., 1995. Efficient estimation of models for dynamic panel data. Journal ofEconometrics 68, 5—27.

Anderson, T.W., Hsiao, C., 1982. Formulation and estimation of dynamic models using panel data.Journal of Econometrics 18, 47—82.

Arellano, M., Bover, O., 1995. Another look at the instrumental variables estimation of error-component models. Journal of Econometrics 18, 29—51.

Balestra, P., Nerlove, M., 1966. Pooling cross-section and time-series data in the estimation ofa dynamic model: the demand of natural gas. Econometrica 34, 585—612.

Baltagi, B., Hildago, J., Li, Q., 1996. A nonparametric poolability test. Journal of Econometrics 75,345—367.

Bierens, H., 1983. Uniform consistency of kernel estimators of a regression function under generaliz-ed conditions. Journal of American Statistical Association 78, 699—707.

Breusch, T.S., Pagan, A.R., 1980. The lagrange multiplier test and its applications to modelspecification in econometrics. Review of Economics Studies 47, 239—253.

Box, G., Pierce, D., 1970. Distribution of residual autocorrelations in autoregressive and movingaverage time series models. Journal of American Statistical Association 65, 1509—1526.

Breusch, T.S., 1978. Testing for autocorrelation in dynamic linear models. Australian EconomicPapers 17, 334—355.

Delgado, M., Mora, J., 1995. Nonparametric and semiparametric estimation with discrete re-gressors.

Durbin, J., 1970. Testing for serial correlation in least squares regression when some of the regressorsare lagged dependent variables. Econometrica 38, 410—421.

Durbin, J., Watson, G.S., 1950. Testing serial correlation in least squares regression. Biometrica 37,409—428.

Engle, R.F., Grange, C.W.J., Rice, J., Weiss, A., 1986. Semiparametric estimates of the relationshipbetween weather and electricity sales. Journal of the American Statistical Association 81,310—320.

Godfrey, L.G., 1978. Testing for higher order serial correlation in regression equations when theregressors include lagged dependent variables. Econometrica 46, 1303—1310.

Hsiao, C., 1986. Analysis of Panel Data. Cambridge University Press, Cambridge.


Keane, M.P., Runkle, D.E., 1992. On the estimation of panel-data models with serial correlationwhen instruments are not strictly exogenous. Journal of Business and Economics Statistics 10,1—9.

Kniesner, T., Li, Q., 1994. Semiparametric panel data models with dynamic adjustment: Theoreticalconsideration and an application to labor supply. Discussion paper, Department of Economics,Indiana University.

Li, Q., 1996. A note on root-n-consistent semiparametric estimation of partially linear models.Economics Letters 51, 277—285.

Li, Q., Stengos, T., 1996. Semiparametric estimation of partially linear panel data models. Journal ofEconometrics 71, 389—397.

Pesaran, M.H., Smith, R., 1995. Estimating long-run relationship from dynamic heterogeneouspanels. Journal of Econometrics 68, 79—113.

Powell, J.L., Stock, J.H., Stoker, T.M., 1989. Semiparametric estimation of index coefficients.Econometrica 57, 1043—1430.

Robinson, P., 1988. Root-N consistent semiparametric regression. Econometrica 56, 931—954.


Documents

Testing serial correlation in semiparametric panel data models