Wild Bootstrap Research Articles

The wild bootstrap for few (treated) clusters

SummaryInference based on cluster‐robust standard errors in linear regression models, using either the Student's t‐distribution or the wild cluster bootstrap, is known to fail when the number of treated clusters is very small. We propose a family of new procedures called the subcluster wild bootstrap, which includes the ordinary wild bootstrap as a limiting case. In the case of pure treatment models, where all observations within clusters are either treated or not, the latter procedure can work remarkably well. The key requirement is that all cluster sizes, regardless of treatment, should be similar. Unfortunately, the analogue of this requirement is not likely to hold for difference‐in‐differences regressions. Our theoretical results are supported by extensive simulations and an empirical example.

Structural breaks in panel data: Large number of panels and short length time series

ABSTRACTThe detection of (structural) breaks or the so called change point problem has drawn increasing attention from the theoretical, applied economic and financial fields. Much of the existing research concentrates on the detection of change points and asymptotic properties of their estimators in panels when N, the number of panels, as well as T, the number of observations in each panel are large. In this paper we pursue a different approach, i.e., we consider the asymptotic properties when N→∞ while keeping T fixed. This situation is typically related to large (firm-level) data containing financial information about an immense number of firms/stocks across a limited number of years/quarters/months. We propose a general approach for testing for break(s) in this setup. In particular, we obtain the asymptotic behavior of test statistics. We also propose a wild bootstrap procedure that could be used to generate the critical values of the test statistics. The theoretical approach is supplemented by numerous simulations and by an empirical illustration. We demonstrate that the testing procedure works well in the framework of the four factors CAPM model. In particular, we estimate the breaks in the monthly returns of US mutual funds during the period January 2006 to February 2010 which covers the subprime crises.

A LOCAL GAUSSIAN BOOTSTRAP METHOD FOR REALIZED VOLATILITY AND REALIZED BETA

This article introduces a local Gaussian bootstrap method useful for the estimation of the asymptotic distribution of high-frequency data-based statistics such as functions of realized multivariate volatility measures as well as their asymptotic variances. The new approach consists of dividing the original data into nonoverlapping blocks ofMconsecutive returns sampled at frequencyh(whereh−1denotes the sample size) and then generating the bootstrap observations at each frequency within a block by drawing them randomly from a mean zero Gaussian distribution with a variance given by the realized variance computed over the corresponding block.Our main contributions are as follows. First, we show that the local Gaussian bootstrap is first-order consistent when used to estimate the distributions of realized volatility and realized betas under assumptions on the log-price process which follows a continuous Brownian semimartingale process. Second, we show that the local Gaussian bootstrap matches accurately the first four cumulants of realized volatility up too(h), implying that this method provides third-order refinements. This is in contrast with the wild bootstrap of Gonçalves and Meddahi (2009,Econometrica77(1), 283–306), which is only second-order correct. Third, we show that the local Gaussian bootstrap is able to provide second-order refinements for the realized beta, which is also an improvement of the existing bootstrap results in Dovonon, Gonçalves, and Meddahi (2013,Journal of Econometrics172, 49–65) (where the pairs bootstrap was shown not to be second-order correct under general stochastic volatility). In addition, we highlight the connection between the local Gaussian bootstrap and the local Gaussianity approximation of continuous semimartingales established by Mykland and Zhang (2009,Econometrica77, 1403–1455) and show the suitability of this bootstrap method to deal with the new class of estimators introduced in that article. Lastly, we provide Monte Carlo simulations and use empirical data to compare the finite sample accuracy of our new bootstrap confidence intervals for integrated volatility with the existing results.

Assessing the role of components of life satisfaction

ABSTRACTFactors affecting subjective well-being and their gross effect measurement have been widely studied. However, how people derive utility from these factors has not been fully explored. This article provides a way to decompose the gross effect of major determinants on life satisfaction into direct and indirect effects and make inference for the latter. Because the indirect effect is nonlinear in parameters associated with different models in an equation system, it creates a problem in estimating the standard error. Besides, the categorical nature in survey data further introduces bias to the covariance estimation even asymptotically. To address these issues without knowing the form of heteroskedasticity under the null hypothesis, we propose to extend the wild bootstrap procedure in this particular context. Its robustness against various data properties is validated via several simulation experiments. Using three recent waves of World Values Survey, we find that the relative importance of life control has significantly increased recently, and endowing citizens with the right to enjoy their freedoms and liberties is a more effective policy in raising life satisfaction than promoting national income.

BOOTSTRAP-ASSISTED UNIT ROOT TESTING WITH PIECEWISE LOCALLY STATIONARY ERRORS

In unit root testing, a piecewise locally stationary process is adopted to accommodate nonstationary errors that can have both smooth and abrupt changes in second- or higher-order properties. Under this framework, the limiting null distributions of the conventional unit root test statistics are derived and shown to contain a number of unknown parameters. To circumvent the difficulty of direct consistent estimation, we propose to use the dependent wild bootstrap to approximate the nonpivotal limiting null distributions and provide a rigorous theoretical justification for bootstrap consistency. The proposed method is compared through finite sample simulations with the recolored wild bootstrap procedure, which was developed for errors that follow a heteroscedastic linear process. Furthermore, a combination of autoregressive sieve recoloring with the dependent wild bootstrap is shown to perform well. The validity of the dependent wild bootstrap in a nonstationary setting is demonstrated for the first time, showing the possibility of extensions to other inference problems associated with locally stationary processes.

The wild bootstrap for multivariate Nelson\u2013Aalen estimators

We rigorously extend the widely used wild bootstrap resampling technique to the multivariate Nelson–Aalen estimator under Aalen’s multiplicative intensity model. Aalen’s model covers general Markovian multistate models including competing risks subject to independent left-truncation and right-censoring. This leads to various statistical applications such as asymptotically valid confidence bands or tests for equivalence and proportional hazards. This is exemplified in a data analysis examining the impact of ventilation on the duration of intensive care unit stay. The finite sample properties of the new procedures are investigated in a simulation study.

Evaluation of inter-site bias and variance in diffusion-weighted MRI.

An understanding of the bias and variance of diffusion weighted magnetic resonance imaging (DW-MRI) acquisitions across scanners, study sites, or over time is essential for the incorporation of multiple data sources into a single clinical study. Studies that combine samples from various sites may be introducing confounding due to site-specific artifacts and patterns. Differences in bias and variance across sites may render the scans incomparable, and, without correction, any inferences obtained from these data are misleading. We present an analysis of the bias and variance of scans of the same subjects across different sites and evaluate their impact on statistical analyses. In previous work, we presented a simulation extrapolation (SIMEX) technique for bias estimation as well as a wild bootstrap technique for variance estimation in metrics obtained from a Q-ball imaging (QBI) reconstruction of empirical high angular resolution diffusion imaging (HARDI) data. We now apply those techniques to data acquired from 5 healthy volunteers on 3 independent scanners under closely matched acquisition protocols. The bias and variance of GFA measurements were estimated on a voxel-wise basis for each scan and compared across study sites to identify site-specific differences. Further, we provide model recommendations that can be used to determine the extent of the impact of bias and variance as well as aspects of the analysis to account for these differences. We include a decision tree to help researchers determine if model adjustments are necessary based on the bias and variance results.

A wild bootstrap approach for the Aalen-Johansen estimator.

We suggest a wild bootstrap resampling technique for nonparametric inference on transition probabilities in a general time-inhomogeneous Markov multistate model. We first approximate the limiting distribution of the Nelson-Aalen estimator by repeatedly generating standard normal wild bootstrap variates, while the data is kept fixed. Next, a transformation using a functional delta method argument is applied. The approach is conceptually easier than direct resampling for the transition probabilities. It is used to investigate a non-standard time-to-event outcome, currently being alive without immunosuppressive treatment, with data from a recent study of prophylactic treatment in allogeneic transplanted leukemia patients. Due to non-monotonic outcome probabilities in time, neither standard survival nor competing risks techniques apply, which highlights the need for the present methodology. Finite sample performance of time-simultaneous confidence bands for the outcome probabilities is assessed in an extensive simulation study motivated by the clinical trial data. Example code is provided in the web-based Supplementary Materials.

Nonlinear measurement errors models subject to partial linear additive distortion

We study nonlinear regression models when the response and predictors are unobservable and distorted in a multiplicative fashion by partial linear additive models (PLAM) of some observed confounding variables. After approximating the additive nonparametric components in the PLAM via polynomial splines and calibrating the unobserved response and unobserved predictors, we develop a semi-parametric profile nonlinear least squares procedure to estimate the parameters of interest. The resulting estimators are shown to be asymptotically normal. To construct confidence intervals for the parameters of interest, an empirical likelihood-based statistic is proposed to improve the accuracy of the associated normal approximation. We also show that the empirical likelihood statistic is asymptotically chi-squared. Moreover, a test procedure based on the empirical process is proposed to check whether the parametric regression model is adequate or not. A wild bootstrap procedure is proposed to compute $p$-values. Simulation studies are conducted to examine the performance of the estimation and testing procedures. The methods are applied to re-analyze real data from a diabetes study for an illustration.

Testing for parameter instability in predictive regression models

We consider tests for structural change, based on the SupF and Cramer–von-Mises type statistics of Andrews (1993) and Nyblom (1989), respectively, in the slope and/or intercept parameters of a predictive regression model where the predictors display strong persistence. The SupF type tests are motivated by alternatives where the parameters display a small number of breaks at deterministic points in the sample, while the Cramer–von-Mises alternative is one where the coefficients are random and slowly evolve through time. In order to allow for an unknown degree of persistence in the predictors, and for both conditional and unconditional heteroskedasticity in the data, we implement the tests using a fixed regressor wild bootstrap procedure. The asymptotic validity of the bootstrap tests is established by showing that the asymptotic distributions of the bootstrap parameter constancy statistics, conditional on the data, coincide with those of the asymptotic null distributions of the corresponding statistics computed on the original data, conditional on the predictors. Monte Carlo simulations suggest that the bootstrap parameter stability tests work well in finite samples, with the tests based on the Cramer–von-Mises principle seemingly the most useful in practice. An empirical application to U.S. stock returns data demonstrates the practical usefulness of these methods.

Estimated Wold Representation and Spectral-Density-Driven Bootstrap for Time Series

Summary The second-order dependence structure of purely non-deterministic stationary processes is described by the coefficients of the famous Wold representation. These coefficients can be obtained by factorizing the spectral density of the process. This relationship together with some spectral density estimator is used to obtain consistent estimators of these coefficients. A spectral-density-driven bootstrap for time series is then developed which uses the entire sequence of estimated moving average coefficients together with appropriately generated pseudoinnovations to obtain a bootstrap pseudo-time-series. It is shown that if the underlying process is linear and if the pseudoinnovations are generated by means of an independent and identically distributed wild bootstrap which mimics, to the extent necessary, the moment structure of the true innovations, this bootstrap proposal asymptotically works for a wide range of statistics. The relationships of the proposed bootstrap procedure to some other bootstrap procedures, including the auto-regressive sieve bootstrap, are discussed. It is shown that the latter is a special case of the spectral-density-driven bootstrap, if a parametric auto-regressive spectral density estimator is used. Simulations investigate the performance of the new bootstrap procedure in finite sample situations. Furthermore, a real life data example is presented.

Testing Linear Factor Pricing Models with Individual Securities in Japan: Application of Shrinkage Estimation

This study proposes a multivariate test for linear factor asset pricing models when the number of assets, N, is larger than the time dimension of returns, T. We extend the exact test proposed by Gibbons et al. (1989) to obtain a nonsingular covariance matrix with fewer estimation errors in the case of T < N. We apply the shrinkage estimation to the covariance matrix of the idiosyncratic error and obtain the sampling distribution of the proposed test statistics by using the fixed-design wild bootstrap to address the conditional heteroskedasticity and cross-sectional correlation. Our Monte-Carlo evidence shows that the proposed test statistics have a satisfying size and power. By using the proposed test statistics, we find that the Tokyo Stock Price Index is not mean-variance efficient most of the time from 1985 to 2015 compared with the tangency portfolio constructed from the individual securities of the Tokyo Stock Exchange first section, while efficiency hypothesis of the market portfolio in the second section is not rejected during periods of Asian financial crisis and the global financial crisis.

Testing for Bubbles in Cryptocurrencies with Time-Varying Volatility

The recent evolution of cryptocurrencies has been characterized by bubble-like behavior and extreme volatility. While it is difficult to assess an intrinsic value to a specific cryptocurrency, one can employ recently proposed bubble tests that rely on recursive applications of classical unit root tests. This paper extends this approach to the case where volatility is time varying, assuming a deterministic long-run component that may take into account a decrease of unconditional volatility when the cryptocurrency matures with a higher market dissemination. Volatility also includes a stochastic short-run component to capture volatility clustering. The wild bootstrap is shown to correctly adjust the size properties of the bubble test, which retains good power properties. In an empirical application using eleven of the largest cryptocurrencies and the CRIX index, the general evidence in favor of bubbles is confirmed, but much less pronounced than under constant volatility.

Adaptive Wild Bootstrap Tests for a Unit Root With Non‐Stationary Volatility

Recent research has emphasised that permanent changes in the innovation variance (caused by structural shifts or an integrated volatility process) lead to size distortions in conventional unit root tests. Cavaliere and Taylor (2008) show how these size distortions may be resolved using the wild bootstrap. In this paper, we first derive the asymptotic power envelope for the unit root testing problem when the nonstationary volatility process is known. Next, we show that under suitable conditions, adaptation with respect to the volatility process is possible, in the sense that nonparametric estimation of the volatility process leads to the same asymptotic power envelope. Implementation of the resulting test involves cross-validation and the wild bootstrap. A Monte Carlo experiment shows that the asymptotic results are reflected in finite sample properties, and an empirical analysis of real exchange rates illustrates the applicability of the proposed procedures.

Does market capitalization matters? Tests of weak-form efficient market hypothesis for Thai stock market

This study contributes to the literature of stock market efficiency by examining the weak-form efficient hypothesis for Thai stock market, especially after the subprime mortgage crisis in which the restricted financial policies issued by Thai government has been implemented. Thus, the full sample and sub-sample windows covering pre- and post-subprime financial crisis periods are investigated. The newly developed multiple variance ratio tests based on the wild bootstrap together with the Lo-MacKinlay and the Chow-Denning variance ratio tests are employed for testing the hypothesis. In addition, sub-indices based upon market capitalisation are categorised for further investigation on the impact of firm-sized on efficiency. The results reveal that, for the market level, the Stock Exchange of Thailand exhibits the market efficiency during the period of pre- and post-subprime mortgage crisis. For the sub-indices point of view, however, the small-capitalisation index provides some evidences against a weak-form efficiency.

Significant Alphas in Real Estate Funds: An Empirical Comparison of Alternative Estimators

Executive SummaryReal estate returns recurrently show heteroscedasticity. Using Jensen's alpha as a measure of risk-adjusted returns, we compare test statistic sensitivity to alternative estimates of the standard errors. Utilizing several robust estimators, we find the wild bootstrap consistently provides the most conservative result in real estate mutual funds and REITs. Surprisingly, the Newey-West standard error increases the percentage of REITs exhibiting significant alphas. Sensitivity to specification error in the model is examined. Explanatory variables failed to systematically attenuate significant alphas. When using the wild boot-strapped HC3 standard errors, significant alphas in REITs are no greater than random chance. Our results suggest appropriate adjustment for heteroscedasticity in real estate returns would minimize the potential for erroneous interpretation.

The Wild Bootstrap with a 'Small' Number of 'Large' Clusters

This paper studies the properties of the wild bootstrap-based test proposed in Cameron et al. (2008) for testing hypotheses about the coefficients in a linear regression model with clustered data. Cameron et al. (2008) provide simulations that suggest this test works well even in settings with as few as five clusters, but existing theoretical analyses of its properties all rely on an asymptotic framework in which the number of clusters is “large. In contrast to these analyses, we employ an asymptotic framework in which the number of clusters is “small, but the number of observations per cluster is “large. In this framework, we provide conditions under which an unstudentized version of the test is valid in the sense that it has limiting rejection probability under the null hypothesis that does not exceed the nominal level. Importantly, these conditions require, among other things, certain homogeneity restrictions on the distribution of covariates. In contrast, we establish that a studentized version of the test may only over-reject the null hypothesis by a “small amount in the sense that it has limiting rejection probability under the null hypothesis that does not exceed the nominal level by more than an amount that decreases exponentially with the number of clusters. We obtain results qualitatively similar to those for the studentized version of the test for closely related \score bootstrap-based tests, which permit testing hypotheses about parameters in nonlinear models. We illustrate the relevance of our theoretical results for applied work via a simulation study and empirical application.

Inference for Local Distributions at High Sampling Frequencies: A Bootstrap Approach

We study inference for the local innovations of It\^o semimartingales. Specifically, we construct a resampling procedure for the empirical CDF of high-frequency innovations that have been standardized using a nonparametric estimate of its stochastic scale (volatility) and truncated to rid the effect of ``large jumps. Our locally dependent wild bootstrap (LDWB) accommodate issues related to the stochastic scale and jumps as well as account for a special block-wise dependence structure induced by sampling errors. We show that the LDWB replicates first and second-order limit theory from the usual empirical process and the stochastic scale estimate, respectively, as well as an asymptotic bias. Moreover, we design the LDWB sufficiently general to establish asymptotic equivalence between it and and a nonparametric local block bootstrap, also introduced here, up to second-order distribution theory. Finally, we introduce LDWB-aided Kolmogorov-Smirnov tests for local Gaussianity as well as local von-Mises statistics, with and without bootstrap inference, and establish their asymptotic validity using the second-order distribution theory. The finite sample performance of CLT and LDWB-aided local Gaussianity tests are assessed in a simulation study as well as two empirical applications. Whereas the CLT test is oversized, even in large samples, the size of the LDWB tests is accurate, even in small samples. The empirical analysis verifies this pattern, in addition to providing new insights about the distributional properties of equity indices, commodities, exchange rates and popular macro finance variables.

Nonparametric model checks of single-index assumptions

Semiparametric single-index assumptions are convenient and widely used dimen\-sion reduction approaches that represent a compromise between the parametric and fully nonparametric models for regressions or conditional laws. In a mean regression setup, the SIM assumption means that the conditional expectation of the response given the vector of covariates is the same as the conditional expectation of the response given a scalar projection of the covariate vector. In a conditional distribution modeling, under the SIM assumption the conditional law of a response given the covariate vector coincides with the conditional law given a linear combination of the covariates. Several estimation techniques for single-index models are available and commonly used in applications. However, the problem of testing the goodness-of-fit seems less explored and the existing proposals still have some major drawbacks. In this paper, a novel kernel-based approach for testing SIM assumptions is introduced. The covariate vector needs not have a density and only the index estimated under the SIM assumption is used in kernel smoothing. Hence the effect of high-dimensional covariates is mitigated while asymptotic normality of the test statistic is obtained. Irrespective of the fixed dimension of the covariate vector, the new test detects local alternatives approaching the null hypothesis slower than $n^{-1/2}h^{-1/4},$ where $h$ is the bandwidth used to build the test statistic and $n$ is the sample size. A wild bootstrap procedure is proposed for finite sample corrections of the asymptotic critical values. The small sample performances of our test compared to existing procedures are illustrated through simulations.

A powerful wild bootstrap diagnosis of panel unit roots under linear trends and time-varying volatility

Noting that standard panel unit root tests (PURTs) are not reliable in the presence of time varying volatility, recent papers have suggested a few heteroskedasticity–robust PURTs. These tests do not remain pivotal, however, if they are applied on detrended data. Building on recent asymptotic results for bootstrap approximations (Smeekes and Urbain in A multivariate invariance principle for modified wild bootstrap methods with an application to unit root testing, 2014), we provide simulation evidence that recursive detrending followed by a wild bootstrap correction leads to very good size precision for a variety of PURTs designed for testing against panel stationarity formalized with a cross sectionally homogenous autoregressive parameter. With respect to power, the bootstrap variant of the original nonrobust test of Levin et al. (J Econom 108(1):1–24, 2002) turns out to be the most effective among alternative tests. At the implementation side, performing the wild bootstrap correction using the so-called Rademacher distribution is found to be the most effective for both size and power properties of panel unit root inference by means of wild bootstrap based critical values. The empirical illustration shows that GDP per capita is best characterized as a unit root process.