2. Consider the moving average process of order q, MA(q) xt...

2. Consider the moving average process of order q, MA(q) xt = et + ?1et-1 + ?2et-2 + ? + ?qet-q where {et, t = 0,1,2, …} is an i.i.d. sequence with zero mean and variance ?e^2. (1) Find E(xt), Var(xt), Cov(xt, xt+h). (2) Find corr(xt, xt+h). (3) Is {xt} covariance stationary? (4) Is {xt} weakly dependent?

Transcript

00:01 Hello students, here in the given problem given the xt term which is the et plus alpha 1 et minus 1 plus alpha 2 into et minus 2 up to alpha q et minus q where the mean is zero and the variance is sigma e square.

00:15 So first find out the expected value of xt here expected value of xt here put the value of xt where the term is then since we can say that the et terms are id with zero mean.

00:29 So, the expected value of the et is equal to zero therefore, the similarly the all the term are zero that is expected value of et minus 1 expected value of et minus 2 are also zero.

00:42 Therefore, here expected value of xt is equal to zero therefore, the final answer is given.

00:47 Now second one to find the variance of xt here put the values of the xt term here variance of et plus alpha 1 et minus 1 up to alpha q et minus q here we can say that the et terms are also iid with variance sigma e square and they are uncorrelated with each other.

01:06 So due to being white nodes therefore, here take the variance of xt here the sigma square e take the common therefore, 1 plus alpha 1 square plus alpha 2 square up to dash alpha square q.

01:20 This is a final answer for the variance of xt.

01:23 Now find out the covariance of xt and xt plus h here the term is zero for all h is not equal to zero.

01:30 This is because the moving average process is linear combination of the et terms at different time points and the et terms are uncorrelated with each other for the different point.

01:41 So, we can say that the covariance term is zero for all the h is not equal to zero.

01:47 Now to find out the correlation here the formula for the correlation is covariance of xt and xt plus h divided by square root of variance of xt into square root of variance of xt plus h here the covariance term is zero...

Question

Please give Ace some feedback