The PANEL Procedure

Panel Data Poolability Test

Subsections:

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 N and T denote the number of cross sections and time periods, then the number of observations is $n=NT$.[7] Let k 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 $ q=(N-1)(k+1)$. 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 $ q=(N-1)k$. 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 N and T, 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 N linear regressions.

LR Test

Zellner (1962) also proved that the likelihood ratio test for null hypothesis of poolability can be based on the $F$ 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 $H_{0}$, $LR$ is asymptotically distributed as a chi-square with q degrees of freedom.


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