The Yule-Walker equations for the given AR(2) process are:
These equations relate the autocovariances of the process to its parameters.
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To derive the Yule-Walker equations for the given process defined by
we first identify the coefficients of the AR process, which are and . In the case of an AR(2) process, the Yule-Walker equations relate the autocovariances of the process.
The first Yule-Walker equation is:
Substituting our coefficients gives us:
Rearranging this yields:
The second Yule-Walker equation is:
Substituting our values, we have:
Now, substituting :
To add these fractions, we need a common denominator, which is 63:
In summary, we have:
These are the Yule-Walker equations for the given process.