I have a time series and want to use AIC / BIC to decide which of the following model is most appropriate:
- A) AR(1), no constant with Gaussian innovation term
- B) AR(2), no constant with Gaussian innovation term
- C) AR(1), no constant with Student t innovation term
- D) AR(2), no constant with Student t innovation term
What is the correct number of parameters to use in AIC / BIC for the models above?
I found in the Matlab doc the following explanation for an ARMA(p,q) model with Gaussian distribution: "Calculate the BIC for each fitted model. The number of parameters in a model is p + q + 1 (for the AR and MA coefficients, and constant term)."
What I do not understand is why there is no parameter added for the variance of the Gaussian distribution which is also estimated. In particular if the innovation term is Student-t distributed, I assume that the additional 'degree of freedom' parameter of the student t-distribution needs to be considered in AIC / BIC?
I intuitively would have chosen a number of parameters of 2 for A, 3 for B, 3 for C and 4 for D, but it may also be 1 for A, 2 for B, 2 for C and 3 for D if the variance is not counted as parameter (as in the Matlab example).