A Simple Approach to the Parametric Estimation of Potentially Nonstationary Diffusions

A simple and robust approach is proposed for the parametric estimation of scalar homogeneous stochastic di¤erential equations. We specify a parametric class of di¤usions and estimate the parameters of interest by minimizing criteria based on the integrated squared di¤erence between kernel estimates of the drift and di¤usion functions and their parametric counterparts. The procedure does not require simulations or approximations to the true transition density and has the simplicity of standard nonlinear least-squares methods in discrete-time. A complete asymptotic theory for the parametric estimates is developed. The limit theory relies on in…ll and long span asymptotics and is robust to deviations from stationarity, requiring only recurrence. Keywords: Di¤usion, Drift, Local time, Parametric estimation, Semimartingale, Stochastic differential equation. JEL Classi…cation: C14, C22 A preliminary version of this paper entitled “Accelerated asymptotics for di¤usion model estimation”was written for the Cowles Foundation Conference “New Developments in Time Series Econometrics,”Yale University, October 23-24, 1999. We are grateful to an anonymous Associate Editor and two anonymous Referees for their valuable comments. We thank Robert de Jong, Ron Gallant, Eric Renault, and seminar participants at the NBER Working Group on empirical methods in macroeconomics and …nance (Summer Institute 2000), the 2000 Econometrics Society World Conference in Seattle, the University of Chicago and the Université de Montréal for useful discussions. Bandi thanks the Graduate School of Business of the University of Chicago for …nancial support. Phillips thanks the NSF for support under Grant Nos. SBR 97-30295, SES 00-92509 and SES 04-142254.