In this exercise you will create some simulated data and try out estimation and testing methods. Use your software to create a new data set, or "workfile," with \(N=100\) observations. All modern software has functions, called random number generators, to create uniformly distributed and normally distributed random values. Follow these steps.

1. Create \(X 2=1+5 \times U 1\), where \(U 1\) is a random number between zero and one.

2. Create \(X 3=1+5 \times U 2\), where \(U 2\) is another random number between zero and one.

3. Create \(E=\sqrt{\exp (2+0.6 X 2)} \times Z\), where \(Z \sim N(0,1)\).

4. Create \(Y=5+4 X 2+E\)

You should now have 100 values for \(Y, X 2\), and \(X 3\). Your results should be different from your classmates, and your results might change from one experiment to the next. To prevent this from happening, you can set the random number's "seed." See your software documentation for instructions.

a. Regress \(Y\) on \(X 2\) and \(X 3\) and obtain conventional OLS standard errors. Compare the estimated coefficients to the true values of the regression parameters, \(\beta_{1}=5, \beta_{2}=4, \beta_{3}=0\). Do the \(t\)-values suggest that the coefficients are significantly different from 0 at the \(5 \%\) level?

b. Calculate the least squares residuals \(\hat{e}\) from the OLS estimation in (a) and regress \(\hat{e}^{2}\) on \(X 2\) and \(X 3\). What evidence, if any, do you find for the presence of heteroskedasticity?

c. Regress \(Y\) on \(X 2\) and \(X 3\) and obtain robust standard errors. Compare these to the conventional standard errors in (a).

d. Assume the heteroskedasticity pattern is \(\sigma^{2} X 2^{2}\). Obtain GLS estimates with conventional and robust standard errors. Are the GLS parameter estimates closer to the true parameter values or not? Which set of standard errors should be used?

e. Assume the multiplicative heteroskedasticity model \(\exp \left(\alpha_{1}+\alpha_{2} X 2+\alpha_{3} X 3\right)\). Obtain FGLS estimates with conventional and robust standard errors. Are the FGLS estimates closer to the true parameter values than the GLS or OLS estimates? Which set of standard errors should be used?