PANEL Procedure

Tests of Poolability

You can obtain tests for poolability across cross sections by specifying the POOLTEST option in the MODEL statement. The null hypothesis of poolability assumes homogeneous slope coefficients.

F Test

For the unrestricted model, run a regression for each cross section and save the sum of squared residuals as normal upper S normal upper S normal upper E Subscript u i. Sum over all cross sections to obtain normal upper S normal upper S normal upper E Subscript u Baseline equals sigma-summation Underscript i equals 1 Overscript upper N Endscripts normal upper S normal upper S normal upper E Subscript u i. For the restricted model, save the sum of squared residuals for each cross section as normal upper S normal upper S normal upper E Subscript r i. Sum over all cross sections to obtain normal upper S normal upper S normal upper E Subscript r Baseline equals sigma-summation Underscript i equals 1 Overscript upper N Endscripts normal upper S normal upper S normal upper E Subscript r i. 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. Let k be the number of regressors except the constant. The degrees of freedom for the unrestricted model is d f Subscript u Baseline equals upper M minus upper N left-parenthesis k plus 1 right-parenthesis. If the constant is restricted to be the same, the degrees of freedom for the restricted model is d f Subscript r Baseline equals upper M minus k minus 1 and the number of restrictions is q equals left-parenthesis upper N minus 1 right-parenthesis left-parenthesis k plus 1 right-parenthesis. If the restricted model is the fixed one-way model, the degrees of freedom is d f Subscript r Baseline equals upper M minus k minus upper N and the number of restrictions is q equals left-parenthesis upper N minus 1 right-parenthesis k. So the F test is

upper F equals StartFraction left-parenthesis normal upper S normal upper S normal upper E Subscript r Baseline minus normal upper S normal upper S normal upper E Subscript u Baseline right-parenthesis slash q Over normal upper S normal upper S normal upper E Subscript u Baseline slash d f Subscript u Baseline EndFraction tilde upper F left-parenthesis q comma d f Subscript u Baseline right-parenthesis

For large N and T, you can use a chi-square distribution to approximate the limiting distribution, namely, q upper F right-arrow chi squared left-parenthesis q right-parenthesis. The test is the same as the Chow test (Chow 1960) extended to N linear regressions.

Likelihood Ratio (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 upper L upper R equals minus 2 log left-brace left-parenthesis 1 plus q upper F slash d f Subscript u Baseline right-parenthesis Superscript negative upper M slash 2 Baseline right-brace. Because upper L upper R equals q upper F plus upper O left-parenthesis n Superscript negative 1 Baseline right-parenthesis, under the null hypothesis LR is asymptotically distributed as chi-square with q degrees of freedom.

Last updated: June 19, 2025