Nonlinear Regressions with Integrated Time Series

An asymptotic theo~yis developed for nonlinear regression with integrated processes.
The models allow for nonlinear effects from unit root time series and therefore deal with
the case of parametric nonlinear cointegration. The theory covers integrable and asymptotically
homogeneous functions. Sufficient conditions for weak consistency are given and
a limit distribution theory is provided. The rates of convergence depend on the properties
of the nonlinear regression function, and are shown to be as slow as n'14 for integrable
functions, and to be generally polynomial in n'12 for homogeneous functions. For
regressions with integrable functions, the limiting distribution theo~yis mixed normal with
mixing variates that depend on the sojourn time of the limiting Brownian motion of the
integrated process.
KEYWORDS:Functionals of Brownian motion, integrated process, local time, mixed
normal limit theo~yn, onlinear regression, occupation density.
1. INTRODUCTION AND HEURISTIC IDEAS
THE ASYMPTOTIC THEORY OF NONLINEAR REGRESSION plays a central role in
econometrics, underlying models as diverse as simultaneous equations systems
and discrete choice. In the context of time series applications, a longstanding
restriction on the range of potential applications has been the availability of
suitable strong laws or central limit theorems, effectively restricting attention to
models with stationary or weakly dependent data. While it is well known (e.g.,
Wu (1981)) that consistent estimation does not rely on assumptions of stationarity
or weak dependence, the development of a limit distribution theory has been
hamstrung by such restrictions for a very long time.