How to model an AR process with exogenous shocks?

The model is roughly as follows:

δit=ϕ⋅δi,t−1+β⋅Xit+wit,wit∼N(0,σ2)

My idea is to modify the existing AR model, for example by replacing the innovation term

ϵt∼N(0,σ2) with ϵt∼N(X⋅β,σ2)

But I’m not sure how to adjust the AR model to implement this functionality.

Alternatively, is there a simpler or more elegant way to achieve this?

You can do this with the pymc_extras.statespace, using a combination of AR and Exogenous components. See an example here, though it’s a bit out of date and might need some updating.

I also want to add support for exogenous regressors directly in the BayesianSARMIA model, but haven’t gotten around to it yet. Help welcome :slight_smile:

I’m currently working on state space models as part of a potential GSoC project, so this topic is quite relevant.

While my main focus would be on the Dynamic Factor Models (DFM), I might be able to look into AR processes with exogenous shocks as well, time permitting. If I have some bandwidth later in the summer, I’d be happy to explore this further or collaborate on it.