Instrumental Variables Estimation and Weak-Identification-Robust Inference Based on a Conditional Quantile Restriction

Abstract

Extending the L1-IV estimator proposed by Sakata (1997Sakata ( , 2007, we develop a method, named the ρ τ -IV estimator, to estimate structural equations based on the conditional quantile restriction imposed on the error terms. We study its asymptotic behavior and show how to make statistical inferences on the regression parameters. Given practical importance of weak identification, a highlight of the paper is proposal of a test robust to the weak identification. The statistics used in our method can be viewed as a natural counterpart of the Anderson and Rubin's (1949) statistic in the ρτ -IV estimation. As is the case for the Anderson-Rubin test in the standard IV regression, our test can have a low power, when there many instruments excluded from the regression equation. We show how we could avoid such problem 1 by taking for instruments a small number of linear combinations of all available instruments that are suitably selected based on data.

Quantile models with endogeneity

“…Chernozhukov and Hansen (2008) show that confidence region in equation (5.23) is valid when the model parameters are strongly identified and remains valid when the model is weakly identified or even unidentified. Marmer and Sakata (2012) provide a similar procedure and result for their procedure as well. Jun (2008) provides yet a different approach to performing weak-identificationrobust inference in models defined by conditions (2.3).…”

“…This representation of the identification region Q is implicit. Nevertheless, statistical inference about q ∈ Q can be based on (2.5) and can be carried out in practice using weak-identification robust inference as described in Chernozhukov and Hansen (2008), Marmer and Sakata (2012), Jun (2008), Santos (2012), or Chernozhukov, Hansen, and Jansson (2009). Under conditions that yield point identification, these regions collapse to a singleton, and the aforementioned weak-identification-robust inference procedures retain their validity.…”

“…Chernozhukov and Hansen (2006) and Chernozhukov and Hansen (2008) provide conditions under which the resulting estimator for α(τ ) and β(τ ) is consistent and asymptotically normal and provide a consistent variance estimator. Marmer and Sakata (2012) provide a similar multi-step algorithm that circumvents the same numeric problems using the objective function of Sakata (2007).…”