AMEMIYA'S FORM OF THE WEIGHTED LEAST SQUARES ESTIMATOR

Amemiya's estimator is a weighted least squares estimator of the regression coefficients in a linear model with heteroscedastic errors. It is attractive because the heteroscedasticity is not parametrized and the weights (which depend on the error covariance matrix) are estimated nonparametrically. This paper derives an asymptotic expansion for Amemiya's form of the weighted least squares estimator, and uses it to discuss the effects of estimating the weights, of the number of iterations, and of the choice of the initial estimate. The paper also discusses the special case of normally distributed errors and clarifies the particular consequences of assuming normality.     

Efficient Estimation and Robust Inference of Linear Regression Models in the Presence of Heteroscedastic Errors and High Leverage Points

It is common for linear regression models that the error variances are not the same for all observations and there are some high leverage data points. In such situations, the available literature advocates the use of heteroscedasticity consistent covariance matrix estimators (HCCME) for the testing of regression coefficients. Primarily, such estimators are based on the residuals derived from the ordinary least squares (OLS) estimator that itself can be seriously inefficient in the presence of heteroscedasticity. To get efficient estimation, many efficient estimators, namely the adaptive estimators are available but their performance has not been evaluated yet when the problem of heteroscedasticity is accompanied with the presence of high leverage data. In this article, the presence of high leverage data is taken into account to evaluate the performance of the adaptive estimator in terms of efficiency. Furthermore, our numerical work also evaluates the performance of the robust standard errors based on this efficient estimator in terms of interval estimation and null rejection rate (NRR).     

Bayesian Interpretations of Heteroskedastic Consistent Covariance Estimators Using the Informed Bayesian Bootstrap

This article provides Bayesian interpretations for White's heteroskedastic consistent (HC) covariance estimator, and various modifications of it, in linear regression models. An informed Bayesian bootstrap provides a useful framework.     

Confidence Corridors for Multivariate Generalized Quantile Regression

We focus on the construction of confidence corridors for multivariate nonparametric generalized quantile regression functions. This construction is based on asymptotic results for the maximal deviation between a suitable nonparametric estimator and the true function of interest, which follow after a series of approximation steps including a Bahadur representation, a new strong approximation theorem, and exponential tail inequalities for Gaussian random fields. As a byproduct we also obtain multivariate confidence corridors for the regression function in the classical mean regression. To deal with the problem of slowly decreasing error in coverage probability of the asymptotic confidence corridors, which results in meager coverage for small sample sizes, a simple bootstrap procedure is designed based on the leading term of the Bahadur representation. The finite-sample properties of both procedures are investigated by means of a simulation study and it is demonstrated that the bootstrap procedure considerably outperforms the asymptotic bands in terms of coverage accuracy. Finally, the bootstrap confidence corridors are used to study the efficacy of the National Supported Work Demonstration, which is a randomized employment enhancement program launched in the 1970s. This article has supplementary materials online.     

Performance of Kibria's Method for the Heteroscedastic Ridge Regression Model: Some Monte Carlo Evidence

It is common for a linear regression model that the error terms display some form of heteroscedasticity and at the same time, the regressors are also linearly correlated. Both of these problems have serious impact on the ordinary least squares (OLS) estimates. In the presence of heteroscedasticity, the OLS estimator becomes inefficient and the similar adverse impact can also be found on the ridge regression estimator that is alternatively used to cope with the problem of multicollinearity. In the available literature, the adaptive estimator has been established to be more efficient than the OLS estimator when there is heteroscedasticity of unknown form. The present article proposes the similar adaptation for the ridge regression setting with an attempt to have more efficient estimator. Our numerical results, based on the Monte Carlo simulations, provide very attractive performance of the proposed estimator in terms of efficiency. Three different existing methods have been used for the selection of biasing parameter. Moreover, three different distributions of the error term have been studied to evaluate the proposed estimator and these are normal, Student's t and F distribution.     

Statistical inference for a semiparametric measurement error regression model with heteroscedastic errors

Efficient inference for regression models requires that the heteroscedasticity be taken into account. We consider statistical inference under heteroscedasticity in a semiparametric measurement error regression model, in which some covariates are measured with errors. This paper has multiple components. First, we propose a new method for testing the heteroscedasticity. The advantages of the proposed method over the existing ones are that it does not need any nonparametric estimation and does not involve any mismeasured variables. Second, we propose a new two-step estimator for the error variances if there is heteroscedasticity. Finally, we propose a weighted estimating equation-based estimator (WEEBE) for the regression coefficients and establish its asymptotic properties. Compared with existing estimators, the proposed WEEBE is asymptotically more efficient, avoids undersmoothing the regressor functions and requires less restrictions on the observed regressors. Simulation studies show that the proposed test procedure and estimators have nice finite sample performance. A real data set is used to illustrate the utility of our proposed methods.     

On maximum likelihood estimation of the semi-parametric Cox model with time-varying covariates

Including time-varying covariates is a popular extension to the Cox model and a suitable approach for dealing with non-proportional hazards. However, partial likelihood (PL) estimation of this model has three shortcomings: (i) estimated regression coefficients can be less accurate in small samples with heavy censoring; (ii) the baseline hazard is not directly estimated and (iii) a covariance matrix for both the regression coefficients and the baseline hazard is not easily produced.We address these by developing a maximum likelihood (ML) approach to jointly estimate regression coefficients and baseline hazard using a constrained optimisation ensuring the latter''s non-negativity. We demonstrate asymptotic properties of these estimates and show via simulation their increased accuracy compared to PL estimates in small samples and show our method produces smoother baseline hazard estimates than the Breslow estimator.Finally, we apply our method to two examples, including an important real-world financial example to estimate time to default for retail home loans. We demonstrate using our ML estimate for the baseline hazard can give much clearer corroboratory evidence of the ‘humped hazard’, whereby the risk of loan default rises to a peak and then later falls.     

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.     

Inference under Heteroscedasticity of Unknown Form Using an Adaptive Estimator

The heteroscedasticity consistent covariance matrix estimators are commonly used for the testing of regression coefficients when error terms of regression model are heteroscedastic. These estimators are based on the residuals obtained from the method of ordinary least squares and this method yields inefficient estimators in the presence of heteroscedasticity. It is usual practice to use estimated weighted least squares method or some adaptive methods to find efficient estimates of the regression parameters when the form of heteroscedasticity is unknown. But HCCM estimators are seldom derived from such efficient estimators for testing purposes in the available literature. The current article addresses the same concern and presents the weighted versions of HCCM estimators. Our numerical work uncovers the performance of these estimators and their finite sample properties in terms of interval estimation and null rejection rate.     

On the error term in bahadur's representation of an order statistic

Bahadur (1966) presented a representation of an order statistic, giving its asymptotic distribution and the rate of convergence, under weak assumptions on the density function of the parent distribution. In this paper we consider the mean(squared) deviation of the error term in Bahadur’s approximation of the q th sample quantile (q_n ). We derive a uniform bound on the mean (squared) deviation of q_n , not depending on the value of q. An application of the given result provides the corresponding result for a kernel type estimator of the q th quantile.     

Comparison of estimators for heteroskedastic models

Ordinary least squares (OLS) yield inefficient parameter estimates and inconsistent estimates of the covariance matrix in case of heteroskedastic errors. Robinson's adaptive estimator and the Cragg estimator avoid any explicit parameterization of heteroskedasticity, and reduce the danger of misspecification. A small Monte Carlo experiment is performed to compare the behavior of the adaptive estimator with the performance of the Cragg estimator. The Monte Carlo experiment includes simulations of the Generalized Least Squares (GLS) estimator. Indeed, an interesting question is how more sophisticated techniques, like the adaptive estimator, compare with GLS when the latter relies on an incorrect specification of the heteroskedastic process. It turns out that the regression parameters, when estimated adaptively, display small mean squared errors and great efficiency in case of medium or high heteroskedasticity. The covariance matrix, instead, is better estimated by the Cragg estimator or by GLS based on a misspecified error term, since the adaptive estimator overpredicts the standard errors of the regression parameters.     

Effects of the estimation of covariance matrix parameters in the generalized multivariate linear model

We Consider the generalized multivariate linear model and assume the covariance matrix of the p x 1 vector of responses on a given individual can be represented in the general linear structure form described by Anderson (1973). The effects of the use of estimates of the parameters of the covariance matrix on the generalized least squares estimator of the regression coefficients and on the prediction of a portion of a future vector, when only the first portion of the vector has been observed, are investigated. Approximations are derived for the covariance matrix of the generalized least squares estimator and for the mean square error matrix of the usual predictor, for the practical case where estimated parameters are used.     

Small sample behavior of a robust heteroskedasticity consistent covariance matrix estimator

In heteroskedastic regression models, the least squares (OLS) covariance matrix estimator is inconsistent and inference is not reliable. To deal with inconsistency one can estimate the regression coefficients by OLS, and then implement a heteroskedasticity consistent covariance matrix (HCCM) estimator. Unfortunately the HCCM estimator is biased. The bias is reduced by implementing a robust regression, and by using the robust residuals to compute the HCCM estimator (RHCCM). A Monte-Carlo study analyzes the behavior of RHCCM and of other HCCM estimators, in the presence of systematic and random heteroskedasticity, and of outliers in the explanatory variables.     

Alternative estimators of the common regression matrix in two GMANOVA models under weighted quadratic losses

We consider the problem of estimating the common regression matrix of two GMANOVA models with different unknown covariance matrices under certain type of loss functions which include a weighted quadratic loss function as a special case. We consider a class of estimators, which contains the Graybill–Deal-type estimator proposed by Sugiura and Kubokawa (Ann. Inst. Statist. Math. 40 (1988) 119), and we give its risk representation via Kubokawa and Srivastava's (Ann. Statist. 27 (1999) 600; J. Multivariate Anal. 76 (2001) 138) identities when the error matrices follow the elliptically contoured distributions. Using the method similar to an approximate minimization of the unbiased risk estimate due to Stein (Studies in the Statistical Theory of Estimation, vol. 74, Nauka, Leningrad, 1977, p. 4), we obtain an alternative estimator to the Graybill–Deal-type estimator which was given under the normality assumption. However, it seems difficult to evaluate the risk of our proposed estimator analytically because of complex nature of its risk function. Instead, we conduct a Monte-Carlo simulation to evaluate the performance of our proposed estimator. The results indicate that our proposed estimator compares favorably with the Graybill–Deal-type estimator.     

Selecting Regressors for Prediction Using PRESS and White t Statistics

It is shown that selecting regressors based on the prediction error sum of squares is asymptotically related to hypothesis testing with White's (1980) heteroscedasticity-consistent covariance matrix. A simulation experiment suggests that this asymptotic relation may be useful. Illustrative examples are also given.     

Efficient improved estimation of the parameters in two seemingly unrelated regression models

For a system of two seemingly unrelated regression equations, this paper proposes a two-stage covariance improved estimator of the regression coefficients. The new estimator is shown to uniformly dominate the present estimators in terms of generalized mean square error criterion. In addition, we also propose the exact generalized mean square error of new estimator.     

Approaches to regression analysis with multiple measurements from individual sampling units

Application of ordinary least-squares regression to data sets which contain multiple measurements from individual sampling units produces an unbiased estimator of the parameters but a biased estimator of the covariance matrix of the parameter estimates. The present work considers a random coefficient, linear model to deal with such data sets: this model permits many senses in which multiple measurements are taken from a sampling unit, not just when it is measured at several times. Three procedures to estimate the covariance matrix of the error term of the model are considered. Given these, three procedures to estimate the parameters of the model and their covariance matrix are considered; these are ordinary least-squares, generalized least-squares, and an adjusted ordinary least-squares procedure which produces an unbiased estimator of the covariance matrix of the parameters with small samples. These various procedures are compared in simulation studies using three examples from the biological literature. The possibility of testing hypotheses about the vector of parameters is also considered. It is found that all three procedures for regression estimation produce estimators of the parameters with bias of no practical consequence, Both generalized least-squares and adjusted ordinary least-squares generally produce estimators of the covariance matrix of the parameter estimates with bias of no practical consequence, while ordinary least-squares produces a negatively biased estimator. Neither ordinary nor generalized least-squares provide satisfactory hypothesis tests of the vector of parameter estimates. It is concluded that adjusted ordinary least-squares, when applied with either of two of the procedures used to estimate the error coveriance matrix, shows promise for practical application with data sets of the nature considered here.     

The estimation of treatment effect for the censored bivariate data under the univariate censoring

This paper establishes a nonparametric estimator for the treatment effect on censored bivariate data under unvariate censoring. This proposed estimator is based on the one from Lin and Ying(1993)'s nonparametric bivariate survival function estimator, which is itself a generalized version of Park and Park(1995)' quantile estimator. A Bahadur type representation of quantile functions were obtained from the marginal survival distribution estimator of Lin and Ying' model. The asymptotic property of this estimator is shown below and the simulation studies are also given     

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.     

Estimated Generalized Least Squares for Random Coefficient Regression Models

ABSTRACT. This paper considers a general class of random coefficient regression (RCR) models to represent pooled cross-sectional and time series data. A new method is given to estimate the covariance matrix of the error component in these RCR models. Also, the asymptotic and small sample properties of the estimated generalized least squares estimator of the regression coefficient vector are established. Procedures for testing a linear restriction on the mean vector of the random coefficients are derived. Finally, a test for non-randomness in the RCR model is devised, and the asymptotic distribution of the test statistic is obtained.