30
Apr 17

## Nonstationary processes 2

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

Example 2. Random walk. Consider

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

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 $t\ne s.$

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

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

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

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

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) $t$ takes all positive integer values and in (4) $t$ is some positive integer. Then $k$ can be taken equal to $t$, so that

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

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

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

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: $Ey_t=0$. However, the second is violated (use (2)):

$Var(y_t)=Ey_t^2=E(u_1+...+u_{t-1}+u_t)(u_1+...+u_{t-1}+u_t)$ $=Eu_1^2+...+Eu_{t-1}^2+Eu_t^2=t\sigma^2,$

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