Simple Bootstrap Procedure Research Articles

Unified specification tests in partially linear quantile regression models

We propose specification tests for parametric quantile regression models versus semiparametric alternatives over a continuum of quantile levels. The test statistics are constructed as continuous functionals of a quantile-marked residual process. We show that using an orthogonal projection on the tangent space of nuisance parameters at each quantile index delivers unified asymptotic properties for tests based on different estimators. Consistency of the tests and asymptotic power under a sequence of local alternatives converging to the null at a parametric rate are also discussed. We propose a simple multiplier bootstrap procedure to carry out the tests, whose nominal levels are well approximated in our simulation study for modest sample sizes.

Partially identifying competing risks models: An application to the war on cancer

Competing risks models for discretely measured durations are partially identifying due to the unknown dependence structure between risks and the discrete nature of the outcome. This article develops a highly tractable bounds approach for underlying distributions of latent durations by exploiting the discreteness. Bounds are obtained from a system of nonlinear (in)equalities. I devise a sequential solution method that requires much less computational burden than existing methods. Asymptotic properties of bound estimators and a simple bootstrap procedure are provided. I apply the proposed approach to re-evaluate trends in cancer mortality extending the data studied in Honoré and Lleras-Muney (2006). Estimated patterns differ from the original findings.

An Adapted Loss Function for Censored Quantile Regression

ABSTRACTIn this article, we study a novel approach for the estimation of quantiles when facing potential right censoring of the responses. Contrary to the existing literature on the subject, the adopted strategy of this article is to tackle censoring at the very level of the loss function usually employed for the computation of quantiles, the so-called “check” function. For interpretation purposes, a simple comparison with the latter reveals how censoring is accounted for in the newly proposed loss function. Subsequently, when considering the inclusion of covariates for conditional quantile estimation, by defining a new general loss function the proposed methodology opens the gate to numerous parametric, semiparametric, and nonparametric modeling techniques. To illustrate this statement, we consider the well-studied linear regression under the usual assumption of conditional independence between the true response and the censoring variable. For practical minimization of the studied loss function, we also provide a simple algorithmic procedure shown to yield satisfactory results for the proposed estimator with respect to the existing literature in an extensive simulation study. From a more theoretical prospect, consistency and asymptotic normality of the estimator for linear regression are obtained using several recent results on nonsmooth semiparametric estimation equations with an infinite-dimensional nuisance parameter, while numerical examples illustrate the adequateness of a simple bootstrap procedure for inferential purposes. Lastly, an application to a real dataset is used to further illustrate the validity and finite sample performance of the proposed estimator. Supplementary materials for this article are available online.

THE FACTOR-LASSO AND K-STEP BOOTSTRAP APPROACH FOR INFERENCE IN HIGH-DIMENSIONAL ECONOMIC APPLICATIONS

We consider inference about coefficients on a small number of variables of interest in a linear panel data model with additive unobserved individual and time specific effects and a large number of additional time-varying confounding variables. We suppose that, in addition to unrestricted time and individual specific effects, these confounding variables are generated by a small number of common factors and high-dimensional weakly dependent disturbances. We allow that both the factors and the disturbances are related to the outcome variable and other variables of interest. To make informative inference feasible, we impose that the contribution of the part of the confounding variables not captured by time specific effects, individual specific effects, or the common factors can be captured by a relatively small number of terms whose identities are unknown. Within this framework, we provide a convenient inferential procedure based on factor extraction followed by lasso regression and show that the procedure has good asymptotic properties. We also provide a simplek-step bootstrap procedure that may be used to construct inferential statements about the low-dimensional parameters of interest and prove its asymptotic validity. We provide simulation evidence about the performance of our procedure and illustrate its use in an empirical application.

Variance estimation for integrated population models

State-space models are widely used in ecology. However, it is well known that in practice it can be difficult to estimate both the process and observation variances that occur in such models. We consider this issue for integrated population models, which incorporate state-space models for population dynamics. To some extent, the mechanism of integrated population models protects against this problem, but it can still arise, and two illustrations are provided, in each of which the observation variance is estimated as zero. In the context of an extended case study involving data on British Grey herons, we consider alternative approaches for dealing with the problem when it occurs. In particular, we consider penalised likelihood, a method based on fitting splines and a method of pseudo-replication, which is undertaken via a simple bootstrap procedure. For the case study of the paper, it is shown that when it occurs, an estimate of zero observation variance is unimportant for inference relating to the model parameters of primary interest. This unexpected finding is supported by a simulation study.

Confidence Corridors for Multivariate Generalized Quantile Regression

We focus on the construction of confidence corridors for multivariate nonparametric generalized quantile regression functions. This construction is based on asymptotic results for the maximal deviation between a suitable nonparametric estimator and the true function of interest, which follow after a series of approximation steps including a Bahadur representation, a new strong approximation theorem, and exponential tail inequalities for Gaussian random fields. As a byproduct we also obtain multivariate confidence corridors for the regression function in the classical mean regression. To deal with the problem of slowly decreasing error in coverage probability of the asymptotic confidence corridors, which results in meager coverage for small sample sizes, a simple bootstrap procedure is designed based on the leading term of the Bahadur representation. The finite-sample properties of both procedures are investigated by means of a simulation study and it is demonstrated that the bootstrap procedure considerably outperforms the asymptotic bands in terms of coverage accuracy. Finally, the bootstrap confidence corridors are used to study the efficacy of the National Supported Work Demonstration, which is a randomized employment enhancement program launched in the 1970s. This article has supplementary materials online.

Asymptotics and bootstrap for random-effects panel data transformation models

ABSTRACTThis article investigates the asymptotic properties of quasi-maximum likelihood (QML) estimators for random-effects panel data transformation models where both the response and (some of) the covariates are subject to transformations for inducing normality, flexible functional form, homoskedasticity, and simple model structure. We develop a QML-type procedure for model estimation and inference. We prove the consistency and asymptotic normality of the QML estimators, and propose a simple bootstrap procedure that leads to a robust estimate of the variance-covariance (VC) matrix. Monte Carlo results reveal that the QML estimators perform well in finite samples, and that the gains by using the robust VC matrix estimate for inference can be enormous.

A Joint Specification Test for Response Probabilities in Unordered Multinomial Choice Models

Estimation results obtained by parametric models may be seriously misleading when the model is misspecified or poorly approximates the true model. This study proposes a test that jointly tests the specifications of multiple response probabilities in unordered multinomial choice models. The test statistic is asymptotically chi-square distributed, consistent against a fixed alternative and able to detect a local alternative approaching to the null at a rate slower than the parametric rate. We show that rejection regions can be calculated by a simple parametric bootstrap procedure, when the sample size is small. The size and power of the tests are investigated by Monte Carlo experiments.

Bootstrap multi-step forecasts of non-Gaussian VAR models

In this paper, we establish the asymptotic validity and analyse the finite sample performance of a simple bootstrap procedure for constructing multi-step multivariate forecast densities in the context of non-Gaussian unrestricted VAR models. This bootstrap procedure avoids the backward representation, and, as a consequence, can be used to obtain multivariate forecast densities in, for example, VARMA or VAR-GARCH models. In the context of bivariate stationary VAR(p) models, we show that its finite sample properties are comparable to those of alternatives based on the backward representation. The bootstrap procedure is also implemented in a VAR-DCC model which lacks a backward representation. Finally, joint forecast densities of US quarterly inflation, unemployment and GDP growth are obtained.

A general method for third-order bias and variance corrections on a nonlinear estimator

Motivated by a recent study of Bao and Ullah (2007a) on finite sample properties of MLE in the pure SAR (spatial autoregressive) model, a general method for third-order bias and variance corrections on a nonlinear estimator is proposed based on stochastic expansion and bootstrap. Working with concentrated estimating equation simplifies greatly the high-order expansions for bias and variance; a simple bootstrap procedure overcomes a major difficulty in analytically evaluating expectations of various quantities in the expansions. The method is then studied in detail using a more general SAR model, with its effectiveness in correcting bias and improving inference fully demonstrated by extensive Monte Carlo experiments. Compared with the analytical approach, the proposed approach is much simpler and has a much wider applicability. The validity of the bootstrap procedure is formally established. The proposed method is then extended to the case of more than one nonlinear estimator.

HAC robust trend comparisons among climate series with possible level shifts

Comparisons of trends across climatic data sets are complicated by the presence of serial correlation and possible step‐changes in the mean. We build on heteroskedasticity and autocorrelation robust methods, specifically the Vogelsang–Franses (VF) nonparametric testing approach, to allow for a step‐change in the mean (level shift) at a known or unknown date. The VF method provides a powerful multivariate trend estimator robust to unknown serial correlation up to but not including unit roots. We show that the critical values change when the level shift occurs at a known or unknown date. We derive an asymptotic approximation that can be used to simulate critical values, and we outline a simple bootstrap procedure that generates valid critical values and p‐values. Our application builds on the literature comparing simulated and observed trends in the tropical lower troposphere and mid‐troposphere since 1958. The method identifies a shift in observations around 1977, coinciding with the Pacific Climate Shift. Allowing for a level shift causes apparently significant observed trends to become statistically insignificant. Model overestimation of warming is significant whether or not we account for a level shift, although null rejections are much stronger when the level shift is included. © 2014 The Authors. Environmetrics published by John Wiley & Sons, Ltd.

A score‐type test for heterogeneity in zero‐inflated models in a stratified population

We propose a score-type statistic to evaluate heterogeneity in zero-inflated models for count data in a stratified population, where heterogeneity is defined as instances in which the zero counts are generated from two sources. Evaluating heterogeneity in this class of models has attracted considerable attention in the literature, but existing testing procedures have primarily relied on the constancy assumption under the alternative hypothesis. In this paper, we extend the literature by describing a score-type test to evaluate homogeneity against general alternatives that do not neglect the stratification information under the alternative hypothesis. The limiting null distribution of the proposed test statistic is a mixture of chi-squared distributions that can be well approximated by a simple parametric bootstrap procedure. Our numerical simulation studies show that the proposed test can greatly improve efficiency over tests of heterogeneity that ignore the stratification information. An empirical application to dental caries data in early childhood further shows the importance and practical utility of the methodology in using the stratification profile to detect heterogeneity in the population.

Specification analysis of linear quantile models

This paper introduces a nonparametric test for the correct specification of a linear conditional quantile function over a continuum of quantile levels. These tests may be applied to assess the validity of post-estimation inferences regarding the effect of conditioning variables on the distribution of outcomes. We show that the use of an orthogonal projection on the tangent space of nuisance parameters at each quantile index both improves power and facilitates the simulation of critical values via the application of a simple multiplier bootstrap procedure. Monte Carlo evidence and an application to the empirical analysis of age–earnings curves are included.

Bootstrapping continuous-time autoregressive processes

We develop a bootstrap procedure for Levy-driven continuous-time autoregressive (CAR) processes observed at discrete regularly-spaced times. It is well known that a regularly sampled stationary Ornstein–Uhlenbeck process [i.e. a CAR(1) process] has a discrete-time autoregressive representation with i.i.d. noise. Based on this representation a simple bootstrap procedure can be found. Since regularly sampled CAR processes of higher order satisfy ARMA equations with uncorrelated (but in general dependent) noise, a more general bootstrap procedure is needed for such processes. We consider statistics depending on observations of the CAR process at the uniformly-spaced times, together with auxiliary observations on a finer grid, which give approximations to the derivatives of the continuous time process. This enables us to approximate the state-vector of the CAR process which is a vector-valued CAR(1) process, and whose sampled version, on the uniformly-spaced grid, is a multivariate AR(1) process with i.i.d. noise. This leads to a valid residual-based bootstrap which allows replication of CAR $$(p)$$ processes on the underlying discrete time grid. We show that this approach is consistent for empirical autocovariances and autocorrelations.

The Hybrid Wild Bootstrap for Time Series

We introduce a new and simple bootstrap procedure for general linear processes, called the hybrid wild bootstrap. The hybrid wild bootstrap generates frequency domain replicates of the periodogram that imitate asymptotically correct the first- and second-order properties of the ordinary periodogram including its weak dependence structure at different frequencies. As a consequence, the hybrid wild bootstrapped periodogram succeeds in approximating consistently the distribution of statistics that can be expressed as functionals of the periodogram, including the important class of spectral means for which all so far existing frequency domain bootstrap methods generally fail. Moreover, by inverting the hybrid wild bootstrapped discrete Fourier transform, pseudo-observations in the time domain are obtained. The generated time domain pseudo-observations can be used to approximate correctly the random behavior of statistics, the distribution of which depends on the first-, second-, and, to some extent, on the fourth-order structure of the underlying linear process. Thus, the proposed hybrid wild bootstrap procedure applied to general time series overcomes several of the limitations of standard linear time domain bootstrap methods.

A Comparison of Alternative Approaches to Supremum-Norm Goodness of Fit Tests with Estimated Parameters

Goodness of fit tests based on empirical processes have nonstandard limiting distributions when the null hypothesis is composite — that is, when parameters of the null model are estimated. Several solutions to this problem have been suggested, including the calculation of adjusted critical values for these nonstandard distributions and the transformation of the empirical process such that statistics based on the transformed process are asymptotically distribution-free. The approximation methods proposed by Durbin (1985) can be applied to compute appropriate critical values for tests based on supremum-norm statistics. The resulting tests have quite accurate size, a fact which has gone unrecognized in the econometrics literature. Some justification for this accuracy lies in the similar features that Durbin’s approximation methods share with the theory of extrema for Gaussian random fields and for Gauss-Markov processes. These adjustment techniques are also related to the transformation methodology proposed by Khmaladze (1981) through the score function of the parametric model. Simulation experiments suggest that these two testing strategies are roughly comparable to one another and more powerful than a simple bootstrap procedure.

Testing Conditional Independence via Rosenblatt Transforms

This paper investigates the problem of testing conditional independence of Y and Z given λθ(X) for some unknown θ ∈ Θ ⊂ Rd, for a parametric function λθ(·). For instance, such a problem is relevant in recent literatures of heterogeneous treatment effects and contract theory. First, this paper finds that using Rosenblatt transforms in a certain way, we can construct a class of tests that are asymptotically pivotal and asymptotically unbiased against √n-converging Pitman local alternatives. The asymptotic pivotalness is convenient especially because the asymptotic critical values remain invariant over different estimators of the unknown parameter θ. Even when tests are asymptotically pivotal, however, it is often the case that simulation methods to obtain asymptotic critical values are yet unavailable or complicated, and hence this paper suggests a simple wild bootstrap procedure. A special case of the proposed testing framework is to test the presence of quantile treatment effects in a program evaluation data set. Using the JTPA training data set, we investigate the validity of nonexperimental procedures for inferences about quantile treatment effects of the job training program.

Using bootstrap methods to obtain nonnormality robust Chow prediction tests

This paper emphasizes the sensitivity to nonnormality of the standard Chow test for predictive failure. Based on well established asymptotic arguments, a simple double bootstrap procedure is proposed, evaluated and found to be robust to nonnormality.

Bootstrap Tests for Distributional Treatment Effects in Instrumental Variable Models

This article considers the problem of assessing the distributional consequences of a treatment on some outcome variable of interest when treatment intake is (possibly) nonrandomized, but there is a binaryinstrument available for the researcher. Such a scenario is common in observational studies and in randomized experiments with imperfect compliance. One possible approach to this problem is to compare the counterfactual cumulative distribution functions of the outcome with and without the treatment. This article shows how to estimate these distributions using instrumental variable methods and a simple bootstrap procedure is proposed to test distributional hypotheses, such as equality of distributions, first-order and second-order stochastic dominance. These tests and estimators are applied to the study of the effects of veteran status on the distribution of civilian earnings. The results show a negative effect of military service during the Vietnam era that appears to be concentrated on the lower tail of the distribution of earnings. First-order stochastic dominance cannot be rejected by the data.

Controlling the significance levels of prediction error tests for linear regression models

This paper provides evidence on problems associated with using standard tests for predictive failure when the errors of a linear regression model are not normally distributed. The ability of a simple bootstrap procedure to give a useful degree of control over the significance levels is examined.