Autoregressive–moving-average model

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 , 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. Autoregressive model The notation AR(p) refers to the autoregressive model of order p. The AR(p) model is written as where are parameters and the random variable is white noise, usually independent and identically distributed (i.i.d.) normal random variables. 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 are not stationary because the root of lies within the unit circle. Moving-average model The notation MA(q) refers to the moving average model of order q: where the are the parameters of the model, is the expectation of (often assumed to equal 0), and the , ,... are again i.i.d. white noise error terms that are commonly normal random variables. 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.