What kind of data is your time series?

I'd guess that Sibel is referring to some financial data (specifically the financial markets such as FOREX or stocks) but she can confirm it.

No, I have daily date of "sales", let's say the amount spent in a backery

For my study, we did some interventions, so the aim of my analysis is:

- model the data of the pre-situation

- forecast the post-situation

- see if there is a break, interruption (sometimes called interrupted TS in health studies) that can be dedicated to the invention (in the direction we wanted)

Firstly, an ARMA(2,1) model for differentiated data is equivalent to an ARIMA(2,1,1) model for the original series.

On the other hand, I think you could delete observations corresponding to Saturday and Sunday. Then, the seasonal period would be 5 instead of 7. Of course, you might estimate a seasonal ARIMA model. However, to avoid overdifferencing, you could apply seasonal unit root tests.

Perhaps, an structural time series model could be a better choice in order to approximate the level and seasonal variations.

There is no data during Saturday and Sunday, thus this series is a time-non homogenous series. Deriving ACF from this is questionable? because lags are not constant for all the data in the given time series.

But still, you can use ARMA(2,1) for the already differentiated data or ARIMA(2,1,1) for the original data.

You should use ARIMA(2,1,1). According to the rule first we plot the TS then ACF and PCF graph to check the stationary of data. From this you have found that if the data series value p=2, d=1 and q=1 then it will stationary. Therefore ARIMA(2,1,1)is applied to the original data series. To make the Seasonal data stationary you have make difference with 4,6 or 12 according to the seasonal effect as identified from the ACF and PCF of original data. after seasonal difference again plot the ACF and PCF and select the ARIMA(P,D,Q). from this identify the significant seasonal and non-seasonal lag and find ARIMA(p,d,q)(P,D,Q). you can go through the book

Forecasting Method and Applications

by Spyros G. Makridakis, Rob J. Hyndman

@ Mr, Kotosz, Oh sorry for that. But if it is daily data a collected for more than a year then can we not identify the seasonality?

@ Balazs: yes, I check for stationarity. The original data is non stationary, that is why I toke the first D. I don't know what seasonal unit root test are, so no I haven't checked that.

And yes: now it is the pre-situation, I know the date when the intervention started so the purpose is to forecast the sales and then compare with the actual data to detect a break. And yes, indeed I can use a dummy variable to indicate if it is PRE or POST

@ José Juan: ok, I really have to figure out those seasonal unit root tests you mention!

So, what I will do:

- seasonal unit root tests: what and why and results?

- seasonal ARIMA (2,1,1) model for a seasonlity of 7 days

Thank you all!

A source I can recommand: Enders, W. (2004). Applied Econometric Time Series, 2nd edition

@Balazs: I use STATA. Thank you very much for your comments. I read the section on "seasonality" of Enders, and now I do understand what you mean with seasonal differencing! The model I will use is of this format: ARIMA(p,d,q) (P,D,Q)s with s: period of data, so 7 in my case.

@Balazs: I cannot do a seasonal unit root test in Stata, I found 2 codes:

one for HEGY but it doesn't allow gaps in the data but I have gaps! the other one is only for quarterly data but I have daily data!

Any suggestions are welcome...

Yes, that is what I did but HEGY is still not an option...

I will try an ARIMA model, with AR/MA components or with multiplicative seasonality (like Enders descirbes). Thank you again!

Hi

As far as I know, you cannot pass to ARIMA without dealing with the stationarity (unit root test), or seasonal unit root test in your case. Why? because the "I" between AR and MA stands for the order of integration, in other words the filter that you should apply to make the time series stationary. When dealing with seasonal data there is a risk to overdifferenciate the time series when applying the (1-Ls) filter. The paper of HEGY highlighted this issue and states that the appropriate filter depends on the number of seasonal unit roots present in the time series. Now, which software can you use to conduct this analysis? As fas as I know, for quarterly and monthly data STATA and R (HEGY package) provide this test. For higher frequncies such as weekly and dialy, I'm afraid to tell you that you should write a code for that based on the paper of HEGY.

Best regards

There is an article writen by Haldrup et al (2007) (http://www.tandfonline.com/doi/abs/10.1198/073500106000000459#.UZC4PcpsO_o) that deals partially with this issue using periodic models which are a generalization of SARIMA models ..hope this would help

Thank you for the reaction. I'm read the article.

As Balazs already wrote, my problem is that I have daily data and only a 5 days-week and in Stata I don't know how to define my time series for a 5 days week and apply unit root tests...

Now I know, there is an option "business calender" but it only has 1 format so it is best to create your own business calender depending on the public holidays of your country.