Improper use of the ordinary least squares estimator in the switching regression model

The paper considers the consequences of incorrectly using the ordinary least squares estimator, when the true but unknown model is a switching regression. Bias and mean square error express ons are given for slope and residual variance estimators. Except for in very specialized cases the estimators are biased. A numerical exarnple illustrates some of the issues raised and provides a conpelison between the ordinary least squares and maximum likelihood estimators.     

Two-stage estimation of inequality-constrained marginal linear models with longitudinal data

Inequality-constrained regression models have received increased attention in longitudinal analysis during recent years. Regression parameters are usually obtained from iteration algorithms. An analytical formulae of the estimators cannot be provided. Therefore, the asymptotic behavior of estimators has not been fully clarified yet. This paper presents a TS estimation (TS for short) and the asymptotic distribution of the estimators. Simulations are conducted to compare constrained TS estimation, constrained ordinary least squares (OLS) estimation and TS estimation in terms of sample bias, sample mean-square error (MSE) and sample variance of the estimators.     

A new estimator combining the ridge regression and the restricted least squares methods of estimation

It is well-known in the literature on multicollinearity that one of the major consequences of multicollinearity on the ordinary least squares estimator is that the estimator produces large sampling variances, which in turn might inappropriately lead to exclusion of otherwise significant coefficients from the model. To circumvent this problem, two accepted estimation procedures which are often suggested are the restricted least squares method and the ridge regression method. While the former leads to a reduction in the sampling variance of the estimator, the later ensures a smaller mean square error value for the estimator. In this paper we have proposed a new estimator which is based on a criterion that combines the ideas underlying these two estimators. The standard properties of this new estimator have been studied in the paper. It has also been shown that this estimator is superior to both the restricted least squares as well as the ordinary ridge regression estimators by the criterion of mean sauare error of the estimator of the regression coefficients when the restrictions are indeed correct. The conditions for superiority of this estimator over the other two have also been derived for the situation when the restrictions are not correct.     

Estimation of parameters in a generalized GMANOVA model based on an outer product analogy and least squares

This paper investigates estimation of parameters in a combination of the multivariate linear model and growth curve model, called a generalized GMANOVA model. Making analogy between the outer product of data vectors and covariance yields an approach to directly do least squares to covariance. An outer product least squares estimator of covariance (COPLS estimator) is obtained and its distribution is presented if a normal assumption is imposed on the error matrix. Based on the COPLS estimator, two-stage generalized least squares estimators of the regression coefficients are derived. In addition, asymptotic normalities of these estimators are investigated. Simulation studies have shown that the COPLS estimator and two-stage GLS estimators are alternative competitors with more efficiency in the sense of sample mean, standard deviations and mean of the variance estimates to the existing ML estimator in finite samples. An example of application is also illustrated.     

A Simple Approach to Inference in Random Coefficient Models

Random coefficient regression models have been used to analyze cross-sectional and longitudinal data in economics and growth-curve data from biological and agricultural experiments. In the literature several estimators, including the ordinary least squares and the estimated generalized least squares (EGLS), have been considered for estimating the parameters of the mean model. Based on the asymptotic properties of the EGLS estimators, test statistics have been proposed for testing linear hypotheses involving the parameters of the mean model. An alternative estimator, the simple mean of the individual regression coefficients, provides estimation and hypothesis-testing procedures that are simple to compute and teach. The large sample properties of this simple estimator are shown to be similar to that of the EGLS estimator. The performance of the proposed estimator is compared with that of the existing estimators by Monte Carlo simulation.     

Estimation of linear models for field experiments

General mixed linear models for experiments conducted over a series of sltes and/or years are described. The ordinary least squares (OLS) estlmator is simple to compute, but is not the best unbiased estimator. Also, the usuaL formula for the varlance of the OLS estimator is not correct and seriously underestimates the true variance. The best linear unbiased estimator is the generalized least squares (GLS) estimator. However, t requires an inversion of the variance-covariance matrix V, whlch is usually of large dimension. Also, in practice, V is unknown.

We presented an estlmator [Vcirc] of the matrix V using the estimators of variance components [for sites, blocks (sites), etc.]. We also presented a simple transformation of the data, such that an ordinary least squares regression of the transformed data gives the estimated generalized least squares (EGLS) estimator. The standard errors obtained from the transformed regression serve as asymptotic standard errors of the EGLS estimators. We also established that the EGLS estlmator is unbiased.

An example of fitting a linear model to data for 18 sites (environments) located in Brazil is given. One of the site variables (soil test phosphorus) was measured by plot rather than by site and this established the need for a covariance model such as the one used rather than the usual analysis of variance model. It is for this variable that the resulting parameter estimates did not correspond well between the OLS and EGLS estimators. Regression statistics and the analysis of variance for the example are presented and summarized.     

Ridge Regression for Estimation of Transition Probabilities from Aggregate Data

When data from several independent Markov chains are aggregated over each time point, least square estimation of transition probabilities faces the problem of multi-collinearity. We propose here an estimation procedure which involves use of ridge regression for the ordinary least square estimators. Performance of this estimator is then compared with that of the ordinary least squares.     

An evaluation of biased estimators of regression coefficients—A simulation study

Different versions of generalized and ordinary ridge estimators and shrinkage estimators of regression coefficients are studied in comparison with least squares estimators using simulations. The results show that some of the biased estimators considered are better than the least squares estimator in general and the improvement is substantial in some cases.     

Alternative estimators in logistic regression when the data are collinear

Logistic regression using conditional maximum likelihood estimation has recently gained widespread use. Many of the applications of logistic regression have been in situations in which the independent variables are collinear. It is shown that collinearity among the independent variables seriously effects the conditional maximum likelihood estimator in that the variance of this estimator is inflated in much the same way that collinearity inflates the variance of the least squares estimator in multiple regression. Drawing on the similarities between multiple and logistic regression several alternative estimators, which reduce the effect of the collinearity and are easy to obtain in practice, are suggested and compared in a simulation study.     

Robust Sparse Regression with High-Breakdown Value

Penalized least squares estimators are sensitive to the influence of outliers like the ordinary least squares estimator. We propose a sparse regression estimator for robust variable selection and estimation based on a robust initial estimator. It is proven that our estimator has at least the same breakdown value as the initial estimator. Numerical examples are presented to illustrate our method.     

Estimation of regression models with equi-correlated responses when some observations on the response variable are missing

The present article deals with the problem of estimation of parameters in a linear regression model when some data on response variable is missing and the responses are equi-correlated. The ordinary least squares and optimal homogeneous predictors are employed to find the imputed values of missing observations. Their efficiency properties are analyzed using the small disturbances asymptotic theory. The estimation of regression coefficients using these imputed values is also considered and a comparison of estimators is presented.     

Combining ridge and principal component regression:a money demand illustration

In this paper we examine a class of estimators which includes ordinary least squares, ordinary ridge regression, and principal components regression. It is demonstrated that gains exist from jointly utilizing the PCR and ORR techniques.A money demand example is utilized to illustrate alternative estimators.     

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 generalized least squares estimation of the weibull distribution

A simple estimation procedure, based on the generalized least squares method, for the parameters of the Weibull distribution is described and investigated. Through a simulation study, this estimation technique is compared with maximum likelihood estimation, ordinary least squares estimation, and Menon's estimation procedure; this comparison is based on observed relative efficiencies (that is, the ratio of the Cramer-Rao lower bound to the observed mean squared error). Simulation results are presented for samples of size 25. Among the estimators considered in this simulation study, the generalized least squares estimator was found to be the "best" estimator for the shape parameter and a close competitor to the maximum likelihood estimator of the scale parameter.     

A new estimator for the unit root

We compare the ordinary least squares, weighted symmetric, modified weighted symmetric (MWS), maximum likelihood, and our new modification for least squares (MLS) estimator for first-order autoregressive in the case of unit root using Monte Carlo method. The Monte Carlo study sheds some light on how well the estimators and the predictors perform on different samples sizes. We found that MLS estimator is less biased and has less mean squared error (MSE) than any other estimators, and MWS predictor error performs well, in the sense of MSE, than any other predictors’ methods. The sample percentiles for the distribution of the τ statistic for the first, second, and third periods in the future, for alternative estimators, are reported to know if it agrees with those of normal distribution or not.     

Adaptive estimation of vector autoregressive models with time-varying variance: Application to testing linear causality in mean

Linear vector autoregressive (VAR) models where the innovations could be unconditionally heteroscedastic are considered. The volatility structure is deterministic and quite general, including breaks or trending variances as special cases. In this framework we propose ordinary least squares (OLS), generalized least squares (GLS) and adaptive least squares (ALS) procedures. The GLS estimator requires the knowledge of the time-varying variance structure while in the ALS approach the unknown variance is estimated by kernel smoothing with the outer product of the OLS residual vectors. Different bandwidths for the different cells of the time-varying variance matrix are also allowed. We derive the asymptotic distribution of the proposed estimators for the VAR model coefficients and compare their properties. In particular we show that the ALS estimator is asymptotically equivalent to the infeasible GLS estimator. This asymptotic equivalence is obtained uniformly with respect to the bandwidth(s) in a given range and hence justifies data-driven bandwidth rules. Using these results we build Wald tests for the linear Granger causality in mean which are adapted to VAR processes driven by errors with a nonstationary volatility. It is also shown that the commonly used standard Wald test for the linear Granger causality in mean is potentially unreliable in our framework (incorrect level and lower asymptotic power). Monte Carlo experiments illustrate the use of the different estimation approaches for the analysis of VAR models with time-varying variance innovations.     

Adaptive Estimation Using Weighted Least Squares

This paper proposes an adaptive estimator that is more precise than the ordinary least squares estimator if the distribution of random errors is skewed or has long tails. The adaptive estimates are computed using a weighted least squares approach with weights based on the lengths of the tails of the distribution of residuals. Smaller weights are assigned to those observations that have residuals in the tails of long-tailed distributions and larger weights are assigned to observations having residuals in the tails of short-tailed distributions. Monte Carlo methods are used to compare the performance of the proposed estimator and the performance of the ordinary least squares estimator. The estimates that were studied in this simulation include the difference between the means of two populations, the mean of a symmetric distribution, and the slope of a regression line. The adaptive estimators are shown to have lower mean squared errors than those for the ordinary least squares estimators for short-tailed, long-tailed, and skewed distributions, provided the sample size is at least 20. The ordinary least squares estimator has slightly lower mean squared error for normally distributed errors. The adaptive estimator is recommended for general use for studies having sample sizes of at least 20 observations unless the random errors are known to be normally distributed.     

Alternative parameter estimators based upon grouped data

The problem of estimating population parameters based upon grouped data is considered and several alternative estimation schemes such as the method of scoring, least lines, least squares, minimum chi square, and a method of approximating method of moments and maximum likelihood estimators are considered. These estimators are compared with maximum likelihood and method of moments estimators based upon individual observations using a Monte Carlo study where the parent population is characterized by a gamma distribution. An application of these techniques to fitting a gamma distribution to 1970-74 census income data is considered.     

Parameter Estimation Through Weighted Least-Squares Rank Regression with Specific Reference to the Weibull and Gumbel Distributions

Probability plots are often used to estimate the parameters of distributions. Using large sample properties of the empirical distribution function and order statistics, weights to stabilize the variance in order to perform weighted least squares regression are derived. Weighted least squares regression is then applied to the estimation of the parameters of the Weibull, and the Gumbel distribution. The weights are independent of the parameters of the distributions considered. Monte Carlo simulation shows that the weighted least-squares estimators outperform the usual least-squares estimators totally, especially in small samples.     

Superiority of the r–k Class Estimator Over Some Estimators In A Linear Model

In regression analysis, to overcome the problem of multicollinearity, the r ? k class estimator is proposed as an alternative to the ordinary least squares estimator which is a general estimator including the ordinary ridge regression estimator, the principal components regression estimator and the ordinary least squares estimator. In this article, we derive the necessary and sufficient conditions for the superiority of the r ? k class estimator over each of these estimators under the Mahalanobis loss function by the average loss criterion. Then, we compare these estimators with each other using the same criterion. Also, we suggest to test to verify if these conditions are indeed satisfied. Finally, a numerical example and a Monte Carlo simulation are done to illustrate the theoretical results.