Complete and complete moment convergence with applications to the EV regression models

In this paper, the complete convergence and complete moment convergence for arrays of rowwise m-extended negatively dependent (m-END, for short) random variables are established. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for m-END random variables is also achieved. By using the results that we established, we further investigate the strong consistency of the least square estimator in the simple linear errors-in-variables models, and provide some simulations to verify the validity of our theoretical results.     

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.     

Equivalence of two tests in varying coefficient partially linear errors in variable model with missing responses

This paper considers two tests on varying coefficient partially linear errors-in-variables models (VCPLM-EV) with missing responses under the linear constraint. The restricted estimator for the parametric component is derived and proven to share asymptotically normal distribution. In order to test the linear constraint, two statistics based on the profile Lagrange multiplier method and the corrected residual sum of squares method respectively, are proposed. It is of interest to obtain that the magnitudes of the two statistics are equal exactly and follow the asymptotical chi-square distribution. This reveals a new type of Wilk’s phenomenon in VCPLM-EV models with missing response. Finally, some numerical examples are carried out to illustrate relevant performances.     

SUR Approach for IV Estimation of Canonical Contagion Models

Seemingly unrelated regression (SUR) method is applied to the instrumental variable (IV) estimation of the canonical contagion models. A finite sample Monte Carlo experiment shows that the resulting estimator, IV-SUR estimator, is substantially better than the existing IV estimator in terms of both bias and mean squares error under diverse circumstance of instrument, conditional heteroscedasticity, and cross-section correlation.     

Errors in variables: consistent adjusted least squares (cals) estimation

A single equation errors-in-variables model is considered. Exact restrictions on the parameters in the model are assumed to be available such that the model is just-identified. A Consistent Adjusted Least Squares (CALS) estimator for this model is proposed and its asymptotic distribution is given. Special cases are given as illustrations. CALS is identical to the Method of Moments (MM), and to Maximum Likelihood (ML) under the structural interpretation. Under the functional interpretation it is identical to ML in cases where the latter method is consistent.     

A combined estimator in the simple errors-in-variables model

A new general method of combining estimators is proposed in order to obtain an estimator with “improved” small sample properties. It is based on a specification test statistic and incorporates some well-known methods like preliminary testing. It is used to derive an alternative estimator for the slope in the simple errors-in-variables model, combining OLS and the modified instrumental variable estimator by Fuller. Small sample properties of the new estimator are investigated by means of a Monte Carlo study.     

Unconditional Quantile Treatment Effects Under Endogeneity

This article develops estimators for unconditional quantile treatment effects when the treatment selection is endogenous. We use an instrumental variable (IV) to solve for the endogeneity of the binary treatment variable. Identification is based on a monotonicity assumption in the treatment choice equation and is achieved without any functional form restriction. We propose a weighting estimator that is extremely simple to implement. This estimator is root n consistent, asymptotically normally distributed, and its variance attains the semiparametric efficiency bound. We also show that including covariates in the estimation is not only necessary for consistency when the IV is itself confounded but also for efficiency when the instrument is valid unconditionally. An application of the suggested methods to the effects of fertility on the family income distribution illustrates their usefulness. Supplementary materials for this article are available online.     

Polynomial regression with errors in the variables

A polynomial functional relationship with errors in both variables can be consistently estimated by constructing an ordinary least squares estimator for the regression coefficients, assuming hypothetically the latent true regressor variable to be known, and then adjusting for the errors. If normality of the error variables can be assumed, the estimator can be simplified considerably. Only the variance of the errors in the regressor variable and its covariance with the errors of the response variable need to be known. If the variance of the errors in the dependent variable is also known, another estimator can be constructed.     

Testing parameter significance in instrumental variables probit estimators: some simulation

Binary choice models that contain endogenous regressors can now be estimated routinely using modern software. Each of the two packages, Stata 11 [Stata Statistical Software: Release 11, StataCorp LP, College Station, TX, 2009] and Limdep 9 [Econometric Software Inc., Plainview, NY, 2008], contains two estimators that can be used to estimate such a model. This paper compares the performance of maximum likelihood, Newey's Amemiya's generalized least-squares (AGLS) estimator, an instrumental variables plug-in estimator and others in samples of sizes 200 and 1000 using simulation. Specifically, this paper focuses on tests of parameter significance under various degrees of instrument strength and severity of endogeneity. Although the maximum-likelihood estimator performs well in large samples, there is some evidence that the more computationally robust AGLS estimator may perform better in smaller samples when instruments are weak. It also appears that instruments in endogenous probit estimation need to be even stronger than when used in linear instrumental variables (IV) estimation.     

On the simple linear regression model with correlated measurement errors

The simple linear regression model with measurement error has been subject to much research. In this work we will focus on this model when the error in the explanatory variable is correlated with the error in the regression equation. Specifically, we are interested in the comparison between the ordinary errors-in-variables estimator of the regression coefficient β$\beta$ and the estimator that takes account of the correlation between the errors. Based on large sample approximations, we compare the estimators and find that the estimator that takes account of the correlation should be preferred in most situations. We also compare the estimators in small sample situations. This is done by stochastic simulation. The results show that the estimators behave quite similarly in most of the simulated situations, but that the ordinary errors-in-variables estimator performs considerably worse than the estimator that takes account of the correlation for certain parameter combinations. In addition, we look briefly into the bias introduced by ignoring correlated errors when computing sample correlations, and in predictions.     

Nonparametric Additive Instrumental Variable Estimator: A Group Shrinkage Estimation Perspective

In this article, we study a nonparametric approach regarding a general nonlinear reduced form equation to achieve a better approximation of the optimal instrument. Accordingly, we propose the nonparametric additive instrumental variable estimator (NAIVE) with the adaptive group Lasso. We theoretically demonstrate that the proposed estimator is root-n consistent and asymptotically normal. The adaptive group Lasso helps us select the valid instruments while the dimensionality of potential instrumental variables is allowed to be greater than the sample size. In practice, the degree and knots of B-spline series are selected by minimizing the BIC or EBIC criteria for each nonparametric additive component in the reduced form equation. In Monte Carlo simulations, we show that the NAIVE has the same performance as the linear instrumental variable (IV) estimator for the truly linear reduced form equation. On the other hand, the NAIVE performs much better in terms of bias and mean squared errors compared to other alternative estimators under the high-dimensional nonlinear reduced form equation. We further illustrate our method in an empirical study of international trade and growth. Our findings provide a stronger evidence that international trade has a significant positive effect on economic growth.     

Estimation in autoregressive models with surrogate data and validation data

Time-series data are often subject to measurement error, usually the result of needing to estimate the variable of interest. Generally, however, the relationship between the surrogate variables and the true variables can be rather complicated compared to the classical additive error structure usually assumed. In this article, we address the estimation of the parameters in autoregressive models in the presence of function measurement errors. We first develop a parameter estimation method with the help of validation data; this estimation method does not depend on functional form and the distribution of the measurement error. The proposed estimator is proved to be consistent. Moreover, the asymptotic representation and the asymptotic normality of the estimator are also derived, respectively. Simulation results indicate that the proposed method works well for practical situation.     

Factor analysis regression

In the presence of multicollinearity the literature points to principal component regression (PCR) as an estimation method for the regression coefficients of a multiple regression model. Due to ambiguities in the interpretation, involved by the orthogonal transformation of the set of explanatory variables, the method could not yet gain wide acceptance. Factor analysis regression (FAR) provides a model-based estimation method which is particularly tailored to overcome multicollinearity in an errors-in-variables setting. In this paper two feasible versions of a FAR estimator are compared with the OLS estimator and the PCR estimator by means of Monte Carlo simulation. While the PCR estimator performs best in cases of strong and high multicollinearity, the Thomson-based FAR estimator proves to be superior when the regressors are moderately correlated.     

Instrumental variable estimation of factor models with possibly many variables

In this paper, we consider the instrumental variables (IV) estimation of factor models. In the psychometrics literature, although the two-stage least squares (2SLS) estimator is routinely used in IV estimation of factor models, alternative estimators have been proposed in the econometrics literature. Therefore, in this paper, we compare the performance of these alternative IV estimators in the context of factor models. Monte Carlo simulation results reveal that the HLIM/HFUL estimator by Hausman et al. (2012 Hausman, J., W. K. Newey, T. Woutersen, J. C. Chao, and N. R. Swanson. 2012. Instrumental variable estimation with heteroskedasticity and many instruments. Quantitative Economics 3:211–55. [Google Scholar]) outperforms the 2SLS estimator and performs best in many cases.     

Bias From Censored Regressors

We study the bias that arises from using censored regressors in estimation of linear models. We present results on bias in ordinary least aquares (OLS) regression estimators with exogenous censoring and in instrumental variable (IV) estimators when the censored regressor is endogenous. Bound censoring such as top-coding results in expansion bias, or effects that are too large. Independent censoring results in bias that varies with the estimation method—attenuation bias in OLS estimators and expansion bias in IV estimators. Severe biases can result when there are several regressors and when a 0–1 variable is used in place of a continuous regressor.     

Semiparametric Estimators for Limited Dependent Variable (LDV) Models with Endogenous Regressors

This article reviews semiparametric estimators for limited dependent variable (LDV) models with endogenous regressors, where nonlinearity and nonseparability pose difficulties. We first introduce six main approaches in the linear equation system literature to handle endogenous regressors with linear projections: (i) ‘substitution’ replacing the endogenous regressors with their projected versions on the system exogenous regressors x, (ii) instrumental variable estimator (IVE) based on E{(error) × x} = 0, (iii) ‘model-projection’ turning the original model into a model in terms of only x-projected variables, (iv) ‘system reduced form (RF)’ finding RF parameters first and then the structural form (SF) parameters, (v) ‘artificial instrumental regressor’ using instruments as artificial regressors with zero coefficients, and (vi) ‘control function’ adding an extra term as a regressor to control for the endogeneity source. We then check if these approaches are applicable to LDV models using conditional mean/quantiles instead of linear projection. The six approaches provide a convenient forum on which semiparametric estimators in the literature can be categorized, although there are a few exceptions. The pros and cons of the approaches are discussed, and a small-scale simulation study is provided for some reviewed estimators.     

Conditional inference in finite population sampling under a calibration model

Several estimators, including the classical and the regression estimators of finite population mean, are compared, both theoretically and empirically, under a calibration model, where the dependent variable(y), and not the independent variable(x), can be observed for all units of the finite population. It is shown asymptotically that when conditioned on x, the bias of the classical estimator may be much smaller than that of the regression estimators; whereas when conditioned on y, the regression estimator may have much smaller conditional bias than the classical estimator. Since all the y's(not x's) can be observed, it seems appropriate to make comparison under the conditional distribution of each estimator with y fixed. In this case, the regression estimator has smaller variance, smaller conditional bias, and the conditional coverage probability closer to its nominal level     

OPTIMAL IV ESTIMATION OF SYSTEMS WITH STOCHASTIC REGRESSORS AND VAR DISTURBANCES WITH APPLICATIONS TO DYNAMIC SYSTEMS

This paper considers the general problem of Feasible Generalized Least Squares Instrumental Variables (FGLS IV) estimation using optimal instruments. First we summarize the sufficient conditions for the FGLS IV estimator to be asymptotically equivalent to an optimal GLS IV estimator. Then we specialize to stationary dynamic systems with stationary VAR errors, and use the sufficient conditions to derive new moment conditions for these models. These moment conditions produce useful IVs from the lagged endogenous variables, despite the correlation between errors and endogenous variables. This use of the information contained in the lagged endogenous variables expands the class of IV estimators under consideration and thereby potentially improves both asymptotic and small-sample efficiency of the optimal IV estimator in the class. Some Monte Carlo experiments compare the new methods with those of Hatanaka (1976). For the DGP used in the Monte Carlo experiments, asymptotic efficiency is strictly improved by the new IVs, and experimental small-sample efficiency is improved as well.     

OPTIMAL IV ESTIMATION OF SYSTEMS WITH STOCHASTIC REGRESSORS AND VAR DISTURBANCES WITH APPLICATIONS TO DYNAMIC SYSTEMS

This paper considers the general problem of Feasible Generalized Least Squares Instrumental Variables (FGLS IV) estimation using optimal instruments. First we summarize the sufficient conditions for the FGLS IV estimator to be asymptotically equivalent to an optimal GLS IV estimator. Then we specialize to stationary dynamic systems with stationary VAR errors, and use the sufficient conditions to derive new moment conditions for these models. These moment conditions produce useful IVs from the lagged endogenous variables, despite the correlation between errors and endogenous variables. This use of the information contained in the lagged endogenous variables expands the class of IV estimators under consideration and thereby potentially improves both asymptotic and small-sample efficiency of the optimal IV estimator in the class. Some Monte Carlo experiments compare the new methods with those of Hatanaka (1976). For the DGP used in the Monte Carlo experiments, asymptotic efficiency is strictly improved by the new IVs, and experimental small-sample efficiency is improved as well.     

Least Trimmed Squares Estimator in the Errors-in-Variables Model

We propose a robust estimator in the errors-in-variables model using the least trimmed squares estimator. We call this estimator the orthogonal least trimmed squares (OLTS) estimator. We show that the OLTS estimator has the high breakdown point and appropriate equivariance properties. We develop an algorithm for the OLTS estimate. Simulations are performed to compare the efficiencies of the OLTS estimates with the total least squares (TLS) estimates and a numerical example is given to illustrate the effectiveness of the estimate.