PRACTITIONERS' CORNER*: Using Gaussian Estimators Robustly
| Date | 01 February 1988 |
| Published date | 01 February 1988 |
| DOI | http://doi.org/10.1111/j.1468-0084.1988.mp50001008.x |
| Author | P. M. Robinson |
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 50,1(1988)
0305-9049 S3.00
PRACTITIONERS' CORNER*
Using Gaussian Estimators Robustly
P. M. Robinson'
I. INTRODUCTION
Parametric and semiparametric modelling of economic observables,
represented as a sequence Yi, ..., y of N observations on a p-vector-valued
random variable y,, often takes the form of conditional first and second
moment restrictions only: a subset of q or more elements of the expectations
of y, and y,y conditional on the information set up to time t, t = 1, ..., N, are
expressed as known functions of a q-dimensional parameter, whose value 00
is unknown. Because the Gaussian distribution is characterized by its first two
moments, inference on 00 can be conducted via a Gaussian pseudo-likelihood
namely a function that would be the likelihood were the y, conditionally
Gaussian. Although optimization of this pseudo-likelihood only
exceptionally provides closed-form estimation of 00 (as in the case of multiple
linear regression models), its simple algebraic form facilitates construction of
formulae for the first-order conditions for a maximum and for the Hessian.
Correspondingly, the computations are relatively tractable, and computer
packages are available for computing Gaussian estimators of a variety of
models. A further desirable property of Gaussian estimation is statistical:
under much broader assumptions than conditional Gaussianity of the y,, even
in the absence of conditional symmetry and under mild moment conditions,
Gaussian estimators are typically consistent, indeed '/Ñ-consistent with a
limiting Gaussian distribution after centring the estimator at 00 and íÑ-
forming, indicating non-zero asymptotic first-order efficiency relative to the
actual maximum likelihood estimator (mle). We emphasize that 'Gaussian
estimation' does not embrace estimation of probit models, models for
markets in disequilibrium with unknown sample separation, and related
models involving sample selectivity or truncation, based on conditionally
Gaussian latent dependent variables, where the likelihood does not have the
form of a Gaussian density, and in fact such estimators are generally
inconsistent when Gaussianity does not hold.
To form standard errors of Gaussian estimators, or to use them as the basis
for interval estimators or test statistics, their asymptotic covariance matrix
'This article is based on research funded by the Economic and Social Research Council
(ESRC) reference number: B00232156. I am grateful to David Hendry and a referee for
comments on earlier versions.
97
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