The Bootstrap Estimation In Time Series

Abstract

Time series, a special case in dependent data sequence, is widely used in many fields. In time series, linear process models are quite popularly used. General form of linear process indicates the time dependence property of time series, AR(p), MA(q) and ARMA(p, q) models are all linear process models. In this report, simulations are based on the simplest models of these linear process models, such as AR(1), MA(1) and ARMA(1,1) models. AR(1)-SEASON, which is developed based on AR(1) model by changing the weight of residuals, is also considered in this report. To deal with dependent data sequence, common methods which aim to deal with independent data are no longer accurate to do inference. For dependent data, a conventional method involves consistent estimation of the long run variance, for example, Andrews [2]. However, in Andrews method, it might be hard to determine the bandwidth. As an alternative, bootstrap methods can be used to approximate the limiting distribution. Block based bootstrap methods, such as moving block bootstrap, non-overlapping block bootstrap and circular block bootstrap, can be used for dependent data. Stationary bootstrap, which is with flexible block length following xiii a geometric distribution with parameter pS, has also been proved to be consistent. AR-Sieve bootstrap aims to construct a fitted model of AR(p) and resampling the data with the fitted model. In our simulations, we compare finite sample confidence interval coverage rates. We also consider these bootstrap methods with Andrews estimation of variance [2] and simulations results show that with the help of Andrews estimation, the estimations are more accurate. A further discussion of determining an optimal block length for AR(1) model is also mentioned in our report.