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.     

Numerical evaluation of tests based on different heteroskedasticity-consistent covariance matrix estimators

This article considers the issue of performing tests in linear heteroskedastic models when the test statistic employs a consistent variance estimator. Several different estimators are considered, namely: HC0, HC1, HC2, HC3, and their bias-adjusted versions. The numerical evaluation is performed using numerical integration methods; the Imhof algorithm is used to that end. The results show that bias-adjustment of variance estimators used to construct test statistics delivers more reliable tests when they are performed for the HC0 and HC1 estimators, but the same does not hold for the HC3 estimator. Overall, the most reliable test is the HC3-based one.     

The Extent of Gross Errors Eliminated by Robust Multiple Linear Regressions

Robust estimation methods can effectively eliminate the influence of gross errors on parameter estimation. However, the extent of gross errors eliminated (EGEE) by robust estimation methods is far-reaching. This article presents a new approach to determine EGEE by robust estimation method. Taking multiple linear regressions (2–5) as examples, simulation experiments were conducted to compare the EGEE of 14 frequently used robust estimation methods. This article confirms several additional efficient robust estimation methods for dealing with multiple linear regressions, as well as the minimum number of observations needed to eliminate gross errors in certain ranges completely.     

Robust Inference in Generalized Linear Models

Robust inference on the parameters in generalized linear models is performed using the weighted likelihood method. Two cases are considered: a case with replicated observations and a case with a single observation of the dependent variable for each combination of the explanatory variables. The first case is common in the design of experiments, while the second case arises in observational studies. Theoretical and computational results on real datasets are presented and compared with other existing techniques.     

Testing Inference in Inflated Beta Regressions under Model Misspecification

We consider testing inference in inflated beta regressions subject to model misspecification. In particular, quasi-z tests based on sandwich covariance matrix estimators are described and their finite sample behavior is investigated via Monte Carlo simulations. The numerical evidence shows that quasi-z testing inference can be considerably more accurate than inference made through the usual z tests, especially when there is model misspecification. Interval estimation is also considered. We also present an empirical application that uses real (not simulated) data.     

A sequence of improved standard errors under heteroskedasticity of unknown form

The linear regression model is commonly used by practitioners to model the relationship between the variable of interest and a set of explanatory variables. The assumption that all error variances are the same (homoskedasticity) is oftentimes violated. Consistent regression standard errors can be computed using the heteroskedasticity-consistent covariance matrix estimator proposed by White (1980). Such standard errors, however, typically display nonnegligible systematic errors in finite samples, especially under leveraged data. Cribari-Neto et al. (2000) improved upon the White estimator by defining a sequence of bias-adjusted estimators with increasing accuracy. In this paper, we improve upon their main result by defining an alternative sequence of adjusted estimators whose biases vanish at a much faster rate. Hypothesis testing inference is also addressed. An empirical illustration is presented.     

Variational Inference of Linear Regression with Nonzero Prior Means

In this article, we employ the variational Bayesian method to study the parameter estimation problems of linear regression model, wherein some regressors are of Gaussian distribution with nonzero prior means. We obtain an analytical expression of the posterior parameter distribution, and then propose an iterative algorithm for the model. Simulations are carried out to test the performance of the proposed algorithm, and the simulation results confirm both the effectiveness and the reliability of the proposed algorithm.     

On Using Linear Regressions in Welfare Economics

This article consists of two parts. The first part shows that the ordinary least squares regression coefficient is a weighted average of slopes between adjacent sample points. When applied to a linear regression, with income as the independent variable, the regression coefficient depends heavily on the slopes of high-income groups. The weight of the highest income decile may well exceed that of the other nine deciles. This may be undesirable, especially if the regression is used for welfare analysis, because the marginal propensities to consume attributed to the commodities are determined by the high-income groups. The second part of the article proposes alternative estimators, the extended Gini estimators, that enable investigators to control the weighting scheme and to incorporate their social views into the weighting scheme of the estimators     

Statistical Inference in Orthogonal Regression for Three-Part Compositional Data Using a Linear Model with Type-II Constraints

Orthogonal regression is a proper tool to analyze relations between two variables when three-part compositional data, i.e., three-part observations carrying relative information (like proportions or percentages), are under examination. When linear statistical models with type-II constraints (constraints involving other parameters besides the ones of the unknown model) are employed for estimating the parameters of the regression line, approximate variances and covariances of the estimated line coefficients can be determined. Moreover, the additional assumption of normality enables to construct confidence domains and perform hypotheses testing. The theoretical results are applied to a real-world example.     

A General Class of Estimators for the Linear Regression Model Affected by Collinearity and Outliers

In this article, a general class of estimators for the linear regression model affected by outliers and collinearity is introduced and studied in some detail. This class of estimators combines the theory of light, maximum entropy, and robust regression techniques. Our theoretical findings are illustrated through a Monte Carlo simulation study.     

缺失偏态数据下线性回归模型的统计推断

研究缺失偏态数据下线性回归模型的参数估计问题,针对缺失偏态数据,为克服样本分布扭曲缺点和提高模型的回归系数、尺度参数和偏度参数的估计效果,提出了一种适合偏态数据下线性回归模型中缺失数据的修正回归插补方法.通过随机模拟和实例研究,并与均值插补、回归插补、随机回归插补方法比较,结果表明所提出的修正回归插补方法是有效可行的.     

Robust Estimation of Multivariate Linear Model Based on Depth Weighted Mean and Scatter

Based on the projection depth weighted mean and scatter estimation of the joint distribution of (x, y), we introduce a robust estimator of the regression coefficients for the multivariate linear model. The new estimator possesses desirable properties including affine invariance, Fisher consistency, and asymptotic normality. Also, we study the robustness of the estimator in terms of breakdown point and influence function. Extensive simulation studies are performed to investigate the finite sample behavior of robustness and efficiency. The methodology is illustrated with a real data example.     

A Note on Asymmetry and Robustness in Linear Regression

We discuss the assumption of symmetry in robust linear regression. It is important to distinguish between the intercept term and the slope parameters. Ordinary robust regression requires no assumption of symmetry when interest lies in slope parameters; computer programs, confidence intervals, standard errors, and so forth do not change because the errors are asymmetric. The situation is radically different for bounded-influence estimators. With the exception of the Mallows class, these estimators are inconsistent for slope when the errors are asymmetric.     

A performance counterexample of Billor–Chatterjee–Hadi procedure and an improvement proposal for robust regression

The BCH procedure introduced by Billor, Chatterjee, and Hadi for fitting linear models was found to be inefficient for y-outliers in the presence of a high perturbation level. We propose to modify the first step of the BCH procedure, so that the robust distances are computed on the matrix Z = (y, X) of the basic subset. The performance of the present note procedure (PNP), as compared to the BCH procedure and the ordinary least-square (OLS) method, was studied by processing several datasets used in the literature for robust regression and by performing a Monte Carlo experiment. PNP performs better particularly with datasets having high perturbation.     

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.     

Bayesian Estimation of the Number of Change Points in Simple Linear Regression Models

A Bayesian approach is considered to detect the number of change points in simple linear regression models. A normal-gamma empirical prior for the regression parameters based on maximum likelihood estimator (MLE) is employed in the analysis. Under mild conditions, consistency for the number of change points and boundedness between the estimated location and the true location of the change points are established. The Bayesian approach to the detection of the number of change points is suitable whether the switching simple regression is continuous or discontinuous. Some simulation results are given to confirm the accuracy of the proposed estimator.     

Quasi-Minimax Estimation in the Linear Model with Measurement Errors

In this article, we consider quasi-minimax estimation in the linear regression model where some covariates are measured with additive errors. When measurement errors are directly ignored the minimax risk of the resulting estimator can be large. By correcting the attenuation we propose a penalized quadratic risk function. A simulation study is conducted to illustrate the performance of the proposed estimators.     

Bias of S 2 in Linear Regressions with Dependent Errors

Bounds are given for the expected value of the estimator of the error variance in linear regressions, when the errors are dependent or heteroscedastic. The bounds are valid irrespective of the covariance structure between the errors. Necessary and sufficient conditions to attain the bounds are supplied.     

Dimension Reduction in Regressions through Weighted Variance Estimation

Because sliced inverse regression (SIR) using the conditional mean of the inverse regression fails to recover the central subspace when the inverse regression mean degenerates, sliced average variance estimation (SAVE) using the conditional variance was proposed in the sufficient dimension reduction literature. However, the efficacy of SAVE depends heavily upon the number of slices. In the present article, we introduce a class of weighted variance estimation (WVE), which, similar to SAVE and simple contour regression (SCR), uses the conditional variance of the inverse regression to recover the central subspace. The strong consistency and the asymptotic normality of the kernel estimation of WVE are established under mild regularity conditions. Finite sample studies are carried out for comparison with existing methods and an application to a real data is presented for illustration.     

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.