please help solve the following
Then a consistent estimator of B is4. (20 points) In estimating the bivariate regression model Mi = Britei, with e, iid. with mean 0 and variance of, you face the problem that you cannot perfectly observe r;. You have access to a random sample of information {y, 2/}g, where where u, is i.i.d. with mean 0 and variance o (2, E, and v can all be considered independently distributed with respect to one another). The variables y, and z are expressed as deviations from their sample means. If you have information that of = 1 (from an earlier study), can you form a consistent estimator of B? If so, what is it? Answer: Substituting, we have at least under the assumption that a is i.id. with mean 0 (which is by definition) and common variance o2. Since we know that of = 1, we should form the estimator of a2 by(c) If Bit = Be, t = 1, 2, for all i, define an unbiased estimator of B,, if one exists. If one does exist. is it consistent (as N - co)?3. (30 points) Consider the estimation of the linear regression model Bit = PirTi + Eit, t = 1,2, where Eit = mi + Wit, where up is independently and identically distributed with mean 0 and variance o? for all (i, t), and where E(mira,12) #0, for all i. You have access to information from a random sample of N individuals who are observed at the two points in time, t = 1 and 2. The information you have is {ya, yo, a, cohen. For each i, za # 22. All variables are expressed as deviations from their respective sample means. (a) If By = B for all (i, t), define an unbiased and consistent estimator of S, if one exists. (b) If Bar = 8, for all t, define an unbiased estimator of 8,, if one exists. If one does exist, is your estimator consistent (as N - co)?(b) Define the maximum likelihood estimator for a, and denote this estimator by a. (c) If the correctly measured data, {}, were available to you, we know that the maximum likelihood estimator of or is & = N/ St Discuss why your estimator a is not generally equal to a in any finite sample. Is it true that plim a = plim a?(b) Define a consistent estimator of / given the available data, if one exists. 2. (30 points) A population of unemployed job searchers have completed spell lengths that follow a negative exponential distribution, that is, the p.d.f. and c.d.f. of the spell length distributions are f (t) = aexp(-at), t>0, a >0. F(t) = 1-exp(-at). In a survey, currently unemployed respondents are asked to report how many months they have been searching for work, where their responses are limited to the values +* E {0, 1,2, ..}. Assume that someone unemployed for less than 1/2 of a month answers with f* = 0, and that for those unemployed more than 1/2 month, their response is equal to the integer #* if their true duration is greater than or equal to (" - .5 and less than f* + .5, that is, The data available to you are {{}N,. (a) Can the administrative records be used to consistently estimate o? Answer this by first writing down the log likelihood function.1. (20 points) Consider the regression model Me = Put-1 + en, 1 =...,-2, -1,0, 1, 2,..., where 61 = We + 0104-1 + 0201-2, and where w is i.i.d. with mean 0 and variance o,, and both 6, and 62 are unknown parameters that are not equal to 0. You have observations on the y process from period t = 1 through t = T, that is, {or}-1. Each we is measured as a deviation from the sample mean over the period 1, ..., T, so that -1 3 = 0. (a) The Ordinary Least Squares (OLS) estimator of S in this sample of size T is BT 21 2#1-190 Show that By is not consistent, that is, plim By * B. T-+DO
