Specification Tests

The TSCSREG Procedure

Specification Tests

The TSCSREG procedure outputs the results of one specification test for fixed effects and one specification test for random effects.

For fixed effects, let ${{\beta}_{f} }$ be the n dimensional vector of fixed effects parameters. The specification test reported is the conventional F-statistic for the hypothesis ${{\beta}_{f} =0}$ . The F-statistic with n, M-K degrees of freedom is computed as

$\hat{{\beta}}_{f} \hat{S}^{-1}_{f} \hat{{\beta}}_{f}/ n$

where ${\hat{S}_{f} }$ is the estimated covariance matrix of the fixed effects parameters.

Hausman's (1978) specification test or m-statistic can be used to test hypotheses in terms of bias or inconsistency of an estimator. This test was also proposed by Wu (1973) and further extended in Hausman and Taylor (1982). Hausman's m-statistic is as follows.

Consider two estimators, ${\hat{{\beta}}_{a}}$ and ${\hat{{\beta}}_{b}}$ , which under the null hypothesis are both consistent, but only ${\hat{{\beta}}_{a}}$ is asymptotically efficient. Under the alternative hypothesis, only ${\hat{{\beta}}_{b}}$ is consistent. The m-statistic is

$m = (\hat{{\beta}}_{b} - \hat{{\beta}}_{a})^{'} (\hat{S}_{b} - \hat{S}_{a})^{-} (\hat{{\beta}}_{b} - \hat{{\beta}}_{a})$

where ${\hat{S}_{b}}$ and ${\hat{S}_{a}}$ are consistent estimates of the asymptotic covariance matrices of ${\hat{{\beta}}_{b}}$ and ${\hat{{\beta}}_{a}}$ . Then m is distributed ${{\chi}^2}$ with k degrees of freedom, where k is the dimension of ${\hat{{\beta}}_{a}}$ and ${\hat{{\beta}}_{b}}$ .

In the random effects specification, the null hypothesis of no correlation between effects and regressors implies that the OLS estimates of the slope parameters are consistent and inefficient but the GLS estimates of the slope parameters are consistent and efficient. This facilitates a Hausman specification test. The reported ${{\chi}^2}$ statistic has degrees of freedom equal to the number of slope parameters.

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