For my practice . please answer in detail

2.1 Trended vs. differenced regression Consider a linear model with a linearly trending regressor: 3/ = a + At+ Et . where the sequence , is independently and identically distributed according to some distribution D with mean zero and variance o'. The object of interest is 9. 1. Write out the OLS estimator A of # in deviations form and find its asymptotic distribution. 2. A researcher suggests removing the trending regressor by taking differences to obtain Me - 91-1 = P + ex - 81-1 and then estimating A by OLS. Write out the OLS estimator A of A and find its asymptotic distribution. 3. Compare the estimators # and # in terms of asymptotic efficiency. 2.2 Long run variance for AR(1) Often one needs to estimate the long-run variance Vie _ lim V of a stationary sequence are, that satisfies the restriction Ble|2,] =0. Derive a compact expression for Vie in the case when er and 2, follow independent scalar AR(1) processes. For this example, propose a method to consistently estimate Ve, and show your estimator's consistency. 2.3 Asymptotics of averages of AR(1) and MA(1) Let It be a martingale difference sequence with respect to its own past, and let all conditions for the CLT be satisfied: VTir = T-V/?EL, n - N(0, o'). Let now yr = pyr-1 + It and & = Inter-1, where Ipl