Abstract and Keywords
Nonparametric and semiparametric estimation and hypothesis testing methods have been intensively studied for cross-sectional independent data and weakly dependent time series data. However, many important macroeconomics and financial data are found to exhibit stochastic trend and/or deterministic trend, and the trend can be nonlinear in nature; see a nonlinear growth model studied by Granger, Inoue, and Morin (1997). While a linear model may provide a decent approximation to a nonlinear model for weakly dependent data, the linearization can result in severely biased approximation to a nonlinear model with nonstationary data. For example, Park and Phillips (1999) derived limit results for nonlinearly transformed integrated time series whose sample average converges at different rates depending on the form of the nonlinear transformation functions. Therefore, it is utterly important to explore nonlinear data-generating mechanisms or nonlinear relations when persistent nonstationary time series are to be analyzed. This chapter reviews some of the recent theoretical developments in nonparametric and semiparametric techniques applied to nonstationary or near nonstationary variables. First, this chapter reviews some of the existing works on extending the I(0), I(1), and cointegrating relation concepts defined in a linear model to a nonlinear framework, and it points out some difficulties in providing satisfactory answers to extend the concepts of I(0), I(1), and cointegration to nonlinear models with persistent time series data. Second, the chapter reviews kernel estimation and hypothesis testing for nonparametric and semiparametric autoregressive and cointegrating models to explore unknown nonlinear relation among I(1) or near I(1) process(es). The asymptotic mixed-normal results of kernel estimation generally replace asymptotic normality results usually obtained for weakly dependent data. We also discuss kernel estimation of semiparametric varying coefficient regression models with correlated but not cointegrated data. Finally, we discuss the concept of co-summability introduced by Berengner-Rico and Gonzalo (2012), which provides an extension of cointegration concept to nonlinear time series data.