The asymptotic equivalence between the iterated improved 2sls estimator and the 3sls estimator

In this paper we show that the 3SLS estimator of a system of equations is asymptotically equivalent to an iterative 2SLS estimator applied to each equation, augmented with the residuals from the other equations. This result is a natural extension of Telser (1964).     

Stein-like 2SLS estimator

ABSTRACT

Maasoumi (1978 Maasoumi, E. (1978). A modified Stein-like estimator for the reduced form coefficients of simultaneous equations. Econometrica 46:695–703.[Crossref], [Web of Science ®] , [Google Scholar]) proposed a Stein-like estimator for simultaneous equations and showed that his Stein shrinkage estimator has bounded finite sample risk, unlike the three-stage least square estimator. We revisit his proposal by investigating Stein-like shrinkage in the context of two-stage least square (2SLS) estimation of a structural parameter. Our estimator follows Maasoumi (1978 Maasoumi, E. (1978). A modified Stein-like estimator for the reduced form coefficients of simultaneous equations. Econometrica 46:695–703.[Crossref], [Web of Science ®] , [Google Scholar]) in taking a weighted average of the 2SLS and ordinary least square estimators, with the weight depending inversely on the Hausman (1978 Hausman, J. A. (1978). Specification tests in econometrics. Econometrica 46:1251–1271.[Crossref], [Web of Science ®] , [Google Scholar]) statistic for exogeneity. Using a local-to-exogenous asymptotic theory, we derive the asymptotic distribution of the Stein estimator and calculate its asymptotic risk. We find that if the number of endogenous variables exceeds 2, then the shrinkage estimator has strictly smaller risk than the 2SLS estimator, extending the classic result of James and Stein (1961 James W, ., Stein, C. M. (1961). Estimation with quadratic loss. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability 1:361–380. [Google Scholar]). In a simple simulation experiment, we show that the shrinkage estimator has substantially reduced finite sample median squared error relative to the standard 2SLS estimator.     

On the cohen-sackrowitz estimator of a common mean

The paper reconsider certain estimators proposed by COHENand SACKROWITZ[Ann.Statist.(1974)2,1274-1282,Ann.Statist.4,1294]for the common mean of two normal distributions on the basis of independent samples of equal size from the two populations. It derives the ncecessary and sufficient condition for improvement over the first sample mean, under squared error loss, for any member of a class containing these. It shows that the estimator proposded by them for simultaneous improvement over botyh sample means has the desired property if and only if the common size of the samples is at least nine. The requirement is milder than that for any other estimator at the present state of knolwledge and may be constrasted with their result which implies the desired property of the estimator only if the common size of the samples is at least fifteen. Upper bounds for variances if the estimators derived by them are also improved     

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.     

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.     

Pmc-optimal nonparametric quantile estimator

According to Pitman's Measure of Closeness, if T₁and T₂are two estimators of a real parameter $[d], then T₁is better than T₂if Po[d]{\T₁-o[d] < \T₂-0[d]\} > 1/2 for all 0[d]. It may however happen that while T₁is better than T₂and T₂is better than T₃, T₃is better than T₁. Given q ? (0,1) and a sample X₁, X₂, ..., X_nfrom an unknown F ? F, an estimator T* = T*(X₁,X₂...X_n)of the q-th quantile of the distribution F is constructed such that P_F{\F(T*)-q\ <[d] \F(T)-q\} >[d] 1/2 for all F?F and for all T€T, where F is a nonparametric family of distributions and T is a class of estimators. It is shown that T* =X_j:n'for a suitably chosen jth order statistic.     

A precise estimator for the log-normal mean

The log-normal distribution is frequently encountered in applications. The uniformly minimum variance unbiased (UMVU) estimator for the log-normal mean is given explicitly by a formula found by Finney in 1941. In contrast to this the most commonly used estimator for a log-normal mean is the sample mean. This is possibly due to the complexity of the formula given by Finney. A modified maximum likelihood estimator which approximates the UMVU estimator is derived here. It is sufficiently simple to be implemented in elementary spreadsheet applications. An elementary approximate formula for the root-mean-square error of the suggested estimator and the UMVU estimator is presented. The suggested estimator is compared with the sample mean, the maximum likelihood, and the UMVU estimators by Monte Carlo simulation in terms of root-mean-square error.     

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.     

A consistent nonparametric density estimator for the deconvolution problem

The problem of nonparametric estimation of a probability density function when the sample observations are contaminated with random noise is studied. A particular estimator f?_n(x) is proposed which uses kernel-density and deconvolution techniques. The estimator f?_n(x) is shown to be uniformly consistent, and its appearance and properties are affected by constants M_n and h_n which the user may choose. The optimal choices of M_n and h_n depend on the sample size n, the noise distribution, and the true distribution which is being estimated. Particular selections for M_n and h_n which minimize upper-bound functions of the mean squared error for f?_n(x) are recommended.     

A pre-test like estimator dominating the least-squares method

We develop a pre-test type estimator of a deterministic parameter vector β$\mathit{\beta }$ in a linear Gaussian regression model. In contrast to conventional pre-test strategies, that do not dominate the least-squares (LS) method in terms of mean-squared error (MSE), our technique is shown to dominate LS when the effective dimension is greater than or equal to 4. Our estimator is based on a simple and intuitive approach in which we first determine the linear minimum MSE (MMSE) estimate that minimizes the MSE. Since the unknown vector β$\mathit{\beta }$ is deterministic, the MSE, and consequently the MMSE solution, will depend in general on β$\mathit{\beta }$ and therefore cannot be implemented. Instead, we propose applying the linear MMSE strategy with the LS substituted for the true value of β$\mathit{\beta }$ to obtain a new estimate. We then use the current estimate in conjunction with the linear MMSE solution to generate another estimate and continue iterating until convergence. As we show, the limit is a pre-test type method which is zero when the norm of the data is small, and is otherwise a non-linear shrinkage of LS.     

Combining the unrestricted estimators into a single estimator and a simulation study on the unrestricted estimators

The purpose of this paper is to combine several regression estimators (ordinary least squares (OLS), ridge, contraction, principal components regression (PCR), Liu, r?k and r?d class estimators) into a single estimator. The conditions for the superiority of this new estimator over the PCR, the r?k class, the r?d class, β?(k, d), OLS, ridge, Liu and contraction estimators are derived by the scalar mean square error criterion and the estimators of the biasing parameters for this new estimator are examined. Also, a numerical example based on Hald data and a simulation study are used to illustrate the results.     

On the performance of the Jackknifed Liu-type estimator in linear regression model

In this paper, we are proposing a modified jackknife Liu-type estimator (MJLTE) that was created by combining the ideas underlying both the Liu-type estimator (LTE) and the jackknifed Liu-type estimator (JLTE). We will also present the necessary and sufficient conditions for superiority of the MJLTE over the LTE and JLTE, in terms of mean square error matrix criterion. Finally, a real data example and a Monte Carlo simulation are also given to illustrate theoretical results.     

A new modified Jackknifed estimator for the Poisson regression model

The Poisson regression is very popular in applied researches when analyzing the count data. However, multicollinearity problem arises for the Poisson regression model when the independent variables are highly intercorrelated. Shrinkage estimator is a commonly applied solution to the general problem caused by multicollinearity. Recently, the ridge regression (RR) estimators and some methods for estimating the ridge parameter k in the Poisson regression have been proposed. It has been found that some estimators are better than the commonly used maximum-likelihood (ML) estimator and some other RR estimators. In this study, the modified Jackknifed Poisson ridge regression (MJPR) estimator is proposed to remedy the multicollinearity. A simulation study and a real data example are provided to evaluate the performance of estimators. Both mean-squared error and the percentage relative error are considered as the performance criteria. The simulation study and the real data example results show that the proposed MJPR method outperforms the Poisson ridge regression, Jackknifed Poisson ridge regression and the ML in all of the different situations evaluated in this paper.     

On the weighted mixed Liu-type estimator under unbiased stochastic restrictions

In this article, we introduce the weighted mixed Liu-type estimator (WMLTE) based on the weighted mixed and Liu-type estimator (LTE) in linear regression model. We will also present necessary and sufficient conditions for superiority of the weighted mixed Liu-type estimator over the weighted mixed estimator (WME) and Liu type estimator (LTE) in terms of mean square error matrix (MSEM) criterion. Finally, a numerical example and a Monte Carlo simulation is also given to show the theoretical results.     

A Bayes-type estimator for the Bradley-Terry model for paired comparison

A Bayes-type estimator is proposed for the worth parameter π_i and for the treatment effect parameter ln π_i in the Bradley-Terry Model for paired comparison. In contrast to current Bayes estimators which require iterative numberical calculations, this estimator has a closed form expression. This estimation technique is also extended to obtain estimators for the Luce Multiple Comparison Model. An application of this technique to a 2³ factorial experiment with paired comparisons is presented.     

Finite sample moments of a bootstrap estimator of the james-stein rule

The finite sample moments of the bootstrap estimator of the James-Stein rule are derived and shown to be biased. Analytical results shed some light upon the source of bias and suggest that the bootstrap will be biased in other settings where the moments of the statistic of interest depends on nonlinear functions of the parameters of its distribution.     

Finite sample moments of a bootstrap estimator of the james-stein rule

The finite sample moments of the bootstrap estimator of the James-Stein rule are derived and shown to be biased. Analytical results shed some light upon the source of bias and suggest that the bootstrap will be biased in other settings where the moments of the statistic of interest depends on nonlinear functions of the parameters of its distribution.     

Developing a restricted two-parameter Liu-type estimator: A comparison of restricted estimators in the binary logistic regression model

In the context of estimating regression coefficients of an ill-conditioned binary logistic regression model, we develop a new biased estimator having two parameters for estimating the regression vector parameter β when it is subjected to lie in the linear subspace restriction Hβ = h. The matrix mean squared error and mean squared error (MSE) functions of these newly defined estimators are derived. Moreover, a method to choose the two parameters is proposed. Then, the performance of the proposed estimator is compared to that of the restricted maximum likelihood estimator and some other existing estimators in the sense of MSE via a Monte Carlo simulation study. According to the simulation results, the performance of the estimators depends on the sample size, number of explanatory variables, and degree of correlation. The superiority region of our proposed estimator is identified based on the biasing parameters, numerically. It is concluded that the new estimator is superior to the others in most of the situations considered and it is recommended to the researchers.     

Mean square error and efficiency of the least squares estimator over interval constraints

We derive the mean square error of an interval constrained least squares estimator (INCLS) for a regression model. We then show that the INCLS estimator dominates, in mean square error, the unconstrained least squares estimator provided the regression residuais are normally distri'iiuted and Ynat Yrie imposed coii-

straint is satisfied or nearly satisfied.     

A new heteroskedasticity-consistent covariance matrix estimator and inference under heteroskedasticity

To solve the heteroscedastic problem in linear regression, many different heteroskedasticity-consistent covariance matrix estimators have been proposed, including HC0 estimator and its variants, such as HC1, HC2, HC3, HC4, HC5 and HC4m. Each variant of the HC0 estimator aims at correcting the tendency of underestimating the true variances. In this paper, a new variant of HC0 estimator, HC5m, which is a combination of HC5 and HC4m, is proposed. Both the numerical analysis and the empirical analysis show that the quasi-t inference based on HC5m is typically more reliable than inferences based on other covariance matrix estimators, regardless of the existence of high leverage points.