Wild Bootstrap Research Articles

Bootstrap Inference for Group Factor Models

Abstract Andreou et al. (2019) have proposed a test for common factors based on canonical correlations between factors estimated separately from each group. We propose a simple bootstrap test that avoids the need to estimate the bias and variance of the canonical correlations explicitly and provide high-level conditions for its validity. We verify these conditions for a wild bootstrap scheme similar to the one proposed in Gonçalves and Perron (2014). Simulation experiments show that this bootstrap approach leads to null rejection rates closer to the nominal level in all of our designs compared to the asymptotic framework.

A goodness-of-fit test for functional time series with applications to Ornstein-Uhlenbeck processes

High-frequency financial data can be collected as a sequence of time-ordered curves, such as intraday prices. The Functional Data Analysis (FDA) framework offers a powerful approach to uncover information embedded in the shape of the daily paths, often unavailable from classical statistical methods. A novel goodness-of-fit test for autoregressive Hilbertian (ARH) models is introduced, imposing only the Hilbert-Schmidt condition on the autocorrelation operator. The test statistic is formulated in terms of a Cramér–von Mises norm, with calibration achieved via a wild bootstrap resampling procedure. A simulation study examines the test's finite-sample performance in terms of power and size. Furthermore, a new specification test for diffusion models, including Ornstein-Uhlenbeck processes, is proposed, illustrated with an application to intraday currency exchange rates. Specifically, a two-stage methodology is proffered: firstly, the relationship between functional samples and their lagged values is assessed using an ARH(1) model; second, under linearity, a functional F-test is conducted..

Robust Permutation Tests in Linear Instrumental Variables Regression

This paper develops permutation versions of identification-robust tests in linear instrumental variables regression. Unlike the existing randomization and rank-based tests in which independence between the instruments and the error terms is assumed, the permutation Anderson-Rubin (AR), Lagrange Multiplier (LM) and Conditional Likelihood Ratio (CLR) tests are asymptotically similar and robust to conditional heteroskedasticity under standard exclusion restriction i.e. the orthogonality between the instruments and the error terms. Moreover, when the instruments are independent of the structural error term, the permutation AR tests are exact, hence robust to heavy tails. As such, these tests share the strengths of the rank-based tests and the wild bootstrap AR tests. Numerical illustrations corroborate the theoretical results.

World energy futures market efficiency and its determinants; evidence from white noise test based on block-wise wild bootstrap approach

ABSTRACT The study examines the time-varying efficiency of nine energy futures traded across different exchanges with varied trade settlement methods and pricing currency using the novel white noise test based on block-wsie wild bootstrap approach. The advantage of using this approach is that it relaxes the assumption of serial independence and the martingale difference sequence. Time-varying efficiency is observed for all nine energy futures consistent with the Adoptive Market Hypothesis (AMH). The three most (least) efficient energy futures based on efficiency ratio are Coal, Gasoline and NaturalGas (WTI, Brent and OmanCrude). Quantile regression analysis shows that the higher the level of illiquidity furthers the level of efficiency towards inefficiency (high level of efficiency) in the case of a less (more) efficient market. A mixed impact of volatility is observed across energy futures. No differential impact is found due to exchange, method of settlement and pricing currency. The robustness of the results is investigated, policy implications are discussed, and further research is recommended.

Bootstrap inference for linear time-varying coefficient models in locally stationary time series

Time-varying coefficient models can capture evolving relationships. However, constructing asymptotic confidence bands for coefficient curves in these models is challenging due to slow convergence rates and the presence of various nuisance parameters. A residual-based sieve bootstrap method has recently been proposed to address these issues. While it successfully produces confidence bands with accurate empirical coverage, its applicability is restricted to strictly stationary processes. We introduce a new bootstrap scheme, the local blockwise wild bootstrap (LBWB), that allows for locally stationary processes. The LBWB can replicate the distribution of the parameter estimates while automatically accounting for nuisance parameters. An extensive simulation study reveals the superior performance of the LBWB compared to various benchmark approaches. It also shows the potential applicability of the LBWB in broader scenarios, including time-varying cointegrating models. We then examine herding effects in the Chinese renewable energy market using the proposed methods. Our findings strongly support the presence of herding behaviors before 2016, aligning with earlier studies. However, contrary to previous research, we find no significant evidence of herding between around 2018 and 2021. Online supplementary materials are available for this article.

Shapley Curves: A Smoothing Perspective

This article fills the limited statistical understanding of Shapley values as a variable importance measure from a nonparametric (or smoothing) perspective. We introduce population-level Shapley curves to measure the true variable importance, determined by the conditional expectation function and the distribution of covariates. Having defined the estimand, we derive minimax convergence rates and asymptotic normality under general conditions for the two leading estimation strategies. For finite sample inference, we propose a novel version of the wild bootstrap procedure tailored for capturing lower-order terms in the estimation of Shapley curves. Numerical studies confirm our theoretical findings, and an empirical application analyzes the determining factors of vehicle prices.

Test for the mean of high-dimensional functional time series

The one-sample test and two-sample test for the mean of high-dimensional functional time series are considered in this study. The proposed tests are built on the dimension-wise max-norm of the sum of squares of diverging projections. The null distribution of the test statistics is investigated using normal approximation, and the asymptotic behavior under the alternative is studied. The approach is robust to the cross-series dependence of unknown forms and magnitude. To approximate the critical values, a blockwise wild bootstrap method for functional time series is employed. Both fully and partially observed data are analyzed in theoretical research and numerical studies. Evidence from simulation studies and an IT stock data case study demonstrates the usefulness of the test in practice. The proposed methods have been implemented in a R package.

Estimation of Optimal Hedge Ratio: A Wild Bootstrap Approach

This paper proposes a new approach to estimating the minimum variance hedge ratio (MVHR) based on the wild bootstrap and evaluates the approach using a spectrum of conservative to aggressive alternative hedging strategies associated with the percentiles of the MVHR’s bootstrap distribution. This approach is suggested to be more informative and effective relative to the conventional method of hedging solely based on a single-point estimate. Furthermore, the percentile-based MVHRs are robust to influential outliers, non-normality, and unknown forms of heteroskedasticity. The bootstrap percentile-based hedging strategies’ effectiveness is compared with those from the naïve method and the asymmetric DCC-GARCH model for a range of financial assets and commodities. The bootstrap percentile-based hedging technique is identified to outperform its alternatives in terms of hedging effectiveness, downside risk, and return variability, suggesting its superiority to other methods in both the literature and in practice.

High-dimensional multivariate analysis of variance via geometric median and bootstrapping.

The geometric median, which is applicable to high-dimensional data, can be viewed as a generalization of the univariate median used in 1-dimensional data. It can be used as a robust estimator for identifying the location of multi-dimensional data and has a wide range of applications in real-world scenarios. This paper explores the problem of high-dimensional multivariate analysis of variance (MANOVA) using the geometric median. A maximum-type statistic that relies on the differences between the geometric medians among various groups is introduced. The distribution of the new test statistic is derived under the null hypothesis using Gaussian approximations, and its consistency under the alternative hypothesis is established. To approximate the distribution of the new statistic in high dimensions, a wild bootstrap algorithm is proposed and theoretically justified. Through simulation studies conducted across a variety of dimensions, sample sizes, and data-generating models, we demonstrate the finite-sample performance of our geometric median-based MANOVA method. Additionally, we implement the proposed approach to analyze a breast cancer gene expression dataset.

The Inefficiency of Bitcoin and the COVID-19 Pandemic

From the perspectives of asset pricing, market outreach, regulatory framework, and investor ethos, crypto markets differ substantially from traditional financial markets. Given these fundamental differences, it is interesting to examine how the notions of market efficiency apply to crypto markets, especially because the arguments of the efficient market hypothesis (Fama, 1970) and the adaptive market hypothesis (Lo, 2004, and Lo, 2008) were originally developed in the context of traditional financial markets. Research on the informational efficiency of crypto markets has attracted increasing attention in recent years. This paper examines the evolving efficiency of Bitcoin, the leading cryptocurrency, especially during the period when economies around the world were devastated by the Covid-19 pandemic. For a newly-emerged cryptocurrency with a market that is essentially global, a global shock like the Covid-19 pandemic presents ideal conditions for assessing how efficiency evolves as the market faces a series of shocks. Employing a fixed-length rolling window approach, this paper carries out the following tests: the automatic portmanteau test of Escanciano and Lobato (2009), the wild bootstrap automatic variance ratio test proposed by Kim (2009), the generalized spectral test of Escaciano and Valesco (2006), and the test proposed by Dominguez and Lobato (2003). The results provide evidence of episodes of inefficiency in a market that is efficient over extended periods. The inefficiency index constructed in this study shows that the Bitcoin market went through proportionately longer and more frequent episodes of inefficiency during the Covid-19 period. This is in line with the adaptive market hypothesis and has practical significance for investors and regulators.

Robust change-point detection for functional time series based on U-statistics and dependent wild bootstrap

The aim of this paper is to develop a change-point test for functional time series that uses the full functional information and is less sensitive to outliers compared to the classical CUSUM test. For this aim, the Wilcoxon two-sample test is generalized to functional data. To obtain the asymptotic distribution of the test statistic, we prove a limit theorem for a process of U-statistics with values in a Hilbert space under weak dependence. Critical values can be obtained by a newly developed version of the dependent wild bootstrap for non-degenerate 2-sample U-statistics.

Reprint of: Out-of-sample tests for conditional quantile coverage: An application to Growth-at-Risk

This paper proposes tests for out-of-sample comparisons of interval forecasts based on parametric conditional quantile models. The tests rank the distance between actual and nominal conditional coverage with respect to the set of conditioning variables from all models, for a given loss function. We propose a pairwise test to compare two models for a single predictive interval. The set-up is then extended to a comparison across multiple models and/or intervals. The limiting distribution varies depending on whether models are strictly non-nested or overlapping. In the latter case, degeneracy may occur. We establish the asymptotic validity of wild bootstrap based critical values across all cases. An empirical application to Growth-at-Risk (GaR) uncovers situations in which a richer set of financial indicators are found to outperform a commonly-used benchmark model when predicting downside risk to economic activity.

Distributional imputation for the analysis of censored recurrent events.

Longitudinal clinical trials for which recurrent events endpoints are of interest are commonly subject to missing event data. Primary analyses in such trials are often performed assuming events are missing at random, and sensitivity analyses are necessary to assess robustness of primary analysis conclusions to missing data assumptions. Control-based imputation is an attractive approach in superiority trials for imposing conservative assumptions on how data may be missing not at random. A popular approach to implementing control-based assumptions for recurrent events is multiple imputation (MI), but Rubin's variance estimator is often biased for the true sampling variability of the point estimator in the control-based setting. We propose distributional imputation (DI) with corresponding wild bootstrap variance estimation procedure for control-based sensitivity analyses of recurrent events. We apply control-based DI to a type I diabetes trial. In the application and simulation studies, DI produced more reasonable standard error estimates than MI with Rubin's combining rules in control-based sensitivity analyses of recurrent events.

ENCOMPASSING TESTS FOR NONPARAMETRIC REGRESSIONS

We set up a formal framework to characterize encompassing of nonparametric models through the$L^2$distance. We contrast it to previous literature on the comparison of nonparametric regression models. We then develop testing procedures for the encompassing hypothesis that are fully nonparametric. Our test statistics depend on kernel regression, raising the issue of bandwidth’s choice. We investigate two alternative approaches to obtain a “small bias property” for our test statistics. We show the validity of a wild bootstrap method. We empirically study the use of a data-driven bandwidth and illustrate the attractive features of our tests for small and moderate samples.

Resampling-based confidence intervals and bands for the average treatment effect in observational studies with competing risks

The g-formula can be used to estimate the treatment effect while accounting for confounding bias in observational studies. With regard to time-to-event endpoints, possibly subject to competing risks, the construction of valid pointwise confidence intervals and time-simultaneous confidence bands for the causal risk difference is complicated, however. A convenient solution is to approximate the asymptotic distribution of the corresponding stochastic process by means of resampling approaches. In this paper, we consider three different resampling methods, namely the classical nonparametric bootstrap, the influence function equipped with a resampling approach as well as a martingale-based bootstrap version, the so-called wild bootstrap. For the latter, three sub-versions based on differing distributions of the underlying random multipliers are examined. We set up a simulation study to compare the accuracy of the different techniques, which reveals that the wild bootstrap should in general be preferred if the sample size is moderate and sufficient data on the event of interest have been accrued. For illustration, the resampling methods are further applied to data on the long-term survival in patients with early-stage Hodgkin’s disease.

Testing Many Zero Restrictions in a High Dimensional Linear Regression Setting

We propose a test of many zero parameter restrictions in a high dimensional linear iid regression model with k ≫ n regressors. The test statistic is formed by estimating key parameters one at a time based on many low dimension regression models with nuisance terms. The parsimoniously parametrized models identify whether the original parameter of interest is or is not zero. Estimating fixed low dimension sub-parameters ensures greater estimator accuracy, it does not require a sparsity assumption nor therefore a regularized estimator, it is computationally fast compared to, e.g., de-biased Lasso, and using only the largest in a sequence of weighted estimators reduces test statistic complexity and therefore estimation error. We provide a parametric wild bootstrap for p-value computation, and prove the test is consistent and has non-trivial n / { ln ( n ) M n } -local-to-null power where M n is the l ∞ covariate fourth moment.

Linear mixed models and latent growth curve models for group comparison studies contaminated by outliers.

The linear mixed model (LMM) and latent growth model (LGM) are frequently applied to within-subject two-group comparison studies to investigate group differences in the time effect, supposedly due to differential group treatments. Yet, research about LMM and LGM in the presence of outliers (defined as observations with a very low probability of occurrence if assumed from a given distribution) is scarce. Moreover, when such research exists, it focuses on estimation properties (bias and efficiency), neglecting inferential characteristics (e.g., power and type-I error). We study power and type-I error rates of Wald-type and bootstrap confidence intervals (CIs), as well as coverage and length of CIs and mean absolute error (MAE) of estimates, associated with classical and robust estimations of LMM and LGM, applied to a within-subject two-group comparison design. We conduct a Monte Carlo simulation experiment to compare CIs and MAEs under different conditions: data (a) without contamination, (b) contaminated by within-subject outliers, (c) contaminated by between-subject outliers, and (d) both contaminated by within- and between-subject outliers. Results show that without contamination, methods perform similarly, except CIs based on S, a robust LMM estimator, which are slightly less close to nominal values in their coverage. However, in the presence of both within- and between-subject outliers, CIs based on robust estimators, especially S, performed better than those of classical methods. In particular, the percentile CI with the wild bootstrap applied to the robust LMM estimators outperformed all other methods, especially with between-subject outliers, when we found the classical Wald-type CI based on the t statistic with Satterthwaite approximation for LMM to be highly misleading. We provide R code to compute all methods presented here. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Generalized Spectral Tests for Multivariate Martingale Difference Hypotheses

This study proposes new generalized spectral tests for multivariate martingale difference hypotheses, specifically geared toward high-dimensionality scenarios where the dimension of the time series is comparable or even larger than the sample size in practice. We develop an asymptotic theory and a valid wild bootstrapping procedure for the new test statistics, in which the dimension of the time series is fixed. We demonstrate that a bias-reduced version of the test statistics effectively addresses the high-dimensionality concerns. Comprehensive Monte Carlo simulations reveal that the bias-reduced statistic performs substantially better than its competitors. The application to testing the efficient market hypothesis on the U.S. stock market illustrates the usefulness of our proposal.

Dynamic efficiency in MENA stock markets during COVID-19 outbreak and vaccines

PurposeThis paper examines the time-varying efficiency of nine major Middle East and North Africa (MENA) stock markets namely Egypt, Bahrain, UAE, Jordan, Saudi Arabia, Oman, Qatar, Morocco and Tunisia during times of COVID-19 pandemic outbreak and vaccines.Design/methodology/approachThe authors use two econometric approaches: (1) autocorrelation tests including the wild bootstrap automatic variance ratio test, the automatic portmanteau test and the Generalized spectral test, and (2) a non-Bayesian generalized least squares-based time-varying model with statistical inferences.FindingsThe results show that the degree of stock market efficiency of Egyptian, Bahraini, Saudi, Moroccan and Tunisian stock markets is influenced by the COVID-19 pandemic crisis. Furthermore, the authors find a tendency toward efficiency in most of the MENA markets after the announcement of the COVID-19's vaccine approval. Finally, the Jordanian, Omani, Qatari and UAE stock markets remain globally efficient during the three sub-periods of the COVID-19 pandemic outbreak.Originality/valueThe results have important implications for asset allocations and financial risk management. Portfolio managers may maximize the benefit of arbitrage opportunities by taking strategic long and short positions in these markets during downward trend periods. Policymakers should implement the action plans and reforms to protect the stock markets from global shocks and ensure the stability of the stock markets.

Long Short-Term Memory-Based Twin Support Vector Regression for Probabilistic Load Forecasting.

A probabilistic load forecast that is accurate and reliable is crucial to not only the efficient operation of power systems but also to the efficient use of energy resources. In order to estimate the uncertainties in forecasting models and nonstationary electric load data, this study proposes a probabilistic load forecasting model, namely BFEEMD-LSTM-TWSVRSOA. This model consists of a data filtering method named fast ensemble empirical model decomposition (FEEMD) method, a twin support vector regression (TWSVR) whose features are extracted by deep learning-based long short-term memory (LSTM) networks, and parameters optimized by seeker optimization algorithms (SOAs). We compared the probabilistic forecasting performance of the BFEEMD-LSTM-TWSVRSOA and its point forecasting version with different machine learning and deep learning algorithms on Global Energy Forecasting Competition 2014 (GEFCom2014). The most representative month data of each season, totally four monthly data, collected from the one-year data in GEFCom2014, forming four datasets. Several bootstrap methods are compared in order to determine the best prediction intervals (PIs) for the proposed model. Various forecasting step sizes are also taken into consideration in order to obtain the best satisfactory point forecasting results. Experimental results on these four datasets indicate that the wild bootstrap method and 24-h step size are the best bootstrap method and forecasting step size for the proposed model. The proposed model achieves averaged 46%, 11%, 36%, and 44% better than suboptimal model on these four datasets with respect to point forecasting, and achieves averaged 53%, 48%, 46%, and 51% better than suboptimal model on these four datasets with respect to probabilistic forecasting.