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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.

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$N_t$ 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

\begin{align} var(D_i) & = P_t(1-P_t) \\ & = P_t\times P_c \\ & = \frac{N_t}{N} \times \frac{N_c}{N} \\ & = \frac{N_t N_c}{N^2} \\ \end{align}

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