Nonstationary Nonlinearity: An Outlook for New Opportunities

In this paper, we look for new opportunities that can be exploited using some of the recent developments on the theory of nonlinear models with integrated time series. Heuristic introductions on the basic tools and asymptotics are followed by the opportunities in three different directions: in data generation, in mean and in volatility. In the direction of data generation, we investigate the nonlinear transformations of random walks. It is shown in particular that they can generate stationary long memory as well as bounded nonstationarity and leptokurticity, which we commonly observe in many of economic and financial data. We then discuss how the nonlinear mean relationships between integrated processes can be appropriately formulated, interpreted and estimated within the regression framework. Both the nonlinear least squares regression and the nonparametric kernel regression are considered. Such formulations allow us to explore the nonlinear and nonparametric cointegration, which may be used in modelling the nonlinear and nonparametric longrun relationships among various economic and financial time series. Finally, a stochastic volatility model with the conditional variance specified as a nonlnear function of a random walk is examined. Established are various time series properties of the model, which are shown to be largely consistent with the observed characteristics of many time series data.