# Time Series: weakly stationary or non-constant ACF?

#### Jameson

Staff member
I am taking an introductory course in Time Series and our initial study of ARMA processes has proven to be challenging for me. The math we are asked to do is quite simple but recognizing various attributes is tricky.

We are given 5 time series and are asked to label them from a given list of various types. I've correctly identified all but 2. The two left to use are:

1) Weakly stationary
2) Non-constant auto correlation function

I've looked up the definition for weakly stationary and it's far beyond what we have been given to use in lectures, so I imagine there is something simple I should notice about them versus strictly stationary. I still can't find any information on a non-constant ACF and would greatly appreciate any links to materials.

Here are the two series in question:

z7:

z10:

And here are some other useful time series analysis results to possibly use:

z7:

z:10

Sorry for all the graphs but this is a portion of the output I've produced. I'm coding all of this in SAS, which is a language I'm pretty comfortable in so can adjust the above in various ways if necessary.

Any comments on the nature of weak-stationary and/or non-constant ACF series? Any comments on the attached images?

Thank you!

Jameson

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Stationary means that the moving average of the time series is more or less constant.
Non-constant or non-stationary is he opposite.
When analyzing the characteristics of a time series, first we will want to identify global trends (such as seasons) and then we will want to eliminate global trends before further analyzing the series. If such trends are not present to begin with, that makes our job a little easier.

In practice this is for instance relevant if we want to decide how much of an article to keep in stock.
Typically we will plan ahead for Christmas, when people buy a lot.
Beyond that global seasonal trend, we want to know how much demand varies, so we can keep enough in stock to serve most customers, but not so much that we're needlessly paying holding costs.

From the SAS user's guide:

The noise (or residual) series for an ARMA model must be stationary, which means that both the expected values of the series and its autocovariance function are independent of time.

The standard way to check for nonstationarity is to plot the series and its autocorrelation function. You can visually examine a graph of the series over time to see if it has a visible trend or if its variability changes noticeably over time. If the series is nonstationary, its autocorrelation function will usually decay slowly.

If I look at the ACF graphs, I can see that the z10 series decreases fairly quick and would therefore be weak-stationary.
The ACF of the z7 series decreases slowly and would therefore be non-constant.

#### Jameson

Staff member
If I look at the ACF graphs, I can see that the z10 series decreases fairly quick and would therefore be weak-stationary.
The ACF of the z7 series decreases slowly and would therefore be non-constant.
Hmm, I'm not sure I understand this. I agree that the ACF of z7 takes longer to move within the error bands than z10 but how does this relate to the average? Unfortunately our professor never really defined "non-constant ACF". You're saying that "non-constant" and "non-stationary" are essentially the same, correct? That sounds reasonable.

With an AR(1) process, the ACF will decrease exponentially in a similar way to z7 so I don't see how this alone shows us that it's a non-stationary process. If unprompted I could even see an argument that z7 is a AR(1) process since the ACF decreases exponentially and the PACF has a sharp cut off after lag 1. I know that this isn't supposed to be the answer though so how can I distinguish it from an AR(1)?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Hmm, I'm not sure I understand this. I agree that the ACF of z7 takes longer to move within the error bands than z10 but how does this relate to the average? Unfortunately our professor never really defined "non-constant ACF". You're saying that "non-constant" and "non-stationary" are essentially the same, correct? That sounds reasonable.

With an AR(1) process, the ACF will decrease exponentially in a similar way to z7 so I don't see how this alone shows us that it's a non-stationary process. If unprompted I could even see an argument that z7 is a AR(1) process since the ACF decreases exponentially and the PACF has a sharp cut off after lag 1. I know that this isn't supposed to be the answer though so how can I distinguish it from an AR(1)?
Ultimately it is not about the ACF - it is about the time series, whether it contains global trend or not.
Perhaps you can make moving average graphs of, say, around 20 time units?
Those should show global trends.

#### Jameson

Staff member
Ultimately it is not about the ACF - it is about the time series, whether it contains global trend or not.
Perhaps you can make moving average graphs of, say, around 20 time units?
Those should show global trends.
My professor suggested chunking of the series into smaller bits. Is this what you mean? There are 200 observations so I could split that into 4 regions: 1 to 50, 51 to 100, 101 to 150 and 151 to 200.

#### Klaas van Aarsen

##### MHB Seeker
Staff member
My professor suggested chunking of the series into smaller bits. Is this what you mean? There are 200 observations so I could split that into 4 regions: 1 to 50, 51 to 100, 101 to 150 and 151 to 200.
Not quite.
I'm talking about a moving average series.
From wiki:
A moving average is commonly used with time series data to smooth out short-term fluctuations and highlight longer-term trends or cycles.

Create a new series with the same number of samples.
Make each point the average of the 20 points surrounding it.
Graph it.

I expect SAS to support it, although I'm not familiar with SAS.

#### Jameson

Staff member
I can try to do this. We haven't gone over this in class so I'm fairly sure we aren't expected to use this to solve the problems. Basically all we've covered are the ideas of stationary versus non-stationary (mean or variance) and using the ACF plus PACF graphs to correctly identify AR(1), MA(1) and ARMA(1,1) processes.

According to him a MA(1) should have one significant spike in the ACF graph and gradually decrease to 0 in the PACF (the opposite of an AR(1)).

I'll see what I can do about the coding.

While I'm doing that I have an observation about the two graphs in the OP. The mean for z10 seems much more concrete than the one for z7. In fact, z10 could almost be a stationary process so it makes sense to call it the weak stationary one. I am trying to imagine an ACF graph that is non-stationary and have no idea what that would look like. Maybe it would decrease to 0 and then go back up again?