A Test for Nonlinearity of Time Series with Infinite Variance

A heavy tailed time series that can be represented as an innnite moving average has the property that the sample autocorrelation function (ACF) at lag h converges in probability to a constant (h), although the mathematical correlation typically does not exist. For many nonlinear heavy tailed models, however, the sample ACF at lag h converges in distribution to a nondegenerate random variable. In this paper, a test for (non)linearity of a given innnite variance time series is constructed, based on subsample stability of the sample ACF. The test is applied to several real and simulated datasets. 1. Introduction An important question in time series analysis is how to t models to data which exhibit non-standard features such as long range dependence, nonlinearity and heavy tails. For example, many datasets from elds such as telecommunications, nance and economics appear to be compatible with the assumption of heavy tailed marginal distributions. In the traditional setting of stationary time series with nite variance, every purely nondeterministic process can be represented as a linear process driven by an uncorrelated input sequence. For such processes, the autocorrelation function can be well approximated by that of a nite order ARMA(p,q) model. So from a second order point of view, linear processes are suucient for data analysis. For stationary time series with innnite variance, the class of linear models does not appear to be suuciently rich and exible for modeling purposes. There are few, if any, succesful ts of heavy tailed MA(1) models to real, non-simulated data.