Asymptotic Refinements Research Articles

Asymptotic refinements of a misspecification-robust bootstrap for generalized method of moments estimators

I propose a nonparametric iid bootstrap that achieves asymptotic refinements for t tests and confidence intervals based on GMM estimators even when the model is misspecified. In addition, my bootstrap does not require recentering the moment function, which has been considered as critical for GMM. Regardless of model misspecification, the proposed bootstrap achieves the same sharp magnitude of refinements as the conventional bootstrap methods which establish asymptotic refinements by recentering in the absence of misspecification. The key idea is to link the misspecified bootstrap moment condition to the large sample theory of GMM under misspecification of Hall and Inoue (2003). Two examples are provided: combining data sets and invalid instrumental variables.

Asymptotic Refinements of a Misspecification-Robust Bootstrap for Generalized Method of Moments Estimators

I propose a nonparametric iid bootstrap that achieves asymptotic refinements for t tests and confidence intervals based on the generalized method of moments (GMM) estimators even when the model is misspecified. In addition, my bootstrap does not require recentering the bootstrap moment function, which has been considered as critical for GMM. Regardless of model misspecification, the proposed bootstrap achieves the same sharp magnitude of refinements as the conventional bootstrap methods which establish asymptotic refinements by recentering in the absence of misspecification. The key idea is to link the misspecified bootstrap moment condition to the large sample theory of GMM under misspecification of Hall and Inoue (2003, Journal of Econometrics 114, 361-394). Examples of possibly misspecified moment condition models with Monte Carlo simulation results are provided: (i) Combining data sets, and (ii) invalid instrumental variables.

The Block-Block Bootstrap for Time Series

This article proves that the block-block bootstrap of Andrews (2004) can be helpful to provide asymptotic refinements for the GMM estimator when autocorrelation structures of moment functions are unknown (i.e., incorporating the HAC covariance matrix) and when we allow for statistics that are inefficient. The asymptotic refinements of this block-block bootstrap in the time series context are shown to exist with the use of less restricted kernels than in the block bootstrap in Inoue and Shintani (2006), since they do not require to have a characteristic exponent larger than 2. The procedure allows to apply in practice kernels that guarantee that the HAC covariance matrix estimator is positive semidefinite, and to get asymptotic refinements at the same time.

Nonparametric Bootstrap for Quasi-Likelihood Ratio Tests

We introduce a nonparametric block bootstrap approach for Quasi-Likelihood Ratio type tests of nonlinear restrictions. Our method applies to extremum estimators, such as quasi-maximum likelihood and generalized method of moments estimators. Unlike existing parametric bootstrap procedures for Quasi-Likelihood Ratio type tests, our procedure constructs bootstrap samples in a fully nonparametric way. We study the higher order properties of our nonparametric block bootstrap and show the asymptotic refinements implied with respect to the standard asymptotic theory. Our approach delivers the same higher order properties of the nonparametric block bootstrap methods introduced in Andrews (2002) and Kim (2003) in relation to Wald and Lagrange Multiplier tests, respectively. Monte Carlo simulations confirm the accuracy of our bootstrap procedure.

Guest Editorial: Workshop on Bayesian Econometric Methods

3rd RCEA Bayesian workshop was held in Rimini in July, 2009. workshop opened with a debate, chaired by Gael Martin, between a leading Bayesian econometrician, Christian Robert, and a leading classical econometrician, Russell Davidson, on the relative virtues of the Bayesian and classical (or frequentist) approaches. pleasantly civil debate was conducted under the topic The 21st Century Belongs to Bayes. workshop also brought together a variety of both classical and Bayesian econometricians and statisticians, with a view to participants exchanging information on developments in their specific fields of research. two papers of this volume one classical in approach and one Bayesian, with insights into classical approaches partially reflect this purpose, with many more papers having been presented at the workshop itself. In his paper, Russell Davidson investigates issues in bootstrap testing using, as an example, testing for a unit root in an autoregressive moving average (ARMA) (1,1) process. He focuses on the situation when the root of the MA polynomial is close to minus one. Size distortions in the bootstrap tests result when testing in this situation, due to the near cancellation of the unit AR root with the MA root. Davidson proposes estimators based on nonlinear least squares that are faster to compute than, but not quite as efficient as, the maximum likelihood estimator. These estimators are slower to compute than those proposed by Galbraith and Zinde-Walsh (1994) and (1997), but far more efficient than the Galbraith and Zinde-Walsh estimators. Further, Davidson proposes a new bootstrap procedure that is computationally less demanding than the double bootstrap of Beran (1988). Davidson produces all of his results without recourse to asymptotic theory or asymptotic refinements of the testing procedure. Polasek, Sellner and Llano consider the problem of estimation and prediction in spatial models when data measurements are taken at different degrees of aggregation and where some observations are missing at one or more levels of aggregation. They adapt the procedure of Chow and Lin (1971), which was developed for time series data with observations on

Bootstrapping Realized Volatility

We propose bootstrap methods for a general class of nonlinear transformations of realized volatility which includes the raw version of realized volatility and its logarithmic transformation as special cases. We consider the independent and identically distributed (i.i.d.) bootstrap and the wild bootstrap (WB), and prove their first-order asymptotic validity under general assumptions on the log-price process that allow for drift and leverage effects. We derive Edgeworth expansions in a simpler model that rules out these effects. The i.i.d. bootstrap provides a second-order asymptotic refinement when volatility is constant, but not otherwise. The WB yields a second-order asymptotic refinement under stochastic volatility provided we choose the external random variable used to construct the WB data appropriately. None of these methods provides third-order asymptotic refinements. Both methods improve upon the first-order asymptotic theory in finite samples.

Accurately sized test statistics with misspecified conditional homoskedasticity

We study the finite-sample performance of test statistics in linear regression models where the error dependence is of unknown form. With an unknown dependence structure, there is traditionally a trade-off between the maximum lag over which the correlation is estimated (the bandwidth) and the amount of heterogeneity in the process. When allowing for heterogeneity, through conditional heteroskedasticity, the correlation at far lags is generally omitted and the resultant inflation of the empirical size of test statistics has long been recognized. To allow for correlation at far lags, we study the test statistics constructed under the possibly misspecified assumption of conditional homoskedasticity. To improve the accuracy of the test statistics, we employ the second-order asymptotic refinement in Rothenberg [Approximate power functions for some robust tests of regression coefficients, Econometrica 56 (1988), pp. 997–1019] to determine the critical values. The simulation results of this paper suggest that when sample sizes are small, modelling the heterogeneity of a process is secondary to accounting for dependence. We find that a conditionally homoskedastic covariance matrix estimator (when used in conjunction with Rothenberg's second-order critical value adjustment) improves test size with only a minimal loss in test power, even when the data manifest significant amounts of heteroskedasticity. In some specifications, the size inflation was cut by nearly 40% over the traditional heteroskedasticity and autocorrelation consistent (HAC) test. Finally, we note that the proposed test statistics do not require that the researcher specify the bandwidth or the kernel.

Bootstrap-Based Improvements for Inference with Clustered Errors

Abstract Researchers have increasingly realized the need to account for within-group dependence in estimating standard errors of regression parameter estimates. The usual solution is to calculate cluster-robust standard errors that permit heteroskedasticity and within-cluster error correlation, but presume that the number of clusters is large. Standard asymptotic tests can over-reject, however, with few (five to thirty) clusters. We investigate inference using cluster bootstrap-t procedures that provide asymptotic refinement. These procedures are evaluated using Monte Carlos, including the example of Bertrand, Duflo, and Mullainathan (2004). Rejection rates of 10% using standard methods can be reduced to the nominal size of 5% using our methods.

Bootstrap-Based Improvements for Inference with Clustered Errors

Microeconometrics researchers have increasingly realized the essential need to account for any within-group dependence in estimating standard errors of regression parameter estimates. The typical preferred solution is to calculate cluster-robust or sandwich standard errors that permit quite general heteroskedasticity and within-cluster error correlation, but presume that the number of clusters is large. In applications with few (5-30) clusters, standard asymptotic tests can over-reject considerably. We investigate more accurate inference using cluster bootstrap-t procedures that provide asymptotic refinement. These procedures are evaluated using Monte Carlos, including the much-cited di¤erences-in-di¤erences example of Bertrand, Mullainathan and Duflo (2004). In situations where standard methods lead to rejection rates in excess of ten percent (or more) for tests of nominal size 0:05, our methods can reduce this to five percent. In principle a pairs cluster bootstrap should work well, but in practice a Wild cluster bootstrap performs better.

A bootstrap theory for weakly integrated processes

Abstract This paper develops a bootstrap theory for models including autoregressive time series with roots approaching to unity as the sample size increases. In particular, we consider the processes with roots converging to unity with rates slower than n - 1 . We call such processes weakly integrated processes. It is established that the bootstrap relying on the estimated autoregressive model is generally consistent for the weakly integrated processes. Both the sample and bootstrap statistics of the weakly integrated processes are shown to yield the same normal asymptotics. Moreover, for the asymptotically pivotal statistics of the weakly integrated processes, the bootstrap is expected to provide an asymptotic refinement and give better approximations for the finite sample distributions than the first order asymptotic theory. For the weakly integrated processes, the magnitudes of potential refinements by the bootstrap are shown to be proportional to the rate at which the root of the underlying process converges to unity. The order of boostrap refinement can be as large as o ( n - 1 / 2 + ϵ ) for any ϵ > 0 . Our theory helps to explain the actual improvements observed by many practitioners, which are made by the use of the bootstrap in analyzing the models with roots close to unity.

Bootstrapping GMM estimators for time series

This paper considers the bootstrap for the GMM estimator of overidentified linear models when autocorrelation structures of moment functions are unknown. When moment functions are uncorrelated after finite lags, Hall and Horowitz, [1996. Bootstrap critical values for tests based on generalized method of moments estimators. Econometrica 64, 891–916] showed that errors in the rejection probabilities of the bootstrap tests are o(T-1). However, this rate cannot be obtained with the HAC covariance matrix estimator since it converges at a nonparametric rate. By incorporating the HAC covariance matrix estimator in the Edgeworth expansion of the distribution, we show that the bootstrap provides asymptotic refinements when the characteristic exponent of the kernel function is greater than two.

The Block-Block Bootstrap: Improved Asymptotic Refinements

The asymptotic refinements attributable to the block bootstrap for time series are not as large as those of the nonparametric iid bootstrap or the parametric bootstrap. One reason is that the independence between the blocks in the block bootstrap sample does not mimic the dependence structure of the original sample. This is the join-point problem. In this paper, we propose a method of solving this problem. The idea is not to alter the block bootstrap. Instead, we alter the original sample statistics to which the block bootstrap is applied. We introduce block statistics that possess join-point features that are similar to those of the block bootstrap versions of these statistics. We refer to the application of the block bootstrap to block statistics as the block–block bootstrap. The asymptotic refinements of the block–block bootstrap are shown to be greater than those obtained with the block bootstrap and close to those obtained with the nonparametric iid bootstrap and parametric bootstrap.

Bootstrap Unit Root Tests

We consider the bootstrap unit root tests based on finite order autoregressive integrated models driven by iid innovations, with or without deterministic time trends. A general methodology is developed to approximate asymptotic distributions for the models driven by integrated time series, and used to obtain asymptotic expansions for the Dickey-Fuller unit root tests. The second-order terms in their expansions are of stochastic orders O p (n -1/4 ) and O p (n -1/2 ), and involve functionals of Brownian motions and normal.random variates. The asymptotic expansions for the bootstrap tests are also derived and compared with those of the Dickey-Fuller tests. We show in particular that the bootstrap offers asymptotic refinements for the Dickey-Fuller tests, i.e., it corrects their second-order errors. More precisely, it is shown that the critical values obtained by the bootstrap resampling are correct up to the second-order terms, and the errors in rejection probabilities are of order o(n- 1/2 ) if the tests are based upon the bootstrap critical values. Through simulations, we investigate how effective is the bootstrap correction in small samples.

Higher-order asymptotic refinements for score tests in proper dispersion models

Abstract This paper develops second-order asymptotic theory for score tests in proper dispersion models without imposing known dispersion. Our results can be used to provide an Edgeworth expansion for the test statistic or to obtain a Bartlett-corrected statistic. The latter is a modified version of the original statistic distributed as chi-squared with an error of order O (n −3/2 ) , n being the sample size, which is an improvement over Rao's original score statistic which has a chi-squared distribution with error of order O (n −1 ) . We also show that the formulae we obtain generalize a number of previously published results.

Bootstrap Methods for Median Regression Models

The least-absolute-deviations (LAD) estimator for a median-regression model does not satisfy the standard conditions for obtaining asymptotic refinements through use of the bootstrap because the LAD objective function is not smooth. This paper overcomes this problem by smoothing the objective function. The smoothed estimator is asymptotically equivalent to the standard LAD estimator. With bootstrap critical values, the rejection probabilities of symmetrical t and X 2 tests based on the smoothed estimator are correct through O(n -γ ) under the null hypothesis, where γ<1 but can be arbitrarily close to 1. In contrast, first-order asymptotic approximations make errors of size O(n -γ ). These results also hold for symmetrical t and X 2 tests for censored median regression models.

Bootstrap Critical Values for Tests Based on Generalized-Method-of-Moments Estimators

Monte Carlo experiments have shown that tests based on generalized-method-ofmoments estimators often have true levels that differ greatly from their nominal levels when asymptotic critical values are used. This paper gives conditions under which the bootstrap provides asymptotic refinements to the critical values of t tests and the test of overidentifying restrictions. Particular attention is given to the case of dependent data. It is shown that with such data, the bootstrap must sample blocks of data and that the formulae for the bootstrap versions of test statistics differ from the formulae that apply with the original data. The results of Monte Carlo experiments on the numerical performance of the bootstrap show that it usually reduces the errors in level that occur when critical values based on first-order asymptotic theory are used. The bootstrap also provides an indication of the accuracy of critical values obtained from first-order asymptotic theory.

Discussion of the Papers by Hinkley and Diciccio and Romano

Discussion of the Papers by Hinkley and Diciccio and Romano