Suppose that an outcome variable \(y_{i j}=\beta_{1}+\beta_{2} x_{i j}+e_{i j}, i=1, \ldots, N ; j=1, \ldots, N_{i}\). Assume \(E\left(e_{i j} \mid \mathbf{X}\right)=0\) and \(\operatorname{var}\left(e_{i j} \mid \mathbf{X}\right)=\sigma^{2}\). One illustration is \(y_{i j}=\) the \(i\) th farm's production on the \(j\) th acre of land, with each farm consisting of \(N_{i}\) acres. The variable \(x_{i j}\) is the amount of an input, labor or fertilizer, used by the \(i\) th farm on the \(j\) th acre.

a. Suppose that we do not have data on each individual acre, but only aggregate, farm-level data, \(\sum_{j=1}^{N_{i}} y_{i j}=y_{A i}, \quad \sum_{j=1}^{N_{i}} x_{i j}=x_{A i}\). If we specify the linear model \(y_{A i}=\beta_{1}+\beta_{2} x_{A i}+e_{A i}, i=1, \ldots, N\), what is the conditional variance of the random error?

b. Suppose that we do not have data on each individual acre, but only average data for each farm, \(\sum_{j=1}^{N_{i}} y_{i j} / N_{i}=\bar{y}_{i}, \sum_{j=1}^{N_{i}} x_{i j} / N_{i}=\bar{x}_{i}\). If we specify the linear model \(\bar{y}_{i}=\beta_{1}+\beta_{2} \bar{x}_{i}+\bar{e}_{i}, i=1, \ldots, N\), what is the conditional variance of the random error?

c. Suppose the outcome variable is binary. For example, suppose \(y_{i j}=1\) if a crop shows evidence of blight on the \(j\) th acre of the \(i\) th farm, and \(y_{i j}=0\) otherwise. In this case \(\sum_{j=1}^{N_{i}} y_{i j} / N_{i}=p_{i}\), where \(p_{i}\) is the sample proportion of acres that show the blight on the \(i\) th farm. Suppose the probability of the \(i\) th farm showing blight on a particular acre is \(P_{i}\). If we specify the linear model \(\bar{y}_{i}=\beta_{1}+\beta_{2} \bar{x}_{i}+\bar{e}_{i}, \quad i=1, \ldots, N\), what is the conditional variance of the random error?