A Correction for Local Biasedness of the Wald and Null Wald Tests
| Author | Kim‐Leng Goh,Maxwell L. King |
| Date | 01 August 1999 |
| DOI | http://doi.org/10.1111/1468-0084.00137 |
| Published date | 01 August 1999 |
A CORRECTION FOR LOCAL BIASEDNESS OF THE
WALD AND NULL WALD TESTS
Kim-Leng Goh and Maxwell L. Kingy
I. INTRODUCTION
It is well known that uniformly most powerful tests cannot exist against
two-sided alternative hypotheses without the restriction to only those tests
which are locally unbiased. A locally unbiased test is one for which power
is not below size near the null hypothesis. The Wald test of a single
restriction is a popular test that is asymptotically locally most powerful
unbiased but typically biased in small samples. This latter point was
demonstrated by Peers (1971) who derived the power function to O(nÿ1=2)
for a simple null hypothesis against a class of two-sided composite
alternative hypotheses where nis the sample size. Hayakawa (1975) consid-
ered the asymptotic expansion of the distribution of the Wald statistic under
a sequence of alternatives converging to a composite null hypothesis at rate
nÿ1=2, and reached the same conclusion.
Our concern in this paper is with testing a restriction on a single
parameter against a two-sided alternative using a small sample. In such
circumstances, it is not unusual to ®nd that the power of the Wald test drops
below its size on one side of the null hypothesis while rising on the other
side. Also, the Wald test based on asymptotic critical values is known to
have poor ®nite-sample size properties and bootstrap methods have been
advocated to ®nd ®nite-sample critical values in order to get the size of the
test right (see, e.g., Horowitz and Savin, 1992). We take this bootstrapping
approach a step further to ®nd an additional ®nite-sample `critical value'
(which we refer to as a correction factor) that ensures the power curve is
approximately symmetrical in the neighbourhood of the null hypothesis in
order to overcome local biasedness.
Another problem with the Wald test is the possibility that its power can
®rst increase and then decline to zero as the true model moves further and
further away from the null hypothesis. Empirical evidence of this possible
non-monotonic behaviour of the Wald test has been reported by Hauck and
Donner (1977), Vaeth (1985), Mantel (1987), Nelson and Savin (1988,
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 61, 3 (1999)
0305-9049
435
#Blackwell Publishers Ltd, 1999. Published by Blackwell Publishers, 108 Cowley Road, Oxford
OX4 1JF,UK and 350 Main Street, Malden, MA 02148, USA.
yWe would like to thank Anindya Banerjee for helpful comments and acknowledge partial
®nancial support from an ARC grant. This paper was written while the ®rst author was on study
leave from the University of Malaya.
1990) and Laskar and King (1997). Goh and King (1997) and Laskar and
King (1997) have demonstrated that the null Wald test based on a sugges-
tion of Mantel (1987) can help solve this problem. We show that in small
samples, the null Wald test can also suffer from local biasedness which can
be corrected for using our suggested procedure.
Given the problems associated with the Wald test, one may want to
consider an alternative procedure such as the likelihood ratio (LR) test.
However, the LR test is also beset with problems of local biasedness in
some instances. Theoretical results have been reported by Hayadawa (1975)
and Harris and Peers (1980), and this paper provides some further empirical
evidence. The asymptotic approximations work better for the LR test com-
pared to the Wald test, but not always satisfactorily and therefore the
bootstrap also needs to be employed to size-correct the test (see McManus
et al., 1994). We do not propose a bias-corrected LR test, as this involves
more technical and computing intricacies than the bias- and size-corrected
Wald test because both constrained and unconstrained maximum likelihood
(ML) estimators are required in computing the LR test. As we will see later,
the simplicity of the local-biasedness corrective procedure lies in the fact
that only unconstrained ML estimates are needed to compute the Wald and
null Wald tests.
The plan of this paper is as follows. Section II outlines the new procedure
for rectifying local biasedness in the Wald and null Wald tests. A Monte
Carlo experiment, designed to examine the small-sample properties of the
various tests considered for a linear as well as a non-liner restriction, is
reported in Section III. The ®nal section contains some concluding remarks.
II. CORRECTION FOR LOCAL BIASEDNESS
Let yt,t1, ...,n, denote nindependent observations distributed with
density f(ytjxt,è) which depends on a vector of exogenous variables xtand
an unknown k31 parameter vector è. Let è(â,ã9)9, where âis a single
parameter of interest and ãtherefore is a vector of nuisance parameters.
Suppose it is desired to test
H0:h(â)0 against Ha:h(â)6 0
where h(â) is a continuously differentiable function of âonly, that can be
linear or non-linear. We consider the class of hypotheses where âassumes a
unique value, say â0, for both type of restrictions under H0. The Wald test is
based on the statistic
W(h(^
â))2A(^
è)ÿ1(1)
where A(è)H(â)V(è)H(â)9,H(â)@h(â)=@è9,V(è) is the asymptotic
variance-covariance matrix for ^
èand parameters marked with ^represent
their unconstrained ML estimators.
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