Autoregressive moving-average model: Difference between revisions

Revision as of 06:49, 3 September 2022

Revision as of 00:46, 6 December 2022

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:<math> X_t = \sum_{i=1}^p \varphi_i X_{t-i}+ \varepsilon_t</math>

where <math>\varphi_1, \ldots, \varphi_p</math> are [[parameter]]s, <math>c</math> is a constant, and the random variable <math>\varepsilon_t</math> is [[white noise]], usually [[Independent and identically distributed random variables|independent and identically distributed]] (i.i.d.) [[Normal distribution|normal random variables]].<ref>{{Cite book |last=Box |first=George E. P. |url=https://www.worldcat.org/oclc/28888762 |title=Time series analysis : forecasting and control |date=1994 |publisher=Prentice Hall |others=Gwilym M. Jenkins, Gregory C. Reinsel |isbn=0-13-060774-6 |edition=3rd |___location=Englewood Cliffs, N.J. |pages=54 |language=en |oclc=28888762}}</ref><ref>{{Cite book |last=Shumway |first=Robert H. |url=https://www.worldcat.org/oclc/42392178 |title=Time series analysis and its applications |date=2000 |publisher=Springer |others=David S. Stoffer |isbn=0-387-98950-1 |___location=New York |pages=90-91 |language=en |oclc=42392178}}</ref>

In order for the model to remain [[Stationary process|stationary]], the roots of its [[Autoregressive model#Characteristic polynomial|characteristic polynomial]] must lie outside of the unit circle. For example, processes in the AR(1) model with <math>|\varphi_1| \ge 1</math> are not stationary because the root of <math>1 - \varphi_1B = 0</math> lies within the unit circle.<ref>{{Cite book |last=Box |first=George E. P. |url=https://www.worldcat.org/oclc/28888762 |title=Time series analysis : forecasting and control |last2=Jenkins |first2=Gwilym M. |last3=Reinsel |first3=Gregory C. |date=1994 |publisher=Prentice Hall |others= |isbn=0-13-060774-6 |edition=3rd |___location=Englewood Cliffs, N.J. |pages=54-55 |language=en |oclc=28888762}}</ref>