Consider the simple regression model \(y_{i}=\beta_{1}+\beta_{2} x_{i 2}+e_{i}\). Suppose \(N=5\) and the values of \(x_{i 2}\) are \((1,2,3,4,5)\). Let the true values of the parameters be \(\beta_{1}=1, \beta_{2}=1\). Let the true random error values, which are never known in reality, be \(e_{i}=(1,-1,0,6,-6)\).

a. Calculate the values of \(y_{i}\).

b. The OLS estimates of the parameters are \(b_{1}=3.1\) and \(b_{2}=0.3\). Compute the least squares residual, \(\hat{e}_{1}\), for the first observation, and \(\hat{e}_{4}\), for the fourth observation. What is the sum of all the least squares residuals? In this example, what is the sum of the true random errors? Is the sum of the residuals always equal to the sum of the random errors? Explain.

c. It is hypothesized that the data are heteroskedastic with the variance of the first three random errors being \(\sigma_{1}^{2}\), and the variance of the last two random errors being \(\sigma_{2}^{2}\). We regress the squared residuals \(\hat{e}_{i}^{2}\) on the indicator variable \(z_{i}\), where \(z_{i}=0, i=1,2,3\) and \(z_{i}=1, i=4,5\). The overall model \(F\)-statistic value is 12.86. Does this value provide evidence of heteroskedasticity at the \(5 \%\) level of significance? What is the \(p\)-value for this \(F\)-value (requires computer)?

d. \(R^{2}=0.8108\) from the regression in (c). Use this value to carry out the LM (Breusch-Pagan) test for heteroskedasticity at the \(5 \%\) level of significance. What is the \(p\)-value for this test (requires computer)?

e. We now regress \(\ln \left(\hat{e}_{i}^{2}\right)\) on \(z_{i}\). The estimated coefficient of \(z_{i}\) is 3.81. We discover that the software reports using only \(N=4\) observations in this calculation. Why?

f. In order to carry out feasible generalized least squares using information from the regression in part (e), we first create the transformed variables \(\left(y_{i}^{*}, x_{i 1}^{*}, x_{i 2}^{*}\right)\). List the values of the transformed observations for \(i=1\) and \(i=4\).\