My data, its ACF, PACF looks like thisenter image description here The data is daily data and has period of 365 days though with varying mean. I tried fitting ARIMA (0,1,1) model (following suggestion from ESACF), it cannot pass the Ljung-Box test and the residual distribution does not have normal distribution.

When to take into account seasonality I add seasonal differencing identify var=x(1,365) the data looks like this- enter image description here

When I fit (0,1,1) (0,1,1)_365 seasonal ARIMA model to it, the residual diagnostic again cannot pass the Ljung-Box test. I also tried taking logarithm of x, but no luck there either.

No matter which model I fit, It cannot pass the Ljung-Box test

Any suggestion on how to model this data so the residuas would be white noise?

  • $\begingroup$ Your series is clearly effected by deterministic structure such as pulses ,perhaps holiday effects, perhaps week of the year , perhaps month of the year , perhaps day-of-the month etc.. ...If you post your data I will try and be more specific.;Post it as a column , specifying the beginning date and country of origin. $\endgroup$
    – IrishStat
    Commented Nov 7, 2016 at 2:10
  • $\begingroup$ Thank you so much for you kind offer. dropbox.com/s/wcvgv6x85brytni/data.xlsx?dl=0 would be the link to the excel file with data and date. Also the data is water inflow of a reservoir (country of origin USA), so holidays are not relevant $\endgroup$
    – Orpia
    Commented Nov 7, 2016 at 3:23

1 Answer 1


I took your 1461 observations (4 years) into AUTOBOX ( a software package that I have helped to develop) and obtained the ACF of the original series enter image description here . Standard (i.e. almost always naive ) model identification strategy assumes no outliers/level shifts/seasonal pulses/special causes are present. AUTOBOX developed a first-pass model that identified a plethora of pulse outliers for the months of September and October for the first 2 years.

We have seen retails sales buildup starting about Nov 15th of each year and lasts about 60 days requiring model structure to deal with this recurring phenomenon. In a similar way I suggested that a similar variable be considered/ used for a 75 day window stating at 9/1 for each of the first 2 years.

The fully resolved model combining this "special variable" , month of the year and ARIMA structure and allowances for identified pulses is here in two images enter image description here and enter image description here . The statistical summary is here enter image description here . The residuals from the model are plotted here enter image description here with an ACF here enter image description here. The Actual/Fit and Forecast graph is here enter image description here with the Actual and Cleansed presented here enter image description here

Finally the ARIMA model (in concert with other structure) is essentially a (0,1,0)(0,0,0) as the AR(1) coefficient is nearly 1.0 .


I added one series call M_DYN175 and placed 0's everywhere except for 9/1/09 and 9/1/2010 . For example this is a snapshot around 9/1/09 . AUTOBOX has a feature that enables a "long polynomial" .. in this case up to lag75 providing a shortcut to estimate 75 individual coefficients .In this example some 46 were found to be statistically significant and the others deleted. You can effect the same solution by creating 46 individual pulse indicators (0/1) in your regression model. Now as to how PULSE VARIABLES (OUTLIERS) were found this is accomplished by enabling Intervention Detection procedures following Tsay http://onlinelibrary.wiley.com/doi/10.1002/for.3980070102/abstract ( a feature of AUTOBOX and elsewhere ).

enter image description here

  • $\begingroup$ I have accepted the answer and upvoted it. Unfortunately since I do have enough reputation, it will not be shown publicly. I just need one explanation. Could you explain how you formulated and included the special variable from the images that you have provided? Thank you. $\endgroup$
    – Orpia
    Commented Nov 7, 2016 at 19:45
  • $\begingroup$ and how did you include allowance for pulses? $\endgroup$
    – Orpia
    Commented Nov 7, 2016 at 19:54
  • $\begingroup$ If you wish to get more specific details (beyond the normal interest of the group) you can contact me. $\endgroup$
    – IrishStat
    Commented Nov 7, 2016 at 20:38
  • $\begingroup$ Actually I have asked another question in this forum, stats.stackexchange.com/questions/245280/… can you please look into it if you don't mind? $\endgroup$
    – Orpia
    Commented Nov 10, 2016 at 21:13
  • $\begingroup$ You have so so so so many questions !..... why don't u call me so we can have a dialogue .. I will be glad to clear up ur questions $\endgroup$
    – IrishStat
    Commented Nov 10, 2016 at 21:23

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