Reconsider the data file inequality used in Exercise 9.26 and the model in part (a) of that exercise but include the median marginal tax rate for the upper \(1 \%\) of incomes (TAX). We are interested in whether the marginal tax rate is a useful instrument for reducing inequality. The resulting model is

a. Estimate this equation using data for Canada. Obtain both conventional and HAC standard errors. Compare the \(95 \%\) interval estimates for \(\alpha_{2}\) from each of the standard errors.

b. Use an LM test with a 5\% significance level and three lagged residuals to test for autocorrelation in the errors of the equation estimated in part (a). What do you conclude about the use of HAC standard errors in part (a)?

c. Estimate a parameter \(ho\) by applying OLS to the equation \(\hat{e}_{t}=ho \hat{e}_{t-1}+\hat{v}_{t}\) where \(\hat{e}_{t}\) are the least squares residuals from part (a). What assumption is being made when you estimate this equation?

d. Transform each of the variables in the original equation using a transformation of the form \(x_{t}^{*}=x_{t}-\) \(\hat{ho} x_{t-1}\) and apply OLS to the transformed variables. Compute both conventional and HAC standard errors. Find the resulting 95\% interval estimates for \(\alpha_{2}\). Compare them with each other and with those found in part (a).

e. Use an LM test with a 5\% significance level and three lagged residuals to test for autocorrelation in the errors of the equation estimated in part (d). What do you conclude about the use of HAC standard errors in part (d)?

f. For each of the equations estimated in parts (a) and (d), discuss whether the exogeneity assumption required for consistent estimation of \(\alpha_{2}\) is likely to be satisfied.

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SHARE, = +TAX, + 3 YEAR, + YEAR + e