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Time Series Analysis - ARIMA models - ARMA(p,q) process

[Home] [Up] [Basics] [AR(1) process] [AR(2) process] [AR(p) process] [MA(1) process] [MA(2) process] [MA(q) process] [ARMA(1,1) process] [Wold's decomp.] [Non stationarity] [Differencing] [Behavior] [Inverse Autocorr.] [Unit Root Tests] [ARMA(p,q) process]


i. The ARMA(p,q) process

The general ARMA(p,q) can be defined by

Time Series Analysis - ARIMA models - ARMA(p,q) process

(V.I.1-167)

or alternatively in MA(¥) notation

(V.I.1-168)

or in AR(¥) notation

(V.I.1-169)

where p(B) = 1/y(B).

The stationarity conditions depend on the AR part: the roots of f(B) = 0 must be larger than 1. The invertibility conditions only depend on the MA part: the roots of q(B) = 0 must also be larger than 1.

The theoretical ACF and PACF patterns are deduced from the so-called difference equations

(V.I.1-170)

and Cramer's rule applied to the Yule-Walker equations.
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AR(1) process
AR(2) process
AR(p) process
MA(1) process
MA(2) process
MA(q) process
ARMA(1,1) process
Wold's decomp.
Non stationarity
Differencing
Behavior
Inverse Autocorr.
Unit Root Tests
ARMA(p,q) process
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