Frisch-Waugh-Lovell Theorem: Different standard errors

The Frisch-Waugh-Lovell theorem states that the b1 vector (and e vector) in Y = X_1 * b_1 + X_2 * b_2 + e are equal to the b* vector (and e* vector) in M_2 * Y = M_2 * X_1 * b* + e*, with M_2 = I - X_2 * (X_2' * X_2)^-1 * X_2', which cleans out the effect of X_2 on Y and X_1. So in short, b_1 = b* and e = e*.

Seeing that these coefficients are unchanged in the partial and multiple regression, I wonder how it is then possible that the standard error of the b_1 coefficient is unequal to the standard error of the b* coefficient, more specifically, the standard error of b_1 (Multiple regression) is always larger than the standard error of b* (Partial regression).

If b_1 = b*, then shouldn't it always hold that their standard errors are also equal?

• Their true standard errors are equal, but you have to adjust the degrees of freedom in the formula for the FWL SEs to account for the fact that more predictors are in the model than appear to the model fitting program. After this adjustment the SEs should be equal.
– Noah
Jul 15 at 14:50