Editing Autoregressive moving-average model

{{Short description|Statistical model used in time series analysis}}
{{redirect|ARMA model||ARMA (disambiguation)}}

In the [[statistics|statistical]] analysis of [[time series]], '''autoregressive–moving-average''' ('''ARMA''') '''models''' are a way to describe a [[stationary stochastic process|(weakly) stationary stochastic process]] using [[AR model|autoregression]] (AR) and a [[MA model|moving average]] (MA), each with a polynomial. They are a tool for understanding a series and predicting future values. AR involves regressing the variable on its own lagged (i.e., past) values. MA involves modeling the [[errors and residuals in statistics|error]] as a [[linear combination]] of error terms occurring contemporaneously and at various times in the past. The model is usually denoted ARMA(''p'', ''q''), where ''p'' is the order of AR and ''q'' is the order of 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]].

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

== Mathematical formulation ==

=== Autoregressive model ===
{{Main|Autoregressive model}}
The notation AR(''p'') refers to the autoregressive model of order ''p''. The AR(''p'') model is written 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 [[parameter]]s 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 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 |last1=Box |first1=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 |isbn=0-13-060774-6 |edition=3rd |location=Englewood Cliffs, N.J. |pages=54–55 |language=en |oclc=28888762}}</ref>

The [[augmented Dickey–Fuller test]] can assesses the stability of an [[intrinsic mode function]] and trend components. For stationary time series, the ARMA models can be used, while for non-stationary series, [[Long short-term memory]] models can be used to derive abstract features. The final value is obtained by reconstructing the predicted outcomes of each time series.{{Citation needed|date=April 2025}}

=== Moving average model ===
{{Main|Moving-average model}}
The notation MA(''q'') refers to the moving average model of order ''q'':

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

where the <math>\theta_1,...,\theta_q</math> are the parameters of the model, <math>\mu</math> is the expectation of <math>X_t</math> (often assumed to equal 0), and <math>\varepsilon_1</math>, ..., <math>\varepsilon_t</math> are i.i.d. white noise error terms that are commonly normal random variables.<ref>{{Cite book |last1=Box |first1=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 |isbn=978-1-118-67492-5 |edition=5th |location=Hoboken, New Jersey |pages=53 |language=en |oclc=908107438}}</ref>

=== ARMA model ===
The notation ARMA(''p'', ''q'') refers to the model with ''p'' autoregressive terms and ''q'' moving-average terms. This model contains the AR(''p'') and MA(''q'') models,<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=98 |language=en |oclc=42392178}}</ref>

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

=== In terms of lag operator ===
In some texts, the models is specified using the [[lag operator]] ''L''.  In these terms, the AR(''p'') model is given by

:<math> \varepsilon_t = \left(1 - \sum_{i=1}^p \varphi_i L^i\right) X_t =  \varphi (L) X_t\,</math>

where <math>\varphi</math> represents the polynomial

:<math> \varphi (L) = 1 - \sum_{i=1}^p \varphi_i L^i.\,</math>

The MA(''q'') model is given by

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

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

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

Finally, the combined ARMA(''p'', ''q'') model is given by

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

or more concisely,

:<math> \varphi(L) X_t = \theta(L) \varepsilon_t \, </math>

or

:<math> \frac{\varphi(L)}{\theta(L)}X_t = \varepsilon_t \, .</math>
This is the form used in [[George Box|Box]], [[Gwilym M. Jenkins|Jenkins]] & Reinsel.<ref>{{cite book |last1=Box |first1=George |title=Time Series Analysis: Forecasting and Control |last2=Jenkins |first2=Gwilym M. |last3=Reinsel |first3=Gregory C. |publisher=Prentice-Hall |year=1994 |isbn=0130607746 |edition=Third}}</ref> 

Moreover, starting summations from  <math> i=0 </math> and setting <math> \phi_0 = -1 </math> and <math> \theta_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>

== Spectrum ==
The [[spectral density]] of an ARMA process is<math display="block">S(f) = \frac{\sigma^2}{2\pi} \left\vert \frac{\theta(e^{-if})}{\phi(e^{-if})} \right\vert^2</math>where <math>\sigma^2</math> is the [[variance]] of the white noise, <math>\theta</math> is the characteristic polynomial of the moving average part of the ARMA model, and <math>\phi</math> is the characteristic polynomial of the autoregressive part of the ARMA model.<ref>{{Cite book |last=Rosenblatt |first=Murray |url=https://www.worldcat.org/oclc/42061096 |title=Gaussian and non-Gaussian linear time series and random fields |date=2000 |publisher=Springer |isbn=0-387-98917-X |location=New York |pages=10 |language=en |oclc=42061096}}</ref><ref>{{Cite book |last=Wei |first=William W. S. |url=https://www.worldcat.org/oclc/18166355 |title=Time series analysis : univariate and multivariate methods |date=1990 |publisher=Addison-Wesley Pub |isbn=0-201-15911-2 |location=Redwood City, Calif. |pages=242–243 |language=en |oclc=18166355}}</ref>

== Fitting models ==

===Choosing ''p'' and ''q''===

An appropriate value of ''p'' in the ARMA(''p'', ''q'') model can be found by plotting the [[partial autocorrelation function]]s. Similarly, ''q'' can be estimated by using the [[autocorrelation function]]s. Both ''p'' and ''q'' can be determined simultaneously using extended autocorrelation functions (EACF).<ref>{{Cite web|last=Missouri State University|title=Model Specification, Time Series Analysis|url=http://people.missouristate.edu/songfengzheng/Teaching/MTH548/Time%20Series-ch06.pdf}}</ref> Further information can be gleaned by considering the same functions for the residuals of a model fitted with an initial selection of ''p'' and ''q''.

Brockwell & Davis recommend using [[Akaike information criterion]] (AIC) for finding ''p'' and ''q''.<ref>{{cite book |last1=Brockwell |first1=P. J. |last2=Davis |first2=R. A. |title=Time Series: Theory and Methods |edition=2nd |publisher=Springer |location=New York |year=2009 |page=273 |isbn=9781441903198 }}</ref> Another option is the [[Bayesian information criterion]] (BIC).

===Estimating coefficients===

After choosing ''p'' and ''q,'' ARMA models can be fitted by [[least squares]] regression to find the values of the parameters which minimize the error term.  It is good practice to find the smallest values of ''p'' and ''q'' which provide an acceptable fit to the data.  For a pure AR model, the [[AR model#Calculation of the AR parameters|Yule-Walker equations]] may be used to provide a fit.

ARMA outputs are used primarily to forecast (predict), and not to infer causation as in other areas of econometrics and regression methods such as OLS and 2SLS.

=== Software implementations ===
* In [[R (programming language)|R]], standard package <code>stats</code> has function <code>arima</code>, documented in [http://search.r-project.org/R/library/stats/html/arima.html ARIMA Modelling of Time Series]. Package [https://cran.r-project.org/web/packages/astsa/index.html <code>astsa</code>] has an improved script called <code>sarima</code> for fitting ARMA models (seasonal and nonseasonal) and <code>sarima.sim</code> to simulate data from these models. Extension packages contain related and extended functionality: package <code>tseries</code> includes the function <code>arma()</code>, documented in [http://finzi.psych.upenn.edu/R/library/tseries/html/arma.html "Fit ARMA Models to Time Series"]; package[https://cran.r-project.org/web/packages/fracdiff <code>fracdiff</code>] contains <code>fracdiff()</code> for fractionally integrated ARMA processes; and package [https://cran.r-project.org/web/packages/forecast/index.html <code>forecast</code>] includes <code>auto.arima</code> for selecting a parsimonious set of ''p, q''.  The CRAN task view on [https://cran.r-project.org/web/views/TimeSeries.html Time Series] contains links to most of these.
* [[Mathematica]] has a complete library of time series functions including ARMA.<ref>[http://www.wolfram.com/products/applications/timeseries/features.html Time series features in Mathematica] {{webarchive |url=https://web.archive.org/web/20111124032002/http://www.wolfram.com/products/applications/timeseries/features.html |date=November 24, 2011 }}</ref>
* [[MATLAB]] includes functions such as [http://www.mathworks.com/help/econ/arma-model.html <code>arma</code>], [http://www.mathworks.com/help/ident/ref/ar.html <code>ar</code>] and [http://www.mathworks.com/help/ident/ref/arx.html <code>arx</code>] to estimate autoregressive, exogenous autoregressive and ARMAX models.  See [http://www.mathworks.com/help/ident/ug/estimating-ar-and-arma-models.html System Identification Toolbox] and [http://www.mathworks.com/help/econ/arima.estimate.html Econometrics Toolbox] for details.
* [[Julia_(programming_language) | Julia]] has community-driven packages that implement fitting with an ARMA model such as [https://github.com/joefowler/ARMA.jl <code>arma.jl</code>].
* Python has the <code>statsmodels</code>[http://statsmodels.sourceforge.net/ S] package which includes many models and functions for time series analysis, including ARMA. Formerly part of the [[scikit-learn]] library, it is now stand-alone and integrates well with [[Pandas (software)|Pandas]].
* [[PyFlux]] has a Python-based implementation of ARIMAX models, including Bayesian ARIMAX models.
* [[IMSL Numerical Libraries]] are libraries of numerical analysis functionality including ARMA and ARIMA procedures implemented in standard programming languages like C, Java, C# .NET, and Fortran.
* [[gretl]] can estimate ARMA models, as mentioned [http://constantdream.wordpress.com/2008/03/16/gnu-regression-econometrics-and-time-series-library-gretl/ here]
* [[GNU Octave]] extra package [http://octave.sourceforge.net/ <code>octave-forge</code>] supports AR models.
* [[Stata]] includes the function <code>arima</code>. for ARMA and [[Autoregressive integrated moving average|ARIMA]] models.
* [[SuanShu]] is a Java library of numerical methods that implements univariate/multivariate ARMA, ARIMA, ARMAX, etc models, documented in [http://www.numericalmethod.com/javadoc/suanshu/ "SuanShu, a Java numerical and statistical library"].
* [[SAS (software)|SAS]] has an econometric package, ETS, that estimates ARIMA models. [https://web.archive.org/web/20110930032431/http://support.sas.com/rnd/app/ets/proc/ets_arima.html See details].

== History and interpretations ==
The general ARMA model was described in the 1951 thesis of [[Peter Whittle (mathematician)|Peter Whittle]], who used mathematical analysis ([[Laurent series]] and [[Fourier analysis]]) and statistical inference.<ref>{{cite book |last=Hannan |first=Edward James |author-link=Edward James Hannan |title=Multiple time series |publisher=John Wiley and Sons |year=1970 |series=Wiley series in probability and mathematical statistics |location=New York}}</ref><ref>{{cite book |author=Whittle, P. |title=Hypothesis Testing in Time Series Analysis |publisher=Almquist and Wicksell |year=1951}}

{{cite book |author=Whittle, P. |title=Prediction and Regulation |publisher=English Universities Press |year=1963 |isbn=0-8166-1147-5}}

: Republished as: {{cite book |author=Whittle, P. |title=Prediction and Regulation by Linear Least-Square Methods |publisher=University of Minnesota Press |year=1983 |isbn=0-8166-1148-3}}</ref> ARMA models were popularized by a 1970 book by [[George E. P. Box]] and Jenkins, who expounded an iterative ([[Box–Jenkins]]) method for choosing and estimating them. This method was useful for low-order polynomials (of degree three or less).<ref>{{harvtxt|Hannan|Deistler|1988|loc=p. 227}}: {{cite book |last1=Hannan |first1=E. J. |author-link=Edward James Hannan |title=Statistical theory of linear systems |last2=Deistler |first2=Manfred |publisher=John Wiley and Sons |year=1988 |series=Wiley series in probability and mathematical statistics |location=New York}}</ref>

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

In [[digital signal processing]], ARMA is represented as a digital filter with white noise at the input and the ARMA process at the output.

== Applications ==
ARMA is appropriate when a system is a function of a series of unobserved shocks (the MA or moving average part) as well as its own behavior.  For example, stock prices may be shocked by fundamental information as well as exhibiting technical trending and [[mean reversion (finance)|mean-reversion]] effects due to market participants.{{Citation needed|date=November 2018}}

== Generalizations ==

There are various generalizations of ARMA. '''Nonlinear''' AR (NAR), nonlinear MA (NMA) and nonlinear ARMA (NARMA) model nonlinear dependence on past values and error terms. [[vector autoregression|Vector AR]] (VAR) and vector ARMA (VARMA) model '''multivariate''' time series. [[Autoregressive integrated moving average]] (ARIMA) models non-stationary time series (that is, whose mean changes over time). [[Autoregressive conditional heteroskedasticity]] (ARCH) models time series where the variance changes. Seasonal ARIMA (SARIMA or periodic ARMA) models '''periodic''' variation.  [[Autoregressive fractionally integrated moving average]] (ARFIMA, or Fractional ARIMA, FARIMA) model time-series that exhibits '''long memory'''. Multiscale AR (MAR) is indexed by the nodes of a [[Tree (graph theory)|tree]] instead of integers.

=== {{anchor|ARMAX}}Autoregressive–moving-average model with exogenous inputs (ARMAX) === <!-- This section is linked from [[ARMAX]], so if you change the title, please also change the corresponding link in the ARMAX page -->

The notation ARMAX(''p'', ''q'', ''b'') refers to a model with ''p'' autoregressive terms, ''q'' moving average terms and ''b'' exogenous inputs terms. The last term is a linear combination of the last ''b'' terms of a known and external time series <math>d_t</math>. It is given by:

:<math> X_t = \varepsilon_t +  \sum_{i=1}^p \varphi_i X_{t-i} + \sum_{i=1}^q \theta_i \varepsilon_{t-i} + \sum_{i=1}^b \eta_i d_{t-i}.\,</math>
where <math>\eta_1, \ldots, \eta_b</math> are the '''''parameters''''' of the exogenous input <math>d_t</math>.

Some nonlinear variants of models with exogenous variables have been defined: see for example [[Nonlinear autoregressive exogenous model]].

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 <math>m_t</math> incorporates all exogenous (or independent) variables:
: <math>m_t = c + \sum_{i=0}^b \eta_i d_{t-i}.\,</math>

== See also ==
* [[Autoregressive integrated moving average]] (ARIMA)
* [[Exponential smoothing]]
* [[Linear predictive coding]]
* [[Predictive analytics]]
* [[Infinite impulse response]]
* [[Finite impulse response]]

{{More footnotes|date=August 2010}}

== References ==
{{Reflist}}
== Further reading ==
* {{cite book |last=Mills |first=Terence C. |title=Time Series Techniques for Economists |publisher=Cambridge University Press |year=1990 |isbn=0521343399 |url-access=registration |url=https://archive.org/details/timeseriestechni0000mill }}
* {{cite book |last1=Percival |first1=Donald B. |first2=Andrew T. |last2=Walden |title=Spectral Analysis for Physical Applications |publisher=Cambridge University Press |year=1993 |isbn=052135532X }}
* {{citation| first1= C. | last1= Francq | first2= J.-M. | last2= Zakoïan | chapter= Recent results for linear time series models with non independent innovations | pages= 241–265 | title= Statistical Modeling and Analysis for Complex Data Problems | editor1-first= P. |editor1-last= Duchesne | editor2-first=B. | editor2-last= Remillard | publisher= Springer | year= 2005| citeseerx= 10.1.1.721.1754 }}.
* [https://link.springer.com/book/10.1007/978-3-319-52452-8 Shumway, R.H. and Stoffer, D.S. (2017). ''Time Series Analysis and Its Applications with R Examples''. Springer. DOI: 10.1007/978-3-319-52452-8] 

{{Stochastic processes}}
{{Statistics|analysis}}

{{DEFAULTSORT:Autoregressive-Moving-Average Model}}
[[Category:Autocorrelation]]