I need some guidance with this exercise about regressions, OLS formula, and Two Stage Least Squares:

1. Regression Anatomy Dopey is a regression elf in training. Dopey is assigned the job of estimating the OLS coefficient from the model: yi = Bo + Piri + Bywit ci where w, is a vector of other controls. Dopey has had regression elf training in the Regression Anatomy formula (and the Frisch Waugh theorem), but makes a mistake. He tries to estimate #1 in two steps. First, he regresses r, on wit Ti = Not Twit Hi Then he regresses y; on wit yi = do + own + vi Then he estimates 81 by regressing i; on u;: Dopey Cov(Di, mi ) Var(in) Caitlin, the head of the regression elves, looks at what Dopey has done and tells him to go back and do his job over again. (a) What is the "right" formula for the OLS estimate of 31, according to the Regression Anatomy formula (and the Frisch Waugh theorem)? (b) Show that Dopey's way of doing it gets the same answer. (Hint: Look at the FOC for OLS for the "right" formula and for Dopey's way. Show that the same value of 8, satisfies both equations.) 2. Two Stage Least Squares (2SLS) and Instrumental Variables Consider a 2SLS system, where the causal model of interest is: Hi = Bo+ Bilitui and an instrumental variable 2, is available, with Elau,] = 0 (a) Write out the "first stage" population regression model for ri. (b) Write out the "reduced form" population regression model for yi. (c) Show that the residual in the reduced form model is uncorrelated with 2; (i.e. if we call the residual vi, then Elev;] =0. (Hint: using the model for y and the first stage model, find an expression for the residual of the reduced form.)