# Proof for the sampling variance of the Neyman Estimator

I'm going through Imbens and Rubin's new book and I just for the life of me can't figure out 1 minor detail in their proof for the sampling variance of the Neyman estimator $\bar{Y}^{obs}_{t} - \bar{Y}^{obs}_{c}$.

The proof is here - you don't have to go down but a few lines to get to where I'm stuck.

I just can't derive on my own why $Var(D_{i}) = \frac{N_{c}N_{t}}{N^2}$.

Could someone lay out how its derived? Basic question, but significant effort and online searching has proven fruitless.

• I was able to find a more detailed and clear explanation online here This should help you. Aug 30, 2021 at 2:01

$N_t$ and $N_c$ are fixed by the design, with $N_t +N_c=N$.
Define $P_t = \dfrac{N_t}{N}$ and $P_c = \dfrac{N_c}{N} = 1-P_t$.
$W_i =1$ for treated observations, $=0$ for controls. The key fact is that the $W_i$ are independent and follow a Bernoulli distribution with index $P_t$.
The text gives the centered treatment indicators. $$D_i = W_i - \frac{N_t}{N} = W_i - P_t$$
$P_t$ is non-random, so $D_i$ has the same variance as $W_i$, and