# Regression: Potential Ouctomes Frameworkw with Heterogeneous Treatment Effect

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

• What is the formula you got for the conditional variance..? Commented Sep 20, 2019 at 4:03
• I got that its the sum of the variances sigma_0^2 and sigma_1^2. Commented Sep 20, 2019 at 4:14
• 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? Commented Sep 20, 2019 at 4:29
• Thanks! I think Ive got it now! Cheers! Commented Sep 20, 2019 at 15:06

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