The HPPANEL Procedure

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.