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.