$q(\boldsymbol{\beta}_{j}|\boldsymbol{\beta}_{j-1},\mathbf{t}_{j})$ using the conditional expectations

$$\begin{aligned}
\mathbf{m}_{j} & =E_{\eta_{j}}\left[E\left[\boldsymbol{\beta}_{j}|\eta_{j},\boldsymbol{\beta}_{j-1},\mathbf{t}_{1:j-1}\right]|\mathbf{t}_{j}\right],\end{aligned}$$

$$\begin{aligned}
\mathbf{C}_{j} & =E_{\eta_{j}}\left[V\left[\boldsymbol{\beta}_{j}|\eta_{j},\boldsymbol{\beta}_{j-1},\mathbf{t}_{1:j-1}\right]|\mathbf{t}_{j}\right]+\nonumber \\
 & \,\,\,V_{\eta_{j}}\left[E\left[\boldsymbol{\beta}_{j}|\eta_{j},\boldsymbol{\beta}_{j-1},\mathbf{t}_{1:j-1}\right]|\mathbf{t}_{j}\right].\label{eq:moments of PFP prop}\end{aligned}$$

The inner expectation of (\[eq:moments of PFP prop\]) are computed from a joint (degenerate) prior of $\eta_{j}$ and $\boldsymbol{\beta}_{j}$, $$\left(\begin{array}{c}
\eta_{j}\\
\boldsymbol{\beta}_{j}
\end{array}\right)|\boldsymbol{\beta}_{j-1},\mathbf{t}_{1:j-1}\sim N_{P+2}\left(\left(\begin{array}{c}
\boldsymbol{\mathbf{z}}^{\prime}\boldsymbol{\beta}_{j-1}\\
\boldsymbol{\beta}_{j-1}
\end{array}\right),\,\,\left[\begin{array}{cc}
\boldsymbol{\mathbf{z}}^{\prime}\hat{\boldsymbol{U}}_{j}\boldsymbol{\mathbf{z}} & \boldsymbol{\mathbf{z}}^{\prime}\mathbf{\hat{U}}_{j}\\
\hat{\boldsymbol{U}}_{j}\boldsymbol{\mathbf{z}} & \hat{\mathbf{U}}_{j}
\end{array}\right]\right).\label{eq:Degenerate prior}$$ This method is known as linear Bayes and was proposed by @west1985dynamic, and is guaranteed to work as long as the posterior of $\lambda_{j}$ is twice differentiable with respect to $\eta_{j}$. For more details see Appendix \[sec:Propal-distributions details\].

Similarly, particles from the backward and the smoothing filters are respectively proposed from $$\begin{aligned}
\tilde{q}\left(\boldsymbol{\beta}_{j}|\tilde{\boldsymbol{\beta}}_{j+1},\mathbf{t}_{j}\right) & \simeq N\left(\tilde{\boldsymbol{m}}_{j},\,\tilde{\boldsymbol{C}}_{j}\right),\nonumber \\
\bar{q}\left(\boldsymbol{\beta}_{j}|\boldsymbol{\beta}_{j-1},\mathbf{t}_{j},\tilde{\boldsymbol{\beta}}_{j+1}\right) & \simeq N\left(\bar{\boldsymbol{m}}_{j},\,\bar{\boldsymbol{C}}_{j}\right),\label{eq:smoothing proposal distribution}\end{aligned}$$ where $\tilde{\boldsymbol{m}}_{j}=(1-\phi)\hat{\boldsymbol{\mu}}_{j}+\phi\tilde{\boldsymbol{\beta}}_{j+1}$, $\tilde{\boldsymbol{C}}_{j}=(1-\phi)\hat{\boldsymbol{\Sigma}}_{j}$, $\bar{\boldsymbol{m}}_{j}=(1-\phi)\boldsymbol{m}_{j}+\phi\tilde{\boldsymbol{\beta}}_{j+1}$ and $\boldsymbol{\bar{C}}_{j}=(1-\phi)\boldsymbol{C}_{j}$. $\hat{\boldsymbol{\mu}}_{j}$ and $\hat{\boldsymbol{\Sigma}}_{j}$ are defined in (\[eq:particle approx of the posterior estimate\]) and $\boldsymbol{m}_{j}$ and $\boldsymbol{C}_{j}$ are the moments of the proposal for the forward filter defined in (\[eq:moments of PFP prop\]); further details are provided in Appendix \[sec:Details-on-the backward and smoothing proposals\].

Model comparison and prediction\[subsec:Model-comparison-and prediction\]
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The inference methodology developed here relies on three key model choices: i) the discount factor, ii) the interval partition and iii) the covariates used in the model. Inference 