It can be shown that the theoretically useful form of the OLS estimator of \(\beta_{1}\) in the simple linear regression model \(y_{i}=\beta_{1}+\beta_{2} x_{i 2}+e_{i}\) is \(b_{1}=\beta_{1}+\sum\left(-\bar{x} w_{i}+N^{-1}\right) e_{i}=\sum v_{i} e_{i}\), where \(v_{i}=\left(-\bar{x} w_{i}+N^{-1}\right)\) and \(w_{i}=\left(x_{i}-\bar{x}\right) / \sum\left(x_{i}-\bar{x}\right)^{2}\). Using this formula consider the simple treatment effect model \(y_{i}=\beta_{1}+\beta_{2} d_{i}+e_{i}\). Suppose that \(d_{i}=1\) or \(d_{i}=0\) indicating that a treatment is given to a randomly selected individual or not. The dependent variable \(y_{i}\) is the outcome variable. See the discussion of the difference estimator in Section 7.5.1. Suppose that \(N_{1}\) individuals are given the treatment and \(N_{0}\) individuals in the control group are not given the treatment. Let \(N=N_{0}+N_{1}\) be the total number of observations.

a. Show that when \(d_{i}=0, v_{i}=1 / N\) and that when \(d_{i}=1, v_{i}=0\).

b. Derive \(\operatorname{var}\left(b_{1} \mid \mathbf{d}\right)\) under the assumption of homoskedastic errors, \(\operatorname{var}\left(e_{i} \mid \mathbf{d}\right)=\sigma^{2}\). What is an unbiased estimator of \(\operatorname{var}\left(b_{1} \mid \mathbf{d}\right)\) in this case?

c. Derive \(\operatorname{var}\left(b_{1} \mid \mathbf{d}\right)\) under the assumption of heteroskedastic errors, \(\operatorname{var}\left(e_{i} \mid d_{i}=1\right)=\sigma_{1}^{2}\) and \(\operatorname{var}\left(e_{i} \mid d_{i}=0\right)=\sigma_{0}^{2}\). What is an unbiased estimator of \(\operatorname{var}\left(b_{1} \mid \mathbf{d}\right)\) in this case?