2008 Edition

# ARMA Models

Reference work entry
DOI: https://doi.org/10.1007/978-0-387-32833-1_14

ARMA models (sometimes called Box-Jenkins models) are autoregressive moving average models used in time series analysis. The autoregressive part, denoted AR, consists of a finite linear combination of previous observations. The moving average part, MA, consists of a finite linear combination in t of the previous values for a white noise (a sequence of mutually independent and identically distributed random variables).

## MATHEMATICAL ASPECTS

1. 1.

AR model (autoregressive)

In an autoregressive process of order p, the present observation yt is generated by a weighted mean of the past observations up to the pth period. This takes the following form:
\begin{aligned} AR (1) \colon y_t &= \theta_1 y_{t-1} + \varepsilon_t\:,\\ AR (2) \colon y_t &= \theta_1 y_{t-1} + \theta_2 y_{t-2} +\varepsilon_t\:,\\ \vdots\\ AR (p) \colon y_t &= \theta_1 y_{t-1} + \theta_2 y_{t-2} + \ldots\\ &\quad + \theta_p y_{t-p} + \varepsilon_t\:, \end{aligned}
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## REFERENCES

1. 1.
Box, G.E.P., Jenkins, G.M.: Time Series Analysis: Forecasting and Control (Series in Time Series Analysis). Holden Day, San Francisco (1970)