Inference Under Heteroskedasticity and Leveraged Data

We evaluate the finite-sample behavior of different heteros-ke-das-ticity-consistent covariance matrix estimators, under both constant and unequal error variances. We consider the estimator proposed by Halbert White (HC0), and also its variants known as HC2, HC3, and HC4; the latter was recently proposed by Cribari-Neto (2004 Cribari-Neto , F. ( 2004 ). Asymptotic inference under heteroskedasticity of unknown form . Computat. Statist. Data Anal. 45 : 215 – 233 .[Crossref], [Web of Science ®] , [Google Scholar]). We propose a new covariance matrix estimator: HC5. It is the first consistent estimator to explicitly take into account the effect that the maximal leverage has on the associated inference. Our numerical results show that quasi-t inference based on HC5 is typically more reliable than inference based on other covariance matrix estimators.     

A Note on Bayesian Analyses of Capture–Recapture Data with Perfect Recaptures

The present article deals with the problem of misspecifying the disturbance-covariance matrix as scalar, when it is locally non scalar. We consider a family of shrinkage estimators based on OLS estimator and compare its asymptotic properties with the properties of OLS estimator. We proposed a similar family of estimators based on FGLS and compared its asymptotic properties with the shrinkage estimator based on OLS under a Pitman's drift process. The effect of misspecifying the disturbances covariance matrix was analyzed with the help of a numerical simulation.     

A general central limit theorem and a subsampling variance estimator for α‐mixing point processes

We establish a central limit theorem for multivariate summary statistics of nonstationary α‐mixing spatial point processes and a subsampling estimator of the covariance matrix of such statistics. The central limit theorem is crucial for establishing asymptotic properties of estimators in statistics for spatial point processes. The covariance matrix subsampling estimator is flexible and model free. It is needed, for example, to construct confidence intervals and ellipsoids based on asymptotic normality of estimators. We also provide a simulation study investigating an application of our results to estimating functions.     

Selection of Variables in Multivariate Regression Models for Large Dimensions

The Akaike information criterion, AIC, and Mallows’ C _p statistic have been proposed for selecting a smaller number of regressors in the multivariate regression models with fully unknown covariance matrix. All of these criteria are, however, based on the implicit assumption that the sample size is substantially larger than the dimension of the covariance matrix. To obtain a stable estimator of the covariance matrix, it is required that the dimension of the covariance matrix is much smaller than the sample size. When the dimension is close to the sample size, it is necessary to use ridge-type estimators for the covariance matrix. In this article, we use a ridge-type estimators for the covariance matrix and obtain the modified AIC and modified C _p statistic under the asymptotic theory that both the sample size and the dimension go to infinity. It is numerically shown that these modified procedures perform very well in the sense of selecting the true model in large dimensional cases.     

A Class of Improved Heteroskedasticity-Consistent Covariance Matrix Estimators

Abstract

The heteroskedasticity-consistent covariance matrix estimator proposed by White [White, H. A. (1980 White, H. A. 1980. Heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48: 817–838. [Crossref], [Web of Science ®] , [Google Scholar]). Heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 48:817–838], also known as HC0, is commonly used in practical applications and is implemented into a number of statistical software. Cribari–Neto et al. [Cribari–Neto, F., Ferrari, S. L. P., Cordeiro, G. M. (2000 Cribari–Neto, F., Ferrari, S. L. P. and Cordeiro, G. M. 2000. Improved heteroscedasticity–consistent covariance matrix estimators. Biometrika, 87: 907–918. [Crossref], [Web of Science ®] , [Google Scholar]). Improved heteroscedasticity–consistent covariance matrix estimators. Biometrika 87:907–918] have developed a bias-adjustment scheme that delivers bias-corrected White estimators. There are several variants of the original White estimator that are also commonly used by practitioners. These include the HC1, HC2, and HC3 estimators, which have proven to have superior small-sample behavior relative to White's estimator. This paper defines a general bias-correction mechamism that can be applied not only to White's estimator, but to variants of this estimator as well, such as HC1, HC2, and HC3. Numerical evidence on the usefulness of the proposed corrections is also presented. Overall, the results favor the sequence of improved HC2 estimators.     

Feasible robust estimator in restricted semiparametric regression models based on the LTS approach

Under some nonstochastic linear restrictions based on either additional information or prior knowledge in a semiparametric regression model, a family of feasible generalized robust estimators for the regression parameter is proposed. The least trimmed squares (LTS) method proposed by Rousseeuw as a highly robust regression estimator is a statistical technique for fitting a regression model based on the subset of h observations (out of n) whose least-square fit possesses the smallest sum of squared residuals. The coverage h may be set between n/2 and n. The LTS estimator involves computing the hyperplane that minimizes the sum of the smallest h squared residuals. For practical purpose, it is assumed that the covariance matrix of the error term is unknown and thus feasible estimators are replaced. Then, we develop an algorithm for the LTS estimator based on feasible methods. Through the Monte Carlo simulation studies and a real data example, performance of the feasible type of robust estimators is compared with the classical ones in restricted semiparametric regression models.     

Adaptive robust estimation in joint mean–covariance regression model for bivariate longitudinal data

The estimation of the covariance matrix is important in the analysis of bivariate longitudinal data. A good estimator for the covariance matrix can improve the efficiency of the estimators of the mean regression coefficients. Furthermore, the covariance estimation itself is also of interest, but it is a challenging job to model the covariance matrix of bivariate longitudinal data due to the complex structure and positive definite constraint. In addition, most of existing approaches are based on the maximum likelihood, which is very sensitive to outliers or heavy-tail error distributions. In this article, an adaptive robust estimation method is proposed for bivariate longitudinal data. Unlike the existing likelihood-based methods, the proposed method can adapt to different error distributions. Specifically, at first, we utilize the modified Cholesky block decomposition to parameterize the covariance matrices. Secondly, we apply the bounded Huber's score function to develop a set of robust generalized estimating equations to estimate the parameters both in the mean and the covariance models simultaneously. A data-driven approach is presented to select the parameter c in the Huber's score function, which can ensure that the proposed method is robust and efficient. A simulation study and a real data analysis are conducted to illustrate the robustness and efficiency of the proposed approach.     

Approximate inference in heteroskedastic regressions: A numerical evaluation

The commonly made assumption that all stochastic error terms in the linear regression model share the same variance (homoskedasticity) is oftentimes violated in practical applications, especially when they are based on cross-sectional data. As a precaution, a number of practitioners choose to base inference on the parameters that index the model on tests whose statistics employ asymptotically correct standard errors, i.e. standard errors that are asymptotically valid whether or not the errors are homoskedastic. In this paper, we use numerical integration methods to evaluate the finite-sample performance of tests based on different (alternative) heteroskedasticity-consistent standard errors. Emphasis is placed on a few recently proposed heteroskedasticity-consistent covariance matrix estimators. Overall, the results favor the HC4 and HC5 heteroskedasticity-robust standard errors. We also consider the use of restricted residuals when constructing asymptotically valid standard errors. Our results show that the only test that clearly benefits from such a strategy is the HC0 test.     

Estimation of the Length Distribution of Marine Populations in the Gaussian-multinomial Setting using the Method of Moments

In this paper, the problem of estimation of the length distribution of marine populations in the Gaussian-multinomial model is considered. For the purpose of the mean and covariance parameter estimation, the method of moments estimators are developed. That is, minimum variance linear unbiased estimator for the mean frequency vector is derived and a consistent estimator for the covariance matrix of the length observations is presented. The usefulness of the proposed estimators is illustrated with an analysis of real cod length measurement data.     

Three estimators for the poisson regression model with measurement errors

We consider two consistent estimators for the parameters of the linear predictor in the Poisson regression model, where the covariate is measured with errors. The measurement errors are assumed to be normally distributed with known error variance σ _u ² . The SQS estimator, based on a conditional mean-variance model, takes the distribution of the latent covariate into account, and this is here assumed to be a normal distribution. The CS estimator, based on a corrected score function, does not use the distribution of the latent covariate. Nevertheless, for small σ _u ² , both estimators have identical asymptotic covariance matrices up to the order of σ _u ² . We also compare the consistent estimators to the naive estimator, which is based on replacing the latent covariate with its (erroneously) measured counterpart. The naive estimator is biased, but has a smaller covariance matrix than the consistent estimators (at least up to the order of σ _u ² ).     

Heteroskedasticity-consistent covariance matrix estimation:white's estimator and the bootstrap ∗

This paper considers the issue of estimating the covariance matrix of ordinary least squares estimates in a linear regression model when heteroskedasticity is suspected. We perform Monte Carlo simulation on the White estimator, which is commonly used in.

empirical research, and also on some alternatives based on different bootstrapping schemes. Our results reveal that the White estimator can be considerably biased when the sample size is not very large, that bias correction via bootstrap does not work well, and that the weighted bootstrap estimators tend to display smaller biases than the White estimator and its variants, under both homoskedasticity and heteroskedasticity. Our results also reveal that the presence of (potentially) influential observations in the design matrix plays an important role in the finite-sample performance of the heteroskedasticity-consistent estimators.     

k-Step shape estimators based on spatial signs and ranks

In this paper, the shape matrix estimators based on spatial sign and rank vectors are considered. The estimators considered here are slight modifications of the estimators introduced in Dümbgen (1998) and Oja and Randles (2004) and further studied for example in Sirkiä et al. (2009). The shape estimators are computed using pairwise differences of the observed data, therefore there is no need to estimate the location center of the data. When the estimator is based on signs, the use of differences also implies that the estimators have the so called independence property if the estimator, that is used as an initial estimator, has it. The influence functions and limiting distributions of the estimators are derived at the multivariate elliptical case. The estimators are shown to be highly efficient in the multinormal case, and for heavy-tailed distributions they outperform the shape estimator based on sample covariance matrix.     

Small-area estimation by combining time-series and cross-sectional data

A model involving autocorrelated random effects and sampling errors is proposed for small-area estimation, using both time-series and cross-sectional data. The sampling errors are assumed to have a known block-diagonal covariance matrix. This model is an extension of a well-known model, due to Fay and Herriot (1979), for cross-sectional data. A two-stage estimator of a small-area mean for the current period is obtained under the proposed model with known autocorrelation, by first deriving the best linear unbiased prediction estimator assuming known variance components, and then replacing them with their consistent estimators. Extending the approach of Prasad and Rao (1986, 1990) for the Fay-Herriot model, an estimator of mean squared error (MSE) of the two-stage estimator, correct to a second-order approximation for a small or moderate number of time points, T, and a large number of small areas, m, is obtained. The case of unknown autocorrelation is also considered. Limited simulation results on the efficiency of two-stage estimators and the accuracy of the proposed estimator of MSE are présentés.     

Robust regression through robust covariances

This paper discusses the estimation of regression parameters after summarizing the data by a covariance matrix of the concatenated vector of explanatory variables and response variable. A robust estimate of the covariance matrix leads to a robust regression estimator. An M-estimator at the covariance estimation step is studied in the paper, and the resulting regression estimator is compared to a few previously proposed robust regression estimators.     

A test of the mean square error criterion for linear admissible estimators

A test for choosing between a linear admissible estimator and the least squares estimator (LSE) is developed. A characterization of linear admissible estimators useful for comparing estimators is presented and necessary and sufficient conditions for superiority of a linear admissible estimator over the LS estimetor is derived for the test. The test is based on the MSE matrix superiority, but also new resl?!ts concerning covariance matrix comparisons of linear estimators are derived. Further,shown that the test of Toro - Vizcarrondo and Wailace applies iioi only the restricted least squares estimators but also to certain estimators outside this class.     

Wavelet threshold based on Stein's unbiased risk estimators of restricted location parameter in multivariate normal

In this paper, the problem of estimating the mean vector under non-negative constraints on location vector of the multivariate normal distribution is investigated. The value of the wavelet threshold based on Stein''s unbiased risk estimators is calculated for the shrinkage estimator in restricted parameter space. We suppose that covariance matrix is unknown and we find the dominant class of shrinkage estimators under Balance loss function. The performance evaluation of the proposed class of estimators is checked through a simulation study by using risk and average mean square error values.     

Caliper pair-matching on a continuous variable in case-control studies

The use of matched pairs has been criticized as being less efficient than estimators based on random samples. This paper compares the mean square error of an analysis of covariance estimator based on random samples with two estimators based on caliper matched pairs. The first of these is a simple mean difference estimator and the second a regression estimator suggested by Rubin (1973b). Under conditions which commonly occur in epidemiologic case-control studies, both of the matched pair estimators can have smaller mean square errors than analysis o f covariance estimator. When there is a weak relationship between the matching and response variate, the mean difference estimator has a lower mean square error than the regression estimator.     

Regularized covariance matrix estimation under the common principal components model

The common principal components (CPC) model provides a way to model the population covariance matrices of several groups by assuming a common eigenvector structure. When appropriate, this model can provide covariance matrix estimators of which the elements have smaller standard errors than when using either the pooled covariance matrix or the per group unbiased sample covariance matrix estimators. In this article, a regularized CPC estimator under the assumption of a common (or partially common) eigenvector structure in the populations is proposed. After estimation of the common eigenvectors using the Flury–Gautschi (or other) algorithm, the off-diagonal elements of the nearly diagonalized covariance matrices are shrunk towards zero and multiplied with the orthogonal common eigenvector matrix to obtain the regularized CPC covariance matrix estimates. The optimal shrinkage intensity per group can be estimated using cross-validation. The efficiency of these estimators compared to the pooled and unbiased estimators is investigated in a Monte Carlo simulation study, and the regularized CPC estimator is applied to a real dataset to demonstrate the utility of the method.     

Robust linear discriminant analysis using S‐estimators

The authors consider a robust linear discriminant function based on high breakdown location and covariance matrix estimators. They derive influence functions for the estimators of the parameters of the discriminant function and for the associated classification error. The most B‐robust estimator is determined within the class of multivariate S‐estimators. This estimator, which minimizes the maximal influence that an outlier can have on the classification error, is also the most B‐robust location S‐estimator. A comparison of the most B‐robust estimator with the more familiar biweight S‐estimator is made.     

Artificial neural networks for some macroeconomic series: A first report

There is a tendency for the true variability of feasible GLS estimators to be understated by asymptotic standard errors. For estimation of SUR models, this tendency becomes more severe in large equation systems when estimation of the error covariance matrix, C, becomes problematic. We explore a number of potential solutions involving the use of improved estimators for the disturbance covariance matrix and bootstrapping. In particular, Ullah and Racine (1992) have recently introduced a new class of estimators for SUR models that use nonparametric kernel density estimation techniques. The proposed estimators have the same structure as the feasible GLS estimator of Zellner (1962) differing only in the choice of estimator for C. Ullah and Racine (1992) prove that their nonparametric density estimator of C can be expressed as Zellner's original estimator plus a positive definite matrix that depends on the smoothing parameter chosen for the density estimation. It is this structure of the estimator that most interests us as it has the potential to be especially useful in large equation systems.

Atkinson and Wilson (1992) investigated the bias in the conventional and bootstrap estimators of coefficient standard errors in SUR models. They demonstrated that under certain conditions the former were superior, but they caution that neither estimator uniformly dominated and hence bootstrapping provides little improvement in the estimation of standard errors for the regression coefficients. Rilstone and Veal1 (1996) argue that an important qualification needs to be made to this somewhat negative conclusion. They demonstrated that bootstrapping can result in improvements in inferences if the procedures are applied to the t-ratios rather than to the standard errors. These issues are explored for the case of large equation systems and when bootstrapping is combined with improved covariance estimation.