Hi all, I'm not sure how the theoretical mean and variance was obtained for the MA and AR processes? is a formula used in this case and where can this be found in the notes? Thanks in advance!
Hello In Section 2.3 of Chapter 14, the asymptotic results are given for the estimators of autocorrelations of an MA(q) process for lags larger than q. Here we have an MA(1) process and so this result holds for the estimator of rho_2 as 2 > q = 1. For the AR(1) model, we have to use the fact that the estimator is consistent meaning that the asymptotic mean is given by the true value (ie we just need to work out the lag 2 autocorrelation). Hope this helps! Andy
Thanks Andy! I understand the variance portion now. However, not too sure about the theoretical mean for both the AR and MA process. Where can I find the formula for the theoretical mean for these processes in the course notes? Thanks!
Great! For the MA process we can again use the asymptotic result, which states the asymptotic mean is 0. For the AR process we use the fact that the estimator is consistent, meaning the asymptotic mean is given by the true value, here \( \rho_2 \). We can work this out in R using the ARMAacf function. From a paper A perspective, calculation of \( \gamma_k \) and \( \rho_k \) for an AR(1) process are covered in Section 3 of Chapter 13. Hope this helps Andy
Hi Andy, Could you please help me understand the explanation for mean and variance of AR process, I am not able to understand why have we considered "asymptotic mean is given by the true value" and variance of AR process is not given in examiners report. Thanks in advance!
Hi Priyanka Rho^~(2) is a consistent estimator, which means that the asymptotic mean of the estimator is the true value, rho(2). There is no general result presented in the Core Reading for the asymptotic distribution of rho^~(k) for an AR process, hence why the variance is not mentioned. Hope this helps! Andy
Hi, thank you so much! could you please point the section in core reading which states this about AR process?
Hi Priyanka Are you referring to the result about the estimators being consistent? I don't believe that is explicitly mentioned in the Core Reading, at least I can't find it. It is something we mention in our notes at the bottom of page 20 in Chapter 14 (2022 version). Andy
