yi2 − yi1 wi2 − wi1 ui2 − ui1
Δyi Δwi Δui.
If EΔwiΔui 0, that is, the change in treatment status is uncorrelated with changes in the idiosyncratic errors, then OLS applied to (13) is consistent. ∙ If wi1 0 for all i, the OLS estimate is
̂ Δȳtreat − Δȳcontrol,
which is a DD estimate except that we different the means of the same units over time. ∙ With many time periods and arbitrary treatment patterns, we can use
yit t wit xit ci uit, t 1, . . . , T,