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...

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