Improved Bootstrap Methods for Time Series Models.

Abstract

The bootstrap methods are used extensively in statistical analysis of econometric and time series models. In small sample situations, where the asymptotic theory does not work well to approximate the unknown sampling distribution of a statistic, bootstrap methods are used as an alternative. The idea is that the observed sample contains useful information about the population characteristics and resampling from it can give a good approximation to the sampling distribution. Therefore, bootstrap approximations can provide better small sample performance than those obtained from asymptotic theory. The main focus of the present work is the application of bootstrap methods to time series models. The current study has three dimensions. The first part is concerned with the construction of bootstrap prediction intervals for autoregressive fractionally integrated moving-average processes which is a special class of long memory time series. For linear short-range dependent time series, the bootstrap based prediction interval is a good nonparametric alternative to those constructed under parameter assumptions. In the long memory case, we use AR-sieve bootstrap which approximates the data generating process of a given long memory time series by a finite order autoregressive process and resample the residuals. For the construction of prediction intervals, we applied two sieve bootstrap algorithms. A simulation study is conducted to examine and compare the performance of these AR-sieve bootstrap procedures. We use four different values of the long memory parameter d. For the purpose of illustration a real data example is also presented. In second part of this work, we propose two bootstrap procedures to construct prediction intervals for ARFIMA-GARCH models. The first method is based on the model based bootstrap, in which the order of the model is assumed to be known. The second bootstrap method is based on the idea of approximating the ARFIMA part by an AR model. In modeling the ARFIMA-GARCH model, the first step is to determine the order of ARFIMA part. Determination of the order of ARFIMA model is a complicated task. To simplify the model building procedure, we approximate the ARFIMA part of the ARFIMA-GARCH model by an AR(p) model and fit an AR-GARCH model instead of ARFIMA-GARCH model. The third part of this thesis is based on testing goodness-of-fit in Autoregressive fractionally integrated moving-average models with conditional hetroscedasticity. We extend the applicability of Hong’s and power transformed Hong’s test statistics as goodness-of-fit tests in ARFIMA-GARCH models where the structural form of GARCH model is unknown. Simulation study is performed to assess the size and power performance of both tests.