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

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


  • $ \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.