ASYMPTOTICS FOR NONLINEAR TRANSFORMATIONS OF INTEGRATED TIME SERIES

An asymptotic theory for stochastic processes generated from nonlinear transformations of nonstationary integrated time series is developed. Various nonlinear functions of integrated series such as ARIMA time series are studied, and the asymptotic distributions of sample moments of such functions are obtained and analyzed. The transformations considered in the paper include a variety of functions that are used in practical nonlinear statistical analysis. It is shown that their asymptotic theory is quite different from that of integrated processes and stationary time series. When the transformation function is exponentially explosive, for instance, the convergence rate of sample functions is path-dependent. In particular, the convergence rate depends not only on the size of the sample, but also on the realized sample path. Some brief applications of these asymptotics are given to illustrate the effects of nonlinearly transformed integrated processes on regression. The methods developed in the paper are useful in a project of greater scope concerned with the development of a general theory of nonlinear regression for nonstationary time series.

[1]  Kai Lai Chung,et al.  A Course in Probability Theory , 1949 .

[2]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[3]  H. Jeffreys.  The Theory of Probability , 1922 .

[4]  P. Hall,et al.  Martingale Limit Theory and its Application. , 1984 .

[5]  Peter C. B. Phillips,et al.  Optimal Inference in Cointegrated Systems , 1991 .

[6]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[7]  C. Granger,et al.  Co-integration and error correction: representation, estimation and testing , 1987 .

[8]  Peter C. B. Phillips,et al.  Nonlinear Regressions with Integrated Time Series , 2001 .

[9]  Jacques Akonom,et al.  Comportement asymptotique du temps d'occupation du processus des sommes partielles , 1993 .

[10]  P. Phillips,et al.  Statistical Inference in Regressions with Integrated Processes: Part 1 , 1988, Econometric Theory.

[11]  P. Phillips.  Time series regression with a unit root , 1987 .

[12]  Peter C. B. Phillips,et al.  An Asymptotic Theory of Bayesian Inference for Time Series , 1996 .

[13]  P. Jeganathan.  Some Aspects of Asymptotic Theory with Applications to Time Series Models , 1995, Econometric Theory.

[14]  Victor Solo,et al.  Asymptotics for Linear Processes , 1992 .

[15]  D. Pollard.  Convergence of stochastic processes , 1984 .

[16]  L. Horváth,et al.  Weighted Approximations in Probability and Statistics , 1993 .

[17]  P. Phillips.  Multiple Regression with Integrated Time Series , 1987 .

[18]  P. Hall,et al.  Martingale Limit Theory and Its Application , 1980 .