Bootstrapping rank statistics

The bootstrap, which provides powerful approximations for many classes of statistics, is studied for simple linear rank statistics employing bounded and smooth score functions. To verify consistency we view a rank statistic as a statistic induced by a statistical functional ψ which is evaluated at a pair of dependent signed measures. Thus, we can apply the von Mises method to verify asymptotic results for the bootstrap. The strong consistency of the bootstrap distribution estimator is derived for the bootstrap based on resampling from the original data. Further, the residual bootstrap is studied. The accuracy of the bootstrap approximations for small sample sizes is studied by simulations. The simulations indicate that the bootstrap provides better results than a normal approximation.     

A Comparison of Sequential and Information‐based Methods for Determining the Co‐integration Rank in Heteroskedastic VAR Models

In this article, we investigate the behaviour of a number of methods for estimating the co‐integration rank in VAR systems characterized by heteroskedastic innovation processes. In particular, we compare the efficacy of the most widely used information criteria, such as Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) , with the commonly used sequential approach of Johansen [Likelihood‐based Inference in Cointegrated Vector Autoregressive Models (1996)] based around the use of either asymptotic or wild bootstrap‐based likelihood ratio type tests. Complementing recent work done for the latter in Cavaliere, Rahbek and Taylor [Econometric Reviews (2014) forthcoming], we establish the asymptotic properties of the procedures based on information criteria in the presence of heteroskedasticity (conditional or unconditional) of a quite general and unknown form. The relative finite‐sample properties of the different methods are investigated by means of a Monte Carlo simulation study. For the simulation DGPs considered in the analysis, we find that the BIC‐based procedure and the bootstrap sequential test procedure deliver the best overall performance in terms of their frequency of selecting the correct co‐integration rank across different values of the co‐integration rank, sample size, stationary dynamics and models of heteroskedasticity. Of these, the wild bootstrap procedure is perhaps the more reliable overall as it avoids a significant tendency seen in the BIC‐based method to over‐estimate the co‐integration rank in relatively small sample sizes.     

Bootstrap testing for the null of no cointegration in a threshold vector error correction model

We develop a test for the linear no cointegration null hypothesis in a threshold vector error correction model. We adopt a sup-Wald type test and derive its null asymptotic distribution. A residual-based bootstrap is proposed, and the first-order consistency of the bootstrap is established. A set of Monte Carlo simulations shows that the bootstrap corrects size distortion of asymptotic distribution in finite samples, and that its power against the threshold cointegration alternative is significantly greater than that of conventional cointegration tests. Our method is illustrated with used car price indexes.     

A bootstrap-assisted spectral test of white noise under unknown dependence

To test for the white noise null hypothesis, we study the Cramér-von Mises test statistic that is based on the sample spectral distribution function. Since the critical values of the test statistic are difficult to obtain, we propose a blockwise wild bootstrap procedure to approximate its asymptotic null distribution. Using a Hilbert space approach, we establish the weak convergence of the difference between the sample spectral distribution function and the true spectral distribution function, as well as the consistency of bootstrap approximation under mild assumptions. Finite sample results from a simulation study and an empirical data analysis are also reported.     

The power of bootstrap and asymptotic tests

The bootstrap discrepancy measures the difference in rejection probabilities between a bootstrap test and one based on the true distribution. The order of magnitude of the bootstrap discrepancy is the same under the null hypothesis and under non-null processes described by Pitman drift. If the test statistic is not an exact pivot, critical values depend on which data-generating process (DGP) is used to determine the null distribution. We propose using the DGP which minimizes the bootstrap discrepancy. We also show that, under an asymptotic independence condition, the power of both bootstrap and asymptotic tests can be estimated cheaply by simulation.     

Likelihood inference in some finite mixture models

Parametric mixture models are commonly used in applied work, especially empirical economics, where these models are often employed to learn for example about the proportions of various types in a given population. This paper examines the inference question on the proportions (mixing probability) in a simple mixture model in the presence of nuisance parameters when sample size is large. It is well known that likelihood inference in mixture models is complicated due to (1) lack of point identification, and (2) parameters (for example, mixing probabilities) whose true value may lie on the boundary of the parameter space. These issues cause the profiled likelihood ratio (PLR) statistic to admit asymptotic limits that differ discontinuously depending on how the true density of the data approaches the regions of singularities where there is lack of point identification. This lack of uniformity in the asymptotic distribution suggests that confidence intervals based on pointwise asymptotic approximations might lead to faulty inferences. This paper examines this problem in details in a finite mixture model and provides possible fixes based on the parametric bootstrap. We examine the performance of this parametric bootstrap in Monte Carlo experiments and apply it to data from Beauty Contest experiments. We also examine small sample inferences and projection methods.     

An improved bootstrap test of stochastic dominance

We propose a new method of testing stochastic dominance that improves on existing tests based on the standard bootstrap or subsampling. The method admits prospects involving infinite as well as finite dimensional unknown parameters, so that the variables are allowed to be residuals from nonparametric and semiparametric models. The proposed bootstrap tests have asymptotic sizes that are less than or equal to the nominal level uniformly over probabilities in the null hypothesis under regularity conditions. This paper also characterizes the set of probabilities so that the asymptotic size is exactly equal to the nominal level uniformly. As our simulation results show, these characteristics of our tests lead to an improved power property in general. The improvement stems from the design of the bootstrap test whose limiting behavior mimics the discontinuity of the original test’s limiting distribution.     

Locally Most Powerful and Other Rank Tests for Independence – with a Contaminated Weighted Alternative

Several authors in the literature have attempted the quantification of the concept of stochastic dependence for bivariate distribution. Two weighted rank tests for testing independence against a weighted contamination alternative is proposed and their distributional properties are studied. We also derived a locally most powerful rank test for the alternative setting. The rank tests proposed are shown to be asymptotic locally most powerful for specific distributions.     

A nonparametric test for equality of distributions with mixed categorical and continuous data

In this paper we consider the problem of testing for equality of two density or two conditional density functions defined over mixed discrete and continuous variables. We smooth both the discrete and continuous variables, with the smoothing parameters chosen via least-squares cross-validation. The test statistics are shown to have (asymptotic) normal null distributions. However, we advocate the use of bootstrap methods in order to better approximate their null distribution in finite-sample settings and we provide asymptotic validity of the proposed bootstrap method. Simulations show that the proposed tests have better power than both conventional frequency-based tests and smoothing tests based on ad hoc smoothing parameter selection, while a demonstrative empirical application to the joint distribution of earnings and educational attainment underscores the utility of the proposed approach in mixed data settings.     

A consistent model specification test with mixed discrete and continuous data

In this paper we propose a nonparametric kernel-based model specification test that can be used when the regression model contains both discrete and continuous regressors. We employ discrete variable kernel functions and we smooth both the discrete and continuous regressors using least squares cross-validation (CV) methods. The test statistic is shown to have an asymptotic normal null distribution. We also prove the validity of using the wild bootstrap method to approximate the null distribution of the test statistic, the bootstrap being our preferred method for obtaining the null distribution in practice. Simulations show that the proposed test has significant power advantages over conventional kernel tests which rely upon frequency-based nonparametric estimators that require sample splitting to handle the presence of discrete regressors.     

Asymptotic distributions of impulse response functions in short panel vector autoregressions

This paper establishes the asymptotic distributions of the impulse response functions in panel vector autoregressions with a fixed time dimension. It also proves the asymptotic validity of a bootstrap approximation to their sampling distributions. The autoregressive parameters are estimated using the GMM estimators based on the first differenced equations and the error variance is estimated using an extended analysis-of-variance type estimator. Contrary to the time series setting, we find that the GMM estimator of the autoregressive coefficients is not asymptotically independent of the error variance estimator. The asymptotic dependence calls for variance correction for the orthogonalized impulse response functions. Simulation results show that the variance correction improves the coverage accuracy of both the asymptotic confidence band and the studentized bootstrap confidence band for the orthogonalized impulse response functions.     

Parametric bootstrap tests for continuous and discrete distributions

Statistical procedures based on the estimated empirical process are well known for testing goodness of fit to parametric distribution families. These methods usually are not distribution free, so that the asymptotic critical values of test statistics depend on unknown parameters. This difficulty may be overcome by the utilization of parametric bootstrap procedures. The aim of this paper is to prove a weak approximation theorem for the bootstrapped estimated empirical process under very general conditions, which allow both the most important continuous and discrete distribution families, along with most parameter estimation methods. The emphasis is on families of discrete distributions, and simulation results for families of negative binomial distributions are also presented.     

Nonparametric estimation in case of endogenous selection

This paper addresses the problem of estimation of a nonparametric regression function from selectively observed data when selection is endogenous. Our approach relies on independence between covariates and selection conditionally on potential outcomes. Endogeneity of regressors is also allowed for. In the exogenous and endogenous case, consistent two-step estimation procedures are proposed and their rates of convergence are derived. Pointwise asymptotic distribution of the estimators is established. In addition, bootstrap uniform confidence bands are obtained. Finite sample properties are illustrated in a Monte Carlo simulation study and an empirical illustration.     

Asymptotic and bootstrap tests for linearity in a TAR-GARCH(1,1) model with a unit root

This paper derives the limiting distribution of the Lagrange Multiplier (LM) test for threshold nonlinearity in a TAR model with GARCH errors when one of the regimes contains a unit root. It is shown that the asymptotic distribution is nonstandard and depends on nuisance parameters that capture the degree of conditional heteroskedasticity and non-Gaussian nature of the process. We propose a bootstrap procedure for approximating the exact finite-sample distribution of the test for linearity and establish its asymptotic validity.     

Blockwise bootstrap wavelet in nonparametric regression model with weakly dependent processes

In this paper, we suggest a blockwise bootstrap wavelet to estimate the regression function in the nonparametric regression models with weakly dependent processes for both designs of fixed and random. We obtain the asymptotic orders of the biases and variances of the estimators and establish the asymptotic normality for a modified version of the estimators. We also introduce a principle to select the length of data block. These results show that the blockwise bootstrap wavelet is valid for general weakly dependent processes such as α-mixing, φ-mixing and ρ-mixing random variables.     

Higher order properties of the wild bootstrap under misspecification

We examine the higher order properties of the wild bootstrap (Wu, 1986) in a linear regression model with stochastic regressors. We find that the ability of the wild bootstrap to provide a higher order refinement is contingent upon whether the errors are mean independent of the regressors or merely uncorrelated with them. In the latter case, the wild bootstrap may fail to match some of the terms in an Edgeworth expansion of the full sample test statistic. Nonetheless, we show that the wild bootstrap still has a lower maximal asymptotic risk as an estimator of the true distribution than a normal approximation, in shrinking neighborhoods of properly specified models. To assess the practical implications of this result we conduct a Monte Carlo study contrasting the performance of the wild bootstrap with a normal approximation and the traditional nonparametric bootstrap.     

Subsampling inference for the mean in the heavy-tailed case

In this article, asymptotic inference for the mean of i.i.d. observations in the context of heavy-tailed distributions is discussed. While both the standard asymptotic method based on the normal approximation and Efron's bootstrap are inconsistent when the underlying distribution does not possess a second moment, we propose two approaches based on the subsampling idea of Politis and Romano (1994) which will give correct answers. The first approach uses the fact that the sample mean, properly standardized, will under some regularity conditions have a limiting stable distribution. The second approach consists of subsampling the usual t-statistic and is somewhat more general. A simulation study compares the small sample performance of the two methods. Received: December 1998     

Testing the parametric form of the volatility in continuous time diffusion models—a stochastic process approach

We present new tests for the form of the volatility function which are based on stochastic processes of the integrated volatility. We prove weak convergence of these processes to centered processes whose conditional distributions are Gaussian. In the case of testing for a constant volatility the limiting process are standard Brownian bridges. As a consequence an asymptotic distribution free test and bootstrap tests (for testing of a general parametric form) can easily be implemented. It is demonstrated that the new tests are more than the currently available procedures. The new approach is also demonstrated by means of a simulation study.     

A consistent nonparametric test of parametric regression functional form in fixed effects panel data models

We propose a consistent test for a linear functional form against a nonparametric alternative in a fixed effects panel data model. We show that the test has a limiting standard normal distribution under the null hypothesis, and show that the test is a consistent test. We also establish the asymptotic validity of a bootstrap procedure which is used to better approximate the finite sample null distribution of the test statistic. Simulation results show that the proposed test performs well for panel data with a large number of cross-sectional units and a finite number of observations across time.     

Stationary bootstrapping for common mean change detection in cross-sectionally dependent panels

Stationary bootstrapping is applied to a CUSUM test for common mean break detection in cross-sectionally correlated panel data. Asymptotic null distribution of the bootstrapped test is derived, which is the same as that of the original CUSUM test depending on cross-sectional correlation parameter. A bootstrap test using the CUSUM test with bootstrap critical values is proposed and its asymptotic validity is proved. Finite sample Monte-Carlo simulation shows that the proposed test has reasonable size while other existing tests have severe size distortion under cross-section correlation. The simulation also shows good power performance of the proposed test against non-cancelling mean changes. The simulation also shows that the theoretically justified stationary bootstrapping CUSUM test has comparable size and power relative to other, theoretically unjustified, moving block or tapered block bootstrapping CUSUM tests.