PRACTITIONERS’ CORNER: Double Length Artificial Regressions†
| Author | Russell Davidson,James G. MacKinnon |
| Published date | 01 May 1988 |
| Date | 01 May 1988 |
| DOI | http://doi.org/10.1111/j.1468-0084.1988.mp50002008.x |
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 50,2(1988)
0305-9049 $3.00
PRACTITIONERS' CORNER
Double Length Artificial Regressionsl
Russell Davidson and James G. MacKinnon
I. INTRODUCTION
In recent years applied econometricians have become fajìiliar with the idea
that 'artificial' regressions may provide a convenient way to compute many
test statistics. Such regressions are run after the model of interest has been
estimated, using constructed variables which depend on parameter estimates
and on the hypotheses to be tested. Test statistics may then be computed in
several ways, perhaps as the explained sum of squares from the artificial
regression or as n (the sample size) times the uncentred R 2, both of which are
often derived as variants of the LM statistic, or perhaps as ordinary t or F
statistics calculated from the artificial regression.
Three families of artificial regressions are widely used in applied work.
The best-known is the Gauss-NewtOn family, which can be used to test the
parameters of a univariate or multivariate nonlinear regression function. In
the univariate case, this family simply involves regressing the residuals from
the restricted model (that is, the model in which some of the parameters are
estimated subject to the restrictions to be tested) on the derivatives of the
regression function with respect to all of the parameters of the unrestricted
model; see Section III. An early and important application in econometrics of
tests based on the Gauss-Newton family was to testing linear regression
models with lagged dependent variables for serial correlation (Durbin (1970),
Godfrey (1978)), but there have subsequently been a great many other
applications (see Pagan (1984)). For a general discussion of these tests, see
Engle(1982a).
A second widely-used family of artificial regressions is also applicable only
to regression models, and is useful when one wants to test for some form of
heteroskedasticity. In the univariate case, this family simply involves regress-
ing squared residuals on certain regressors with which they should be asymp-
totically uncorrelated under the null hypothesis of homoskedasticity. Tests
for various forms of heteroskedasticity which utilize this family of artificial
regressions include those suggested by Breusch and Pagan (1 979), White
(1980), Koenker(1981)and Engle(1982b).
t This research was supported, in part, by grants from the Social Sciences and Humanities
Research Council of Canada. We are gräteful to David Hendry for comments on earlier
versions.
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