Dow Jones Index, GARCH(1,1) and change-points

Many econometric time series data sets, such as log returns of stocks, exhibit evidence of the so called stylized facts. Namely it is generally observed that the data itself is uncorrelated with heavy tails, but the squared data has significant autocorrelation. For such data sets, there appears to be little or no linear information in the past about the future values of the series. Thus the class of Autoregressive Integrated moving average models (ARIMA) are not appropriate. However, there does in general appear to be information in past values of the squared data about future values of the squared data. This allows for modeling of the conditional variance as a function of the observed past. One choice for a class of models able to incorporate the stylized facts are the generalized autoregressive conditional heteroskedastic (GARCH) family. In practice the GARCH(1,1) process is used most often.; This work reviews what is known about the GARCH(1,1) model and investigates the appropriateness of a GARCH(1,1) model for daily Dow Jones Index stock returns (DWJ). A GARCH(1,1) model is fit to the DWJ series. Based on visual inspection of the return data, there may be one or more changepoints in the process governing the data. We use a recent test proposed by Berkes, Horvath, and Kokoszka in order to locate possible change-points in the DWJ data. Although the test is designed to detect a single changepoint in a GARCH process, the test is applied sequentially in an attempt to find multiple changepoints. It is found that a single GARCH(1,1) model cannot be fitted for the DWJ series from January 1997 through December 2006. Sequential changepoint testing indicates a single changepoint in the data. In order to assess the reliability of these conclusions, a simulation study is used to investigate the properties of the changepoint test. The change-point test performs quite well in terms of type I error, but its power is small. Overall a better test for change-point detection in GARCH(1,1) processes would be welcome.