The HPPANEL Procedure (Experimental)

Linear Hypothesis Testing

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

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

where

  • $\hat{\mb {u}}_{*}$is the residual vector from the restricted regression

  • $\hat{\mb {u}}$ is the residual vector from the unrestricted regression

  • $J$ is the number of restrictions

  • $M - K$ are the degrees of freedom, $M$ is the number of observations, and $K$ is the number of parameters in the model

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

The Wald test statistic is

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

The likelihood ratio is

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

The LaGrange multiplier test statistic is

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

where $JF$ represents the number of restrictions multiplied by the result of the $F$ test.

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