First of all, let me apologise if this question has been resolved elsewhere on the site / the net. I have been researching a methodology for IV Quantile regression with continuous endogenous variables, and have concluded that the following is probably the best method to use. I am still fairly new to econometrics, and so would like the opinion of someone more knowledgeable than myself as to whether or not my methodology is correct - hence me taking to the forums! My methodology is:
1) Estimate the first stage of the regression, i.e. estimate the endogenous variable, with OLS, by running a regression of the endog.var on instruments as normal. Use bootstrapping here to ensure correct standard errors.
2) Run a quantile regression using predicted values of the endogenous variables in place of the originals, as normal with IV estimation. Again, use bootstrapping for correct standard errors.
Is my approach valid? Most of the literature I've come across on my quest seems to be relating to endogenous binary variables, not continuous ones. If my approach is not valid, do you know of anywhere I could find an explanation of a valid IV Qreg methodology that is sufficiently simple for an amateur statistician to understand?
Thank you very much for your help and expertise.