The PANEL Procedure

Linear Hypothesis Testing

For a linear hypothesis of the form R ${\beta }=\mb {r}$ where $\mb {R}$ is ${\mi {J} {\times } \mi {K} }$ and $\mb {r}$ is ${\mi {J} {\times } 1}$ , the $\mi {F}$ -statistic with ${\mi {J}, \mi {M}-\mi {K} }$ degrees of freedom is computed as

$(\mb {R} {\beta }-\mb {r} )^{} [\mb {R} \hat{\mb {V} } {\mb {R} ’}]^{-1}(\mb {R} {\beta }-\mb {r} )$

However, it is also possible to write the $\mi {F}$ statistic as

$\mi {F} = \frac{(\hat{\mb {u}}^{}_{*}\hat{\mb {u}}_{*}- \hat{\mb {u}}^{}\hat{\mb {u}} )/J}{\hat{\mb {u}}^{}\hat{\mb {u}}/(M - K)}$

where

$\hat{\mb {u}}_{*}$ is the residual vector from the restricted regression
$\hat{\mb {u}}$ is the residual vector from the unrestricted regression
$\mi {J}$ is the number of restrictions
$\mi {(M - K)}$ are the degrees of freedom, $\mi {M}$ is the number of observations, and $\mi {K}$ is the number of parameters in the model

The Wald, likelihood ratio (LR) and Lagrange multiplier (LM) tests are all related to the $\mi {F}$ test. You use this relationship of the $\mi {F}$ test to the likelihood ratio and Lagrange multiplier tests. The Wald test is calculated from its definition.

The Wald test statistic is:

$\mi {W} = (\mb {R} {\beta }-\mb {r} )^{} [\mb {R} \hat{\mb {V} } {\mb {R} ’}]^{-1}(\mb {R} {\beta }-\mb {r} )$

The advantage of calculating Wald in this manner is that it enables you to substitute a heteroscedasticity-corrected covariance matrix for the matrix $\mb {V}$ . PROC PANEL makes such a substitution if you request the HCCME option in the MODEL statement.

The likelihood ratio is:

$\mi {LR} = \mi {M} \ln {\left[1 + \frac{1}{\mi {M - K}} \mi {JF} \right]}$

The Lagrange multiplier test statistic is:

$\mi {LM} = \mi {M}\left[\frac{\mi {JF}}{\mi {M - K + JF}}\right]$

where JF represents the number of restrictions multiplied by the result of the $\mi {F}$ test.

Note that only the Wald is changed when the HCCME option is selected. The LR and LM tests are unchanged.

The distribution of these test statistics is the $\chi ^{2}$ with degrees of freedom equal to the number of restrictions imposed ( $\mi {J}$ ). The three tests are asymptotically equivalent, but they have differing small sample properties. Greene (2000, p. 392) and Davidson and MacKinnon (1993, pg. 456-458) discuss the small sample properties of these statistics.