Statistics for Environmental Engineers

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The first-order autoregressive process zt= фzt-1+ a, is valid for all values of t. If we replace t with t -1, we get zt-i = фzt-2 + at-1. Substituting this into the original expression gives:


Zt = ф( фZt-2 + a,-1) + at = a, + Ф2,-1 + ф2z,-2 If we repeat this substitution к — 1 times, we get:


z, = a, + fL-1 + фa,-2 + » + ф la,-(k-1) + ф z,-k


Because | ф | < 1, the last term at some point will disappear and zt can be expressed as an infinite series of present and past terms of a decaying white noise series:


z, = a, + ^1 -1 + ф a,-2 + ф a,-3 +


This is an infinite moving average process. This important result allows us to express an infinite moving average process in the parsimonious form of an AR(1) model that has only one parameter.


In a similar manner, the finite moving average process described by an MA(1) model can be written as a infinite autoregressive series:


z, = a, 01Z,-1 — 62z,-2 — 63z,-3—-

Mixed Autoregressive-Moving Average Processes


We saw that the finite first-order autoregressive model could be expressed as an infinite series moving average model, and vice versa. We should express a first-order model with one parameter, either 6 in the moving average form or ф in the autoregressive form, rather than have a large series of parameters. As the model becomes more complicated, the parsimonious parameterization may call for a combination of moving average and autoregressive terms. This is the so-called mixed autoregressive-moving average process. The abbreviated name for this process is ARMA(p,q), where p and q refer to the number of terms in each part of the model:

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