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{{redirect|ARMA model||ARMA (disambiguation)}}
In the [[statistics|statistical]] analysis of [[time series]], '''autoregressive–moving-average''' ('''ARMA''') '''models''' are a way to describe
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]].
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ARMA models can be estimated by using the [[Box–Jenkins method]].
==
=== Autoregressive model ===
{{Main|Autoregressive model}}
The notation AR(''p'') refers to the autoregressive model of order ''p''. The AR(''p'') model is written as
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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 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.
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.
The [[augmented Dickey–Fuller test]] can assesses the stability of
=== Moving
{{Main|Moving-average model}}
The notation MA(''q'') refers to the moving average model of order ''q'':
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:<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.
=== 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.
▲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>
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|series=Wiley series in probability and mathematical statistics|year=1970|___location=New York|publisher=John Wiley and Sons}}</ref><ref>{{cite book|title=Hypothesis Testing in Time Series Analysis|author=Whittle, P.|publisher=Almquist and Wicksell|year=1951}}▼
{{cite book|title=Prediction and Regulation|author=Whittle, P.|publisher=English Universities Press|year=1963|isbn=0-8166-1147-5}}▼
:Republished as: {{cite book|title=Prediction and Regulation by Linear Least-Square Methods|author=Whittle, P.|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|last2=Deistler|first2=Manfred|title=Statistical theory of linear systems|series=Wiley series in probability and mathematical statistics|year=1988|___location=New York|publisher=John Wiley and Sons}}</ref>▼
The ARMA model is essentially an [[infinite impulse response]] filter applied to white noise, with some additional interpretation placed on it.▼
▲== Specification 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
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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
In [[digital signal processing]], the ARMA model is represented as a digital filter with white noise at the input and the ARMA process at the output.▼
== Fitting models ==
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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]],
* [[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,
* [[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>[https://statsmodels.sourceforge.net/
* [[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]]
* [[Stata]] includes the function <code>arima</code>.
* [[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 ==
▲== Spectrum ==
▲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
▲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>
▲{{cite book
▲: Republished as: {{cite book |author=Whittle, P. |title=Prediction and Regulation by Linear Least-Square Methods
▲
▲In [[digital signal processing]],
== Applications ==
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== 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
The notation ARMAX(''p'', ''q'', ''b'') refers to
▲=== {{anchor|ARMAX}}Autoregressive–moving-average model with exogenous inputs model (ARMAX model) === <!-- 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 the model with ''p'' autoregressive terms, ''q'' moving average terms and ''b'' exogenous inputs terms. This model contains the AR(''p'') and MA(''q'') models and 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>
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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
: <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:
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== 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 |url=https://archive.org/details/spectralanalysis0000perc |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]
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