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I would like to know whether I can compare instrumental variables quantile estimates, for instance using the the Chernozhukov and Hansen (2006) IVQR estimator (or quantile regression with fixed effects), to standard quantile regressions via the Hausman test in order to test for endogeneity at different quantiles of the outcome distribution. Due to the different properties of the quantile and the expectations operators I am not sure whether this comparison can be made in the same way.

Is someone aware of a paper that extends the Hausman test to the quantile regression setting or are there reasons for why this is not feasible?

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