A Bayesian Multinomial Probit MODEL FOR THE ANALYSIS OF PANEL CHOICE DATA

Abstract

A new Bayesian multinomial probit model is proposed for the analysis of panel choice data. Using a parameter expansion technique, we are able to devise a Markov Chain Monte Carlo algorithm to compute our Bayesian estimates efficiently. We also show that the proposed procedure enables the estimation of individual level coefficients for the single-period multinomial probit model even when the available prior information is vague. We apply our new procedure to consumer purchase data and reanalyze a well-known scanner panel dataset that reveals new substantive insights. In addition, we delineate a number of advantageous features of our proposed procedure over several benchmark models. Finally, through a simulation analysis employing a fractional factorial design, we demonstrate that the results from our proposed model are quite robust with respect to differing factors across various conditions.

Appendices

Appendix 1: Derivation of the Full Conditional Distributions for the Proposed Model

1.1 Proof of Equation (6)

Since $\varvec{\beta }_{h}$ now denotes the reduced $(k-1)$-dimensional vector of parameters, Equation (2) becomes

$$\begin{aligned} {{\mathbf {y}}}_{ht} =\tilde{{\mathbf {X}}}_{ht} \varvec{\beta }_{h} +\varvec{\varepsilon }_{ht} , \end{aligned}$$

where $\tilde{{\mathbf {X}}}_{ht}$ is the reduced matrix of ${{\mathbf {X}}}_{ht}$ by deleting its first column. Then, the full conditional distribution of ${{\mathbf {y}}}_{ht}$ is

$$\begin{aligned} \pi ({{\mathbf {y}}}_{ht} |\hbox {all}\,\,\hbox {others})\propto \mathrm{{exp}}\left\{ -\frac{1}{2}({{\mathbf {y}}}_{ht} -\tilde{{\mathbf {X}}}_{ht} \varvec{\beta }_{h} )^{{\prime }}\varvec{R}^{-1}({{\mathbf {y}}}_{ht} -\tilde{{\mathbf {X}}}_{ht} \varvec{\beta }_{h} )\right\} I(\varvec{y}_{ht[I_{ht} ]} >\mathrm{{max}}(\varvec{y}_{ht[-I_{ht} ]} )). \end{aligned}$$

So $({{\mathbf {y}}}_{ht} |\hbox {all}\,\,\hbox {others})$ follows a truncated normal distribution, $TN\left( {\tilde{{\mathbf {X}}}_{ht} \varvec{\beta }_{h} ,\varvec{R}} \right) ,$where the truncation is such that $\varvec{y}_{ht[I_{ht} ]} >\mathrm{{max}}(\varvec{y}_{ht[-I_{ht} ]} )$. The $j$th component of ${{\mathbf {y}}}_{ht} ,\varvec{y}_{ht[j]}$, has a univariate truncated normal distribution conditional on all other components of ${{\mathbf {y}}}_{ht}$ and other parameters. Let $\varvec{D}_j$ be a matrix that switches the first and the $j$th components of ${{\mathbf {y}}}_{ht}$:

$$\begin{aligned} \varvec{D}_j \tilde{{\mathbf {X}}}_{ht} \varvec{\beta }_{h} \mathop {=}\limits ^{\Delta } \left( {{\begin{array}{l} {\mu _j } \\ {\mu _{-j} } \\ \end{array} }} \right) , \hbox {and}\,\,\varvec{D}_j \varvec{RD}_{j^{{\prime }}} \mathop {=}\limits ^{\Delta } \left( {{\begin{array}{ll} {\hbox {r}_{\mathrm{jj}} }&{} {r_{j12} } \\ {r_{j21} }&{} {r_{j22} } \\ \end{array} }} \right) . \end{aligned}$$

Then,

$$\begin{aligned} \varvec{y}_{ht[j]} |\hbox {all}\,\,\hbox {others}\sim TN(\mu _j +r_{j12} r_{j22}^{-1} (\varvec{y}_{ht[-j]} -\varvec{\mu }_{-j} ),r_{jj} -r_{j12} r_{j22}^{-1} r_{j21} ), \end{aligned}$$

where the truncation is given by

$$\begin{aligned} \left\{ {{\begin{array}{l} {\varvec{y}_{htj} \in \left( {\hbox {max}\left( {\varvec{y}_{ht\left[ {-j} \right] } } \right) ,+\infty } \right) ,\quad \hbox {if}\quad j =I_{ht} ;} \\ {\varvec{y}_{htj} \in \left( {-\infty ,y_{ht\left[ {I_{ht} } \right] } } \right) ,\quad \hbox {if}\quad j\ne I_{ht} .} \\ \end{array} }} \right. \end{aligned}$$

1.2 Proof of Equation (7)

The full conditional distribution of $\varvec{\beta }_{h}$ is

$$\begin{aligned}&\!\! {\pi (\varvec{\beta }_{h} |\hbox {all}\,\,\hbox {others})} \\&\!\!\propto {\mathrm{{exp}}\left\{ -\frac{1}{2} \mathop \sum \limits _{t=1}^{T_h } \left[ ({{\mathbf {y}}}_{ht} -\tilde{{\mathbf {X}}}_{ht} \varvec{\beta }_{h} )^{{\prime }}\varvec{R}^{-1}({{\mathbf {y}}}_{ht} -\tilde{{\mathbf {X}}}_{ht} \varvec{\beta }_{h} )\right] -\frac{1}{2}(\varvec{\beta }_{h} -\varvec{\Delta }\varvec{Z}_h )^{{\prime }}\varvec{V}_\beta ^{-1} (\varvec{\beta }_{h} -\varvec{\Delta }\varvec{Z}_h )\right\} }\\&\!\!\propto {\mathrm{{exp}}\left\{ -\frac{1}{2}\left[ \varvec{\beta }_{h}^{\prime } \,\, \left( \mathop \sum \limits _t (\tilde{{\mathbf {X}}}_{ht} ^{{\prime }}\varvec{R}^{-1}\tilde{{\mathbf {X}}}_{ht} )+\varvec{V}_\beta ^{-1} \right) \varvec{\beta }_{h} -2\,\,\varvec{\beta }_{h}^{\prime } \left( \mathop \sum \limits _t (\tilde{{\mathbf {X}}}_{ht} ^{{\prime }}\varvec{R}^{-1}{{\mathbf {y}}}_{ht} )+\varvec{V}_\beta ^{-1} \varvec{\Delta }\varvec{Z}_h \right) \right] \right\} .} \end{aligned}$$

So, $\varvec{\beta }_{h} |\hbox {all}\,\,\hbox {others}\sim N(\varvec{\beta }_{h}^0 ,\varvec{V}_\beta ^h )$, where

$$\begin{aligned} \varvec{V}_\beta ^h =\left( \mathop \sum \nolimits _t(\tilde{{\mathbf {X}}} _{ht} ^{{\prime }}\varvec{R}^{-1}\tilde{{\mathbf {X}}}_{ht} )+\varvec{V}_\beta ^{-1} \right) ^{-1}\,\,\hbox {and}\,\,\varvec{\beta }_{h}^0 =\varvec{V}_\beta ^h \left( \mathop \sum \nolimits _t (\tilde{{\mathbf {X}}}_{ht} ^{{\prime }}\varvec{R}^{-1}{{\mathbf {y}}}_{ht} )+\varvec{V}_\beta ^{-1} \varvec{\Delta }\varvec{Z}_h \right) . \end{aligned}$$

1.3 Proof of Equation (8)

The full conditional distribution of $\varvec{\Delta }$ is

$$\begin{aligned}&\pi (\varvec{\Delta }|\hbox {all}\,\,\hbox {others})\propto {\mathrm{{etr}}\left\{ -\frac{1}{2}\varvec{V}_\beta ^{-1} [(\varvec{B}-\varvec{\Delta }\varvec{Z})(\varvec{B}- \varvec{\Delta }\varvec{Z})^{{\prime }}+(\varvec{\Delta }-\varvec{\Delta }_0 )\varvec{A}_d (\varvec{\Delta }-\varvec{\Delta }_0 )^{{\prime }}]\right\} }\\&{\propto } \,\, {\mathrm{{etr}}\{\varvec{V}_\beta ^{-1} [\varvec{\Delta }(\varvec{ZZ}^{{\prime }}+\varvec{A}_d )\varvec{\Delta }^{{\prime }}-2\varvec{\Delta }(\varvec{ZB}^{{\prime }}+\varvec{A}_d \varvec{\Delta }_0^{\prime } )]\},\,\,} \end{aligned}$$

here etr refers to an exponential function of the trace of (a matrix).

So, $\varvec{\Delta }|\hbox {all}\,\,\hbox {others}\sim MN((\varvec{BZ}^{{\prime }}+\varvec{\Delta }_0 \varvec{A}_d )(\varvec{ZZ}^{{\prime }}+\varvec{A}_d )^{-1},\varvec{V}_\beta ,(\varvec{ZZ}^{{\prime }}+\varvec{A}_d )^{-1})$, where

$$\begin{aligned} \varvec{B}=[\varvec{\beta } _1 ,\ldots ,\varvec{\beta } _H ] \quad \hbox {and}\quad \varvec{Z}=[\varvec{Z}_1 ,\ldots ,\varvec{Z}_H ]. \end{aligned}$$

1.4 Proof of Equation (9)

The full conditional distribution of $\varvec{V}_\beta ^{-1}$ is

$$\begin{aligned}&{\pi (\varvec{V}_\beta ^{-1} |\hbox {all}\,\,\hbox {others})\propto |\varvec{V}_\beta ^{-1} |^{H/2}\mathrm{{etr}}\left\{ -\frac{1}{2}\varvec{V}_\beta ^{-1} (\varvec{B}-\varvec{\Delta }\varvec{Z}) (\varvec{B}-\varvec{\Delta }\varvec{Z})^{{\prime }}\right\} } \\&{\times |\varvec{V}_\beta ^{-1} |^{l/2}\mathrm{{etr}}\left\{ -\frac{1}{2}\varvec{V}_\beta ^{-1} (\varvec{\Delta }-\varvec{\Delta }_0 )\varvec{A}_d (\varvec{\Delta }-\varvec{\Delta }_0 )^{{\prime }}\right\} |\varvec{V}_\beta ^{-1} |^{\frac{v-k}{2}}\mathrm{{etr}}\left\{ -\frac{1}{2}\varvec{V}_\beta ^{-1} \varvec{V}^{-1}\right\} } \end{aligned}$$

which is proportional to

$$\begin{aligned} |\varvec{V}_\beta ^{-1} |^{\frac{v+H+l-k}{2}}\mathrm{{etr}}\left\{ -\frac{1}{2}\varvec{V}_\beta ^{-1} [(\varvec{B}-\varvec{\Delta }\varvec{Z})(\varvec{B}- \varvec{\Delta }\varvec{Z})^{{\prime }}+(\varvec{\Delta }- \varvec{\Delta }_0 )\varvec{A}_d (\varvec{\Delta }-\varvec{\Delta }_0 )^{{\prime }}+\varvec{V}^{-1}]\right\} . \end{aligned}$$

So, $\varvec{V}_\beta ^{-1} |\hbox {all}\,\,\hbox {others}\sim W(v+H+l,\varvec{V}_n )$, where

$$\begin{aligned} \varvec{V}_n =[(\varvec{B}-\varvec{\Delta }\varvec{Z})(\varvec{B}- \varvec{\Delta }\varvec{Z})^{{\prime }}+(\varvec{\Delta }-\varvec{\Delta }_0 )\varvec{A}_d (\varvec{\Delta }-\varvec{\Delta }_0 )^{{\prime }}+\varvec{V}^{-1}]^{-1}. \end{aligned}$$

Appendix 2: A Parameter Expansion Algorithm for Sampling the Correlation Matrix

1.1 Stage I: Parameter Expanded Reparameterization

We define the following one-to-one mapping from $\left\{ {{{\mathbf {y}}}_{ht} ,\varvec{R}} \right\}$ to $\left\{ {{{\mathbf {y}}}_{ht}^*,\varvec{\varSigma } } \right\}$ to facilitate making random draws of the correlation matrix:

$$\begin{aligned} \left\{ {{\begin{array}{l} {{{\mathbf {y}}}_{ht} =\tilde{{\mathbf {X}}} _{ht} \varvec{\beta }_{h} +\varvec{D}^{-1/2}{{\mathbf {y}}}_{ht}^*} \\ {\varvec{R}=\varvec{D}^{-1/2}\varvec{\varSigma } \varvec{D}^{-1/2}} \\ \end{array} }} \right. \quad h=1,\ldots ,\hbox {H}\quad \hbox {and} \quad t=1,\ldots ,T_h \end{aligned}$$

(20)

where $\varvec{\Sigma }$ is a positive definite matrix, $\mathop \sum \nolimits _{h=1}^H \,\,\mathop \sum \nolimits _{t=1}^{T_h } (\varvec{y}_{ht[j]}^{*} )^{2}=1\hbox {for}\,\,\hbox {any}\,\,j=1,...,m$, and $\varvec{D}$ is a diagonal matrix. Note that the constraints, $\mathop \sum \nolimits _{h=1}^H\,\,\mathop \sum \nolimits _{t=1}^{T_h } (\varvec{y}_{ht[j]}^{*} )^{2}=1\hbox {for}\,\,\hbox {any}\,\,j=1,...,m$, are needed to make the transformation in (20) a one-to-one mapping. Given $\varvec{\beta }_{h} ,$ the step that draws ${{\mathbf {y}}}_{ht}$ implicitly draws ${{\mathbf {y}}}_{ht}^{*}$ and $\varvec{D}$ because $\varvec{D}_{jj}^{1/2} =[\mathop \sum \nolimits _{h=1}^H \mathop \sum \nolimits _{t=1}^{T_h }(\varvec{y}_{ht[j]} -\tilde{{\mathbf {X}}}_{ht[j]} \varvec{\beta }_{h} )^{2}]^{-1/2}$, where $\varvec{D}_{jj}$ is the jth element of $\varvec{D}$ and $\tilde{{\mathbf {X}}}_{ht[j]}$ is the jth row of $\tilde{{\mathbf {X}}}_{ht}$. Thus, one can derive the joint conditional distribution of (${{\mathbf {y}}}_{ht}^*,\varvec{\varSigma } |\hbox {all}\,\,\hbox {others})$ from that of (${{\mathbf {y}}}_{ht} ,\varvec{R}|\hbox {all}\,\,\hbox {others})$. When the prior distribution of $\varvec{R}$ is $\pi (\varvec{R})\propto |\varvec{R}|^{ -(m+1)/2}$, the full conditional distribution of $\varvec{\Sigma }$ is

$$\begin{aligned} \pi \left( {\varvec{\Sigma }|\hbox {all}\,\,\hbox {others}} \right) \propto \mathop \prod \nolimits _{ht} \pi \,\,\, ({{\mathbf {y}}}_{ht}^*,\varvec{\varSigma } |\hbox {all}\,\,\hbox {others})\propto |\varvec{\Sigma }|^{-(\mathop \sum \nolimits _h T_h +m+1)/2}\mathrm{{exp}}\left\{ -\frac{1}{2}tr(\varvec{S}\varvec{\Sigma }^{-1})\right\} \nonumber \\ \end{aligned}$$

(21)

where $\varvec{S}=\mathop \sum \nolimits _{h=1}^H \mathop \sum \nolimits _{t=1}^{T_h } {{\mathbf {y}}}_{ht}^*{{\mathbf {y}}}_{ht} ^{*{\prime }}$. The expression in (21) is proportional to a Wishart distribution and so $\varvec{\Sigma }^{-1}|\hbox {all}\,\,\hbox {others}\sim \,\,W(\mathop \sum \nolimits _h \varvec{T}_h ,\,\,\varvec{S})$.

1.2 Stage II: Parameter Expanded Metropolis–Hastings

After we have obtained a random deviate of $\varvec{\Sigma }$ from (21) Stage I, we can transform it into a correlation matrix using $\varvec{R}^{{*}}=\varvec{D}^{-1/2}\varvec{\Sigma }\varvec{D}^{-1/2}$. Since $\varvec{R}^{{*}}$ is obtained based on the candidate prior $\pi (\varvec{R}^{*})\propto |\varvec{R}^{*}|^{-(m+1)/2}$, it is accepted in a Metropolis–Hasting step with probability $\upalpha$, where, at iteration n+1, $\alpha =min\{1,\mathrm{{exp}}[\frac{m+1}{2}(log|\varvec{R}^{{*}}|-log|\varvec{R}^{(n)}|)]\}$.

Appendix 3

1.1 Proof of Theorem 1

Since the joint posterior distribution is

$$\begin{aligned} \pi (\varvec{Y},\varvec{B},\varvec{\Delta },\varvec{R}, \varvec{V}_{\varvec{\beta }} |\varvec{I},\varvec{X},\varvec{Z})\!&= \! {\left[ \mathop \prod \nolimits _{h=1}^H \pi (\varvec{\beta }_{\varvec{h}} |\varvec{I},\varvec{X},\varvec{Z},\varvec{y}_{\varvec{h}} ,\varvec{\Delta },\varvec{R},\varvec{V}_{\varvec{\beta }} )\right] \pi (\varvec{\Delta }|\varvec{I},\varvec{X},\varvec{Z},\varvec{Y},\varvec{R},\varvec{V}_{\varvec{\beta }} )}\\&\,\,\times \,\, \pi \left( {\varvec{Y}|\varvec{I},\varvec{X},\varvec{Z},\varvec{R},\varvec{V}_{\varvec{\beta }} } \right) \pi \left( {\varvec{R}|\varvec{I},\varvec{X},\varvec{Z},\varvec{V}_{\varvec{\beta }} } \right) \pi \left( {\varvec{V}_{\varvec{\beta }} | \varvec{I},\varvec{X},\varvec{Z}} \right) , \end{aligned}$$

it is a proper probability distribution if each of the posterior distributions on the right-hand side of the above equation is proper. To establish the result, we make use of the likelihood function of our model when $T_h =1$:

$$\begin{aligned}&L(\varvec{Y,B,\Delta ,R,V}_{\varvec{\beta }} |\varvec{I,X,Z})\propto \mathop \prod \limits _{h=1}^H \,\,[I(\varvec{y}_{h[I_h ]} >\mathrm{{max}}(\varvec{y}_{h[-I_h ]} ))]|\varvec{R}|^{-{\mathrm{H}}/2}|\varvec{V}_\beta |^{-{\mathrm{H}}/2}\\&\times \,\,\mathrm{{exp}}\left\{ -\frac{1}{2}\mathop \sum \limits _{h=1}^H \,\,((\varvec{y}_h -\tilde{{\mathbf {X}}}_h \varvec{\beta }_{h} )^{{\prime }}\varvec{R}^{-1}(\varvec{y}_h -\tilde{{\mathbf {X}}}_h \varvec{\beta }_{h} )+(\varvec{\beta }_{h} -\varvec{\Delta }\varvec{Z}_h )^{{\prime }}\varvec{V}_\beta ^{-1} (\varvec{\beta }_{h} -\varvec{\Delta }\varvec{Z}_h ))\right\} , \end{aligned}$$

where $\varvec{X}=[\tilde{{\mathbf {X}}}_1 ,\ldots ,\tilde{{\mathbf {X}}}_H ]$ and $\varvec{Y}=[\varvec{y}_1 ^{\prime },\ldots ,\varvec{y}_H ^{{\prime }}]^{{\prime }}$.

• For $h=1,\ldots ,H$,

$$\begin{aligned}&{\pi (\varvec{\beta }_{h} |\varvec{I,X,Z,Y,\Delta ,R},\varvec{V}_\beta )=\pi (\varvec{\beta }_{h} |\varvec{X,Z,Y,\Delta ,R},\varvec{V}_\beta )}\\&\propto {exp\left\{ -\frac{1}{2}[(\varvec{y}_h -\tilde{{\mathbf {X}}}_h \varvec{\beta }_{h} )^{{\prime }}\varvec{R}^{-1}(\varvec{y}_h -\tilde{{\mathbf {X}}}_h \varvec{\beta }_{h} )+(\varvec{\beta }_{h} -\varvec{\Delta }\varvec{Z}_h )^{{\prime }}\varvec{V}_\beta ^{-1} (\varvec{\beta }_{h} -\varvec{\Delta }\varvec{Z}_h )]\right\} .} \end{aligned}$$

So, $\varvec{\beta }_{h} |\varvec{I,X,Z,Y,\Delta ,R},\varvec{V}_\beta \sim N(\varvec{b}_h^0 ,\varvec{V}_\beta ^{\hbox {new}} )$, where

$$\begin{aligned} \varvec{V}_\beta ^{\hbox {new}} =(\tilde{{\mathbf {X}}}_h^{\prime }\varvec{R}^{-1}\tilde{{\mathbf {X}}}_h +\varvec{V}_\beta ^{-1} )^{-1}\hbox {and} \quad \varvec{b}_h^0 =\varvec{V}_\beta ^{\hbox {new}} (\tilde{{\mathbf {X}}}_h^{\prime }\varvec{R}^{-1}\varvec{y}_h +\varvec{V}_\beta ^{-1} \varvec{\Delta }\varvec{Z}_h ). \end{aligned}$$

Thus, $\pi (\varvec{\beta }_{h} |\varvec{I,X,Z,Y,\Delta ,R,}\varvec{V}_\beta )$ is proper.

• Let $\varvec{\eta }$ be the vectorization of $\varvec{\Delta }$, and $\varvec{\Upsilon }_h = \tilde{{\mathbf {X}}}_h (\varvec{E}_{k-1} \otimes \varvec{Z}_h )^{{\prime }}$, where $\varvec{E}_{k-1}$ is the identity matrix with dimension $(k-1)\times (k-1)$ and $\otimes$ denotes the Kronecker product. $\pi (\varvec{\Delta }|\varvec{I,X,Z,Y,R,}\varvec{V}_\beta )$ is equivalent to $\pi (\varvec{\eta } |\varvec{I,X,Z,Y,R,}\varvec{V}_\beta )$, where

$$\begin{aligned}&\pi (\varvec{\eta } |\varvec{I,X,Z,Y,R},\varvec{V}_\beta )=\pi (\varvec{\eta } |\varvec{X,Z,Y,R,}\varvec{V}_{\varvec{\beta }} )\propto L(\varvec{\eta } |\varvec{X,Z,Y,R,}\varvec{V}_\beta )\pi (\varvec{\eta } |\varvec{V}_{\varvec{\beta }} ) \\&\propto {\mathrm{{exp}}\left\{ -\frac{1}{2}\mathop \sum \nolimits _{h=1}^H \,\,[(\varvec{y}_h -\varvec{\Upsilon }_h \varvec{\eta } )^{{\prime }}(\tilde{{\mathbf {X}}}_h \varvec{V}_{\varvec{\beta }} \tilde{{\mathbf {X}}}_h ^{{\prime }}+\varvec{R})^{-1}(\varvec{y}_h -\varvec{\Upsilon }_h \varvec{\eta } )]\right\} } \\&\,\,{\times \,\, \mathrm{{exp}}\left\{ -\frac{1}{2}(\varvec{\eta } -\overline{\varvec{d}})^{{\prime }}(\varvec{V}_\beta \otimes \varvec{A}_d^{-1} )^{-1}(\varvec{\eta } -\overline{\varvec{d}} )\right\} } \\&\propto \mathrm{{exp}}\left\{ -\frac{1}{2}\left[ \varvec{\eta } ^{{\prime }}((\varvec{V}_\beta ^{-1} \otimes \varvec{A}_d )+\mathop \sum \nolimits _{h=1}^H \,\,[\varvec{\Upsilon }_h ^{{\prime }}(\tilde{{\mathbf {X}}}_h \varvec{V}_\beta \tilde{{\mathbf {X}}}_h ^{{\prime }}+\varvec{R})^{-1}\varvec{\Upsilon }_h ])\varvec{\eta }\right. \right. \\&\left. \left. -2\varvec{\eta } ^{{\prime }}\left( \mathop \sum \nolimits _{h=1}^H \,\,[\varvec{\Upsilon }_h ^{{\prime }}(\tilde{{\mathbf {X}}}_h \varvec{V}_\beta \tilde{{\mathbf {X}}}_h ^{{\prime }}+\varvec{R})^{-1}\varvec{y}_h ]+(\varvec{V}_\beta ^{-1} \otimes \varvec{A}_d )\overline{\varvec{d}} \right) \right] \right\} . \end{aligned}$$

Typically, one lets $\varvec{A}_d$ in the prior distribution to approach a zero matrix to represent vague information. Then, in the limit, the above expression goes to

$$\begin{aligned} \mathrm{{exp}}\{\frac{-1}{2}[\varvec{\eta } ^{{\prime }}(\mathop \sum \nolimits _{h=1}^H \,\,[\varvec{\Upsilon }_h ^{{\prime }}(\tilde{{\mathbf {X}}}_h \varvec{V}_\beta \tilde{{\mathbf {X}}}_h ^{{\prime }}+\varvec{R})^{-1}\varvec{\Upsilon }_h ])\varvec{\eta } -2\varvec{\eta }^{{\prime }}(\mathop \sum \nolimits _{h=1}^H \,\,[\varvec{\Upsilon }_h ^{{\prime }}(\tilde{{\mathbf {X}}}_h \varvec{V}_\beta \tilde{{\mathbf {X}}}_h ^{{\prime }}+\varvec{R})^{-1}\varvec{y}_h ])]\}. \end{aligned}$$

Thus, $\varvec{\eta }|\varvec{I,X,Z,Y,R,}\varvec{V}_\beta \sim N(\varvec{d}_0 ,\varvec{V}_\eta )$, where

$$\begin{aligned} \varvec{V}_\eta&= \left[ \mathop \sum \nolimits _{h=1}^H \,\,(\varvec{\Upsilon }_h ^{{\prime }}(\tilde{{\mathbf {X}}}_h \varvec{V}_\beta \tilde{{\mathbf {X}}}_h ^{{\prime }}+\varvec{R})^{-1}\varvec{\Upsilon }_h )\right] ^{-1}\\&\quad \hbox {and}\quad \varvec{d}_0 =\varvec{V}_\eta \mathop \sum \nolimits _{h=1}^H \,\,(\varvec{\Upsilon }_h ^{{\prime }}(\tilde{{\mathbf {X}}}_h \varvec{V}_\beta \tilde{{\mathbf {X}}}_h ^{{\prime }}+\varvec{R})^{-1}\varvec{y}_h ). \end{aligned}$$

So, the posterior distribution of $\varvec{\eta } |\varvec{I,X,Z,Y,R,}\varvec{V}_\beta$ is proper.

• Let

$$\begin{aligned} \varvec{\Upsilon }=\left( {{\begin{array}{l} {\varvec{\Upsilon }_1 } \\ \vdots \\ {\varvec{\Upsilon }_H } \\ \end{array} }} \right) \quad \hbox {and}\quad \varvec{\Psi }=\left( {{\begin{array}{lll} {(\tilde{{\mathbf {X}}}_1 \varvec{V}_\beta \tilde{{\mathbf {X}}}_1 ^{{\prime }}+\varvec{R})^{-1}} &{} &{} \\ &{} \ddots &{} \\ &{} &{} {(\tilde{{\mathbf {X}}}_H \varvec{V}_\beta \tilde{{\mathbf {X}}}_H ^{{\prime }}+\varvec{R})^{-1}} \\ \end{array} }} \right) . \end{aligned}$$

Since

$$\begin{aligned}&{\pi (\varvec{Y}|\varvec{I},\varvec{X},\varvec{Z},\varvec{R},\varvec{V}_\beta )}\\&{\propto \mathrm{{exp}}\left\{ -\frac{1}{2}\varvec{Y}^{{\prime }}(\varvec{\Psi }-\varvec{ \Psi \Upsilon }(\varvec{\Upsilon }^{\prime }\varvec{\Psi \Upsilon })^{-1}\varvec{\Upsilon }^{\prime }\varvec{\Psi })\varvec{Y}\right\} \mathop \prod \nolimits _h \,\,[I\{\varvec{y}_{h[I_h ]} >\mathrm{{max}}(\varvec{y}_{h[-I_h ]} )\}].}\\ \end{aligned}$$

So, $\varvec{Y}|\varvec{I,X,Z,R,}\varvec{V}_\beta \sim \,\,TN(0,(\varvec{\Psi }-\varvec{\Psi \Upsilon }(\varvec{\Upsilon }^{\prime }\varvec{\Psi \Upsilon })^{-1}\varvec{\Upsilon }^{\prime }\varvec{\Psi })^{-1})$ with truncation restriction on $I\{\varvec{y}_{h[I_h ]} >\varvec{y}_{h[-I_h ]} \}$ for any $h=1,\ldots ,H.$ Thus, the posterior distribution of $\varvec{Y}|\varvec{I,X,Z,R,}\varvec{V}_\beta$ is proper.

• Since the space of $m$-dimensional correlation matrix is convex and compact (Rousseeuw & Molenberghs, 1994), the prior distribution we specify on $\varvec{R}$ is a proper distribution. Thus, the conditional posterior distribution $\pi (\varvec{R}|\varvec{I,X,Z,}\varvec{V}_\beta )$ is also proper.

• Similar to the argument given above, because the prior distribution on $\varvec{V}_\beta ^{-1}$ is Wishart which is a proper distribution, the conditional distribution of $\varvec{V}_\beta |\varvec{I,X,Z}$ is proper.

Appendix 4

1.1 Proof of Theorem 2

When $T_h =1,$ the joint posterior distribution is

$$\begin{aligned} \pi (\varvec{Y,B,}\varvec{\Delta },\varvec{\Lambda },\varvec{V}_{\varvec{\beta }} |\varvec{I,X,Z})\!&= \! \left[ \!\mathop \prod \nolimits _{h=1}^H \,\,\pi (\varvec{\beta }_{h} |\varvec{I,X,Z,}\varvec{y}_{\varvec{h}} ,\varvec{\Delta },\varvec{\Lambda },\varvec{V}_{\varvec{\beta }} )\!\right] \pi (\varvec{\Delta }|\varvec{I,X,Z,Y},\varvec{\Lambda },\varvec{V}_{\varvec{\beta }}) \\&\,\, \times \pi \left( {\varvec{\Lambda }\hbox {|}\varvec{I,X,Z,Y,}\varvec{V}_{\varvec{\beta }} } \right) \pi \left( {\varvec{Y}\hbox {|}\varvec{I,X,Z},\varvec{V}_{\varvec{\beta }} } \right) \pi \left( {\varvec{V}_{\varvec{\beta }} \hbox {|}\varvec{I,X,Z}} \right) . \end{aligned}$$

We will show that the conditional distribution $\pi (\varvec{\Lambda }|\varvec{I,X,Z,Y,}\varvec{V}_\beta )$ is improper. Recall that $\varvec{\Lambda }=\mathrm{{diag}}(1,\sigma _2^2 ,\ldots ,\sigma _m^2 )$, and we let

$$\begin{aligned} \varvec{\Psi }^{{*}}=\left( {{\begin{array}{lll} {(\tilde{{\mathbf {X}}}_1 \varvec{V}_\beta \tilde{{\mathbf {X}}}_{1{\prime }} +\varvec{\Lambda })^{-1}} &{} &{}\\ &{} \ddots &{} \\ &{} &{} {(\tilde{{\mathbf {X}}}_H \varvec{V}_\beta \tilde{{\mathbf {X}}}_{H^{{\prime }}} +\varvec{\Lambda })^{-1}} \end{array} }} \right) . \end{aligned}$$

Since the conditional distribution is proportional to the prior of $\varvec{\Lambda }$ multiplied by the corresponding likelihood function, we have (see the proof of Theorem 1)

$$\begin{aligned}&{\pi (\varvec{\Lambda }|\varvec{I,X,Z,Y,}\varvec{V}_\beta )} \\&= \varvec{C}_1 \mathop \prod \limits _{j=2}^m \,\,\left[ (\sigma _j^{-2} )^{\nu -1}\mathrm{{exp}}\left\{ -\frac{\sigma _j^{-2} }{v_j }\right\} \right] \left( \frac{|\varvec{\Psi }^{{*}}|}{|\varvec{\Upsilon }^{{\prime }}\varvec{\Psi }^{{*}}\varvec{\Upsilon }|}\right) ^{\frac{1}{2}}\\&exp\left\{ -\frac{1}{2}\varvec{Y}^{\prime }(\varvec{\Psi }^{*}-\varvec{\Psi }^{*}\varvec{\Upsilon }(\varvec{\Upsilon }^{\prime }\varvec{\Psi }^{*}\varvec{\Upsilon })^{-1}\varvec{\Upsilon }^{\prime }\varvec{\Psi }^{*})\varvec{Y}\right\} , \end{aligned}$$

where $\varvec{C}_1$ is the normalizing constant. Again, we set the variance of the Gamma distribution to be large to represent vague information. So, in the limit as we let $\nu \rightarrow 0$ and $v_j \rightarrow \infty$, the above posterior distribution becomes

$$\begin{aligned}&{\pi (\varvec{\Lambda }|\varvec{I,X,Z,Y,}\varvec{V}_\beta )} \\&= {\varvec{C}_1 \left[ \mathop \prod \nolimits _{j=2}^m \,\,(\sigma _j^{-2} )^{-1}\right] \left( \frac{|\varvec{\Psi }^{*}|}{|\varvec{\Upsilon }^{\prime }\varvec{\Psi }^{*}\varvec{\Upsilon }|}\right) ^{\frac{1}{2}}\mathrm{{exp}}\left\{ -\frac{1}{2}\varvec{Y}^{\prime }(\varvec{\Psi }^{*}-\varvec{\Psi }^{*}\varvec{\Upsilon }(\varvec{\Upsilon }^{\prime }\varvec{\Psi }^{*}\varvec{\Upsilon })^{-1}\varvec{\Upsilon }^{\prime }\varvec{\Psi }^{*})\varvec{Y}\right\} .} \end{aligned}$$

Let

$$\begin{aligned} {f(\sigma ^{-2})}&= \mathop \prod \nolimits _{j=2}^m \,\,(\sigma _j^{-2} )^{-1} \\ {g(\sigma ^{-2})}&= \left( \frac{|\varvec{\Psi }^{*}|}{|\varvec{\Upsilon }^{\prime }\varvec{\Psi }^{*}\varvec{\Upsilon }|}\right) ^{\frac{1}{2}}\mathrm{{exp}}\left\{ -\frac{1}{2}\varvec{Y}^{\prime }(\varvec{\Psi }^{*}-\varvec{\Psi }^{*}\varvec{\Upsilon }(\varvec{\Upsilon }^{\prime }\varvec{\Psi }^{*}\varvec{\Upsilon })^{-1}\varvec{\Upsilon }^{\prime }\varvec{\Psi }^{*})\varvec{Y}\right\} . \end{aligned}$$

Then,

$$\begin{aligned}&\int \,\,\pi (\varvec{\Lambda }|\varvec{I,X,Z,Y,}\varvec{V}_\beta )\mathrm{{d}}\varvec{\Lambda }\\&= \int \nolimits _0^\infty \,\,\cdots \int \nolimits _0^\infty \,\,\pi (\sigma _2^{-2} ,\ldots ,\sigma _m^{-2} |\varvec{I,X,Z,Y,}\varvec{V}_\beta )\mathrm{{d}}\sigma _2^{-2} \ldots \mathrm{{d}}\sigma _m^{-2} \\&\ge \int \nolimits _w^\infty \,\,\ldots \int \nolimits _w^\infty \,\,\pi (\sigma _2^{-2} ,\ldots ,\sigma _m^{-2} |\varvec{I,X,Z,Y,}\varvec{V}_\beta )\mathrm{{d}}\sigma _2^{-2} \ldots \mathrm{{d}}\sigma _m^{-2} ,\,\,\quad (\forall w>0)\\&= \varvec{C}_1 \int \nolimits _w^\infty \,\,\ldots \int \nolimits _w^\infty \,\,f(\sigma ^{-2})g(\sigma ^{-2})\mathrm{{d}}\sigma _2^{-2} \ldots \mathrm{{d}}\sigma _m^{-2} . \end{aligned}$$

Let $F$ be the region defined by $\{\sigma _j^{-2} >w,\forall j=2,\ldots ,m\}$. It is obvious that $g(\sigma ^{-2})>0$ over $F$ with the only possible exception when $\sigma _j^{-2}$ approaches infinity. However, we show that $g(\sigma ^{-2})\ne 0$ in the limiting case. First, when $\sigma _j^{-2} \rightarrow \infty \,\,\left( {j=2,\ldots ,m} \right) , \quad (\tilde{{\mathbf {X}}}_j \varvec{V}_\beta \tilde{{\mathbf {X}}}_j ^{{\prime }}+\varvec{\Lambda })^{-1}\rightarrow (\tilde{{\mathbf {X}}}_j \varvec{V}_\beta \tilde{{\mathbf {X}}}_j ^{{\prime }}+\ddot{\varvec{E}})^{-1}$, where $\ddot{E}$ is a $m\times m$ zero matrix except that its $(1,1)$ component is $1$. Then,

$$\begin{aligned} \varvec{\Psi }^{*}\rightarrow \left( \begin{array}{lll} (\tilde{{\mathbf {X}}}_\mathbf{1} \varvec{V}_{\varvec{\beta }} \tilde{{\mathbf {X}}}_\mathbf{1} ^{\prime }+\ddot{\varvec{E}})^{-1} &{} {} \ddots &{} \\ &{} &{} (\tilde{{\mathbf {X}}}_{\varvec{H}} \varvec{V}_{\varvec{\beta }} \tilde{{\mathbf {X}}}_{\varvec{H}} ^{\prime }+\ddot{\varvec{E}})^{-1} \\ {} &{} {} &{} {} \\ \end{array}\right) \mathop {=}\limits ^{\Delta } \varvec{\Psi }_\infty ^{*} . \end{aligned}$$

Therefore,

$$\begin{aligned} g(\sigma ^{-2})\rightarrow \left( \frac{|\varvec{\Psi }_\infty ^{*} |}{|\varvec{\Upsilon }^{\prime }\varvec{\Psi }_\infty ^*\varvec{\Upsilon }|}\right) ^{\frac{1}{2}}\mathrm{{exp}}\left\{ -\frac{1}{2}\varvec{Y}^{\prime }(\varvec{\Psi }_\infty ^*-\varvec{\Psi }_\infty ^*\varvec{\Upsilon }(\varvec{\Upsilon }^{\prime }\varvec{\Psi }_\infty ^*\varvec{\Upsilon })^{-1}\varvec{\Upsilon }^{\prime }\varvec{\Psi }_\infty ^*)\varvec{Y}\right\} >0. \end{aligned}$$

Hence, min$_{\sigma ^{-2}\in F} g(\sigma ^{-2})=m_0$, where $0<m_0 <\infty$. Thus

$$\begin{aligned} \int \pi (\varvec{\Lambda }|\varvec{I,X,Z,Y,}\varvec{V}_\beta )d\varvec{\Lambda }&\ge {\varvec{C}_1 \int \nolimits _w^\infty \,\,\ldots \int \nolimits _w^\infty \,\,f(\sigma ^{-2})g(\sigma ^{-2})\mathrm{{d}}\sigma _2^{-2} \ldots \mathrm{{d}}\sigma _m^{-2} }\\&\ge {m_0 \varvec{C}_1 \int \nolimits _w^\infty \,\,\cdots \int \nolimits _w^\infty \,\,\left[ \mathop \prod \nolimits _{j=2}^m \,\,(\sigma _j^{-2} )^{-1}\right] \mathrm{{d}}\sigma _2^{-2} \ldots \mathrm{{d}}\sigma _m^{-2} }\\&= {m_0 \varvec{C}_1 \mathop \prod \nolimits _{j=2}^m \,\,[\hbox {ln}\sigma _j^{-2} |_w^\infty ]=\infty .} \end{aligned}$$

Because $\int \pi (\varvec{\Lambda }|\varvec{I,X,Z,Y,}\varvec{V}_\beta )d\varvec{\Lambda }$ does not exist, $\pi (\varvec{\Lambda }|\varvec{I,X,Z,Y,}\varvec{V}_\beta )$ is not a proper distribution, and so the joint posterior distribution is improper.