The quest for nonlinearity in time series

In this chapter, we review the problem of testing for nonlinearity in time series. First, we discuss the definition and the properties of linear processes and the implications that such properties have on the operational strand. Then, we present and review a tentative classification of the various tests that can be found both in the time series and in the nonlinear dynamics literature. Two main factors contributed to the production of a plethora of alternatives for assessing nonlinearity in time series: the first factor is the intrinsic asymmetry between the linear and the nonlinear realm. In fact, there can be departures from linearity in various directions as nonlinear phenomena possess a virtually infinite richness of features. Among such features we can mention irreversibility, nonuniform predictability, noise amplification/ suppression, phase synchronization, noise-induced phenomena, sensitivity to initial conditions, and so on. The second factor is the multidisciplinary nature of the problem. Indeed, the problem of characterizing the various aspects of nonlinear processes is shared among different disciplines, such as Statistics, Econometrics, Nonlinear Dynamics, Biology, and Engineering. The review is by no means exhaustive and reflects the personal inclinations of the author