4.15 Extra Questions

1. 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?

2. 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?

3. Suppose you are interested in testing whether an extra year of education increases earnings by the same amount for men and women.

   1. Propose a regression and strategy for this sort of test.

   2. Suppose you also want to control for experience in conducting this test, how would do it?

4. 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?

5. 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 \(X\)).

   1. 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.

   2. Using this expression for \(V\), explain how to calculate standard errors for an estimate of \(\beta_1\) in a simple linear regression.

   3. Explain how to construct a t-statistic for testing \(H_0: \beta_1=0\) under homoskedasticity.

   4. Explain how to contruct a p-value for \(\beta_1\) under homoskedasticity.

   5. Explain how to construct a 95% confidence interval for \(\beta_1\) under homoskedasticity.