# Autoregressive–moving-average model

ARMAautoregressive moving average modelautoregressive moving averageARMAXautoregressive moving-averageautoregressive–moving-averageauto-regressive moving average system identificationauto-regressive or moving average modelAutoregressive moving average (ARMA)Autoregressive moving average with exogenous inputs (ARMAX)
In the statistical analysis of time series, autoregressive–moving-average (ARMA) models provide a parsimonious description of a (weakly) stationary stochastic process in terms of two polynomials, one for the autoregression (AR) and the second for the moving average (MA).wikipedia
84 Related Articles

### Autoregressive model

autoregressiveautoregressionAutoregressive process
In the statistical analysis of time series, autoregressive–moving-average (ARMA) models provide a parsimonious description of a (weakly) stationary stochastic process in terms of two polynomials, one for the autoregression (AR) and the second for the moving average (MA).
Together with the moving-average (MA) model, it is a special case and key component of the more general autoregressive–moving-average (ARMA) and autoregressive integrated moving average (ARIMA) models of time series, which have a more complicated stochastic structure; it is also a special case of the vector autoregressive model (VAR), which consists of a system of more than one interlocking stochastic difference equation in more than one evolving random variable.

### Moving-average model

Moving average modelmoving averagemoving average process
In the statistical analysis of time series, autoregressive–moving-average (ARMA) models provide a parsimonious description of a (weakly) stationary stochastic process in terms of two polynomials, one for the autoregression (AR) and the second for the moving average (MA).
Together with the autoregressive (AR) model, the moving-average model is a special case and key component of the more general ARMA and ARIMA models of time series, which have a more complicated stochastic structure.

### Time series

time series analysistime-seriestime-series analysis
In the statistical analysis of time series, autoregressive–moving-average (ARMA) models provide a parsimonious description of a (weakly) stationary stochastic process in terms of two polynomials, one for the autoregression (AR) and the second for the moving average (MA).
Combinations of these ideas produce autoregressive moving average (ARMA) and autoregressive integrated moving average (ARIMA) models.

### Box–Jenkins method

Box–Jenkins Box–Jenkins approachBox–Jenkins analysis
ARMA models can be estimated by using the Box–Jenkins method.
In time series analysis, the Box–Jenkins method, named after the statisticians George Box and Gwilym Jenkins, applies autoregressive moving average (ARMA) or autoregressive integrated moving average (ARIMA) models to find the best fit of a time-series model to past values of a time series.

### Stationary process

stationarynon-stationarystationarity
In the statistical analysis of time series, autoregressive–moving-average (ARMA) models provide a parsimonious description of a (weakly) stationary stochastic process in terms of two polynomials, one for the autoregression (AR) and the second for the moving average (MA).
Other examples of a discrete-time stationary process with continuous sample space include some autoregressive and moving average processes which are both subsets of the autoregressive moving average model.

### Gwilym Jenkins

Jenkins, GwilymGwilym M. JenkinsJenkins
The general ARMA model was described in the 1951 thesis of Peter Whittle, Hypothesis testing in time series analysis, and it was popularized in the 1970 book by George E. P. Box and Gwilym Jenkins.
He is most notable for his pioneering work with George Box on autoregressive moving average models, also called Box–Jenkins models, in time-series analysis.

### Lag operator

backshift operatorlagLag or backshift operator
In some texts the models will be specified in terms of the lag operator L.
Polynomials of the lag operator can be used, and this is a common notation for ARMA (autoregressive moving average) models.

### Autoregressive integrated moving average

ARIMAAutoregressive integrated moving average modelAutoregressive integrated moving average (ARIMA)
See also autoregressive conditional heteroskedasticity (ARCH) models and autoregressive integrated moving average (ARIMA) models.
In statistics and econometrics, and in particular in time series analysis, an autoregressive integrated moving average (ARIMA) model is a generalization of an autoregressive moving average (ARMA) model.

### Autoregressive conditional heteroskedasticity

GARCHARCHARCH model
See also autoregressive conditional heteroskedasticity (ARCH) models and autoregressive integrated moving average (ARIMA) models.
The ARCH model is appropriate when the error variance in a time series follows an autoregressive (AR) model; if an autoregressive moving average (ARMA) model is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH) model.

### Predictive analytics

predictiveCARTpredictive analysis
The Box–Jenkins methodology (1976) developed by George Box and G.M. Jenkins combines the AR and MA models to produce the ARMA (autoregressive moving average) model, which is the cornerstone of stationary time series analysis.

### Exponential smoothing

basic exponential smoothingDouble exponential smoothingexponential

### Statistics

statisticalstatistical analysisstatistician
In the statistical analysis of time series, autoregressive–moving-average (ARMA) models provide a parsimonious description of a (weakly) stationary stochastic process in terms of two polynomials, one for the autoregression (AR) and the second for the moving average (MA).

### Peter Whittle (mathematician)

Peter WhittleWhittle, Peter
The general ARMA model was described in the 1951 thesis of Peter Whittle, Hypothesis testing in time series analysis, and it was popularized in the 1970 book by George E. P. Box and Gwilym Jenkins. The general ARMA model was described in the 1951 thesis of Peter Whittle, who used mathematical analysis (Laurent series and Fourier analysis) and statistical inference.

### Errors and residuals

residualserror termresidual
The MA part involves modeling the error term as a linear combination of error terms occurring contemporaneously and at various times in the past.

### Linear combination

linear combinationslinearly combined(finite) left ''R''-linear combinations
The MA part involves modeling the error term as a linear combination of error terms occurring contemporaneously and at various times in the past.

### Parameter

parametersparametricargument
where are parameters, c is a constant, and the random variable is white noise.

### White noise

whitenoisestatic
where are parameters, c is a constant, and the random variable is white noise.

### Laurent series

Laurent expansion theoremLaurent power seriesfield of Laurent series
The general ARMA model was described in the 1951 thesis of Peter Whittle, who used mathematical analysis (Laurent series and Fourier analysis) and statistical inference.

### Fourier analysis

FourierFourier synthesisanalyse the output wave into its constituent harmonics
The general ARMA model was described in the 1951 thesis of Peter Whittle, who used mathematical analysis (Laurent series and Fourier analysis) and statistical inference.

### Independent and identically distributed random variables

independent and identically distributedi.i.d.iid
The error terms are generally assumed to be independent identically distributed random variables (i.i.d.) sampled from a normal distribution with zero mean: ~ N(0,σ 2 ) where σ 2 is

### Normal distribution

normally distributedGaussian distributionnormal
The error terms are generally assumed to be independent identically distributed random variables (i.i.d.) sampled from a normal distribution with zero mean: ~ N(0,σ 2 ) where σ 2 is

### George E. P. Box

George BoxGeorge Edward Pelham BoxBox, George E. P.
The general ARMA model was described in the 1951 thesis of Peter Whittle, Hypothesis testing in time series analysis, and it was popularized in the 1970 book by George E. P. Box and Gwilym Jenkins.

### Partial autocorrelation function

Partial autocorrelationPACF
Finding appropriate values of p and q in the ARMA(p,q) model can be facilitated by plotting the partial autocorrelation functions for an estimate of p, and likewise using the autocorrelation functions for an estimate of q.

### Autocorrelation

autocorrelation functionserial correlationautocorrelated
Finding appropriate values of p and q in the ARMA(p,q) model can be facilitated by plotting the partial autocorrelation functions for an estimate of p, and likewise using the autocorrelation functions for an estimate of q.

### Akaike information criterion

AICAIC-basedAICc
Brockwell & Davis recommend using Akaike information criterion (AIC) for finding p and q.