Control Variables, Discrete Instruments, and Identification of Structural Functions

Control variables provide an important means of controlling for endogeneity in econometric models with nonseparable and/or multidimensional heterogeneity. We allow for discrete instruments, giving identification results under a variety of restrictions on the way the endogenous variable and the control variables affect the outcome. We consider many structural objects of interest, such as average or quantile treatment effects. We illustrate our results with an empirical application to Engel curve estimation.Comment: 37 pages, 4 figure

Nonparametric Estimation of Labor Supply Functions Generated by Piece Wise Linear Budget Constraints

The basic idea in this paper is that labor supply can be viewed as a function of the entire budget set, so that one way to account non-parametrically for a nonlinear budget set is to estimate a nonparametric regression where the variable in the regression is the budget set. In the special case of a linear budget constraint, this estimator would be the same as nonparametric regression on wage and nonlabor income. Nonlinear budget sets will in general be charac-terized by many variables. An important part of the estimation method is a procedure to reduce the dimensionality of the regression problem. It is of interest to see if nonparametrically estimated labor supply functions support the result of earlier studies using parametric methods. We therefore apply parametric and nonparametric labor supply functions to calculate the effect of recent Swedish tax reform. Qualitatively the nonparametric and parametric labor supply functions give the same results. Recent tax reform in Sweden hasincreased labor supply by a small but economically important amount.Nonparametric estimation; labor supply; nonlinear budget constraints; tax reform

Heterogenous Coefficients, Discrete Instruments, and Identification of Treatment Effects

Multidimensional heterogeneity and endogeneity are important features of a wide class of econometric models. We consider heterogenous coefficients models where the outcome is a linear combination of known functions of treatment and heterogenous coefficients. We use control variables to obtain identification results for average treatment effects. With discrete instruments in a triangular model we find that average treatment effects cannot be identified when the number of support points is less than or equal to the number of coefficients. A sufficient condition for identification is that the second moment matrix of the treatment functions given the control is nonsingular with probability one. We relate this condition to identification of average treatment effects with multiple treatments.Comment: 15 page

GMM with many weak moment conditions

Using many moment conditions can improve efficiency but makes the usual GMM inferences inaccurate. Two step GMM is biased. Generalized empirical likelihood (GEL) has smaller bias but the usual standard errors are too small. In this paper we use alternative asymptotics, based on many weak moment conditions, that addresses this problem. This asymptotics leads to improved approximations in overidentified models where the variance of the derivative of the moment conditions is large relative to the squared expected value of the moment conditions and identification is not too weak. We obtain an asymptotic variance for GEL that is larger than the usual one and give a "sandwich" estimator of it. In Monte Carlo examples we find that this variance estimator leads to a better Gaussian approximation to t-ratios in a range of cases. We also show that Kleibergen (2005) K statistic is valid under these asymptotics. We also compare these results with a jackknife GMM estimator, finding that GEL is asymptotically more efficient under many weak moments.GMM, Continuous Updating, Many Moments, Variance Adjustment

Instrumental variables estimation with flexible distributions

Instrumental variables are often associated with low estimator precision. This paper explores efficiency gains which might be achievable using moment conditions which are nonlinear in the disturbances and are based on flexible parametric families for error distributions. We show that these estimators can achieve the semiparametric efficiency bound when the true error distribution is a member of the parametric family. Monte Carlo simulations demonstrate low efficiency loss in the case of normal error distributions and potentially significant efficiency improvements in the case of thick-tailed and/or skewed error distributions

Nonseparable Multinomial Choice Models in Cross-Section and Panel Data

Multinomial choice models are fundamental for empirical modeling of economic choices among discrete alternatives. We analyze identification of binary and multinomial choice models when the choice utilities are nonseparable in observed attributes and multidimensional unobserved heterogeneity with cross-section and panel data. We show that derivatives of choice probabilities with respect to continuous attributes are weighted averages of utility derivatives in cross-section models with exogenous heterogeneity. In the special case of random coefficient models with an independent additive effect, we further characterize that the probability derivative at zero is proportional to the population mean of the coefficients. We extend the identification results to models with endogenous heterogeneity using either a control function or panel data. In time stationary panel models with two periods, we find that differences over time of derivatives of choice probabilities identify utility derivatives "on the diagonal," i.e. when the observed attributes take the same values in the two periods. We also show that time stationarity does not identify structural derivatives "off the diagonal" both in continuous and multinomial choice panel models.Comment: 23 page

Identification and Estimation of Triangular Simultaneous Equations Models Without Additivity

This paper investigates identification and inference in a nonparametric structural model with instrumental variables and non-additive errors. We allow for non-additive errors because the unobserved heterogeneity in marginal returns that often motivates concerns about endogeneity of choices requires objective functions that are non-additive in observed and unobserved components. We formulate several independence and monotonicity conditions that are sufficient for identification of a number of objects of interest, including the average conditional response, the average structural function, as well as the full structural response function. For inference we propose a two-step series estimator. The first step consists of estimating the conditional distribution of the endogenous regressor given the instrument. In the second step the estimated conditional distribution function is used as a regressor in a nonlinear control function approach. We establish rates of convergence, asymptotic normality, and give a consistent asymptotic variance estimator.

Estimation with many instrumental variables

Using many valid instrumental variables has the potential to improve efficiency but makes the usual inference procedures inaccurate. We give corrected standard errors, an extension of Bekker (1994) to nonnormal disturbances, that adjust for many instruments. We find that this adujstment is useful in empirical work, simulations, and in the asymptotic theory. Use of the corrected standard errors in t-ratios leads to an asymptotic approximation order that is the same when the number of instrumental variables grow as when the number of instruments is fixed. We also give a version of the Kleibergen (2002) weak instrument statistic that is robust to many instruments.

Automatic Debiased Machine Learning of Causal and Structural Effects

Many causal and structural effects depend on regressions. Examples include average treatment effects, policy effects, average derivatives, regression decompositions, economic average equivalent variation, and parameters of economic structural models. The regressions may be high dimensional. Plugging machine learners into identifying equations can lead to poor inference due to bias and/or model selection. This paper gives automatic debiasing for estimating equations and valid asymptotic inference for the estimators of effects of interest. The debiasing is automatic in that its construction uses the identifying equations without the full form of the bias correction and is performed by machine learning. Novel results include convergence rates for Lasso and Dantzig learners of the bias correction, primitive conditions for asymptotic inference for important examples, and general conditions for GMM. A variety of regression learners and identifying equations are covered. Automatic debiased machine learning (Auto-DML) is applied to estimating the average treatment effect on the treated for the NSW job training data and to estimating demand elasticities from Nielsen scanner data while allowing preferences to be correlated with prices and income