Conditional heteroscedasticity has been often used in modelling and understanding the variability of statistical data. Under a general setup which includes the nonlinear time series model as a special case, we propose an e cient and adaptive method for estimating the conditional variance. The basic idea is to apply a local linear regression to the squared residuals. We demonstrate that without knowing the regression function, we can estimate the conditional variance asymptotically as well as if the regression were given. This asymptotic result, established under the assumption that the observations are made from a strictly stationary and absolutely regular process, is also veri ed via simulation. Further, the asymptotic result paves the way for adapting an automatic bandwidth selection scheme. An application with nancial data illustrates the usefulness of the proposed techniques.

We apply the local linear regression technique for estimation of functional-cefficient regression models for time series data. The models include threshold autoregressive models (Tong 1990) and functional-coefficient autoregressive models (Chen and Tsay 1993) as special cases but with the added advantages such as depicting finer structure of the underlying dynamics and better post-sample forecasting performance. We have also proposed a new bootstrap test for the goodness of fit models and a bandwidth selector based on newly defined cross-validatory estimation for the expected forecasting errors. The proposed methodology is data-analytic and is of appreciable flexibility to analyze complex and multivariate nonlinear structures without suffering from the "curse of dimensionality". The asymptotic properties of the proposed estimators are investigated under the alpha-mixing condition. Both simulated and real data examples are used for illustration.