16. Flexible Time Series Analysis

Wolfgang Härdle and Rolf Tschernig
18 November 2003

In this chapter we present nonparametric methods and available quantlets for nonlinear modelling of univariate time series. A general nonlinear time series model for an univariate stochastic process $ \{Y_t\}_{t=1}^T $ is given by the heteroskedastic nonlinear autoregressive (NAR) process

$\displaystyle Y_t = f(Y_{t-i_1},Y_{t-i_2},\ldots,Y_{t-i_m}) + \sigma(Y_{t-i_1},Y_{t-i_2},\ldots,Y_{t-i_m})\xi_t,$ (16.1)

where $ \{\xi_t\}$ denotes an i.i.d. noise with zero mean and unit variance and $ f(\cdot)$ and $ \sigma(\cdot)$ denote the conditional mean function and conditional standard deviation with lags $ i_1,\ldots,i_m$, respectively. In practice, the conditional functions $ f(\cdot)$ and $ \sigma(\cdot)$ as well as the number of lags $ m$ and the lags itself $ i_1,\ldots,i_m$ are unknown and have to be estimated.

In Section 16.1 we discuss nonparametric estimators for the conditional mean function of nonlinear autoregressive processes of order one. While this case has been most intensively studied in theory, in practice models with several lags are often more appropriate. Section 16.2 covers the estimation of the latter, including the selection of appropriate lags. For all models we discuss methods of bandwidth selection which aim at an optimal trade-off between variance and bias of the presented estimators.

Both sections contain practical examples. The corresponding quantlets for fitting nonlinear autoregressive processes of order one are contained in the quantlib smoother . A number of quantlets for fitting higher order models are found in the third party quantlib tp/cafpe/cafpe .

Although obvious we would like to mention that in the following we only discuss methods for which quantlets are available. For an overview of alternative methods and models we would like to refer the reader to the surveys of Tjøstheim (1994) or Härdle et al. (1997).