Editing Autoregressive model (section)

==Definition==

The notation <math>AR(p)</math> indicates an autoregressive model of order ''p''. The AR(''p'') model is defined as

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

where <math>\varphi_1, \ldots, \varphi_p</math> are the ''parameters'' of the model, and <math>\varepsilon_t</math> is [[white noise]].<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 |access-date=2022-09-03 |archive-date=2023-04-16 |archive-url=https://web.archive.org/web/20230416160928/https://www.worldcat.org/title/42392178 |url-status=live }}</ref> This can be equivalently written using the [[backshift operator]] ''B'' as

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

so that, moving the summation term to the left side and using [[polynomial notation]], we have

:<math>\phi [B]X_t= \varepsilon_t</math>

An autoregressive model can thus be viewed as the output of an all-[[pole (complex analysis)|pole]] [[infinite impulse response]] filter whose input is white noise.

Some parameter constraints are necessary for the model to remain [[Stationary process#Weak or wide-sense stationarity|weak-sense stationary]].  For example, processes in the AR(1) model with <math>|\varphi_1 | \geq 1</math> are not stationary. More generally, for an AR(''p'') model to be weak-sense stationary, the roots of the polynomial <math>\Phi(z):=\textstyle 1 - \sum_{i=1}^p \varphi_i z^{i}</math> must lie outside the [[unit circle]], i.e., each (complex) root <math>z_i</math> must satisfy <math>|z_i |>1</math> (see pages 89,92 <ref>{{cite book |last1=Shumway |first1=Robert H. |last2=Stoffer |first2=David |title=Time series analysis and its applications : with R examples |date=2010 |publisher=Springer |isbn=978-1441978646 |edition=3rd}}</ref>).