Optimal convergence rates,Bahadur representation,and asymptotic normality of partitioning estimators

This paper studies the asymptotic properties of partitioning estimators of the conditional expectation function and its derivatives. Mean-square and uniform convergence rates are established and shown to be optimal under simple and intuitive conditions. The uniform rate explicitly accounts for the effect of moment assumptions, which is useful in semiparametric inference. A general asymptotic integrated mean-square error approximation is obtained and used to derive an optimal plug-in tuning parameter selector. A uniform Bahadur representation is developed for linear functionals of the estimator. Using this representation, asymptotic normality is established, along with consistency of a standard-error estimator. The finite-sample performance of the partitioning estimator is examined and compared to other nonparametric techniques in an extensive simulation study.     

A bootstrap algorithm for testing cointegration rank in VAR models in the presence of stationary variables

In this paper, a bootstrap algorithm for a reduced rank vector autoregressive (VAR) model which also includes stationary regressors, is analyzed. It is shown that the bootstrap distribution for estimating the rank converges to the distribution derived from the usual asymptotic framework. Because the asymptotic distribution will typically depend on unknown parameters, bootstrap distributions are of considerable interest in this context. The result of an application and some Monte Carlo experiments are also presented.     

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.     

On the Applicability of the Sieve Bootstrap in Time Series Panels*

In this article, we investigate the validity of the univariate autoregressive sieve bootstrap applied to time series panels characterized by general forms of cross‐sectional dependence, including but not restricted to cointegration. Using the final equations approach we show that while it is possible to write such a panel as a collection of infinite order autoregressive equations, the innovations of these equations are not vector white noise. This causes the univariate autoregressive sieve bootstrap to be invalid in such panels. We illustrate this result with a small numerical example using a simple DGP for which the sieve bootstrap is invalid, and show that the extent of the invalidity depends on the value of specific parameters. We also show that Monte Carlo simulations in small samples can be misleading about the validity of the univariate autoregressive sieve bootstrap. The results in this article serve as a warning about the practical use of the autoregressive sieve bootstrap in panels where cross‐sectional dependence of general form may be present.     

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.     

Bootstrap of a linear model with AR-error structure

We consider a linear model with autoregressive error structure. It is shown that with probability one the distribution of the two-stage GLS estimator admits a bootstrap approximation. In a simulation study it is demonstrated that the bootstrap outperforms the normal approximation if the innovation variables are heavily correlated.     

Bootstrap and Asymptotic Tests of Long-run Relationships in Cointegrated Systems

Hypothesis testing on cointegrating vectors based on the asymptotic distributions of the test statistics are known to suffer from severe small sample size distortion. In this paper an alternative bootstrap procedure is proposed and evaluated through a Monte Carlo experiment, finding that the Type I errors are close to the nominal signficance levels but power might be not entirely adequate. It is then shown that a combined test based on the outcomes of both the asymptotic and the bootstrap tests will have both correct size and low Type II error, therefore improving the currently available procedures.     

ON BOOTSTRAPPING A CLASS OF DIFFERENTIABLE STATISTICAL FUNCNONALS WITH APPLICATIONS TO L– AND M– ESTIMATES

EFRON'S (1979) "bootstrap" method is justified for a class of differentiable statistical functionals which includes many L – and M –statistics. For illustration, a Monte Carlo study for the trimmed means is also included.     

Bootstrap based goodness-of-fit-tests

Summary Let ℱ={F _θ} be a parametric family of distribution functions, and denote withF _n the empirical d.f. of an i.i.d. sample. Goodness-of-fit tests of a composite hypothesis (contained in ℱ) are usually based on the so-called estimated empirical process. Typically, they are not distribution-free. In such a situation the bootstrap offers a useful alternative. It is the purpose of this paper to show that this approximation holds with probability one. A simulation study is included which demonstrates the validity of the bootstrap for several selected parametric families.     

Bias reduction when data are rounded

Analytical bias reduction methods are developed for univariate rounded data for the first time. Extensions are given to rounding of multivariate data, and to smooth functionals of several distributions. As a by‐product, we give for the first time the relation between rounded and unrounded multivariate cumulants. Estimators obtained by analytical bias reduction are compared with bootstrap and jackknife estimators by simulation.     

Median unbiased forecasts for highly persistent autoregressive processes

This paper considers the construction of median unbiased forecasts for near-integrated autoregressive processes. It derives the appropriately scaled limiting distribution of the deviation of the forecast from the true conditional mean. The dependence of the limiting distribution on nuisance parameters precludes the use of the standard asymptotic and bootstrap methods for bias correction. We propose a bootstrap method that generates samples backward in time and approximates the median function of the predictive distribution on a grid of values for the nuisance parameter. The method can be easily adapted to approximate any quantile of the conditional predictive distribution.     

A note on finite sample breakdown points of projection based multivariate location and scatter statistics

In this note, general results of finite sample breakdown point are obtained for two classes of projection based location and scatter statistics: the Stahel-Donoho statistics and the Maronna-Yohai statistics. It is shown that these projection based location and scatter statistics can achieve the maximum breakdown point of affine equivariant multivariate location and scatter statistics. General relationships between the finite sample breakdown point of these statistics and the uniform finite sample breakdown point of the sample median and a modified sample median absolute deviation are formally established. Received: May 1999     

Unit root testing via the stationary bootstrap

A nonparametric, residual-based stationary bootstrap procedure is proposed for unit root testing in a time series. The procedure generates a pseudoseries which mimics the original, but ensures the presence of a unit root. Unlike many others in the literature, the proposed test is valid for a wide class of weakly dependent processes and is not based on parametric assumptions on the data-generating process. Large sample theory is developed and asymptotic validity is shown via a bootstrap functional central limit theorem. The case of a least squares statistic is discussed in detail, including simulations to investigate the procedure's finite sample performance.     

On bootstrapping panel factor series

This paper studies the asymptotic validity of sieve bootstrap for nonstationary panel factor series. Two main results are shown. Firstly, a bootstrap Invariance Principle is derived pointwise in i$i$, obtaining an upper bound for the order of truncation of the AR polynomial that depends on n$n$ and T$T$. Consistent estimation of the long run variances is also studied for (n,T)→∞$(n,T)\to \infty$. Secondly, joint bootstrap asymptotics is also studied, investigating the conditions under which the bootstrap is valid. In particular, the extent of cross sectional dependence which can be allowed for is investigated. Whilst we show that, for general forms of cross dependence, consistent estimation of the long run variance (and therefore validity of the bootstrap) is fraught with difficulties, however we show that “one-cross-sectional-unit-at-a-time” resampling schemes yield valid bootstrap based inference under weak forms of cross-sectional dependence.     

BOOTSTRAPPING THE TRACE STATISTIC IN VAR MODELS: MONTE CARLO RESULTS AND APPLICATIONS

ABSTRACT This paper investigates through Monte Carlo experiments both size and power properties of a bootstrapped trace statistic in two prototypical DGPs. The Monte Carlo results indicate that the ordinary bootstrap has similar size and power properties as inference procedures based on asymptotic critical values. Considering empirical size, the stationary bootstrap is found to provide a uniform improvement over the ordinary bootstrap if the dynamics is underspecified. The use of the stationary bootstrap as a diagnostic tool is suggested. In two illustrative examples this seems to work, and again it appears that the bootstrap incorporates the finite-sample correction required for the asymptotic critical values to apply.     

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.     

Bootstrapping sequential change-point tests for linear regression

Bootstrap methods for sequential change-point detection procedures in linear regression models are proposed. The corresponding monitoring procedures are designed to control the overall significance level. The bootstrap critical values are updated constantly by including new observations obtained from the monitoring. The theoretical properties of these sequential bootstrap procedures are investigated, showing their asymptotic validity. Bootstrap and asymptotic methods are compared in a simulation study, showing that the studentized bootstrap tests hold the overall level better especially for small historic sample sizes while having a comparable power and run length.     

Nonparametric tests for conditional symmetry in dynamic models

This article proposes omnibus tests for conditional symmetry around a parametric function in a dynamic context. Conditional moments may not exist or may depend on the explanatory variables. Test statistics are suitable functionals of the empirical process of residuals and explanatory variables, whose limiting distribution under the null is nonpivotal. The tests are implemented with the assistance of a bootstrap method, which is justified assuming very mild regularity conditions on the specification of the center of symmetry and the underlying serial dependence structure. Finite sample properties are examined by means of a Monte Carlo experiment.     

Bootstrapping cointegrating regressions

In this paper, we consider bootstrapping cointegrating regressions. It is shown that the method of bootstrap, if properly implemented, generally yields consistent estimators and test statistics for cointegrating regressions. For the cointegrating regression models driven by general linear processes, we employ the sieve bootstrap based on the approximated finite-order vector autoregressions for the regression errors and the first differences of the regressors. In particular, we establish the bootstrap consistency for OLS method. The bootstrap method can thus be used to correct for the finite sample bias of the OLS estimator and to approximate the asymptotic critical values of the OLS-based test statistics in general cointegrating regressions. The bootstrap OLS procedure, however, is not efficient. For the efficient estimation and hypothesis testing, we consider the procedure proposed by Saikkonen [1991. Asymptotically efficient estimation of cointegration regressions. Econometric Theory 7, 1–21] and Stock and Watson [1993. A simple estimator of cointegrating vectors in higher order integrating systems. Econometrica 61, 783–820] relying on the regression augmented with the leads and lags of differenced regressors. The bootstrap versions of their procedures are shown to be consistent, and can be used to do asymptotically valid inferences. A Monte Carlo study is conducted to investigate the finite sample performances of the proposed bootstrap methods.