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Time Series Analysis - ARIMA Models - MA(1) process

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

e. The MA(1) process

The definition of the MA(1) process is given by

Time Series Analysis - ARIMA Models - MA(1) process


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

eq. (V.I.1-46) and (V.I.1-45) we obtain


Therefore the pattern of the theoretical ACF is


Note that from eq. (5.I.1-141) it follows that


This implies that there exist at least two MA(1) processes which generate the same theoretical ACF.

Since an MA process consists of a finite number of y weights it follows that the process is always stationary. However, it is necessary to impose the so-called invertibility restrictions such that the MA(q) process can be rewritten into a AR() model.



On using the Yule-Walker equations and eq. (V.I.1-141) it can be shown that the theoretical PACF is


Hence the theoretical PACF is dominated by an exponential function which decreases.

The theoretical ACF and PACF for the MA(1) are illustrated in figure (V.I.1-4).

(figure V.I.1-4)

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