Autoregressive moving-average model: Difference between revisions

Revision as of 11:38, 2 January 2021

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Some authors, including [[George Box|Box]], [[Gwilym M. Jenkins|Jenkins]] & Reinsel use a different convention for the autoregression coefficients.<ref>{{cite book |first=George |last=Box |first2=Gwilym M. |last2=Jenkins |first3=Gregory C. |last3=Reinsel |title=Time Series Analysis: Forecasting and Control |edition=Third |publisher=Prentice-Hall |year=1994 |isbn=0130607746 }}</ref> This allows all the polynomials involving the lag operator to appear in a similar form throughout. Thus the ARMA model would be written as

:<math> \left(1 - \sum_{i=1}^p \phi_i L^i\right) X_t = \left(1 + \sum_{i=1}^q \theta_i L^i\right) \varepsilon_t \, .</math>

Moreover, if we set <math>X_0 \phi_0 = -1 </math> and <math> \varepsilon_0theta_0 = 1 </math>, then we get an even more elegant formulation:

<math> -\sum_{i=0}^p \phi_i L^i \; X_t = \sum_{i=0}^q \theta_i L^i \; \varepsilon_t \, .</math>