GMM estimation in partial linear models with endogenous covariates causing an over-identified problem

ABSTRACT

We study partial linear models where the linear covariates are endogenous and cause an over-identified problem. We propose combining the profile principle with local linear approximation and the generalized moment methods (GMM) to estimate the parameters of interest. We show that the profiled GMM estimators are root? n consistent and asymptotically normally distributed. By appropriately choosing the weight matrix, the estimators can attain the efficiency bound. We further consider variable selection by using the moment restrictions imposed on endogenous variables when the dimension of the covariates may be diverging with the sample size, and propose a penalized GMM procedure, which is shown to have the sparsity property. We establish asymptotic normality of the resulting estimators of the nonzero parameters. Simulation studies have been presented to assess the finite-sample performance of the proposed procedure.     

Local Linear GMM Estimation of Functional Coefficient IV Models With an Application to Estimating the Rate of Return to Schooling

We consider the local linear generalized method of moment (GMM) estimation of functional coefficient models with a mix of discrete and continuous data and in the presence of endogenous regressors. We establish the asymptotic normality of the estimator and derive the optimal instrumental variable that minimizes the asymptotic variance-covariance matrix among the class of all local linear GMM estimators. Data-dependent bandwidth sequences are also allowed for. We propose a nonparametric test for the constancy of the functional coefficients, study its asymptotic properties under the null hypothesis as well as a sequence of local alternatives and global alternatives, and propose a bootstrap version for it. Simulations are conducted to evaluate both the estimator and test. Applications to the 1985 Australian Longitudinal Survey data indicate a clear rejection of the null hypothesis of the constant rate of return to education, and that the returns to education obtained in earlier studies tend to be overestimated for all the work experience.     

Double filter instrumental variable estimation of panel data models with weakly exogenous variables

In this article, we propose instrumental variables (IV) and generalized method of moments (GMM) estimators for panel data models with weakly exogenous variables. The model is allowed to include heterogeneous time trends besides the standard fixed effects (FE). The proposed IV and GMM estimators are obtained by applying a forward filter to the model and a backward filter to the instruments in order to remove FE, thereby called the double filter IV and GMM estimators. We derive the asymptotic properties of the proposed estimators under fixed T and large N, and large T and large N asymptotics where N and T denote the dimensions of cross section and time series, respectively. It is shown that the proposed IV estimator has the same asymptotic distribution as the bias corrected FE estimator when both N and T are large. Monte Carlo simulation results reveal that the proposed estimator performs well in finite samples and outperforms the conventional IV/GMM estimators using instruments in levels in many cases.     

Improving estimation for beta regression models via EM-algorithm and related diagnostic tools

In this paper we propose an alternative procedure for estimating the parameters of the beta regression model. This alternative estimation procedure is based on the EM-algorithm. For this, we took advantage of the stochastic representation of the beta random variable through ratio of independent gamma random variables. We present a complete approach based on the EM-algorithm. More specifically, this approach includes point and interval estimations and diagnostic tools for detecting outlying observations. As it will be illustrated in this paper, the EM-algorithm approach provides a better estimation of the precision parameter when compared to the direct maximum likelihood (ML) approach. We present the results of Monte Carlo simulations to compare EM-algorithm and direct ML. Finally, two empirical examples illustrate the full EM-algorithm approach for the beta regression model. This paper contains a Supplementary Material.     

Bootstrapping stochastic regression models under homoskedasticity: wild bootstrap vs. pairs bootstrap

We investigate by simulation how the wild bootstrap and pairs bootstrap perform in t and F tests of regression parameters in the stochastic regression model, where explanatory variables are stochastic and not given and there exists no heteroskedasticity. The wild bootstrap procedure due to Davidson and Flachaire [The wild bootstrap, tamed at last, Working paper, IER#1000, Queen's University, 2001] with restricted residuals works best but its dominance is not strong compared to the result of Flachaire [Bootstrapping heteroskedastic regression models: wild bootstrap vs. pairs bootstrap, Comput. Statist. Data Anal. 49 (2005), pp. 361–376] in the fixed regression model where explanatory variables are fixed and there exists heteroskedasticity.     

Selection of terms in random coefficient regression models

The selection of suitable terms in random coefficient regression models is a challenging problem to practitioners. Although many techniques, ranging from those with a theoretical flavour to those with an exploratory spirit, have been proposed for such purposes, no particular one may be considered as a paradigm. In fact, many authors advocate that they should be used in a complementary way. We consider exploratory methods based on fitting standard regression models to the individual response profiles or to the rows of the sample within-units covariance matrix (for balanced data) that may serve as additional tools in the process of selecting an appropriate model. We evaluate the performance of the proposal via a simulation study and consider applications to two examples in the field of Biostatistics.     

Adaptive estimation of non-linear regression models

This paper summarizes from an econometric perspective recent work by statisticians on adaptive estimation. It also presents new findings concerning the adaptive estimability of non-linear regression models.     

Empirical likelihood estimation for linear regression models with AR(p) error terms with numerical examples

Linear regression models are useful statistical tools to analyze data sets in different fields. There are several methods to estimate the parameters of a linear regression model. These methods usually perform under normally distributed and uncorrelated errors. If error terms are correlated the Conditional Maximum Likelihood (CML) estimation method under normality assumption is often used to estimate the parameters of interest. The CML estimation method is required a distributional assumption on error terms. However, in practice, such distributional assumptions on error terms may not be plausible. In this paper, we propose to estimate the parameters of a linear regression model with autoregressive error term using Empirical Likelihood (EL) method, which is a distribution free estimation method. A small simulation study is provided to evaluate the performance of the proposed estimation method over the CML method. The results of the simulation study show that the proposed estimators based on EL method are remarkably better than the estimators obtained from CML method in terms of mean squared errors (MSE) and bias in almost all the simulation configurations. These findings are also confirmed by the results of the numerical and real data examples.     

Quantile regression estimation of partially linear additive models

In this paper, we consider the estimation of partially linear additive quantile regression models where the conditional quantile function comprises a linear parametric component and a nonparametric additive component. We propose a two-step estimation approach: in the first step, we approximate the conditional quantile function using a series estimation method. In the second step, the nonparametric additive component is recovered using either a local polynomial estimator or a weighted Nadaraya–Watson estimator. Both consistency and asymptotic normality of the proposed estimators are established. Particularly, we show that the first-stage estimator for the finite-dimensional parameters attains the semiparametric efficiency bound under homoskedasticity, and that the second-stage estimators for the nonparametric additive component have an oracle efficiency property. Monte Carlo experiments are conducted to assess the finite sample performance of the proposed estimators. An application to a real data set is also illustrated.     

Second-order least squares estimation of censored regression models

This paper proposes the second-order least squares estimation, which is an extension of the ordinary least squares method, for censored regression models where the error term has a general parametric distribution (not necessarily normal). The strong consistency and asymptotic normality of the estimator are derived under fairly general regularity conditions. We also propose a computationally simpler estimator which is consistent and asymptotically normal under the same regularity conditions. Finite sample behavior of the proposed estimators under both correctly and misspecified models are investigated through Monte Carlo simulations. The simulation results show that the proposed estimator using optimal weighting matrix performs very similar to the maximum likelihood estimator, and the estimator with the identity weight is more robust against the misspecification.     

LASSO and shrinkage estimation in Weibull censored regression models

In this paper we address the problem of estimating a vector of regression parameters in the Weibull censored regression model. Our main objective is to provide natural adaptive estimators that significantly improve upon the classical procedures in the situation where some of the predictors may or may not be associated with the response. In the context of two competing Weibull censored regression models (full model and candidate submodel), we consider an adaptive shrinkage estimation strategy that shrinks the full model maximum likelihood estimate in the direction of the submodel maximum likelihood estimate. We develop the properties of these estimators using the notion of asymptotic distributional risk. The shrinkage estimators are shown to have higher efficiency than the classical estimators for a wide class of models. Further, we consider a LASSO type estimation strategy and compare the relative performance with the shrinkage estimators. Monte Carlo simulations reveal that when the true model is close to the candidate submodel, the shrinkage strategy performs better than the LASSO strategy when, and only when, there are many inactive predictors in the model. Shrinkage and LASSO strategies are applied to a real data set from Veteran's administration (VA) lung cancer study to illustrate the usefulness of the procedures in practice.     

On the first order regression procedure of estimation for incomplete regression models

This article discusses some properties of the first order regression method for imputation of missing values on an explanatory variable in linear regression model and presents an estimation strategy based on hypothesis testing. This work was carried out before Professor V.K. Srivastava passed away in 2001. The author is grateful to the referees for their illuminating comments on an earlier draft of this paper.     

Cluster-based multivariate outlier identification and re-weighted regression in linear models

A cluster methodology, motivated by a robust similarity matrix is proposed for identifying likely multivariate outlier structure and to estimate weighted least-square (WLS) regression parameters in linear models. The proposed method is an agglomeration of procedures that begins from clustering the n-observations through a test of ‘no-outlier hypothesis’ (TONH) to a weighted least-square regression estimation. The cluster phase partition the n-observations into h-set called main cluster and a minor cluster of size n?h. A robust distance emerge from the main cluster upon which a test of no outlier hypothesis’ is conducted. An initial WLS regression estimation is computed from the robust distance obtained from the main cluster. Until convergence, a re-weighted least-squares (RLS) regression estimate is updated with weights based on the normalized residuals. The proposed procedure blends an agglomerative hierarchical cluster analysis of a complete linkage through the TONH to the Re-weighted regression estimation phase. Hence, we propose to call it cluster-based re-weighted regression (CBRR). The CBRR is compared with three existing procedures using two data sets known to exhibit masking and swamping. The performance of CBRR is further examined through simulation experiment. The results obtained from the data set illustration and the Monte Carlo study shows that the CBRR is effective in detecting multivariate outliers where other methods are susceptible to it. The CBRR does not require enormous computation and is substantially not susceptible to masking and swamping.     

Estimation of nonseparable models with censored dependent variables and endogenous regressors

In this article we develop a nonparametric estimator for the local average response of a censored dependent variable to endogenous regressors in a nonseparable model where the unobservable error term is not restricted to be scalar and where the nonseparable function need not be monotone in the unobservables. We formalize the identification argument put forward in Altonji, Ichimura, and Otsu (2012 Altonji, J. G., Ichimura, H., Otsu, T. (2012). Estimating derivatives in nonseparable models with limited dependent variables. Econometrica 80:1701–1719.[Crossref], [Web of Science ®] , [Google Scholar]), construct a nonparametric estimator, characterize its asymptotic property, and conduct a Monte Carlo investigation to study its small sample properties. Identification is constructive and is achieved through a control function approach. We show that the estimator is consistent and asymptotically normally distributed. The Monte Carlo results are encouraging.     

Equivalence of maximum likelihood estimation and iterative two-stage estimation for seemingly unrelated regression models

The seemingly unrelated regression model is viewed in the context of repeated measures analysis. Regression parameters and the variance-covariance matrix of the seemingly unrelated regression model can be estimated by using two-stage Aitken estimation. The first stage is to obtain a consistent estimator of the variance-covariance matrix. The second stage uses this matrix to obtain the generalized least squares estimators of the regression parameters. The maximum likelihood (ML) estimators of the regression parameters can be obtained by performing the two-stage estimation iteratively. The iterative two-stage estimation procedure is shown to be equivalent to the EM algorithm (Dempster, Laird, and Rubin, 1977) proposed by Jennrich and Schluchter (1986) and Laird, Lange, and Stram (1987) for repeated measures data. The equivalence of the iterative two-stage estimator and the ML estimator has been previously demonstrated empirically in a Monte Carlo study by Kmenta and Gilbert (1968). It does not appear to be widely known that the two estimators are equivalent theoretically. This paper demonstrates this equivalence.     

Asymptotic results for estimation and testing variances in regression models

Usually the variance of independent observations resulting from a linear or a nonlinear relationship is estimated by the Least-Squares residual estimator. In this paper its asymptotic properties are investigated. Further the asymptotic behaviour of tests for one-sided hypotheses on the variance is studied. The paper splits into two parts, the first one concerned with linear and the second one with nonlinear models.     

Effect of Berkson measurement error on parameter estimates in Cox regression models

We study the effect of additive and multiplicative Berkson measurement error in Cox proportional hazard model. By plotting the true and the observed survivor function and the true and the observed hazard function dependent on the exposure one can get ideas about the effect of this type of error on the estimation of the slope parameter corresponding to the variable measured with error. As an example, we analyze the measurement error in the situation of the German Uranium Miners Cohort Study both with graphical methods and with a simulation study. We do not see a substantial bias in the presence of small measurement error and in the rare disease case. Even the effect of a Berkson measurement error with high variance, which is not unrealistic in our example, is a negligible attenuation of the observed effect. However, this effect is more pronounced for multiplicative measurement error.     

Efficient estimation of non parametric simultaneous equations models

This paper defines a new procedure to efficiently estimate non parametric simultaneous equations models. The proposed estimation procedure exploits the additive structure and achieves oracle efficiency without the knowledge of unobserved error terms. Furthermore, simulation results show that our new estimator outperforms the existing estimator in terms of mean squared error.