Essays in Nonparamentric Estimation with Instrumental Variables

This thesis deals with the broad problem of causality and endogeneity in econometrics when the function of interest is estimated nonparametrically. It explores this problem in two separate frameworks. In the cross sectional, iid setting, it considers the estimation of a nonlinear additively separable model, in which the regression function depends on an endogenous explanatory variable. Endogeneity is, in this case, broadly defined. It can relate to reverse causality (the dependent variable can also affects the independent regressor) or to simultaneity (the error term contains information that can be related to the explanatory variable). Identification and estimation of the regression function is performed using the method of instrumental variables. In the time series context, it studies the implications of the assumption of exogeneity in a regression type model in continuous time. In this model, the state variable depends on its past values, but also on some external covariates and the researcher is interested in the nonparametric estimation of both the conditional mean and the conditional variance functions. This first chapter deals with the latter topic. In particular, we give sufficient conditions under which the researcher can make meaningful inference in such a model. It shows that noncausality is a sufficient condition for exogeneity if the researcher is not willing to make any assumption on the dynamics of the covariate process. However, if the researcher is willing to assume that the covariate process follows a simple stochastic differential equation, then the assumption of noncausality becomes irrelevant. Chapters two to four are instead completely devoted to the simple iid model. The function of interest is known to be the solution of an inverse problem. In the second chapter, this estimation problem is considered when the regularization is achieved using a penalization on the L2-norm of the function of interest (so-called Tikhonov regularization). We derive the properties of a leave-one-out cross validation criterion in order to choose the regularization parameter. In the third chapter, coauthored with Jean-Pierre Florens, we extend this model to the case in which the dependent variable is not directly observed, but only a binary transformation of it. We show that identification can be obtained via the decomposition of the dependent variable on the space spanned by the instruments, when the residuals in this reduced form model are taken to have a known distribution. We finally show that, under these assumptions, the consistency properties of the estimator are preserved. Finally, chapter four, coauthored with Frédérique Fève and Jean-Pierre Florens, performs a numerical study, in which the properties of several regularization techniques are investigated. In particular, we gather data-driven techniques for the sequential choice of the smoothing and the regularization parameters and we assess the validity of wild bootstrap in nonparametric instrumental regressions.