Nonlinear models for strongly dependent processes with financial applications

This paper is motivated by recent evidence that many univariate economic and financial time series have both nonlinear and long memory characteristics. Hence, this paper considers a general nonlinear, smooth transition regime autoregression which is embedded within a strongly dependent, long memory process. A time domain MLE$MLE$ with simultaneous estimation of the long memory, linear AR$AR$ and nonlinear parameters is shown to have desirable asymptotic properties. The Bayesian and Hannan–Quinn information criteria are shown to provide consistent model selection procedures. The paper also considers an alternative two step estimator where the original time series is fractionally filtered from an initial semi-parametric estimate of the long memory parameter. Simulation evidence indicates that the time domain MLE$MLE$ is generally superior to the two step estimator. The paper also includes some applications of the methodology and estimation of a fractionally integrated, nonlinear autoregressive-ESTAR$ESTAR$ model to forward premium and real exchange rates.