Optimal Instruments in High Dimensional models

In an increasing number of empirical studies, the dimensionality measured e.g. as the number of variables in a model, can be very large. Two instances of large dimensional models are the linear regression with a large number of covariates and the estimation of a regression function with many instrumental variables. We will recall these examples in the talk by means of some economic examples. We also recall why classical least square or IV estimators behaves poorly in such large dimensional regression problems.

An appropriate setting to analyze high dimensional problems is provided by a functional linear model, in which the covariates are a vector in Rp for large p (p can tend to infinity). More generally we consider that covariates belongs to some Hilbert space. We also consider the case where covariates are endogenous and assume the existence of instrumental variables (that are functional as well).

In this talk we show that estimating the regression function is a linear ill-posed inverse problem, with a known but data-dependent operator. Our first contribution is to analyse the rate of convergence of the Tikhonov regularized estimator, when we premultiply the problem by an instrument dependent operator. This extends the technology of Generalized Method of Moments to functional (GMM) to high dimensional (or functional) data. We then discuss the optimal choice of the premultiplication operator and propose an extension of the notion of “weak instrument” to this nonparametric framework. A central limit theorem is also established on the inner product of the estimator. The good finite-sample performance of the resulting nonparametric estimator is

also showed through simulations.

This is a joint work with Jean-Pierre Florens (University of Toulouse 1).