Additive Nonparametric Instrumental Regressions: A Guide to Implementation

Abstract

We present a review on the implementation of regularization methods for the estimation of additive nonparametric regression models with instrumental variables. We consider various versions of Tikhonov, Landweber-Fridman and Sieve (Petrov-Galerkin) regularization. We review data-driven techniques for the sequential choice of the smoothing and the regularization parameters. Through Monte Carlo simulations, we discuss the finite sample properties of each regularization method for different smoothness properties of the regression function. Finally, we present an application to the estimation of the Engel curve for food in a sample of rural households in Pakistan, where a partially linear specification is described that allows one to embed other exogenous covariates.

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“…However, the smallest eigenvalues of the matrix can get arbitrarily close to zero and therefore, in practice, the direct inversion of may lead to an explosive, noncontinuous solution. For more intuition, Centorrino et al (); Horowitz () also provides an excellent review of ill‐posedness and regularization in economics.To bypass the ill‐posedness, we regularize , which effectively entails choosing a regularization (tuning) parameter to make the problem well posed. In essence, the regularization procedure replaces with its continuous transformation to rule out explosive solutions.…”

“…In this paper, we adopt an alternative Landweber–Fridman regularization technique, with its primary appeal being that it is iterative, thereby not requiring direct inversion of a large‐dimensional matrix , which is of particular importance to us given large values that n takes in our application. Recent applications of the Landweber–Fridman regularization include Centorrino (), Centorrino et al (), Florens et al (), and Centorrino et al ().The sample analogue of the identifying equation (Equation ) is which defines the estimator of ψ as a solution of this large‐dimensional system of equation. In light of the ill‐posed inverse problem, the system in Equation is expected to be almost singular in the finite sample, which is why we are to regularize it.To obtain consistent estimates of r , and to be used in Equation , we employ series regressions.…”

“…This yields the estimator of r : where is the matrix of basis functions constructed by stacking up in ascending order of index i first then index t ; the vector y is constructed analogously but using y it .To estimate and , recall that they are conditional expectation operators and hence can be estimated using linear smoothers. Following Centorrino et al (), we approximate via which is essentially a projection matrix onto the space of Z =[ WX ], once we fix L n . Analogously, the approximation of can be obtained from the polynomial series regression of on ( W it , M it ), which yields …”

“…However, the smallest eigenvalues of the matrix can get arbitrarily close to zero and therefore, in practice, the direct inversion of may lead to an explosive, noncontinuous solution. For more intuition, Centorrino et al (); Horowitz () also provides an excellent review of ill‐posedness and regularization in economics.…”

“…In this paper, we adopt an alternative Landweber–Fridman regularization technique, with its primary appeal being that it is iterative, thereby not requiring direct inversion of a large‐dimensional matrix , which is of particular importance to us given large values that n takes in our application. Recent applications of the Landweber–Fridman regularization include Centorrino (), Centorrino et al (), Florens et al (), and Centorrino et al ().…”