Raising the bar

Nonstationary processes 2: here is a second example that provides some ground for intuition.

Example 2. Random walk. Consider

(1) $latex y_t= y_{t-1}+u_t$

where $latex u_t$ is white noise:

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

(1) is an example of a dynamic model. It does not allow us to investigate properties of $latex y_t$ directly because there is a reference to the process at moment $latex t-1$, which is an unknown itself. To get rid of $latex y_{t-1}$, we use the procedure called recurrent substitution. (1) is assumed to hold for all $latex t$, so for the previous period it looks like this:

(3) $latex y_{t-1}= y_{t-2}+u_{t-1}.$

Plugging (3) in (1) we get $latex y_t=y_{t-2}+u_{t-1}+u_t.$ After doing this $latex k$ times we obtain

(4) $latex y_t=y_{t-k}+u_{t-k+1}+...+u_{t-1}+u_t.$

In Example 1 the range of time moments didn't matter because the model wasn't dynamic. Here we have to assume that in (1) $latex t$ takes all positive integer values and in (4) $latex t$ is some positive integer. Then $latex k$ can be taken equal to $latex t$, so that

$latex y_t=y_0+u_1+...+u_{t-1}+u_t.$

As $latex t\rightarrow\infty$, the initial value $latex y_0$ doesn't matter much. For simplicity, we assume that it is zero. Then we see that (1) implies

$latex y_t=u_1+...+u_{t-1}+u_t.$

This representation is free from references to unknown variables, and can be easily used to study the properties of the process under consideration. For example, the first condition of a stationary process is satisfied: $latex Ey_t=0$. However, the second is violated (use (2)):

$latex Var(y_t)=Ey_t^2=E(u_1+...+u_{t-1}+u_t)(u_1+...+u_{t-1}+u_t)$

$latex =Eu_1^2+...+Eu_{t-1}^2+Eu_t^2=t\sigma^2,$

which changes with $latex t$. Thus, under our assumptions (1) is an example of a nonstationary process.