Whittle Likelihood Estimation of Nonlinear Autoregressive Models With Moving Average Residuals

Abstract

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.

• Asymptotic theory
• Correlated residuals
• Moving average process
• Nonlinear time series
• Spectral analysis.

Additional information

Notes on contributors

Tianhao Wang

Tianhao Wang is Research Assistant, National University of Singapore, Singapore (E-mail: [email protected]). Yingcun Xia is Professor of Statistics, National University of Singapore, Singapore and University of Electronic Science and Technology of China, China (E-mail: [email protected]). The authors thank an associate editor and two referees for their thoughtful comments. YC Xia’s research is partially supported by National Natural Science Foundation of China: 71371095, and NUS grant: R-155-000-145-112, Singapore.

Yingcun Xia

Tianhao Wang is Research Assistant, National University of Singapore, Singapore (E-mail: [email protected]). Yingcun Xia is Professor of Statistics, National University of Singapore, Singapore and University of Electronic Science and Technology of China, China (E-mail: [email protected]). The authors thank an associate editor and two referees for their thoughtful comments. YC Xia’s research is partially supported by National Natural Science Foundation of China: 71371095, and NUS grant: R-155-000-145-112, Singapore.