Time Series: Question on Stationarity

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SUMMARY

The discussion centers on the concept of stationarity in time series analysis, specifically addressing the relationship between lag and joint distribution. Participants clarify that in a stationary process, the joint distribution of does not depend on the specific time points but rather on the time difference. It is established that the joint distributions can differ based on the lag, as demonstrated through examples like random walks. The key takeaway is that stationarity is more accurately described in terms of differences, such as Y_{t+a} - Y_t, rather than the joint distribution itself.

PREREQUISITES
  • Understanding of time series analysis concepts
  • Familiarity with joint distributions in probability theory
  • Knowledge of stationary processes in statistics
  • Basic grasp of random walks and their properties
NEXT STEPS
  • Study the properties of stationary time series in detail
  • Learn about the concept of autocorrelation in time series
  • Explore the implications of non-stationarity in time series forecasting
  • Investigate the use of differencing techniques to achieve stationarity
USEFUL FOR

Statisticians, data analysts, and researchers working with time series data who need to understand the implications of stationarity and its effects on joint distributions.

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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