I have been experimenting with using an augmented state space in which I store / memorise previous states as new variables at the bottom of the state vector when performing discrete Kalman filtering.
I can then approximate a high-order discrete model by passing the lagged terms down through the state vector after each time step.
My initial results seem to work well, so I was wondering why the Markov assumption is so firmly stated in the filtering literature. Besides the massive increase in dimensionality of the state, are there any possible side effects I should look out for, or reasons why this approach is not generalisable?