Econometric Theory, 27, 2011, 792–843. doi:10.1017/S0266466610000502 SIMULTANEOUS SPECIFICATION TESTING OF MEAN AND VARIANCE STRUCTURES IN NONLINEAR TIME SERIES REGRESSION SONG XI CHEN Peking University JITI GAO The University of Adelaide This paper proposes a nonparametric simultaneous test for parametric specification of the conditional mean and variance functions in a time series regression model. The test is based on an empirical likelihood (EL) statistic that measures the goodness of fit between the parametric estimates and the nonparametric kernel estimates of the mean and variance functions. A unique feature of the test is its ability to distribute natural weights automatically between the mean and the variance components of the goodness-of-fit measure. To reduce the dependence of the test on a single pair of smoothing bandwidths, we construct an adaptive test by maximizing a standardized version of the empirical likelihood test statistic over a set of smoothing bandwidths. The test procedure is based on a bootstrap calibration to the distribution of the em- pirical likelihood test statistic. We demonstrate that the empirical likelihood test is able to distinguish local alternatives that are different from the null hypothesis at an optimal rate. 1. INTRODUCTION Let { ( X t , Y t ) :1 ≤ t ≤ n} be a sequence of weakly dependent stationary observa- tions satisfying a nonparametric regression model of the form Y t = m 1 ( X t ) + σ( X t ) e t , t = 1, 2,..., n (1.1) We thank Peter C.B. Phillips, the editor, Yuichi Kitamura, the associate editor, and two referees for their construc- tive and insightful comments and suggestions, which have improved the presentation of the paper. We also thank Ming Li and Isabel Casas Villalba for their valuable computational assistance. Chen acknowledges the financial sup- port from National Science Foundation grants SES-0518904 and DMS-0604533, and Gao acknowledges the finan- cial support by Australian Research Council Discovery Grants under grant numbers DP0558602 and DP0879088. Address correspondence to Jiti Gao, School of Economics, University of Adelaide, Adelaide SA 5005, Australia; e-mail: jiti.gao@adelaide.edu.au. 792 c  Cambridge University Press 2011