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=0Eu_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


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


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)):



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

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