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I actually found the problem. The higher moments IV approach requires the endogenous regressors to be non-normal. In my example above, the logs of $Z_1$ and $Z_2$ are almost normal. If I change rlnorm(N,0,1) + 10 in the data generating process to e.g. rlnorm(N,0,1.5) + 10 also the logs of $Z_1$ and $Z_2$ remain non-normal and the IVs for the log-log model ...

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The OLS regression of $Y$ on $X$ decomposes dependent variable $Y$ into orthogonal components $$Y = \hat{Y} + e$$ in $\mathbb{R}^n$, where $\hat{Y} = X \hat{\beta}$ is the fitted value and $e$ the residuals. It's a basic fact of linear algebra that this orthogonal decomposition remains the same, for all regressor matrix $X$ with the same column space. The ...

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Yes, in fact, you have to use the same instruments for $X_1$, $X_2$, and $X_1X_2$. Just to be clear. You need as least as many instruments as endogenous variables, i.e., three in this case counting the interaction. However, two would suffice plus the interaction of those two instruments. All instruments are used in all first stages.

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