A Simple Artificial Regression Based Test of the Goodness of Fit of Binary Choice Models
| DOI | http://doi.org/10.1111/1468-0084.00180 |
| Date | 01 August 2000 |
| Author | Anthony Murphy |
| Published date | 01 August 2000 |
practitioners corner
A SIMPLE ARTIFICIAL REGRESSION BASED TEST
OF THE GOODNESS OF FIT OF BINARY CHOICE
MODELS
Anthony Murphy
I. INTRODUCTION
Very often one wishes to examine the goodness of ®t of binary choice
models. Informal tests generally involve comparing actual and predicted
aggregate choice shares when the sample data set is divided into disjoint
cells or strata. Intuitively, if the model ®ts `well' then the average prediction
error in the cells should be `small'. Goodness of ®t tests along these lines
have been considered by Andrews (1988a, 1988b) and Horowitz (1985),
inter alia. These authors consider quite general models so the derivations of
the test statistics are rather complicated and more general than required
here.
In this paper a simple arti®cial regression based test of the goodness of ®t
of a binary choice model, which has been estimated by the maximum
likelihood method, is derived. The conditional moment (CM) framework of
Newey (1985) and Tauchen (1985) is used to simplify the derivation of the
test statistic. Both Newey (1985) and Tauchen (1985) suggest calculating
the CM test statistics using outer product gradient (OPG) based arti®cial
regressions. Although arti®cial regression based test statistics are very
attractive from the computational point of view, it is well known that the
empirical size of OPG based Lagrange multiplier and information matrix
tests tends to be far higher than the nominal size of tests resulting in
unreliable inferences, even in quite large samples. See Davidson and
MacKinnon (1984) and Orme (1990), inter alia. In this paper, an alternative,
non-OPG based arti®cial regression is used to calculate an asymptotically
ef®cient form of the goodness of ®t test statistic. In this alternative
approach, the information matrix is not approximated by the outer product
of the matrix of contributions to the score. Instead it is calculated as the
expectation of this outer product which generally results in a much better
small sample performance.
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 62, 3 (2000)
0305-9049
445
#Blackwell Publishers Ltd, 2000. Published by Blackwell Publishers, 108 Cowley Road, Oxford
OX4 1JF,UK and 350 Main Street, Malden, MA 02148, USA.
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