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A skew-normal dynamic linear model and Bayesian forecasting

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Abstract

Dynamic linear models are typically developed assuming that both the observational and system distributions are normal. In this work, we relax this assumption by considering a skew-normal distribution for the observational random errors, providing thus an extension of the standard normal dynamic linear model. Full Bayesian inference is carried out using the hierarchical representation of the model. The inference scheme is led by means of the adaptation of the Forward Filtering Backward sampling and the usual MCMC algorithms to perform the inference. The proposed methodology is illustrated by a simulation study and applied to the condition factor index of male and female anchovies off northern Chile. These indexes have not been studied in a dynamic linear model framework.

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Acknowledgements

We thank the Instituto de Fomento Pesquero (IFOP, Valparaíso, Chile), for providing the biological information used in this work; and to Francisco Torres-Avilés (R.I.P) and José S. Romeo for all their useful comments on an early version of this paper. Arellano-Valle’s research was supported by Grant FONDECYT (Chile) Nro. 1120121 and 1150325. We are sincerely grateful to the two anonymous reviewers for their comments and suggestions that greatly improved an early version of this manuscript. All R codes used in this paper are available in bitbucket repository: https://bitbucket.org/vate01/sndlm_v2.

Author information

Authors and Affiliations

  1. Departamento de Estadística, Pontificia Universidad Católica de Chile, Santiago, Chile

    Reinaldo B. Arellano-Valle & Abel Valdebenito

  2. Instituto de Estadística, Universidad de Valparaíso, Avenida Gran Bretaña 1111, 2360102, Valparaíso, Chile

    Javier E. Contreras-Reyes

  3. Telefónica I+D, Compañía de Telecomunicaciones de Chile S.A., Santiago, Chile

    Freddy O. López Quintero

Authors
  1. Reinaldo B. Arellano-Valle
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  2. Javier E. Contreras-Reyes
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  3. Freddy O. López Quintero
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  4. Abel Valdebenito
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Corresponding author

Correspondence to Javier E. Contreras-Reyes.

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Appendix: Proofs

Appendix: Proofs

Theorem 1

Proof

The proof is by induction and using some well-known multivariate normal distributional properties, as the following marginal–conditional relation:

$$\begin{aligned} \phi _k({\varvec{x}} \mid {\varvec{\mu }},{\varvec{\Sigma }})\phi _m({\varvec{y}} \mid {\varvec{\eta }}+ {\varvec{A}}({\varvec{x}} - {\varvec{\mu }}),{\varvec{\Psi }})= & {} \phi _k({\varvec{x}} \mid {\varvec{\mu }}+ {\varvec{\Sigma }}{\varvec{A}}'{\varvec{\Omega }}^{-1}({\varvec{y}}-{\varvec{\eta }}),\nonumber \\&{\varvec{\Sigma }}- {\varvec{\Sigma }}{\varvec{A}}'{\varvec{\Omega }}^{-1}{\varvec{A}}{\varvec{\Sigma }})\phi _m({\varvec{y}} \mid {\varvec{\eta }},{\varvec{\Omega }}),\qquad \end{aligned}$$
(32)

where \({\varvec{\Omega }}={\varvec{A}}{\varvec{\Sigma }}{\varvec{A}}' + {\varvec{\Psi }}\). In fact, suppose that the first step is true. Then, by the conditional independence assumptions considered in the Step 2, the evolutive prior distribution is

$$\begin{aligned} p({\varvec{\theta }}_t \mid {\varvec{D}}_{t-1} )= & {} \int _{\mathbb {R}^p} p({\varvec{\theta }}_t \mid {\varvec{\theta }}_{t-1} ) p({\varvec{\theta }}_{t-1} \mid {\varvec{D}}_{t-1})d{\varvec{\theta }}_{t-1} \\\propto & {} \int _{\mathbb {R}^p} \phi _p ({\varvec{\theta }}_t \mid {\varvec{G}}_t{\varvec{\theta }}_{t-1} ,{\varvec{W}}_t) \phi _p ({\varvec{\theta }}_{t-1} \mid {\varvec{m}}_{t-1|t-1}, {\varvec{C}}_{t-1|t-1})\\&\times \,\Phi _{t-1} ( {\varvec{\Lambda }}_{t-1|t-1} ({\varvec{\theta }}_{t-1} -{\varvec{m}}_{t-1|t-1}) + {\varvec{\tau }}_{t-1|t-1} \mid {\varvec{\Gamma }}_{t-1|t-1} ) d{\varvec{\theta }}_{t-1} \\= & {} \phi _p ({\varvec{\theta }}_t | {\varvec{m}}_{t|t-1} , {\varvec{C}}_{t|t-1})\\&\times \,\int _{\mathbb {R}^p} \phi _p ({\varvec{\theta }}_{t-1}-{\varvec{m}}_{t-1|t-1}\mid {\varvec{C}}_{t-1|t-1} {\varvec{G}}_t' {\varvec{C}}_{t|t-1}^{-1} ({\varvec{\theta }}_t - {\varvec{m}}_{t|t-1}),\\&{\varvec{C}}_{t-1|t-1}- {\varvec{C}}_{t-1|t-1} {\varvec{G}}_t' {\varvec{C}}_{t|t-1}^{-1}{\varvec{G}}_t{\varvec{C}}_{t-1|t-1})\\&\times \,\Phi _{t-1} ( {\varvec{\Lambda }}_{t-1|t-1} ({\varvec{\theta }}_{t-1} -{\varvec{m}}_{t-1|t-1}) \\&+\, {\varvec{\tau }}_{t-1|t-1} \mid {\varvec{\Gamma }}_{t-1|t-1}) d{\varvec{\theta }}_{t-1} \,\, (\text{ by }\,(32))\\= & {} \phi _p ({\varvec{\theta }}_t \mid {\varvec{m}}_{t|t-1} , {\varvec{C}}_{t|t-1})\\&\times \,\Phi _{t-1} ( {\varvec{\Lambda }}_{t-1|t-1}{\varvec{C}}_{t-1|t-1} {\varvec{G}}_t' {\varvec{C}}_{t|t-1}^{-1} ({\varvec{\theta }}_t - {\varvec{m}}_{t|t-1})\\&+\, {\varvec{\tau }}_{t-1|t-1} \mid {\varvec{\Gamma }}_{t-1|t-1} + {\varvec{\Lambda }}_{t-1|t-1} ({\varvec{C}}_{t-1|t-1} \\&-\, {\varvec{C}}_{t-1|t-1} {\varvec{G}}_t' {\varvec{C}}_{t|t-1}^{-1}{\varvec{G}}_t{\varvec{C}}_{t-1|t-1}){\varvec{\Lambda }}_{t-1|t-1}')\,\,(\mathrm{by}\,(7)), \end{aligned}$$

which is the kernel of the pdf of \({\varvec{\theta }}_t \mid {\varvec{D}}_{t-1} \sim SUN_{p,t-1}\big ( {\varvec{m}}_{t|t-1}, {\varvec{C}}_{t|t-1}, {\varvec{\Lambda }}_{t|t-1},{\varvec{\tau }}_{t|t-1},{\varvec{\Gamma }}_{t|t-1}\big )\). To proof the Step 3 in the sequential inference scheme, we have by Eq. (32) and the relation \({\varvec{m}}_{t|t}-{\varvec{m}}_{t|t-1}= {\varvec{C}}_{t|t-1}{\varvec{F}}_t {\varvec{Q}}_t^{-1}({\varvec{y}}_t - {\varvec{f}}_t)\) that

$$\begin{aligned} p({\varvec{y}}_t \mid {\varvec{D}}_{t-1})= & {} \int _{\mathbb {R}^p} p({\varvec{y}}_t \mid {\varvec{\theta }}_t , {\varvec{D}}_{t-1}) p({\varvec{\theta }}_t \mid {\varvec{D}}_{t-1}) d{\varvec{\theta }}_t \\\propto & {} \int _{\mathbb {R}^p} \phi _r\left( {\varvec{y}}_t \mid {\varvec{F}}'_t {\varvec{\theta }}_t -\sqrt{\frac{2}{\pi }}{\varvec{\Delta }}_t, {\varvec{V}}_t\right) \Phi _1\left( {\varvec{\Lambda }}_t \left( {\varvec{y}}_t - {\varvec{F}}'_t{\varvec{\theta }}_t +\sqrt{\frac{2}{\pi }}{\varvec{\Delta }}_t\right) \right) \\&\times \, \phi _p({\varvec{\theta }}_t \mid {\varvec{m}}_{t|t-1} , {\varvec{C}}_{t|t-1})\Phi _{t-1}({\varvec{\Lambda }}_{t|t-1}({\varvec{\theta }}_t - {\varvec{m}}_{t|t-1}) \\&+\, {\varvec{\tau }}_{t|t-1} \mid {\varvec{\Gamma }}_{t|t-1}) d{\varvec{\theta }}_t \\= & {} \phi _r ({\varvec{y}}_t \mid {\varvec{f}}_t, {\varvec{Q}}_t) \int _{\mathbb {R}^p} \phi _p ({\varvec{\theta }}_t \mid {\varvec{m}}_{t|t}, {\varvec{C}}_{t|t})\Phi _t\left( {\varvec{\Lambda }}_{t\mid t} ({\varvec{\theta }}_t - {\varvec{m}}_{t|t}) \right. \\&\left. +\, {\varvec{\tau }}_{t\mid t}\mid {\varvec{\Gamma }}_{t\mid t} \right) d{\varvec{\theta }}_t, \end{aligned}$$

where

$$\begin{aligned} {\varvec{\Lambda }}_{t\mid t}=\left( \begin{array}{c} -{\varvec{\Lambda }}_t {\varvec{F}}'_t \\ {\varvec{\Lambda }}_{t|t-1} \end{array}\right) ,\,\,{\varvec{\tau }}_{t\mid t}= \left( \begin{array}{c} {\varvec{\Lambda }}_t {\varvec{V}}_t{\varvec{Q}}^{-1}_t({\varvec{y}}_t - {\varvec{f}}_t)\\ {\varvec{\Lambda }}_{t|t-1} {\varvec{C}}_{t|t-1}{\varvec{F}}'_t{\varvec{Q}}^{-1}_t ({\varvec{y}}_t-{\varvec{f}}_t)+ {\varvec{\tau }}_{t|t-1} \end{array}\right) \end{aligned}$$

and \({\varvec{\Gamma }}_{t\mid t} = \mathrm{diag}(1,{\varvec{\Gamma }}_{t|t-1})\). Thus, by Eq. (7) we have

$$\begin{aligned} p({\varvec{y}}_t \mid {\varvec{D}}_{t-1})\propto & {} \phi _r ({\varvec{y}}_t \mid {\varvec{f}}_t, {\varvec{Q}}_t) \Phi _t({\varvec{\tau }}_{t \mid t} \mid {\varvec{\Gamma }}_{t|t} + {\varvec{\Lambda }}_{t\mid t} {\varvec{C}}_{t|t}{\varvec{\Lambda }}'_{t\mid t} )\\= & {} \phi _r ({\varvec{y}}_t \mid {\varvec{f}}_t, {\varvec{Q}}_t) \Phi _t \left( \left( \begin{array}{c} {\varvec{\Lambda }}_t{\varvec{V}}_t{\varvec{Q}}_t^{-1} \\ {\varvec{\Lambda }}_{t|t-1}{\varvec{C}}_{t|t-1}{\varvec{F}}_t'{\varvec{Q}}_t^{-1} \\ \end{array}\right) ({\varvec{y}}_t-{\varvec{f}}_t) + \left( \begin{array}{l} 0\\ {\varvec{\tau }}_{t|t-1} \end{array}\right) \mid \right. \\&\left. {\varvec{\Gamma }}_{t|t} + {\varvec{\Lambda }}_{t\mid t} {\varvec{C}}_{t|t}{\varvec{\Lambda }}'_{t\mid t} \phantom {\begin{array}{c} {\varvec{\Lambda }}_t{\varvec{V}}_t{\varvec{Q}}_t^{-1} \\ {\varvec{\Lambda }}_{t|t-1}{\varvec{C}}_{t|t-1}{\varvec{F}}_t'{\varvec{Q}}_t^{-1} \\ \end{array}}\right) , \end{aligned}$$

which corresponds to the kernel of the pdf of \( {\varvec{Y}}_t | {\varvec{D}}_{t-1} \sim SUN_{r,t}({\varvec{f}}_t , {\varvec{Q}}_t , \tilde{{\varvec{\Lambda }}}_t, \tilde{{\varvec{\tau }}}_t,\tilde{{\varvec{\Gamma }}}_t )\). To prove the Step 4 in the sequential inference scheme, we proceed as follows

$$\begin{aligned} p({\varvec{\theta }}_t \mid {\varvec{y}}_t , {\varvec{D}}_{t-1} )\propto & {} p({\varvec{y}}_t \mid {\varvec{\theta }}_t) p({\varvec{\theta }}_t \mid {\varvec{D}}_{t-1})\\= & {} \phi _r\left( {\varvec{y}}_t \mid {\varvec{F}}'_t {\varvec{\theta }}_t-\sqrt{\frac{2}{\pi }}{\varvec{\Delta }}_t,{\varvec{V}}_t\right) \phi _p({\varvec{\theta }}_t \mid {\varvec{m}}_{t|t-1} , {\varvec{C}}_{t|t-1})\\&\times \, \Phi _1\left( {\varvec{\Lambda }}_t \left( {\varvec{y}}_t - {\varvec{F}}'_t{\varvec{\theta }}_t+\sqrt{\frac{2}{\pi }}{\varvec{\Delta }}_t\right) \right) \\&\times \,\Phi _{t-1}({\varvec{\Lambda }}_{t|t-1}({\varvec{\theta }}_t - {\varvec{m}}_{t|t-1}) + {\varvec{\tau }}_{t|t-1} \big | {\varvec{\Gamma }}_{t|t-1})\\\propto & {} \phi _p ({\varvec{\theta }}_t \mid {\varvec{m}}_{t|t-1} + {\varvec{C}}_{t|t-1}{\varvec{F}}_t {\varvec{Q}}_t^{-1}({\varvec{y}}_t - {\varvec{f}}_t), {\varvec{C}}_{t|t-1}\\&-\, {\varvec{C}}_{t|t-1}{\varvec{F}}_t{\varvec{Q}}_t^{-1} {\varvec{F}}_t'{\varvec{C}}_{t|t-1}) \Phi _1\left( {\varvec{\Lambda }}_t \left( {\varvec{y}}_t - {\varvec{F}}'_t{\varvec{\theta }}_t + \sqrt{\frac{2}{\pi }}{\varvec{\Delta }}_t\right) \right) \\&\times \, \Phi _{t-1}({\varvec{\Lambda }}_{t|t-1}({\varvec{\theta }}_t - {\varvec{m}}_{t|t-1} ) + {\varvec{\tau }}_{t|t-1} \mid {\varvec{\Gamma }}_{t|t-1})\,\,(\text{ by } \text{ Eq. }\,32)\\= & {} \phi _p ({\varvec{\theta }}_t \mid {\varvec{m}}_{t|t}, {\varvec{C}}_{t|t})\Phi _1(- {\varvec{\Lambda }}_t {\varvec{F}}'_t ( {\varvec{\theta }}_t -{\varvec{m}}_{t|t}) + {\varvec{\Lambda }}_t {\varvec{V}}_t {\varvec{Q}}^{-1}_t ({\varvec{y}}_t - {\varvec{f}}_t))\\&\times \, \Phi _{t-1}({\varvec{\Lambda }}_{t|t-1}({\varvec{\theta }}_t - {\varvec{m}}_{t|t}) + {\varvec{\Lambda }}_{t|t-1}({\varvec{m}}_{t|t} - {\varvec{m}}_{t|t-1}) + {\varvec{\tau }}_{t|t-1} \mid {\varvec{\Gamma }}_{t|t-1})\\= & {} \phi _p ({\varvec{\theta }}_t | {\varvec{m}}_{t|t}, {\varvec{C}}_{t|t})\Phi _t\left( \left( \begin{array}{l} -{\varvec{\Lambda }}_t{\varvec{F}}_t' \\ {\varvec{\Lambda }}_{t|t-1} \\ \end{array} \right) ({\varvec{\theta }}_t -{\varvec{m}}_{t|t})\right. \\&\left. +\,\left( \begin{array}{c} {\varvec{\Lambda }}_t{\varvec{V}}_t{\varvec{Q}}_t^{-1}({\varvec{y}}_t-{\varvec{f}}_t) \\ {\varvec{\Lambda }}_{t|t-1}{\varvec{C}}_{t|t-1}{\varvec{F}}_t {\varvec{Q}}_t^{-1} ({\varvec{y}}_t - {\varvec{f}}_t)+ {\varvec{\tau }}_{t|t-1} \\ \end{array} \right) \mid \left( \begin{array}{ll} 1 &{} \quad {\varvec{0}}\\ {\varvec{0}} &{}\quad {\varvec{\Gamma }}_{t|t-1}\end{array}\right) \right) , \end{aligned}$$

where the relations \({\varvec{\Lambda }}_t ({\varvec{y}}_t -{\varvec{F}}'_t {\varvec{\theta }}_t + \sqrt{\frac{2}{\pi }}{\varvec{\Delta }}_t ) = - {\varvec{\Lambda }}_t {\varvec{F}}'_t ( {\varvec{\theta }}_t -{\varvec{m}}_{t|t}) + {\varvec{\Lambda }}_t {\varvec{V}}_t {\varvec{Q}}^{-1}_t ({\varvec{y}}_t - {\varvec{f}}_t)\) and \({\varvec{m}}_{t|t}-{\varvec{m}}_{t|t-1}= {\varvec{C}}_{t|t-1}{\varvec{F}}_t {\varvec{Q}}_t^{-1}({\varvec{y}}_t - {\varvec{f}}_t)\) are used. This last expression is the density kernel of \({\varvec{\theta }}_{t}\mid {\varvec{D}}_{t} \sim SUN_{p,t}({\varvec{m}}_{t|t} , {\varvec{C}}_{t|t} , {\varvec{\Lambda }}_{t|t}, {\varvec{\tau }}_{t|t},{\varvec{\Gamma }}_{t|t} )\), concluding thus the proof. \(\square \)

Theorem 2

Proof

Since \({\varvec{\theta }}_{t-k} \mid {\varvec{D}}_{t-k} \sim SUN_{p,t-k} ({\varvec{m}}_{t-k|t-k},{\varvec{C}}_{t-k|t-k},{\varvec{\Lambda }}_{t-k|t-k},{\varvec{\tau }}_{t-k|t-k},{\varvec{\Gamma }}_{t-k|t-k})\) and \({\varvec{\theta }}_{t-k+1} \mid {\varvec{\theta }}_{t-k}, {\varvec{D}}_{t-k} \sim N_p ({\varvec{G}}_{t-k+1} {\varvec{\theta }}_{t-k} ,{\varvec{W}}_{t-k+1})\), we have by Eq. (32)

$$\begin{aligned} p({\varvec{\theta }}_{t-k} \mid {\varvec{\theta }}_{t-k+1}, {\varvec{D}}_{t-k} )\propto & {} p({\varvec{\theta }}_{t-k} | {\varvec{D}}_{t-k}) p({\varvec{\theta }}_{t-k+1} | {\varvec{\theta }}_{t-k} , {\varvec{D}}_{t-k}) \\\propto & {} \phi _p({\varvec{\theta }}_{t-k} \mid {\varvec{m}}_{t-k\mid t-k},{\varvec{C}}_{t-k\mid t-k} )\\&\times \, \phi _p ({\varvec{\theta }}_{t-k+1}\mid {\varvec{G}}_{t-k+1} {\varvec{\theta }}_{t-k} , {\varvec{W}}_{t-k+1})\\&\times \, \Phi _{t-k} ({\varvec{\Lambda }}_{t-k\mid t-k}({\varvec{\theta }}_{t-k} - {\varvec{m}}_{t-k\mid t-k}) + {\varvec{\tau }}_{t-k\mid t-k}\mid {\varvec{\Gamma }}_{t-k\mid t-k}) \\\propto & {} \phi _p ({\varvec{\theta }}_{t-k} \mid {\varvec{m}}_{t-k\mid t-k} + {\varvec{B}}_{t-k} ({\varvec{\theta }}_{t-k+1} - {\varvec{m}}_{t-k+1\mid t-k}),\\&{\varvec{C}}_{t-k\mid t-k} - {\varvec{B}}_{t-k} {\varvec{C}}_{t-k+1\mid t-k} {\varvec{B}}'_{t-k})\\&\times \,\Phi _{t-k} ({\varvec{\Lambda }}_{t-k\mid t-k}({\varvec{\theta }}_{t-k} - {\varvec{m}}_{t-k\mid t-k}) + {\varvec{\tau }}_{t-k\mid t-k}\mid {\varvec{\Gamma }}_{t-k\mid t-k})\\= & {} \phi _p ({\varvec{\theta }}_{t-k} \mid {\varvec{h}}_{t}(k), {\varvec{H}}_{t}(k))\Phi _{t-k} ({\varvec{\Lambda }}_{t-k\mid t-k}({\varvec{\theta }}_{t-k} - {\varvec{h}}_{t}(k))\\&+\, {\varvec{\Lambda }}_{t-k\mid t-k}({\varvec{h}}_{t}(k) - {\varvec{m}}_{t-k\mid t-k})+{\varvec{\tau }}_{t-k\mid t-k}\mid {\varvec{\Gamma }}_{t-k\mid t-k}), \end{aligned}$$

where \({\varvec{h}}_{t}(k) - {\varvec{m}}_{t-k\mid t-k}= {\varvec{B}}_{t-k} ({\varvec{\theta }}_{t-k+1} - {\varvec{m}}_{t-k+1\mid t-k})\). Thus, this last result corresponds to the density kernel of the distribution given in Eq. (20). \(\square \)

Theorem 3

Proof

We need to solve the following integral

$$\begin{aligned} p({\varvec{\theta }}_{t-k} \mid {\varvec{D}}_t)= & {} \int _{\mathbb {R}^p} p({\varvec{\theta }}_{t-k} \mid {\varvec{\theta }}_{t-k+1} , {\varvec{D}}_t) p ({\varvec{\theta }}_{t-k+1}\mid {\varvec{D}}_t) d{\varvec{\theta }}_{t-k+1}. \end{aligned}$$

For this, note first that \(p({\varvec{\theta }}_{t-k} \mid {\varvec{\theta }}_{t-k+1} , {\varvec{D}}_t) = p({\varvec{\theta }}_{t-k} \mid {\varvec{\theta }}_{t-k+1} , {\varvec{D}}_{t-k})\), where by Theorem 2 \({\varvec{\theta }}_{t-k} \mid {\varvec{\theta }}_{t-k+1}, {\varvec{D}}_{t-k} \sim SUN_{p,t-k} ({\varvec{h}}_{t}(k),{\varvec{H}}_{t}(k),{\varvec{\Lambda }}_t(k),{\varvec{\tau }}_t(k), {\varvec{\Gamma }}_t(k))\) and by induction \({\varvec{\theta }}_{t-k+1}\mid {\varvec{D}}_t \sim \)\(SUN_{p,t} ({\varvec{m}}_{t-k+1|t},{\varvec{C}}_{t-k+1|t},{\varvec{\Lambda }}_{t-k+1|t},{\varvec{\tau }}_{t-k+1|t}, {\varvec{\Gamma }}_{t-k+1|t})\).

Hence, from the definitions of \({\varvec{h}}_{t}(k),\)\({\varvec{H}}_{t}(k)\), \(\Lambda _t(k)\), \({\varvec{\tau }}_t(k)\) and \({\varvec{\Gamma }}_t(k)\), the relations \({\varvec{\Lambda }}_{t-k+1|t-k}={\varvec{\Lambda }}_{t-k|t-k}{\varvec{B}}_{t-k}\), \({\varvec{\tau }}_{t-k+1|t-k}={\varvec{\tau }}_{t-k|t-k}\), \({\varvec{\Gamma }}_{t-k+1|t-k}={\varvec{\Gamma }}_t(k)+{\varvec{\Lambda }}_t(k){\varvec{H}}_t(k){\varvec{\Lambda }}_t(k)'\), by applying the Eqs. (32) and (7), we have

$$\begin{aligned} p({\varvec{\theta }}_{t-k} \mid {\varvec{D}}_t)\propto & {} \int _{\mathbb {R}^p}\frac{\phi _p ({\varvec{\theta }}_{t-k} \mid {\varvec{h}}_{t}(k), {\varvec{H}}_{t}(k))\Phi _{t-k} ({\varvec{\Lambda }}_t(k)({\varvec{\theta }}_{t-k} - {\varvec{h}}_{t}(k)) + {\varvec{\tau }}_t(k) \mid {\varvec{\Gamma }}_t(k))}{\Phi _{t-k} ({\varvec{\tau }}_t(k) \mid {\varvec{\Gamma }}_t(k)+{\varvec{\Lambda }}_t(k){\varvec{H}}_{t}(k){\varvec{\Lambda }}'_t(k))}\\&\times \, \phi _p ({\varvec{\theta }}_{t-k+1} \mid {\varvec{m}}_{t-k+1|t},{\varvec{C}}_{t-k+1|t})\\&\times \, \Phi _{t} ({\varvec{\Lambda }}_{t-k+1|t}({\varvec{\theta }}_{t-k+1} - m_{t-k+1|t}) + {\varvec{\tau }}_{t-k+1|t} \mid {\varvec{\Gamma }}_{t-k+1|t}) d{\varvec{\theta }}_{t-k+1}\\= & {} \phi _{p}({\varvec{\theta }}_{t-k}\mid {\varvec{m}}_{t-k|t},{\varvec{C}}_{t-k|t})\\&\times \, \Phi _{t-k} ({\varvec{\Lambda }}_{t-k|t-k}({\varvec{\theta }}_{t-k} - {\varvec{m}}_{t-k|t-k}) + {\varvec{\tau }}_{t-k\mid t-k}\mid {\varvec{\Gamma }}_{t-k\mid t-k}) \\&\times \, \int _{\mathbb {R}^p} \phi _p ({\varvec{\theta }}_{t-k+1}-{\varvec{m}}_{t-k+1|t} \mid {\varvec{C}}_{t-k+1|t}{\varvec{B}}_{t-k}'{\varvec{C}}_{t-k|t}^{-1}({\varvec{\theta }}_{t-k} - {\varvec{m}}_{t-k|t}),\\&{\varvec{S}}_{t-k+1|t})\Phi _{k-1} \left( {\varvec{\Lambda }}^{(2)}_{t-k+1|t}({\varvec{\theta }}_{t-k+1} - {\varvec{m}}_{t-k+1|t}) + {\varvec{\tau }}^{(2)}_{t-k+1|t} \mid {\varvec{\Gamma }}^{(2)}_{t-k+1|t} \right) \\&\times \, \Phi _1(-{\varvec{\Lambda }}_{t-k+1}{\varvec{F}}'_{t-k+1}({\varvec{\theta }}_{t-k+1} - {\varvec{m}}_{t-k+1|t})+{\varvec{\Lambda }}_{t-k+1} ({\varvec{y}}_{t-k+1} \\&-\, {\varvec{F}}'_{t-k+1}{\varvec{m}}_{t-k+1|t})) d{\varvec{\theta }}_{t-k+1}\\= & {} \phi _{p}({\varvec{\theta }}_{t-k}\mid {\varvec{m}}_{t-k|t},{\varvec{C}}_{t-k|t})\\&\times \, \Phi _{t-k} ({\varvec{\Lambda }}_{t-k|t-k}({\varvec{\theta }}_{t-k} - {\varvec{m}}_{t-k|t-k}) + {\varvec{\tau }}_{t-k\mid t-k}\mid {\varvec{\Gamma }}_{t-k\mid t-k})\\&\times \, \int _{\mathbb {R}^p} \phi _p ({\varvec{\theta }}_{t-k+1}-{\varvec{m}}_{t-k+1|t} \mid {\varvec{C}}_{t-k+1|t}{\varvec{B}}_{t-k}'{\varvec{C}}_{t-k|t}^{-1}({\varvec{\theta }}_{t-k} - {\varvec{m}}_{t-k|t}),\\&{\varvec{S}}_{t-k+1|t}) \Phi _k\left( \left( \begin{array}{c} -{\varvec{\Lambda }}_{t-k+1}{\varvec{F}}'_{t-k+1}\\ {\varvec{\Lambda }}^{(2)}_{t-k+1|t} \end{array}\right) ({\varvec{\theta }}_{t-k+1}-{\varvec{m}}_{t-k+1|t})\right. \\&\left. +\, \left( \begin{array}{l} {\varvec{\Lambda }}_{t-k+1}({\varvec{y}}_{t-k+1}-{\varvec{F}}'_{t-k+1}{\varvec{m}}_{t-k+1|t}+\sqrt{\frac{2}{\pi }}\Delta _{t-k+1})\\ {\varvec{\tau }}^{(2)}_{t-k+1|t} \end{array}\right) \mid \left( \begin{array}{ll} 1&{}\quad {\varvec{0}}\\ {\varvec{0}} &{}\quad {\varvec{\Gamma }}_{t-k+1|t} \end{array}\right) \right) d{\varvec{\theta }}_{t-k+1}\\= & {} \phi _{p}({\varvec{\theta }}_{t-k}\mid {\varvec{m}}_{t-k|t},{\varvec{C}}_{t-k|t})\\&\times \,\Phi _{t-k} ({\varvec{\Lambda }}_{t-k|t-k}({\varvec{\theta }}_{t-k} - {\varvec{m}}_{t-k|t-k}) + {\varvec{\tau }}_{t-k\mid t-k}\mid {\varvec{\Gamma }}_{t-k\mid t-k})\\&\times \, \Phi _k\left( \left( \begin{array}{c} -{\varvec{\Lambda }}_{t-k+1}{\varvec{F}}'_{t-k+1}\\ {\varvec{\Lambda }}^{(2)}_{t-k+1|t} \end{array}\right) {\varvec{C}}_{t-k+1|t}{\varvec{B}}_{t-k}{\varvec{C}}_{t-k|t}^{-1}({\varvec{\theta }}_{t-k} - {\varvec{m}}_{t-k|t})\right. \\&\left. +\, \left( \begin{array}{l} {\varvec{\Lambda }}_{t-k+1}({\varvec{y}}_{t-k+1}-{\varvec{F}}'_{t-k+1}{\varvec{m}}_{t-k+1|t}+\sqrt{\frac{2}{\pi }}\Delta _{t-k+1}) \\ {\varvec{\tau }}^{(2)}_{t-k+1|t} \end{array}\right) \mid {\varvec{\Gamma }}_{t-k|t} \right) \\= & {} \phi _{p}({\varvec{\theta }}_{t-k}\mid {\varvec{m}}_{t-k|t},{\varvec{C}}_{t-k|t}) \Phi _t ({\varvec{\Lambda }}_{t-k|t}({\varvec{\theta }}_{t-k} - {\varvec{m}}_{t-k|t}) + {\varvec{\tau }}_{t-k|t} \mid {\varvec{\Gamma }}_{t-k|t}). \end{aligned}$$

Thus, we have the kernel of the \(SUN_{p,t}({\varvec{m}}_{t-k|t} , {\varvec{C}}_{t-k|t}, {\varvec{\Lambda }}_{t-k|t}, {\varvec{\tau }}_{t-k|t},{\varvec{\Gamma }}_{t-k|t} )\) pdf, which concludes the proof. \(\square \)

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Arellano-Valle, R.B., Contreras-Reyes, J.E., Quintero, F.O.L. et al. A skew-normal dynamic linear model and Bayesian forecasting. Comput Stat 34, 1055–1085 (2019). https://doi.org/10.1007/s00180-018-0848-1

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