Quasi-maximum likelihood estimation and testing for nonlinear models with endogenous explanatory variables

I propose a quasi-maximum likelihood framework for estimating nonlinear models with continuous or discrete endogenous explanatory variables. Joint and two-step estimation procedures are considered. The joint procedure is a quasi-limited information maximum likelihood procedure, as one or both of the log likelihoods may be misspecified. The two-step control function approach is computationally simple and leads to straightforward tests of endogeneity. In the case of discrete endogenous explanatory variables, I argue that the control function approach can be applied with generalized residuals to obtain average partial effects. I show how the results apply to nonlinear models for fractional and nonnegative responses.     

Two-step estimation of panel data models with censored endogenous variables and selection bias

This paper presents some two-step estimators for a wide range of parametric panel data models with censored endogenous variables and sample selection bias. Our approach is to derive estimates of the unobserved heterogeneity responsible for the endogeneity/selection bias to include as additional explanatory variables in the primary equation. These are obtained through a decomposition of the reduced form residuals. The panel nature of the data allows adjustment, and testing, for two forms of endogeneity and/or sample selection bias. Furthermore, it incorporates roles for dynamics and state dependence in the reduced form. Finally, we provide an empirical illustration which features our procedure and highlights the ability to test several of the underlying assumptions.     

Bias corrections for two-step fixed effects panel data estimators

This paper introduces large-T bias-corrected estimators for nonlinear panel data models with both time invariant and time varying heterogeneity. These models include systems of equations with limited dependent variables and unobserved individual effects, and sample selection models with unobserved individual effects. Our two-step approach first estimates the reduced form by fixed effects procedures to obtain estimates of the time varying heterogeneity underlying the endogeneity/selection bias. We then estimate the primary equation by fixed effects including an appropriately constructed control variable from the reduced form estimates as an additional explanatory variable. The fixed effects approach in this second step captures the time invariant heterogeneity while the control variable accounts for the time varying heterogeneity. Since either or both steps might employ nonlinear fixed effects procedures it is necessary to bias adjust the estimates due to the incidental parameters problem. This problem is exacerbated by the two-step nature of the procedure. As these two-step approaches are not covered in the existing literature we derive the appropriate correction thereby extending the use of large-T bias adjustments to an important class of models. Simulation evidence indicates our approach works well in finite samples and an empirical example illustrates the applicability of our estimator.     

Homogeneity and endogeneity in systems of demand equations

In estimating systems of demand equations one of the right-hand-side explanatory variables, expenditure, may be endogenous in the sense that it is correlated with the equation error. If the assumption of homogeneity of degree zero in prices and nominal income is imposed on the system, it turns out it is still possible to estimate the parameters of the system even when expenditure is endogenous. The estimation procedure is simple requiring just one additional ordinary least squares regression.The paper also demostrates that a model in which homogeneity is tested with expenditure assumed exogenous is exactly equivalent to a model in which the exogeneity of expenditure is tested with homogeneity imposed. Previous tests of demand systems which have rejected the homogeneity postulate might therefore be reinterpreted instead as rejecting the hypothesis of exogeneity of expenditure with homogeneity of degree zero in prices and nominal income taken as given.     

Instrumental variable estimation in the presence of many moment conditions

This paper develops shrinkage methods for addressing the “many instruments” problem in the context of instrumental variable estimation. It has been observed that instrumental variable estimators may behave poorly if the number of instruments is large. This problem can be addressed by shrinking the influence of a subset of instrumental variables. The procedure can be understood as a two-step process of shrinking some of the OLS coefficient estimates from the regression of the endogenous variables on the instruments, then using the predicted values of the endogenous variables (based on the shrunk coefficient estimates) as the instruments. The shrinkage parameter is chosen to minimize the asymptotic mean square error. The optimal shrinkage parameter has a closed form, which makes it easy to implement. A Monte Carlo study shows that the shrinkage method works well and performs better in many situations than do existing instrument selection procedures.     

Endogeneity in quantile regression models: A control function approach

This paper considers a linear triangular simultaneous equations model with conditional quantile restrictions. The paper adjusts for endogeneity by adopting a control function approach and presents a simple two-step estimator that exploits the partially linear structure of the model. The first step consists of estimation of the residuals of the reduced-form equation for the endogenous explanatory variable. The second step is series estimation of the primary equation with the reduced-form residual included nonparametrically as an additional explanatory variable. This paper imposes no functional form restrictions on the stochastic relationship between the reduced-form residual and the disturbance term in the primary equation conditional on observable explanatory variables. The paper presents regularity conditions for consistency and asymptotic normality of the two-step estimator. In addition, the paper provides some discussions on related estimation methods in the literature.     

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.     

Simple solutions to the initial conditions problem in dynamic,nonlinear panel data models with unobserved heterogeneity

I study a simple, widely applicable approach to handling the initial conditions problem in dynamic, nonlinear unobserved effects models. Rather than attempting to obtain the joint distribution of all outcomes of the endogenous variables, I propose finding the distribution conditional on the initial value (and the observed history of strictly exogenous explanatory variables). The approach is flexible, and results in simple estimation strategies for at least three leading dynamic, nonlinear models: probit, Tobit and Poisson regression. I treat the general problem of estimating average partial effects, and show that simple estimators exist for important special cases. Copyright © 2005 John Wiley & Sons, Ltd.     

Testing the exogeneity specification in the complete dynamic simultaneous equation model

It is shown that in the complete dynamic simultaneous equation model exogenous variables cause endogenous variables in the sense of Granger (1969) and satisfy the criterion of econometric exogeneity discussed by Sims (1977a), but that the stationarity assumptions invoked by Granger and Sims are not necessary for this implication. Inference procedures for testing each implication are presented and a new joint test of both implications isderived. Detailed attention is given to estimation and testing when the error vector of the final form of the complete dynamic simultaneous equation model is both singular and serially correlated. The theoretical points of the paper are illustrated by testing the exogeneity specification in a small macroeconometric model.     

IV models of ordered choice

This paper studies single equation instrumental variable models of ordered choice in which explanatory variables may be endogenous. The models are weakly restrictive, leaving unspecified the mechanism that generates endogenous variables. These incomplete models are set, not point, identifying for parametrically (e.g. ordered probit) or nonparametrically specified structural functions. The paper gives results on the properties of the identified set for the case in which potentially endogenous explanatory variables are discrete. The results are used as the basis for calculations showing the rate of shrinkage of identified sets as the number of classes in which the outcome is categorised increases.     

Censored regression models with unobserved,stochastic censoring thresholds

The ‘Tobit’ model is a useful tool for estimation of regression models with truncated or limited dependent variables, but it requires a threshold which is either a known constant or an observable and independent variable. The model presented here extends the Tobit model to the censored case where the threshold is an unobserved and not necessarily independent random variable. Maximum likelihood procedures can be employed for joint estimation of both the primary regression equation and the parameters of the distribution of that random threshold.     

Semiparametric three‐step estimation methods for simultaneous equation systems

This paper proposes a new method for estimating a structural model of labour supply in which hours of work depend on (log) wages and the wage rate is considered endogenous. The main innovation with respect to other related estimation procedures is that a nonparametric additive structure in the hours of work equation is permitted. Though the focus of the paper is on this particular application, a three‐step methodology for estimating models in the presence of the above econometric problems is described. In the first step the reduced form parameters of the participation equation are estimated by a maximum likelihood procedure adapted for estimation of an additive nonparametric function. In the second step the structural parameters of the wage equation are estimated after obtaining the selection‐corrected conditional mean function. Finally, in the third step the structural parameters of the labour supply equation are estimated using local maximum likelihood estimation techniques. The paper concludes with an application to illustrate the feasibility, performance and possible gain of using this method. Copyright © 2005 John Wiley & Sons, Ltd.     

A Social Interactions Model with Endogenous Friendship Formation and Selectivity

This paper analyzes the endogeneity bias problem caused by associations of members within a network when the spatial autoregressive (SAR) model is used to study social interactions. When there are unobserved factors that affect both friendship decisions and economic outcomes, the spatial weight matrix (sociomatrix; adjacency matrix) in the SAR model, which represents the structure of a friendship network, might correlate with the disturbance term of the model, and consequently result in an endogenous selection problem in the outcomes. We consider this problem of selection bias with a modeling approach. In this approach, a statistical network model is adopted to explain the endogenous network formation process. By specifying unobserved components in both the network model and the SAR model, we capture the correlation between the processes of network and outcome formation, and propose a proper estimation procedure for the system. We demonstrate that the estimation of this system can be effectively done by using the Bayesian method. We provide a Monte Carlo experiment and an empirical application of this modeling approach on the friendship networks of high school students and their interactions on academic performance in the Add Health data. Copyright © 2015 John Wiley & Sons, Ltd.     

Some remarks on measurement errors and the identification of panel data models*

We demonstrate that despite the common worry about the possible correlations between the unobserved individual effects and the explanatory variables in panel data models the likelihood approach can provide a unified framework towards the study of the identification of a panel data model subject to measurement errors. In fact, it can also serve as a basis for deriving efficient estimation methods.     

存在自相关时自回归模型的估计和检验

讨论存在自相关情况下自回归模型中随机解释变量的内生性,指出目前的计量经济理论所存在的问题,证明了随机误差存在自相关情况下一阶自回归模型和高阶自回归模型的随机解释变量与随机误差都不相关,同时改进了自回归模型的估计和检验方法。     

Binary response panel data models with sample selection and self‐selection

We consider estimating binary response models on an unbalanced panel, where the outcome of the dependent variable may be missing due to nonrandom selection, or there is self‐selection into a treatment. In the present paper, we first consider estimation of sample selection models and treatment effects using a fully parametric approach, where the error distribution is assumed to be normal in both primary and selection equations. Arbitrary time dependence in errors is permitted. Estimation of both coefficients and partial effects, as well as tests for selection bias, are discussed. Furthermore, we consider a semiparametric estimator of binary response panel data models with sample selection that is robust to a variety of error distributions. The estimator employs a control function approach to account for endogenous selection and permits consistent estimation of scaled coefficients and relative effects.     

Common correlated effect cross‐sectional dependence corrections for nonlinear conditional mean panel models

This paper provides an approach to estimation and inference for nonlinear conditional mean panel data models, in the presence of cross‐sectional dependence. We modify Pesaran's (Econometrica, 2006, 74(4), 967–1012) common correlated effects correction to filter out the interactive unobserved multifactor structure. The estimation can be carried out using nonlinear least squares, by augmenting the set of explanatory variables with cross‐sectional averages of both linear and nonlinear terms. We propose pooled and mean group estimators, derive their asymptotic distributions, and show the consistency and asymptotic normality of the coefficients of the model. The features of the proposed estimators are investigated through extensive Monte Carlo experiments. We also present two empirical exercises. The first explores the nonlinear relationship between banks' capital ratios and riskiness. The second estimates the nonlinear effect of national savings on national investment in OECD countries depending on countries' openness.     

The presence of an error term does not preclude causal inference in regression: a comment on Krause (2012)

A recent article by Krause (Qual Quant, doi:10.1007/s11135-012-9712-5, Krause (2012)) maintains that: (1) it is untenable to characterize the error term in multiple regression as simply an extraneous random influence on the outcome variable, because any amount of error implies the possibility of one or more omitted, relevant explanatory variables; and (2) the only way to guarantee the prevention of omitted variable bias and thereby justify causal interpretations of estimated coefficients is to construct fully specified models that completely eliminate the error term. The present commentary argues that such an extreme position is impractical and unnecessary, given the availability of specialized techniques for dealing with the primary statistical consequence of omitted variables, namely endogeneity, or the existence of correlations between included explanatory variables and the error term. In particular, the current article discusses the method of instrumental variable estimation, which can resolve the endogeneity problem in causal models where one or more relevant explanatory variables are excluded, thus allowing for accurate estimation of effects. An overview of recent methodological resources and software for conducting instrumental variables estimation is provided, with the aim of helping to place this crucial technique squarely in the statistical toolkit of applied researchers.     

Semiparametric Bayesian inference in smooth coefficient models

We describe procedures for Bayesian estimation and testing in cross-sectional, panel data and nonlinear smooth coefficient models. The smooth coefficient model is a generalization of the partially linear or additive model wherein coefficients on linear explanatory variables are treated as unknown functions of an observable covariate. In the approach we describe, points on the regression lines are regarded as unknown parameters and priors are placed on differences between adjacent points to introduce the potential for smoothing the curves. The algorithms we describe are quite simple to implement—for example, estimation, testing and smoothing parameter selection can be carried out analytically in the cross-sectional smooth coefficient model.     

Autocorrelation and dynamic methodology with an application to wage determination models

A demonstration is provided of rigorous, statistical methodology whereby both the type and order of an error process can be identified in dynamic, single equation econometric models. The paper relies heavily upon maximum likelihood estimation, nested likelihood ratio tests and the overfitting or exponentially weighted procedure for model selection. An application of the methodology to a class of quarterly wage determination models is included.