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). 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.
Given a time series of data Xt , 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 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.
The notation AR(p) refers to the autoregressive model of order p. The AR(p) model is written
Some constraints are necessary on the values of the parameters so that the model remains stationary. For example, processes in the AR(1) model with |?1| >= 1 are not stationary.
The notation MA(q) refers to the moving average model of order q:
where the ?1, ..., ?q are the parameters of the model, ? is the expectation of (often assumed to equal 0), and the , ,... are again, white noise error terms.
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,
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. 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).
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,?2) where ?2 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.
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
where represents the polynomial
The MA(q) model is given by
where ? represents the polynomial
Finally, the combined ARMA(p, q) model is given by
or more concisely,
Some authors, including Box, Jenkins & Reinsel use a different convention for the autoregression coefficients. This allows all the polynomials involving the lag operator to appear in a similar form throughout. Thus the ARMA model would be written as
Moreover, if we set and , then we get an even more elegant formulation:
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. 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.
This section needs expansion. You can help by adding to it. (March 2017)
ARMA models in general can be, after choosing p and q, 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 Yule-Walker equations may be used to provide a fit.
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 Xt on past values and the error terms ?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).
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 . It is given by:
where are the parameters of the exogenous input .
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" or "independent" variables. Care must be taken when interpreting the output of those packages, because the estimated parameters usually (for example, in R and gretl) refer to the regression:
where mt incorporates all exogenous (or independent) variables:
This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. (August 2010) (Learn how and when to remove this template message)