Autoregressive moving-average model: Difference between revisions

Revision as of 20:47, 26 August 2022

Revision as of 23:13, 27 August 2022

Line 4:

In the [[statistics|statistical]] analysis of [[time series]], '''autoregressive–moving-average''' ('''ARMA''') '''models''' provide a parsimonious description of a [[stationary stochastic process|(weakly) stationary stochastic process]] in terms of two polynomials, one for the [[AR model|autoregression]] (AR) and the second for the [[MA model|moving average]] (MA). The general ARMA model was described in the 1951 thesis of [[Peter Whittle (mathematician)|Peter Whittle]], ''Hypothesis testing in time series analysis'', and it was popularized in the 1970 book by [[George E. P. Box]] and [[Gwilym Jenkins]].

Given a time series of data ''X''<submath>''t''X_t</submath> , the ARMA model is a tool for understanding and, perhaps, predicting future values in this series. The AR part involves regressing the variable on its own lagged (i.e., past) values. The MA part involves modeling the [[errors and residuals in statistics|error term]] as a [[linear combination]] of error terms occurring contemporaneously and at various times in the past. The model is usually referred to as the ARMA(''p'',''q'') model where ''p'' is the order of the AR part and ''q'' is the order of the MA part (as defined below).

ARMA models can be estimated by using the [[Box–Jenkins method]].

Line 24:

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

where the θ<sub>1</submath>\theta_1, ..., θ<sub>''q''\theta_q</submath> are the parameters of the model, μ<math>\mu</math> 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 ==

Line 55:

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

where θ<math>\theta</math> represents the polynomial

:<math> \theta(L)= 1 + \sum_{i=1}^q \theta_i L^i .\,</math>

Line 110:

== Generalizations ==

The dependence of ''X''<submath>''t''X_t</submath> on past values and the error terms ε_t is assumed to be linear unless specified otherwise. If the dependence is nonlinear, the model is specifically called a ''nonlinear moving average'' (NMA), ''nonlinear autoregressive'' (NAR), or ''nonlinear autoregressive–moving-average'' (NARMA) model.

Autoregressive–moving-average models can be generalized in other ways. See also [[autoregressive conditional heteroskedasticity]] (ARCH) models and [[autoregressive integrated moving average]] (ARIMA) models. If multiple time series are to be fitted then a vector ARIMA (or VARIMA) model may be fitted. If the time-series in question exhibits long memory then fractional ARIMA (FARIMA, sometimes called ARFIMA) modelling may be appropriate: see [[Autoregressive fractionally integrated moving average]]. If the data is thought to contain seasonal effects, it may be modeled by a SARIMA (seasonal ARIMA) or a periodic ARMA model.

Line 129:

Statistical packages implement the ARMAX model through the use of "exogenous" (that is, independent,) variables. Care must be taken when interpreting the output of those packages, because the estimated parameters usually (for example, in [[R (programming language)|R]]<ref name="R.stats.arima">[http://search.r-project.org/R/library/stats/html/arima.html ARIMA Modelling of Time Series], R documentation</ref> and [[gretl]]) refer to the regression:

: <math> X_t - m_t = \varepsilon_t + \sum_{i=1}^p \varphi_i (X_{t-i} - m_{t-i}) + \sum_{i=1}^q \theta_i \varepsilon_{t-i}.\,</math>

where ''m<submath>tm_t</submath>'' incorporates all exogenous (or independent) variables:

: <math>m_t = c + \sum_{i=0}^b \eta_i d_{t-i}.\,</math>