4.15 Extra Questions
Suppose you run the following regression \[\begin{align*} Earnings = \beta_0 + \beta_1 Education + U \end{align*}\] with \(\mathbb{E}[U|Education] = 0\). How do you interpret \(\beta_1\) here?
Suppose you run the following regression \[\begin{align*} Earnings = \beta_0 + \beta_1 Education + \beta_2 Experience + \beta_3 Female + U \end{align*}\] with \(\mathbb{E}[U|Education, Experience, Female] = 0\). How do you interpret \(\beta_1\) here?
Suppose you are interested in testing whether an extra year of education increases earnings by the same amount for men and women.
Propose a regression and strategy for this sort of test.
Suppose you also want to control for experience in conducting this test, how would do it?
Suppose you run the following regression \[\begin{align*} \log(Earnings) = \beta_0 + \beta_1 Education + \beta_2 Experience + \beta_3 Female + U \end{align*}\] with \(\mathbb{E}[U|Education, Experience, Female] = 0\). How do you interpret \(\beta_1\) here?
A common extra condition (though somewhat old-fashioned) is to impose homoskedasticity. Homoskedasticity says that \(\mathbb{E}[U^2|X] = \sigma^2\) (i.e., the variance of the error term does not change across different values of \(X\)).
Under homoskedasticity, the expression for \(V\) in (4.7) simplifies. Provide a new expression for \(V\) under homoskedasticity. Hint: you will need to use the law of iterated expectations.
Using this expression for \(V\), explain how to calculate standard errors for an estimate of \(\beta_1\) in a simple linear regression.
Explain how to construct a t-statistic for testing \(H_0: \beta_1=0\) under homoskedasticity.
Explain how to contruct a p-value for \(\beta_1\) under homoskedasticity.
Explain how to construct a 95% confidence interval for \(\beta_1\) under homoskedasticity.