Time Series: Question on Stationarity

This means that as the difference between times increases, the joint distributions remain the same. However, if the actual values at specific times are changed, the joint distribution may not be the same.
  • #1
roadworx
21
0
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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  • #2
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.
 
  • #3
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.
 

Related to Time Series: Question on Stationarity

1. What is a time series?

A time series is a set of data points collected at regular intervals over time. These data points are usually recorded in chronological order and can be used to analyze trends and patterns over time.

2. What is stationarity in time series?

Stationarity refers to the statistical properties of a time series remaining constant over time. This means that the mean, variance, and autocorrelation of the data do not change significantly over time.

3. Why is stationarity important in time series analysis?

Stationarity is important because most time series analysis techniques assume that the data is stationary. If the data is not stationary, the results of the analysis may be misleading or inaccurate. Stationary time series are also easier to model and forecast.

4. How can we test for stationarity in a time series?

There are several statistical tests that can be used to check for stationarity in a time series, such as the Augmented Dickey-Fuller test, the Kwiatkowski-Phillips-Schmidt-Shin test, and the Phillips-Perron test. These tests look for certain patterns in the data, such as constant mean and variance, to determine if the time series is stationary.

5. What can we do if a time series is not stationary?

If a time series is not stationary, we can try to make it stationary by using techniques such as differencing, detrending, or transforming the data. These methods aim to remove trends and seasonal patterns from the data to make it more stationary. If these techniques are not successful, we may need to use different time series analysis methods that are suitable for non-stationary data.

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