Babii, Andrii (2017) Essays on econometrics models with endogeneity. Toulouse School of Economics (Toulouse).

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Abstract

Le but du présent travail est d’analyser quelques problèmes d’estimation et d’inférence dans les modèles avec endogénéité. Le premier chapitre se concentre sur les problèmes d’inférence dans les modèles mal-posés. On construit les intervalles de confiance uniformes pour le modèle non-paramétrique instrumental, les modèles de régression fonctionnelle linéaire, et le modèle de déconvolution des densités. Le second chapitre propose un type particulier de modèle de régression linéaire à variables instrumentales avec une composante endogène de grande dimension, appelé la régression instrumentale fonctionnelle linéaire (FLIR). Nous montrons que sous de faibles conditions de régularité, l’identification de ce modèle peut être obtenue à travers une seule variable instrumentale réelle qu’on peut obtenir l’indentification dans cette modèle avec la variable instrumentale réelle. Deux estimateurs basés sur la régularisation de Tikhonov et de sieve sont étudiés. Le chapitre 3, écrit en collaboration avec Jean-Pierre Florens, propose un test d’hypothèse de la séparabilité de la régression avec endogénéité. Le dernier chapitre, écrit en collaboration avec Jean-Pierre Florens est consacré à l’estimation et l’inférence pour les modèles mal-posés avec l’identification faible. Nous démontrons que dans ce contexte, les estimateurs d’une large classe, comprenant l’estimateur de Tikhonov, Landweber-Fridman, et la coupure spectrale, convergent vers la meilleure approximation de la fonction structurelle.

This thesis consists of four independent chapters on estimation, inference, and testing in nonparametric and high-dimensional econometric models with endogneity. The first chapter provides novel methods for inference in a very general class of ill-posed models in econometrics, encompassing the nonparametric instrumental regression, different functional regressions, and density deconvolution. This chapter addresses the problem of construction of uniform confidence sets for the parameter of interest estimated with Tikhonov regularization. It is shown that it is not possible to develop inferential methods directly based on the uniform central limit theorem. To circumvent this difficulty two approaches that lead to valid confidence sets are developed. Expected diameters and coverage properties are studied uniformly over a large class of models (i.e. constructed confidence sets are honest). Finally, using Monte Carlo simulations and considering application to Engel curves, it is demonstrated that introduced confidence sets have reasonable width and coverage properties in samples commonly used in applications. In the second chapter I study a variation of the linear IV regression model with high-dimensional endogenous component, called the functional linear instrumental regression (FLIR). The distinguishing feature of the model is that it can handle high-dimensionality, without relying on sparsity restrictions. It is shown that identification in this model can be achieved with a single real-valued instrumental variable under the linear completeness condition. Two estimators based on the Tikhonov and sieve regularizations are studied. Upper bounds on the mean-integrated squared errors and corresponding convergence rates are obtained. In the third chapter in collaboration with Jean-Pierre Florens, we develop a uniform asymptotic expansion for the empirical distribution function of residuals in the nonparametric IV regression. Such expansion opens a door for building a broad range of residual-based specification tests in nonparametric IV models. Building on obtained result, we develop a test for the separability of unobservable in econometric models with endogeneity. The test is based on verifying the independence condition between residuals of the NPIV estimator and the instrument and can distinguish between the non-separable and the separable specification under endogeneity. The last fourth chapter studies estimation and inference in non-identified and/or weakly identified ill-posed inverse models. We show that in the case of identification failures, a very general family of continuously-regularized estimators is consistent for the best approximation of the parameter of interest and obtain L2 and L∞ convergence rates for this general class of regularization schemes, including Tikhonov, iterated Tikhonov, Landweber-Fridman, and spectral 1 cut-off. Unlike in the identified case, estimation of the operator has non-negligible impact on the estimation and inference. We develop inferential methods for linear functionals in potentially non-identified models. Lastly, we demonstrate the discontinuity in the asymptotic distribution in case of weak identification. In particular, the estimator has a degenerate U-process type behavior, in the extreme case of weak identification.