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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.

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  • $\begingroup$ 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. And what interpretational issues exist for non-normal predictors in regression? $\endgroup$ – Heteroskedastic Jim Oct 3 '18 at 12:51
  • $\begingroup$ This is useful, I hadn't thought of simulating to explore. Thanks. If you drop this in an answer I'll accept it. Results showed there wasn't an impact on estimates. $\endgroup$ – robin.datadrivers Oct 3 '18 at 15:29
  • $\begingroup$ Done, thanks. I'm still curious about the interpretational issues that exist for non-normal predictors in regression? $\endgroup$ – Heteroskedastic Jim Oct 3 '18 at 15:32
  • $\begingroup$ Since coefficients are marginal changes, sometimes I think a marginal change for a variable that may have a non-normal distribution may not make sense for certain contexts. A lumpy or multi-modal distribution, for example, may make more sense categorical or ordinal. Sometimes logging a non-normal predictor makes an interpretation as an elasticity easier. I don't think it is required for statistical validity, so speaking mostly from my experience working with different audiences. $\endgroup$ – robin.datadrivers Oct 3 '18 at 16:21
  • $\begingroup$ oh, I see, you meant it as an issue of specification. $\endgroup$ – Heteroskedastic Jim Oct 3 '18 at 16:53
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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.

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