ARMA Models

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

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© Springer-Verlag 2008