Time series analysis - residuals arent stationary

[ltotheb]

Hey everyone,
i'm struggling with modelling some times series to get residuals with white noise characteristics. I use SPSS for ARIMA modelling and exponential smoothing and Gretl for stationary testing with the Augmented-Dickey-Fuller and the KPSS-Test.

My workflow:
At first I use Gretl to test, if I need to differenciate or to transform my time series, to get stationary data. Gretl uses the ADF- and the KPSS-test to check, if the time series is stationary or not.
My second step is to check for autocorrelations and partial autocorrelations in the differenciated time series with SPSS and try to remove them by subtracting a suitable ARIMA model from the time series. I also always check, if I get residuals with lower autocorrelations and partial autocorrelations by using an exponential smoothing method with the time series modeler.
My last step is than to use Gretl and SPSS again to check if the residuals of my time series are stationary and have no significant autocorrelations and partial autocorrelations or in other words, to test if the time series are white noise processes, so I can cross correlate them afterwards.

My problem now is that I have used a ARIMA (1,1,1) model and a simple exponential smoothing model for modelling a specific time series, but in both cases I get residuals that aren’t stationary, indicated by a non-significant ADF-Test in Gretl. I’ve tried several other models, but in all cased I got non-stationary residuals. Does somebody has an idea, how I can get rid of this problem?

Thanks in advance.

PS: In the attached files you see the plots of my raw time series and of the residuals I get after modelling my time series with a ARIMA(1,1,1) and a simple exponential smoothing model. I also attached the PACF, the ACF and the results of the ADF-tests.

 

[noetsi]

I don't know how you combine exponential smoothing with ARIMA since they make very different assumptions. I will stick to commenting on ARIMA. First, I would run a box ljung test on the residuals to see if you have serial correlation. Its not clear to me you have run this test. Second, I would run ADF and KPSS on your residuals. If they are non-stationary then you probably have another unit root. Again it is not clear to me you are doing this or not.

I don't know the software you use, I use SAS and R. If you use R I would run auto.arima in the forecast package and see what it says your ARIMA model is.

 

[ltotheb]

I installed R, run the autoarima function and got an ARIMA(0,1,3) model that produces stationary residuals without any seriel correlations, so thank you a lot, noetsi!

The questions that remains is, if I can even combine exponential models and ARIMA models as I do. My goal is to get white noise residuals, so I always try out, if a exponential model or a ARIMA model fits the time series better to achieve that goal. Isn't that possible?

 

[noetsi]

I don't know how you could or why you would want to do this. They operate from totally different logic I prefer exponential smoothing (ESM) because they are easier to do and have been shown to generate good results while being robust to violations of assumptions. In its original form they had no known distribution and were not what most would think of as statistics, they operate by a totally different mechanism and arguably make none of the ARIMA assumptions. They can be non-linear in form which ARIMA can not be.

If you are using auto.arima. you need to use the Box-Ljung test (which I assume you have from your comments).

 

[ltotheb]

I know that those models work with very different approaches, but isnt it possible to compare those models by how good they fit my data?

Yes, I used the auto.arima function with the Box-Ljung test and the tests for stationary afterwards.

 

[noetsi]

I know that those models work with very different approaches, but isnt it possible to compare those models by how good they fit my data?

Yes, I used the auto.arima function with the Box-Ljung test and the tests for stationary afterwards.

I use a MAPE to choose among ESM models and I am going to add ARIMA to the mix of models considered. So I would say yes if I understand your question. But you have to be careful in how you build these. ESM does not assume stationarity in many of its forms unlike ARIMA.