# What are the assumptions of ARIMA/Box-Jenkins modeling for forecasting time series?

What are the assumptions of ARIMA/Box-Jenkins modeling for forecasting time series?

1. There are no known/suspected predictor variables
2. There are no level shifts
3. There are no deterministic time trends of the form $1,2,3,...,t$
4. There are no seasonal dummies
5. There are no one time anomalies
6. The model parameters are constant over time
7. The error process is homoscedastic (constant) over time

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.

• Thanks @Irishstat when you say error process is homoscedastic, hoes this reffer to residuals ? – forecaster Nov 23 '13 at 0:19
• Yes the residuals need to have a constant variance. – IrishStat Nov 23 '13 at 12:57
• ARIMA (the question here) rather than ARMA does not rule out all time trends??? – Nick Cox Nov 23 '13 at 14:18
• @Nick within the framework of ARIMA models we can have – IrishStat Nov 24 '13 at 1:01
• @Nick within the framework of ARIMA models we can have [1-B]Y= constant + [THETHA/PHI]*A where the constant is a steady state differential thus there is a deterministic trend and as you point and the model has a trend component. I was referring to trend variables of the form (1,2,3,4....t1) which introduce another kind of deterministic trend variable.It would be possible to have the need for multiple trend variables of this type to correctly model a time series.For example a series like 1,2,3,4,5,7,9,11,13,15,17,20,23,26,29,32 with some additional random error would need 3 of these variables. – IrishStat Nov 24 '13 at 1:14

For the "pure" ARIMA models,

1. That the time-series involved are weakly stationary or Integrated of some order (which implies restrictions on the values of the unknown coefficients, as well as their constancy).

2. That all observed time series are combinations of white noises only, and perhaps a constant.

Moreover, the very fact that you use the abbreviation "ARIMA", implies in itself that

1. There are no other predictors (in which case you would have an "ARIMA-X" model) and

2. The relations are exclusively linear (to indicate the possibility of non-linear modelling, you should abbreviate to "NARIMA").

• Thanks @Alecos, are there any assumptions on Residuals of Arima ? – forecaster Nov 23 '13 at 0:09
• I am not sure what you mean - "residuals" are the "left-overs" of the estimation procedure. We do not make assumptions on residuals, since their properties are a consequence of the assumptions made related to the initial specification. Do you by any chance meant "error terms"? – Alecos Papadopoulos Nov 23 '13 at 0:17
• yes, residuals refers to the left overs of the estimation procedure. Should they be normally distributed without any structure ? Is this one of the requirements of ARIMA modeling? I'm sorry if I'm not clear. Thanks for your help. – forecaster Nov 23 '13 at 1:15
• see irishstat response below, the residuals should be homoscedastic. – forecaster Nov 27 '13 at 1:27
• Homoskedasticity is unrelated to normality. Weak stationarity (assumption 1 in my answer) implies homoskedasticity. – Alecos Papadopoulos Nov 27 '13 at 2:16