Stationary processes: Here we consider an example that appeared in the Econometrics exams of the University of London.
Example 3. Consider the process defined by
where is white noise:
(2) , for all and for all
In Economics, the error is treated as a shock in the current period. So in (1) sustains a shock in period . There is no reason to put a coefficient in front of (or, you could say, the coefficient is 1). Besides, remembers the shock from the previous period, and the aftereffect of this shock is measured by the coefficient .
Observation. Before we proceed with the analysis of this model, it is a good idea to elucidate the techniques of working with white noise. For this, we can rewrite the conditions on the second moments of errors from (2) as
if and if
In words, the expected value of a product of any error with itself is and expected values of products of different errors disappear. This fact is used many times in Econometrics, in particular, here. I am mentioning it here because some students have problems with it even in the second semester.
First stationarity condition. Obviously,
(3) for all .
Second stationarity condition. Variance does not depend on time:
because only products have nonzero expectations.
Third stationarity condition. Using the observation above, let us look at the expectation
The error from the first parenthesis does not have a match in the second one, while the error from the second parenthesis does not have a match in the first one. Thus, (5) equals
Further, if the distance between points is larger than one, then in
expected values of all products will be zero. Equations (4), (6)-(8) are summarized as follows:
if , if , and if .
Recalling the first stationarity condition, we can rewrite this as
if , if , and if
Thus the covariance is indeed a function of only .
The process we studied is called a moving average of order 1 and denoted MA(1).