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This question came up as part of the practice problems in the Econometrics course I am taking. Its is the following. In the potential outcomes framework with heterogeneous (non-constant) treatment effect, write the error as:

$$u_i = (1-x_i)u_i(0)+x_iu_i(1)$$.

$$\sigma_0^2 = Var[u_i(0)] \;\text{ and }\; \sigma_1^2 = Var[u_i(1)].$$ Assume Random Assignment.

  1. Find $Var[u_i|x_i]$
  2. When is this value constant?

My attempt at the problem was to take the conditional variance of $u_i$ as defined above. Where Im having troubles is how do I breakup the variance to further simplify...

Any help or hints would be greatly appreciated.

Cheers, Groot99

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  • $\begingroup$ What is the formula you got for the conditional variance..? $\endgroup$
    – baruuum
    Sep 20, 2019 at 4:03
  • $\begingroup$ I got that its the sum of the variances sigma_0^2 and sigma_1^2. $\endgroup$
    – Groot99
    Sep 20, 2019 at 4:14
  • $\begingroup$ Hm...The "conditional" variance should depend on $x_i$. Think about what the variance would be if $x_i = 1$ and what the variance would be if $x_i = 0$. Are they the same? $\endgroup$
    – baruuum
    Sep 20, 2019 at 4:29
  • $\begingroup$ Thanks! I think Ive got it now! Cheers! $\endgroup$
    – Groot99
    Sep 20, 2019 at 15:06

1 Answer 1

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When $x_i =0$, $u_i = u_i(0)$, so $Var[u_i|x_i=0]=Var[u_i(0)]=\sigma^2_0$.

When $x_i =1$, $u_i = u_i(1)$, so $Var[u_i|x_i=1]=Var[u_i(1)]=\sigma^2_1$.

So, $Var[u_i|x_i] = (1-x_i)\sigma_0^2 + x_i \sigma_1^2$.

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