Applying outlier adjustment using student's t distribution in a state-space model I'm exploring performing outlier adjustment in a state-space model by using student's $t$ distribution. The gist of the problem is formulated as follows:
$$ \begin{align*}
y_t^* &= u_t + o_t - o_{t-1}, \\
u_t &= \rho_1 u_{t-1} + \rho_2 u_{t-2} + \eta_{t}, \\
o_t &\sim t_\nu (\sigma_o).
\end{align*}
$$
where $y_t^*$ is a time series of observables, modeled as the sum of $u_t$ and the one-period change in $o_t$, where $o_t$ is the additive outlier component. We let $u_t$ follow the $AR(2)$ process, and $o_t$ follows i.i.d. $t$ distribution.
After consulting some literature, I decided to model the $t$-distributed variables using scale mixtures; i.e., the measurement equation above is equivalent to
$$ \begin{align*} 
y_t^* & = u_t + \sqrt{\varphi_t} z_t - \sqrt{\varphi_{t-1}} z_{t-1},\qquad z\sim N(0, \sigma_o^2) \text{ and } \varphi\sim IG.
\end{align*} $$
I then cast the above system in the state-space format, with the following measurement equation:
$$ y_t^* = \begin{pmatrix} \sqrt{\varphi_t} & -\sqrt{\varphi_{t-1}} & 1 & 0 \end{pmatrix} \begin{pmatrix} z_t \\ z_{t-1} \\ u_t \\ u_{t-1} \end{pmatrix},$$ and the following transition system
$$ \begin{pmatrix} z_t \\ z_{t-1} \\ u_t \\ u_{t-1} \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & \rho_1 & \rho_2 \\ 0 & 0 & 1 & 0  \end{pmatrix} \begin{pmatrix} z_{t-1} \\ z_{t-2} \\ u_{t-1} \\ u_{t-2} \end{pmatrix} + \begin{pmatrix} \eta_z \\ 0 \\ \eta_u \\ 0 \end{pmatrix}, $$
where $\eta_z \sim N(0, \sigma_o^2)$.
To draw the outlier component and the hyperparameters, I thought the following Gibbs sampling procedure might be appropriate:

*

*Conditional on $\varphi_t^{j-1}$ and $\sigma_o^{j-1}$, use simulation smoother to draw $z_t$ and $u_t$; then it's trivial to compute $o_t^j = \sqrt{\varphi_t} z_t$;

*Given $o_t^j$ and conditional on $\varphi_t^{j-1}$, draw $\sigma_o^j$.

*Conditional on $\sigma_o^j$ and $\nu^{j-1}$, draw $\varphi_t^j$.

*Draw $\nu^j$.

*Increment $j$ by 1.

I’m somewhat new to Gibbs sampling and non-Gaussian state-space models. I'm wondering whether there are obvious theoretical issues with this setup.
 A: I would recommend you read

West, Mike, and Jeff Harrison. Bayesian forecasting and dynamic models. Springer Science & Business Media, 2006


are there obvious theoretical issues with this setup?
Yes, a few.
A state space model is defined by an observation equation and a state equation. Terminology for these two equations varies across Kalman filter, linear dynamical systems, and dynamic linear models literature.
Using your notation, the observation measurement $y_t^*$ is typically a composite of state $u_t$ and measurement error $o_t$:
$$ y_t^* = u_t + o_t ~.$$
$o_t$ and $o_s$ are independent for $s \neq t$. It is not clear what measurement process would require the equation you provided.

model the t-distributed variables using scale mixtures
No.
The $t$-distribution is used to model normally distributed values when you (the modeler) do not know the variance of the values.  The $t$-distribution quickly converges to a normal distribution with increasing number of observations as the modeler "learns" the variance of the measurements.
Inverse-gamma is useful to combine a prior distribution with observed data, forming a more precise posterior distribution.
You certainly do not need both. Modeling unknown measurement noise is explained in West & Harrison.

In your description, variance of observation noise $o_t$ is unknown and variance of evolution noise $\nu_t$ is known. West & Harrison address this scenario in one chapter.
West & Harrison covers AR process modeling; and, shows mixture models for the detection and handling of outliers in state modeling.  Observation noise $o_t$ is modeled to arise from a mixture of "typical" and "outlier" distributions.  For instance, maybe typical measurement error is $o_t \sim N(0,V)$ and outlier measurement error is $o_t \sim N(0,100 V)$.

Good luck.  I think you will make good progress using the book and a simple mixture model.
