Page 1

A Simple Heteroskedasticity and

Nonnormality Robust F-Test for

Individual E¤ects?

Chris D. Orme

Economics, School of Social Sciences, University of Manchester

and

Takashi Yamagata

Faculty of Economics, University of Cambridge

February 14, 2007

Abstract

This paper employs …rst order asymptotic theory in order to es-

tablish the asymptotic distribution of the F-test statistic for individ-

ual e¤ects, under non-normality and possible heteroskedasticity of the

errors, when N ! 1 (the number of cross-sections) and T is …xed

(the number of time periods). Whilst asymptotically valid under ho-

moskedasticity, the usual F-test and random e¤ects test procedures will

be asymptotically over-sized under heteroskedasticity. Both, however,

be easily re-scaled to provide an asymptotically valid test procedures

which exhibit the same relative power properties as those described in

Orme and Yamagata (2006).

1 Introduction

In a previous paper, Orme and Yamagata (2006) added to the already large

literature on (and text book treatment of) the analysis of variance testing,

by establishing that, in a linear panel data model, the standard F-test for

individual e¤ects remains asymptotically valid (large N; …xed T) under non-

normality of the error term. Moreover, their (local) asymptotic analysis,

supported by Monte Carlo evidence, showed that under (pure) local random

e¤ects both the F-test and random e¤ects test (RE-test) will have similar

power whilst under local …xed e¤ects, or random e¤ects which are correlated

?Address correspondence to: Chris D.Orme, Economics, School of Social Sciences, Uni-

versity of Manchester, Manchester M13 9PL, UK. Email: chris.orme@manchester.ac.uk

1

Page 2

with the regressors, the RE-test procedure will have lower asymptotic power

than the F-test procedure.

Their analysis, however, did not cover the case of heteroskedastic er-

rors in the linear model (although it did allow for heteroskedastic individual

e¤ects). This paper addresses that omission and …nds that the F-test and

RE-test statistics (hereafter denoted FNand RN; respectively) are no longer

asymptotically valid, although a straightforward transformation of each re-

covers asymptotically validity. This transformation, or correction, involves

simple functions of the pooled model’s residuals (i.e., the restricted residuals)

of which there are a number of asymptotically valid choices. Sampling exper-

iments suggest that the transformation which exploits the zero-correlation

property of the errors, provides closer agreement between nominal and ac-

tual signi…cance levels than that which does not take into account the zero-

correlation structure of the errors. The use of restricted residuals is advo-

cated by Godfrey and Orme (2004) and Davidson and MacKinnon (1985),

in a slightly di¤erent setting, who report reliable sampling performance of

heteroskedasticity robust tests of linear restrictions in the linear model when

employing restricted residuals in the construction of asymptotically robust

standard errors.1

The heteroskedastic-robust tests for individual e¤ects so constructed,

retain the qualitative properties that were reported by Orme and Yamagata

(2006). Speci…cally, (i) under (pure) local random e¤ects, both tests have

the same asymptotic power; and (ii) under local …xed e¤ects, or random

e¤ects which are correlated with the regressors, the RE-test procedure will

have lower asymptotic power than the F-test procedure.

The plan of this paper is as follows. Section 2 introduces the notation

and test statistics. Assumptions are introduced in Section 3, justifying the

ensuing asymptotic analysis which characterises the asymptotic behaviour

of the F-test statistic, including its relationship with the RE-test statistic

under the null and local alternatives. All proofs of the main propositions

are relegated to the Appendix. Section 4 illustrates the main …ndings by

reporting the results of a small Monte Carlo study and Section 5 concludes.

2The Notation, Model and Test Statistics

2.1 Notation and Model

In the standard linear panel data model, individual e¤ects for cross section

i are introduced as follows

yi= ?i?T+ Xi?1+ui;i = 1;:::;N (1)

1As Wooldridge (2002, p.265) points out, standard tests for individual e¤ects essen-

tially test for non-zero correlation in the errors; thus, constructing auto-correlation robust

procdures would appear to be counter productive.

2

Page 3

where yi= (yi1;:::;yiT)0, ui= (ui1;:::;uiT)0, ?T is a (T ? 1) vector of ones,

and Xi= (xi1;:::;xiT)0a (T ? K) matrix. The innovations, uit; have zero

mean and uniformly bounded variances, 0 < ?2

the ?iare the individual e¤ects. By stacking the N equations of (1), the

model for all individuals becomes

i< 1; for all i and t; and

y = D? + X?1+u, (2)

where y = (y0

? = (?1;:::;?N)0is a (N ? 1) vector, D = [IN? ?T] is a (NT ? N) ma-

trix, X = (X0

rank. Thus, for the purposes of the current exposition, xit= (xit1;:::;xitK)0;

(K ? 1); contains no time invariant regressors, in particular a constant term

corresponding to an overall intercept. In the context of …xed e¤ects this

allows estimation of all the unknown regression parameters,

follows.

In general, de…ne the projection matrices, PB= B(B0B)?1B0and MB=

INT?PB; for any (NT ? S) matrix B of full column rank, with~B = MDB

being the residual matrix from a multivariate least squares regression of B on

D which is, of course, the within transformation. For example, conformably

with X;~X = (~X0

xi= T?1PT

~?1= (X0MDX)?1X0MDy

= (~X0~X)?1~X0~ y

1;:::;y0

N)0and u = (u0

1;:::;u0

N)0are both (NT ? 1) vectors,

1;:::;X0

N)0is a (NT ? K) matrix, and [D;X] has full column

??0;?0

1

?0; as

1;:::;~XN)0, where~Xihas rows (xit? xi)0; i = 1;:::;N; and

t=1xitand similarly for ~ y:2Then the …xed e¤ects (least squares

dummy variable) estimator of ?1in (2) is given by

(3)

and the corresponding estimator of ? is

~ ? = (D0MXD)?1D0MXy.(4)

The null model of no individual e¤ects is the pooled regression model of

y

=?0?NT+ X?1+ u,

Z? + u;

(5)

=

where Z =[?NT;X]; with zit = (1;xit1;:::;xitK)0= fzitjg; j = 1;:::;K +

1; which delivers the ordinary least squares estimator^? =

(Z0Z)?1Z0y:

The standard F-test for …xed e¤ects requires estimation of both (2),

treating the ?ias unknown parameters, and (5) whilst the standard RE-test

only requires estimation of (5). In order to provide a framework in which

to investigate the limiting behaviour of the F-test and RE-test statistics,

?^?0;^?0

1

?0

=

2To see this, note that PD =?IN ? T?1?T?0

T

?.

3

Page 4

under both …xed and random e¤ects, the individual e¤ects are assumed to

have the form ? = ?0?N+ ?; ? = (?1;:::;?N)0. Fixed e¤ects correspond

to the ?i; i = 1;:::;N; being …xed unknown parameters (or, equivalently,

?1? 0 with ?0and ?i; i = 2;:::;N; being the …xed unknown parameters).

The case of random e¤ects is accommodated when the ?i, i = 1;:::;N are

random variables. Equations (1) and (2) will be employed to characterise

the data generation process, with the restrictions of H0: ? = ?1?Nproviding

the null model of no individual e¤ects (notice that ? = 0 belongs to this

set of restrictions). Speci…cally, when considering the alternative of …xed

e¤ects, the (N ? 1) restrictions placed on (2) are H0 : H? = 0; where

H =[?N?1;?IN?1]; whilst for random e¤ects the null is H0: var(?i) = 0:

2.2The Test Statistics

2.2.1The F-test statistic

Consider, …rst, the standard F-test for …xed e¤ects which is based on the

statistic

FN=(RSSR? RSSU)=(N ? 1)

RSSU=(N(T ? 1) ? K)

where RSSR= ^ u0^ u is the restricted sum of squares (from the pooled regres-

sion (5)) with ^ u = MZy; and RSSU= ~ u0~ u is the unrestricted sum of squares

(from the …xed e¤ects regression (2)) with ~ u = M~ X~ y, the residual vector

from regressing ~ y on~X: If normality, homoskedasticity and strong exogene-

ity were imposed such that, conditional on X; ui? N(0;?2IT); i = 1;:::;N;

then a standard F-test would be exact. The case of non-normality, but

homoskedasticity, was dealt with by Orme and Yamagata (2006). In this

paper, that analysis is extended to cover possible heteroskedasticity. It is

shown that, with T …xed and N ! 1; !N

mal distribution, where !N is a scaling factor such that !N

homoskedasticity.

,(6)

pN (FN? 1) has a limit nor-

p

! 1 under

2.2.2

De…ne vi=PT

structed as

The Random E¤ects test statistic

t=1uitand A = A0= ?T?0

s6=tuituis= u0(IN? A)u: Then, the usual RE-test statistic is con-

s

2(T ? 1)

which has a limit standard normal distribution, as N ! 1; under H0: and

homoskedasticity. By directly analysing the FN, and its asymptotic rela-

tionship with RN, it is readily established that !NRNhas a limit standard

normal distribution, under H0:

T?IT; so thatPN

?^ u0(IN? A)^ u

i=1v2

i?PN

i=1

PT

t=1u2

it=

P

i

P

t

P

RN=

NT

^ u0^ u

?

(7)

4

Page 5

3Asymptotic Properties of FN

In this section we describe the properties of FN; under both local …xed

and random e¤ects, by (i) deriving its asymptotic distribution, and (ii) es-

tablishing its asymptotic relationship with RN. In the subsequent analysis

asymptotic theory is employed in which N ! 1 and T is …xed. To facilitate

this, the following assumptions are made, which are of the sort found in, for

example, White (2001, p.120):

Assumption 1:

(i) fXi;uigN

(ii) E (uijXi) = 0:

Assumption 2:

h

1;:::;K + 1; and all i = 1;:::;N;

(ii) var?N?1=2Z0u?

?

de…nite;

h

1;:::;K + 1; and all i = 1;:::;N;

i=1is an independent sequence;

(i) Ejzisjuitj2+?i

? ? < 1 for some ? > 0; all s;t = 1;:::;T; j =

= N?1PN

?

i=1E (Z0

iuiu0

iZi) is uniformly positive de…-

nite;

(iii) varN?1=2~X0u

= N?1PN

i=1E

?~X0

iuiu0

i~Xi

?

is uniformly positive

(iv) Ejzitjzislj2+?i

? ? < 1 for some ? > 0; all t = 1;:::;T; j;l =

(v) E (Z0Z=N) is uniformly positive de…nite;

?~X0~X=N

Assumption 3:

(vi) E

?

is uniformly positive de…nite.

(i) E (uiu0

(ii) E?u2

(iii) E

i) = ?2

?= ?4

iIT; 0 < ?2

i< 1; for all i = 1;:::;N;

itu2

juitj4+?i

1;:::;N;

N= N?1PN

Assumption 4:

isi; for all i and t 6= s;

h

? ? < 1 for some ? > 0; all t = 1;:::;T; and all i =

(iv) ? ?2

i=1?2

iis uniformly positive.

(i) ?i= ?0+

?i

N1=4; i = 1;:::;N;

5

Page 6

(ii) the ?iare independent, with E j?ij2+?? ? < 1 for some ? > 0; and

all i = 1;:::;N;

(iii) N?1PN

Assumption 1 (i) re‡ects independent sampling of cross-section units

and 1(ii) imposes a strong exogeneity assumption on Xi; implying that

E(~X0

iui) = 0 and thus ruling out (for example) lagged dependent variables:

Together with Assumption 2, we obtain consistency and asymptotic normal-

ity of both the pooled and …xed e¤ects least squares regression estimators

(^? and~?1; respectively) and corresponding error variance estimators. This

follows directly the approach of White (2001, Exercise 6.8). and allows for

heteroskedastic disturbances; note that Assumption 2(i) and (iv) imply that

h

from justifying the consistency of “White” standard errors in this context,

Assumptions 2(i) and (iv) also ensure consistency of the estimated scaling

factor which transforms the FNand RN:

If Assumption 1 (ii) is weakened to E (X0

(zero contemporaneous correlation),~?1is not guaranteed to be consistent

and, when it is inconsistent, the F-test statistic is asymptotically invalid any-

way, even under normality; for example, in the presence of lagged dependent

variables - see the discussion in Wooldridge (2002, Sections 10.5 and 11.1).

Notice that Assumption 3(i) and (ii) allow cross-sectional heteroskedastic-

ity but constrains the elements of both fuitgT

uncorrelated. Assumptions 3(i)-(iv), justi…es the limit distribution obtained

in Proposition 1 below. In particular, although the assumptions ensure that

? ?2

Nis O(1); at most, it is not required to converge to a particular limit.

Assumption 4 characterises the alternative data generation process and

permits the investigation of asymptotic power, under local individual ef-

fects, by restricting the test criteria under consideration to be Op(1) with

well de…ned limit distributions. As well as …xed e¤ects (with the ?ibeing

non-stochastic) it also accommodates local heteroskedastic random e¤ects:

If the "iare also distributed independently of Xi; then we have “pure” ran-

dom e¤ects whilst if the "iare correlated with Xithen we have “correlated”

random e¤ects. (As pointed out by Wooldridge (2002, p.252), in micro-

econometric applications of panel data models with individual e¤ects, the

term …xed e¤ect is generally used to mean correlated random e¤ects, rather

than ?ibeing strictly non-stochastic.)

In the analysis that follows, it will be useful to note that ?0D0D? = T?0?

and Z0D? = TPN

E[Z0Z=N]! 0; where convergence is understood to be element by ele-

ment, with E [Z0Z=N] = O(1); and uniformly positive de…nite, although it

need not converge to any particular limit, and~Z0~Z=N ? E[~Z0~Z=N]

i=1E??2

i

?is uniformly positive; where ?0= (?1;:::;?N):

both Ej~ xisjuitj2+?i

and E

h

j~ xitj~ xislj2+?i

are uniformly bounded. Apart

iui) = 0; or even E (xituit) = 0

t=1and?u2

it

?T

t=1to be serially

i=1? zi?i = T?Z0?; where?Z; (N ? K + 1); has rows ? z0

it: Furthermore, Assumptions 2(iv)-(vi) ensures that Z0Z=N ?

as

i=

T?1PT

t=1z0

as

! 0,

6

Page 7

with E[~Z0~Z=N] = O(1), at most. Together with Assumption 4,?Z0?=N ?

E[?Z0?=N]

0; with E??0?=N?= O(1) and uniformly positive. Again, observe that both

as

! 0; with E[?Z0?=N] = O(1); at most, and ?0?=N ?E??0?=N?as

E??0?=N?and ? ?2

3.1The Asymptotic Distribution of FN

!

Nare O(1); at most, but need not necessarily converge.

The asymptotic distribution of FN, under non-normality, is given by follow-

ing proposition:

Proposition 1 Under model (2) and Assumptions 1 to 4,

Op(1); with

pN (FN? 1) =

pN (FN? 1) =

1

pN

[u0(IN? A)u]

(T ? 1) ? ?2

N

+?N

? ?2

N

+ op(1)

and

8

:

<

? ?2

N

q

N?1PN

i=1?4

i

9

;

=

pN (FN? 1) ?

?N

q

N?1PN

i=1?4

i

d

! N

?

0;

2T

T ? 1

?

where ?N= O(1) and is de…ned by

?N

=E

??0

1?1

N

?

= ?N? ?0

N??1

N?N? 0

?1

=

D? ? Z??1

N?N;

?N= E [Z0Z=N], ?N= E [Z0D?=N]; ?N= E??0D0D?=N?:

Proof. See Appendix.

Notice that, for example, ?Nneed not necessarily converge, but it will be

O(1); at most, and ?N??0D0MZD?

e¤ects with ? = ?1?N; ?N ? 0; as it should be (this includes the case of

? = 0), and !N

! N(0;1): Here, !N =

for all N; but under homoskedasticity of the errors, E?u2

that !N

! 1:

As exploited by Orme and Yamagata (2006) in the homoskedastic case,

it is easy to show that if ?Nhas an F distribution with n1= N ? 1 and

n2 = N(T ? 1) ? K degrees of freedom, then ??

N

as

! 0: In the special case of no individual

pN (FN? 1)

d

? ?2

N

q

N?1PN

it

i=1?4

i

? 1

?= ?2; it follows

p

N=

q

N(T?1)

2T

(?N? 1) s

7

Page 8

N(0;1); or approximately for large N; ?N

As N

?

1;

2T

N(T ? 1)

?

. Therefore,

in the heteroskedastic case and under the null (where ?N= 0); we can write

!?

N

pN (FN? 1)

d

! N

?

0;

2T

T ? 1

?

or, approximately,

!?

NfFN? 1g + 1

As F (n1;n2):

where !?

Nis such that !?

N? !N

p

! 0:

Remark 1 Since !N ? 1; for all N; it follows that the usual F statistic,

FN; whilst being asymptotically valid under homoskedasticity will be asymp-

totically oversized under heteroskedasticity.

The question is how to …nd such a ~ !N: This is given by the following

Proposition.

Proposition 2 De…ne ^ ?2= ^ u0^ u=(NT ? K ? 1); ^ ui = yi? Zi^? =f^ uitg;

~ ui= ~ yi?~Xi~?1= f~ uitg: Under model (2) and Assumptions 1 to 4,

1. ^ ?2? ? ?2

2. N?1PN

3. N?1PN

Proof. See Appendix.

N

p

! 0;

?P

i=1

i=1

t

P

t6=s^ u2

t6=s^ uit^ uis

?2

? 2T (T ? 1)1

N

PN

i=1?4

i

p

! 0; and,

P

t

P

it^ u2

is? T (T ? 1)1

N

PN

i=1?4

i

p

! 0:

From this analysis it follows that asymptotically valid choices for !?

include

N

^ !(1)

N

=

^ ?2

r

1

2NT (T ? 1)

PN

i=1(^ u0

iA^ ui)2

;(8)

^ !(2)

N

=

^ ?2

r

1

NT (T ? 1)

PN

i=1^ e0

iA^ ei

;(9)

where ei =

test statistics can then be constructed as ^ !j

Finally, notice that, under the null

?u2

it

?

and riAri =PP

t6=sritris; with ri = fritg: Robust F

NfFN? 1g + 1; j = 1;2:

!N

r

T ? 1

2T

pN(FN? 1)

d

! N (0;1):

8

Page 9

Under pure local random e¤ects, ?N= ?T

?N = TE

N

= T?2: Then, with a slight adaptation of the proof, we

obtain the following Corollary to Proposition 1:

N

PN

i=1E ["i? zi] = 0; so that

h

?0?

i

Corollary 1 Under the alternative of (pure) local random e¤ects, and under

the assumptions of Proposition 1,

!N

pN (FN? 1) ?

T?2

q

N?1PN

i=1?4

i

d

! N

?

0;

2T

T ? 1

?

Therefore, the F-test will have non-trivial asymptotic local power against

random e¤ects. In fact, a stronger result will be established in Section 3.2

which shows that, under (pure) local random e¤ects, the F-test statistic is

proportional (asymptotically) to the RE-test statistic, and will thus possess

the same asymptotic power. However, under “correlated” local random

e¤ects the F-test will possess higher asymptotic power than the RE-test.

3.2The Relationship with RN

Under the null of no individual e¤ects, it is readily shown that

?^ u0(IN? A)^ u

1

pN

^ u0^ u=NT

?

=

1

pN

1

pN

?u0MZ(IN? A)MZu

u0(IN? A)u

? ?2

N

u0M0

Zu=NT

?

=+ op(1)

using the fact that N?1=2u0MZ(IN? A)MZu = N?1=2u0(IN? A)u +

op(1) and u0MZu=NT ? ? ?2

s

2NT (T ? 1)

r

2T

r

2T

N

p

! 0: From Proposition 1, therefore, we have

?^ u0(IN? A)^ u

T ? 1

pN

(T ? 1) ? ?2

T ? 1

RN

=

1

^ u0^ u=NT

?

=

1

u0(IN? A)u

N

+ op(1)

=

pN (FN? 1) + op(1);

under the null. Moreover, !NRN

same scaling factor that is applied to the F-test statistic is applied to the

RE test statistic in order to construct an asymptotically valid procedure.

Notice, again, that without such an adjustment the standard RE test will

be asymptotically over-sized.

The following proposition extends this result to the case of local individ-

ual e¤ects (…xed or random).

d

! N(0;1); under the null, so that the

9

Page 10

Proposition 3 Under model (2) and Assumptions 1 to 4,

(r

2T

RN=

(T ? 1)

)pN [FN? 1] ?

s

T

2(T ? 1)

?N

? ?2

N

+ op(1)

where ?N= O(1) de…ned by

?N

=E

??0

N?N

2?2

N

?

= ?0

N??1

N~?N??1

N?N? 0

?2

=

~Z??1

~?N = E[~Z0~Z=N]; and the limit distribution of

Proposition 1.

Proof. See Appendix.

pN [FN? 1] is given by

Again, ?Nneed not converge, but it is O(1) and ?N??0D0Z(Z0Z)?1(~Z0~Z)(Z0Z)?1Z0D?

0: As with Proposition 1, ?N? 0 obtains under H0: ? = ?1?N; as it should,

since (Z0Z)?1Z0D? = (?1;00)0and the top-left, (1;1); element of~Z0~Z is

0. As discussed above, under the alternative of (pure) local random e¤ects

?N= 0; and we obtain the following Corollary:

N

as

!

Corollary 2 Under the alternative of (pure) local random e¤ects, and under

the assumptions of Proposition 1,

(r

2

!NRN?

T (T ? 1)

)

?2

q

N?1PN

(FN? 1) are asymptotically equivalent,

i=1?4

i

d

! N (0;1):

Since RN and F?

both the RE and F-test procedures will have identical asymptotic power

functions, under (pure) local random e¤ects. However, under local …xed

e¤ects or random e¤ects which are correlated with Xi; the F-test can have

greater asymptotic power. In particular, when individual e¤ects are corre-

lated with the mean values of the regressors, ?N6= 0 and is O(1); implying

?N> 0 so that a test based on RN (but suitably robust to heteroskedas-

ticity) should have lower asymptotic local power than one based on FN:

This makes intuitive sense, since FNis designed to test for individual e¤ects

which are correlated with ? zi; whereas RNis constructed on the assumption

that the individual e¤ects are uncorrelated with all regressor values. The

importance of distinguishing between individual e¤ects which are correlated

or uncorrelated with regressors, rather than simply labelling them …xed or

random, is discussed by Wooldridge (2002, Section 10.2).

In the next section, the preceding analysis is supported by the results of

a small Monte Carlo experiment which illustrates the asymptotic robustness

of the F-test to non-normality/heteroskedasticity and its power properties

relative to the RE-test.

N=

q

N(T?1)

2T

10

Page 11

4Monte Carlo Simulation

In order to shed light on the relevance of the preceding asymptotic analy-

sis for …nite sample behaviour, this section reports the results of a small

Monte Carlo experiment which investigates the sampling behaviour of the

test statistics considered above under a variety of error distributions using

N = 20;50;100, T = 5. The model employed is

yit= ?i+

3

X

j=1

zit;j?j+ uit, uit= ?i"it

where zit;1= 1, zit;2is drawn from a uniform distribution on (1;31) inde-

pendently for i and t, and zit;3is generated following Nerlove (1971), such

that

zit;3= 0:1t + 0:5zit?1;3+ ?it,

where the value zi0;3is chosen as 5+10?i0, and ?it(and ?i0) is drawn from

the uniform distribution on (?0:5;0:5) independently for i and t, in order to

avoid any normality in regressors. These regressor values are held …xed over

replications. Also, observe that the regression design is not quadratically

balanced.3

Table 1 shows the largest value of hs=?h, where hs is the sth

diagonal elements of PZ, s = 1;2;:::;NT and?h is the average of hs, and the

number of leverage points, where hs=?h > 2,4con…rming that the regressors

used are not quadratically balanced.

Without loss of generality, the coe¢cients are set as ?j= 1 for j =

1;2;3 and the error terms, uit, are all iid(0;1) in the experiments. They

are drawn from the following distributions and standardised: (i) standard

normal distribution (SN); (ii) Student’s t distribution with 5 degrees of

freedom (t(5)); (iii) uniform distribution over the unit interval (UN); (iv)

mixture normal distribution from either N (?1;1) and N (1;1) with equal

probability of 0:5 (MN); (v) log-normal distribution (LN); and, (vi) chi-

square distribution with 2 degrees of freedom (?2(2)).

The sampling behaviour of three tests are investigated using 5000 repli-

cations of sample data and employing a nominal 5% signi…cance level based

on the predictions of the asymptotic theory presented in Section ??.

The following test statistics are considered. The !?

tion 3.1.

Nare de…ned in Sec-

1. Modi…ed F-Test statistics (denoted F in Tables)

F?

N= !?

N(FN? 1) + 1;

3The results of Ali and Sharma (1996) show that, with a quadratically balanced design,

the e¤ects of non-normality on the F-test for linear restrictions in the linear model is

minimal. Hence the Monte Carlo design guards against that possibility.

4See Belsley et al. (1980) for the discussion of leverage points.

11

Page 12

where

FN=(RSSR? RSSU)=(N ? 1)

RSSU=(N(T ? 1) ? K)

;

which is the F-Test statistic. Use these in conjunction with critical

vales from an F distribution with n1and n2degrees of freedom, respec-

tively, where n1= N ?1 and n2= N(T ?1)?K. That is, reject H0if

FN> cN;?; where Pr(? > cN;?) = ?; for chosen ?; and ? ? F (n1;n2)

2. One sided (positive) Modi…ed F-Test statistics (denoted F(N) in Ta-

bles)

q

in conjunction with (right-hand) critical values from a N (0;1) dis-

!?

N

N(T?1)

2T

(FN? 1)

tribution. That is, reject H0 if !?

Pr(Z > z?) = ?; for chosen ?; and Z ? N (0;1).

3. One sided (positive) Modi…ed Random E¤ects Test Statistic (denoted

R(N) in Tables)

!?

NRN

N

q

N(T?1)

2T

(FN? 1) > z?; where

where

RN=

s

NT

2(T ? 1)

?^ u0(IN? A)^ u

^ u0^ u

?

which is the One sided (positive) Random E¤ects Test Statistic, under

homoskedastic assumption, in conjunction with (right-hand) critical

values from a N (0;1) distribution. That is, reject H0if !?

where z?de…ned in 2.

NRN> z?;

4. Modi…ed Random e¤ects statistic (denoted R(F) in Tables)

(s

N (T ? 1)

2T

)

!?

NRN+ 1

in conjunction with critical vales from an F distribution with n1and

n2 degrees of freedom, respectively, where n1 = N ? 1 and n2 =

N(T ? 1) ? K. That is, reject H0if FN> cN;?; where cN;?is de…ned

in 1.

Four versions of the above each test statistics, associated with the choices

of !?

N, are considered. First version uses

!?

N= 1

12

Page 13

which gives the tests under homoskedastic assumption. The second choice

is based on the true information ?2

i, namely

!?

N= !N=

? ?2

N

q

N?1PN

N, ^ !(2)

i=1?4

i

which give the benchmark results. Observe that under homoskedasticity,

!N = 1. Other choices of !?

(9), respectively.

We allow both individual e¤ects and errors to have quite general patterns

of heteroskedasticity. Note that our assumptions (under local alternatives)

allow for the ?ito be heteroskedastic. Furthermore, it would be useful to

investigate the behaviour of the hetero-robust procedures when in fact the

idiosyncratic errors are homoskedastic.

We consider three speci…cations of ?i:

Nare ^ !(1)

N, which are de…ned by (8) and

1. Homoskedasticity

?i= ? = 1

2. Heteroskedasticity1

?i

=?1; i = 1;:::;n1

=?2; i = n1+ 1;:::;N

with n1= N=2, ?1=p0:2 and ?2=p1:8.

3. Heteroskedasticity2

?i=

p

?i(vi0+ 0:5)

where vi0is correlated with zit;3, ?i(?) is the inverse of the cumulative

distribution function of the chi-squared distribution with degrees of

freedom c, divided by c. In addition, since (vi0+0:5) ? iid U(0;1), ?i?

iid ?2(c)=c, which has mean 1 and variance 2=c, so it is easy to control

the degree of heteroskedasticity by changing c. c = 2 (var(?2

has been examined.

i) = 1)

Employing the standardised distributions that we used before,

hp

where the 'iare iid N(0;1), gi(? zi) = ?0

being overall average of zit, s being the standard deviation of ?0

R2is from the regression of (10). With this set up, the variance of inside of

the square brackets is always unity across designs.

The following choices for R2and ?iare made:

?i= ?i

R2gi(? zi) +

p

1 ? R2'i

i

(10)

3(? zi? ? z)=s with ?3 = (1;1;1)0, ? z

3? zi, and the

13

Page 14

Design 1 ?i= 0; R2= 0.

This is a simple null model speci…cation, with ?i? 0.

Design 2 ?i= v?, R2= 1:0:

This is simple …xed e¤ects speci…cation (given that the zitare …xed

over replications). Putting R2= 0:5 bears comparison with the fol-

lowing four speci…cations:

Design 3 ?i= v?; R2= 0

This is a pure (homoskedastic) random e¤ects speci…cation.

To control the power, we consider the values v2

?= 0:1.

4.1Results

Tables 1-3 provide the results of the above experiments. As predicted by

the previous asymptotic theory, the non-heteroskedastic-robust tests tend to

reject the null hypothesis too much under heteroskedasticity. Table 1 reports

the rejections rates of the various tests under the null of no individual e¤ects.

To begin with, let us look at the size of the benchmark test statistics (with

!?

N= !N). Across experiments, F and R(N) tests have correct size. Among

the choice of feasible !?

benchmark tests, and the size distortion becomes smaller as N increases. As

the asymptotic analysis predicts, the power of the …xed e¤ects tests is much

higher than that of the random e¤ects tests when the individual e¤ects are

correlated with regressors (see Table 2). On the other hand, and again as

predicted, the power of these tests becomes very similar when the individual

e¤ects and regressors are uncorrelated (see Table 3).

N, the tests based on ^ !(2)

Ngive most close size of the

5Conclusions

This paper has provided an asymptotic analysis of the sampling behaviour of

the standard F-test statistic for …xed e¤ects, in a static panel data model,

under non-normality and heteroskedasticty of the error terms, and …xed

time periods. A simple transformation of the commonly cited F and RE

tests (using a simple function of restricted residuals) provides asymptot-

ically valid test procedures, when employed in conjunction with the usual

F and standard normal critical values (respectively), and these procedures

exhibit good agreement between nominal and actual signi…cance levels in

sampling experiments. The second contribution is to show that the asymp-

totic relationship between the heteroskedastic robust F-test and the RE-test

statistics, carries over from the homoskedastic case. That is, under (pure)

local random e¤ects, they share the same asymptotic power, whilst under

14

Page 15

local …xed (or correlated) individual e¤ects the heteroskedastic robust F-

test enjoys higher asymptotic power. Again, these theoretical …ndings are

supported by Monte Carlo evidence.

References

[1] Ali, M. M. & S. C. Sharma (1996) Robustness to nonnormality of re-

gression F-tests. Journal of Econometrics 71, 175-205.

[2] Belsley, D. A., E. Kuh & R. E. Welsch (1980) Regression Diagnostics.

New York: John Wiley & Sons.

[3] Davidson, R., MacKinnon, J.G., (1985). Heteroskedasticity-robust tests

in regression directions. Annales de l’INSEE 59/60, 183-218.

[4] Godfrey, L.G. and C.D. Orme, (2004) Controlling the Finite Sample

Signi…cance Levels of heteroskedasticity-Robust Tests of Several Linear

Restrictions on Regression Coe¢cients. Economics Letters, 82, 281-287.

[5] Nerlove, M. (1971) Further evidence on the estimation of dynamic eco-

nomic relations from a time-series of cross-sections. Econometrica 39,

359-382.

[6] Orme, C.D., and T. Yamagata, (2006). The Asymptotic Distribution

of the F-Test Statistic for Individual E¤ects. Econometrics Journal, 9,

404-422.

[7] White, H. (2001) Asymptotic Theory for Econometricians. Revised Edi-

tion. Academic Press.

[8] Wooldridge, J. M. (2002) Econometric Analysis of Cross Section and

Panel Data. MIT Press.

15