Suppose I am trying to estimate the effect of R&D investment $x_i$ on patents per worker $y_i$ for firms $i$.
Suppose I use as an instrument a R&D subsidy lottery ($z_i = 1$ if won). Winning the lottery grants an R&D subsidy chosen uniformly at random from $[0,\$2S]$ while losing gives a subsidy of $\bar{S}$, where $0<\bar{S}<S$.
Is this a valid instrument? My immediate thought is that the instrument doesn't satisfy the no-defiers assumption necessary for the LATE proof; since winning the lottery increases R&D investment for some relative to the losers, but decreases it for others. On the other hand, the average subsidy for winners $S$ is greater than that of losers $\bar{S}$.
Thanks!
