Minimum Chi-Square and Three-Stage Least Squares in Fixed Effects Models.


Two approaches to estimation and testing of fixed effects models are
commonly found in the econometrics literature. The first involves variations on
instrumental variables. The second, a Minimum Chi-Square (MCS) procedure
introduced by Chamberlain, minimizes a quadratic form in the difference between
unrestricted regression coefficients and the restrictions implied by the fixed
effects model. This paper is concerned with the relationship between Three-Stage
Least Squares (3SLS) and MCS. A 3SLS equivalent of the MCS estimator is
presented and, in the usual case wherein the time varying error component has a
scalar covariance matrix, 3SLS is shown to simplify to the conventional
deviations from means estimator. Furthermore, the corresponding over-
identification test statistic is the degrees of freedom times the R2 from a
regression of residuals on all leads and lags of right hand side variables. The
relationship between MCS and some recently introduced efficient instrumental
variables procedures is also considered.
An empirical example from the literature on life-cycle labor supply is used
to illustrate properties of 3SLS procedures for panel data under alternative
assumptions regarding residual covariance. Estimated labor supply elasticities
and standard errors appear to be insensitive to these assumptions. In contrast,
the over-identification test statistics are found to be substantially smaller
when residuals are allowed to be intertemporally correlated and heteroscedastic.
At conventional levels of significance, however, even the smallest of the test
statistics leads to rejection of the over-identifying restrictions implicit in
the labor supply models.

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Journal of Business & Economic Statistics, Vol. 9, No. 3, Jul.y, 1991
Working Papers