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.
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
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.