Wild Bootstrap Research Articles

General tests of the Markov property in multi-state models.

Multi-state models for event history analysis most commonly assume the process is Markov. This article considers tests of the Markov assumption that are applicable to general multi-state models. Two approaches using existing methodology are considered; a simple method based on including time of entry into each state as a covariate in Cox models for the transition intensities and a method involving detecting a shared frailty through a stratified Commenges-Andersen test. In addition, using the principle that under a Markov process the future rate of transitions of the process at times $t > s$ should not be influenced by the state occupied at time $s$, a new class of general tests is developed by considering summaries from families of log-rank statistics where patients are grouped by the state occupied at varying initial time $s$. An extended form of the test applicable to models that are Markov conditional on observed covariates is also derived. The null distribution of the proposed test statistics are approximated by using wild bootstrap sampling. The approaches are compared in simulation and applied to a dataset on sleeping behavior. The most powerful test depends on the particular departure from a Markov process, although the Cox-based method maintained good power in a wide range of scenarios. The proposed class of log-rank statistic based tests are most useful in situations where the non-Markov behavior does not persist, or is not uniform in nature across patient time.

A goodness‐of‐fit test for the functional linear model with functional response

AbstractThe functional linear model with functional response (FLMFR) is one of the most fundamental models to assess the relation between two functional random variables. In this article, we propose a novel goodness‐of‐fit test for the FLMFR against a general, unspecified, alternative. The test statistic is formulated in terms of a Cramér–von Mises norm over a doubly projected empirical process which, using geometrical arguments, yields an easy‐to‐compute weighted quadratic norm. A resampling procedure calibrates the test through a wild bootstrap on the residuals and the use of convenient computational procedures. As a sideways contribution, and since the statistic requires a reliable estimator of the FLMFR, we discuss and compare several regularized estimators, providing a new one specifically convenient for our test. The finite sample behavior of the test is illustrated via a simulation study. Also, the new proposal is compared with previous significance tests. Two novel real data sets illustrate the application of the new test.

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 union of information sets across 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.

A statistical analysis of time trends in atmospheric ethane

Ethane is the most abundant non-methane hydrocarbon in the Earth’s atmosphere and an important precursor of tropospheric ozone through various chemical pathways. Ethane is also an indirect greenhouse gas (global warming potential), influencing the atmospheric lifetime of methane through the consumption of the hydroxyl radical (OH). Understanding the development of trends and identifying trend reversals in atmospheric ethane is therefore crucial. Our dataset consists of four series of daily ethane columns. As with many other decadal time series, our data are characterized by autocorrelation, heteroskedasticity, and seasonal effects. Additionally, missing observations due to instrument failure or unfavorable measurement conditions are common in such series. The goal of this paper is therefore to analyze trends in atmospheric ethane with statistical tools that correctly address these data features. We present selected methods designed for the analysis of time trends and trend reversals. We consider bootstrap inference on broken linear trends and smoothly varying nonlinear trends. In particular, for the broken trend model, we propose a bootstrap method for inference on the break location and the corresponding changes in slope. For the smooth trend model, we construct simultaneous confidence bands around the nonparametrically estimated trend. Our autoregressive wild bootstrap approach, combined with a seasonal filter, is able to handle all issues mentioned above (we provide R code for all proposed methods on https://www.stephansmeekes.nl/code.).

Performance and Market Maturity in Mutual Funds: Is Real Estate Different?

Despite the lack of convincing evidence that active investment fund managers add value, the number of actively-managed US mutual funds has increased substantially over the last 25 years. While non-sector diversified mutual funds have received much attention, sector funds, except real estate mutual funds (REMFs), have not. In this paper, we provide new and more robust evidence on the performance of active REMFs compared to all actively managed mutual funds. We use the Carhart four-factor model with an additional liquidity factor as a risk-adjusted benchmark. We use wild bootstrap methods to deal with small samples, non-normality and heteroscedasticity, and we control for the false discovery of significant results. For portfolios of fund types, we find evidence of both significant outperformance and underperformance, net of fees, during 1992-2016. We consider non-overlapping five-year and three-year periods and find very limited evidence of persistent outperformance. For individual funds, we find that, for both sector and diversified funds, net of fees, only 0.79% are skilled. We find persistence in skills for only two individual fund managers of diversified funds. We investigate the effects of the outsourcing of management and of team versus individual management. Outsourcing has no effect on performance of non-RE sector funds but, for cap-based funds and style-based funds, it has a negative effect. There is some evidence that this may also be true for REMFs. Team management has no effect for any types of funds. Overall, we conclude that REMFs are generally no different from other sector funds.

Recursive nonparametric regression estimation for dependent strong mixing functional data

In the present paper, we extend the work of Slaoui (Stat Sin 30:417–437, 2020) in the case of strong mixing data. Since, we are interested in nonparametric regression estimation, we focus on well adapted dependence structures based on mixing type conditions. We study the properties of these regression estimators and compare them with the nonparametric non-recursive regression estimator. The bias, variance and mean squared error are computed explicitly. We showed that using a selected wild bootstrap bandwidth procedure and a special stepsize, our proposed recursive regression estimators allowed us to obtain quite similar results compared to the non-recursive regression estimator under $$\alpha $$ -mixing condition in terms of estimation error and much better in terms of computational costs.

The Environmental Kuznets Curve across Australian states and territories

Abstract We test the Environmental Kuznets Curves (EKC) hypothesis for a panel of eight Australian states and territories using non-parametric panel estimation over the period 1990 to 2017. A feature of our non-parametric estimation method is that it allows carbon dioxide (CO2) emissions to evolve over time in the form of an unknown functional form with confidence bands constructed using a wild bootstrapping method. This framework has the advantage that we can plot the time-varying relationship between Gross State Product (GSP) and CO2 emissions. The non-parametric local linear estimates for GSP are consistent with a conventional inverted U-shaped EKC, which peaks in 2010 and declines thereafter. The peak coincides with a significant policy shift in Australia's commitment to reducing CO2 emissions, whereby, in 2010, the Australian Government, and subsequently the states and territories, pledged to reduce CO2 emissions to 5% below 2000 levels by 2020 under the United Nations Framework Convention on Climate Change (UNFCCC) (the Cancun Agreement).

Bootstrap tests for structural breaks when the regressors and the serially correlated error term are unstable

AbstractStructural break problems in linear regression coefficients are often accompanied by the instabilities of the regressors and/or the serially correlated error term such as structural changes in their marginal distributions. In these circumstances, we find that existing tests as well as Hansen (2000)'s fixed regressor bootstrap have serious size distortions, although the latter is designed to overcome the unstable regressors. To tackle this problem, we propose a method that combines the fixed regressor bootstrap and the sieve wild bootstrap. We show that the sieve wild bootstrap asymptotically replicates a broad set of serially correlated unstable error processes. Using that the fixed regressor bootstrap is designed to capture the instability of the regressors, the mixture of the two is then shown to provide correct asymptotic critical values of the existing tests in various feasible nonstandard cases. Monte Carlo experiments show significant improvements in size and power.

Adaptive Inference in Heteroscedastic Fractional Time Series Models

We consider estimation and inference in fractionally integrated time series models driven by shocks which can display conditional and unconditional heteroscedasticity of unknown form. Although the standard conditional sum-of-squares (CSS) estimator remains consistent and asymptotically normal in such cases, unconditional heteroscedasticity inflates its variance matrix by a scalar quantity, , thereby inducing a loss in efficiency relative to the unconditionally homoscedastic case, λ = 1. We propose an adaptive version of the CSS estimator, based on nonparametric kernel-based estimation of the unconditional volatility process. We show that adaptive estimation eliminates the factor λ from the variance matrix, thereby delivering the same asymptotic efficiency as that attained by the standard CSS estimator in the unconditionally homoscedastic case and, hence, asymptotic efficiency under Gaussianity. Importantly, the asymptotic analysis is based on a novel proof strategy, which does not require consistent estimation (in the sup norm) of the volatility process. Consequently, we are able to work under a weaker set of assumptions than those employed in the extant literature. The asymptotic variance matrices of both the standard and adaptive CSS (ACSS) estimators depend on any weak parametric autocorrelation present in the fractional model and any conditional heteroscedasticity in the shocks. Consequently, asymptotically pivotal inference can be achieved through the development of confidence regions or hypothesis tests using either heteroscedasticity-robust standard errors and/or a wild bootstrap. Monte Carlo simulations and empirical applications illustrate the practical usefulness of the methods proposed.

Hisse Senedi Piyasalarının Zayıf Form Piyasa Etkinliğinin Küresel Ölçekte Karşılaştırılması: G-20 Üyeleri Üzerine Ampirik Bir Çalışma

This paper analyzes stock markets of G-20 members in terms of comparing return predictability, in other words, weak-form market efficiency using stock indexes weekly data of the G-20 members between 07.06.2009 and 09.02.2020. As a result of the wild bootstrap automatic variance ratio test analysis developed by Kim (2009), it is found that the stock markets of Brazil, South Africa, and Germany were weak-form efficient in the date range within the scope of the study, so the returns are unpredictable, while the weak-form efficiency (return predictability) of other markets are time-varying. In addition to Brazil, South Africa, and Germany, there is a very low chance to estimate returns with historical price movements or returns in Russia, France, Italy, United States, United Kingdom, and Canada stock markets. It is also found that the return predictability periods of Japan, Australia, China, Saudi Arabia, and especially Mexico’s stock markets are higher than other markets and the chance of success in estimating returns by using historical price information in these markets is quite high.

Standard Errors for Nonparametric Regression

This paper proposes five pointwise consistent and asymptotic normal estimators of the asymptotic variance function of the Nadaraya-Watson kernel estimator for nonparametric regression. The proposed estimators are constructed based on the first-stage nonparametric residuals, and their asymptotic properties are established under the assumption that the same bandwidth sequences are used throughout, which mimics what researchers do in practice while making derivations more complicated instead. A limited Monte Carlo experiment demonstrates that the proposed estimators possess smaller pointwise variability in small samples than the pair and wild bootstrap estimators which are commonly used in practice.

WEAK-IDENTIFICATION ROBUST WILD BOOTSTRAP APPLIED TO A CONSISTENT MODEL SPECIFICATION TEST

We present a new robust bootstrap method for a test when there is a nuisance parameter under the alternative, and some parameters are possibly weakly or nonidentified. We focus on a Bierens (1990, Econometrica 58, 1443–1458)-type conditional moment test of omitted nonlinearity for convenience. Existing methods include the supremum p-value which promotes a conservative test that is generally not consistent, and test statistic transforms like the supremum and average for which bootstrap methods are not valid under weak identification. We propose a new wild bootstrap method for p-value computation by targeting specific identification cases. We then combine bootstrapped p-values across polar identification cases to form an asymptotically valid p-value approximation that is robust to any identification case. Our wild bootstrap procedure does not require knowledge of the covariance structure of the bootstrapped processes, whereas Andrews and Cheng’s (2012a, Econometrica 80, 2153–2211; 2013, Journal of Econometrics 173, 36–56; 2014, Econometric Theory 30, 287–333) simulation approach generally does. Our method allows for robust bootstrap critical value computation as well. Our bootstrap method (like conventional ones) does not lead to a consistent p-value approximation for test statistic functions like the supremum and average. Therefore, we smooth over the robust bootstrapped p-value as the basis for several tests which achieve the correct asymptotic level, and are consistent, for any degree of identification. They also achieve uniform size control. A simulation study reveals possibly large empirical size distortions in nonrobust tests when weak or nonidentification arises. One of our smoothed p-value tests, however, dominates all other tests by delivering accurate empirical size and comparatively high power.

벡터자기회귀모형의 잔차 검정을 위한 와일드 붓스트랩 Ljung-Box 검정

본 논문에서는 벡터자기회귀모형 적합 후 잔차 검정을 위한 Ljung-Box(LB) 검정통계량에 와일드 붓스트랩을 적용하는 연구를 진행하였다. 일반적인 시계열 자료와는 달리 조건부 이분산성이 존재하는 다변량 금융시계열 자료에서는 LB 검정통계량의 제 1종의 오류가 주어진 유의수준보다 매우 크게 나타난다. 이를 극복하기 위해 벡터자기회귀모형에 와일드 붓스트랩을 이용한 LB 검정법을 제안하였다. 모의실험을 실행한 결과 와일드 붓스트랩 LB 검정법의 제 1종의 오류가 주어진 유의수준을 잘 만족하고 있음을 확인하였다.

A MAX-CORRELATION WHITE NOISE TEST FOR WEAKLY DEPENDENT TIME SERIES

This article presents a bootstrapped p-value white noise test based on the maximum correlation, for a time series that may be weakly dependent under the null hypothesis. The time series may be prefiltered residuals. The test statistic is a normalized weighted maximum sample correlation coefficient $ \max _{1\leq h\leq \mathcal {L}_{n}}\sqrt {n}|\hat {\omega }_{n}(h)\hat {\rho }_{n}(h)|$, where $\hat {\omega }_{n}(h)$ are weights and the maximum lag $ \mathcal {L}_{n}$ increases at a rate slower than the sample size n. We only require uncorrelatedness under the null hypothesis, along with a moment contraction dependence property that includes mixing and nonmixing sequences. We show Shao’s (2011, Annals of Statistics 35, 1773–1801) dependent wild bootstrap is valid for a much larger class of processes than originally considered. It is also valid for residuals from a general class of parametric models as long as the bootstrap is applied to a first-order expansion of the sample correlation. We prove the bootstrap is asymptotically valid without exploiting extreme value theory (standard in the literature) or recent Gaussian approximation theory. Finally, we extend Escanciano and Lobato’s (2009, Journal of Econometrics 151, 140–149) automatic maximum lag selection to our setting with an unbounded lag set that ensures a consistent white noise test, and find it works extremely well in controlled experiments.

Bootstrap procedures for detecting multiple persistence shifts in heteroskedastic time series

This article proposes new bootstrap procedures for detecting multiple persistence shifts in a time series driven by non‐stationary volatility. The assumed volatility process can accommodate discrete breaks, smooth transition variation as well as trending volatility. We develop wild bootstrap sup‐Wald tests of the null hypothesis that the process is either stationary [I(0)] or has a unit root [I(1)] throughout the sample. We also propose a sequential procedure to estimate the number of persistence breaks based on ordering the regime‐specific bootstrap p‐values. The asymptotic validity of the advocated procedures is established both under the null of stability and a variety of persistence change alternatives. A comparison with existing tests that assume homoskedasticity illustrates the finite sample improvements offered by our methods. An application to OECD inflation rates highlights the empirical relevance of the proposed approach and weakens the case for persistence change relative to existing procedures.

Dynamic inference in general nested case-control designs.

Nested case-control designs are attractive in studies with a time-to-event endpoint if the outcome is rare or if interest lies in evaluating expensive covariates. The appeal is that these designs restrict to small subsets of all patients at risk just prior to the observed event times. Only these small subsets need to be evaluated. Typically, the controls are selected at random and methods for time-simultaneous inference have been proposed in the literature. However, the martingale structure behind nested case-control designs allows for more powerful and flexible non-standard sampling designs. We exploit that structure to find simultaneous confidence bands based on wild bootstrap resampling procedures within this general class of designs. We show in a simulation study that the intended coverage probability is obtained for confidence bands for cumulative baseline hazard functions. We apply our methods to observational data about hospital-acquiredinfections.

The Art Market: What Do We Know About Returns?

We examine the annual returns based on auction data for two groups of artists (Surrealists and Impressionists) and two individual artists (Picasso and Renoir) using hedonic pricing models in combination with a wild bootstrap statistical technique. With this approach we estimate confidence intervals for such returns; we also estimate confidence intervals for the correlations with returns of other type of assets, and risk-return metrics. We find that the confidence intervals associated with these figures of merit are so wide that it is difficult, if not impossible, to derive absolute conclusions or to make meaningful comparisons, with the behavior of other assets. We also observe that relying on single-point estimates of the above-mentioned metrics – without accounting for the corresponding confidence intervals – can lead to erroneous interpretations regarding art-market returns. Moreover, our results suggest that previous studies regarding art market returns, their correlation with broader market indices, and their risk-return profiles, should be re-examined as they were based on single-point estimates of the relevant metrics. These findings might be of interest to researchers who use hedonic pricing models in the analysis of other infrequently traded assets as the number of sales/observations is likely to be rather low.

Testing for episodic predictability in stock returns

Standard tests based on predictive regressions estimated over the full available sample data have tended to find little evidence of predictability in stock returns. Recent approaches based on the analysis of subsamples of the data suggest in fact that predictability where it occurs might exist only within so-called “pockets of predictability” rather than across the entire sample. However, these methods are prone to the criticism that the subsample dates are endogenously determined such that the use of standard critical values appropriate for full sample tests will result in incorrectly sized tests leading to spurious findings of stock returns predictability. To avoid the problem of endogenously-determined sample splits, we propose new tests derived from sequences of predictability statistics systematically calculated over subsamples of the data. Specifically, we will base tests on the maximum of such statistics from sequences of forward and backward recursive, rolling, and double-recursive predictive subsample regressions. We develop our approach using the over-identified instrumental variable-based predictability test statistics of Breitung and Demetrescu (2015). This approach is based on partial-sum asymptotics and so, unlike many other popular approaches including, for example, those based on Bonferroni corrections, can be readily adapted to implementation over sequences of subsamples. We show that the limiting null distributions of our proposed test statistics depend in general on whether the putative predictor is strongly or weakly persistent and on any heteroskedasticity present (indeed on any time-variation present in the unconditional variance matrix of the innovations), the latter even if the subsample statistics are based on heteroskedasticity-robust standard errors. As a consequence, we develop fixed regressor wild bootstrap implementations of the tests which we demonstrate to be first-order asymptotically valid. Finite sample behaviour against a variety of temporarily predictable processes is considered. An empirical application to US stock returns illustrates the usefulness of the new predictability testing methods we propose.

Bootstrapping Laplace Transforms of Volatility

This paper studies inference for the realized Laplace transform (RLT) of volatility in a fixed-span setting using bootstrap methods. Specifically, since standard wild bootstraps provide inconsistent inference, we propose local Gaussian (LG) and modified wild (MW) bootstrap procedures, and establish their first-order asymptotic validity. Moreover, motivated by its superior finite sample performance in simulations, we use Edgeworth expansions to show that the LG inference achieves second-order asymptotic refinements. To further broaden the scope of the bootstraps, we provide new Laplace transform-based estimators of the spot variance as well as the covariance, correlation and beta between two semimartingales, and adapt our inference procedures to the requisite scenario. We establish central limit theory for our estimators and show first-order asymptotic validity of their associated bootstraps. Not surprisingly, a Monte Carlo study shows that the LG bootstrap outperforms the MW bootstrap and existing (first-order) feasible inference theory in finite samples. Moreover, it demonstrates that our new spot measure estimators and inference procedures are very accurate. Finally, we illustrate the use of the new methods by examining the volatility of, and the coherence between, stocks and bonds during the large equity sell-off in December 2018.

A modified bootstrap for kernel-based specification test with heavy-tailed data

This paper provides a new resampling strategy to improve the finite sample performance of a nonparametric kernel-based specification test in the presence of heavy-tailed error terms. Based on the test statistic of Li and Wang (1998), we propose to generate the bootstrapped samples using a modified wild bootstrap. This new method matches all moments of the error terms if the error has a symmetric distribution and matches the first and all even moments when error distribution is asymmetric around zero. This new resampling method has better finite sample performance than the traditional one when the distribution of the error terms is symmetric and heavy-tailed.