Discussion on the review

May 24, 2011 This review of bootstrap methods for time series is most welcome especially coming from two key figures in the development of these methods. We would like to complement their exposition by focusing on some further issues of current interest. Block bootstrap methods for time series data have been most intensively studied under the assumption of stationarity and mixing. An important example is the stationary bootstrap of Politis and Romano (1994). As detailed in the review, this method is a variation of the standard block bootstrap that manages to create bootstrap series that are strictly stationary. This property not only replicates the strict stationarity of the original time series, but it also makes the method easy to analyze theoretically. However, some rather involved Mean Square Error calculations of Lahiri (1999) seemed to imply that the stationary bootstrap has poor asymptotic efficiency relative to the usual block bootstrap in the sample mean case. As a result, practitioners have been understandably reluctant in endorsing the stationary bootstrap after the publication of Lahiri’s (1999) paper. It was quite a surprise when Nordman (2009) discovered an error in Lahiri’s (1999) calculations; as it turns out, the stationary bootstrap has identical asymptotic accuracy as the block bootstrap with non-overlapping blocks; the latter is well-known to be only slightly less efficient than the full-overlap block bootstrap. These new findings are expected to rekindle the interest of researchers since the stationarity of bootstrap sample paths is a very powerful tool. For example, functional data are of current interest in the 21st century; see the review by McMurry and Politis (2011). Suffices to note that Politis and Romano (1994b) were able to prove the asymptotic validity of the stationary bootstrap for the sample mean of functional data almost twenty years ago. Such a result does not seem to be available for the regular block bootstrap even to date. The assumptions of strict stationarity and strong mixing can be too strong for many applications in economics and finance. First, for many of these applications, the time span is rather long, making it unlikely that stationarity holds. A more appropriate assumption is to allow for some time series heterogeneity (for instance, in the form of time varying moments). Similarly, mixing is too strong a dependence condition to be broadly applicable. Andrews (1994) gives an example of a simple AR(1) model that fails to be strong mixing. More generally, although measurable functions of mixing processes are themselves mixing when the function depends on a finite number of lagged values of the mixing process, this is not necessarily the case when the observed time series depends on the infinite history of an underlying mixing process. For instance, popular ARCH (Engle, 1982) and GARCH (Bollerslev, 1986) processes, which are a function of the entire past history of a fundamental driving innovation, are known to be mixing only when