When we want to calculate the autocovariance function at lag 1 (for example), i.e. γ1, are both of these definitions correct? γ1=Cov(X_t,X_t+1) γ1=Cov(X_t,X_t-1) If they are both correct, which one is it recommended to use? Does one get us to the answer more quickly than the other depending on how the Time Series X_t is defined? Thank you
They are both correct and you could use either. However, when finding the autocovariance for AR or ARMA we replace only one of them. It is essential that you replace the highest one (so \(X_{t+1}\) in the first one and \(X_t\) in the second one as otherwise you will have two problems: the covariance of white noise terms with \(X_t\)'s will no longer be zero and the equations you obtain won't be solvable. I prefer the second one as replacing \(X_t\) is easier than replacing \(X_{t+1}\) as nothing needs to change from the defining equation. This is the version used in our solutions (ASET) and most of the examiners solutions.
