Compute and derive...

1. Suppose that we have a sample {Yi} , which is constructed by Yi = Di te, i=1, ...,n where { D;} is a set of dummy variables, i.e. D; = 1 with probability p and D: = 0 with probability 1 - p, and {} is an iid sample of Normal (1, g?) and they are independent of each other. Here note that o, p, At, and o' are unknown parameters. (a) Compute mean, variance, and moment generating function of Y. (b) Consider the linear regression of Y, on the constant 1 and Di. i. Obtain the OLS estimate, say /1, of the coefficient of the constant, say P1- ii. Derive the probability limit of B, in terms of the unknown param- eters. iii. Derive the asymptotic distribution of 31. (c) Now let's consider MLE. i. Derive the likelihood function for {Y}. ii. Derive the likelihood function for {Yi} given {Di}. iii. Obtain the MLE of the unknown parameters (except p) using the conditional likelihood in (ii). iv. Obtain the MLE of 1, which is defined in (b), and compare it with the answer in (b)