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An **A**uto**r**egressive **M**oving **A**verage Process of order (p,q) is a stochastic process defined by the following equation:

X_{t} = µ + φ_{1} * X_{t-1} + ... + φ_{p} * X_{t-p} + ε_{t} + ψ_{1} * ε_{t-1} + ... + ψ_{q} * ε_{t-q}, **(1)**

where ε

X’_{t} = X_{t} – µ / (1 – φ_{1} - ... - φ_{p}).

The standard notation for X

1 - φ_{1} * λ - ... - φ_{p} * λ^{p} = 0

must lie outside the unit circle on the complex plane.

A pure autoregressive process of order p is ARMA(p,0). It is denoted with AR(p) oftentimes. A pure moving average process of order q is ARMA(0,q). It is denoted with MA(q). Both autoregressive and moving average terms in equation (1) ensure correlation of values of X

Assume we are looking at a stochastic process that can be captured well within the ARMA framework. Then the optimal order (p,q) is informally researched using the observed patters in the autocorrelation and partial autocorrelation functions. The impulse response function can be of great help as well, especially if the structure of the optimal ARMA process is complex: large values of p and q, negative coefficients, etc. After several candidate models have been identified, they are estimated. The best model is then identified using 1-2 model selection criteria. Examples of model selection criteria are AIC, BIC and cross-validation.

Typically, ARMA processes are estimated using one of the variations of the maximum likelihood method. Another estimation alternative is the generalized method of moments. In the case of AR processes the method of moments degenerates into simple Yule-Walker equations. The Yule-Walker equations are linear in the model coefficients.

ARMA processes have wide-spread use in economics, finance and engineering.

Greene, W. H. (2011). Econometric Analysis (7th ed). Upper Saddle River, NJ: Prentice Hall.

Hamilton, J. D. (1994). Time Series Analysis. Princeton University Press.

Brockwell, P. J., & Davis, R. A. (1991). Time Series: Theory and Methods (2nd ed). New York: Springer.

Wei, W. W. S. (1990). Time Series Analysis: Univariate and Multivariate Methods. Redwood City, CA: Addison Wesley.

Tsay, R. S. (2005). Analysis of Financial Time Series. New Jersey: Wiley-Interscience.

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