Nonparametric Instrumental Variable Estimation Under Monotonicity

Abstract

The ill‐posedness of the nonparametric instrumental variable (NPIV) model leads to estimators that may suffer from poor statistical performance. In this paper, we explore the possibility of imposing shape restrictions to improve the performance of the NPIV estimators. We assume that the function to be estimated is monotone and consider a sieve estimator that enforces this monotonicity constraint. We define a constrained measure of ill‐posedness that is relevant for the constrained estimator and show that, under a monotone IV assumption and certain other mild regularity conditions, this measure is bounded uniformly over the dimension of the sieve space. This finding is in stark contrast to the well‐known result that the unconstrained sieve measure of ill‐posedness that is relevant for the unconstrained estimator grows to infinity with the dimension of the sieve space. Based on this result, we derive a novel non‐asymptotic error bound for the constrained estimator. The bound gives a set of data‐generating processes for which the monotonicity constraint has a particularly strong regularization effect and considerably improves the performance of the estimator. The form of the bound implies that the regularization effect can be strong even in large samples and even if the function to be estimated is steep, particularly so if the NPIV model is severely ill‐posed. Our simulation study confirms these findings and reveals the potential for large performance gains from imposing the monotonicity constraint.

Testing for the presence of measurement error in Stata

“…Unlike H 0 , this is a restriction that depends only on observables and can directly be tested without making any parametric assumptions about how the conditional mean of Y depends on X * . The test by Delgado and González Manteiga (2001) introduced in the next subsection and implemented in the new command, dgmtest, for instance, directly tests the restriction (5). Because of the above argument, it can be interpreted as a test of the original null of interest, the null of no measurement error in (1).…”

Testing for the presence of measurement error in Stata

“…when the conditional mean E(Y |X * = x * ) is a production, cost, or utility function. Examples can be found in Matzkin (1994), Olley and Pakes (1996), Cunha et al (2010), Blundell et al (2012Blundell et al ( , 2016, Kasy (2014), Wilhelm (2015), Hoderlein et al (2016), Chetverikov and Wilhelm (2017), among many others.…”

An adaptive test of stochastic monotonicity

“…Such a test is useful in many economic applications, for example for testing monotone IV assumptions (e.g. Kasy (2014), Hoderlein, Holzmann, Kasy, and Meister (2016), Chetverikov and Wilhelm (2017), Wilhelm (2017)) and for testing identifying assumptions (e.g. Matzkin (1994), Lewbel and Linton (2007), Banerjee, Mukherjee, and Mishra (2009)).…”