The bootstrap is a method for estimating the distribution of an estimator or test statistic by resampling one’s data. It amounts to treating the data as if they were the population for the purpose of evaluating the distribution of interest. Under mild regularity conditions, the bootstrap yields an approximation to the distribution of an estimator or test statistic that is at least as accurate as the

Sparked by Efron’s seminal paper, the decade of the 1980s was a period of active research on bootstrap methods for independent data— mainly i.i.d. or regression set-ups. By contrast, in the 1990s much research was directed towards resampling dependent data, for example, time series and random fields. Consequently, the availability of valid nonparametric inference procedures based on resampling and/or subsampling has freed practitioners from the necessity of resorting to simplifying assumptions such as normality or linearity that may be misleading.

The blockwise bootstrap is a modification of Efron's bootstrap designed to give correct results for dependent stationary observations. One drawback of the method is that it depends critically on a block length which had to be chosen by the user. Here we propose a fully data-driven method to select this block length. It is based on the equivalence of the blockwise bootstrap variance to a lag weight estimator of a spectral density at the origin. The relevant spectral density is the one of the process given by the influence function of the statistic to be bootstrapped. In this equivalence the block length is the inverse of the bandwidth. We thus apply a recently developed local bandwidth selection procedure to the time series given by the estimated influence function. Simulations show that this procedure gives good results in a wide range of situations.

The assessment of the performance of learners by means of benchmark experiments is an established exercise. In practice, benchmark studies are a tool to compare the performance of several competing algorithms for a certain learning problem. Cross-validation or resampling techniques are commonly used to derive point estimates of the performances which are compared to identify algorithms with good properties. For several benchmarking problems, test procedures taking the variability of those point estimates into account have been suggested. Most of the recently proposed inference procedures are based on special variance estimators for the cross-validated performance. We introduce a theoretical framework for inference problems in benchmark experiments and show that standard statistical test procedures can be used to test for differences in the performances. The theory is based on well defined distributions of performance measures which can be compared with established tests. To demonstrate the usefulness in practice, the theoretical results are applied to regression and classification benchmark studies based on artificial and real world data.

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We study a bootstrap method for stationary real-valued time series, which is based on the method of sieves. We restrict ourselves to autoregressive sieve bootstraps. Given a sample X1;:::;X n from a linear process fX tg t2 Z, we approximate the underlying process by an autoregressive model with order p = p(n), where p(n)!1;p(n) =o(n) as the sample size n!1. Based on such a model a bootstrap process fX t g t2 Z is constructed from which one can draw samples of any size. We giveanovel result which says that with high probability,such a sieve bootstrap process fX t g t2 Z satis es a new type of mixing condition. This implies that many results for stationary, mixing sequences carry over to the sieve bootstrap process. As an example we derive a functional central limit theorem under a bracketing condition. Key words and phrases. AR(1), ARMA, autoregressive approximation, bracketing, convex

New Keynesian Phillips Curves (NKPC) have been extensively used in the analysis of monetary policy, but yet there are a number of issues of concern about how they are estimated and then related to the underlying macroeconomic theory. The first is whether such equations are identified. To check identification requires specifying the process for the forcing variables (typically the output gap) and solving the model for inflationintermsoftheobservables. Inpractice,theequationisestimatedby GMM, relying on statistical criteria to choose instruments. This may result in failure of identification or weak instruments. Secondly, the NKPC is usually derived as a part of a DSGE model, solved by log-linearising around a steady state and the variables are then measured in terms of deviations from the steady state. In practice the steady states, e.g. for output, are usually estimated by some statistical procedure such as the Hodrick-Prescott (HP) filter that might not be appropriate. Thirdly, there are arguments that other variables, e.g. interest rates, foreign inflation and foreign output gaps should enter the Phillips curve. This paper examines these three issues and argues that all three benefit from a global perspective. The global perspective provides additional instruments to alleviate the weak instrument problem, yields a theoretically consistent measure of the steady state and provides a natural route for foreign inflation or output gap to enter the NKPC. Keywords: Global VAR (GVAR), identification, New Keynesian Phillips Curve, Trend-Cycle decomposition.

In this paper we deal with stationary autoregressive processes of finite or infinite but unknown order. Under fairly general assumptions we derive the asymptotic consistency of a usual residual bootstrap procedure for smooth functions of the empirical autocovariance and autocorrelation. Especially the order of the fitted autoregressive model is allowed to be data-dependent. Supplementary to the usual residual bootstrap we consider a wild bootstrap procedure. Some remarks concerning the asymptotic accuracy of the two proposed bootstrap procedures and a simulation study conclude the paper.

In recent years, there has been increasing interest in nonparametric bootstrap inference for economic time series. Nonparametric resampling techniques help protect against overly optimistic inference in time series models of unknown structure. They are particularly useful for evaluating the fit of dynamic economic models in terms of their spectra, impulse responses, and related statistics, because they do not require a correctly specified economic model. Notwithstanding the potential advantages of nonparametric bootstrap methods, their reliability in small samples is questionable. In this paper, we provide a benchmark for the relative accuracy of several nonparametric resampling algorithms based on ARMA representations of four macroeconomic time series. For each algorithm, we evaluate the effective coverage accuracy of impulse response and spectral density bootstrap confidence intervals for standard sample sizes. We find that the autoregressive sieve approach based on the encompassing model is most accurate. However, care must be exercised in selecting the lag order of the autoregressive approximation.

Abstract. Ever since its introduction, the bootstrap has provided both a powerful set of solutions for practical statisticians, and a rich source of theoretical and methodological problems for statistics. In this article, some recent developments in bootstrap methodology are reviewed and discussed. After a brief introduction to the bootstrap, we consider the following topics at varying levels of detail: the use of bootstrapping for highly accurate parametric inference; theoretical properties of nonparametric bootstrapping with unequal probabilities; subsampling and the m out of n bootstrap; bootstrap failures and remedies for superefficient estimators; recent topics in significance testing; bootstrap improvements of unstable classifiers and resampling for dependent data. The treatment is telegraphic rather than exhaustive. Key words and phrases: Bagging, bootstrap, conditional inference, empirical strength probability, parametric bootstrap, subsampling, superefficient