Autoregressive–moving-average model

The notation AR(p) refers to the autoregressive model of order p. The AR(p) model is written as

${\displaystyle X_{t}=\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\varepsilon _{t}}$

where ${\displaystyle \varphi _{1},\ldots ,\varphi _{p}}$ are parameters and the random variable ${\displaystyle \varepsilon _{t}}$ is white noise, usually independent and identically distributed (i.i.d.) normal random variables.[1][2]

In order for the model to remain stationary, the roots of its characteristic polynomial must lie outside of the unit circle. For example, processes in the AR(1) model with ${\displaystyle |\varphi _{1}|\geq 1}$ are not stationary because the root of ${\displaystyle 1-\varphi _{1}B=0}$ lies within the unit circle.[3]

The notation MA(q) refers to the moving average model of order q:

${\displaystyle X_{t}=\mu +\varepsilon _{t}+\sum _{i=1}^{q}\theta _{i}\varepsilon _{t-i}\,}$

where the ${\displaystyle \theta _{1},...,\theta _{q}}$ are the parameters of the model, ${\displaystyle \mu }$ is the expectation of ${\displaystyle X_{t}}$ (often assumed to equal 0), and the ${\displaystyle \varepsilon _{t}}$, ${\displaystyle \varepsilon _{t-1}}$,... are again i.i.d. white noise error terms that are commonly normal random variables.[4]

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,[5]

${\displaystyle X_{t}=\varepsilon _{t}+\sum _{i=1}^{p}\varphi _{i}X_{t-i}+\sum _{i=1}^{q}\theta _{i}\varepsilon _{t-i}.\,}$

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.[6][7] 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).[8]

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

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

${\displaystyle \varepsilon _{t}=\left(1-\sum _{i=1}^{p}\varphi _{i}L^{i}\right)X_{t}=\varphi (L)X_{t}\,}$

where ${\displaystyle \varphi }$ represents the polynomial

${\displaystyle \varphi (L)=1-\sum _{i=1}^{p}\varphi _{i}L^{i}.\,}$

The MA(q) model is given by

${\displaystyle X_{t}=\left(1+\sum _{i=1}^{q}\theta _{i}L^{i}\right)\varepsilon _{t}=\theta (L)\varepsilon _{t},\,}$

where ${\displaystyle \theta }$ represents the polynomial

${\displaystyle \theta (L)=1+\sum _{i=1}^{q}\theta _{i}L^{i}.\,}$

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

${\displaystyle \left(1-\sum _{i=1}^{p}\varphi _{i}L^{i}\right)X_{t}=\left(1+\sum _{i=1}^{q}\theta _{i}L^{i}\right)\varepsilon _{t}\,,}$

or more concisely,

${\displaystyle \varphi (L)X_{t}=\theta (L)\varepsilon _{t}\,}$

or

${\displaystyle {\frac {\varphi (L)}{\theta (L)}}X_{t}=\varepsilon _{t}\,.}$

The spectral density of an ARMA process is

$${\displaystyle S(f)={\frac {\sigma ^{2}}{2\pi }}\left\vert {\frac {\theta (e^{-if})}{\phi (e^{-if})}}\right\vert ^{2}}$$

where ${\displaystyle \sigma ^{2}}$ is the variance of the white noise, ${\displaystyle \theta }$ is the characteristic polynomial of the moving average part of the ARMA model, and ${\displaystyle \phi }$ is the characteristic polynomial of the autoregressive part of the ARMA model.[13][14]

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 effects due to market participants.

The dependence of ${\displaystyle X_{t}}$ 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.

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.

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).

• Autoregressive integrated moving average (ARIMA)
• Exponential smoothing
• Linear predictive coding
• Predictive analytics
• Infinite impulse response
• Finite impulse response

• Mills, Terence C. (1990). Time Series Techniques for Economists. Cambridge University Press. ISBN 0521343399.
• Percival, Donald B.; Walden, Andrew T. (1993). Spectral Analysis for Physical Applications. Cambridge University Press. ISBN 052135532X.
• Francq, C.; Zakoïan, J.-M. (2005), "Recent results for linear time series models with non independent innovations", in Duchesne, P.; Remillard, B. (eds.), Statistical Modeling and Analysis for Complex Data Problems, Springer, pp. 241–265, CiteSeerX 10.1.1.721.1754.
• 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