1. Comment on: "Treating regressors r in a mean regression as random variables rather than fixed numbers simplifies further analysis, since then the observations (r,, y,) may be treated as IID across i". 2. A labor economist argues: "It is more plausible to think of my regressors as random rather than fixed. Look at education, for example. A person chooses her level of education, thus it is random. Age may be misreported, so it is random too. Even gender is random, because one can get a sex change operation done." Comment on this pearl. 3. Consider a linear mean regression y = 2'8 + e. E[elx] = 0, where a, instead of being IID across i, depends on i through an unknown function p as z, = p(i) + u,, where u, are IID independent of e,. Show that the OLS estimator of S is still unbiased. Let {we), be a strictly stationary and ergodic stochastic process with zero mean and finite variance. (i) Define B = 2 = U1 - Byt-1+ so that we can write It = Byt -1 + 1. Show that the error n, satisfies Blue] = 0 and Cut, y-1] = 0. (ii) Show that the OLS estimator & from the regression of yr on yo-1 is consistent for B. (ini) Show that, without further assumptions, up is serially correlated. Construct an example with serially correlated up. (iv) A 1994 paper in the Journal of Econometrics leads with the statement: "It is well known that in linear regression models with lagged dependent variables, ordinary least squares (OLS) estimators are inconsistent if the errors are autocorrelated". This statement, or a slight variation of it, appears in virtually all econometrics textbooks. Reconcile this statement with your findings from parts (ii) and (ifi). Suppose one has a random sample of n observations from the linear regression model 1= at Br+ yz + e, where e has mean zero and variance o' and is independent of (x, 2). 1. What is the conditional variance of the best linear conditionally (on the r and = samples) unbiased estimator 0 of 0 - a+ Be + 7.. where c, and c, are some given constants? 2. Obtain the limiting distribution of Write your answer as a function of the means, variances and correlations of z, 2 and e and of the constants o, B. 7. Cr. ca, assuming that all moments are finite. 3. For which value of the correlation coefficient between a and z is the asymptotic variance minimized for given variances of e and z? 4. Discuss the relationship of the result of part 3 with the problem of multicollinearity