Bootstrapping Noncausal Autoregressions: With Applications to Explosive Bubble Modeling

In this article, we develop new bootstrap-based inference for noncausal autoregressions with heavy-tailed innovations. This class of models is widely used for modeling bubbles and explosive dynamics in economic and financial time series. In the noncausal, heavy-tail framework, a major drawback of asymptotic inference is that it is not feasible in practice as the relevant limiting distributions depend crucially on the (unknown) decay rate of the tails of the distribution of the innovations. In addition, even in the unrealistic case where the tail behavior is known, asymptotic inference may suffer from small-sample issues. To overcome these difficulties, we propose bootstrap inference procedures using parameter estimates obtained with the null hypothesis imposed (the so-called restricted bootstrap). We discuss three different choices of bootstrap innovations: wild bootstrap, based on Rademacher errors; permutation bootstrap; a combination of the two (“permutation wild bootstrap”). Crucially, implementation of these bootstraps do not require any a priori knowledge about the distribution of the innovations, such as the tail index or the convergence rates of the estimators. We establish sufficient conditions ensuring that, under the null hypothesis, the bootstrap statistics estimate consistently particular conditionaldistributions of the original statistics. In particular, we show that validity of the permutation bootstrap holds without any restrictions on the distribution of the innovations, while the permutation wild and the standard wild bootstraps require further assumptions such as symmetry of the innovation distribution. Extensive Monte Carlo simulations show that the finite sample performance of the proposed bootstrap tests is exceptionally good, both in terms of size and of empirical rejection probabilities under the alternative hypothesis. We conclude by applying the proposed bootstrap inference to Bitcoin/USD exchange rates and to crude oil price data. We find that indeed noncausal models with heavy-tailed innovations are able to fit the data, also in periods of bubble dynamics. Supplementary materials for this article are available online.     

Two‐Sample Bootstrap Hypothesis Tests for Three‐Dimensional Labelled Landmark Data

Abstract. We investigate resampling methodologies for testing the null hypothesis that two samples of labelled landmark data in three dimensions come from populations with a common mean reflection shape or mean reflection size‐and‐shape. The investigation includes comparisons between (i) two different test statistics that are functions of the projection onto tangent space of the data, namely the James statistic and an empirical likelihood statistic; (ii) bootstrap and permutation procedures; and (iii) three methods for resampling under the null hypothesis, namely translating in tangent space, resampling using weights determined by empirical likelihood and using a novel method to transform the original sample entirely within refection shape space. We present results of extensive numerical simulations, on which basis we recommend a bootstrap test procedure that we expect will work well in practise. We demonstrate the procedure using a data set of human faces, to test whether humans in different age groups have a common mean face shape.     

Resampling schemes with low resampling intensity and their applications in testing hypotheses

The paper explores statistical features of different resampling schemes under low resampling intensity. The original sample is considered in a very general framework of triangular arrays, without independence or equally distributed assumptions, although improvements under such conditions are also provided. We show that low resampling schemes have very interesting and flexible properties, providing new insights into the performance of widely used resampling methods, including subsampling, two-sample unbalanced permutation statistics or wild bootstrap. It is shown that, under regularity assumptions, resampling tests with critical values derived by the appertaining low resampling procedures are asymptotically valid and there is no loss of power compared with the power function of an ideal (but unfeasible) parametric family of tests. Moreover we show that in several contexts, including regression models, they may act as a filter for the normal part of a limit distribution, turning down the influence of outliers.     

Bootstrap tests for variance components in generalized linear mixed models

In many applications of generalized linear mixed models to clustered correlated or longitudinal data, often we are interested in testing whether a random effects variance component is zero. The usual asymptotic mixture of chi‐square distributions of the score statistic for testing constrained variance components does not necessarily hold. In this article, the author proposes and explores a parametric bootstrap test that appears to be valid based on its estimated level of significance under the null hypothesis. Results from a simulation study indicate that the bootstrap test has a level much closer to the nominal one while the asymptotic test is conservative, and is more powerful than the usual asymptotic score test based on a mixture of chi‐squares. The proposed bootstrap test is illustrated using two sets of real‐life data obtained from clinical trials. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Some solutions to the multivariate Behrens–Fisher problem for dissimilarity‐based analyses

The essence of the generalised multivariate Behrens–Fisher problem (BFP) is how to test the null hypothesis of equality of mean vectors for two or more populations when their dispersion matrices differ. Solutions to the BFP usually assume variables are multivariate normal and do not handle high‐dimensional data. In ecology, species' count data are often high‐dimensional, non‐normal and heterogeneous. Also, interest lies in analysing compositional dissimilarities among whole communities in non‐Euclidean (semi‐metric or non‐metric) multivariate space. Hence, dissimilarity‐based tests by permutation (e.g., PERMANOVA, ANOSIM) are used to detect differences among groups of multivariate samples. Such tests are not robust, however, to heterogeneity of dispersions in the space of the chosen dissimilarity measure, most conspicuously for unbalanced designs. Here, we propose a modification to the PERMANOVA test statistic, coupled with either permutation or bootstrap resampling methods, as a solution to the BFP for dissimilarity‐based tests. Empirical simulations demonstrate that the type I error remains close to nominal significance levels under classical scenarios known to cause problems for the un‐modified test. Furthermore, the permutation approach is found to be more powerful than the (more conservative) bootstrap for detecting changes in community structure for real ecological datasets. The utility of the approach is shown through analysis of 809 species of benthic soft‐sediment invertebrates from 101 sites in five areas spanning 1960 km along the Norwegian continental shelf, based on the Jaccard dissimilarity measure.     

协整参数的自举推断

 针对完全修正最小二乘(full-modified ordinary least square,简称FMOLS）估计方法,给出一种协整参数的自举推断程序,证明零假设下自举统计量与检验统计量具有相同的渐近分布。关于检验功效的研究表明,虽然有约束自举的实际检验水平表现良好,但如果零假设不成立,自举统计量的分布是不确定的,因而其经验分布不能作为检验统计量精确分布的有效估计。实际应用中建议使用无约束自举,因为无论观测数据是否满足零假设,其自举统计量与零假设下检验统计量都具有相同的渐近分布。最后,利用蒙特卡洛模拟对自举推断和渐近推断的有限样本表现进行比较研究。     

Testing for Linear Regression Relationships in Randomly Right-Censored Varying-Coefficient Models

A test statistic is constructed to test linear relationships in randomly right-censored varying-coefficient models. A residual-based bootstrap procedure is employed to derive the p-value of the test. The performance of the test is examined by extensive simulations. The simulation results show that the bootstrap estimate of the null distribution of the test statistic is approximately valid and the test method with the residual-based bootstrap works satisfactorily for at least moderate censoring rates of the response. Furthermore, the proposed test is applied to the Stanford heart transplant data for exploring a linear regression relationship between the logrithm of the survival time and the age of the patients.     

Extensions of Permutation Solutions to Test for Treatment Effects in Replicated Single-Case Alternation Experiments with Multivariate Response

Single-case experiments are frequently used to do research involving a clinical intervention, since large-n trials are often impractical in clinical research. In order to investigate a possible difference in the effect of the treatments considered in the study, nonparametric instruments are valid tools; in particular, permutation solutions work well when we wish to assess differences in treatment effects. We present an extension of a permutation solution to the multivariate response case and to the case of replicated single-case experiments. A simulation study shows that the approach is both reliable under the null hypothesis and powerful under the alternative. At the end, we present the results of an application to two real experiments.     

Bootstrap and permutation rank tests for proportional hazards under right censoring

We address the testing problem of proportional hazards in the two-sample survival setting allowing right censoring, i.e., we check whether the famous Cox model is underlying. Although there are many test proposals for this problem, only a few papers suggest how to improve the performance for small sample sizes. In this paper, we do exactly this by carrying out our test as a permutation as well as a wild bootstrap test. The asymptotic properties of our test, namely asymptotic exactness under the null and consistency, can be transferred to both resampling versions. Various simulations for small sample sizes reveal an actual improvement of the empirical size and a reasonable power performance when using the resampling versions. Moreover, the resampling tests perform better than the existing tests of Gill and Schumacher and Grambsch and Therneau . The tests’ practical applicability is illustrated by discussing real data examples.

     

BAHADUR SLOPES FOR COMPARISONS OF TESTS BASED ON RESAMPLING

Let X₁,…, X_n be random variables symmetric about θ from a common unknown distribution F_θ(x) =F(x–θ). To test the null hypothesis H₀:θ= 0 against the alternative H₁:θ > 0, permutation tests can be used at the cost of computational difficulties. This paper investigates alternative tests that are computationally simpler, notably some bootstrap tests which are compared with permutation tests. Of these the symmetrical bootstrap-f test competes very favourably with the permutation test in terms of Bahadur asymptotic efficiency, so it is a very attractive alternative.     

A Permutation Solution to Test for Treatment Effects in Alternation Design Single-Case Experiments

Research involving a clinical intervention is normally aimed at testing the treatment effects on a dependent variable, which is assumed to be a relevant indicator of health or quality-of-life status. In much clinical research large-n trials are in fact impractical because the availability of individuals within well-defined categories is limited in this application field. This makes it more and more important to concentrate on single-case experiments. The goal with these is to investigate the presence of a difference in the effect of the treatments considered in the study. In this setting, valid inference generally cannot be made using the parametric statistical procedures that are typically used for the analysis of clinical trials and other large-n designs. Hence, nonparametric tools can be a valid alternative to analyze this kind of data. We propose a permutation solution to assess treatment effects in single-case experiments within alternation designs. An extension to the case of more than two treatments is also presented. A simulation study shows that the approach is both reliable under the null hypothesis and powerful under the alternative, and that it improves the performance of a considered competitor. In the end, we present the results of a real case application.     

A general theory of hypothesis testing based on rankings

In nonparametric statistics, a hypothesis testing problem based on the ranks of the data gives rise to two separate permutation sets corresponding to the null and to the alternative hypothesis, respectively. A modification of Critchlow's unified approach to hypothesis testing is proposed. By defining the distance between permutation sets to be the average distance between pairs of permutations, one from each set, various test statistics are derived for the multi-sample location problem and the two-way layout. The asymptotic distributions of the test statistics are computed under both the null and alternative hypotheses. Some comparisons are made on the basis of the asymptotic relative efficiency.     

Testing nonparametric statistical functionals with applications to rank tests

The present paper discusses how nonparametric tests can be deduced from statistical functionals. Efficient and asymptotically most powerful maximin tests are derived. Their power function is calculated under implicit alternatives given by the functional for one – and two – sample testing problems. It is shown that the asymptotic power function does not depend on the special implicit direction of the alternatives but only on quantities of the functional. The present approach offers a nonparametric principle how to construct common rank tests as the Wilcoxon test, the log rank test, and the median test from special two-sample functionals. In addition it is shown that studentized permutation tests yield asymptotically valid tests for certain extended null hypotheses given by functionals which are strictly larger than the common i.i.d. null hypothesis. As example tests concerning the von Mises functional and the Wilcoxon two-sample test are treated.     

Sunset Salvo

The Wilcoxon—Mann—Whitney test enjoys great popularity among scientists comparing two groups of observations, especially when measurements made on a continuous scale are non-normally distributed. Triggered by different results for the procedure from two statistics programs, we compared the outcomes from 11 PC-based statistics packages. The findings were that the delivered p values ranged from significant to nonsignificant at the 5% level, depending on whether a large-sample approximation or an exact permutation form of the test was used and, in the former case, whether or not a correction for continuity was used and whether or not a correction for ties was made. Some packages also produced pseudo-exact p values, based on the null distribution under the assumption of no ties. A further crucial point is that the variant of the algorithm used for computation by the packages is rarely indicated in the output or documented in the Help facility and the manuals. We conclude that the only accurate form of the Wilcoxon—Mann—Whitney procedure is one in which the exact permutation null distribution is compiled for the actual data.     

Bootstrap Prediction Intervals for Factor Models

We propose bootstrap prediction intervals for an observation h periods into the future and its conditional mean. We assume that these forecasts are made using a set of factors extracted from a large panel of variables. Because we treat these factors as latent, our forecasts depend both on estimated factors and estimated regression coefficients. Under regularity conditions, asymptotic intervals have been shown to be valid under Gaussianity of the innovations. The bootstrap allows us to relax this assumption and to construct valid prediction intervals under more general conditions. Moreover, even under Gaussianity, the bootstrap leads to more accurate intervals in cases where the cross-sectional dimension is relatively small as it reduces the bias of the ordinary least-squares (OLS) estimator.     

Statistical Inference on the Parametric Component in Partially Linear Spatial Autoregressive Models

A statistical test procedure is proposed to check whether the parameters in the parametric component of the partially linear spatial autoregressive models satisfy certain linear constraint conditions, in which a residual-based bootstrap procedure is suggested to derive the p-value of the test. Some simulations are conducted to assess the performance of the test and the results show that the bootstrap approximation to the null distribution of the test statistic is valid and the test is of satisfactory power. Furthermore, a real-world example is given to demonstrate the application of the proposed test.     

On cramér-von mises tests of goodness of fit with censored data

The paper considers the goodness of fit tests with right censored data or doubly censored data. The Fredholm Integral Equation (FIE) method proposed by Ren (1993) is implemented in the simulation studies to estimate the null distribution of the Cramér-von Mises test statistics and the asymptotic covariance function of the self-consistent estimator for the lifetime distribution with right censored data or doubly censored data. We show that for fixed alternatives, the bootstrap method does not estimate the null distribution consistently for doubly censored data. For the right censored case, a comparison between the performance of FIE and the η out of η bootstrap shows that FIE gives better estimation for the null distribution. The application of FIE to a set of right censored Channing House data and to a set of doubly censored breast cancer data is presented.     

Block bootstrap testing for changes in persistence with heavy-tailed innovations

This article considers the detection of changes in persistence in heavy-tailed series. We adopt a Dickey–Fuller-type ratio statistic and derive its null asymptotic distribution of test statistic. We find that the asymptotic distribution depends on the stable index, which is often typically unknown and difficult to estimate. Therefore, the block bootstrap method is proposed to detect changes without estimating κ. The empirical sizes and power values are investigated to show that the block bootstrap test is valid. Finally, the validity of the method is demonstrated by analyzing the exchange rate of RMB and US dollars.     

On Robustness of Model-Based Bootstrap Schemes in Nonparametric Time Series Analysis

Theory in time series analysis is often developed under the assumption of finite-dimensional models for the data generating process. Whereas corresponding estimators such as those of a conditional mean function are reasonable even if the true dependence mechanism is more complex, it is usually necessary to capture the whole dependence structure asymptotically for the bootstrap to be valid. In contrast, we show that certain simplified bootstrap schemes which imitate only some aspects of the time series are consistent for quantities arising in nonparametric statistics. To this end, we generalize the well-known "whitening by windowing" principle to joint distributions of nonparametric estimators of the autoregression function. Consequently, we obtain that model-based nonparametric bootstrap schemes remain valid for supremum-type functionals as long as they mimic those finite-dimensional joint distributions consistently which determine the quantity of interest. As an application, we show that simple regression-type bootstrap schemes can be applied for the determination of critical values for nonparametric tests of parametric or semiparametric hypotheses on the autoregression function in the context of a general process.     

The local fractional bootstrap

We introduce a bootstrap procedure for high‐frequency statistics of Brownian semistationary processes. More specifically, we focus on a hypothesis test on the roughness of sample paths of Brownian semistationary processes, which uses an estimator based on a ratio of realized power variations. Our new resampling method, the local fractional bootstrap, relies on simulating an auxiliary fractional Brownian motion that mimics the fine properties of high‐frequency differences of the Brownian semistationary process under the null hypothesis. We prove the first‐order validity of the bootstrap method, and in simulations, we observe that the bootstrap‐based hypothesis test provides considerable finite‐sample improvements over an existing test that is based on a central limit theorem. This is important when studying the roughness properties of time series data. We illustrate this by applying the bootstrap method to two empirical data sets: We assess the roughness of a time series of high‐frequency asset prices and we test the validity of Kolmogorov's scaling law in atmospheric turbulence data.