## 5.16 Answers to Some Extra Questions

$$\beta_1$$ is how much $$Earnings$$ increase on average when $$Education$$ increases by one year holding $$Experience$$ and $$Female$$ constant.

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 $$Female$$.

1. Starting from (5.7)

\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.