ARIMA models for time series forecasting


Notes on nonseasonal ARIMA models (pdf file)

Slides on seasonal and nonseasonal ARIMA models (pdf file)

Introduction to ARIMA: nonseasonal models
Identifying the order of differencing in an ARIMA model
Identifying the numbers of AR or MA terms in an ARIMA model
Estimation of ARIMA models
Seasonal differencing in ARIMA models
Seasonal random walk: ARIMA(0,0,0)x(0,1,0)
Seasonal random trend: ARIMA(0,1,0)x(0,1,0)

General seasonal models: ARIMA (0,1,1)x(0,1,1) etc.
Summary of rules for identifying ARIMA models
ARIMA models with regressors
The mathematical structure of ARIMA models (pdf file)


Summary of rules for identifying ARIMA models

Identifying the order of differencing and the constant:

Identifying the numbers of AR and MA terms:

Identifying the seasonal part of the model:

*A caveat about long-term forecasting in general: linear time series models such as ARIMA and exponential smoothing models predict the more distant future by making a series of one-period-ahead forecasts and plugging them in for unknown future values as they look farther ahead. For example, a 2-period-ahead forecast is computed by treating the 1-period-ahead forecast as if it were data and then applying the same forecasting equation. This step can be repeated any number of times in order to forecast as far into the future as you want, and the method also yields formulas for computing theoretically-appropriate confidence intervals around the longer-term forecasts. However, the models are identified and optimized based on their one-period-ahead forecasting performance, and rigid extrapolation of them may not be the best way to forecast many periods ahead (say, more than one year when working with monthly or quarterly business data), particularly when the modeling assumptions are at best only approximately satisfied (which is nearly always the case). If one of your objectives is to generate long-term forecasts, it would be good to also draw on other sources of information during the model selection process and/or to optimize the parameter estimates for multi-period forecasting if your software allows it and/or use an auxiliary model (possibly one that incorporates expert opinion) for long-term forecasting.