yule walker | Data Stories

Yule Walker Equation in Matrix Form

If we write and above equation for k=1, 2, . . ., n and use the fact that ρ(k) = ρ(-k), we can write it in a matrix form.

Using the data we have we can estimate values of ρ (auto correlation coefficients)

acf() routine in R gives us that

Using values of ρ we can then estimate values of Φ (parameters of AR process)

Above is an example for AR process

We can solve these equation for values of Φ1, Φ2 and Φ3

Reference:

maq

Below are the plots for AR(2) process
Depending upon the value of phi1 and phi2 ACF has alternative positive and negative values

Writing AR(p) process as MA process by substituting values of X(t-1). And yes phi is constant, we don’t need phi1, phi2 anymore.

Mean, variance and auto-correlation of AR(p) process, we have assumed Z = Norm(0, sigma2)

It is a method of solving difference equation in recursive relation
We first obtained auxiliary equation (also known as characteristic equation) which is polynomial and find root of that
Using these root we get weighted geometric series and find weights using some initial condition
We had learned in mathematics that this way of solving difference equation also related to solving differential equations
In the course they had solved it for Fibonacci series and root had come out to be golden ratio
For AR(p) ACF comes out to be difference equation, solving which can give us ACF for different values of lag

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