Consider the regression model y = 0 + 1x1 + 2x2 + u, with E(u) = 0.

Consider the regression model y = β0 + β1x1 + β2x2 + u, with E(u) = 0. You may assume that the regressors are non-stochastic. Suppose the error has a normal distribution and exhibits heteroskedasticity, i.e., Var(ui) varies with x1i and/or x2i , for all i. You are provided with a random sample {(yi , x1i , x2i)} n i=1 .

1. Discuss how, with the help of heteroskedasticity robust standard errors, you would test the null hypothesis H (a) 0 : β1 = 1 against the one-sided alternative H (a) A : β1 > 1 when you use OLS to estimate β1. Explain the need for robust standard errors. 

2.  Suppose you are told that V ar (ui) = c 2x 2 1i where c is an unknown constant and x1i = 0 6 for all i. Discuss how this knowledge about the form of heteroskedasticity allows you to (i) obtain a more efficient estimator of β1 and (ii) hence conduct a test of the null hypothesis H0 : β1 = 0 that is more powerful.

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