Chapter 2

In the potential outcomes framework with heterogeneous (nonconstant) treatment effect, write the error as$$u_i=(1-x_i)u_i(0)+x_i{u}_i(1).$$Let ${\sigma }_0^2=\mathrm{Var}[u_i(0)]$ and ${\sigma }_1^2=\mathrm{Var}[u_i(1)].$ Assume random assignment. i. Find $\mathrm{Var}\left(u_i|x_i\right)$.ii. When is $\mathrm{Var}\left(u_i|x_i\right)$ constant?

Let $x$ be a binary explanatory variable and suppose $P(x=1)=\rho$ for $0<\rho <1$. i. If you draw a random sample of size $n$, find the probability-call it ${\gamma }_n-$ that Assumption $\mathrm{SLR}.3$ fails. [Hint: Find the probability of observing all zeros or all ones for the $x_i.$ ] Argue that ${\gamma }_n\to 0$ as $n\to \mathrm{\infty }$. ii. If $\rho =0.5,$ compute the probablity in part (i) for $n=10$ and $n=100.$ Discuss. iii. Do the calculations from part (ii) with $\rho =0.9.$ How do your answers compare with part (ii)?