Hi, When doing the Autocoviariance calculations (i.e cov(Xt,Xt-1)). How do we know when to expand the time series out on both sides of the comma. For example I notice that sometimes if we have a random TS -> Xt = 2Xt-1 + 2Xt-2 + et, in the solutions we might have the first step for calculating the autocovariance function as: cov(2Xt-1 + 2Xt-2 + et, Xt-1). However other times, it might be written out as expanded terms such as: cov(2Xt-1 + 2Xt-2 + et, 2Xt-2 + 2Xt-3 + et). PS. I made up this TS so not sure if it works here. I thought we would only expand out the terms when working with a MA(q) TS. The thing that confused me was the solutions for CT6 Sept 2017 Q10 part iii. The ACF here has expanded out terms and i don't believe this is an MA model.
Hi I have a simple rule which I teach for this: For MA models, replace both bits of the cov function with the defining equation For AR models, replace only the first bit and leave the second as Xt For ARMA models, replace only the first bit and leave the second as Xt Also, make sure you go back in time with the second bit (ie the Xt) otherwise you will tie yourself up in knots. CT6 Sep 17 Q10(iii) is doing something slightly different because we are working with delta_Xt (ie Xt - Xt-1), so the full formula for the series is never put into the formula, we are instead expressing the gamma function of delta_Xt in terms of the gamma function of Yt. Hope this helps, Dave
