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I would like to know if models like the one below exist/make sense and, if so, how they are called.

\begin{equation} X_t \sim \mathcal{Pois} ( \lambda_t ) \\ \lambda_t = \mu_1 + \alpha_1 X_{t-1} + \beta_1 \lambda_{t-1} + \gamma_1 Y_{t} + \delta_1\ Y_{t-1}\\ Y_t = \mu_2 + \alpha_2 Y_{t-1} + \beta_2 \epsilon_{t-1} + \gamma_2 X_t + \delta_2 X_{t-1} + \epsilon_{t} \\ \epsilon_t \sim \mathcal{N}(0,1) \end{equation}

The one above is similar to a dynamic Poisson model and the one below is similar to an ARMA(1,1).
I think the structure resembles a VAR but I would like to know if it is feasible/acceptable having two different distribution (Poisson and Normal).

A possibly very naive usage would be for example the modelling of volatility ($Y$) and the number of transactions ($X$) together (to check if a high number of transactions helps in forecasting volatilities and vice versa, just to give an idea).

Richard Hardy
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pietrosan
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  • There are no time subscripts on the left hand side. Is that intentional? The equation for $\lambda$ does not seem to have an error term, so in that sense it does not look like an equation of a VAR model. – Richard Hardy Jun 26 '21 at 16:33
  • corrected, thanks. I have not put an error term in the first equation since usually this model (autocorrelated poisson or INGARCH) have no error term (if you don't consider the last 2 terms) – pietrosan Jun 26 '21 at 16:45
  • Do you want to keep the model in the structural form, i.e. with contemporaneous dependencies? Also, I suppose $\epsilon_t$s are uncorrelated or even i.i.d. – Richard Hardy Jun 26 '21 at 17:11
  • Yes, also beacuse I think that excluding the terms with $\gamma$ there is no more need of a multivariate model. The $\epsilon_t$ that I have in mind are uncorrelated but in the transactions-volatility example they can possibly follow a GARCH (but it is not crucial ) – pietrosan Jun 26 '21 at 19:04
  • A structural VAR always has a reduced form. You do not change the model by rewriting it in another representation (structural --> reduced), but you could make the interpretation easier to follow. In other words, if there is no need for a multivariate model of the reduced form, then there hardly can be a need for the model in its structural form. – Richard Hardy Jun 26 '21 at 20:03
  • Not sure to have completely understood your point (structural --> reduced is ok). I think that my idea to call it kind of VAR model it's been misleading. What i was trying to say is that the main problem here is given by $\gamma_1 Y_t$ and $\gamma_2 X_t$ because if one consider only lagged values as in a normal VAR one could simply estimate $Y_{t-1}$ and then use it as a covariate to estimate $\lambda_t$ and so on treating them as two univariate models. – pietrosan Jun 26 '21 at 20:42

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