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

b. The AR(1) process

The AR(1) process is defined as

(V.I.1-83)

where W_t is a stationary time series, e_t is a white noise error term, and F_t 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

(V.I.1-84)

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

(V.I.1-85)

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

(V.I.1-86)

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

(V.I.1-87)

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

(V.I.1-88)

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

(V.I.1-89)

and

(V.I.1-90)

(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

(V.I.1-91)

For a general AR(p) model the solutions of

(V.I.1-92)

for which

(V.I.1-93)

must be satisfied in order to obtain stationarity.