# Coupled Multivariate Regression and AR(1) Process for the Covariates

I am wondering how to fit the following model:

I have a standard multivariate regression, and a set of covariates X. I want to couple my regression equation with an AR(1) process for the dynamics of the covariates, that is, rather than estimating y given the covariates, I want to estimate first the covariate from the AR(1) dynamics, and then given the estimated covariate estimate y via multivariate regression. This is basically a flavor of a state space model, but there are no latent variables, we have two time series, Y for the dependent variables, X for the covariates, with the extra layer that the covariates follow some dynamics.

How can I fit such a model?

The model looks like this:

• Hi: I think it would be helpful if you wrote the model so it's easier to see what's happening. Commented Aug 1 at 16:56
• I added the model equations. Commented Aug 1 at 17:11
• Hi ksheen: I hate to keep throwing questions back at you but do you happen to mean that $\epsilon_t$ is AR(1) ( obviously, rather than $X_t$ ) because, in that case, you are correct that you wouldn't need a state space representation. But, if you really do mean that $X_t$ is AR(1), then I need to think about that case. I think it needs a state space representation but I want to see if I'm missing something. Commented Aug 2 at 3:38
• ksheen: I've been looking at the equations you provided and I don't see how that is not the standard KF ? For simplicity's sake, let's assume that everything is a scalar. Then, the state is $X_t$ which happens to have AR(1) dynamics. The AR(1) dynamics are represented by the state equation for $X_t$. Then, for the observation equation, $Y_t$ is a linear function of the state plus an error term. So, I could be not understanding something but I think this is standard KF ? ( harvey, harrison and west etc ) Commented Aug 2 at 3:46
• That is exactly my question, would a Kalman filter apply in this case? my understanding is that KF is for when you have unobserved states, but here we have the observed dataset X Commented Aug 2 at 4:02