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Periodic autoregressive models with closed skew-normal innovations

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Abstract

This paper is concerned with the estimation problem of a periodic autoregressive model with closed skew-normal innovations. The closed skew-normal (CSN) distribution has some useful properties similar to those of the Gaussian distribution. Maximum likelihood (ML), Maximum a posteriori (MAP) and Bayesian approaches are proposed and compared in order to estimate the model parameters. For the Bayesian approach, the Gibbs sampling algorithm and for computing the ML and MAP estimations, the expectation–maximization algorithms are performed. The simulation studies are then conducted to compare the frequentist average losses of competing estimators and to study the asymptotic properties of the given estimators. The proposed model and methods developed in this paper are also applied to a real time series. The accuracy of the CSN and Gaussian models is compared by cross validation criterion.

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Authors and Affiliations

  1. Department of Statistics, Shiraz University, Shiraz, Iran

    T. Manouchehri & A. R. Nematollahi

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  1. T. Manouchehri
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  2. A. R. Nematollahi
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Correspondence to A. R. Nematollahi.

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Appendices

Appendix A: Proof of Theorem 3.1

In order to prove of Theorem 3.1, we need some preliminary definitions and properties.

Definition 1

(Truncated multivariate normal). If \( \varvec{W} \sim N_{q} \left( {\varvec{\mu},{\varvec{\Sigma}}} \right) \) and \( \varvec{U} = \left\{ {\begin{array}{*{20}l} \varvec{W} & {if\quad \varvec{W} \ge \varvec{c}} \\ {\mathbf{0}} & {if\quad ~\varvec{W} < \varvec{c}} \\ \end{array} } \right. \) where \( \varvec{W} \ge \varvec{c} \) means \( W_{j} \ge c_{j} , j = 1, \ldots ,q \), then the density function of \( \varvec{U} \) is:

$$ f\left( {\varvec{u};\varvec{\mu},\varvec{\varSigma},\varvec{c}} \right) = \varPhi_{q}^{ - 1} \left( {0;\varvec{c} -\varvec{\mu},\varvec{\varSigma}} \right)\varphi_{q} \left( {\varvec{u};\varvec{\mu},\varvec{\varSigma}} \right) , \varvec{u} \ge \varvec{c}. $$

\( \varvec{U} \) is truncated multivariate normal denote by \( \varvec{U} \sim N_{q}^{\varvec{c}} \left( {\varvec{\mu},{\varvec{\Sigma}}} \right) \).

Property 1

If \( \varvec{U} \sim N_{q}^{\varvec{c}} \left( {\varvec{\mu},{\varvec{\Sigma}}} \right) \) then the moment generating function of \( \varvec{U} \) is given by

$$ M_{\varvec{U}} \left( \varvec{t} \right) = \varPhi_{q}^{ - 1} \left( {{\mathbf{0}};\varvec{c} -\varvec{\mu},\varvec{\varSigma}} \right)e^{{\varvec{t^{\prime}\mu } + \frac{1}{2}\varvec{t^{\prime}\varSigma t}}} \varPhi_{q} \left( {\varvec{\varSigma t};\varvec{c} -\varvec{\mu},\varvec{\varSigma}} \right), \varvec{t} \in {\mathcal{R}}^{q} . $$

Property 2

If\( \varvec{Z} \sim CSN_{p,q} \left( {\varvec{\mu},\varvec{\varSigma},\varvec{\varGamma},\varvec{\nu},\varvec{\varDelta}} \right) \), then the moment generative function of\( \varvec{Z} \)is given in González-Farías et al. (2004) as

$$ M_{\varvec{Z}} \left( \varvec{s} \right) = \frac{{\varPhi_{q} \left( {\varvec{\varGamma \varSigma s};\varvec{\nu},\varvec{\varDelta}+ \varvec{\varGamma \varSigma \varGamma^{\prime}}} \right)}}{{\varPhi_{q} \left( {{\mathbf{0}};\varvec{\nu},\varvec{\varDelta}+ \varvec{\varGamma \varSigma \varGamma^{\prime}}} \right)}}\exp \left( {{\mathbf{s^{\prime}}}\varvec{\mu}+ \frac{1}{2}{\mathbf{s^{\prime}}}\varvec{\varSigma s}} \right),\quad {\mathbf{s}} \in {\mathbb{R}}^{p} . $$

Proof of Theorem 3.1

The result of part (a) is proved by using the uniqueness property of the moment generating functions. Note that

$$ \begin{aligned} M_{{\varvec{Y}_{t} }} \left( \varvec{s} \right) & = M_{{\varvec{V}_{t} }} \left( \varvec{s} \right)M_{{\varvec{W}_{t} }} \left( {\varvec{D^{\prime}s}} \right) = e^{{\varvec{s^{\prime}}\left( {\mathop \sum \limits_{j = 1}^{P}\varvec{\varPhi}_{j} \varvec{y}_{t - j} } \right) + \frac{1}{2}\varvec{s}^{'} \varvec{Gs}}} \\ & \quad \times\varPhi_{T}^{ - 1} \left( {{\mathbf{0}};{\mathbf{0}},\varvec{\varLambda}} \right)e^{{\frac{1}{2}\varvec{s}^{\prime} \varvec{D\varLambda D^{\prime}s}}} \varPhi_{T} \left( {\varvec{\varLambda D^{\prime}s};{\mathbf{0}},\varvec{\varLambda}} \right) \\ & = \frac{{\varPhi_{T} \left( {\varvec{\varLambda D^{\prime}s};{\mathbf{0}},\varvec{\varLambda}} \right)}}{{\varPhi_{T} \left( {{\mathbf{0}};{\mathbf{0}},\varvec{\varLambda}} \right)}}e^{{\varvec{s^{\prime}}\left( {\mathop \sum \limits_{j = 1}^{P}\varvec{\varPhi}_{j} \varvec{y}_{t - j} } \right) + \frac{1}{2}\varvec{s}^{'} \left( {\varvec{G} + \varvec{D\varLambda D^{\prime}}} \right)\varvec{s}}} \\ & = \frac{{\varPhi_{T} \left( {\varvec{\varGamma}^{*}\varvec{\varSigma}^{*} \varvec{s};{\mathbf{0}},\varvec{I}_{\varvec{T}} +\varvec{\varGamma}^{*}\varvec{\varSigma}^{*}\varvec{\varGamma}^{{*'}} } \right)}}{{\varPhi_{T} \left( {{\mathbf{0}};{\mathbf{0}},\varvec{I}_{\varvec{T}} +\varvec{\varGamma}^{*}\varvec{\varSigma}^{*}\varvec{\varGamma}^{{*'}} } \right)}}e^{{\varvec{s^{\prime}}\left( {\mathop \sum \limits_{j = 1}^{P}\varvec{\varPhi}_{j} \varvec{y}_{t - j} } \right) + \frac{1}{2}\varvec{s}^{'}\varvec{\varSigma}^{*} \varvec{s}}} = M_{\varvec{Z}} \left( \varvec{s} \right) \\ \end{aligned} $$

where \( \varvec{Z} \sim CSN_{T,T} \left( {\mathop \sum \limits_{j = 1}^{P}\varvec{\varPhi}_{j} \varvec{y}_{t - j} ,\varvec{\varSigma}^{*} ,\varvec{\varGamma}^{*} ,{\mathbf{0}},\varvec{I}_{\varvec{T}} } \right) \).

(b) It is proved by using the linearity property of the multivariate normal distributions.

(c) It can be proved by the following arguments:

$$ \begin{aligned} f\left( {\varvec{W}_{t} |\varvec{Y}_{t} } \right) & = \frac{{f\left( {\varvec{Y}_{t} ,\varvec{W}_{t} } \right)}}{{f\left( {\varvec{Y}_{t} } \right)}} = \frac{{f\left( {\varvec{W}_{t} } \right)f\left( {\varvec{Y}_{t} |\varvec{W}_{t} } \right)}}{{f\left( {\varvec{Y}_{t} } \right)}} \\ & = \frac{{\varPhi_{T}^{ - 1} \left( {{\mathbf{0}};{\mathbf{0}},\varvec{\varLambda}} \right)\varphi_{T} \left( {\varvec{w}_{t} ;{\mathbf{0}},\varvec{\varLambda}} \right)\varphi_{T} \left( {\varvec{y}_{t} ;\mathop \sum \nolimits_{j = 1}^{P}\varvec{\varPhi}_{j} \varvec{y}_{t - j} + \varvec{DW}_{t} ,\varvec{G}} \right)}}{{\varphi_{T} \left( {\varvec{y}_{t} ;\mathop \sum \nolimits_{j = 1}^{P}\varvec{\varPhi}_{j} \varvec{Y}_{t - j} ,\varvec{\varSigma}^{*} } \right)\varPhi_{T} \left( {\varvec{\varGamma}^{*} \left( {\varvec{Y}_{t} - \mathop \sum \nolimits_{j = 1}^{P}\varvec{\varPhi}_{j} \varvec{y}_{t - j} } \right);{\mathbf{0}},\varvec{I}_{T} } \right)\varPhi_{T}^{ - 1} \left( {{\mathbf{0}};{\mathbf{0}},\varvec{I}_{T} } \right)}} \\ & = \frac{{\varphi_{T} \left( {\varvec{w}_{t} ;{\mathbf{0}},\varvec{\varLambda}} \right)\varphi_{T} \left( {\varvec{Y}_{t} ;\mathop \sum \nolimits_{j = 1}^{P}\varvec{\varPhi}_{j} \varvec{y}_{t - j} + \varvec{Dw}_{t} ,\varvec{G}} \right)}}{{\varphi_{T} \left( {\varvec{y}_{t} ;\mathop \sum \nolimits_{j = 1}^{P}\varvec{\varPhi}_{j} \varvec{y}_{t - j} ,\varvec{\varSigma}^{*} } \right)\varPhi_{T} \left( {\varvec{\varGamma}^{*} \left( {\varvec{y}_{t} - \mathop \sum \nolimits_{j = 1}^{P}\varvec{\varPhi}_{j} \varvec{y}_{t - j} } \right);{\mathbf{0}},\varvec{I}_{T} } \right)}} \\ & = C_{1} \left( {\varvec{y}_{t} ,\varvec{\theta}_{1} } \right)\varphi_{q} \left( {\varvec{w}_{t} ;\nu^{*} ,\varLambda^{*} } \right) = C_{2} \left( {\varvec{y}_{t} ,\varvec{\theta}_{1} } \right)\varPhi_{T}^{ - 1} \left( {{\mathbf{0}}; -\varvec{\nu}^{*} ,\varvec{\varLambda}^{*} } \right)\varphi_{q} \left( {\varvec{w}_{t} ;\varvec{\nu}^{*} ,\varvec{\varLambda}^{*} } \right) \\ & = C\left( {\varvec{y}_{t} ,\varvec{\theta}_{1} } \right)N_{T}^{{\mathbf{0}}} \left( {\varvec{w}_{t} ;\varvec{\nu}^{*} ,\varvec{\varLambda}^{*} } \right),\varvec{w}_{t} \ge 0, \\ \end{aligned} $$

where \( {\varvec{\Lambda}}^{*} = \left( {\varvec{\varLambda}^{ - 1} + \varvec{D^{\prime}G}^{ - 1} \varvec{D}} \right)^{ - 1} \), \( \varvec{\nu}^{*} = {\varvec{\Lambda}}^{*} \varvec{D^{\prime}G}^{ - 1} \left( {\varvec{y}_{t} - \mathop \sum \limits_{j = 1}^{P}\varvec{\varPhi}_{j} \varvec{y}_{t - j} } \right) \), and \( C \) is function of parameters \( \varvec{\theta}_{1} = \left( {\varvec{\varPhi}_{j} ,{\varvec{\Lambda}},\varvec{G},\varvec{D}} \right) \) and observed data \( \varvec{y}_{t} \).

The moment generating function of \( \varvec{W}_{t} |\varvec{Y}_{t} \) is given by

$$ M_{{\varvec{W}_{t} |\varvec{Y}_{t} }} \left( s \right) = C\left( {\varvec{Y}_{t} ,\varvec{\theta}_{1} } \right)\frac{{\varPhi_{T} \left( {{\varvec{\Lambda}}^{*} \varvec{s}; -\varvec{\nu}^{*} ,{\varvec{\Lambda}}^{*} } \right)}}{{\varPhi_{T} \left( {{\mathbf{0}}; -\varvec{\nu}^{*} ,{\varvec{\Lambda}}^{*} } \right)}}e^{{\varvec{s}^{\prime}\varvec{\nu}^{*} + \frac{1}{2}\varvec{s^{\prime}}{\varvec{\Lambda}}^{*} \varvec{s}}} , \varvec{s} \in {\mathcal{R}}^{T} , $$

and so

$$ E(\varvec{W}_{t} |\varvec{Y}_{t} ) = \frac{{\partial M_{{\varvec{W}_{t} |\varvec{Y}_{t} }} \left( \varvec{s} \right)}}{{\partial \varvec{s}}}|_{{\varvec{s} = 0}} , $$

where

$$ \begin{aligned} \frac{{\partial M_{{\varvec{W}_{t} |\varvec{Y}_{t} }} \left( \varvec{s} \right)}}{{\partial \varvec{s}}} & = C\left( {\varvec{Y}_{t} ,\varvec{\theta }_{1} } \right)\left\{ {\Lambda ^{*} \frac{{\frac{{\partial \Phi _{T} \left( {\varvec{s}; - \varvec{\nu }^{*} ,{\varvec{\Lambda }}^{*} } \right)}}{{\partial \varvec{s}}}}}{{\Phi _{T} \left( {{\varvec{0}}; - \varvec{\nu }^{*} ,{\varvec{\Lambda }}^{*} } \right)}}e^{{\varvec{s^{\prime}\nu }^{*} + \frac{1}{2}\varvec{s^{\prime}}{\varvec{\Lambda }}^{*} \varvec{s}}} } \right. \\ & \quad \left. { + \frac{{\Phi _{T} \left( {{\varvec{\Lambda }}^{*} \varvec{s}; - \varvec{\nu }^{*} ,{\varvec{\Lambda }}^{*} } \right)}}{{\Phi _{T} \left( {{\varvec{0}}; - \varvec{\nu }^{*} ,{\varvec{\Lambda }}^{*} } \right)}}\left( {\varvec{\nu }^{*} + {\varvec{\Lambda }}^{*} \varvec{s}} \right)e^{{\varvec{s^{\prime}\nu }^{*} + \frac{1}{2}\varvec{s^{\prime}}{\varvec{\Lambda }}^{*} \varvec{s}}} } \right\}. \\ \end{aligned} $$

Therefore

$$ E\left( {\varvec{W}_{t} |\varvec{Y}_{t} } \right) = \frac{{\partial M_{{\varvec{W}_{t} |\varvec{Y}_{t} }} \left( \varvec{s} \right)}}{{\partial \varvec{s}}}|_{{\varvec{s} = 0}} = C\left( {\varvec{Y}_{t} ,\varvec{\theta}_{1} } \right)\left( {\varvec{\nu}^{*} + \varLambda^{*} \xi_{1} } \right), $$

where \( \xi_{1} = \frac{{\frac{{\partial \varPhi_{T} \left( {\varvec{s}; -\varvec{\nu}^{*} ,{\varvec{\Lambda}}^{*} } \right)}}{{\partial \varvec{s}}}}}{{\varPhi_{T} \left( {0; -\varvec{\nu}^{*} ,{\varvec{\Lambda}}^{*} } \right)}}|_{{\varvec{s} = 0}} . \) Also,

$$ E\left( {\varvec{W}_{t}^{'} \varvec{W}_{t} |\varvec{Y}_{t} } \right) = \frac{{\partial^{2} M_{{\varvec{W}_{t} |\varvec{Y}_{t} }} \left( \varvec{s} \right)}}{{\partial \varvec{s}\partial \varvec{s^{\prime}}}}|_{{\varvec{s} = \varvec{s^{\prime}} = {\mathbf{0}}}} , $$

Where

$$ \begin{aligned} &\frac{{\partial^{2} M_{{\varvec{W}_{t} |\varvec{Y}_{t} }} \left( \varvec{s} \right)}}{{\partial \varvec{s}\partial \varvec{s^{\prime}}}} = C\left( {\varvec{Y}_{t} ,\varvec{\theta}_{1} } \right)\left\{ {\varvec{\Lambda}}^{*} \frac{{\frac{{\partial^{2} \varPhi_{T} \left( {\varvec{s}; -\varvec{\nu}^{*} ,{\varvec{\Lambda}}^{*} } \right)}}{{\partial \varvec{s}\partial \varvec{s^{\prime}}}}}}{{\varPhi_{T} \left( {{\mathbf{0}}; -\varvec{\nu}^{*} ,{\varvec{\Lambda}}^{*} } \right)}}e^{{\varvec{s^{\prime}\nu }^{*} + \frac{1}{2}\varvec{s^{\prime}}{\varvec{\Lambda}}^{*} \varvec{s}}} \right. \\ &\quad \left. + {\varvec{\Lambda}}^{*} \frac{{\frac{{\partial \varPhi_{T} \left( {\varvec{s}; -\varvec{\nu}^{*} ,{\varvec{\Lambda}}^{*} } \right)}}{{\partial \varvec{s}}}}}{{\varPhi_{T} \left( {{\mathbf{0}}; -\varvec{\nu}^{*} ,{\varvec{\Lambda}}^{*} } \right)}}\left( {\varvec{\nu}^{*} + {\varvec{\Lambda}}^{*} \varvec{s}} \right)^{'} e^{{\varvec{s^{\prime}\nu }^{*} + \frac{1}{2}\varvec{s^{\prime}}{\varvec{\Lambda}}^{*} \varvec{s}}} \right. \\ &\quad \left. + \left( {\varvec{\Lambda}}^{*} \frac{{\frac{{\partial \varPhi_{T} \left( {\varvec{s}; -\varvec{\nu}^{*} ,{\varvec{\Lambda}}^{*} } \right)}}{{\partial \varvec{s}}}}}{{\varPhi_{T} \left( {{\mathbf{0}}; -\varvec{\nu}^{*} ,{\varvec{\Lambda}}^{*} } \right)}}\left( {\varvec{\nu}^{*} + {\varvec{\Lambda}}^{*} \varvec{s}} \right)^{'} {e^{{\varvec{s^{\prime}\nu }^{*} + \frac{1}{2}\varvec{s^{\prime}}{\varvec{\Lambda}}^{*} \varvec{s}}} } \right)^{{\prime }} \right. \\ &\quad \left. + \frac{{\varPhi_{T} \left( {{\varvec{\Lambda}}^{*} \varvec{s}; -\varvec{\nu}^{*} ,{\varvec{\Lambda}}^{*} } \right)}}{{\varPhi_{T} \left( {{\mathbf{0}}; -\varvec{\nu}^{*} ,{\varvec{\Lambda}}^{*} } \right)}}\left( {\left( {\varvec{\nu}^{*} + {\varvec{\Lambda}}^{*} \varvec{s}} \right)\left( {\varvec{\nu}^{*} + {\varvec{\Lambda}}^{*} \varvec{s}} \right)^{'} e^{{\varvec{s^{\prime}\nu }^{*} + \frac{1}{2}\varvec{s^{\prime}}{\varvec{\Lambda}}^{*} \varvec{s}}} + {\varvec{\Lambda}}^{*} e^{{\varvec{s^{\prime}\nu }^{*} + \frac{1}{2}\varvec{s^{\prime}}{\varvec{\Lambda}}^{*} \varvec{s}}} } \right) \vphantom{\frac{{\frac{{\partial^{2} \varPhi_{T} \left( {\varvec{s}; -\varvec{\nu}^{*} ,{\varvec{\Lambda}}^{*} } \right)}}{{\partial \varvec{s}\partial \varvec{s^{\prime}}}}}}{{\varPhi_{T} \left( {{\mathbf{0}}; -\varvec{\nu}^{*} ,{\varvec{\Lambda}}^{*} } \right)}}}\right\} \hfill \\ \end{aligned} $$

Therefore

$$ \begin{aligned} E\left( {\varvec{W}_{t}^{'} \varvec{W}_{t} |\varvec{Y}_{t} } \right)& = \frac{{\partial^{2} M_{{\varvec{W}_{t} |\varvec{Y}_{t} }} \left( \varvec{s} \right)}}{{\partial \varvec{s}\partial \varvec{s^{\prime}}}}|_{{\varvec{s} = \varvec{s^{\prime}} = {\mathbf{0}}}} \\ & \quad = C\left( {\varvec{Y}_{t} ,\varvec{\theta}_{1} } \right)\left( {{\varvec{\Lambda}}^{*} \xi_{2} + {\varvec{\Lambda}}^{*} \xi_{1}\varvec{\nu}^{*'} + \left( {{\varvec{\Lambda}}^{*} \xi_{1}\varvec{\nu}^{*'} } \right)^{'} +\varvec{\nu}^{*}\varvec{\nu}^{*'} + {\varvec{\Lambda}}^{*} } \right),\end{aligned} $$

where \( \xi_{2} = \frac{{\frac{{\partial^{2} \varPhi_{T} \left( {\varvec{s}; -\varvec{\nu}^{*} ,{\varvec{\Lambda}}^{*} } \right)}}{{\partial \varvec{s}\partial \varvec{s^{\prime}}}}}}{{\varPhi_{T} \left( {0; -\varvec{\nu}^{*} ,{\varvec{\Lambda}}^{*} } \right)}}|_{{\varvec{s} = \varvec{s^{\prime}} = 0}} . \)

Appendix B. Algorithm CJJ

Step 1 Compute \( \varvec{\nu}_{{\left( {k + 1} \right)}} = \varvec{I}_{T} \) and \( \varvec{m}_{{\left( {k + 1} \right)}} =\varvec{\alpha}_{k} {\varvec{\Delta}}_{k}^{ - 1/2}\varvec{\varPhi}_{0k}^{\varvec{'}} \left( {\varvec{Y}_{ - P} - \varvec{Z}_{ - P} {\varvec{\Phi}}_{k} } \right).\varvec{ } \) Simulate \( \varvec{W}_{k + 1} \) from a multivariate truncated normal with mean \( \varvec{m}_{{\left( {k + 1} \right)}} \) and \( T \times T \) variance–covariance matrix \( \varvec{\nu}_{k + 1} \).

Step 2 Select a \( T \)-dimention random vector \( \varvec{V}_{1} \) with elements \( v_{1i} = z_{1i} /( {\mathop \sum \limits_{j} z_{1j}^{2} } )^{1/2} \), where, \( z_{1i} \), \( 1 \le i \le T \) are \( iid \sim N\left( {0,1} \right) \). Generate \( \lambda_{1} \sim N\left( {0,1} \right) \) and set \( {\varvec{\Upsilon}}_{1} =\varvec{\varPhi}_{k} + \lambda_{1} \varvec{V}_{1} \). Compute

$$ \tau_{k + 1} = { \log }\left\{ {\pi \left( {{\varvec{\Upsilon}}_{1} |\varvec{W}_{k + 1} } \right)} \right\} - { \log }\left\{ {\pi \left( {\varvec{\varPhi}_{k} |\varvec{W}_{k + 1} } \right)} \right\} $$

Simulate \( u_{1} \sim {\text{Unif}}\left( {0,1} \right) \). If \( u_{1} \le { \hbox{min} }\left( {1,{ \exp }\left( {\tau_{k + 1} } \right)} \right) \)., let \( \varvec{\varPhi}_{k + 1} = {\varvec{\Upsilon}}_{1} \). Otherwise, let \( \varvec{\varPhi}_{k + 1} =\varvec{\varPhi}_{k} \).

Step 3 Decompose \( {\varvec{\Sigma}}_{k} = \varvec{ODO^{\prime}} \), where, \( \varvec{D} = {\text{diag}}( {d_{1} , \ldots ,d_{T} }) \), \( d_{1} \ge d_{2} \ge \ldots \ge d_{T} \), and \( \varvec{OO^{\prime}} = \varvec{I} \). Let \( d_{i}^{*} = { \log }\left( {d_{i} } \right) \), \( \varvec{D}^{*} = {\text{diag}}\left( {d_{1}^{*} , \ldots ,d_{T}^{*} } \right) \) and \( {\varvec{\Sigma}}_{k}^{*} = \varvec{OD}^{*} \varvec{O^{\prime}}. \)

Select a random symmetric \( T \times T \) matrix \( \varvec{V}_{2} \) with elements \( v_{2ij} = z_{2ij} /( {\mathop \sum \nolimits_{l \le m} z_{2lm}^{2} } )^{1/2} \), where, \( z_{2ij} \), \( 1 \le i \le j \le T \times \left( {T + 1} \right)/2 \), are \( iid \sim N\left( {0,1} \right) \). (the other elements of \( \varvec{V}_{2} \) are defined by symmetry).

Generate \( \lambda_{2} \sim N\left( {0,1} \right) \) and set \( {\varvec{\Upsilon}}_{2} = {\varvec{\Sigma}}_{k}^{*} + \lambda_{2} \varvec{V}_{2} \). Decompose \( {\varvec{\Upsilon}}_{2} = \varvec{QC}^{*} \varvec{Q^{\prime}} \), where, \( \varvec{C}^{*} = diag( {c_{1}^{*} , \ldots ,c_{T}^{*} }) \), \( c_{1}^{*} \ge c_{2}^{*} \ge \ldots \ge c_{T}^{*} \), and \( \varvec{QQ^{\prime} } = \varvec{I} \). Compute

$$ \tau_{k + 1} = { \log }\left\{ {\pi \left( {{\varvec{\Upsilon}}_{2} |\varvec{W}_{k + 1} ,\varvec{\varPhi}_{k + 1} } \right)} \right\} - { \log }\left\{ {\pi \left( {{\varvec{\Sigma}}_{k}^{*} |\varvec{W}_{k + 1} ,\varvec{\varPhi}_{k + 1} } \right)} \right\} $$

Simulate \( u_{2} \sim {\text{Unif}}\left( {0,1} \right) \). If \( u_{2} \le { \hbox{min} }\left( {1,exp\left( {\tau_{k + 1} } \right)} \right) \), let \( {\varvec{\Sigma}}_{k + 1}^{*} = {\varvec{\Upsilon}}_{2} \), \( \varvec{C} = {\text{diag}}\left( {e^{{c_{1}^{*} }} , \ldots ,e^{{c_{T}^{*} }} } \right) \) and \( {\varvec{\Sigma}}_{k + 1} = \varvec{QCQ^{\prime}} \). Otherwise, let \( {\varvec{\Sigma}}_{k + 1}^{*} = {\varvec{\Sigma}}_{k}^{*} \) and \( {\varvec{\Sigma}}_{k + 1} = {\varvec{\Sigma}}_{k} \).

Step 4 Select a \( T \)-dimention random vector \( \varvec{V}_{3} \) with elements \( v_{3i} = z_{3i} /( {\mathop \sum \limits_{j} z_{3j}^{2} } )^{1/2} \), where, \( z_{3i} \), \( 1 \le i \le T \) are \( iid \sim N\left( {0,1} \right) \). Generate \( \lambda_{3} \sim N\left( {0,1} \right) \) and set \( {\varvec{\Upsilon}}_{3} =\varvec{\alpha}_{k} + \lambda_{3} \varvec{V}_{3} \). Compute

$$ \tau_{k + 1} = { \log }\left\{ {\pi \left( {{\varvec{\Upsilon}}_{3} |\left( {\varvec{W}_{k + 1} ,{\varvec{\Sigma}}_{k + 1} ,\varvec{\varPhi}_{k + 1} } \right)} \right)} \right\} - { \log }\left\{ {\pi \left( {\varvec{\alpha}_{k} |\left( {\varvec{W}_{k + 1} ,{\varvec{\Sigma}}_{k + 1} ,\varvec{\varPhi}_{k + 1} } \right)} \right)} \right\}. $$

Simulate \( u_{3} \sim {\text{Unif}}\left( {0,1} \right) \). If \( u_{3} \le { \hbox{min} }\left( {1,exp\left( {\tau_{k + 1} } \right)} \right) \), let \( \varvec{\alpha}_{k + 1} = {\varvec{\Upsilon}}_{3} \). Otherwise, let \( \varvec{\alpha}_{k + 1} =\varvec{\alpha}_{k} \).

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Manouchehri, T., Nematollahi, A.R. Periodic autoregressive models with closed skew-normal innovations. Comput Stat 34, 1183–1213 (2019). https://doi.org/10.1007/s00180-019-00893-z

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