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Stochastic process X_{t} is an **A**uto**r**egressive **I**ntegrated **M**oving **A**verage Process of order (p,d,q) if process Y_{t} = Δ^{d} X_{t} is an autoregressive moving average process of order (p,q). Y_{t} must follow the equation:

Y_{t} = µ + φ_{1} Y_{t-1} + ... + φ_{p} Y_{t-p} + ε_{t} + ψ_{1} ε_{t-1} + ... + ψ_{q} ε_{t-q},

where ε

The standard notation for an autoregressive integrated moving average process of order (p,d,q) is ARIMA(p,d,q), while the standard notation for an autoregressive moving average process of order (p,q) is ARMA(p,q). ARMA processes are stationary for many choices of parameters. Therefore, ARIMA processes are used to model processes in real life which are non-stationary but have stationary growth (d=1) or the speed of growth (d=2), or the speed of the speed of growth (d=3), etc.

Modeling is typically done in two stages. First, process X

In theory, several values of d can be tried provided they ensure stationarity of the differenced series. The optimal value can then be identified using the model selection criteria. In practice, in most cases the smallest d ensuring stationarity is used.

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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