Does it make sense to conduct an instrumental variables model where the endogenous variable of interest is continuous but not normally distributed? I know for normal regression purposes, there is no requirement that the independent variables be normally distributed (though there can be interpretation issues). However, with an IV model, if you use two-stage least squares for example, the first stage equation would have the endogenous variable as the dependent variable. While one isn't necessarily interpreting the coefficients and examining the standard errors and p-values for that stage, would we still need to assume normality? I plan on bootstrapping standard errors, if that would help ease some of the restrictions.
Well, normality should remain irrelevant for estimating effects in TSLS. LS has no normality assumption. Normality can affect inference. But it is normality of errors that we care about. Are the residuals from the first model non-normal? What about sample size? You can try a simulation for a sample size like yours with a non normal error for the endogenous predictor conditional on the instrument and covariates. My hunch is it does not matter so much given moderate sample size.