4.16 Answers to Some Extra Questions
Answer to Question 2
\(\beta_1\) is how much \(Earnings\) increase on average when \(Education\) increases by one year holding \(Experience\) and \(Female\) constant.
Answer to Question 3
Run the regression \[\begin{align*} Earnings &= \beta_0 + \beta_1 Education + \beta_2 Female + \beta_3 Education \times Female + U \end{align*}\] and test (e.g., calculate a t-statistic and check if it is greater than 1.96 in absolute value) if \(\beta_3=0\).
You can run the following regression: \[\begin{align*} Earnings &= \beta_0 + \beta_1 Education + \beta_2 Female \\ & \hspace{25pt} + \beta_3 Education \times Female + \beta_4 Experience + U \end{align*}\] Here, you would still be interested in \(\beta_3\). If you thought that the return to experience varied for men and women, you might also include an interaction term involving \(Experience\) and \(Female\).
Partial Answer to Question 5
- Starting from (4.7)
\[\begin{align*} V &= \mathbb{E}\left[ \frac{(X - \mathbb{E}[X])^2 U^2}{\mathrm{var}(X)^2} \right] \\ &= \frac{1}{\mathrm{var}(X)^2} \mathbb{E}[(X-\mathbb{E}[X])^2 U^2] \\ &= \frac{1}{\mathrm{var}(X)^2} \mathbb{E}\big[(X-\mathbb{E}[X])^2 \mathbb{E}[U^2|X] \big] \\ &= \frac{1}{\mathrm{var}(X)^2} \mathbb{E}[(X-\mathbb{E}[X])^2 \sigma^2 ] \\ &= \frac{\sigma^2}{\mathrm{var}(X)^2} \mathbb{E}[(X-\mathbb{E}[X])^2] \\ &= \frac{\sigma^2}{\mathrm{var}(X)^2} \mathrm{var}(X) \\ &= \frac{\sigma^2}{\mathrm{var}(X)} \end{align*}\]
where
the second equality holds because \(\mathrm{var}(X)^2\) is non-random and can come out of the expectation,
the third equality uses the law of iterated expectations,
the fourth equality holds by the condition of homoskedasticity,
the fifth equality holds because \(\sigma^2\) is non-random and can come out of the expectation,
the sixth equality holds by the definition of variance, and
the last equality holds by canceling \(\mathrm{var}(X)\) in the numerator with one of the \(\mathrm{var}(X)\)’s in the denominator.