10:09 | 0.2KB/s HASPERE Cattaneo... 1 Question 1: Non-linear Least Squares Let {(x):1in} be a random sample with y; ER and x; Rd. Suppose that Ex=(x) and Vy, x,]=0(x,), 29 where the (link) function : RR is known, but the (heteroskedasticity) function o: RR++ and the parameter B E Rd are unknown. Consider the M-estimator: " 1 3 arg min - p(x(3)). BER n 1-1 under appropriate regularity conditions required below. For example, you may assume as many integrability and differentiability conditions (e.g., on p(-)) as needed. 1. Give sufficient conditions so that B = arg minger E[{y - p(x3))]; Are there conditions under which 3, can be written in closed form? 2. Give sufficient conditions so that (8-8)N(0, Vo). Identify precisely the form of the asymptotic variance V Rxd 3. Propose an estimator V of Vo, and give sufficient conditions so that V, Vo Construct an asymptotic 95% confidence interval for Show that if 2(x)=2 an unknown constant, then " o, with || || the Euclidean norm. simplifies. Using this result, propose an estimator of Vo and give sufficient conditions so that V = Vo. 4. Suppose that x, ~N((x),0). Show that BB, where B is the MLE of 3- Derive the MLE of o2. Does it coincide with the one proposed in part 37 5. Show that if the (link) function (-) is unknown, then 3, is not identifiable. Can you give conditions so that identifiability of 3, is restored? 6. Suppose that y=1(x-20), with x 18, and {1in} i.i.d. with c.d.f. F(-). Show that this model fits the non-linear least square setup, and give the exact form of p(), 2) and V. Is the homoskedasticity assumption 2(x) = plausible in this case? 1