Let {et: t _ 1, 0, 1, …} be a sequence of independent, identically distributed random vari...

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 γh= Cov(xt, xt+h) for h ≥ 0. [Therefore, γ0= Var(xt).] Show that Corr(xt, xt_h) = γh/ γ0