Coefficient stability alone is not evidence: a coefficient that
barely moves when controls are added is uninformative unless the
controls also moved the R-squared. Oster's adjustment ties the two
together through delta, the ratio of selection on
unobservables to selection on observables.
Arguments
- beta_short
Treatment coefficient without controls.
- beta_long
Treatment coefficient with controls.
- R_short
R-squared without controls.
- R_long
R-squared with controls; must exceed
R_short.- R_max
R-squared of the hypothetical regression on treatment plus all observed and unobserved controls.
- delta
Proportional-selection coefficient.
Details
Formula: \(beta* = beta_long - delta (beta_short - beta_long)
(R_max - R_long) / (R_long - R_short)\). Also returned is
\(delta_star = beta_long (R_long - R_short) /
((beta_short - beta_long) (R_max - R_long))\), the value of
delta at which beta* is exactly zero.