Local Polynomial Prediction and Volatility Estimation in Financial Time Series

Abstract

Chaos and nonlinear theory has significant impact on the analysis of economic and financial time series. Nonlinearity plays an important role in explaining the empirical features of asymmetric business cycles, clustered volatility, and regime switching in finance data. In this Chapter, we will focus the popular local polynomial prediction method and its applications to chaotic time series prediction and financial volatility estimation. Volatility and conditional covariance estimation is important in many aspects of modern finance theory. We introduce a nonparametric volatility model, called local ARCH, and propose a weighted least square method for goodness of fit. The statistical theory is based on a martingale regression framework developed in Lu (1999a,b), which includes a wide variety of nonlinear time series models, such as nonlinear autoregression, ARCH, and nonlinear vector autoregressive models. The daily AOL stock data is used as an example to illustrate the developed techniques. First, we apply the nonlinear regression procedure to model the spread-volume relationship—We find a nice power-law relationship in all appropriate periods after discovering that the spurious nonlinearity in the overall data is due to nonstationarity. We also find a vastly changing structure in GARCH models fitted to different parts of the return rate series based on closing prices. We apply the developed local ARCH theory to a stationary subseries of the return series, and find some encouraging results.