Linear Hypothesis Testing :: SAS/ETS(R) 13.1 User's Guide: High-Performance Procedures

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.

The HPPANEL Procedure (Experimental)

Linear Hypothesis Testing