TIME SERIES BLOCK BOOTSTRAP APPLICATION AND EFFECT OF AGGREGATION AND SYSTEMATIC SAMPLING

Abstract

In this dissertation, we review the basic properties of the bootstrap and time series application. Then we apply parametric bootstrap on three simulated normal i.i.d. samples and nonparametric bootstrap on four real life financial returns. Among the time series bootstrap methods, we look into the specific method called block bootstrap and investigate the block length consideration to properly select a suitable block size for AR(1) model. We propose a new rule of blocking named as Combinatorially-Augmented Block Bootstrap(CABB). We compare the existing block bootstrap and CABB method using the simulated i.i.d. samples, AR(1) time series, and the real life examples. Both methods perform equally well in estimating AR(1) coefficients. CABB produces a smaller standard deviation based on our simulated and empirical studies. We study two procedures of collecting time series, (i) aggregation of a flow variable and (ii) systematic sampling of a stock variable. In these two procedures, we derive theorems that calculate exact equations for $m$ aggregated and $m^{th}$ systematically sampled series of the original AR(1) model. We evaluate the performance of block bootstrap estimation of the parameters of ARMA(1,1) and AR(1) model using aggregated and systematically sampled series. Simulation and real data analyses show that in some cases, the performance of the estimation based on the block bootstrap method for the MA(1) parameter of the ARMA(1,1) model in aggregated series is better than the one without using bootstrap. In an extreme case of stock price movement, which is close to a random walk, the block bootstrap estimate using systematically sampled series is closer to the true parameter, defined as the parameter calculated by the theorem. Specifically, the block bootstrap estimate of the parameter of AR(1) model using the systematically sampled series is closer to phi(n) than that based on the MLE for the AR(1) model. Future research problems include theoretical investigation of CABB, effectiveness of block bootstrap in other time series analyses such as nonlinear or VAR.