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Time Series Analysis - ARIMA models - AR(1) process

[Home] [Up] [Basics] [AR(2) process] [AR(p) process] [MA(1) 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] [AR(1) process]

b. The AR(1) process

The AR(1) process is defined as

Time Series Analysis - ARIMA models - AR(1) process


where Wt is a stationary time series, et is a white noise error term, and Ft is called the forecasting function. Now we derive the theoretical pattern of the ACF of an AR(1) process for identification purposes.

First, we note that (V.I.1-83) may be alternatively written in the form


Second, we multiply the AR(1) process in (V.I.1-83) by Wt-k in expectations form


Since we know that for k = 0 the RHS of eq. (V.I.1-85) may be rewritten as


and that for k > 0 the RHS of eq. (V.I.1-85) is


we may write the LHS of (V.I.1-85) as


From (V.I.1-88) we deduce




(figure V.I.1-1)

We can now easily observe how the theoretical ACF of an AR(1) process should look like. Note that we have already added the theoretical PACF of the AR(1) process since the first partial autocorrelation coefficient is exactly equivalent to the first autocorrelation coefficient.

In general, a linear filter process is stationary if the y(B) polynomial converges.

Remark that the AR(1) process is stationary if the solution for (1 - fB) = 0 is larger in absolute value than 1 (c.q. the roots of y(B) are, in absolute value, less than 1).

This solution is f-1. Hence, if the absolute value of the AR(1) parameter is less than 1, then model is stationary which can be illustrated by the fact that


For a general AR(p) model the solutions of


for which


must be satisfied in order to obtain stationarity.

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AR(2) process
AR(p) process
MA(1) 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
AR(1) process
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