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Asymptotics for Nonlinear Transformations of Integrated Time Series

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Abstract

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.

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File URL: http://cowles.econ.yale.edu/P/cd/d11b/d1182.pdf
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Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1182.

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Length: 27 pages
Date of creation: Jun 1998
Date of revision:
Publication status: Published in Econometric Theory (1999), 15: 260-298
Handle: RePEc:cwl:cwldpp:1182

Note: CFP 980.
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Keywords: Additive functionals of Brownian motion; Brownian motion; integrated process; local time; nonlinear transformation; occupation time; regression asymptotics;

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  1. Peter C.B. Phillips, 1987. "Multiple Regression with Integrated Time Series," Cowles Foundation Discussion Papers 852, Cowles Foundation for Research in Economics, Yale University.
  2. Ross Williams, 2013. "Introduction," Australian Economic Review, The University of Melbourne, Melbourne Institute of Applied Economic and Social Research, vol. 46(4), pages 460-461, December.
  3. Peter C.B. Phillips, 1995. "Unit Root Tests," Cowles Foundation Discussion Papers 1104, Cowles Foundation for Research in Economics, Yale University.
  4. Peter C.B. Phillips, 1988. "Optimal Inference in Cointegrated Systems," Cowles Foundation Discussion Papers 866R, Cowles Foundation for Research in Economics, Yale University, revised Aug 1989.
  5. Peter C.B. Phillips & Victor Solo, 1989. "Asymptotics for Linear Processes," Cowles Foundation Discussion Papers 932, Cowles Foundation for Research in Economics, Yale University.
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