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Licensed Unlicensed Requires Authentication Published by De Gruyter November 18, 2015

Nonparametric Instrumental Variable Estimation in Practice

Philip Shaw, Michael Andrew Cohen and Tao Chen

Abstract

This paper investigates recent developments in the literature on nonparametric instrumental variables estimation and considers the practical importance of the features of these estimators in the context of typically applied econometric models. Our primary focus is on the estimation of econometric models with endogenous regressors, and their marginal effects, without a known functional form. We develop an estimator for the marginal effects and investigate its finite sample performance. We show that when instruments are weak, in the classic sense, the nonparametric estimates of the marginal effect outperforms the classic two-stage least squares estimator, even when the model is correctly specified. When the instruments are strong, we show that the nonparametric estimator for the partial effects is still effective compared to the two-stage least squares estimator even as the number of IVs increases. We also investigate bandwidth choice and find that a rule-of-thumb bandwidth performs relatively well. Whereas cross-validation leads to a better fit when the number of instruments is small, as the number of instruments increases the rule-of-thumb standard actually results in better model fit. In an empirical application we estimate the work-horse aggregate logit demand model, discuss the required nonparametric identification properties, and document the differences between nonparametric and parametric specifications on the estimation of demand elasticities.

JEL codes: C13; C14; C15

Corresponding author: Philip Shaw, Fordham University – Economics, 441 E. Fordham Rd., Bronx, NY 10458, USA, E-mail:

Appendix

It is well known that for a sequence of convergence functions, in general we cannot infer the convergence of the derivative functions. For a detailed discussion, we refer to (Rudin 1976, Chapter 7). As it is desirable to estimate the derivative of the limiting function, we propose the following estimator:

ϕ^(y2,x1)y2:=ϕ^(y2+cn,x1)ϕ^(y2,x1)cn,

for some cn0. Now let’s derive the conditions we would impose on cn such that

limnϕ^n(y2,x1)y2=ϕ(y2,x1)y2,

where

ϕ(y2,x1)y2:=limnϕ(y2+cn,x1)ϕ(y2,x1)cn.

limn(ϕ^n(y2,x1)y2ϕ(y2,x1)y2)=limn(ϕ^(y2+cn,x1)ϕ^(y2,x1)cnlimnϕ(y2+cn,x1)ϕ(y2,x1)cn)=limn(ϕ^(y2+cn,x1)ϕ^(y2,x1)cnϕ(y2+cn,x1)ϕ(y2,x1)cn)=limn[ϕ^(y2+cn,x1)ϕ(y2+cn,x1)][ϕ^(y2,x1)ϕ(y2,x1)]cnlimn|ϕ^(y2+cn,x1)ϕ(y2+cn,x1)|+|ϕ^(y2,x1)ϕ(y2,x1)|cn.

If we use the estimator defined by GS, then by their Proposition 2,

sup|ϕ^(y2+cn,x1)ϕ(y2+cn,x1)|=sup|ϕ^(y2,x1)ϕ(y2,x1)|=OP((logn)nκ),

for some κ>0. Therefore, if we choose cn =O((log n)n−γ) for some γ∈(0, κ), we will obtain

limn(ϕ^n(y2,x1)y2ϕ(y2,x1)y2)=OP(1).

A similar argument could be made for BCK’s estimator, but with a possibly different γ.

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Supplemental Material

The online version of this article (DOI: 10.1515/jem-2013-0002) offers supplementary material, available to authorized users.


Published Online: 2015-11-18
Published in Print: 2016-1-1

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