Trended vs. differenced regression Consider a linear model with a linearly trending regressor: the = 0 + pl. + en, where the sequence e is independently and identically distributed according to some distribution D with mean zero and variance o'. The object of interest is B. 1. Write out the OLS estimator B of f in deviations form and find its asymptotic distribution. 2. A researcher suggests removing the trending regressor by taking differences to obtain 101-11-1=8+61 -61-1 and then estimating # by OLS. Write out the OLS estimator / of S and find its asymptotic distribution. 3. Compare the estimators & and / in terms of asymptotic efficiency. Long run variance for AR(1) Often one needs to estimate the long-run variance of a stationary sequence are, that satisfies the restriction Efez,] = 0. Derive a compact expression for Vs, in the case when of and 2, follow independent scalar AR(1) processes. For this example, propose a method to consistently estimate Ve, and show your estimator's consistency. Asymptotics of averages of AR(1) and MA(1) Let a, be a martingale difference sequence with respect to its own past, and let all conditions for the CLT be satisfied: VTay =71/2Era, " N(0, a'). Let now in = put 1 + me and a = 2 + 03-1, where lol