Essays on semiparametric identification and estimation

• Autores: Telmo juan Pérez Izquierdo
• Directores de la Tesis: Juan Carlos Escanciano Reyero (dir. tes.)
• Lectura: En la Universidad Carlos III de Madrid ( España ) en 2023
• Idioma: inglés
• Tribunal Calificador de la Tesis: Juan M. Rodríguez-Poo (presid.), Carlos Velasco Gómez (secret.), Taisuke Otsu (voc.)
• Programa de doctorado: Programa de Doctorado en Economía por la Universidad Carlos III de Madrid
• Materias:
  • Matemáticas
    • Ciencia de los ordenadores
      • Modelos causales
    • Estadística
      • Estadística analítica
      • Métodos de distribución libre y no paramétrica
  • Ciencias económicas
    • Econometría
      • Modelos econométricos
• Enlaces
  • Tesis en acceso abierto en: e-Archivo
• Resumen

  • The unifying theme of this dissertation is semiparametric identification and estimation. A parameter is semiparametrically identified if the researcher can recover its value from the available data, even if the data does not completely pin down the whole model. This framework covers several parameters of economic interest: treatment effects, marginal effects or counterfactual parameters.

    Chapter 1 of the dissertation specifically studies a counterfactual parameter: the Counterfactual Average Structural Function (CASF). The parameter is an average with respect to a counterfactual distribution (which may be induced by a policy). I give conditions on the counterfactual distribution so that the CASF is identified via a Control Function Assumption. I also examine under which conditions is the CASF root-n estimable (that is, regularly identified).

    The researcher must estimate a Control Function prior to computing the CASF. The Control Function acts then as a regressor in an intermediate step. Estimation of the CASF is thus a problem of generated regressors. Chapter 2 of this dissertation studies estimation of parameters that are (regularly) identified by moment conditions in the presence of generated regressors. It proposes to modify the original moment condition to construct a Locally Robust version. Locally Robust moment conditions are locally insensitive to estimation done in previous steps and, therefore, they allow for regressors to be generated by Machine Learning techniques.

    Whether the CASF is regularly identified or not depends on a finite variance condition. Chapter 3 of the dissertation formally analyzes the link between finite variance and regular identification. It does so in the general framework of a model generated by a (possibly infinite dimensional) parameter living in a normed space. The chapter studies semiparametric identification and root-n estimability in this setup.

    Chapter 1. The determinants of counterfactual identification in the binary choice model with endogenous regressors: The CASF is the Average Structural Function (ASF) averaged with respect to a counterfactual distribution for covariates. While the ASF is irregularly identified (not root-n estimable) for continuous regressors, the CASF may be regularly identified. This chapter shows that, under a control function assumption, the CASF is non-parametrically identified as a weighted average. The weight is given by the likelihood ratio between the counterfactual density and the conditional density of the regressors given the control variables. Using this identifying moment condition, I obtain a necessary condition for regular identification of the CASF. The necessary condition depends on instrument strength, the degree of endogeneity, and the relevance of the regressors (i.e., how sensitive the outcome is to changes in regressors). Moreover, for normal DGPs, the necessary condition maps to a restriction on the structural parameters of the model. This provides further insights about what determines regular identification of the CASF. For instance, I find that if the first-stage R-squared is below a certain threshold, regular identification of the CASF is not possible.

    Chapter 2. Automatic Locally Robust Estimation with Generated Regressors (with Juan Carlos Escanciano): Many economic and causal parameters depend on generated regressors. Examples include structural parameters in models with endogenous variables estimated by Control Functions and in models with sample selection, treatment effect estimation with propensity score matching, and marginal treatment effects. Inference with generated regressors is complicated by the very complex expression for influence functions and asymptotic variances. To address this problem, we propose automatic Locally Robust/debiased GMM estimators in a general setting with generated regressors. Importantly, we allow for the generated regressors to be generated from machine learners, such as Random Forest, Neural Nets, Boosting, and many others. We use our results to construct novel Double-Robust estimators for the CASF and Average Partial Effects in models with endogeneity and sample selection, respectively.

    Chapter 3. Identification and information in models parametrized by a normed space: This chapter generalizes Escanciano (2022)’s identification result and van der Vaart (1991)’s information result to models parametrized by an arbitrary normed space. It provides a general framework to study identification and its regularity (regular if the Fisher Information for the parameter is positive, irregular if it is zero) for a wider range of problems. In particular, this chapter argues that non-parametric estimation of (a) the mean of a known transformation and (b) the density at a point naturally give raise to a Banach tangent space. Using the extended framework, the chapter shows that Fisher Information for estimation of the mean of a transformation is zero if the transformation is not square-integrable. Information for estimation of the density at a point is zero if one considers an absolutely continuous (Lebesgue) density.