is endogeneity a problem in the first-stage regression in a two-stage least square regression? We have this model:
$$D=\alpha_1+\gamma_1 Z + \epsilon_1$$
$$Y=\alpha_2+\gamma_2 D + \epsilon_2$$
Notations are as usual: Y is the outcome, D is the treatment, Z is the instrument
Two conditions for instrument variable to work in the literature: relevance (there is a correlation between Z and D) and exogeneity (no correlation between Z and error term $\epsilon_2$)
However, it seems no one cares about endogeneity in the first stage regression, which is the correlation between Z and $\epsilon_1$. Why is that the case? I have never seen any applied micro paper addressing this.
Yes it is a problem. The first stage itself has to satisfy the same assumptions that standard OLS would and $cov(Z,\epsilon_1)\neq 0$ would violate them (see A Guide to Modern Econometrics by Verbeek).
Furthermore, actually the two conditions you mention are not enough. The instrument should also not be 'weak', that is the first stage should have $F$-statistics above $10$ (as a rule of thumb). Also the instrument should have an effect on $Y$ only through $D$ - the so called exclusion restriction (see Mostly Harmless Econometrics by Angrist and Pischke).