Let's study the relationship between earnings and gender. Suppose that the population regression equation for earnings is Earnings, = 8, + 8, Female, + 8, Bachelors + u, (1) where Bachelors is a dummy variable indicating whether person / has completed a bachelors degree. However, when doing our analysis we (incorrectly) assume that the population regression equation for earnings is simply Earnings, = Ba + 8, Female, + u. (2) That is, we incorrectly omit the dependence of Eamings on Bachelors. It can be shown that you can write the expected value of the OLS coefficient in the incorrect regression as (you don't need to be able to prove this) E[B,] = coulearnings, Femaley (3) r(Female() (a) Insert the "true" population regression in equation (1) into the expression for the expected value of , in equation (3) to derive an expression for the omitted variable bias of f, in terms of the true regression coefficients (8.,. 8,, 8,} and the variances and covariances of female, and bachelors,- (b) Do you think each of the terms in the expression are positive or negative? Explain why? Combining these, is the omitted variable bias overall positive or negative Now let's explore this empirically. We will use the dataset CPS2015.dta. This contains data from the 2015 Current Population Survey. The variables are described in the pdf file CPS2015_Description.pdf.| (c) Regress ahe on female once without the robust option and once with the robust option. What is the difference in the two results? Interpret the coefficient on female. Does your interpretation change with or without the robust option? (d) Graph (twosay scatter) ahe and female. Do you suspect heteroskedasticity? (e) Run the regression implied by the population equation (1) to obtain an estimate of S_ Interpret its magnitude. (f) Calculate the terms in the expression in your answer to part (a) [Hint: running cou, X Y, covariance will calculate the covariances between X and Y']. Plug them in to the expression. Are they consistent with the difference between the coefficients in the regression you ran in part (c) to the regression you ran in part (e)?