Kindly provide correct answers. Thankyou tutors

Let {yt}, be a strictly stationary and ergodic stochastic process with zero mean and finite variance. (i) Define B = C [yt, yt-1] 5.2 V lyt] ut = yt - Byt-1, so that we can write yt = Byt-1 + ut- Show that the error up satisfies E [uf] = 0 and C [ut, yt-1] = 0. (ii) Show that the OLS estimator / from the regression of y on yt-1 is consistent for B. (iii) Show that, without further assumptions, wt is serially correlated. Construct an example with serially correlated ut. (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 (iii).Suppose one has a random sample of n observations from the linear regression model y= 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 z and z samples unbiased estimator 0 of 0 = a+ Bc + YC:. where cr and c, are some given constants? 5.3 2. Obtain the limiting distribution of Vn (8 - 0 ) . Write your answer as a function of the means, variances and correlations of a, z and e and o the constants o, 8, 7, Cr, Cs, 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 x? 4. Discuss the relationship of the result of part 3 with the problem of multicollinearity