(Simpson's Paradox.) You obtain data from UVic on graduate students' grades. You observe student i's GPA, yi , a gender dummy Gi , and a dummy indicating the student is an economics student, Di , where Di = 1 indicates economics and Di = 0 indicates any other department. (a) Present the OLS estimator of the effect of G on y, unconditional on department. (b) Using the FWL theorem, present the OLS estimator of the effect of G on y holding department constant. (c) Is it possible that women on average receive higher grades than men given D = 0 and that women on average receive higher grades than men given D = 1, but also that women receive lower overall average grades? If so, explain this seeming paradox, if not, prove that such an outcome is impossible.

2. Consider a linear regression model with n observations and k regressors. In the notation we developed in class, (a) Prove the (simple, uncentered) R2 is bound on the unit interval. (b) Provide an expression for the centered R2 . Show this statistic can be negative if the model does not include a constant or the equivalent.

3. You have a dataset with n observations and k covariates in which the dependent variable and all of the covariates are measured in 2010 dollars. Your colleague has the same data, except every variable is measured in 2021 dollars. Suppose (for simplicity) the rate of inflation used between 2010 and 2021 was 2.0, that is, that $1 in 2010 is equivalent to $2.00 in 2021. If you estimate the same models by OLS, will you get the same... (a) predicted values? (b) residuals? (c) ESS? (d) TSS? (e) coefficient estimates?