**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:**

- Rule
1: If the series has positive autocorrelations out to a high number of
lags (say, 10 or more), then it probably needs a higher order of
differencing.
- Rule
2: If the lag-1 autocorrelation is zero or negative, or the
autocorrelations are all small and patternless, then the series does
*not*need a higher order of differencing. If the lag-1 autocorrelation is -0.5 or more negative, the series may be overdifferenced.**BEWARE OF OVERDIFFERENCING.** - Rule
3: The optimal order of differencing is often the order of differencing at
which the standard deviation is lowest. (Not always, though. Slightly too
much or slightly too little differencing can also be corrected with AR or
MA terms. See rules 6 and 7.)
- Rule
4: A model with
__no__orders of differencing assumes that the original series is stationary (among other things, mean-reverting). A model with__one__order of differencing assumes that the original series has a constant average trend (e.g. a random walk or SES-type model, with or without growth). A model with__two__orders of total differencing assumes that the original series has a time-varying trend (e.g. a random trend or LES-type model). - Rule
5: A model with
__no__orders of differencing normally includes a constant term (which allows for a non-zero mean value). A model with__two__orders of total differencing normally does__not__include a constant term. In a model with__one__order of total differencing, a constant term should be included if the series has a non-zero average trend.

**Identifying
the numbers of AR and MA terms:**

- Rule
6: If the
__partial autocorrelation function__(PACF) of the differenced series displays a sharp cutoff and/or the lag-1 autocorrelation is__positive__--i.e., if the series appears slightly "underdifferenced"--then consider adding one or more__AR__terms to the model. The lag beyond which the PACF cuts off is the indicated number of AR terms. - Rule
7: If the
__autocorrelation function__(ACF) of the differenced series displays a sharp cutoff and/or the lag-1 autocorrelation is__negative__--i.e., if the series appears slightly "overdifferenced"--then consider adding an__MA__term to the model. The lag beyond which the ACF cuts off is the indicated number of MA terms. - Rule
8: It is possible for an AR term and an MA term to cancel each other's
effects, so if a mixed AR-MA model seems to fit the data, also try a model
with one fewer AR term and one fewer MA term--particularly if the
parameter estimates in the original model require more than 10 iterations
to converge.
**BEWARE OF USING MULTIPLE AR TERMS**__AND__MULTIPLE MA TERMS IN THE SAME MODEL. - Rule
9: If there is a unit root in the AR part of the model--i.e., if the sum
of the AR coefficients is almost exactly 1--you should reduce the number
of AR terms by one and
__increase__the order of differencing by one. - Rule
10: If there is a unit root in the MA part of the model--i.e., if the sum
of the MA coefficients is almost exactly 1--you should reduce the number
of MA terms by one and
__reduce__the order of differencing by one. - Rule
11: If the
__long-term forecasts__* appear erratic or unstable, there may be a unit root in the AR or MA coefficients.

**Identifying
the seasonal part of the model:**

- Rule
12: If the series has a strong and consistent seasonal pattern, then you
__must__use an order of seasonal differencing (otherwise the model assumes that the seasonal pattern will fade away over time). However, never use more than one order of seasonal differencing or more than 2 orders of total differencing (seasonal+nonseasonal). - Rule
13: If the autocorrelation of the appropriately differenced series is
__positive__at lag s, where s is the number of periods in a season, then consider adding an__SAR__term to the model. If the autocorrelation of the differenced series is__negative__at lag s, consider adding an__SMA__term to the model. The latter situation is likely to occur if a seasonal difference has been used, which__should__be done if the data has a stable and logical seasonal pattern. The former is likely to occur if a seasonal difference has__not__been used, which would only be appropriate if the seasonal pattern is__not__stable over time. You should try to avoid using more than one or two seasonal parameters (SAR+SMA) in the same model, as this is likely to lead to overfitting of the data and/or problems in estimation.

***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.