Empirical Likelihood-Based Inference for Nonlinear Di¤usions

Nonparametric methods are becoming popular recently in estimating continuous-time di¤usion models due to their ‡exibility. This paper provides a new approach to constructing point-wise con…dence intervals for nonparametric drift and di¤usion functions via empirical likelihood (EL) in conjunction with local linear smoothing. It serves as a promising competitor to the conventional asymptotic normality-based method, which generally leads to (often seriously) under-covered con…dence intervals in our simulation experiments, especially for the drift function. Limit theories are developed for the EL ratios for both drift and di¤usion functions by means of increasing time span and shrinking observational intervals. Our results apply to both stationary and nonstationary recurrent di¤usion processes. Compared to the asymptotic normality-based method, our approach avoids the di¢ culty of variance estimation by choosing the variance estimator implicitly as an operational nature of EL, and it has the additional advantage of automatic determination of the shape of the resultant con…dence interval. Simulation studies show that for the both drift and di¤usion functions, the EL con…dence intervals are very accurate and robust to bandwidth choices even with moderate sample sizes, and are generally shorter (after con…dence level calibration) than those based on asymptotic normality. I am especially indebted to Peter C. B. Phillips for his invaluable guidance and encouragement. I am grateful to Donald W. K. Andrews, Brendan Beare, Bruce Hansen, Yuichi Kitamura, Arthur Lewbel, Taisuke Otsu, John Rust and Zhijie Xiao for their constructive comments on the original draft. I would also like to thank the participants at seminars at the University of Alberta School of Business, Boston College, University of Maryland at College Park, Yale University, NBER-NSF Time Series Conference at Montréal and Canadian Econometrics Study Group meeting at McGill Univeristy for their useful remarks and suggestion. All errors are mine. Mailing address: Department of Finance and Management Science, School of Business, University of Alberta, Edmonton, Alberta, Canada, T6G 2R6. Email: [email protected].