• Communications for Statistical Applications and Methods
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• v.20 no.1
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• pp.41-52
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• 2013

The stationary bootstrap of Politis and Romano (1994) is adopted to develop prediction intervals of returns and volatilities in a generalized autoregressive heteroskedastic (GARCH)(p, q) model. The stationary bootstrap method is applied to generate bootstrap observations of squared returns and residuals, through an ARMA representation of the GARCH model. The stationary bootstrap estimators of unknown parameters are defined and used to calculate the stationary bootstrap samples of volatilities. Estimates of future values of returns and volatilities in the GARCH process and the bootstrap prediction intervals are constructed based on the stationary bootstrap; in addition, asymptotic validities are also shown.

• Journal of the Korean Mathematical Society
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• v.58 no.1
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• pp.237-264
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• 2021

In this work the stationary bootstrap of Politis and Romano [27] is applied to the empirical distribution function of stationary and associated random variables. A weak convergence theorem for the stationary bootstrap empirical processes of associated sequences is established with its limiting to a Gaussian process almost surely, conditionally on the stationary observations. The weak convergence result is proved by means of a random central limit theorem on geometrically distributed random block size of the stationary bootstrap procedure. As its statistical applications, stationary bootstrap quantiles and stationary bootstrap mean residual life process are discussed. Our results extend the existing ones of Peligrad [25] who dealt with the weak convergence of non-random blockwise empirical processes of associated sequences as well as of Shao and Yu [35] who obtained the weak convergence of the mean residual life process in reliability theory as an application of the association.

• Communications for Statistical Applications and Methods
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• v.24 no.4
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• pp.367-382
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• 2017

We consider an infinite-order long-memory heterogeneous autoregressive (HAR) model, which is motivated by a long-memory property of realized volatilities (RVs), as an extension of the finite order HAR-RV model. We develop bootstrap tests for structural mean or variance changes in the infinite-order HAR model via stationary bootstrapping. A functional central limit theorem is proved for stationary bootstrap sample, which enables us to develop stationary bootstrap cumulative sum (CUSUM) tests: a bootstrap test for mean break and a bootstrap test for variance break. Consistencies of the bootstrap null distributions of the CUSUM tests are proved. Consistencies of the bootstrap CUSUM tests are also proved under alternative hypotheses of mean or variance changes. A Monte-Carlo simulation shows that stationary bootstrapping improves the sizes of existing tests.

• Communications for Statistical Applications and Methods
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• v.22 no.1
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• pp.81-93
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• 2015

Validity of the stationary bootstrap of Politis and Romano (1994) is proved for U-statistics under strong mixing. Weak and strong consistencies are established for the stationary bootstrap of U-statistics. The theory is applied to a symmetry test which is a U-statistic regarding a kernel density estimator. The theory enables the bootstrap confidence intervals of the means of the U-statistics. A Monte-Carlo experiment for bootstrap confidence intervals confirms the asymptotic theory.

• Communications for Statistical Applications and Methods
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• v.20 no.1
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• pp.85-96
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• 2013

A new bootstrap method combined with the stationary bootstrap of Politis and Romano (1994) and the classical residual-based bootstrap is applied to stationary autoregressive (AR) time series models. A stationary bootstrap procedure is implemented for the ordinary least squares estimator (OLSE), along with classical bootstrap residuals for estimated errors, and its large sample validity is proved. A finite sample study numerically compares the proposed bootstrap estimator with the estimator based on the classical residual-based bootstrapping. The study shows that the proposed bootstrapping is more effective in estimating the AR coefficients than the residual-based bootstrapping.

• Communications for Statistical Applications and Methods
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• v.22 no.5
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• pp.463-473
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• 2015

We consider a nonparametric AR(1) model with nonparametric ARCH(1) errors. In order to estimate the unknown function of the ARCH part, we apply the stationary bootstrap procedure, which is characterized by geometrically distributed random length of bootstrap blocks and has the advantage of capturing the dependence structure of the original data. The proposed method is composed of four steps: the first step estimates the AR part by a typical kernel smoothing to calculate AR residuals, the second step estimates the ARCH part via the Nadaraya-Watson kernel from the AR residuals to compute ARCH residuals, the third step applies the stationary bootstrap procedure to the ARCH residuals, and the fourth step defines the stationary bootstrapped Nadaraya-Watson estimator for the ARCH function with the stationary bootstrapped residuals. We prove the asymptotic validity of the stationary bootstrap estimator for the unknown ARCH function by showing the same limiting distribution as the Nadaraya-Watson estimator in the second step.

• Communications for Statistical Applications and Methods
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• v.23 no.2
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• pp.163-177
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• 2016

We consider a jump diffusion process for high-frequency financial asset data. We apply the stationary bootstrapping to construct a bootstrap test for jumps. First-order asymptotic validity is established for the stationary bootstrapping of the jump ratio test under the null hypothesis of no jump. Consistency of the stationary bootstrap test is proved under the alternative of jumps. A Monte-Carlo experiment shows the advantage of a stationary bootstrapping test over the test based on the normal asymptotic theory. The proposed bootstrap test is applied to construct continuous-jump decomposition of the daily realized variance of the KOSPI for the year 2008 of the world-wide financial crisis.

• Communications for Statistical Applications and Methods
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• v.13 no.3
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• pp.745-754
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• 2006

It has been shown that LAD estimates are more efficient than LS estimates when the error distribution is double exponential in AR(1) model. In order to explore the performance of LAD estimates one can use bootstrap approaches. In this paper we consider the efficiencies of bootstrap methods when we apply LAD estimates with highly variable data. Monte Carlo simulation results are given for comparing generalized bootstrap, stationary bootstrap and threshold bootstrap methods.

• Communications for Statistical Applications and Methods
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• v.20 no.3
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• pp.193-197
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• 2013

Almost sure central limit theorems are established for a stationary bootstrap sample mean of strong mixing processes. Both weak and strong consistencies are obtained.

• Journal of the Korean Operations Research and Management Science Society
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• v.38 no.2
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• pp.65-73
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• 2013

In the simulation output analysis, bootstrap method is an applicable resampling technique to insufficient data which are not significant statistically. The moving block bootstrap, the stationary bootstrap, and the threshold bootstrap are typical bootstrap methods to be used for autocorrelated time series data. They are nonparametric methods for stationary time series data, which correctly describe the original data. In the simulation output analysis, however, we may not use them because of the non-stationarity in the data set caused by the trend such as increasing or decreasing. In these cases, we can get rid of the trend by differencing the data, which guarantees the stationarity. We can get the bootstrapped data from the differenced stationary data. Taking a reverse transform to the bootstrapped data, finally, we get the pseudo-samples for the original data. In this paper, we introduce the applicability of bootstrap methods to the time series data having trend, and then verify it through the statistical analyses.