33.5 Negative $R^2$

It’s okay to have negative $R^2$ in the 2nd stage. We care more about consistent coefficient estimates.

$R^2$ has no statistical meaning in instrumental variable regression or 2 or 3SLS

$$R^2 = \frac{MSS}{TSS}$$

where

• MSS = model sum of squares (TSS- RSS)
• TSS = total sum of squares ($\sum(y - \bar{y})^2$)
• RSS = residual sum of squares ($\sum (y - Xb)^2$)

If $TSS > RSS$, then we have negative RSS and negative $R^2$. Since the predicted values of the endogenous variables are different from the endogenous variables themselves, the error that is used to calculate RSS can be different from the error in the second stage, and RSS in the second stage can be less than TSS.