Box–Jenkins method: Difference between revisions

Revision as of 12:24, 19 May 2015

Revision as of 12:28, 19 May 2015

Line 31:

Once stationarity and seasonality have been addressed, the next step is to identify the order (i.e., the ''p'' and ''q'') of the autoregressive and moving average terms. Different authors have different approaches for identifying ''p'' and ''q''. Brockwell and Davis (1991, p. 273) state "our prime criterion for model selection [among ARMA(p,q) models] will be the AICc", i.e. the [[Akaike information criterion]] with correction.

Other authors use the autocorrelation plot and the partial autocorrelation plot. For example, Hyndman & Athanasopoulos suggest the following:

The data may follow an ARIMA(''p'',d,0) model if the ACF and PACF plots of the differenced data show the following patterns:

* the ACF is exponentially decaying or sinusoidal;

* there is a significant spike at lag ''p'' in PACF, but none beyond lag ''p''.

The data may follow an ARIMA(0,d,''q'') model if the ACF and PACF plots of the differenced data show the following patterns:

* the PACF is exponentially decaying or sinusoidal;

* there is a significant spike at lag ''q'' in ACF, but none beyond lag ''q''.<ref>{{cite web|last1=Hyndman|first1=Rob J|last2=Athanasopoulos|first2=George|title=Forecasting: principles and practice|url=https://www.otexts.org/fpp/8/5|accessdate=18 May 2015}}</ref>

====Autocorrelation and partial autocorrelation plots====

Line 78 ⟶ 70:

| Series is not stationary.

|}

*Hyndman there& isAthanasopoulos asuggest significantthe spike at lag ''q'' in ACF, but none beyond lag ''q''.following:<ref>{{cite web|last1=Hyndman|first1=Rob J|last2=Athanasopoulos|first2=George|title=Forecasting: principles and practice|url=https://www.otexts.org/fpp/8/5|accessdate=18 May 2015}}</ref>

:The data may follow an ARIMA(''p'',d,0) model if the ACF and PACF plots of the differenced data show the following patterns:

:* the ACF is exponentially decaying or sinusoidal;

:* there is a significant spike at lag ''p'' in PACF, but none beyond lag ''p''.

:The data may follow an ARIMA(0,d,''q'') model if the ACF and PACF plots of the differenced data show the following patterns:

:* the PACF is exponentially decaying or sinusoidal;

:* there is a significant spike at lag ''q'' in ACF, but none beyond lag ''q''.

In practice, the sample autocorrelation and partial autocorrelation functions are [[random variable]]s and do not give the same picture as the theoretical functions. This makes the model identification more difficult. In particular, mixed models can be particularly difficult to identify. Although experience is helpful, developing good models using these sample plots can involve much trial and error.