The PANEL Procedure

Panel Data Poolability Test

F Test
LR Test

The null hypothesis of poolability assumes homogeneous slope coefficients. An F test can be applied to test for the poolability across cross sections in panel data models.

F Test

For the unrestricted model, run a regression for each cross section and save the sum of squared residuals as $SSE_{u}$ . For the restricted model, save the sum of squared residuals as $SSE_{r}$ . If the test applies to all coefficients (including the constant), then the restricted model is the pooled model (OLS); if the test applies to coefficients other than the constant, then the restricted model is the fixed one-way model with cross-sectional fixed effects. If and denote the number of cross sections and time periods, then the number of observations is .[3] Let be the number of regressors except the constant. The degree of freedom for the unrestricted model is $df_{u} = n-N (k+1)$ . If the constant is restricted to be the same, the degree of freedom for the restricted model is $df_{r}=n-k-1$ and the number of restrictions is . If the restricted model is the fixed one-way model, the degree of freedom is $df_{r}=n-k-N$ and the number of restrictions is . So the F test is

$\begin{equation*} F = \frac{\left(SSE_{r}-SSE_{u}\right)/q}{SSE_{u}/df_{u}}\sim F(q,df_{u}) \end{equation*}$

For large and , you can use a chi-square distribution to approximate the limiting distribution, namely, $qF\implies \chi ^{2}\left(q\right)$ . The error term is assumed to be homogeneous; therefore, $\epsilon \sim \mathcal{N}\left(0,\sigma ^{2}I_{n}\right)$ , and an OLS regression is sufficient. The test is the same as the Chow test (Chow 1960) extended to linear regressions.

LR Test

Zeller (1962) also proved that the likelihood ratio test for null hypothesis of poolability can be based on the statistic. The likelihood ratio can be expressed as $LR = -2log\left(\left(1 + qF/df_{u}\right)^{-NT/2}\right)\implies LR = qF + O\left(n^{-1}\right)$ . Under $H_{0}$ , is asymptotically distributed as a chi-square with degrees of freedom.

^[3]For the unbalanced panel, the number of time series $T_{i}$ might be different. The number of observations needs to be redefined accordingly.