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.