Time Series: Question on Stationarity

AI Thread Summary
Stationarity in time series implies that the joint distribution of values at different times depends on the time difference rather than the specific times. The discussion highlights confusion regarding how lag affects joint distributions, specifically questioning whether the joint distribution of <Yt, Yt+a> remains consistent as 'a' increases. It is clarified that in a stationary process, the joint distribution is indeed based on time differences, meaning that <Yt, Yt+1> could differ from <Yt, Yt+2>. The example of a random walk illustrates that certain joint distributions can have different probabilities based on lag. Ultimately, understanding stationarity requires focusing on the differences between time points rather than the points themselves.
roadworx
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Hi,

I have a question on stationarity in time series.

I basically understand the concept, I think. However, I don't understand why the lag should affect the joint distribution.

For example, the joint distribution of <Yt, Yt+a> should be the same as the joint distribution of <Yp, Yp+a>. If now the a were to increase, the joint distributions should still be the same. Is that correct?

If Yp+a is changed to some other value, say Yp+b, then surely the joint distribution would still be the same in a stationary time series. Does anyone know if this is correct?
 
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I'm not sure precisely what you're asking, but the joint distribution of Y_t and Y_{t+1} might be different from the joint distribution of Y_t and Y_{t+2}. Suppose you have a random walk, where at each step you can go up with probability 1/2, and down with probability 1/2. Then the joint event Y_t = 0, Y_{t+1} = 0 has zero probability, since you never stay in the same place on two consecutive steps. But the event Y_t = 0, Y_{t+2} = 0 has probability 1/2 * P(Y_t=0).

Actually, reading around it seems that you usually talk about stationarity in terms of the _difference_ Y_{t+a} - Y_t, not their joint distribution. In that case as well, Y_{t+1}-Y_t might be distributed differently from Y_{t+2}-Y_t.
 
roadworx said:
Hi,

I have a question on stationarity in time series.

I basically understand the concept, I think. However, I don't understand why the lag should affect the joint distribution.

For example, the joint distribution of <Yt, Yt+a> should be the same as the joint distribution of <Yp, Yp+a>. If now the a were to increase, the joint distributions should still be the same. Is that correct?

If Yp+a is changed to some other value, say Yp+b, then surely the joint distribution would still be the same in a stationary time series. Does anyone know if this is correct?
In simplest terms a stationary process has a joint distribution which depends on the difference of the times, but not on the times themselves.
 

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