Testing for serial correlation against an ARMA(1, 1) process

This article is concerned with tests for serial correlation in time series and in the errors of regression models. In particular, the nonstandard problem of testing for white noise against autoregressive moving average model ARMA(1, 1) alternatives is considered. The likelihood ratio (LR), sup Lagrange multiplier (LM), and exponential average LM and LR tests are shown to be asymptotically admissible for ARMA(1, 1) alternatives. In addition, they are shown to be consistent against all (weakly stationary strong mixing) non-white noise alternatives. Simulation results compare the tests to several tests in the literature. These results show that the LR and Exp-LR infinity tests have very good all-around power properties for nonseasonal alternatives.