Powers and Rock (1999) consider a two-equation model for the effect of coaching on SAT scores:
Xi = 1 if Uiα + δi>0, else Xi = 0;
Yi = cXi + Viβ +σεi.
Here, Xi = 1 if subject i is coached, else Xi = 0. The response variable Yi is subject i's SAT score; Ui and Vi are vectors of personal characteristics for subject i, treated as data. The latent variables (δi, εi ) are IID bivariate normal with mean 0, variance 1, and correlation ρ; they are independent of the U's and V 's. (In this problem, U and V are observable, δ and β are latent.)
(a) Which parameter measures the effect of coaching? Howwould you estimate it?
(b) State the assumptions carefully (including a response schedule, if one is needed). Do you find the assumptions plausible?
(c) Why do Powers and Rock need two equations, and why do they need ρ?
(d) Why can they assume that the disturbance terms have variance 1?