To give a background of the problem: I'm performing a difference-in-differences analysis to estimate the effect of a treatment, say A. I have a panel data with 5 rounds and about 3000 observations in each round. The treatment was introduced between rounds 2 and 3, thus treated rounds are 3,4,5 while untreated rounds are 1,2. The treatment intensity varies at the level of locality, the data for which have been obtained from a separate administrative source and matched with the survey data. The introduction of the treatment was staggered at the level of locality so I also have a measure of years of exposure to treatment. I intend to obtain the difference-in-difference estimator from interaction between treatment dummy,years of exposure, and treatment intensity.
Now the problem is data for some years (corresponding to round 5 of the survey) are missing in the administrative records. For all the localities, treatment intensity data corresponding to round 5 are missing. Out of these localities, some localities have data on treatment intensity at a later year.
I'm considering using the later year's treatment intensity as a proxy for round 5 treatment intensity for the localities available (which would introduce measurement errors), and leaving the rest of the localities missing on treatment intensity in that round. Can this be done?