Linear hypotheses for are expressed in matrix form as
where L is a matrix of coefficients for the linear hypotheses, and c is a vector of constants. The Wald chi-square statistic for testing is computed as
where is the estimated covariance matrix. Under , has an asymptotic chi-square distribution with r degrees of freedom, where r is the rank of .
Let , where is a subset of s regression coefficients. For any vector of length s,
To find such that has the minimum variance, it is necessary to minimize subject to . Let be a vector of 1’s of length s. The expression to be minimized is
where is the Lagrange multiplier. Differentiating with respect to and , respectively, yields
Solving these equations gives
This provides a one degree-of-freedom test for testing the null hypothesis with normal test statistic
This test is more sensitive than the multivariate test specified by the TEST statement
Multivariate: test X1, ..., Xs;
where X
1, …, X
s are the variables with regression coefficients , respectively.