The article explores hypothesis testing in linear models with complex error patterns. Researchers developed higher-order approximations to compare different testing methods for regression coefficients. They focused on models with nonrandom predictors and smooth error covariance functions. The Wald, likelihood ratio, and Lagrange multiplier tests were analyzed and found to be locally equivalent and asymptotically optimal. The likelihood ratio test statistic was shown to be an average of the Wald and Lagrange multiplier statistics under certain conditions. Overall, the three tests were equally powerful for one-dimensional null hypotheses.