Locally Asymptotically Optimal Tests for Autoregressive against Bilinear Serial Dependence

Locally asymptotically most stringent tests for autoregressive against diagonal bilinear time series models are derived. A (restricted) local asymptotic normality property is therefore established for bilinear processes in the vicinity of linear autoregressive ones. The behaviour of the bispectrum under local alternatives of bilinear dependence shows the danger of misinterpreting skewness or kurtosis e ects for nonlinearities. The proposed test statistic is a generalization of the Gaussian Lagrange multiplier statistic considered by Saikkonen and Luukkonen (1988), and is expressed as a closed-form function of the estimated residual spectrum and bispectrum. Its local power is explicitly provided. The local power of the Lagrange multiplier test follows as a particular case.