Autoregressive moving-average model: Difference between revisions

Revision as of 11:10, 26 August 2022

Revision as of 20:47, 26 August 2022

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:<math> X_t = c + \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=Enders |first=Walter |url=https://www.worldcat.org/oclc/52387978 |title=Applied econometric time series |date=2004 |publisher=J. Wiley |isbn=0-471-23065-0 |edition=2nd |___location=Hoboken, NJ |pages=59 |language=en |oclc=52387978}}</ref>

Some constraints are necessary on the values of the parameters so that the model remains [[stationary process|stationary]]. For example, processes in the AR(1) model with <math>|\varphi_1| \ge 1</math> are not stationary.

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:<math> X_t = \mu + \varepsilon_t + \sum_{i=1}^q \theta_i \varepsilon_{t-i}\,</math>

where the θ₁, ..., θ_''q'' are the parameters of the model, μ is the expectation of <math>X_t</math> (often assumed to equal 0), and the <math>\varepsilon_t</math>, <math>\varepsilon_{t-1}</math>,... are again, [[i.i.d. white noise]] error terms that are commonly normal random variables.<ref>{{Cite book |last=Box |first=George E. P. |url=https://www.worldcat.org/oclc/908107438 |title=Time series analysis : forecasting and control |last2=Jenkins |first2=Gwilym M. |last3=Reinsel |first3=Gregory C. |last4=Ljung |first4=Greta M. |date=2016 |publisher=John Wiley & Sons, Incorporated |others= |year= |isbn=978-1-118-67492-5 |edition=5th |___location=Hoboken, New Jersey |pages=53 |language=en |oclc=908107438}}</ref>

== ARMA model ==

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The ARMA model is essentially an [[infinite impulse response]] filter applied to white noise, with some additional interpretation placed on it.

== Note about the error terms ==

The error terms <math>\varepsilon_t</math> are generally assumed to be [[independent identically distributed random variables]] (i.i.d.) sampled from a [[normal distribution]] with zero mean: <math>\varepsilon_t</math> ~ N(0,σ²) where σ² is

the variance. These assumptions may be weakened but doing so will change the properties of the model. In particular, a change to the i.i.d. assumption would make a rather fundamental difference.

== Specification in terms of lag operator ==