GLS Estimator Research Articles

The Box-Cox transformation-of-variables in regression

The application of the Box-Cox transformation to the dependent and independent variables is discussed. Maximum likelihood and iterative GLS estimators are used and bootstrapping is carried out to compare the bootstrap sample variability with the finite sample variability (RMSE) and improve RMSE estimation. The biases of parameter estimators were shown to be substantial in small samples. The standard errors obtained from the Hessian matrix were a poor measure of the finite sample variability. Thet-ratios of the linear parameter estimators may not be normally distributed in small samples.

Ridge regression estimators in the linear regression models with non-spherical errors

The present paper considers a family of ordinary ridge regression estimators in the linear regression model when the disturbances covariance matrix depends upon a few unknown parameters. An asymptotic expansion for the distribution of the ridge regression estimator is developed and under the quadratic loss function its asymptotic risk is compared with that of the feasible GLS estimator.

The logit model and panel data via repeated observations: A clarification and extension of the literature

Large sample results are given for a GLS estimator which is based on a panel data logit model involving repeated observations. The model is such that the sample has two dimensions, say N and T. Our large sample results correspond to the case in which both N and T increase beyond limit. A conjecture relating to a more general case is offered for purposes of stimulating further research.

Introduction of Dual-Class Shares: Some Evidence on Canadian Pro-Rata Distributions

Tests are conducted to assess if abnormal returns or beta nonstationarity are associated with the announcements and effective dates for a sample of Canadian firms. These firms created a new class of common stock by distributing restricted voting ( RV ) shares through the stock split (pro rata) distribution method to current shareholders (holders of single class [ SC ] shares which became voting [ V ] shares). The GLS and SUR estimation techniques and various adjustments are used to deal with possible violations of the assumptions of a dummy-variable version of a two-beta market model for portfolios of V and RV shares, V minus RV shares, V shares only, and RV shares only. Although no statistically significant abnormal returns or beta shifts are found on the announcement and effective dates for such recapitalizations, statistically significant positive abnormal returns are found on the effective dates for the paired differences between V and RV shares. These results differ from those found in Jog and Riding (1986, 1989) for the effective dates.

The behaviour of three weighted least squares estimators for structured means analysis with non‐normal Likert variables

A Monte Carlo framework is employed to examine the effects of non‐normality on three weighted least squares estimators for structured means analysis of Likert scaled variables. The GLS estimator is compared to Browne's (1984) ADF estimator for structured means (Muthen, 1989), and to the CVM estimator (Muthen, 1984) for a severely skewed case. Results show severe lack of robustness for the GLS chi‐square test, not found for ADF. Factor mean estimate bias is virtually non‐existent for both estimators, but some sampling variability bias was found for GLS and ADF with ADF performing slightly worse. The CVM estimator performs similar to ADF with respect to chi‐square and better than ADF with respect to sampling variability. Some problems of weight matrix calculations are noted. A discussion of the problem of mean and covariance structure distortion is discussed, and a case where such distortion exists is studied. Results lead to the conclusion that CVM may be the preferred estimator for structured means analysis of categorical variables.

The error components regression model: conditional relative efficiency comparisons

This paper considers a simple linear regression with two-way error component disturbances and derives the conditional relative efficiency ofany feasible GLS estimator with respect to OLS, true GLS, orany other feasible GLS estimator, conditional on the estimated variance components. This is done at two crucial choices of the x variable. The first choice is where OLS is least efficient with respect to GLS and the second choice is where an arbitrary feasible GLS estimator is least efficient with respect to GLS. Our findings indicate that a better guess of a certain ‘variance components ratio’ leads to better estimates of the regression coefficients.

A note on Park and Heikes' (1983) modified approximate estimator for the first-order moving-average process

This paper extends the approximate transformation procedure developed by Park and Heikes (1983). The extended procedure is appropriate for a moving-average process of any order, and can be used by anyone who has an ordinary least-squares regression package. For a wide range of moving-average parameters, the estimators produced by the procedure are shown to be nearly as efficient as exact-transformation GLS estimators.

Recovering precision from seemingly redundant rounded data

AbstractEconomic time series from the socialist economies of Eastern Europe and the Soviet Union are often not available. Instead, a matrix of rounded growth indexes for different base years is published. Using the constrained least squares and feasible GLS estimators developed in this paper, it is possible to predict the true matrix of growth indexes with greatly improved accuracy.

Consistent Estimation of Cross-Sectional Models in Event Studies

Event studies often include cross-sectional regressions of announcement effects on exogenous variables. If the event is voluntary and investors are rational, then standard OLS and GLS estimators are inconsistent. Consistent ML estimators are constructed for a cross-sectional model of horizontal mergers relating announcement effects to exogenous characteristics of firms and industries. The OLS and ML estimates differ dramatically for bidders but not for targets. The evidence suggests that managers of bidders, but not targets, have valuable private information about the potential synergies from proposed mergers.

On estimating and testing in a linear regression model with autocorrelated errors

In this paper, we examine the small-sample properties of feasible GLS estimators and tests of individual regression coefficients. Bayesian and non-Bayesian estimators for an autocorrelation coefficient (denoted by ϱ) are examined by Monte Carlo experiments. It is shown that when the value of ϱ is large, the Bayesian estimator for ϱ performs better than non-Bayesian estimators for ϱ. When the value of ϱ is large and the sample size is 20, none of the feasible GLS estimators performs well in hypothesis testing. However, when the sample size is 40, the feasible Bayesian GLS estimator performs much better that the feasible non-Bayesian GLS estimators in hypothesis testing.

Exclusion restrictions in instrumental variables equations

In this paper we consider two-stage estimators of parameters of a structural equation in a model with recursive exclusion restrictions on the instrumental variables equations. The estimations considered are simple OLS and GLS estimators after substitution of estimates of the systematic part of the IV equations for the endogenous variables. It is known in the literature that neither imposing the restrictions in the first stage nor ignoring them will in general be more efficient than the alternative. We introduce a class of mixed instrumental variables estimators (MIV) with these possibilities as special cases which yields an estimator which is not only more efficient than the two stage estimators considered in the literature but as efficient as an efficient system estimator like 3SLS.

Testing for Autocorrelation in Dynamic Random Effects Models

This article develops, tests of covariance restrictions after estimating by three-stage least squares a dynamic random effects model from panel data. The asymptotic distribution of covariance matrix estimates under non-normality is obtained. It is shown how minimum chi-square tests for interesting covariance restrictions can be calculated from a generalised linear regression involving the sample autocovariances and dummy variables. Asymptotic efficiency exploiting covariance restrictions can also be attained using a GLS estimator.

Weighted least squares estimation of the linear probability model, revisited

Efficient GLS estimation of the linear probability model is hindered in practice by the possibility that predicted probabilities may yield some negative point variance estimates. This note proposes an alternative estimator may sometimes yield efficiency gains over OLS and will in general be efficient relative to alternative estimators that have been proposed.

An efficient GLS estimator of triangular models with covariance restrictions

A GLS estimator of a triangular model with covariance restrictions is proposed that is asymptotically equivalent to the optimal minimum distance estimator regardless of nonnormality. The estimator used for the error covariance matrix Σ is a matrix-weighted average of the efficient restricted and unrestricted estimators of Σ. The relative weights are determined by a measure of the departure from normality.

Disaggregate Level Evidence on Monetary Neutrality

Existing tests of the neutrality hypothesis focus on aggregate level economic activity. However, failure to examine disaggregate level effects can lead to incorrect inferences concerning anticipated and unanticipated money growth impacts. Disaggregate level testing of neutrality is limited. Also receiving inadequate attention is the impact upon test inferences of the use of in the widely used two-step estimation procedure. This research tests neutrality across eleven manufacturing industries and aggregate GNP, incorporating a correction for generated regressors into GLS estimation. The research finds non-neutrality of anticipated money at the disaggregate level under two general money forecasting specifications.

A Variance Comparison of OLS and Feasible GLS Estimators

Second-order approximations to the variances of OLS and GLS estimators are compared when the covariance matrix is locally nonscalar. Using a result of Rothenberg, the comparison of OLS and GLS variances is shown to be asymptotically equivalent to a weighted mean square error comparison of the error covariance parameter estimators used in those two procedures. When there is only one covariance parameter, this comparison depends only on the noncentrality parameter of a classical hypothesis test for a scalar covariance matrix.

Efficient estimation of limited dependent variable models with endogenous explanatory variables

This paper discusses asymptotically efficient estimation of the parameters of limited dependent variable models with endogenous explanatory variables. General results on asymptotic efficiency of two-stage and Amemiya GLS estimators are derived and used to obtain a simple, asymptotically efficient estimator of the structural coefficients. This estimator can be calculated by applying GLS to estimates of the reduced form coefficients that are obtained by using reduced form residuals as additional explanatory variables. It is also shown that it is possible to obtain asymptotically efficient estimators of the other coefficients by a modified minimum chi-square method.

On Estimating from a More General Time-Series Cum Cross-Section Data Structure

iance matrix ft by 'trial, error and generali zation', and obtains best quadratic unbiased (BQU) estimators of the variance compo nents. This, in turn, provides a feasible GLS estimator of the regression parameters. This note uses the results of Wansbeek and Kap teyn [15, 17] to derive the spectral decom position of ft. This allows a systematic and straightforward derivation of ft-1 and ft~v The latter matrix is used to simplify the com putation of GLS by reducing it into a weighted least squares (WLS) procedure. In addition, alternative BQU estimators of the variance components are proposed based on residual variances of least squares regressions. These OLS regressions are performed on the original model after it has been pre-multiplied by matrices of eigenvectors of ft. In summary, Ghosh's [4] results are reanalyzed in a brief and concise manner. Moreover, the compu tation of GLS is simplified and alternative BQU estimators of the variance components are derived.

Best Median-Unbiased Estimation in Linear Regression with Bounded Asymmetric Loss Functions

We first show that the Generalized Least Squares estimator is the best median unbiased estimator of the regression parameters for quite general loss functions, when the parameter space is unrestricted. Of note is the fact that this result holds without moment restrictions. Thus, the errors may have multivariate Cauchy distribution. Next, we show that a restricted GLS estimator is best median unbiased for a linear combination of the regression parameters, when that linear combination is restricted to lie in an interval. Certain other linear combinations of the parameter vector may be subject to arbitrary additional restrictions. The paper then presents best median unbiased estimators of the error variance sigma-squared, as well as monotone functions of sigma-squared, when the errors are normally distributed. If sigma-squared is constrained to lie in a finite interval, the best estimator is a censored version of its unconstrained counterpart. When sigma-square is constrained only to be positive, the best median unbiased estimator is always larger than the best mean unbiased estimator s-squared, and is approximately equal to s-squared calculated with its degrees of freedom reduced by .66.

Blast Furnace System Identification by Maximum Likelihood Method

This paper explores a scheme of using the global searching techniques to identify a stochastic linear discrete MIMO system, The basic idea of the algorithm consists of maximization of the likelihood function by both parametric and non parametric approach, An approximate GLS estimation is applied to initiate an auto-regressive noise filtering and effectuate the likelihood maximization by consecutive optimum search, The system time shifting characteristics is considered so that more improvement can be obtained. An on-line identification based on recursive computation is proposed. Practical results on dynamic data of blast furnace operation are included.