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Question 4. Point allocation: (a) 10%; (b) 5%; (c) 20%; (d) 15%; (e) 20%; (f) 30% Consider the following stationary and ergodic data generating process for the random variable y: y, = PYL + E, where & is a martingale difference sequence with E(e; ) = o' > 0 and sup, Elj g, ($4 ] 0; 10 0 and yu = 0 if y; 5 0. Will this complicate estimation of A if cq is iid (0,02)? A brief answer will do. (g) Consider the same situation as (f) except o is a parameter to be estimated. Will this compli- cate estimation of /? A brief answer will do.Question 1. Point allocation: (a) 10%; (b) 20%; (c) 20%; (d) 30%; (d) 20%. If y takes only non-negative integer values and has geometric density with parameter A then the density f(y|A) is f (y) = Av(1 + A) (utl), y =0,1, ..., A>0. Furthermore Ely] = / and Vly] = #(1 + #). Here we introduce regressors and suppose that in the true model Ely|x,] = exp(x, 3.), where Bo is an unknown k x 1 parameter vector and x; is a k x 1 nonstochastic regressor vector. We have a random sample (vi, x;), i = 1, ...N. For much of this question we are concerned with the properties of the MLE of S under conditions weaker than correct speciCation of the density. You can apply any laws of large numbers and central limit theorems without formally verifying the necessary assumptions. (a) Give the formula for the objective function Qu() equal to N-1 times the log-likelihood function. (b) Obtain plim QN(B). (c) Given your answer in (b), what assumptions are the essential assumptions to ensure consistency of / that maximizes QN(B). (d) Assuming that the density is correctly speciled, give the limit distribution of B. [Your deriva- tion can be as brief as possible]. (e) Now suppose that the conditional density is misspeciled, the associated conditional variance is misspeciled, but the conditional mean is correctly speciled. Give the formula for a consistent estimate of the asymptotic variance-covariance matrix of B. Your answer and derivation can be as brief as possible]