Heteroskedasticity Of Unknown Form Research Articles

Tests for regression models with heteroskedasticity of unknown form

Evidence is presented on the finite sample performance of tests that are robust to heteroskedasticity. In contrast to previous work, the focus is on testing several restrictions on the coefficients of a linear regression model, rather than on a quasi- t test of a single restriction. Tests based upon different forms of a heteroskedasticity-consistent covariance matrix estimator are examined, as are the relative merits of asymptotic and wild bootstrap critical values. As an alternative to such tests, procedures using the classical F statistic are investigated. These procedures use single and double wild bootstraps to assess the significance of the F statistic. The costs of using heteroskedasticity-robust tests when the errors are actually homoskedastic are discussed.

ROBUST ASYMPTOTIC INFERENCE IN AUTOREGRESSIVE MODELS WITH MARTINGALE DIFFERENCE ERRORS

This paper proposes a GMM-based method for asymptotic confidence interval construction in stationary autoregressive models, which is robust to the presence of conditional heteroskedasticity of unknown form. The confidence regions are obtained by inverting the asymptotic acceptance region of the distance metric test for the continuously updated GMM (CU-GMM) estimator. Unlike the predetermined symmetric shape of the Wald confidence intervals, the shape of the proposed confidence intervals is data-driven owing an estimated sequence of nonuniform weights. It appears that the flexibility of the CU-GMM estimator in downweighting certain observations proves advantageous for confidence interval construction. This stands in contrast to some other generalized empirical likelihood estimators with appealing optimality properties such as the empirical likelihood estimator whose objective function prevents such downweighting. A Monte Carlo simulation study illustrates the excellent small-sample properties of the method for AR models with ARCH errors. The procedure is applied to study the dynamics of the federal funds rate.

EFFICIENCY GAINS OF SYSTEM GMM AND MDE OVER INDIVIDUAL EQUATION ESTIMATION*

In the econometric literature it is known that, under certain conditions, estimating a system of equations together is more efficient than estimating each equation separately. This finding has been proved, however, only under the assumption of a known parametric form of heteroskedasticity (including homoskedasticity) or non-random regressors/instru-ments. This note shows that an analogous finding holds for GMM under heteroskedasticity of unknown form and random regressors/instruments. Specifically, I provide a necessary condition for the efficiency gain of the system GMM over the single-equation GMM. An analogous necessary condition for the efficiency gain is also shown to hold for minimum-distance (or X2) estimation (MDE).

Leverage-adjusted heteroskedastic bootstrap methods

The weighted bootstrap is a bootstrapping scheme that is valid under heteroskedasticity of unknown form. It is often used in linear regression analysis to obtain variance estimates of the ordinary least squares estimators of the linear parameters, especially when heteroskedasticity is suspected to be present in the data. Monte Carlo studies have shown, however, that its finite-sample performance can be poor when the data include points of high leverage. In this paper, we propose alternative bootstrap methods that take into the account the effect of possibly influential observations on the resulting inference via quasi-t tests. Our numerical results show that some of the proposed bootstrapping schemes generally outperform the weighted bootstrap when there are points of high leverage in the data. It is also shown that the potential gains from using the bootstrap inference we propose can be substantial. An application is presented.

Econometric Computing with HC and HAC Covariance Matrix Estimators

This introduction to the R package sandwich is a (slightly) modified version of Zeileis (2004a), published in the Journal of Statistical Software. Data described by econometric models typically contains autocorrelation and/or heteroskedasticity of unknown form and for inference in such models it is essential to use covariance matrix estimators that can consistently estimate the covariance of the model parameters. Hence, suitable heteroskedasticity-consistent (HC) and heteroskedasticity and autocorrelation consistent (HAC) estimators have been receiving attention in the econometric literature over the last 20 years. To apply these estimators in practice, an implementation is needed that preferably translates the conceptual properties of the underlying theoretical frameworks into computational tools. In this paper, such an implementation in the package sandwich in the R system for statistical computing is described and it is shown how the suggested functions provide reusable components that build on readily existing functionality and how they can be integrated easily into new inferential procedures or applications. The toolbox contained in sandwich is extremely flexible and comprehensive, including specific functions for the most important HC and HAC estimators from the econometric literature. Several real-world data sets are used to illustrate how the functionality can be integrated into applications.

Testing for Autocorrelation and Unit Roots in the Presence of Conditional Heteroskedasticity of Unknown Form

The presence of conditional heteroskedasticity invalidates standard autocorrelation tests such as the Durbin-Watson statistic and its many variants, and reduces the power of standard unit root tests like the Dickey-Fuller test. This paper addresses the problem of testing for AR(1) and AR(p) alternatives in the presence of conditional heteroskedasticity, constructs new autocorrelation tests (called g- and gl- tests), and derives an asymptotic theory for the new statistics. The g test is shown to be robust to any form of conditional heteroskedasticity, while the gl test is robust to the joint presence of conditional heteroskedasticity and lagged dependent variables. The paper also develops a GLS-type unit root test that allows for conditional heteroskedasticity. Its limit distribution turns out to be a convex mixture of the Dickey-Fuller distribution and the standard normal distribution. It is shown that the GLS statistic improves the power of the Dickey-Fuller test under condictional heteroskedasticity. Some Monte Carlo simulations with the new tests are reported.

Hedonic price index estimation under mean‐independence of time dummies from quality characteristics

Summary. We estimate hedonic price indices (HPI) for rental offices in Tokyo for the period 1985–1991. We take a partially linear regression (PLR) model, linear in x (year dummies) and nonparametric in z (office quality characteristics), as our main model; the usual linear model is used as well. Since x consists of year dummies, the linearity in x is not a restriction in the PLR model; the only restriction is that of no interaction between x and z. For the PLR model, the HPI are estimated -consistently with a two-stage procedure. For our data, x turns out to be (almost) mean-independent of z. This implies that least squares estimation (LSE) for models with a misspecified function for z is still consistent. The mean-independence also leads to an efficiency result that, under heteroskedasticity of unknown form, the two-stage PLR model estimator is at least as efficient as any LSE for models specifying (rightly or wrongly) the part for z. In addition to these, several interesting practical lessons are noted in doing the two-stage PLR model estimation. First, the cross validation (CV) used in the PLR model literature can fail if the mean-independence is ignored. Second, high order kernels can make the CV criterion function ill behaved. Third, product kernels work as well as spherically symmetric kernels. Fourth, nonparametric specification tests may work poorly due to a sample splitting problem with outliers in the data or due to choosing more than one bandwidth; in this regard, a test suggested by Stute (1997) and Stute et al. (1998) is recommended.

Asymptotic inference under heteroskedasticity of unknown form

We focus on the finite-sample behavior of heteroskedasticity-consistent covariance matrix estimators and associated quasi- t tests. The estimator most commonly used is that proposed by Halbert White. Its finite-sample behavior under both homoskedasticity and heteroskedasticity is analyzed using Monte Carlo methods. We also consider two other consistent estimators, namely: the HC3 estimator, which is an approximation to the jackknife estimator, and the weighted bootstrap estimator. Additionally, we evaluate the finite-sample behavior of two bootstrap quasi- t tests: the test based on a single bootstrapping scheme and the test based on a double, nested bootstrapping scheme. The latter is very computer-intensive, but proves to work well in small samples. Finally, we propose a new estimator, which we call HC4; it is tailored to take into account the effect of leverage points in the design matrix on associated quasi- t tests.

ASYMPTOTICALLY EFFICIENT MEDIAN REGRESSION IN THE PRESENCE OF HETEROSKEDASTICITY OF UNKNOWN FORM

We consider a linear model with heteroskedasticity of unknown form. Using Stone's (1977, Annals of Statistics 5, 595–645) k nearest neighbors (k-NN) estimation approach, the optimal weightings for efficient least absolute deviation regression are estimated consistently using residuals from preliminary estimation. The reweighted least absolute deviation or median regression estimator with the estimated weights is shown to be equivalent to the estimator using the true but unknown weights under mild conditions. Asymptotic normality of the estimators is also established. In the finite sample case, the proposed estimators are found to outperform the generalized least squares method of Robinson (1987, Econometrica 55, 875–891) and the one-step estimator of Newey and Powell (1990, Econometric Theory 6, 295–317) based on a Monte Carlo simulation experiment.

Simple Robust Testing of Regression Hypotheses

IN THIS PAPER WE CONSIDER the problem of hypothesis testing in models with errors that have serial correlation or heteroskedasticity of unknown form. This situation is often encountered in regression models applied to economic time series data. It is a classic textbook result that while ordinary least squares (OLS) estimates of regression parameters remain consistent and asymptotically normal when errors are heteroskedastic or autocorrelated (provided usual regularity conditions hold and no lagged dependent variables are in the model), standard tests are no longer valid. If the true form of serial correlation/heteroskedasticity were known, then generalized least squares (GLS) provides efficient estimates and standard inference can be conducted on the GLS transformed model. But, in practice the form of serial correlation/heteroskedasticity is often unknown, and this has led to the development of techniques that permit valid asymptotic inference without having to specify a model of the serial correlation or heteroskedasticity. The most common approach is to estimate the variance-covariance matrix of the OLS estimates nonparametrically using spectral methods (heteroskedasticity and autocorrelation consistent (HAC) estimators) and construct standard tests using the asymptotic normality of the OLS estimates. HAC estimators have been extensively analyzed in the econometrics literature and important contributions are given by Andrews (1991), Andrews and Monahan (1992), Gallant (1987), Hansen (1992), Newey and West (1987), Robinson (1991, 1998), and White (1984) among others. The benefit of HAC estimator tests is asymptotically valid inference that is robust to general forms of serial correlation/heteroskedasticity in the errors. We propose an alternative method of constructing robust test statistics. We apply a nonsingular data dependent stochastic transformation to the OLS estimates. The asymptotic distribution of the transformed estimates does not depend on nuisance parameters. Then, test statistics that are asymptotically invariant to nuisance parameters (asymptotic pivotal statistics) are constructed. The resulting test statistics have nonstandard asymptotic distributions that only depend on the number of restrictions being tested, and critical values are easy to simulate using standard techniques. The main advantage of our approach compared to standard approaches is that estimates of the variance-covariance matrix are not explicitly required to construct the tests. This is potentially important for two reasons. First, with the exception of the estimator proposed by Robinson (1998),2 consistent nonparametric estimates of variance-covariance matrices in models with serial

A TEST OF AUTOCORRELATION IN THE PRESENCE OF HETEROSKEDASTICITY OF UNKNOWN FORM

This paper develops a test of autocorrelation in the presence of heteroskedasticity of unknown form in the nonlinear regression model. The test statistic is based on the sample autocovariance of the residuals standardized by a nonparametric kernel estimate of the unknown heteroskedasticity function. Under appropriate conditions, the test statistic is shown to have a limiting chi-square distribution. Local power and consistency results for the test are also established. Monte Carlo experiments show that the test has reasonable size performance and generally dominates some of the existing tests in terms of finite-sample power.

Count Data Models with Variance of Unknown Form: An Application to a Hedonic Model of Worker Absenteeism

We examine an econometric model of counts of worker absences due to illness in a sluggishly adjusting hedonic labor market. We compare three estimators that parameterize the conditional variance—least squares, Poisson, and negative binomial pseudo maximum likelihood—to generalized least squares (GLS) using nonparametric estimates of the conditional variance. Our data support the hedonic absenteeism model. Semiparametric GLS coefficients are similar in sign, magnitude, and statistical significance to coefficients where the mean and variance of the errors are specified ex ante. In our data, coefficient estimates are sensitive to a regressor list but not to the econometric technique, including correcting for possible heteroskedasticity of unknown form.

A Robust Heteroskedasticity Consistent Covariance Matrix Estimator

To deal with heteroskedasticity of unknown form, this paper suggests to robustly estimate the regression coefficients and then to implement an heteroskedasticity consistent covariance matrix estimator. The robust regression reduces the sample bias of the heteroskedasticity consistent covariance matrix estimator, and does not require the specification of a functional form for heteroskedasticity. Indeed, it replaces each variance with the squared errors, which are in turn estimated by the squared residuals from the robust regression. A Monte-Carlo study verifies the behavior of the proposed estimator, using real data.

Second order approximation in a linear regression with heteroskedasticity of unknown form

We develop stochastic expansions with remainder , where 0<μ<1/2, for a standardised semiparametric GLS estimator, a standard error, and a studentized statistic, in the linear regression model with heteroskedasticity of unknown form. We calculate the second moments of the truncated expansion, and use these approximations to compare two competing estimators and to define a method of bandwidth choice.

Two-step estimation of heteroskedastic sample selection models

This paper considers two-step estimation of a sample selection model in which there is heteroskedasticity of unknown form in the latent errors. We propose an estimator which uses recent developments in nonparametric regression estimation involving series approximations. The estimator is shown to be consistent and asymptotically normally distributed under reasonable conditions. A small Monte Carlo experiment demonstrates the usefulness of the estimator and highlights the bias inherent in the usual Heckman (1979) estimator when there is heteroskedasticity.

Symmetrically Trimmed Least Squares Estimation for Tobit Models

This paper proposes alternatives to maximum likelihood estimation of the censored and truncated regression models (known to economists as Tobit models). The proposed estimators are based upon symmetric censoring or truncation of the upper tail of the distribution of the dependent variable. Unlike methods based on the assumption of identically distributed Gaussian errors, the estimators are semiparametric, in the sense that they are consistent and asymptotically normal for a wide class of (symmetric) error distributions with heteroskedasticity of unknown form. The paper gives the regularity conditions and proofs of these large sample properties, demonstrates how to construct consistent estimators of the asymptotic covariance matrices, and presents the results of a simulation study for the censored case. Extensions and limitations of the approach are also considered.