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Is the following MA(2) process covariance-stationary?

Is the following MA(2) process covariance-stationary? 


Yt = (1 + 2.4L + 0.8L2) εt, with E ( εt εT) = 

$$ \left\{\begin{array}{c} 1 \text { for } t=\tau \\ 0 \text { otherwise } \end{array}\right. $$


 If so, calculate its auto-covariances.


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Answer #1

Note: L defines the lag in variable for interest, for instance,

Here,

image.png

So

image.png

A process is called covariance-stationary, if:

1) Its mean is independent of time.

2) Variance does not depend on the time

3) image.png, i.e. covariance between two-time points depends only on the size of the interval between them

Let's get the required moments:

1) Mean

Thus, mean is independent of time.

2) Variance

Thus, variance is independent of time too.

3) Auto-Covariances


Thus, autocovariance is independent of time too.

Thus this MA(2) process is covariance-stationary.

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