33 – Applications in Time Series Analysis
Modeling Conditional Variance Functions Using Nonparametric Transfer Function Models
Jun M. Liu
Georgia Southern University
Estimating conditional variance functions is of great importance in practice. A nonparametric method is proposed to estimate conditional variance functions with correlated noise. In this method, polynomial splines are used to approximate the transfer function and the conditional variance function, while the noise is assumed to follow an Autoregressive-Moving Average process. It is shown via simulations that the estimators have the "oracle" property, i.e., the ARMA parameters can be estimated with usual parametric rate of convergence, the conditional variance function estimator behaves as if the transfer function and the ARMA parameters are known, and the transfer function can be estimated as if the conditional variance function and the ARMA parameters are known. Additionally, it is shown that for time series data, it is necessary to model the serial correlation in the noise to achieve optimal efficiency in the nonparametric estimation of both the transfer function and the conditional variance function. By using polynomial splines, this method is not only flexible but also computationally efficient compared with other nonparametric smoothing methods. The asymptotic properties of the estimators are discussed. The usefulness of this model is illustrated through a real data example.