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Time Series Analysis - ARIMA models - ARMA(1,1) process

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

h. The ARMA(1,1) process

On combining an AR(1) and a MA(1) process one obtains an ARMA(1,1) model which is defined as

Time Series Analysis - ARIMA models - ARMA(1,1) process


where Wt is a stationary time series, et is a white noise error component, and Ft is the forecasting function.

Note that the model of (V.I.1-154) may alternatively be written as


such that


in y-weight notation.

The y-weights can be related to the ARMA parameters on using


such that the following is obtained


Also the p-weights can be related to the ARMA parameters on using


such that the following is obtained


From (V.I.1-158) and (V.I.1-160) it can be clearly seen that an ARMA(1,1) is in fact a parsimonious description of either an AR or a MA process with an infinite amount of weights. This does not imply that all higher order AR(p) or MA(q) processes may be written as an ARMA(1,1). Though, in practice an ARMA process (c.q. a mixed model) is, quite frequently, capable of capturing higher order pure-AR p-weights or pure-MA y-weights.

On writing the ARMA(1,1) process as


(which is a difference equation) we may multiply by Wt-k and take expectations. This gives


In case k > 1 the RHS of (V.I.1-162) is zero thus


If k = 0 or if k = 1 then


Hence we obtain


The theoretical ACF is therefore


The theoretical ACF and PACF patterns for the ARMA(1,1) are illustrated in figures (V.I.1-7), (V.I.1-8), and (V.I.1-9).

(figure V.I.1-7)

(figure V.I.1-8)

(figure V.I.1-9)

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AR(1) process
AR(2) process
AR(p) process
MA(1) process
MA(2) process
MA(q) process
ARMA(p,q) process
Wold's decomp.
Non stationarity
Inverse Autocorr.
Unit Root Tests
ARMA(1,1) process
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