Introductory Econometrics 4th Edition by Jeffrey Wooldridge

Exercise 11

Let {et: t _ 1, 0, 1, …} be a sequence of independent, identically distributed random variables with mean zero and variance one. Define a stochastic process by
xt = et - (1/2)et-1 + (1/2) et-2, t = 1, 2,….
(i) Find E(xt) and Var(xt). Do either of these depend on t
(ii) Show that Corr(xt, xt+2)= -1/2 and Corr(xt, xt_2) =1/3. (Hint: It is easiest to use the formula in Problem.)
(iii) What is Corr(xt, xt+h) for h 2
(iv) Is {xt} an asymptotically uncorrelated process
Let {xt: t _ 1, 2, …} be a covariance stationary process and define h= Cov(xt, xt+h) for h 0. [Therefore, 0= Var(xt).] Show that Corr(xt, xt_h) = h/ 0