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

1. 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 \(\beta_3=0\).

2. 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

1. 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.