Non-linear Time Series Analysis

Abstract

Since the early 1980s, there has been a growing interest in stochastic nonlinear dynamical systems of the form x t +1 = f ( x t , x t −1 , …, x t − p ) + σ ( x t ) ε t , where \( {\left\{{x}_t\right\}}_{t=0}^{\infty } \) is a zero mean, covariance stationary process, f : R p +1 → R , σ is the conditional volatility, and \( {\left\{{\varepsilon}_t\right\}}_{t=0}^{\infty } \) is an independent and identically distributed noise process. The major recent developments in nonlinear time series are described using this canonical model: ( a ) representation theory; ( b ) nonparametric modelling; ( c ) ergodic properties; ( d ) piecewise linear models; ( e ) volatility modelling; ( f ) hypothesis testing for linearity and normality; ( g ) forecasting.

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