# Model approach for extending Dynamic Linear Models - Nested Regression Relationships

I have three multivariate random variables $X_t$, $Y_t$, and $Z_t$.

I have been very happily modeling the relationship between $Y_t$ and $X_t$ through a dynamic linear model

$$Y_t = X_t\beta_t + v_t$$ $$\beta_t = \beta_{t-1} + w_t$$

Where $v_t$ and $w_t$ are normally distributed error terms. What I would like to do is extend this model in a natural way (e.g., staying within a state-space bayesian framework - if feasible) but include the relationship with $Z_t$ which I know is related through a linear relation to $\beta_t$. That is: $$Z_t = \alpha_t(\beta_tX_t)_t+\epsilon_t$$

My overall goal is to learn both $\beta_t$ and $\alpha_t$.

To summarize I have two time-varying regression relationships with the following causal pathway: $X \rightarrow Y \rightarrow Z$

I would like some advice regarding what type of models I should now be looking at. Is this the type of thing handled by dynamic bayesian networks?

The short answer is that what I am trying to do is inherently non-linear and poorly handled by much of the tools for bayesian state-space models. That said there is some interesting work on multi-regression models but it seems to have been largely abandoned.

This type of model can be handled quite easily (for dimensional problems) using MCMC though.