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 and on possible extensions and limitations of the estimation approach. Finally, the numerical performance and usefulness of the estimator are illustrated by the results of Monte Carlo experiments and two empirical examples, demand for fish and returns to schooling.

Estimating panel data duration models with censored data

This paper presents a method for estimating a class of panel data duration models, under which an unknown transformation of the duration variable is linearly related to the observed explanatory variables and the unobserved heterogeneity (or frailty) with completely known error distributions. This class of duration models includes a panel data proportional hazards model with fixed effects. The proposed estimator is shown to be n1/2-consistent and asymptotically normal with dependent right censoring. The paper provides some discussions on extending the estimator to the cases of longer panels, multiple states, and endogenous explanatory variables. Some Monte Carlo studies are carried out to illustrate the finite-sample performance of the new estimator.

Uniform confidence bands for functions estimated nonparametrically with instrumental variables

This paper is concerned with developing uniform confidence bands for functions estimated nonparametrically with instrumental variables. We show that a sieve nonparametric instrumental variables estimator is pointwise asymptotically normally distributed. The asymptotic normality result holds in both mildly and severely ill-posed cases. We present an interpolation method to obtain a uniform confidence band and show that the bootstrap can be used to obtain the required critical values. Monte Carlo experiments illustrate the finite-sample performance of the uniform confidence band.

Ability, sorting and wage inequality

In this paper we examine the importance of heterogeneity and self-selection into schooling for the study of inequality. Changes in inequality over time are a combination of price changes, selection bias and composition effects. To distinguish them, we estimate a semiparametric selection model for a sample of white males surveyed (during the 1990s) by the National Longitudinal Survey of Youth, but our results are applicable to broader analyses of inequality. In our data, as college enrollment increases in the economy, average college wages decrease and average high school wages increase, and therefore inequality between college and high school groups decreases. Moreover, selection bias causes us to understate the growth of different measures of the average return to schooling in our sample. It also leads us to understate the increase in wage dispersion at the top of the college wage distribution, and to overstate it at the bottom of the college wage distribution.Comparative advantage, composition effects, local instrumental variables, selection bias, semiparametric estimation, wage distribution.

Trends in quality-adjusted skill premia in the United States, 1960-2000

This paper presents new evidence that increases in college enrollment lead to a decline in the average quality of college graduates between 1960 and 2000, resulting in a decrease of 8 percentage points in the college premium. The standard demand and supply framework (Katz and Murphy, 1992, Card and Lemieux, 2001) can qualitatively account for the trend in the college and age premia over this period, but the quantitative adjustments that need to be made to account for changes in quality are substantial. Furthermore, the standard interpretation of the supply effect can be misleading if the quality of college workers is not controlled for. To illustrate the importance of these adjustments, we reanalyze the problem studied in Card and Lemieux (2001), who observe that the rise in the college premium in the 1980s occurred mainly for young workers, and attribute this to the differential behavior of the supply of skill between the young and the old. Our results show that changes in quality are as important as changes in prices to explain the phenomenon they document.

Does it matter who responded to the survey? Trends in the U.S. gender earnings gap revisited

Blau and Kahn (JOLE, 1997; ILRR, 2006) decomposed trends in the U.S. gender earnings gap into observable and unobservable components using the PSID. They found that the unobservable part contributed significantly not only to the rapidly shrinking earnings gap in the 1980s, but also to the slowing-down of the convergence in the 1990s. In this paper, we extend their framework to consider measurement error due to the use of proxy/representative respondents. First, we document a strong trend of changing gender composition of household-representative respondents toward more females. Second, we estimate the impact of the changing gender composition on Blau and Kahn's decomposition. We find that a non-ignorable portion of changes in the gender gap could be attributed to changes in the self/proxy respondent composition. Specifically, the actual reduction in the gender gap can be smaller than what the estimates without taking into account the measurement error might suggest. We conclude that a careful validation study would be necessary to ascertain the magnitude of the spurious measurement error effects.

Nonparametric instrumental variables estimation of a quantile regression model

We consider nonparametric estimation of a regression function that is identified by requiring a specified quantile of the regression "error" conditional on an instrumental variable to be zero. The resulting estimating equation is a nonlinear integral equation of the first kind, which generates an ill-posed-inverse problem. The integral operator and distribution of the instrumental variable are unknown and must be estimated nonparametrically. We show that the estimator is mean-square consistent, derive its rate of convergence in probability, and give conditions under which this rate is optimal in a minimax sense. The results of Monte Carlo experiments show that the estimator behaves well in finite samples.Statistical inverse, endogenous variable, instrumental variable, optimal rate, nonlinear integral equation, nonparametric regression

Testing a parametric quantile-regression model with an endogenous explanatory variable against a nonparametric alternative

This paper is concerned with inference about a function g that is identified by a conditional quantile restriction involving instrumental variables. The paper presents a test of the hypothesis that g belongs to a finite-dimensional parametric family against a nonparametric alternative. The test is not subject to the ill-posed inverse problem of nonparametric instrumental variables estimation. Under mild conditions, the test is consistent against any alternative model. In large samples, its power is arbitrarily close to 1 uniformly over a class of alternatives whose distance from the null hypothesis is O ( n1/2 ), where n is the sample size. Monte Carlo simulations illustrate the finite-sample performance of the test.Hypothesis test, quantile estimation, instrumental variables, specification

Estimating distributions of potential outcomes using local instrumental variables with an application to changes in college enrollment and wage inequality

This paper extends the method of local instrumental variables developed by Heckman and Vytlacil (1999, 2001, 2005) to the estimation of not only means, but also distributions of potential outcomes. The newly developed method is illustrated by applying it to changes in college enrollment and wage inequality using data from the National Longitudinal Survey of Youth of 1979. Increases in college enrollment cause changes in the distribution of ability among college and high school graduates. This paper estimates a semiparametric selection model of schooling and wages to show that, for fixed skill prices, a 14% increase in college participation (analogous to the increase observed in the 1980s), reduces the college premium by 12% and increases the 90-10 percentile ratio among college graduates by 2%.

Nonparametric identification of accelerated failure time competing risks models

We provide new conditions for identification of accelerated failure time competing risks models. These include Roy models and some auction models. In our set up, unknown regression functions and the joint survivor function of latent disturbance terms are all nonparametric. We show that this model is identified given covariates that are independent of latent errors, provided that a certain rank condition is satisfied. We present a simple example in which our rank condition for identification is verified. Our identification strategy does not depend on identification at infinity or near zero, and it does not require exclusion assumptions. Given our identification, we show estimation can be accomplished using sieves.