Alternative approximations of the bias and MSE of the IV estimator under weak identification with an application to bias correction

We provide analytical formulae for the asymptotic bias (ABIAS) and mean-squared error (AMSE) of the IV estimator, and obtain approximations thereof based on an asymptotic scheme which essentially requires the expectation of the first stage F-statistic to converge to a finite (possibly small) positive limit as the number of instruments approaches infinity. Our analytical formulae can be viewed as generalizing the bias and MSE results of [Richardson and Wu 1971. A note on the comparison of ordinary and two-stage least squares estimators. Econometrica 39, 973–982] to the case with nonnormal errors and stochastic instruments. Our approximations are shown to compare favorably with approximations due to [Morimune 1983. Approximate distributions of k$k$-class estimators when the degree of overidentifiability is large compared with the sample size. Econometrica 51, 821–841] and [Donald and Newey 2001. Choosing the number of instruments. Econometrica 69, 1161–1191], particularly when the instruments are weak. We also construct consistent estimators for the ABIAS and AMSE, and we use these to further construct a number of bias corrected OLS and IV estimators, the properties of which are examined both analytically and via a series of Monte Carlo experiments.     

On the efficient use of the informational content of estimating equations: Implied probabilities and Euclidean empirical likelihood

A number of information-theoretic alternatives to GMM have recently been proposed in the literature. For practical use and general interpretation, the main drawback of these alternatives, particularly in the case of conditional moment restrictions, is that they give up the computational and interpretational simplicity of quadratic optimization. The main contribution of this paper is to analyze the informational content of estimating equations within the unified framework of Chi-square distance. Improved inference by control variables, closed form formulae for implied probabilities and information-theoretic interpretations of continuously updated GMM are discussed in the two cases of unconditional and conditional moment restrictions.     

Functional coefficient instrumental variables models

We consider estimation of nonparametric structural models under a functional coefficient representation for the regression function. Under this representation, models are linear in the endogenous components with coefficients given by unknown functions of the predetermined variables, a nonparametric generalization of random coefficient models. The functional coefficient restriction is an intermediate approach between fully nonparametric structural models that are ill posed when endogenous variables are continuously distributed, and partially linear models over which they have appreciable flexibility. We propose two-step estimators that use local linear approximations in both steps. The first step is to estimate a vector of reduced forms of regression models and the second step is local linear regression using the estimated reduced forms as regressors. Our large sample results include consistency and asymptotic normality of the proposed estimators. The high practical power of estimators is illustrated via both a Monte Carlo simulation study and an application to returns to education.     

Local indirect least squares and average marginal effects in nonseparable structural systems

We study the scope of local indirect least squares (LILS) methods for nonparametrically estimating average marginal effects of an endogenous cause X on a response Y in triangular structural systems that need not exhibit linearity, separability, or monotonicity in scalar unobservables. One main finding is negative: in the fully nonseparable case, LILS methods cannot recover the average marginal effect. LILS methods can nevertheless test the hypothesis of no effect in the general nonseparable case. We provide new nonparametric asymptotic theory, treating both the traditional case of observed exogenous instruments Z and the case where one observes only error-laden proxies for Z.     

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.     

Examining bias in estimators of linear rational expectations models under misspecification

Most rational expectations models involve equations in which the dependent variable is a function of its lags and its expected future value. We investigate the asymptotic bias of generalized method of moment (GMM) and maximum likelihood (ML) estimators in such models under misspecification. We consider several misspecifications, and focus more specifically on the case of omitted dynamics in the dependent variable. In a stylized DGP, we derive analytically the asymptotic biases of these estimators. We establish that in many cases of interest the two estimators of the degree of forward-lookingness are asymptotically biased in opposite direction with respect to the true value of the parameter. We also propose a quasi-Hausman test of misspecification based on the difference between the GMM and ML estimators. Using Monte-Carlo simulations, we show that the ordering and direction of the estimators still hold in a more realistic New Keynesian macroeconomic model. In this set-up, misspecification is in general found to be more harmful to GMM than to ML estimators.     

Estimating a tournament model of intra-firm wage differentials

We consider the estimation of a tournament model with moral hazard (based on Rosen (1986), AER)) when only aggregate data on intra-firm employment levels and salaries are available. Equilibrium restrictions of the model allow us to recover parameters of interest, including equilibrium effort levels in each hierarchical stage of the firm. We illustrate our estimation procedures using data from major retail chains in the US. We find that only a fraction of the wage differential directly compensates workers for higher effort levels, implying that a large portion of the differentials arises to maintain incentives at lower rungs of the retailers.     

Reliable inference for the Gini index

Although attention has been given to obtaining reliable standard errors for the plug-in estimator of the Gini index, all standard errors suggested until now are either complicated or quite unreliable. An approximation is derived for the estimator by which it is expressed as a sum of IID random variables. This approximation allows us to develop a reliable standard error that is simple to compute. A simple but effective bias correction is also derived. The quality of inference based on the approximation is checked in a number of simulation experiments, and is found to be very good unless the tail of the underlying distribution is heavy. Bootstrap methods are presented which alleviate this problem except in cases in which the variance is very large or fails to exist. Similar methods can be used to find reliable standard errors of other indices which are not simply linear functionals of the distribution function, such as Sen’s poverty index and its modification known as the Sen–Shorrocks–Thon index.     

Testing and imposing Slutsky symmetry in nonparametric demand systems

Maximization of utility implies that consumer demand systems have a Slutsky matrix which is everywhere symmetric. However, previous non- and semi-parametric approaches to the estimation of consumer demand systems do not give estimators that are restricted to satisfy this condition, nor do they offer powerful tests of this restriction. We use nonparametric modeling to test and impose Slutsky symmetry in a system of expenditure share equations over prices and expenditure. In this context, Slutsky symmetry is a set of nonlinear cross-equation restrictions on levels and derivatives of consumer demand equations. The key insight is that due to the differing convergence rates of levels and derivatives and due to the fact that the symmetry restrictions are linear in derivatives, both the test and the symmetry restricted estimator behave asymptotically as if these restrictions were (locally) linear. We establish large and finite sample properties of our methods, and show that our test has advantages over the only other comparable test. All methods we propose are implemented with Canadian micro-data. We find that our nonparametric analysis yields statistically significantly and qualitatively different results from traditional parametric estimators and tests.     

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 [Heckman, J., Vytlacil E., 2005. Structural equations, treatment, effects and econometric policy evaluation. Econometrica 73(3), 669–738] 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%.     

Testing the Correlated Random Coefficient Model

The recent literature on instrumental variables (IV) features models in which agents sort into treatment status on the basis of gains from treatment as well as on baseline-pretreatment levels. Components of the gains known to the agents and acted on by them may not be known by the observing economist. Such models are called correlated random coe cient models. Sorting on unobserved components of gains complicates the interpretation of what IV estimates. This paper examines testable implications of the hypothesis that agents do not sort into treatment based on gains. In it, we develop new tests to gauge the empirical relevance of the correlated random coe cient model to examine whether the additional complications associated with it are required. We examine the power of the proposed tests. We derive a new representation of the variance of the instrumental variable estimator for the correlated random coefficient model. We apply the methods in this paper to the prototypical empirical problem of estimating the return to schooling and nd evidence of sorting into schooling based on unobserved components of gains.     

Comparing IV with structural models: What simple IV can and cannot identify

This paper compares the economic questions addressed by instrumental variables estimators with those addressed by structural approaches. We discuss Marschak’s Maxim: estimators should be selected on the basis of their ability to answer well-posed economic problems with minimal assumptions. A key identifying assumption that allows structural methods to be more informative than IV can be tested with data and does not have to be imposed.     

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.     

Bootstrap validity for the score test when instruments may be weak

It is well-known that size adjustments based on bootstrapping the t$t$-statistic perform poorly when instruments are weakly correlated with the endogenous explanatory variable. In this paper, we provide a theoretical proof that guarantees the validity of the bootstrap for the score statistic. This theory does not follow from standard results, since the score statistic is not a smooth function of sample means and some parameters are not consistently estimable when the instruments are uncorrelated with the explanatory variable.     

Generalized empirical likelihood tests in time series models with potential identification failure

We introduce test statistics based on generalized empirical likelihood methods that can be used to test simple hypotheses involving the unknown parameter vector in moment condition time series models. The test statistics generalize those in Guggenberger and Smith [2005. Generalized empirical likelihood estimators and tests under partial, weak and strong identification. Econometric Theory 21 (4), 667–709] from the i.i.d. to the time series context and are alternatives to those in Kleibergen [2005a. Testing parameters in GMM without assuming that they are identified. Econometrica 73 (4), 1103–1123] and Otsu [2006. Generalized empirical likelihood inference for nonlinear and time series models under weak identification. Econometric Theory 22 (3), 513–527]. The main feature of these tests is that their empirical null rejection probabilities are not affected much by the strength or weakness of identification. More precisely, we show that the statistics are asymptotically distributed as chi-square under both classical asymptotic theory and weak instrument asymptotics of Stock and Wright [2000. GMM with weak identification. Econometrica 68 (5), 1055–1096]. We also introduce a modification to Otsu's (2006) statistic that is computationally more attractive. A Monte Carlo study reveals that the finite-sample performance of the suggested tests is very competitive.     

Nonparametric estimation of distributional policy effects

This paper proposes a fully nonparametric procedure to evaluate the effect of a counterfactual change in the distribution of some covariates on the unconditional distribution of an outcome variable of interest. In contrast to other methods, we do not restrict attention to the effect on the mean. In particular, our method can be used to conduct inference on the change of the distribution function as a whole, its moments and quantiles, inequality measures such as the Lorenz curve or Gini coefficient, and to test for stochastic dominance. The practical applicability of our procedure is illustrated via a simulation study and an empirical example.     

A central limit theorem for endogenous locations and complex spatial interactions

We provide a new central limit theorem (CLT) for spatial processes under weak conditions that are plausible for many economic applications in which location is endogenous. In particular, our CLT is designed for problems that have some, but not necessarily all, of the following features: (i) Agents choose the locations of observations to maximize profits, welfare, or some other objective. (ii) The objects that are chosen (e.g., stores or brands) interact with one another. For example, they can be substitutes or complements. (iii) Interaction can be complex. In particular, interaction between i and j need not depend only on the distance between the locations of i and j, but can also depend on distance to or location of other observations k, or possibly on the number of other such observations.