A simple approach for the analysis of generalizea linear mixed models

A broad class of generalized linear mixed models, e.g. variance components models for binary data, percentages or count data, will be introduced by incorporating additional random effects into the linear predictor of a generalized linear model structure. Parameters are estimated by a combination of quasi-likelihood and iterated MINQUE (minimum norm quadratic unbiased estimation), the latter being numerically equivalent to REML (restricted, or residual, maximum likelihood). First, conditional upon the additional random effects, observations on a working variable and weights are derived by quasi-likelihood, using iteratively re-weighted least squares. Second, a linear mixed model is fitted to the working variable, employing the weights for the residual error terms, by iterated MINQUE. The latter may be regarded as a least squares procedure applied to squared and product terms of error contrasts derived from the working variable. No full distributional assumptions are needed for estimation. The model may be fitted with standardly available software for weighted regression and REML.     

The effect of estimation method and sample size in multilevel structural equation modeling

Multilevel structural equation modeling (multilevel SEM) has become an established method to analyze multilevel multivariate data. The first useful estimation method was the pseudobalanced method. This method is approximate because it assumes that all groups have the same size, and ignores unbalance when it exists. In addition, full information maximum likelihood (ML) estimation is now available, which is often combined with robust chi‐squares and standard errors to accommodate unmodeled heterogeneity (MLR). In addition, diagonally weighted least squares (DWLS) methods have become available as estimation methods. This article compares the pseudobalanced estimation method, ML(R), and two DWLS methods by simulating a multilevel factor model with unbalanced data. The simulations included different sample sizes at the individual and group levels and different intraclass correlation (ICC). The within‐group part of the model posed no problems. In the between part of the model, the different ICC sizes had no effect. There is a clear interaction effect between number of groups and estimation method. ML reaches unbiasedness fastest, then the two DWLS methods, then MLR, and then the pseudobalanced method (which needs more than 200 groups). We conclude that both ML(R) and DWLS are genuine improvements on the pseudobalanced approximation. With small sample sizes, the robust methods are not recommended.     

Parametric and Nonparametric Regression in the Presence of Endogenous Control Variables

The aim of this paper is to convey to a wider audience of applied statisticians that nonparametric (matching) estimation methods can be a very convenient tool to overcome problems with endogenous control variables. In empirical research one is often interested in the causal effect of a variable X on some outcome variable Y . With observational data, i.e. in the absence of random assignment, the correlation between X and Y generally does not reflect the treatment effect but is confounded by differences in observed and unobserved characteristics. Econometricians often use two different approaches to overcome this problem of confounding by other characteristics. First, controlling for observed characteristics, often referred to as selection on observables, or instrumental variables regression, usually with additional control variables. Instrumental variables estimation is probably the most important estimator in applied work. In many applications, these control variables are themselves correlated with the error term, making ordinary least squares and two-stage least squares inconsistent. The usual solution is to search for additional instrumental variables for these endogenous control variables, which is often difficult. We argue that nonparametric methods help to reduce the number of instruments needed. In fact, we need only one instrument whereas with conventional approaches one may need two, three or even more instruments for consistency. Nonparametric matching estimators permit     consistent estimation without the need for (additional) instrumental variables and permit arbitrary functional forms and treatment effect heterogeneity.     

A distribution-free least squares estimator for censored linear regression models

This paper describes a method for estimating simultaneously the parameter vector of the systematic component and the distribution function of the random component of a censored linear regression model. The estimator is obtained by minimizing the sum of the squares of the differences between the observed values of the dependent variable and the corresponding expected values of this variable according to the estimated parameter vector and distribution function. The resulting least squares parameter estimator incorporates information on the distribution of the random component of the regression model that is available from the estimation sample. Hence, it may often be more efficient than are parameter estimators that do not use such information. The results of numerical experiments with the least squares estimator tend to support this hypothesis.     

Robust estimation procedures and visual display techniques for a two-way classification model

An important problem in statistics is to study the effect of one or two factors on a dependent variable. This type of problem can be formulated as a regression problem (by using dummy (0,1) variables to represent the levels of factors) and the standard least squares (LS) analysis is well-known. The least absolute value (LAV) analysis is less well known, but certainly is becoming more widely used, especially in exploratory data analysis.The purpose of this report is to present a didactic treatment of visual display methods useful in exploratory data analysis. These visual display techniques (stem- and- leaf, box- and- whisker, and two-way plots) are presented for both the least squares and the least absolute value analyses of a two-way classification model.     

The effects of dynamic feedbacks on LS and MM estimator accuracy in panel data models

The finite sample behavior is analyzed of particular least squares (LS) and a range of (generalized) method of moments (MM) estimators in panel data models with individual effects and both a lagged dependent variable regressor and another explanatory variable. The latter may be affected by lagged feedbacks from the dependent variable too. Asymptotic expansions indicate how the order of magnitude of bias of MM estimators tends to increase with the number of moment conditions exploited. They also provide analytic evidence on how the bias of the various estimators depends on the feedbacks and on other model characteristics such as prominence of individual effects and correlation between observed and unobserved heterogeneity. Simulation results corroborate the theoretical findings and reveal that in small samples of models with dynamic feedbacks none of the techniques examined dominates regarding bias and mean squared error over all parametrizations examined.     

Estimating wage differentials without logarithms

Regression models of wage determination are typically estimated by ordinary least squares using the logarithm of the wage as the dependent variable. These models provide consistent estimates of the proportional impact of wage determinants only under the assumption that the distribution of the error term is independent of the regressors — an assumption that can be violated by the presence of heteroskedasticity, for example. Failure of this assumption is particularly relevant in the estimation of the impact of union status on wages. Alternative wage-equation estimators based on the use of quasi-maximum-likelihood methods are consistent under weaker assumptions about the dependence between the error term and the regressors. They also provide the ability to check the specification of the underlying wage model. Applying this approach to a standard data set, I find that the impact of unions on wages is overstated by a magnitude of 20-30 percent when estimates from log-wage regressions are used for inference.     

On solving bias‐corrected non‐linear estimation equations with an application to the dynamic linear model

In a seminal paper, Mak, Journal of the Royal Statistical Society B, 55, 1993, 945, derived an efficient algorithm for solving non‐linear unbiased estimation equations. In this paper, we show that when Mak's algorithm is applied to biased estimation equations, it results in the estimates that would come from solving a bias‐corrected estimation equation, making it a consistent estimator if regularity conditions hold. In addition, the properties that Mak established for his algorithm also apply in the case of biased estimation equations but for estimates from the bias‐corrected equations. The marginal likelihood estimator is obtained when the approach is applied to both maximum likelihood and least squares estimation of the covariance matrix parameters in the general linear regression model. The new approach results in two new estimators when applied to the profile and marginal likelihood functions for estimating the lagged dependent variable coefficient in the dynamic linear regression model. Monte Carlo simulation results show the new approach leads to a better estimator when applied to the standard profile likelihood. It is therefore recommended for situations in which standard estimators are known to be biased.     

Single-equation estimators and aggregation restrictions when equations have the same sets of regressors

It is well known that linear equations subject to cross-equation aggregation restrictions can be ‘stacked’ and estimated simultaneously. However, if every equation contains the same set of regressors, a number of single-equation estimation procedures can be employed. The applicability of ordinary least squares is widely recognized but the article demonstrates that the class of applicable estimators is much broaders than OLS. Under specified conditions, the class includes instrumental variables, generalized least squares, ridge regression, two-stage least squares, k-class estimators, and indirect least squares. Transformations of the original equations and other related matters are discussed also.     

Random group effects and the precision of regression estimates

When explanatory variable data in a regression model are drawn from a population with grouped structure, the regression errors are often correlated within groups. Error component and random coefficient regression models are considered as models of the intraclass correlation. This paper analyzes several empirical examples to investigate the applicability of random effects models and the consequences of inappropriately using ordinary least squares (OLS) estimation in the presence of random group effects. The principal findings are that the assumption of independent errors is usually incorrect and the unadjusted OLS standard errors often have a substantial downward bias, suggesting a considerable danger of spurious regression.     

Multidimensional skills and the returns to schooling: Evidence from an interactive fixed-effects approach and a linked survey-administrative data set

This paper presents new evidence on returns to schooling based on an interactive fixed-effects framework that allows for multiple unobserved skills with potentially time-varying prices as well as individual-level heterogeneity in returns. This constitutes a substantive generalization of most existing approaches. Our empirical analysis employs a unique linked survey-administrative panel data set on education and earnings. We find average marginal returns to schooling of about 2.8–4.4% relative to least squares/instrumental variable estimates between 7.7% and 12.7%. Omitted ability accounts for a larger fraction of the aggregate least squares bias compared to heterogeneity. We also find considerable heterogeneity in individual returns.     

A Review of Envelope Models

The envelope model was first introduced as a parsimonious version of multivariate linear regression. It uses dimension reduction techniques to remove immaterial variation in the data and has the potential to gain efficiency in estimation and improve prediction. Many advances have taken place since its introduction, and the envelope model has been applied to many contexts in multivariate analysis, including partial least squares, generalised linear models, Bayesian analysis, variable selection and quantile regression, among others. This article serves as a review of the envelope model and its developments for those who are new to the area.     

Censored regression quantiles with endogenous regressors

This paper develops a semiparametric method for estimation of the censored regression model when some of the regressors are endogenous (and continuously distributed) and instrumental variables are available for them. A “distributional exclusion” restriction is imposed on the unobservable errors, whose conditional distribution is assumed to depend on the regressors and instruments only through a lower-dimensional “control variable,” here assumed to be the difference between the endogenous regressors and their conditional expectations given the instruments. This assumption, which implies a similar exclusion restriction for the conditional quantiles of the censored dependent variable, is used to motivate a two-stage estimator of the censored regression coefficients. In the first stage, the conditional quantile of the dependent variable given the instruments and the regressors is nonparametrically estimated, as are the first-stage reduced-form residuals to be used as control variables. The second-stage estimator is a weighted least squares regression of pairwise differences in the estimated quantiles on the corresponding differences in regressors, using only pairs of observations for which both estimated quantiles are positive (i.e., in the uncensored region) and the corresponding difference in estimated control variables is small. The paper gives the form of the asymptotic distribution for the proposed estimator, and discusses how it compares to similar estimators for alternative models.     

Estimating panel data models in the presence of endogeneity and selection

We consider estimation of panel data models with sample selection when the equation of interest contains endogenous explanatory variables as well as unobserved heterogeneity. Assuming that appropriate instruments are available, we propose several tests for selection bias and two estimation procedures that correct for selection in the presence of endogenous regressors. The tests are based on the fixed effects two-stage least squares estimator, thereby permitting arbitrary correlation between unobserved heterogeneity and explanatory variables. The first correction procedure is parametric and is valid under the assumption that the errors in the selection equation are normally distributed. The second procedure estimates the model parameters semiparametrically using series estimators. In the proposed testing and correction procedures, the error terms may be heterogeneously distributed and serially dependent in both selection and primary equations. Because these methods allow for a rather flexible structure of the error variance and do not impose any nonstandard assumptions on the conditional distributions of explanatory variables, they provide a useful alternative to the existing approaches presented in the literature.     

Estimating within‐cluster spillover effects using a cluster randomization with application to knowledge diffusion in rural India

Spillover effects within randomized clusters pose a challenge for identifying impacts of an individualized treatment. The paper proposes a solution. Longitudinal and intra‐household observations are combined in estimating the direct knowledge gain from watching an info‐movie in rural India, while randomized village assignment identifies knowledge sharing. Simulations on synthetic data and econometric tests provide support for the estimation method. We find evidence of information sharing, but far less so for disadvantaged groups, such as illiterate and lower‐caste individuals; these groups rely more on actually seeing the movie. Our results are consistent with sizeable biases in ordinary least squares, matching or instrumental variable impact estimators that ignore within‐cluster spillovers.     

联立方程模型的工具变量方法质疑与商榷

本文质疑联立方程模型前定变量的工具变量性质：前定变量并不保证与当期行为解释变量的相关性,由其构建的工作回归元所完成的分阶段最小二乘估计因而并非两阶段最小二乘估计。建议按照简约式方程构建工作回归元,其具有模型数理逻辑支持下的可替代意义。工作回归元的不同导致结构式方程分阶段最小二乘估计的不同结果,之于恰好识别方程则揭示了业内关于间接最小二乘估计方法的一个误区。     

Simultaneous Parameter Estimation in Exploratory Factor Analysis: An Expository Review

The classical exploratory factor analysis (EFA) finds estimates for the factor loadings matrix and the matrix of unique factor variances which give the best fit to the sample correlation matrix with respect to some goodness-of-fit criterion. Common factor scores can be obtained as a function of these estimates and the data. Alternatively to the classical EFA, the EFA model can be fitted directly to the data which yields factor loadings and common factor scores simultaneously. Recently, new algorithms were introduced for the simultaneous least squares estimation of all EFA model unknowns. The new methods are based on the numerical procedure for singular value decomposition of matrices and work equally well when the number of variables exceeds the number of observations. This paper provides an account that is intended as an expository review of methods for simultaneous parameter estimation in EFA. The methods are illustrated on Harman's five socio-economic variables data and a high-dimensional data set from genome research.     

Optimal nested row-column designs for blocks of size 2×2 under dependence

This paper examines nested 2 ×2 row-column designs when within-block observations are assumed to be dependent. The model considered has fixed block effects, which may also include row and/or column effects. Optimal binary and non-binary designs, constructed from semi-balanced arrays, are given under both generalised and ordinary least squares estimation. It is shown that binary designs are optimal when dependence is low. In general, however, the optimal designs are highly specific to the correlation values. Received: October 1999     

Estimation of limited dependent variable models by ordinary least squares and the method of moments

Under certain conditions, a broad class of qualitative and limited dependent variable models can be consistently estimated by the method of moments using a non-iterative correction to the ordinary least squares estimator, with only a small loss of efficiency compared to maximum likelihood estimation. The class of models is that obtained from a classical multinormal regression by any type of censoring or truncation and includes the tobit, probit, two-limit probit, truncated regression, and some variants of the sample selection models. The paper derives the estimators and their asymptotic covariance matrices.     

What do U.S. biopharmaceutical companies get from patents and research and development spikes for their dynamic corporate performance?

This paper investigates how patents and research and development (R&D) spikes affect the corporate performance of 863 firm-year observations of U.S. biopharmaceutical companies. First, a dynamic data envelopment analysis model is adopted to evaluate the performance of the U.S. biopharmaceutical companies. Then, ordinary least squares regression is used to explore the effects of three patent-related variables (patent counts, citations, and claims) and R&D spikes on corporate performance. This study finds positive impacts of patent counts, citations, and claims on corporate performance. In addition, the results show that R&D spikes have negative contemporaneous effects and time-lagged effects on corporate performance.