Suppose I have $y = \beta_1x_1 + \beta_2x_2$, where $x_1$ is endogenous but $x_2$ is not. I have a relatively weak exogenous instrument for $x_1$ (call this $z$).
If I use instrumental variables, the standard error for $\beta_1$ will be large relative to the OLS estimator (the well known weak instrument problem). What about the standard error for $\beta_2$? Will this be substantially larger, or smaller?