Bootstrap confidence regions computed from autoregressions of arbitrary order

Given a linear time series, e.g. an autoregression of infinite order, we may construct a finite order approximation and use that as the basis for confidence 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 significantly 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 confidence interval. This approach confers second-order accuracy without the need to estimate variance. That offers substantial benefits, since variances of statistics based on time series can be difficult to estimate reliably, and—partly because of the relatively small amount of information contained in a dependent process—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.     

Using the Bootstrap to Test for Symmetry Under Unknown Dependence

This article considers tests for symmetry of the one-dimensional marginal distribution of fractionally integrated processes. The tests are implemented by using an autoregressive sieve bootstrap approximation to the null sampling distribution of the relevant test statistics. The sieve bootstrap allows inference on symmetry to be carried out without knowledge of either the memory parameter of the data or of the appropriate norming factor for the test statistic and its asymptotic distribution. The small-sample properties of the proposed method are examined by means of Monte Carlo experiments, and applications to real-world data are also presented.     

The Wild Bootstrap for Multilevel Models

In this paper, we study the performance of the most popular bootstrap schemes for multilevel data. Also, we propose a modified version of the wild bootstrap procedure for hierarchical data structures. The wild bootstrap does not require homoscedasticity or assumptions on the distribution of the error processes. Hence, it is a valuable tool for robust inference in a multilevel framework. We assess the finite size performances of the schemes through a Monte Carlo study. The results show that for big sample sizes it always pays off to adopt an agnostic approach as the wild bootstrap outperforms other techniques.     

Asymptotic and Bootstrap Inference for AR(∞) Processes with Conditional Heteroskedasticity

The main contribution of this paper is a proof of the asymptotic validity of the application of the bootstrap to AR(∞) processes with unmodelled conditional heteroskedasticity. We first derive the asymptotic properties of the least-squares estimator of the autoregressive sieve parameters when the data are generated by a stationary linear process with martingale difference errors that are possibly subject to conditional heteroskedasticity of unknown form. These results are then used in establishing that a suitably constructed bootstrap estimator will have the same limit distribution as the least-squares estimator. Our results provide theoretical justification for the use of either the conventional asymptotic approximation based on robust standard errors or the bootstrap approximation of the distribution of autoregressive parameters. A simulation study suggests that the bootstrap approach tends to be more accurate in small samples.     

Asymptotic and Bootstrap Inference for AR(∞) Processes with Conditional Heteroskedasticity

The main contribution of this paper is a proof of the asymptotic validity of the application of the bootstrap to AR(∞) processes with unmodelled conditional heteroskedasticity. We first derive the asymptotic properties of the least-squares estimator of the autoregressive sieve parameters when the data are generated by a stationary linear process with martingale difference errors that are possibly subject to conditional heteroskedasticity of unknown form. These results are then used in establishing that a suitably constructed bootstrap estimator will have the same limit distribution as the least-squares estimator. Our results provide theoretical justification for the use of either the conventional asymptotic approximation based on robust standard errors or the bootstrap approximation of the distribution of autoregressive parameters. A simulation study suggests that the bootstrap approach tends to be more accurate in small samples.     

Bootstrap of the linear correlation model

In the linear correlation model one observes an i.i.d. sequence (Xi, F$), i^l, of {p + l)-variate random vectors satisfying Yi=Xif}+ei. In this paper, we show the validity of the bootstrap for the LSE of p under minimal moment assumptions, answering a question posed by D. FREEDMAK     

The Bootstrap and Kriging Prediction Intervals

Kriging is a method for spatial prediction that, given observations of a spatial process, gives the optimal linear predictor of the process at a new specified point. The kriging predictor may be used to define a prediction interval for the value of interest. The coverage of the prediction interval will, however, equal the nominal desired coverage only if it is constructed using the correct underlying covariance structure of the process. If this is unknown, it must be estimated from the data. We study the effect on the coverage accuracy of the prediction interval of substituting the true covariance parameters by estimators, and the effect of bootstrap calibration of coverage properties of the resulting 'plugin' interval. We demonstrate that plugin and bootstrap calibrated intervals are asymptotically accurate in some generality and that bootstrap calibration appears to have a significant effect in improving the rate of convergence of coverage error.     

Bootstrap prediction regions for multivariate autoregressive processes

Two new methods for improving prediction regions in the context of vector autoregressive (VAR) models are proposed. These methods, which are based on the bootstrap technique, take into account the uncertainty associated with the estimation of the model order and parameters. In particular, by exploiting an independence property of the prediction error, we will introduce a bootstrap procedure that allows for better estimates of the forecasting distribution, in the sense that the variability of its quantile estimators is substantially reduced, without requiring additional bootstrap replications. The proposed methods have a good performance even if the disturbances distribution is not Gaussian. An application to a real data set is presented.     

Bootstrap-based bias corrections for INAR count time series

Integer-valued autoregressive (INAR) processes form a very useful class of processes suitable to model time series of counts. Several practically relevant estimators based on INAR data are known to be systematically biased away from their population values, e.g. sample autocovariances, sample autocorrelations, or the dispersion index. We propose to do bias correction for such estimators by using a recently proposed INAR-type bootstrap scheme that is tailor-made for INAR processes, and which has been proven to be asymptotically consistent under general conditions. This INAR bootstrap allows an implementation with and without parametrically specifying the innovations' distribution. To judge the potential of corresponding bias correction, we compare these bootstraps in simulations to several competitors that include the AR bootstrap and block bootstrap. Finally, we conclude with an illustrative data application.     

Bootstrap maximum likelihood for quasi-stationary distributions

Quasi-stationary distributions have many applications in diverse research fields. We develop a bootstrap-based maximum likelihood (BML) method to deal with quasi-stationary distributions in statistical inference. To efficiently implement a bootstrap procedure that can handle the dependence among observations and speed up the computation, a novel block bootstrap algorithm is proposed to accommodate parallel bootstrap. In particular, we select a suitable block length for use with the parallel bootstrap. The estimation error is investigated to show its convergence. The proposed BML is shown to be asymptotically unbiased. Some numerical studies are given to examine the performance of the new algorithm. The advantages are evidenced through a comparison with some competitors and some examples are analysed for illustration.     

Prediction Intervals for Time Series: A Modified Sieve Bootstrap Approach

Traditional Box–Jenkins prediction intervals perform poorly when the innovations are not Gaussian. Nonparametric bootstrap procedures overcome this handicap, but most existing methods assume that the AR and MA orders of the process are known. The sieve bootstrap approach requires no such assumption but produces liberal coverage due to the use of residuals that underestimate the actual variance of the innovations and the failure of the methods to capture variations due to sampling error of the mean. A modified approach, that corrects these deficiencies, is implemented. Monte Carlo simulations results show that the modified version achieves nominal or near nominal coverage.     

New block bootstrap methods: Sufficient and/or ordered

In this study, we propose sufficient time series bootstrap methods that achieve better results than conventional non-overlapping block bootstrap, but with less computing time and lower standard errors of estimation. Also, we propose using a new technique using ordered bootstrapped blocks, to better preserve the dependency structure of the original data. The performance of the proposed methods are compared in a simulation study for MA(2) and AR(2) processes and in an example. The results show that our methods are good competitors that often exhibit improved performance over the conventional block methods.     

Bootstrap estimation of the variance of the error term in monotonic regression models

The variance of the error term in ordinary regression models and linear smoothers is usually estimated by adjusting the average squared residual for the trace of the smoothing matrix (the degrees of freedom of the predicted response). However, other types of variance estimators are needed when using monotonic regression (MR) models, which are particularly suitable for estimating response functions with pronounced thresholds. Here, we propose a simple bootstrap estimator to compensate for the over-fitting that occurs when MR models are estimated from empirical data. Furthermore, we show that, in the case of one or two predictors, the performance of this estimator can be enhanced by introducing adjustment factors that take into account the slope of the response function and characteristics of the distribution of the explanatory variables. Extensive simulations show that our estimators perform satisfactorily for a great variety of monotonic functions and error distributions.     

Parallel Bootstrap and Optimal Subsample Lengths in Smooth Function Models

Parallel bootstrap is an extremely useful statistical method with good performance. In the present study, we introduce a working correlation matrix on the method, which is called parallel bootstrap matrix. We consider some properties of it and the optimal size of the subsample in smooth function models. We also present some performance results of parallel bootstrap estimators, the subsample length selection on the finance time series data.     

Bootstrap for U-statistics: a new approach

Bootstrap for nonlinear statistics like U-statistics of dependent data has been studied by several authors. This is typically done by producing a bootstrap version of the sample and plugging it into the statistic. We suggest an alternative approach of getting a bootstrap version of U-statistics. We will show the consistency of the new method and compare its finite sample properties in a simulation study and by applying both methods to financial data.     

Moving Block Bootstrap for Analyzing Longitudinal Data

In a longitudinal study subjects are followed over time. I focus on a case where the number of replications over time is large relative to the number of subjects in the study. I investigate the use of moving block bootstrap methods for analyzing such data. Asymptotic properties of the bootstrap methods in this setting are derived. The effectiveness of these resampling methods is also demonstrated through a simulation study.     

Optimal Bootstrap Sample Size in Construction of Percentile Confidence Bounds

In traditional bootstrap applications the size of a bootstrap sample equals the parent sample size, n say. Recent studies have shown that using a bootstrap sample size different from n may sometimes provide a more satisfactory solution. In this paper we apply the latter approach to correct for coverage error in construction of bootstrap confidence bounds. We show that the coverage error of a bootstrap percentile method confidence bound, which is of order O ( n ^−2/2) typically, can be reduced to O ( n ⁻¹) by use of an optimal bootstrap sample size. A simulation study is conducted to illustrate our findings, which also suggest that the new method yields intervals of shorter length and greater stability compared to competitors of similar coverage accuracy.     

Bootstrap bandwidth selection method for local linear estimator in exponential family models

Many biological experiments involve data whose distribution belongs to the exponential family. Such data are often analysed using generalised linear models but this method requires specification of the link function which can have strong influence on the resulting estimate. Instead a local method based on quasi-likelihood can be used, but the choice of the smoothing parameter is crucial for its performance. A bootstrap bandwidth selection method is proposed and shown to be consistent. Examples of application to data from biological and psychometric experiments are given.     

Bootstrap methods for stationary functional time series

Bootstrap methods for estimating the long-run covariance of stationary functional time series are considered. We introduce a versatile bootstrap method that relies on functional principal component analysis, where principal component scores can be bootstrapped by maximum entropy. Two other bootstrap methods resample error functions, after the dependence structure being modeled linearly by a sieve method or nonlinearly by a functional kernel regression. Through a series of Monte-Carlo simulation, we evaluate and compare the finite-sample performances of these three bootstrap methods for estimating the long-run covariance in a functional time series. Using the intraday particulate matter (\(\hbox {PM}_{10}\)) dataset in Graz, the proposed bootstrap methods provide a way of constructing the distribution of estimated long-run covariance for functional time series.