What are the assumptions of ARIMA/Box-Jenkins modeling for forecasting time series?
Solved – What are the assumptions of ARIMA/Box-Jenkins modeling for forecasting time series
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I have never seen a model like Box-Jenkins identification process led me to ARIMA(0,1,3) model BUT i had never seen a black swan until I went to Australia. Please post your data as it may suggest the need for
- Intervention Detection leading to including level shifts, local time trends et al
- Time varying parameters
- Time varying error variance
If your data is confidential, simply scale it.
OK having received your data (some 80000 readings), I selected 805 observations starting at point 6287 and obtained.
. A significant change point was detected at period 137 suggesting time-varying parameters. The remaining 668 observations suggest a pdq ARIMA Model (3,0,0) with a level.step shift supporting your preliminary conclusions about lag 3. 
. The Actual/Fit/Forecast graph is 
The Residual Plot 
and the acf of the residuals is 
. Since the acf of the residuals shows strong structure at periods 5 and 10 , 
you might further investigate seasonal structure at lag 5. I hope this helps.
Your series of 116 weekly values is typical of many real-world series that we run into. Identification of a SARIMA model is impeded by the small sample size. The classical ARIMA model identification scheme premises no outliers, no level shifts, no seasonal dummies and no deterministic trends of the form 1,2,3,4,....t . Following is a suggested robust identification procedure which reflects the evolution of ARIMA models that incorporates deterministic structure defined here. Your series evidences a few seasonal deterministic dummies and a time trend and an ARIMA component of the form (1,0,0). The equation is 
. There are 4 weekly indicators ; week 2 ,8 , 11 and 30 . The acf of the errors suggests randomness in the residuals 
and is confirmed by 
a residual plot. Model identification of the need for the 4 seasonal dummies and the time trend was performed following the work of Tsay and others including myself as I helped develop the commercially available software that was used in this analysis. http://www.unc.edu/~jbhill/tsay.pdf. A model summary is presented here .
Best Answer
Most software solutions proceed to ignore all of these assumptions. AUTOBOX a piece of software that I have helped develop identifies and tests and remedies any violations of the above (save 1) leading to a Robust ARIMA solution.