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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

(V.I.1-139)

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

(V.I.1-140)

Therefore the pattern of the theoretical ACF is

(V.I.1-141)

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

(V.I.1-142)

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.

(V.I.1-143)

converges.

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

(V.I.1-144)

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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MA(1) process
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