Whittle Likelihood Estimation of Nonlinear Autoregressive Models With Moving Average Residuals

The Whittle likelihood estimation (WLE) has played a fundamental role in the development of both theory and computation of time series analysis. However, WLE is only applicable to models whose theoretical spectral density function (SDF) is known up to the parameters in the models. In this article, we propose a residual-based WLE, called extended WLE (XWLE), which can estimate models with their SDFs only partially available, including many popular time series models with correlated residuals. Asymptotic properties of XWLE are established. In particular, XWLE is asymptotically equivalent to WLE in estimating linear ARMA models, and is also capable of estimating nonlinear AR models with MA residuals and even with exogenous variables. The finite-sample performances of XWLE are checked by simulated examples and real data analysis.