### Using the TEST Statement to Test Linear Hypotheses

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 .

#### Optimal Weights for the AVERAGE option in the TEST Statement

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 X1, …, Xs are the variables with regression coefficients , respectively.