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I am trying to fit a model where there is a sequence of exogenous "shocks", $X_1, X_2, ..., X_T$, and a AR(1) of these shocks explain $Y_1, Y_2, ..., Y_T$. Specifically,

Data (known):

$X_1, X_2, ..., X_T$

$Y_1, Y_2, ..., Y_T$

Priors (known):

$\phi = 2 \phi^* - 1$, where $\phi^* \sim Beta(\alpha, \beta)$, "persistence" parameter, between $(-1, 1)$

$\beta \sim N(\mu, v)$, "shock strength" parameter

Model:

$E_t = \phi E_{t-1} + \beta X_t$

$Y_t = E_t + \epsilon_t$, $\epsilon_t \sim N(0, 1)$

Unknown parameters:

$\phi$, $\beta$, $E_0$

My question is, what is the easiest way to sample $\beta$ and $\phi$?

It seems that if I am allowed to use $E_{t-1}$ when sampling from $\phi$, it becomes a bit easier, as:

$Y_t = \phi E_{t-1} + \beta X_t + \epsilon$

$\implies \phi E_{t-1} = Y_t - \beta X_t - \epsilon$

$\implies \phi \sim N(\frac{Y_t - \beta X_t}{E_{t-1}}, \frac{1}{E_{t-1}^2})$

However, since $E_t$ is known deterministically from $\phi$, it seems that if I am conditioning on it I must choose a deterministic value for $\phi$.

So I have two questions:

How do I best fit this model using MCMC?

How do I condition on a deterministic auxiliary variable within a Gibbs sampling step?

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