The PANEL Procedure
- Overview
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Getting Started
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Syntax
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Details
Missing Values Computational Resources Restricted Estimates Notation The One-Way Fixed-Effects Model The Two-Way Fixed-Effects Model Balanced Panels Unbalanced Panels Between Estimators Pooled Estimator The One-Way Random-Effects Model The Two-Way Random-Effects Model Parks Method (Autoregressive Model) Da Silva Method (Variance-Component Moving Average Model) Dynamic Panel Estimator Linear Hypothesis Testing Heteroscedasticity-Corrected Covariance Matrices R Square Specification Tests Troubleshooting ODS Graphics The OUTPUT OUT= Data Set The OUTEST= Data Set The OUTTRANS= Data Set Printed Output ODS Table Names
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Examples
- References
| Linear Hypothesis Testing |
For a linear hypothesis of the form R 
where 
is 
and 
is 
, the 
-statistic with 
degrees of freedom is computed as
 |
However, it is also possible to write the statistic as
 |
where
is the residual vector from the restricted regression 
is the residual vector from the unrestricted regression 
is the number of restrictions 
are the degrees of freedom, 
is the number of observations, and 
is the number of parameters in the model
The Wald, likelihood ratio (LR) and Lagrange multiplier (LM) tests are all related to the test. You use this relationship of the 
test to the likelihood ratio and Lagrange multiplier tests. The Wald test is calculated from its definition.
The Wald test statistic is:
 |
The advantage of calculating Wald in this manner is that it enables you to substitute a heteroscedasticity-corrected covariance matrix for the matrix 
. PROC PANEL makes such a substitution if you request the HCCME option in the MODEL statement.
The likelihood ratio is:
 |
The Lagrange multiplier test statistic is:
 |
where JF represents the number of restrictions multiplied by the result of the 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 
with degrees of freedom equal to the number of restrictions imposed (
). 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.