Introductory Econometrics: A Modern Approach 6th Edition by Jeffrey M Wooldridge

Exercise 4

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

Step 1 of 6

(i)

When where is a sequence of independent and identically distributed random variable with mean zero and variance one (for).

This implies

1)

2)

3)

4)