Bootstrap Confidence Regions Computed from Autoregressions of Arbitrary Order

Abstract

Given a linear time series, e.g. an autoregression of in®nite order, we may construct a ®nite order approximation and use that as the basis for bootstrap con®dence regions. The sieve or autoregressive bootstrap, as this method is often called, is generally seen as a competitor with the better-understood block bootstrap approach. However, in the present paper we argue that, for linear time series, the sieve bootstrap has signi®cantly better performance than blocking methods and offers a wider range of opportunities. In particular, since it does not corrupt second-order properties then it may be used in a double-bootstrap form, with the second bootstrap application being employed to calibrate a basic percentile method con®dence interval. This approach confers secondorder accuracy without the need to estimate variance. That offers substantial bene®ts, since variances of statistics based on time series can be dif®cult to estimate reliably, and Ð partly because of the relatively small amount of information contained in a dependent process Ð are notorious for causing problems when used to Studentize. Other advantages of the sieve bootstrap include considerably greater robustness against variations in the choice of the tuning parameter, here equal to the autoregressive order, and the fact that, in contradistinction to the case of the block bootstrap, the percentile t version of the sieve bootstrap may be based on the`raw' estimator of standard error. In the process of establishing these properties we show that the sieve bootstrap is second order correct.(b) when using the percentile t form of the sieve bootstrap, the estimator of the standard error does not need to be adjusted to capture second-order eects and (c) the choice of autoregressive order for the sieve bootstrap is not nearly as critical as the selection of block length for the block bootstrap.Direct attempts by the block bootstrap to replicate the dependence structure of the original time series are accurate only to ®rst order. They fail to reproduce second-order eects because the dependence is corrupted at places where blocks join; see for example Davison and Hall (1993). As a result, when using the percentile t method in conjunction with the block bootstrap to capture second-order features of the distribution of a statistic, an indirect approach must be taken: the standard error estimate must be altered to counteract secondorder inaccuracies. This is unnecessary in the case of the sieve bootstrap, however, since that method preserves the dependence structure to second order. That explains property (b) above. Property (a) follows for essentially the same reason: second-order features of a time series are not consistently estimated by the unadjusted block bootstrap, and so the doublebootstrapped percentile method does not give second-order accuracy in that context, whereas it does when used in conjunction with the sieve bootstrap.The diculty of estimating the variance for general statistics associated with a time series can make percentile t methods unattract...

Bootstrap methods for long-memory processes: a review

“…It is valid to approximate the distribution of n θ/2 (X n − µ) for linear, LRD processes under certain conditions (Kapetanios and Psaradakis, 2006). For short-memory, Bühlmann (1997) and Choi and Hall (2000) illustrated that the AR-sieve bootstrap method can give more accurate distribution estimation in comparison to the block bootstrap methods. However, the condition of the process for the AR-sieve bootstrap is more stringent than the block bootstrap methods, for example the process satisfies the conditions of causality, linearity and often invertibility.…”

“…The AR-sieve bootstrap for weakly dependent time processes was introduced by Kreiss (1992) and has been developed by Bühlmann (1997) and Bickel and Bühlmann (1999). In addition, Choi and Hall (2000) showed that the AR-sieve bootstrap was more powerful than block bootstrap methods under causal linear SRD processes based on the error in the coverage probability of a one-sided confidence interval. Kapetanios and Psaradakis (2006) and Poskitt (2008) applied the AR-sieve bootstrap to causal linear LRD time processes.…”

Heterogeneous Dynamics, Aggregation, and the Persistence of Economic Shocks

“…In order to isolate the impact of an increase in heterogeneity from an increase in average persistence, different degrees of het- (2008). 1 2 Bootstrap methods have also been employed for inference in univariate AR(∞) models, see Kreiss (1997), Choi and Hall (2000) and Gonçalves and Kilian (2004) among others. Extensions to the multivariate case have been proposed by Paparoditis (1996) and Inoue and Kilian (2002).…”

Computational framework for longevity risk management

“…This is done in order to capture the dependency structure of neighbouring observations (Liu and Braun 2010). In the literature, there is considerable evidence that the sieve bootstrap, initially proposed by Kreiss (1992) and Bulhmann (1997), usually outperforms the block bootstrap (Choi and Hall 2000). D'Amato et al (2012) apply a sieve bootstrap on the residuals of the Lee Carter model; they take up the Lee Carter parametric model firstly and then re-sample a particular class of the residuals, the so-called centred residuals, according to the design of the typical autoregressive sieve bootstrap.…”