lets say I have a ARMA(2,2) model, which require lag terms of t-1 and t-2, what will my fitted value be for first (where lag terms of both t-1 and t-2 is missing) and second (where lag terms of t-2 is missing) time position?
Observed that SAS is using the below formula (please correct me if I am wrong):
First fitted value : y1 = constant Second fitted value : y2 =constant + AR(1)y1 + MA(1)e1 Third fitted value : y3 = constant + AR(1)y1 + AR(2)y2 + MA(1)e1 + MA(2)e2 (which is the full formula) where AR(x) and MA(x) is the coefficient for AR and MA terms.
Observed that ARMA model from R is using different method (which I could not decipher yet). But I could not find any proper documentation to support on how should the first few fitted value be computed.
Could anyone advise if there is any research/proofs/guidance/publication available to show the appropriate method to compute the first few fitted value of an ARMA model?
I am still new to this space and this is my first time posting question. Many apologies if there is any lacking in information, and please feel free to post your queries and I will update further. Thank you!
R uses MLE estimation by default, whereas SAS uses conditional lease squares. You can change this in PROC ARIMA with the method=ml option in the estimate statement. Your two models should produce identical estimates. It's generally recommended to always use method=ml in SAS.
proc arima data=have;
identify var=foo;
estimate p=2 q=2 method=ml;
forecast lead=2;
run;