PROC VARMAX: ML estimation, exact log likelihood, and State Space representation for VAR(4)

[tugasakhir]

I am estimating a VAR(4) model. Initially, I used the CML method, but currently, I am trying to learn and explore another estimation method, which is the ML method. This led me to a few questions that I hope someone here can help clarify:

1. Even though I am estimating a pure VAR(4) model (with no MA terms), does PROC VARMAX still compute the exact log likelihood by casting the model into a State Space representation and utilizing the Kalman Filter under the hood when method=ml is specified?
2. If computing the exact log likelihood indeed utilizes a state space formulation, how exactly is the state vector (z_t) constructed for a pure VAR(4) in SAS? I saw in a reference that z_t is defined as z_t = (y't, y'{t-1}, ..., y'{t-(v-1)}, ε't, ε{t-1}, ..., ε'{t-(q-1)})'. Because of this, I am wondering: does the state vector only contain the variables and their lags (y_t, y_t-1, y_t-2, y_t-3), or does it also explicitly incorporate the error/innovation components (e_t) despite the absence of any Moving Average (MA) terms?

Any theoretical insights or references to the SAS documentation would be greatly appreciated

[SASCom1]

Yes, for VAR(p) models, in fact for all VAR, VARX, VARMA, VARMAX models, when method = ML is used, the model is transformed into the state space form and Kalman filtering method is applied.

Yes, for VAR(p) model, the state vector z_t includes only {y_t, y_t-1,.., y_t-(v-1)}, there are no ε_t, ε_t-1, ..., ε_t-(q-1) terms in the state vector.  In the state space form discussed in the documentation in the VARMA and VARMAX modeling framework:

SAS Help Center: VARMA and VARMAX Modeling

when there are no moving average terms, e.g., VAR(p) model, its state space form is a special case of the state space form presented in the VARMA case in the documentation, there are no columns involving moving average parameters in the transition matrix F, and no ε_t, ε_t-1, ..., ε_t-(q-1) terms in the state vector.