Question

Let Xt=0.2Xt−1+Wt​ be an autoregressive process of order 1, where Wt​ is a white noise with mean zero and variance 2.

a) Express it in terms of the Backward Shift operator.
b) Show that the process is stationary.
c) Find the ACF.
d) Obtain the correlogram.

Answer 1

The given autoregressive process is expressed using the Backward Shift operator, proven to be stationary due to the characteristic equation root lying outside the unit circle, and its ACF and correlogram are determined based on the AR(1) process properties.

Step-by-step explanation:

The autoregressive process given is Xt=0.2Xt−1+Wt where Wt is a white noise with mean zero and variance 2.

a. Express it in terms of the Backward Shift operator

The Backward Shift operator B is defined by BXt = Xt-1. The process can be expressed as:
Xt - 0.2BXt = Wt
or
(1 - 0.2B)Xt = Wt.

b. Show that the process is stationary

To show that the process is stationary, the roots of the characteristic equation must lie outside the unit circle. The characteristic equation is 1 - 0.2z = 0, which has a root at z = 5. Since the absolute value of 5 is greater than 1, the process is stationary.

c. Find the ACF

The AutoCorrelation Function (ACF) for an AR(1) process is given by ρ(h) = φ^h where φ is the coefficient of Xt-1. Here ρ(h) = 0.2^h.

d. Obtain the correlogram

The correlogram can be obtained by plotting the ACF ρ(h) against the lag h. The ACF will decay geometrically.