Essays in Econometrics By Alexandre Poirier A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Economics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor James L. Powell, Chair Professor Bryan S. Graham Professor Martin Lettau Professor Demian Pouzo Spring 2013 Essays in Econometrics Copyright 2013 by Alexandre Poirier 2 Abstract Essays in Econometrics by Alexandre Poirier Doctor of Philosophy in Economics University of California, Berkeley Professor James L. Powell, Chair This dissertation consists of two chapters, both contributing to the field of econometrics. The contributions are mostly in the areas of estimation theory, as both chapters develop new estimators and study their properties. They are also both developed for semi-parametric models: models containing both a finite dimensional parameter of interest, as well as infinite dimensional nuisance parameters. In both chapters, we show the estimators’ consistency, asymptotic normality and characterize their asymptotic variance. The second chapter is co-authored with professors Jinyong Hahn, Bryan S. Graham and James L. Powell. In the first chapter, we focus on estimation in a cross-sectional model with indepen- dence restrictions, because unconditional or conditional independence restrictions are used in many econometric models to identify their parameters. However, there are few results about efficient estimation procedures for finite-dimensional parameters under these indepen- dence restrictions. In this chapter, we compute the efficiency bound for finite-dimensional parameters under independence restrictions, and propose an estimator that is consistent, asymptotically normal and achieves the efficiency bound. The estimator is based on a grow- ing number of zero-covariance conditions that are asymptotically equivalent to the inde- pendence restriction. The results are illustrated with four examples: a linear instrumental variables regression model, a semilinear regression model, a semiparametric discrete response model and an instrumental variables regression model with an unknown link function. A Monte-Carlo study is performed to investigate the estimator’s small sample properties and give some guidance on when substantial efficiency gains can be made by using the proposed efficient estimator. In the second chapter, we focus on estimation in a panel data model with correlated random effects and focus on the identification and estimation of various functionals of the random coefficients distributions. In particular, we design estimators for the conditional and unconditional quantiles of the random coefficients distribution. This model allows for irregularly identified panel data models, as in Graham and Powell (2012), where quantiles of the effect are identified by using two subpopulations of “movers” and “stayers”, i.e. those for whom the covariates change by a large amount from one period to another, and those for whom covariates remain (nearly) unchanged. We also consider an alternative asymptotic framework where the fraction of stayers in the population is shrinking with the sample size. The purpose of this framework is to approximate a continuous distribution of covariates 1 where there is an infinitesimal fraction of exact stayers. We also derive the asymptotic variance of the coefficient’s distribution in this framework, and we conjecture the form of the asymptotic variance under a continuous distribution of covariates. The main goal of this dissertation is to expand the choice set of estimators available to applied researchers. In chapter one, the proposed estimator attains the efficiency bound and might allow researchers to gain more precision in estimation, by getting smaller standard errors. In the second chapter, the new estimator allows researchers to estimate quantile effects in a just-identified panel data model, a contribution to the literature. 2 To Stacy i Contents Efficient Estimation in Models with Independence 1 Restrictions 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.3 Estimation Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Computation of Efficiency Bounds . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Unconditional Independence . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 Conditional Independence . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.3 Conditional Independence with Parameter in the Conditioning Variable 14 1.2.4 Independence with Nuisance Function . . . . . . . . . . . . . . . . . 17 1.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3.1 Consistency and Asymptotic Normality . . . . . . . . . . . . . . . . . 22 1.3.2 Example and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.3.3 Feasible GMM Estimation Under Conditional Independence Restrictions 26 1.4 Feasible GMM Estimation Under Independence Restrictions containing Unknown Functions . . . . . . . . . . . . . . . . . . 29 1.5 Monte-Carlo Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.6 Conclusion and Directions for Future Research . . . . . . . . . . . . . . . . . 34 1.7 Proofs of Theorems and additional Lemmas . . . . . . . . . . . . . . . . . . 36 1.7.1 Efficiency Bound Calculations . . . . . . . . . . . . . . . . . . . . . . 36 1.7.2 Consistency under Unconditional Independence . . . . . . . . . . . . 52 1.7.3 Asymptotic Normality under Unconditional Independence . . . . . . 62 1.7.4 Consistency under Conditional Independence . . . . . . . . . . . . . . 66 1.7.5 Asymptotic normality under Conditional Independence . . . . . . . . 69 Estimation of Quantile Effects in Panel Data with 2 Random Coefficients 73 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.2 General Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.2.2 Estimands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 ii 2.3 Additional Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.3.1 Discrete Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.3.2 Just-identification and additional support assumptions . . . . . . . . 77 2.4 Estimation of the ACQE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.4.1 ACQE in the regular case . . . . . . . . . . . . . . . . . . . . . . . . 81 2.4.2 ACQE in the bandwidth case . . . . . . . . . . . . . . . . . . . . . . 84 2.5 Estimation of the UQE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 2.5.1 UQE in the Regular Case . . . . . . . . . . . . . . . . . . . . . . . . 86 2.5.2 UQE in the Bandwidth Case . . . . . . . . . . . . . . . . . . . . . . . 87 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.7 Proofs of Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 References 104 iii Acknowledgments I thank my advisor Jim Powell for his invaluable guidance, dedication and patience over the years. I am grateful for the amount of time he has taken to advise and support me. I have learned a tremendous amount from my conversations with him. I am also very thankful to my other dissertation committee members: Bryan Graham, for his constant support, insights and his help in shaping the direction of my research. Demian Pouzo, for his detailed advice and moral support during the dissertation and job market stage. I also thank Martin Lettau for useful comments and advice on research. I am also indebted to Bryan and Jim for having given me the opportunity to do research with them. I also want to thank Michael Jansson and Denis Nekipelov for useful comments and suggestions. I wish to express a special thanks to Yuriy Gorodnichenko for having been a great and dedicated mentor in my first years as a graduate student. I have also spent much time interacting with fellow graduate students who have helped me in different ways over the years. I especially thank Josh Hausman, Matteo Maggiori, Omar Nayeem, Sebastian Stumpner and James Zuberi. I also acknowledge support from the FQRSC during my graduate studies. I want to thank my parents and brothers, who have supported me in all my decisions from early in my life all the way to finishing this dissertation. Finally, I thank my amazing fiancee Stacy for being supportive throughout my doctoral studies and for being there during both the good and bad times that come with graduate studies. I dedicate this dissertation to her. iv Chapter 1 Efficient Estimation in Models with Independence Restrictions 1.1 Introduction Many econometric models are identified using zero-covariance or mean-independence restric- tions between an unobserved error term and a subset of the explanatory variables. For example, it is common to assume in linear regression models that the error is either uncor- related with the exogenous variables, or uncorrelated with all functions of the exogenous variables. These restrictions identify the parameters of interest, and efficiency bounds for these type of models have been widely studied, for example in the seminal work of Chamber- lain (1987). In other models though, statistical independence of the unobserved error and a subset of the explanatory variables is instead assumed. In most cases, statistical indepen- dence is a stronger restriction than mean-independence, as it implies mean-independence of all measurable functions of the unobserved error with respect to the subset of explanatory variables. Independence assumptions are common in recent strands of the literature includ- ing in non-linear and non-separable models to allow for heterogenous effects, in the potential outcomes framework and in non-linear structural models. In this chapter, we compute the efficiency bound for parameters identified by different types of independence restrictions. The efficiency bound for a parameter θ is the small- est possible asymptotic variance for regular asymptotically linear estimators.1 We use the projection method of Bickel et al. (1993) and Newey (1990c) to compute the bounds. This chapter also derives an estimator that is consistent, asymptotically normal and attains the efficiency bound. We will also highlight the size of the efficiency gains through a Monte Carlo exercise where we evaluate the performance of our estimator. While imposing mean-independence restrictions (i.e. conditional mean restrictions) is commonpracticeineconomics,thestrongerindependenceassumptionisusefultoconsiderfor two reasons. The first is that some models require statistical independence for identification purposes, as is sometimes the case in non-linear semiparametric and nonparametric models. The second reason is that even if mean-independence restrictions are sufficient to identify 1See Bickel et al. (1993) for the formal definition. 1 the parameter of interest, imposing independence can be justified by economic conditions. The additional information included in the independence restriction can potentially be used to derive estimators with smaller asymptotic variances. In either case, using an efficient estimator will ensure that the estimator’s large-sample properties cannot be improved on. We perform efficiency bound calculations for general classes of model where a residual function(Y −Wθ forexample)isindependent(orconditionallyindependent)ofanexogenous variables(X). Wealsoallowforthepresenceofanunknownfunctionintheresidualfunction, andcomputethesemiparametricefficiencyboundforthefinitedimensionalparameterinthat case. The estimator proposed uses a framework similar to that of efficient GMM with two differences. The first is that we use covariance restrictions rather than moment restrictions, which leads to a different optimal weighting matrix. The estimation procedure is based on an increasing number of zero-covariance conditions between some functions of the error and the exogenous variables. Second, the number of restrictions is growing with the sample size and we derive maximal rates at which that number grows to infinity. We will show that by letting the functions considered be in specific classes, independence will be asymptotically equivalent to the zero-covariance conditions, as their number increases. We will further characterize results when the class of function chosen are complex exponential functions, by using facts about characteristic functions. 1.1.1 Related Literature Efficiency bound calculations for mean-independence restrictions were performed in Cham- berlain (1987) and Bickel et al. (1993), and efficient estimators were developed in Newey (1990b) and Newey (1993). For models with the stronger unconditional independence re- strictions, early results can be found in MaCurdy (1982), Newey (1989) and Newey (1990a). MaCurdy(1982)showsthatusingzero-covariancerestrictionsbetweenhighermomentsyields asymptotic efficiency improvements. Both Newey (1989) and Newey (1990a) propose an esti- mator that minimizes a V-statistic based on an approximation to the efficient score. Newey (1989) constructs a locally efficient estimator, meaning that efficiency is achieved if one correctly postulates a parametric family for the unobserved error’s distribution, while the estimator in Newey (1990a) is globally efficient but requires additional assumptions since it nonparametrically estimates the distribution of the error in a first-stage estimate. Hansen et al. (2010) consider a linear instrumental variables system with the instruments inde- pendent from the error term and propose a locally efficient estimator. By contrast, the estimator we propose is globally efficient, and is obtained by minimizing a GMM objective function with an optimal weighting matrix. Manski (1983) also proposed the “closest empir- ical distribution” approach, and Brown and Wegkamp (2002) derived asymptotic properties of estimators based on this approach. Though not considered in this chapter, empirical like- lihood estimators, such as those proposed in Donald et al. (2003) and Donald et al. (2008) for mean-independence restrictions, can also attain efficiency bounds and often exhibit better small-sample properties than the corresponding two-step GMM estimator. An alternative approach for deriving an efficient estimator was proposed in Carrasco and Florens(2000). Theirestimatorisbasedontheestimationofamethodofmomentsestimator 2