I am interested in estimating regime-switching VAR models to a regime setup I don't know the name of. I am hoping that someone can help me out with some references, or if there exists a name for it then tell me that so I know what to search for.
Suppose I have a vector of endogenous variables $\mathbf{y}_t$ and a vector of exogenous variables $\mathbf{x}_t$. Both vectors are observed. Including regime switching in the "usual" setting means that the conditional density of the endogenous vector is given by $$ f(\mathbf{y_t}|s_t=j, \mathbf{x}_t, \mathscr{Y}_{t-1}; \boldsymbol{\alpha}) $$ where $\mathscr{Y}_t=(\mathbf{y}_t, \dots, \mathbf{y}_1, \mathbf{x}_t, \dots, \mathbf{x}_1)$, $s_t$ is the state process such that it is currently in state $j$ and $\boldsymbol{\alpha}$ is a set of parameters for the conditional density.
What I've managed to find is a regime setup such that $s_t$ evolves according to a Markov chain satisfying $$ P\{s_t=j|s_{t-1}=i, s_{t-2}=k, \dots, \mathbf{x}_t, \mathscr{Y}_t\}=P\{s_t=j|s_{t-1}=i\}. $$
However, I want to extend this --- suppose that there is another state process, $s_t^*$, which is the state process governing the exogenous vector $\mathbf{x}_t$. I want the $s_t$ process to evolve according to $$ P\{s_t=j|s_{t-1}=i, s^*_{t-1}=k\}. $$
- Does this have a name? I've found some things that sound interesting (multivariate, composite, double, etc), but not sure they're what I'm after.
- Does anyone have any references to books and/or papers where such a setup is used?
I would really appreciate some help!
By what I understand, you can model your problem with a single larger Markov chain. Say the states of the first markov chain are $R=\{1,\dotsc,m\}$ and of the second are $S=\{1,\dotsc,n\}$. Then you can make a Markov chain taking values over the Cartesian product $X=R\times S$ whose state $x_t=(s_t, s_t^*)$ has the following transition densities:
$$ P(s_t=i,s_k^*=k|s_{t-1}=j, s_{t-1}^*=\ell) = P(s_t=i | s_{t-1}=j, s_{t-1}^*=\ell)P(s_t^*=k | s_{t-1}^*=\ell). $$
Note that the second Markov chain, with state $s^*$, is independent of the first (which is not a true Markov chain as it has this weird exogenous input). This final chain is a regular Markov chain, though, and has no exogenous inputs.