Wald Tests

This describes the computations used by the five regression-based testing instructions: EXCLUDE, SUMMARIZE, TEST, RESTRICT and MRESTRICT. All of these compute the “Wald” test, though many “LM” tests are computed by applying these to auxiliary regressions.

A set of Q linear restrictions on the coefficient matrix \(\beta\) can be written

We list below the formulas for the test statistic (used by all five instructions) and the restricted coefficient vector and covariance matrix (for RESTRICT and MRESTRICT with the CREATE option). The first form is for single equation regressions and the second for most non-linear estimation procedures, systems of equations and regressions with corrected covariance matrices.

In these formulas, \({\left( {{\bf{X'}}{\kern 1pt} {\bf{X}}} \right)^{ - 1}}\) may not precisely be that matrix, but will be its appropriate analogue. \(\Sigma_x\) is the estimated covariance matrix of coefficients. Either matrix is saved by RATS under the name %XX.

Test statistic

\(F\left( {Q,t - K} \right) = \frac{{{{\left( {{\bf{r}} - {\bf{R}}\hat \beta } \right)}^\prime }{\kern 1pt} {\kern 1pt} {{\left[ {{\bf{R}}{{\left( {{\bf{X'}}{\kern 1pt} {\bf{X}}} \right)}^{ - 1}}{\bf{R'}}{\kern 1pt} } \right]}^{ - 1}}\left( {{\bf{r}} - {\bf{R}}\hat \beta } \right)}}{{\left( {Q{{\hat \sigma }^2}} \right)}}\)

\({\chi ^{\rm{2}}}(Q) = {\left( {{\bf{r}} - {\bf{R}}\hat \beta } \right)^\prime }{\kern 1pt} {\kern 1pt} {\left[ {{\bf{R}}{\Sigma _{\bf{x}}}{\bf{R'}}{\kern 1pt} } \right]^{ - 1}}\left( {{\bf{r}} - {\bf{R}}\hat \beta } \right)\)

Restricted coefficient vector

\({\hat \beta _{\bf{R}}} = \hat \beta + {\left( {{\bf{X'}}{\kern 1pt} {\bf{X}}} \right)^{ - 1}}{\bf{R'}}{\kern 1pt} {\left[ {{\bf{R}}{{\left( {{\bf{X'}}{\kern 1pt} {\bf{X}}} \right)}^{ - 1}}{\bf{R'}}} \right]^{ - 1}}\left( {{\bf{r}} - {\bf{R}}\hat \beta } \right)\)

\({\hat \beta _{\bf{R}}} = \hat \beta + {\Sigma _{\bf{x}}}{\bf{R'}}{\kern 1pt} {\kern 1pt} {\left[ {{\bf{R}}{\Sigma _{\bf{x}}}{\bf{R'}}{\kern 1pt} } \right]^{ - 1}}\left( {{\bf{r}} - {\bf{R}}\hat \beta } \right)\)

Restricted covariance matrix

\({\rm{Var}}\left( {{{\hat \beta }_{\bf{R}}}} \right) = {\hat \sigma ^2}\left\{ {{{\left( {{\bf{X'}}{\kern 1pt} {\kern 1pt} {\bf{X}}} \right)}^{ - 1}} - {{\left( {{\bf{X'}}{\kern 1pt} {\kern 1pt} {\bf{X}}} \right)}^{ - 1}}{\bf{R'}}{\kern 1pt} {\kern 1pt} {{\left[ {{\bf{R}}{{\left( {{\bf{X'}}{\kern 1pt} {\kern 1pt} {\bf{X}}} \right)}^{ - 1}}{\bf{R'}}} \right]}^{ - 1}}{\bf{R}}{{\left( {{\bf{X'}}{\kern 1pt} {\kern 1pt} {\bf{X}}} \right)}^{ - 1}}} \right\}\)

\({\rm{Var}}\left( {{{\hat \beta }_{\bf{R}}}} \right) = {\Sigma _{\bf{x}}} - {\Sigma _{\bf{x}}}{\bf{R'}}{\kern 1pt} {\kern 1pt} {\left[ {{\bf{R}}{\Sigma _{\bf{x}}}{\bf{R'}}{\kern 1pt} } \right]^{ - 1}}{\bf{R}}{\Sigma _{\bf{x}}}\)

If the matrix in [...] is not invertible, RATS will, in effect, drop any redundant restrictions and reduce the degrees of freedom appropriately. If RATS makes such an adjustment, you will see a message like

Redundant Restrictions. Using 4 Degrees, not 8