An IV estimator for a functional coefficient model with endogenous discrete treatments

Abstract

We propose instrumental variables (IV) estimators for averaged conditional treatment effects and the parameters upon which they depend in the context of a semiparametric outcome model with endogenous discrete treatment variables. For this model, the treatment impacts are unknown functions of a vector of indices that depend on a finite dimensional parameter vector. We develop the theory for an estimator of these impacts when they are averaged over regions of interest. We prove identification, consistency and $\sqrt{N}$-asymptotic normality of the estimators. We also show that they are efficient under correct model specification. Further, we show that they are robust to misspecification of the propensity score model. In the Monte Carlo study, the estimators perform well over a wide variety of designs covering both correct and incorrect propensity score model specification.

Keywords:

• Semiparametric
• instrumental variables
• functional coefficient models

Acknowledgments

We thank the editor and the referees for the very helpful comments and suggestions. We also thank the seminar participants at Columbia University and New York University for their helpful comments.

Notes

1 The theory extends to the multiple index case under the bias control mechanism that we employ.

2 The estimator for the treatment model discussed here will also handle generalizations of the threshold-crossing model of the form: $$T_i=1\left\{g\left(V_{Ti}\left({\gamma }_0\right),u_i\right)\right\}>0.$$

Such a model would arise, for example, if the error term in a threshold model has multiplicative heteroscedasticity that depends on another index.

3 In the Monte Carlo experiments provided here, we report results for the single index case with s = 2 and window parameter $r=\frac{1}{7.99}.$ In multiple index case we select the stage and window parameter as in Shen and Klein (2019).

4 With a tail assumption on the index density in A4), this trimming function will ensure that the index density converges slowly to zero. This trimming with bounded continuous $X$’s greatly simplifies the exposition. However, with unbounded $X$’s and trimming based on lower and upper non-extreme sample quantiles, all results hold.

5 For the current estimator, if $E(\epsilon {\epsilon }^{\prime }|X_F)=\Sigma \ne {\sigma }_{\epsilon }^{2}I,$ consistency and normality results hold with a modified covariance matrix that is estimable. If the model is appropriately transformed based on an estimate for Σ, then all results apply for the current estimator of the transformed model.

6 This assumption is not needed, but greatly simplifies trimming and uniform convergence arguments. It is convenient for expositional purposes to focus on the main features of the estimator.

7 To simplify trimming arguments we assumed that the X’s were bounded. However, all results extend to the unbounded case with trimming based on lower and upper non-extreme sample quantiles. In the Monte Carlo study, we set these quantiles at the.05 and.95 levels respectively.

8 Consistency requires that the sample moment conditions are uniformly close to these population moment conditions. Theorem 2 provides the result.

9 Letting $B_i\equiv W_i{A}_i$ have k^th element $B_{ik},$ from Cauchy-Schwarz $$\sum _{i=1}^N[t_i(\widehat{q})-t_i\left(q\right)]B_{ik}\le {\left[\frac{1}{N}\sum _{i=1}^N{[t_i(\widehat{q})-t_i\left(q\right)]}^2\right]}^{1/2}{\left[\frac{1}{N}\sum _{i=1}^N{B}_{ik}^2\right]}^{1/2},$$

where $\frac{1}{N}{\displaystyle \sum _{i=1}^N{B}_{ik}}\le \frac{1}{N}{\displaystyle \sum _{i=1}^{N}T}r[B_i{B}_i^{\prime }].$

10 The rate is obtained by bounding ${[t_i(\widehat{q})-t_i(q)]}^2$ from above by a smooth function and then employing a Taylor series expansion.

11 In applications of this lemma, trimming ensures that the index variables are in compact sets.

12 Lemma 2 can be used to prove this result.

13 The proof is simplified under this assumption, but only requires $E(S_{jk}|V_j)$ to be finite.