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.
For the unrestricted model, run a regression for each cross section and save the sum of squared residuals as . For the restricted model, save the sum of squared residuals as . 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 .^{[7]} Let k be the number of regressors except the constant. The degree of freedom for the unrestricted model is . If the constant is restricted to be the same, the degree of freedom for the restricted model is and the number of restrictions is . If the restricted model is the fixed one-way model, the degree of freedom is and the number of restrictions is . So the F test is
For large N and T, you can use a chi-square distribution to approximate the limiting distribution, namely, . The error term is assumed to be homogeneous; therefore, , and an OLS regression is sufficient. The test is the same as the Chow test (Chow, 1960) extended to N linear regressions.
Zellner (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 . Under , is asymptotically distributed as a chi-square with q degrees of freedom.
^{[7] }For the unbalanced panel, the number of time series might be different. The number of observations needs to be redefined accordingly.