Raising the bar Stationary process: example 3, a moving average of order 1

May 17

Stationary processes 2

category: Dougherty Introduction to Econometrics, Econometrics, FN3142 Quantitative Finance, Quantitative Finance, The College Board 1 Comment

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

(1) $y_t=u_t+\theta u_{t-1}$

where $u_t$ is white noise:

(2) $Eu_t=0$ , $Eu_t^2=\sigma^2$ for all $t$ and $Eu_tu_s=0$ for all $t\ne s.$

In Economics, the error $u_t$ is treated as a shock in the current period. So in (1) $y_t$ sustains a shock in period $t$ . There is no reason to put a coefficient in front of $u_t$ (or, you could say, the coefficient is 1). Besides, $y_t$ remembers the shock from the previous period, and the aftereffect of this shock is measured by the coefficient $\theta$ .

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

$Eu_tu_s=\sigma^2$ if $t=s$ and $Eu_tu_s=0$ if $t\ne s.$

In words, the expected value of a product of any error with itself is $\sigma^2$ 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) $Ey_t=0$ for all $t$ .

Second stationarity condition. Variance does not depend on time:

(4) $Var(y_t)=Ey_t^2=E(u_t+\theta u_{t-1})(u_t+\theta u_{t-1})=(1+\theta^2)\sigma^2$

because only products $u_t^2, u_{t-1}^2$ have nonzero expectations.

Third stationarity condition. Using the observation above, let us look at the expectation

(5) $Ey_ty_{t+1}=E(u_t+\theta u_{t-1})(u_{t+1}+\theta u_t).$

The error $u_{t-1}$ from the first parenthesis does not have a match in the second one, while the error $u_{t+1}$ from the second parenthesis does not have a match in the first one. Thus, (5) equals