Abstract
Abstract When the endogenous variables enter non-parametrically into the regression equation standard linear instrumental variables approaches fail. Two existing solutions are the non-parametric instrumental variables (NPIVs) estimators, which are based on a set of conditional moment restrictions (CMRs), and the control function (CF) estimators, which use conditional mean independence (CMI) restrictions. Our first contribution is to show that – similar to CMI – the CMR place shape restrictions on the conditional expectation of the error given the instruments and endogenous variables that are sufficient for identification, and we call our new estimator based on these restrictions the CMR-CF estimator. Our second contribution is to develop an estimator for non-linear and non-parametric settings that can combine both CMR and CMI restrictions, which cannot be done in either the NPIV nor the non-parametric CF setting. This new “Generalized CMR-CF” uses both CMR and CMI restrictions together by allowing the conditional expectation of the structural error to depend on both instruments and control variables. When sieves are used to approximate both the structural function and the CF our estimator reduces to a series of least squares regressions. Our Monte Carlos illustrate that our new estimator performs well across several economic settings.
Highlights
Two existing solutions are the non-parametric instrumental variables (NPIVs) estimators, which are based on a set of conditional moment restrictions (CMRs), and the control function (CF) estimators, which use conditional mean independence (CMI) restrictions
Our first contribution is to show that – similar to CMI – the CMR place shape restrictions on the conditional expectation of the error given the instruments and endogenous variables that are sufficient for identification, and we call our new estimator based on these restrictions the CMR-CF estimator
Our second contribution is to develop an estimator for non-linear and non-parametric settings that can combine both CMR and CMI restrictions, which cannot be done in either the NPIV nor the non-parametric CF setting
Summary
The problem of endogenous regressors in simultaneous equations models has a long history in econometrics and empirical studies. We consider a triangular non-parametric simultaneous equations model with endogeneity to illustrate our main idea. Let xi denote the endogenous regressors and zi denote the instruments. The unknown structural function of interest is f0(xi, zif ), where zi = (zif , zi,−f ) with zif the subset of zi entering the structural function. The model is given as: yi = f0(xi, zi f ) + εi. This work is licensed under the Creative Commons. The econometric problem is further complicated as linear instrumental variable (IV) approaches are no longer consistent because the endogenous variables enter the structural equation non-parametrically
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