Autoregressive moving-average model: Difference between revisions

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{{redirect|ARMA model||ARMA (disambiguation)}}

In the [[statistics|statistical]] analysis of [[time series]], '''autoregressive–moving-average''' ('''ARMA''') '''models''' provideare a parsimoniousway descriptionto ofdescribe a [[stationary stochastic process|(weakly) stationary stochastic process]] in terms of two polynomials, one for theusing [[AR model|autoregression]] (AR) and the second for thea [[MA model|moving average]] (MA), each with a polynomial. TheThey generalare ARMAa modeltool wasfor describedunderstanding ina series and predicting future values. AR involves regressing the 1951variable thesison ofits [[Peterown Whittlelagged (mathematiciani.e., past) values. MA involves modeling the [[errors and residuals in statistics|Petererror]] as a [[linear Whittlecombination]], ''Hypothesisof error terms occurring contemporaneously and at various testingtimes in timethe seriespast. analysisThe model is usually denoted ARMA(''p'', and''q''), itwhere was''p'' popularized inis the 1970order bookof byAR [[Georgeand E.''q'' P.is Box]]the andorder [[Gwilymof Jenkins]]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''_''t'' , 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]].

== AutoregressiveMathematical modelformulation ==

=== 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 = c + \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. |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. |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>

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

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. |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>

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]].

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}}

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.

=== Moving- average model ===

{{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 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_tvarepsilon_1</math>, ..., <math>\varepsilon_{t-1}varepsilon_t</math>, are i.i.d. are again, [[white noise]] error terms that are commonly normal random variables.<ref>{{Cite book |last1=Box |first1=George E. P. |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. |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 = c + \varepsilon_t + \sum_{i=1}^p \varphi_i X_{t-i} + \sum_{i=1}^q \theta_i \varepsilon_{t-i}.\,</math>

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,

=== Specification inIn terms of lag operator ===

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

In some texts, the models will beis specified in terms ofusing the [[lag operator]] ''L''. In these terms, the AR(''p'') model is given by

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|last=Hannan|first=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.

== 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 ==

In some texts the models will be specified in terms of the [[lag operator]] ''L''.

In these terms then 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>

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

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

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:<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>

=== Alternative notation ===

Some authors, including [[George Box|Box]], [[Gwilym M. Jenkins|Jenkins]] & Reinsel use a different convention for the autoregression coefficients.<ref>{{cite book |first=George |last=Box |first2=Gwilym M. |last2=Jenkins |first3=Gregory C. |last3=Reinsel |title=Time Series Analysis: Forecasting and Control |edition=Third |publisher=Prentice-Hall |year=1994 |isbn=0130607746 }}</ref> This allows all the polynomials involving the lag operator to appear in a similar form throughout. Thus the ARMA model would be written as

== Spectrum ==

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

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 |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. |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>

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>

== Fitting models ==

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

FindingAn appropriate valuesvalue of ''p'' and ''q'' in the ARMA(''p'', ''q'') model can be facilitatedfound by plotting the [[partial autocorrelation function]]s. for an estimate ofSimilarly, ''pq'', andcan likewisebe estimated by using the [[autocorrelation function]]s. forBoth an''p'' estimate ofand ''q''. Extended autocorrelation functions (EACF) can be used todetermined simultaneously determineusing pextended andautocorrelation qfunctions (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 |lastlast1=Brockwell |firstfirst1=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 possible choice for order determiningoption is the [[Bayesian information criterion|BIC]] criterion(BIC).

===Estimating coefficients===

{{Expand section|date=March 2017}}

ARMA models in general can be, afterAfter 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 generally considered 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.

=== Implementations in statistics packages ===

* In [[R (programming language)|R]], the ''arima'' function (in standard package ''stats'') is documented in [http://search.r-project.org/R/library/stats/html/arima.html ARIMA Modelling of Time Series]. Extension packages contain related and extended functionality, e.g., the ''tseries'' package includes an ''arma'' function, documented in [http://finzi.psych.upenn.edu/R/library/tseries/html/arma.html "Fit ARMA Models to Time Series"]; the [https://cran.r-project.org/web/packages/fracdiff ''fracdiff'' package] contains ''fracdiff()'' for fractionally integrated ARMA processes; and the [https://cran.r-project.org/web/packages/forecast/index.html ''forecast'' package] includes ''auto.arima'' 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.

=== Software implementations ===

* In [[R (programming language)|R]], the ''arima'' function (in standard package ''<code>stats'')</code> ishas 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,: e.g.,package the ''<code>tseries'' package</code> includes an ''arma''the function <code>arma()</code>, documented in [http://finzi.psych.upenn.edu/R/library/tseries/html/arma.html "Fit ARMA Models to Time Series"]; the package[https://cran.r-project.org/web/packages/fracdiff ''<code>fracdiff'' package</code>] contains ''<code>fracdiff()''</code> for fractionally integrated ARMA processes; and thepackage [https://cran.r-project.org/web/packages/forecast/index.html ''<code>forecast'' package</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>] and, [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 ARautoregressive, ARXexogenous (autoregressive exogenous), 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 more informationdetails.

* [[Julia_(programming_language) | Julia]] has some 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>[[Statsmodels]https://statsmodels.sourceforge.net/ S] Pythonpackage modulewhich includes many models and functions for time series analysis, including ARMA. Formerly part of the [[Scikitscikit-learn]] library, it is now stand-alone and integrates well with [[Pandas (software)|Pandas]]. [http://statsmodels.sourceforge.net/ See here for more details].

* [[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 also estimate ARMA modelmodels, as mentioned [http://constantdream.wordpress.com/2008/03/16/gnu-regression-econometrics-and-time-series-library-gretl/ see here where it's mentioned].

* [[GNU Octave]] can estimate AR models using functions from the extra package [httphttps://octave.sourceforge.net/ <code>octave-forge</code>] supports AR models.

* [[Stata]] includes the function ''<code>arima''</code>. which can estimatefor ARMA and [[Autoregressive integrated moving average|ARIMA]] models. [https://www.stata.com/help.cgi?arima. See here for more details].

* [[SuanShu]] is a Java library of numerical methods, includingthat comprehensive statistics packages, in whichimplements univariate/multivariate ARMA, ARIMA, ARMAX, etc. models are implemented in an object-oriented approach. These implementations are, 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 here for more 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|year=1970 |___location=New York|publisher=John Wiley and Sons}}</ref><ref>{{cite book |author=Whittle, P. |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. |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|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 |lastlast1=Hannan |firstfirst1=E. J. |author-link=Edward James Hannan |title=Statistical theory of linear systems |last2=Deistler |first2=Manfred |titlepublisher=StatisticalJohn theoryWiley ofand linearSons |year=1988 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.

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 ==

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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.

The dependence of ''X''_''t'' 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.

=== {{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 -->

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.

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

Another generalization is the ''multiscale autoregressive'' (MAR) model. A MAR model is indexed by the nodes of a tree, whereas a standard (discrete time) autoregressive model is indexed by integers.

Note that the ARMA model is a '''univariate''' model. Extensions for the multivariate case are the [[vector autoregression]] (VAR) and Vector Autoregression Moving-Average (VARMA).

=== {{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,) 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>

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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 |lastlast1=Percival |firstfirst1=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]

{{Stochastic processes}}