Nonlinearity and temporal dependence

Abstract

a b s t r a c tNonlinearities in the drift and diffusion coefficients influence temporal dependence in diffusion models.We study this link using three measures of temporal dependence: ρ-mixing, β-mixing and α-mixing. Stationary diffusions that are ρ-mixing have mixing coefficients that decay exponentially to zero. When they fail to be ρ-mixing, they are still β-mixing and α-mixing; but coefficient decay is slower than exponential. For such processes we find transformations of the Markov states that have finite variances but infinite spectral densities at frequency zero. The resulting spectral densities behave like those of stochastic processes with long memory. Finally we show how state dependent, Poisson sampling alters the temporal dependence.

Penalized sieve estimation and inference of semi-nonparametric dynamic models: a selective review

“…However, the answer is very likely to be no if the interest rate is modelled as a nonlinear …rst order Markov process or as a discrete time realization of a continuous-time Markov di¤usion process. Another example is in Chen, Hansen and Carrasco (2010). They show that a strictly stationary scalar di¤usion process is always beta-mixing (see de…nition below); but some of the beta-mixing decay rate could be very slow, in which case some of its transformations behave like long memory (in the sense that the spectral density blows up at frequency zero in a manner like long memory in a linear time series).…”

Operator Methods for Continuous-Time Markov Processes

“…In our study of strongly dependent, but stationary, Markov processes, we follow Chen, Hansen, and Carrasco (2003) by using two measures of mixing. Both of these measures have been used extensively in the stochastic process literature.…”

“…At least for scalar diffusions, Chen, Hansen, and Carrasco (2003) show that the exponential decay of the ρ−mixing coefficients is essentially equivalent to the exponential decay of the β−mixing coefficients. When the ρ−mixing coefficients are identically one, however, the β−mixing coefficients will still decay to zero, but at a rate slower than exponential.…”

“…We now display one construction of a nonlinear diffusion model that is strongly dependent. This example is taken from Chen, Hansen, and Carrasco (2003).…”

Long memory via networking

“…Hall (1978), Nelson and Plosser (1982), Perron (1988), Phillips (1987)) aggregation (e.g., Granger (1980), Zafaroni (2004), Abadir and Talmain (2002), Chambers (1998)) learning dynamics (e.g., Alfarano and Lux (2005), Chevillon and Mavroeidis (2011)) nonlinearity (e.g. Chen, Hansen, and Carrasco (2010), Miller and Park (2010)), fractional brownian motion (e.g. Mandelbrot and Ness (1968), Granger and Ding (1996), Comte and Renault (1996), Baillie (1996)), multifractal models (e.g., Calvet and Fisher (2002)), as well as other mechanisms (e.g.…”