For simplicity, I'll offer the simplest of endogenous models. Suppose we have the following scalar model:

$y=\beta x+u$ where $E[ux]\neq 0,\;E[uz]=0\; \text{and}\;E[u^2|z]=\sigma^2$

Clearly we can use the following instrument to identify $\beta$:

$$E[(y-\beta x)z]=0$$ However, suppose we have another moment condition using another variable, $w$: $$E[zw]=0$$ This moment condition doesn't contain any information about $\beta$, so under what conditions would we gain efficiency by using this additional moment condition?

  • $\begingroup$ See: Breusch, T., H. Qian, P. Schmidt, and D. Wyhowski (1999), Redundancy of moment conditions, Journal of Econometrics 91, 89-111. $\endgroup$
    – chan1142
    Dec 20 '16 at 1:23

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