1
$\begingroup$

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?

$\endgroup$
0
$\begingroup$

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. 

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.