Estimating multiple equation hybrid models with endogenous dummy regressors*

Estimating multiple equation hybrid models with endogenous dummy regressors is burdensome if the number of dummies and equations is large. Therefore, after clarifying model structure and identification issues the paper presents a generalization of the Amemiya principle. The new estimator reduces the computational costs and is flexible with regard to different types of variables. Furthermore, it is consistent and asymptotically normal distributed.     

Partial observability in bivariate probit models

This study investigates random utility models in which the observed binary outcome does not reflect the binary choice of a single decision-maker, but rather the joint unobserved binary choices of two decision-makers. Under the usual normality assumptions, the model that arises for the observed binary outcome is not a univariate probit model, but rather a bivariate probit model in which only one of the four possible outcomes is observed. Estimation and identification issues are discussed, and the implications for sample selectivity problems are noted.     

Testing identifying assumptions in bivariate probit models

This paper considers the bivariate probit model's identifying assumptions: linear index specification, joint normality of errors, instrument exogeneity, and relevance. First, we develop sharp testable equalities that detect all possible observable violations of the assumptions. Second, we propose an easy-to-implement testing procedure for the model's validity using existing inference methods for intersection bounds. The test achieves correct empirical size and performs well in detecting violations of the conditions in simulations. Finally, we provide a road map on what to do when the bivariate probit model is rejected, including novel bounds for the average treatment effect that relax the normality assumption.     

Instrumental variables estimators of nonparametric models with discrete endogenous regressors

This paper discusses estimation of nonparametric models whose regressor vectors consist of a vector of exogenous variables and a univariate discrete endogenous regressor with finite support. Both identification and estimators are derived from a transform of the model that evaluates the nonparametric structural function via indicator functions in the support of the discrete regressor. A two-step estimator is proposed where the first step constitutes nonparametric estimation of the instrument and the second step is a nonparametric version of two-stage least squares. Linear functionals of the model are shown to be asymptotically normal, and a consistent estimator of the asymptotic covariance matrix is described. For the binary endogenous regressor case, it is shown that one functional of the model is a conditional (on covariates) local average treatment effect, that permits both unobservable and observable heterogeneity in treatments. Finite sample properties of the estimators from a Monte Carlo simulation study illustrate the practicability of the proposed estimators.     

Inference regarding multiple structural changes in linear models with endogenous regressors

This paper considers the linear model with endogenous regressors and multiple changes in the parameters at unknown times. It is shown that minimization of a Generalized Method of Moments criterion yields inconsistent estimators of the break fractions, but minimization of the Two Stage Least Squares (2SLS) criterion yields consistent estimators of these parameters. We develop a methodology for estimation and inference of the parameters of the model based on 2SLS. The analysis covers the cases where the reduced form is either stable or unstable. The methodology is illustrated via an application to the New Keynesian Phillips Curve for the US.     

Estimation of nonlinear models with mismeasured regressors using marginal information

We consider the estimation of nonlinear models with mismeasured explanatory variables, when information on the marginal distribution of the true values of these variables is available. We derive a semi‐parametric MLE that is shown to be $\sqrt{n}$ consistent and asymptotically normally distributed. In a simulation experiment we find that the finite sample distribution of the estimator is close to the asymptotic approximation. The semi‐parametric MLE is applied to a duration model for AFDC welfare spells with misreported welfare benefits. The marginal distribution of the correctly measured welfare benefits is obtained from an administrative source. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Efficiency bounds for estimating linear functionals of nonparametric regression models with endogenous regressors

Let Y=μ^∗(X)+ε$Y={\mu }^{\ast }\left(X\right)+\epsilon$, where μ^∗${\mu }^{\ast }$ is unknown and E[ε|X]≠0$\mathbb{E}\left[\epsilon \right|X]\ne 0$ with positive probability but there exist instrumental variables W$W$ such that E[ε|W]=0$\mathbb{E}\left[\epsilon \right|W]=0$ w.p.1. It is well known that such nonparametric regression models are generally “ill-posed” in the sense that the map from the data to μ^∗${\mu }^{\ast }$ is not continuous. In this paper, we derive the efficiency bounds for estimating certain linear functionals of μ^∗${\mu }^{\ast }$ without assuming μ^∗${\mu }^{\ast }$ itself to be identified.     

Structural changes in heterogeneous panels with endogenous regressors

This paper extends Pesaran's (Econometrica, 2006, 74, 967–1012) common correlated effects (CCE) by allowing for endogenous regressors in large heterogeneous panels with unknown common structural changes in slopes and error factor structure. Since endogenous regressors and structural breaks are often encountered in empirical studies with large panels, this extension makes Pesaran's CCE approach empirically more appealing. In addition to allowing for slope heterogeneity and cross‐sectional dependence, we find that Pesaran's CCE approach is also valid when dealing with unobservable factors in the presence of endogenous regressors and structural changes in slopes and error factor loadings. This is supported by Monte Carlo experiments.     

Estimation of linear dynamic panel data models with time‐invariant regressors

We present a sequential approach to estimating a dynamic Hausman–Taylor model. We first estimate the coefficients of the time‐varying regressors and subsequently regress the first‐stage residuals on the time‐invariant regressors. In comparison to estimating all coefficients simultaneously, this two‐stage procedure is more robust against model misspecification, allows for a flexible choice of the first‐stage estimator, and enables simple testing of the overidentifying restrictions. For correct inference, we derive analytical standard error adjustments. We evaluate the finite‐sample properties with Monte Carlo simulations and apply the approach to a dynamic gravity equation for US outward foreign direct investment.     

Inference in regression models with many regressors

We investigate the behavior of various standard and modified F$F$, likelihood ratio (LR$LR$), and Lagrange multiplier (LM$LM$) tests in linear homoskedastic regressions, adapting an alternative asymptotic framework in which the number of regressors and possibly restrictions grows proportionately to the sample size. When the restrictions are not numerous, the rescaled classical test statistics are asymptotically chi-squared, irrespective of whether there are many or few regressors. However, when the restrictions are numerous, standard asymptotic versions of classical tests are invalid. We propose and analyze asymptotically valid versions of the classical tests, including those that are robust to the numerosity of regressors and restrictions. The local power of all asymptotically valid tests under consideration turns out to be equal. The “exact” F$F$ test that appeals to critical values of the F$F$ distribution is also asymptotically valid and robust to the numerosity of regressors and restrictions.     

Forecasting in dynamic models with stochastic regressors

When a dependent variable y is related to present and past values of an exogenous variable x in a dynamic regression (distributed lag) model, and when x must be forecast in order to forecast y, necessary and sufficient conditions are derived in order for optimal forecasts of y to possess lower mean square error as a result of including x in the model, relative to forecasting y solely from its own past. The contribution to this forecast MSE reduction of non-invertibility in the lag distribution is assessed. Examples from econometrics and engineering are provided to illustrate the results.     

Estimating a generalized correlation coefficient for a generalized bivariate probit model

In this paper we consider semiparametric estimation of a generalized correlation coefficient in a generalized bivariate probit model. The generalized correlation coefficient provides a simple summary statistic measuring the relationship between the two binary decision processes in a general framework. Our semiparametric estimation procedure consists of two steps, combining semiparametric estimators for univariate binary choice models with the method of maximum likelihood for the bivariate probit model with nonparametrically generated regressors. The estimator is shown to be consistent and asymptotically normal. The estimator performs well in our simulation study.     

Estimating linear models with ordinal qualitative regressors

The theory and application of ordinal qualitative dependent variable models have been given considerable attention in the social science and statistical literatures. Linear models with ordinal qualitative regressors have, however, been neglected. In this paper a simple specification for such models is developed and a consistent asymptotically normal estimator is offered. The estimator is compared to the conventional dummy variable approach using simulated data. The simulation results indicate that substantial gains with regard to bias and efficiency can be achieved relative to estimation using a conventional dummy variable scheme. Said gains appear to increase as the number of ordinal categories increases. The asymptotic properties of the estimator are detailed in the appendices.     

Estimation of Markov regime-switching regression models with endogenous switching

Following Hamilton [1989. A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357–384], estimation of Markov regime-switching regressions typically relies on the assumption that the latent state variable controlling regime change is exogenous. We relax this assumption and develop a parsimonious model of endogenous Markov regime-switching. Inference via maximum likelihood estimation is possible with relatively minor modifications to existing recursive filters. The model nests the exogenous switching model, yielding straightforward tests for endogeneity. In Monte Carlo experiments, maximum likelihood estimates of the endogenous switching model parameters were quite accurate, even in the presence of certain model misspecifications. As an application, we extend the volatility feedback model of equity returns given in Turner et al. [1989. A Markov model of heteroskedasticity, risk, and learning in the stock market. Journal of Financial Economics 25, 3–22] to allow for endogenous switching.     

Usefulness of proxy variables in linear models with stochastic regressors

This paper considers a linear model with two stochastic regressors where one regressor is not observed directly but may be observed with error (proxy variable). The interest is focussed upon the estimation accuracy of the regression coefficient of the observable regressor. The possibility of estimating this coefficient more accurately using a proxy variable in regression instead of omitting the non-observable variable is considered, and an approximate condition for this to occur is given and discussed.     

Bayesian analysis of the ordered probit model with endogenous selection

This paper presents a Bayesian analysis of an ordered probit model with endogenous selection. The model can be applied when analyzing ordered outcomes that depend on endogenous covariates that are discrete choice indicators modeled by a multinomial probit model. The model is illustrated by analyzing the effects of different types of medical insurance plans on the level of hospital utilization, allowing for potential endogeneity of insurance status. The estimation is performed using the Markov chain Monte Carlo (MCMC) methods to approximate the posterior distribution of the parameters in the model.     

Efficient estimation of probit models with correlated errors

Maximum Likelihood (ML) estimation of probit models with correlated errors typically requires high-dimensional truncated integration. Prominent examples of such models are multinomial probit models and binomial panel probit models with serially correlated errors. In this paper we propose to use a generic procedure known as Efficient Importance Sampling (EIS) for the evaluation of likelihood functions for probit models with correlated errors. Our proposed EIS algorithm covers the standard GHK probability simulator as a special case. We perform a set of Monte Carlo experiments in order to illustrate the relative performance of both procedures for the estimation of a multinomial multiperiod probit model. Our results indicate substantial numerical efficiency gains for ML estimates based on the GHK–EIS procedure relative to those obtained by using the GHK procedure.