Instrumental regression in partially linear models

Abstract

We consider the semiparametric regression X t β+φ(Z) where β and φ(•) are unknown slope coefficient vector and function, and where the variables (X, Z) are endogeneous. We propose necessary and sufficient conditions for the identification of the parameters in the presence of instrumental variables. We also focus on the estimation of β. An incorrect parameterization of φ may generally lead to an inconsistent estimator of β, whereas even consistent nonparametric estimators for φ imply a slow rate of convergence of the estimator of β. An additional complication is that the solution of the equation necessitates the inversion of a compact operator that has to be estimated nonparametrically. In general this inversion is not stable, thus the estimation of β is ill-posed. In this paper, a √ n-consistent estimator for β is derived under mild assumptions. One of these assumptions is given by the so-called source condition that is explicitly interprated in the paper. Finally we show that the estimator achieves the semiparametric efficiency bound, even if the model is heteroscedastic. Monte Carlo simulations demonstrate the reasonable performance of the estimation procedure on finite samples.

Doubly robust instrumental variable regression

“…Because it is difficult to find a good parametric model for F (X i ), semiparametric regression that does not require one to specify a parametric model for F (X i ) is a potentially appealing alternative to TSLS. Robinson (1988), Ai and Chen (2003) and Florens, Johannes, and van Bellegem (2005) described approaches to semiparametric IV regression. Robinson (1988) showed that a √ N consistent estimate of α 0 in (1.2) can be obtained, as N → ∞, under certain smoothness conditions; Robinson's method is reviewed below.…”

“…Robinson (1988) showed that a √ N consistent estimate of α 0 in (1.2) can be obtained, as N → ∞, under certain smoothness conditions; Robinson's method is reviewed below. Ai and Chen (2003) considered more general semiparametric problems, and Florens, Johannes, and van Bellegem (2005) focused on the partial linear IV model but allowed X i to be endogenous. The difficulty with these semiparametric approaches is that, even when they are √ N consistent, when X i is of moderate or high dimension relative to the sample size, the semiparametric estimators' finite sample behavior deteriorates because of the curse of dimensionality (see, e.g., Robins and Ritov (1997)).…”

The practice of non‐parametric estimation by solving inverse problems: the example of transformation models

“…3 See Darolles et al (2003) or Hall and Horowitz (2005). 4 See Darolles et al (2003) and Florens et al (2006). This term represents the difference between the true function and the regularized solution of the 'true' problem T ϕ = r ((αI + T * T ) −1 T * T ϕ).…”

Nonparametric estimation of an instrumental regression: A quasi-Bayesian approach based on regularized posterior

“…In fact, the estimation error associated withK (n) is negligible with respect to the other terms in the bias and variance. In the following theorem we focus on the consistency ofÊ α (ϕ|y (n) ); consistency ofμ Theorem 4 Let ϕ * be the true value having generated the data y (n) under model (4) and µ y α be a gaussian measure on L 2 F (Z) with mean and covariance operator defined in (14). Under Assumptions 4 and 5, if α n → 0 and α 2 n n → ∞, we have ||Ê α (ϕ|y (n) ) − ϕ * || 2 → 0 in Pf ,ϕ * ,w -probability and if δ * ∈ Φ β , for some β > 0,…”