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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

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.

2 Responses for "Nonstationary processes 2"

  1. […] White noise is a simplest (after a constant) type of a stationary process. Give the definition, and don't hope […]

  2. […] is some positive integer. However, both this example and the one about random walk show that some condition on the coefficients  will be required for (1) to be stationary. (1) is […]

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