the yule walker equation

[Adriano]

Hello, in the above passage can somebody explain for me how you get the final equation by taking the expectations of both sides?

 

[Lu Shu Kai FRM]

Hi @Adriano ,

You would probably have to provide more context for this particular equation... always provide more details so it is easier for us to answer your question. Just looking at the pattern of the math:

y_t * y_(t - tau) = phi * y_(t - 1) * y_(t - tau) + epsilon_t * y_(t - tau)

→ y_tau = phi * y_(tau - 1)

We can see that y_t * y_(t - tau) reverses the sign of tau and eliminates the t such that y_t * y_(t - tau) gives us y_tau. We can observe similarly for phi * y_(t - 1) * y_(t - tau) reverses the sign of tau and eliminates the t while keeping the -1, giving us y_(tau - 1). We also ignore the epsilon_t * y_(t - tau) term as there is no other y term multiplied to it.

I can't really say for sure what is going on, I am just looking at the pattern and what is going on. That is why the context to math is important.

 

[Adriano]

Hi @lushukai
This is taken from the Bionic Turtles notes here: T2-QA-10-Stationary-TimeSeries p.12
It is still not really clear to me so any further help would be greatly appreciated. Also i believe that in the second equation we dont have "y" anymore but we have gamma

 

[Lu Shu Kai FRM]

Hi @Adriano ,

Thanks, I took a closer look at the notes and realized I said something wrongly above. If you look at the top and bottom equation from your screenshot, y actually changes into gamma (the Greek symbol that looks like y but is not y).

The derivation for what is going on above is quite mathematically intensive and I wouldn't go too much into it (which is probably also the reason why David didn't go too much into it as well - it is not super important in the FRM). I think you just need to know that the Yule-Walker equation makes use of the autoregressive model to build a function for the autocovariance series.

Hope this is helpful!