Optimal tests in AR (m) time series model

In the univariate framework, two problems of testing the nonlinearity are investigated in Hwang and Basawa. The first one is concerned with the testing problem for a nonlinear class contiguous to the AR(1) process. The second one is focused on the testing problem of the ARCH model contiguous to the AR(1) models. In each case, an efficient test of linearity was obtained, the local asymptotic normality (LAN) was proved, an efficient test of linearity was constructed, and the asymptotic power function was derived. All these results were obtained under the assumption where the parameter of the time series model is assumed to be known. In practical situation, this parameter is unspecified and its estimation induces an error that has an impact on the asymptotic limit distribution. A new method for the good evaluation of this error is introduced and investigated in the present article. Consequently, its application allows us to preserve the local asymptotic optimality with the estimated parameter. An extension to testing in class of ARCH models contiguous to p-order autoregressive processes is obtained. The LAN property plays a fundamental role in the present study.