Block Bootstrap Research Articles

Estimating the number and length of episodes in disability using a Markov chain approach

BackgroundMarkov models are a key tool for calculating expected time spent in a state, such as active life expectancy and disabled life expectancy. In reality, individuals often enter and exit states recurrently, but standard analytical approaches are not able to describe this dynamic. We develop an analytical matrix approach to calculating the expected number and length of episodes spent in a state.MethodsThe approach we propose is based on Markov chains with rewards. It allows us to identify the number of entries into a state and to calculate the average length of episodes as total time in a state divided by the number of entries. For sampling variance estimation, we employ the block bootstrap. Two case studies that are based on published literature illustrate how our methods can provide new insights into disability dynamics.ResultsThe first application uses a classic textbook example on prednisone treatment and liver functioning among liver cirrhosis patients. We replicate well-known results of no association between treatment and survival or recovery. Our analysis of the episodes of normal liver functioning delivers the new insight that the treatment reduced the likelihood of relapse and extended episodes of normal liver functioning. The second application assesses frailty and disability among elderly people. We replicate the prior finding that frail individuals have longer life expectancy in disability. As a novel finding, we document that frail individuals experience three times as many episodes of disability that were on average twice as long as the episodes of nonfrail individuals.ConclusionsWe provide a simple analytical approach for calculating the number and length of episodes in Markov chain models. The results allow a description of the transition dynamics that goes beyond the results that can be obtained using standard tools for Markov chains. Empirical applications using published data illustrate how the new method is helpful in unraveling the dynamics of the modeled process.

Incorporating a change-point estimator when bootstrapping the empirical distribution of a stationary process

The moving block bootstrap can be used to determine critical values for test statistics used to detect a change-point in the marginal distribution of a stationary time series. We examine the impact of incorporating an estimator of the change-point when centering the bootstrap blocks and establish conditions under which the bootstrapped test statistics remain stochastically bounded regardless of whether or not a change is present.

Dust storm frequency change in relation to climate drivers

AbstractClimate variability and change in arid regions are important factors in controlling emission, frequency and movement of dust storms. This study provides robust statistical methods including univariate Mann‐Kendall block bootstrapping method and three bivariate trend assessment methods, Covariance Inversion Test, Covariance Sum Test and The Covariance Eigenvalue to detect trends in dust storm frequency across arid regions of Iran in relation to climate variability and trend in recent decades. In this regard, the annual number of dust storms from 25 stations in central arid and semi‐arid regions of Iran were selected. In addition, five major climatic variables including annual rainfall, annual maximum and average wind speed, annual maximum and average temperature were also collected. The univariate trend test indicates both increasing and decreasing trend in dust storm frequency and climate variables. The bivariate trend test shows a strong and statistically significant relationship between trend of climate variables and dust storm frequency for most of the stations across the region. Among climate variables, rainfall change has an inverse impact on dust storm frequency while wind speed and temperature have direct covariance structure with dust storm frequency. The wind speed also seems to be the most effective climate driver on dust storm frequency in arid regions of Iran, followed by temperature. The results also shows that local conditions that are not considered in this study may also play a significant role in dust storm emission in some parts of the region.

Varying Coefficient Panel Data Model with Interactive Fixed Effects

In this paper, we propose a varying coefficient panel data model with unobservable multiple interactive fixed effects that are correlated with the regressors. We approximate each coefficient function by B-spline, and propose a robust nonlinear iteration scheme based on the least squares method to estimate the coefficient functions of interest. We also establish the asymptotic theory of the resulting estimators under certain regularity assumptions, including the consistency, the convergence rate and the asymptotic distribution. Furthermore, we develop a least squares dummy variable method to study an important special case of the proposed model: the varying coefficient panel data model with additive fixed effects. To construct the pointwise confidence intervals for the coefficient functions, a residual-based block bootstrap method is proposed to reduce the computational burden as well as to avoid the accumulative errors. Simulation studies and a real data analysis are also carried out to assess the performance of our proposed methods.

Use of panel time-series data with cross-section dependence in evaluating farmland valuation: a cautionary note

ABSTRACT The note highlights how misspecification of cross-section dependence structure in panel time-series data can lead to erroneous conclusions on farmland valuation. Combining the sample information from time-series and cross-section dimensions by using panel time-series data can improve inference on the net present value hypothesis for farmland. However, cross-section dependence must be addressed to take advantage of the additional information from this type of data. We consider three classes of panel unit root models that account for cross-section dependence through (1) common factor extraction, (2) block bootstrapping and (3) spatial dependence to explore whether farmland values can be explained by their economic fundamentals, given that the appropriate cross-section specification is implemented in testing. Results show that only spatial dependence approach accurately characterizes cross-section dependence in the Iowa panel time-series data, highlighting the importance of model selection when using data with cross-section dependence. Once the econometric model is specified with the underlying spatial cross-section dependence structure, the market valuation of Iowa farmland is mainly determined by fundamentals as predicted by the net present value model.

Reconciling solar forecasts: Probabilistic forecast reconciliation in a nonparametric framework

No forecast is complete without understanding it probabilistically. Previously in “Reconciling solar forecasts: Geographical hierarchy” [Sol. Energy 146 (2017) 276–286], four different point forecast reconciliation techniques were demonstrated using simulated data from 318 photovoltaic systems in California. In this paper, I show how to extend those techniques to probabilistic solar forecasting. More specifically, probabilistic forecast reconciliation is performed in a nonparametric framework through block bootstrapping. As compared to the parametric framework, which requires the base forecasts to be characterized by elliptical distributions, the nonparametric framework is not limited by such assumptions. Probabilistic forecast reconciliation not only provides a description of forecast uncertainty, it could also issue optimal point forecasts based on a directive in the form of a statistical functional. In this regard, there is very little reason to favor point forecast reconciliation, or any point forecasting for that matter, in solar energy meteorology. And probabilistic forecast reconciliation, or more generally, probabilistic solar forecasting, should be made default.

Testing equality of autocovariance operators for functional time series

We consider strictly stationary stochastic processes of Hilbert space‐valued random variables and focus on fully functional tests for the equality of the lag‐zero autocovariance operators of several independent functional time series. A moving block bootstrap (MBB)‐based testing procedure is proposed which generates pseudo random elements that satisfy the null hypothesis of interest. It is based on directly bootstrapping the time series of tensor products which overcomessome common difficulties associated with applications of the bootstrap to related testing problems. The suggested methodology can be potentially applied to a broad range of test statistics of the hypotheses of interest. As an example, we establish validity for approximating the distribution under the null of a test statistic based on the Hilbert–Schmidt distance of the corresponding sample lag‐zero autocovariance operators, and show consistency under the alternative. As a prerequisite, we prove a central limit theorem for the MBB procedure applied to the sample autocovariance operator which is of interest on its own. The finite sample size and power performance of the suggested MBB‐based testing procedure is illustrated through simulations and an application to a real‐life dataset is discussed.

Understanding Machine Learning for Diversified Portfolio Construction by Explainable AI

In this paper, we construct a pipeline to investigate heuristic diversification strategies in asset allocation. We use machine learning concepts (explainable AI) to compare the robustness of different strategies and back out implicit rules for decision making. In a first step, we augment the asset universe (the empirical dataset) with a range of scenarios generated with a block bootstrap from the empirical dataset. Second, we backtest the candidate strategies over a long period of time, checking their performance variability. Third, we use XGBoost as a regression model to connect the difference between the measured performances between two strategies to a pool of statistical features of the portfolio universe tailored to the investigated strategy. Finally, we employ the concept of Shapley values to extract the relationships that the model could identify between the portfolio characteristics and the statistical properties of the asset universe. We test this pipeline for studying risk-parity strategies with a volatility target, and in particular, comparing the machine learning-driven Hierarchical Risk Parity (HRP) to the classical Equal Risk Contribution (ERC) strategy. In the augmented dataset built from a multi-asset investment universe of commodities, equities and fixed income futures, we find that HRP better matches the volatility target, and shows better risk-adjusted performances. Finally, we train XGBoost to learn the difference between the realized Calmar ratios of HRP and ERC and extract explanations. The explanations provide fruitful ex-post indications of the connection between the statistical properties of the universe and the strategy performance in the training set. For example, the model confirms that features addressing the hierarchical properties of the universe are connected to the relative performance of HRP respect to ERC.

Bootstrap seasonal unit root test under periodic variation

Both seasonal unit roots and periodic variation can be prevalent in seasonal data. In the testing of seasonal unit roots under periodic variation, the validity of the existing methods, such as the HEGY test, remains unknown. The behavior of the augmented HEGY test and the unaugmented HEGY test under periodic variation is analyzed. It turns out that the asymptotic null distributions of the HEGY statistics testing the single roots at 1 or −1 when there is periodic variation are identical to the asymptotic null distributions when there is no periodic variation. On the other hand, the asymptotic null distributions of the statistics testing any coexistence of roots at 1, −1,i, or −i when there is periodic variation are non-standard and are different from the asymptotic null distributions when there is no periodic variation. Therefore, when periodic variation exists, HEGY tests are not directly applicable to the joint tests for any concurrence of seasonal unit roots. As a remedy, bootstrap is proposed; in particular, the augmented HEGY test with seasonal independent and identically distributed (iid) bootstrap and the unaugmented HEGY test with seasonal block bootstrap are implemented. The consistency of these bootstrap procedures is established. The finite-sample behavior of these bootstrap tests is illustrated via simulation and prevails over their competitors’. Finally, these bootstrap tests are applied to detect the seasonal unit roots in various economic time series.

A Probabilistic Uncertainty Estimation Method for Turbulence Parameters Measured by Hot-Wire Anemometry in Short-Duration Wind Tunnels

Abstract A robust and complete uncertainty estimation method is developed to quantify the uncertainty of turbulence quantities measured by hot-wire anemometry (HWA) at the inlet of a short-duration turbine test rig. The uncertainty is categorized into two macro-uncertainty sources: the measurement-related uncertainty (the uncertainty of each instantaneous velocity sample) and the uncertainty stemming from the statistical treatment of the time series. The former is addressed by the implementation of a Monte Carlo (MC) method. The latter, which is directly related to the duration of the acquired signal, is estimated using the moving block bootstrap (MBB) method, a nonparametric resampling algorithm suitable for correlated time series. This methodology allows computing the confidence intervals of the spanwise distributions of mean velocity, turbulence intensity, length scales, and other statistical moments at the inlet of the turbine test section.

Block bootstrapping for a panel mean break test

We consider block bootstrappings for panel mean change test of the squared CUSUM test of Horvath and Huskova (J Time Ser Anal 33:631–648, 2012): the circular block bootstrapping and stationary bootstrapping. First order asymptotic null validity of the test is proved under serial and/or cross-sectional correlation. Consistency of the test under an alternative hypothesis is also proved. A Monte-Carlo experiment reveals that the existing tests of Horvath and Huskova (2012) and others have severe size distortions for serially and/or cross-sectionally correlated panels, and the block bootstrappings remedy this size distortion problem. A real data analysis illustrates the proposed method.

Ratio detections for change point in heavy tailed observations

This article modifies the ratio test so that we can detect the change which happens in the latter half observations effectively. Moreover, we propose a new ratio test on the self-normalized numerator and the self-normalized denominator respectively. We establish the asymptotic properties under the null and alternative hypotheses. We also propose a new estimator to locate the early or later change. We check the theoretical validity of the block bootstrap approximation for the modified tests and give the weak convergence of the proposed estimator. Simulations demonstrate the validity of the block bootstrap procedures and the advantage of the proposed estimator. An application on a real data is illustrated for the practicability of the proposed test and the estimators.

Beyond Socially Responsible Investing: Effects of Mission-Driven Portfolio Selection

In their pursuit of value creation, charitable foundations are mission- rather than profit-driven. Therefore, foundations are also mission-driven investors. We explore the effects of mission-driven portfolio selection based on three model foundations, representing common fields of activity in Switzerland. Employing a moving block bootstrap approach, we simulate time series. Based on these model foundations and under the integration of qualitative company rating data, such as environmental, social, and governance-related characteristics (ESG), we find both negative and no significant financial effects of portfolio screening. However, screening portfolios substantially increases mission-driven portfolio quality. Additionally, screening reduces reputational risks and even leptokurtic return characteristics under special consideration of governance issues. After a joint analysis of financial and qualitative factors for portfolios with equity shares of 25% and 50%, we did find strong enough evidence to encourage foundations to implement negative and positive screening criteria. Additionally, we argue that without the integration of mission-based qualitative criteria, for instance, the involvement in business activities contradicting the foundation’s mission, an adequate evaluation of investment opportunities’ desirability is not feasible.

Estimation of long memory parameter in nonparametric regression

This paper considers the estimation of the long memory parameter in nonparametric regression with strongly correlated errors. The key idea is to minimize a unified mean squared error of long memory parameter to select both kernel bandwidth and the number of frequencies used in exact local Whittle estimation. A unified mean squared error framework is more natural because it provides both goodness of fit and measure of strong dependence. The block bootstrap is applied to evaluate the mean squared error. Finite sample performance using Monte Carlo simulations shows the closest performance to the oracle. The proposed method outperforms existing methods especially when dependency and sample size increase. The proposed method is also illustreated to the volatility of exchange rate between Korean Won for US dollar.

Moving block and tapered block bootstrap for functional time series with an application to the $K$-sample mean problem

We consider infinite-dimensional Hilbert space-valued random variables that are assumed to be temporal dependent in a broad sense. We prove a central limit theorem for the moving block bootstrap and for the tapered block bootstrap, and show that these block bootstrap procedures also provide consistent estimators of the long run covariance operator. Furthermore, we consider block bootstrap-based procedures for fully functional testing of the equality of mean functions between several independent functional time series. We establish validity of the block bootstrap methods in approximating the distribution of the statistic of interest under the null and show consistency of the block bootstrap-based tests under the alternative. The finite sample behaviour of the procedures is investigated by means of simulations. An application to a real-life dataset is also discussed.

Levying carbon footprint taxes on animal-sourced foods. A case study in Spain

Abstract The consumption of some kinds of meat contributes significantly to global warming. Hence, dietary changes towards a decreased consumption of these products might lead to reductions in greenhouse gas emissions. Information campaigns can induce a reduction in meat consumption, but they need time since meat is deeply rooted in our diets. Instruments based on changing prices, such as the levying of taxes, stand out as an effective option. This study aims to estimate the greenhouse gas mitigation potential of implementing consumption taxes on some protein-rich foods in Spain. Data from the Spanish panel of household food consumption for the period 2004–2015 and literature data on the carbon footprint of animal sourced foods were used. An almost ideal demand system with the moving blocks bootstrap method was applied to estimate the elasticities and calculate the consequences for the carbon footprint associated with the consumption of seven animal-sourced foods. The results show that taxes on fish can be the most effective at reducing the total carbon footprint and those applied on pork are the least. The results also suggest that high tax rates on the most polluting products do not always lead to the greatest carbon footprint reduction. Further research should incorporate combinations of taxes on different products together with an analysis of potential regressive effects on other socio-economic issues. However, reducing meat GHG emissions is a global challenge which requires inter-disciplinary collaboration together with political actions.

Inference for local distributions at high sampling frequencies: A bootstrap approach

We study inference for the local innovations of Itô 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, in addition to an asymptotic bias. Moreover, we design the LDWB sufficiently general to establish asymptotic equivalence between it 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 is assessed in a simulation study and an empirical application. 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 fine scale distributional properties of innovations to equity indices, commodities and exchange rates.

Local Likelihood Estimation of Complex Tail Dependence Structures, Applied to U.S. Precipitation Extremes

To disentangle the complex nonstationary dependence structure of precipitation extremes over the entire contiguous United States (U.S.), we propose a flexible local approach based on factor copula models. Our subasymptotic spatial modeling framework yields nontrivial tail dependence structures, with a weakening dependence strength as events become more extreme; a feature commonly observed with precipitation data but not accounted for in classical asymptotic extreme-value models. To estimate the local extremal behavior, we fit the proposed model in small regional neighborhoods to high threshold exceedances, under the assumption of local stationarity, which allows us to gain in flexibility. By adopting a local censored likelihood approach, we make inference on a fine spatial grid, and we perform local estimation by taking advantage of distributed computing resources and the embarrassingly parallel nature of this estimation procedure. The local model is efficiently fitted at all grid points, and uncertainty is measured using a block bootstrap procedure. We carry out an extensive simulation study to show that our approach can adequately capture complex, nonstationary dependencies, in addition, our study of U.S. winter precipitation data reveals interesting differences in local tail structures over space, which has important implications on regional risk assessment of extreme precipitation events. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

QUANTILOGRAMS UNDER STRONG DEPENDENCE

We develop the limit theory of the quantilogram and cross-quantilogram under long memory. We establish the sub-root-n central limit theorems for quantilograms that depend on nuisance parameters. We propose a moving block bootstrap (MBB) procedure for inference and establish its consistency, thereby enabling a consistent confidence interval construction for the quantilograms. The newly developed reduction principles for the quantilograms serve as the main technical devices used to derive the asymptotics and establish the validity of MBB. We report some simulation evidence that our methods work satisfactorily. We apply our method to quantile predictive relations between financial returns and long-memory predictors.

Inference on Long-Range Temporal Correlations in Human EEG Data

Detrended Fluctuation Analysis (DFA) is a statistical estimation algorithm used to assess long-range temporal dependence in neural time series. The algorithm produces a single number, the DFA exponent, that reflects the strength of long-range temporal correlations in the data. No methods have been developed to generate confidence intervals for the DFA exponent for a single time series segment. Thus, we present a statistical measure of uncertainty for the DFA exponent in electroencephalographic (EEG) data via application of a moving-block bootstrap (MBB). We tested the effect of three data characteristics on the DFA exponent: (1) time series length, (2) the presence of artifacts, and (3) the presence of discontinuities. We found that signal lengths of ∼5 minutes produced stable measurements of the DFA exponent and that the presence of artifacts positively biased DFA exponent distributions. In comparison, the impact of discontinuities was small, even those associated with artifact removal. We show that it is possible to combine a moving block bootstrap with DFA to obtain an accurate estimate of the DFA exponent as well as its associated confidence intervals in both simulated data and human EEG data. We applied the proposed method to human EEG data to (1) calculate a time-varying estimate of long-range temporal dependence during a sleep-wake cycle of a healthy infant and (2) compare pre- and post-treatment EEG data within individual subjects with pediatric epilepsy. Our proposed method enables dynamic tracking of the DFA exponent across the entire recording period and permits within-subject comparisons, expanding the utility of the DFA algorithm by providing a measure of certainty and formal tests of statistical significance for the estimation of long-range temporal dependence in neural data.