Testing the Information Matrix Equality with Robust Estimators



1 Introduction

White (1982) introduced the information matrix (IM) test as an omnibus
test for misspecification of a parametric model. The test exploits the well-
known property that, at the model, the sum of the Hessian of the log-
likelihood and the outer product of the score has zero expectation. So if,
at parameter estimates, the sample average of the sum of the Hessian and
the outer product of the score differs significantly from zero, this is evidence
against the model. The IM test is typically implemented using maximum
likelihood (ML) estimates of the parameters. In this paper we explore the
potential of replacing the ML estimator with robust estimators. Specific at-
tention is given to the effect on power, conjecturing that unmasking outliers
will lead to an increased power of the IM test. In most cases considered,
using robust estimators effectively increases the power of the test.

Past research on the IM test has mainly focussed on the poor behaviour
of the test under the null hypothesis, and on remedies to overcome this
problem (see e.g. Orme (1990) and Chesher and Spady (1991), among many
others). Considering that the use of bootstrap critical values largely solves
this problem (Horowitz, 1994), we shift our attention to the power of the
IM test.

The standard approach in the literature on the IM test is to substitute
the ML estimator for the unknown parameter in the IM equality. As an
alternative, one can use any estimator which is consistent under the model
specification. When the IM test is used in conjunction with the ML esti-
mator, the test may suffer from the masking effect. Outlying observations
will not be recognised as such (or less so, compared to robust estimators),
and hence the test will have low power against distributions with fatter tails
or when outliers are present. We show that, when robust estimators are
used, the IM test statistic still has a limiting
χ2 distribution under the null
hypothesis. An explicit expression for the asymptotic covariance matrix (
V )
of the indicator vector, to be used in the construction of the test statistic,
is derived.

We analyse the normal location-scale model in detail. It is shown that
V does not depend on the estimator of location. For the ML estimator, as
is well-known, the IM test is the Jarque-Bera (1980) test for skewness and
non-normal kurtosis, and the rank of
V equals 2. We show that, for any



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