PROC PANEL: Linear Hypothesis Testing :: SAS/ETS(R) 9.2 User's Guide

The PANEL Procedure

For a linear hypothesis of the form R $\text{[math]}$ where $\text{[math]}$ is $\text{[math]}$ and $\text{[math]}$ is $\text{[math]}$ , the $\text{[math]}$ -statistic with $\text{[math]}$ degrees of freedom is computed as

$\text{[math]}$

However, it is also possible to write the $\text{[math]}$ statistic as

$\text{[math]}$

where

$\text{[math]}$ is the residual vector from the restricted regression

$\text{[math]}$ is the residual vector from the unrestricted regression

$\text{[math]}$ is the number of restrictions

$\text{[math]}$ are the degrees of freedom

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

The Wald test statistic is:

$\text{[math]}$

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

The likelihood ratio is:

$\text{[math]}$

The Lagrange multiplier test statistic is:

$\text{[math]}$

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 $\text{[math]}$ with degrees of freedom equal to the number of restrictions imposed ( $\text{[math]}$ ). 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.

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