Autoregressive moving average

In the statistical analysis of time series, autoregressive–moving-average (ARMA) models provide a parsimonious description of a in terms of two polynomials, one for the autoregression and the second for the moving average. 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 1971 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 model consists of two parts, an autoregressive (AR) part and a moving average (MA) part. 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 autoregressive part and q is the order of the moving average part (as defined below).

ARMA models can be estimated following the Box–Jenkins approach.

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

where ${\displaystyle \varphi _{1},\ldots ,\varphi _{p}}$ are parameters, ${\displaystyle c}$ is a constant, and the random variable ${\displaystyle \varepsilon _{t}}$ is white noise.

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