Bartlett-corrected tests for heteroskedastic linear models

economics letters ELSEVIER

Economics Letters 48 (1995) 113-118

Bartlett-corrected tests for heteroskedastic linear models F r a n c i s c o C r i b a r i - N e t o a'*, Silvia L.P. F e r r a r i b aDepartment of Economics, Southern Illinois University, Carbondale, IL 62901-4515, USA bDepartamento de Estatistica, Universidade de S~o Paulo, Caixa Postal 20570, $5o Paulo/SP, 01452-990, Brazil

Received 6 June 1994;accepted12 August 1994 Abstract

This paper uses the results in Rothenberg (Econometrica, 1984, 52, 827-842) to derive Bartlett-corrected tests for heteroskedastic linear models. The corrections are applied to three commonly used test statistics, namely: the likelihood ratio, Wald and Lagrange multiplier statistics. JEL classification: C12

1. Introduction

Test statistics having an asymptotic chi-squared distribution are commonly used by applied researchers in economics and other fields. Although the chi-squared distribution is known to approximate well the null distribution function of such test statistics for large sample sizes, there is no guarantee that it provides a reliable approximation in samples of small to moderate sizes. It has even been shown that asymptotically equivalent tests can deliver conflicting inference when applied to the same data set (e.g. Berndt and Savin, 1977; Breusch, 1979; Evans and Savin, 1982; Savin, 1976). It is thus important to obtain corrected tests, i.e. modified testing procedures with better finite-sample behavior. The corrections to the likelihood ratio statistic date from 1937 (Bartlett, 1937) and were developed in full generality in 1956 (Lawley, 1956). Progress in the improvement of Lagrange multiplier and Wald tests has been much slower. Corrections to Lagrange multiplier test statistics were proposed by Cordeiro and Ferrari (1991); see also Chandra and Mukerjee (1991) and Taniguchi (1991). A correction to the Wald statistic for the test of nonlinear restrictions was derived by Ferrari and Cribari-Neto (1993). Corrections to Lagrange multiplier and Wald statistics are commonly referred to as Bartlett-type corrections. A similar literature has focused on size-corrections to critical values obtained as C o r n i s h * Corresponding author. 0165-1765/95/$09.50 t~) 1995 Elsevier Science B.V. All rights reserved SSDI 0165-1765(94)00597-4

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F. Cribari-Neto, S.L.P. Ferrari / Economics Letters 48 (1995) 113-118

Fisher inversions to E d g e w o r t h expansions. A n asymptotic expansion to the null distribution function of the likelihood ratio statistic was given by H a y a k a w a (1977); see also Cordeiro (1987) and Harris (1986). T h e use of i m p r o v e d critical values for likelihood ratio tests is not c o m m o n since, as m e n t i o n e d above, a correction that can be directly applied to test s t a t i s t i c s - k n o w n as the Bartlett c o r r e c t i o n - has long been known. H o w e v e r , given the difficulty of obtaining similar corrections to other testing criteria, m a n y researchers have focused on deriving corrections to critical values. See, for instance, Harris (1985), H a y a k a w a and Puri (1985), Phillips and Park (1988) and R o t h e n b e r g (1984). T h e purpose of this paper is to show how to obtain corrections that can be directly applied to three c o m m o n l y used test statistics (likelihood ratio, Wald and Lagrange multiplier) in the heteroskedastic linear regression model. This is an alternative to the approach in R o t h e n b e r g (1984) who considers corrections to be applied to critical values. T h e plan of the paper is as follows. Section 2 presents the m o d e l and test statistics. In Section 3 we obtain Bartlett-corrected statistics and c o m p a r e the Bartlett-corrected tests with the i m p r o v e d tests based on size-corrected critical values. Section 4 concludes the paper.

2. The model

We consider the linear model y=X/3 +e, where y is an n-vector of responses, X is an n x p matrix of fixed regressors with rank p , / 3 is a p - v e c t o r of u n k n o w n parameters and e is an n-vector of disturbances that has a n o r m a l distribution with m e a n 0 and covariance matrix g2-1, with the elements of g2 being s m o o t h functions of an u n k n o w n d-vector of parameters, 0. It is assumed that X ' g 2 ( O ) X is positive definite for all 0 and that p and d are small relative to n. T h e null hypothesis of interest is H 0 : R/3 = r, where R is a q × p matrix of rank q and r is a q-vector of constants, against a two-sided alternative. Let ~ = (/~', t~)' and 4 = (/~', 0')' d e n o t e the unconstrained and constrained m a x i m u m likelihood estimates of Q = (/3', 0')', respectively, and define ~ = g2(0) and g~ = g2(0). It is a s s u m e d that x/fi(b - O)-*N(O, d A), where

A-1 = [A/j]= [lira (2n)-1 tr(/2 -1 ~ 0 -i S2j)], and x/-~(O - O) = v + Op(n-1), where E(v) =/z/x/ft. T h e likelihood ratio ( L R ) , Wald (W) and Lagrange multiplier ( L M ) statistics can then be written as L R = - 2n [L( ~ - L(~)], W= (Rfi - r)'[R(X' g~X)-IR']-I(R~ LM = (R~ - r)'[R(X'~X)-IR']-I(R~

- r) , - r) ,

respectively, where L is the log-likelihood function and /3 -- ( X ' ~ X ) - l X ' ~ y . problems, these three statistics are asymptotically distributed as X~ u n d e r H 0.

In regular

F. Cribari-Neto, S.L.P. Ferrari / Economics Letters 48 (1995) 113-118

115

3. Bartlett-corrected tests U n d e r certain regularity conditions, R o t h e n b e r g (1984) o b t a i n e d E d g e w o r t h expansions to t h e null distribution functions of LR, W and L M (in w h a t follows d e n o t e d as S 1, S 2 a n d $3, respectively). It is possible to write these expansions as 1

2

Vr[Sj <-x] = Fq(X) + - - Z 3~,fq+2t(x) hi= 1

+ o(n-l)

(1)

j = 1,2,3, w h e r e Fs(. ) and f,(.) d e n o t e the distribution function and the probability density f u n c t i o n of a r a n d o m variable distributed as X 2, respectively, and a

3'11=

3'12----0,

2'

a+b

C

3'21= - - - 2- '

3'22~"=:

a-b

2 '

c 3'32 = "2""

3'31= - - - 2 , Here,

a = t r ( 2 D - C), b = 2tr(B) and c = tr(C) with B = g i E~ XijP'O~[g2-1 _ c = z, E; a , [ c f , + 0 . 5 G tr(Ci)], D = E i Cd, i + E, gj aij(C~Cj + 0.5Dd), w h e r e t h e n o t a t i o n is as follows: g2,. = O0/OOi, Q = [R(X'I2X)-IR'] -1/2, P = X ( X ' I 2 X ) - I R ' Q , C i P'g2,P, O,j= P'[O i j - a , X ( X ' O X ) - I x ' o j - a j X ( X ' O X ) - I x ' a , IP. M a g d a l i n o s (1992) s h o w e d that the e x p a n s i o n in (1) is i n d e e d a valid expansion. W e shall m a k e use of the following relations: xf,,(x)= [ F , , ( x ) - F m + 2 ( x ) ] m / 2 , x2fm(x)= [Fm+2(x ) - Fm+a(X)]m(m + 2 ) / 2 , mfm+2(x ) = Xfm(X ) and m(m + 2)fm+4(x ) = X2fm(X). U s i n g these relations t o g e t h e r with (1) we get

x(x'ax)-'x']gP, =

1{

Pr[Sj<-x]= Fq(x) + n

x

x2

}

3~l q fq(X) + yj2 q(q + 2) fq(X) + o(n -1)

3~2 (Vq+2(x) - Fq+4(x))}_ + o(n -1) = Fq(x) + n1 {~-~1 ~ (Vq(x) - Fq+2(x)) + -~1

2

= Fq(x) + n l~=oajlFq+zl(X) +

°(n-l)

(2)

'

where 3'H a l ° --

2 -

3'12 -- 3'11

a

4 '

3'21 2 -

a+b

a2°a3°-

3'31 2 -

a -b 4

4

an = '

a21-

'

a31-

2

a

=4-'

3'22--721 a+b- c 2 4 ' 3'32 -- 3'31

2

-

a - b + c 4 '

3')/12 a12--

~---0, ')'22

C

a22- - 2 - 4 ' 3'32

a32-

It is possible to show that (see C o r d e i r o a n d Ferrari, 1991, p. 581)

T

C

-

4 "

F. Cribari-Neto, S.L.P. Ferrari / Economics Letters 48 (1995) 113-118

116

S j = Sj 1 -

n i=~=l _ aj, /z/ oj

S'

w h e r e / ~ is the ith m o m e n t about zero of a chi-squared r a n d o m variable with q degrees of f r e e d o m , is distributed as X2q w h e n terms of order smaller than n -1 are ignored. That is, Pr[S 7 -
Sj, =Sj 1 -

\

nq

+ n q ( q + 2 ) Sj

)}

'

j = 1,2,3, have a X2q distribution to order n -1. That is, i m p r o v e d tests can be p e r f o r m e d by c o m p a r i n g the modified statistics

W* = W 1 - \ - ~ -

+ 2nq(q + 2) W

,

c

LM*= LM 1-

2nq

2nq(q + 2) L M

)}

to the tabulated X:q critical values. T h e correction p r o p o s e d above is an alternative to the one suggested by R o t h e n b e r g (1984). His correction is to be applied to the critical values (rather than to the test statistics themselves) yielding Z L R ~" Z a

q-

z w = z,~ 1 + \ - - f ~ + 2nq(q + 2) z,~

( ZLM = Z~, 1+

c -2nq

,

)}

2nq(q + 2) z'~

,

where z~ is the o~th percentile of a X2q distribution. His corrected test then c o m p a r e s the unmodified statistics with the above size-corrected critical values. It should be r e m a r k e d that it is possible to use (2) to obtain m o m e n t s to order n -1 of the three test statistics considered in this paper. For the LR statistic, we have that a E ( L R ) = q +~-ffn + o(n-1), 2a v a r ( L R ) = 2q + --n-- + °(n-l)" Similarly, the first two central m o m e n t s of W are given by

F. Cribari-Neto, S.L.P. Ferrari / Economics Letters 48 (1995) 113-118

E(W) = q +

a+b+c 2n

v a r ( W ) = 2q +

117

+ °(n-a)'

2(a + b + 2c) n + °(n-l)

"

Finally, for the L M statistic we get a-b

E(LM) = q + v a r ( L M ) = 2q +

-c 2n

+ °(n-~) '

2(a - b - 2c) n

+ o(n-a).

Since E(X2q)=q and var(X2q)= 2q, the coefficients a, b and c can be used to m e a s u r e the distance to order n-~ b e t w e e n the m o m e n t s of the test statistics and the m o m e n t s implied by the first-order asymptotic chi-squared null approximation. W h e n a, b and c involve u n k n o w n parameters, as is usually the case, they should be replaced by consistent estimates, and this does not change the order of the approximations. W h e n the null hypothesis involves linear combinations of parameters, it is easier to obtain unrestricted estimates. In this case, a, b and c can be replaced by ~, /~ and 6, w h e r e h = t r ( 2 / ) - (7), /~ = 2tr(/~) and ~ = t r ( C ) , with /~ = B ( O ) , C = C ( O ) and b = D ( O ) .

4. Concluding remarks This paper shows how to obtain corrected likelihood ratio, Wald and Lagrange multiplier statistics for the heteroskedastic linear model. Improved tests based on such modified statistics constitute an alternative to tests based on size-corrected critical values.

Acknowledgements We wish to thank Gauss Cordeiro for useful comments. The financial support of C N P q / Brazil is also gratefully acknowledged.

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