1. (20 points) Consider the regression model M = BULI+ En, = ...-2,-1,0, 1,2,..., where and where my is i.i.d. with mean 0 and variance o, and both , and 6, 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 y, is measured as a deviation from the sample mean over the period 1, ..., T, so that C M = 0. (a) The Ordinary Least Squares (OLS) estimator of 8 in this sample of size T is By = Show that By is not consistent, that is, plim By # 8. T-+002. (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 I* E {0, 1,2,..}. Assume that someone unemployed for less than 1/2 of a month answers with * = 0, and that for those unemployed more than 1/2 month, their response is equal to the integer f* if their true duration is greater than or equal to (" -.5 and less than f* + .5, that is, 0 if 1 C. 5 1= (jiri-osh