# What state-space representation of VARMA is commonly used for fitting

What state-space representation of VARMA is commonly used for fitting?

Is Kalman filter + MLE approach used for fitting VARMA model as a common practice?

Does the choice of which state-space representation to use affect fitting performance?

## BACKGROUND

I'm new to time series modeling.

I'm trying to use a VARMA model to fit some multivariate data. (for now I use artificial data generated using dse)

library(dse)
n = 1000
AR <- array(c(1, .5, .3, 0, .2, .1, 0, .2, .05, 1, .5, .3), c(3,2,2))
MA <- array(c(1, .2, 0, .1, 0, 0, 1, .3), c(2,2,2))
arma <- ARMA(A=AR, B=MA, C=NULL)
result = outputData(simulate(arma, sampleT=n))


I've tried to fit the model by following steps:

1. Convert the model to state-space model using Harvey's approach.
2. Use FKF to kickoff a MLE optim.

The 1st part was pretty natural, I just replace each $\phi$, $\theta$ with their matrix counterparts.

However the 2nd part was tough. FKF's likelihood calculation will 'stall' as soon as Ft become non-positive-semidefinite. (I assume Ft is a covariance matrix)

After that, with doubt in whether my conversion is correct, I tried dse's toSS function, which by it's document:

If the order of the AR polynomial equals or exceeds the MA polynomial (and the input polynomial) then the model is converted by state augmentation.

I have no idea what state augmentation is, and the output matrices looks very different.

I'm seeking for answers and advices, thanks in advance!