16.4 Choosing between fixed and random effects
The choice turns on whether \(\alpha _i\) is correlated with the regressors, which can be tested. Under \(H_0\) of no correlation, both estimators are consistent and random effects is efficient; under the alternative, only fixed effects is consistent. The Hausman statistic of Theorem 13.4.2 applies directly: \[ H = (b_{\text {FE}} - b_{\text {RE}})'\bigl [\widehat \Var (b_{\text {FE}}) - \widehat \Var (b_{\text {RE}})\bigr ]^{-1}(b_{\text {FE}} - b_{\text {RE}}) \xrightarrow {d} \chi ^2_k , \] for the coefficients of the time-varying regressors. Mundlak (1978) gave a regression version and an illuminating interpretation.
Theorem 16.4.1 (Mundlak). In a balanced panel, the least squares coefficient of \(x_{it}\) in the pooled regression of \(y_{it}\) on \(x_{it}\), \(z_i\) and the unit means \(\bar x_i\) equals the fixed effects estimator \(b_{\text {FE}}\).
Proof. By the Frisch–Waugh–Lovell theorem, the coefficient of \(x_{it}\) is obtained by first regressing \(x_{it}\) on the other regressors, a constant, \(z_i\) and \(\bar x_i\), all of which are constant within each unit. Write \(x_{it} = \bar x_i + \tilde x_{it}\). The term \(\bar x_i\) is one of the regressors, and \(\tilde x_{it}\) sums to zero within every unit, so it is orthogonal to any variable that is constant within units. Hence the residual from regressing \(x_{it}\) on these variables is exactly \(\tilde x_{it}\). The coefficient is then \((\sum \tilde x_{it}\tilde x_{it}')^{-1}\sum \tilde x_{it}y_{it}\), and since \(\sum _t\tilde x_{it}\bar y_i = 0\) this is \(b_{\text {FE}}\). □
Mundlak’s model writes \(\alpha _i = \bar x_i'\pi + w_i\), letting the individual effect depend on the unit’s average regressors. Random effects corresponds to \(\pi = 0\), and the \(t\) or Wald test of \(\pi = 0\) in the augmented regression, with cluster-robust standard errors, is a robust version of the Hausman test. Its advantage is that it remains valid when the disturbances are heteroscedastic or serially correlated, when random effects is not efficient and the Hausman variance difference may not even be positive definite. The coefficients of \(z_i\) in Mundlak’s regression are also of interest: they are estimated, unlike in fixed effects, but they are consistent only if \(z_i\) is uncorrelated with the part \(w_i\) of the effect not explained by \(\bar x_i\).
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.