### Type 3 Tests

The following statistics can be used to test the null hypothesis , where is a matrix of known coefficients. Under mild assumptions, each of the following statistics has an asymptotic chi-square distribution with degrees of freedom, where p is the rank of . Let be the maximum likelihood of under the null hypothesis ; that is,

#### Wald’s Statistic

where is the estimated covariance matrix, which can be the model-based covariance matrix or the sandwich covariance matrix (see the section Robust Sandwich Variance Estimate for details).