Analysis of housing price by means of STAR models with neighbourhood effects: a Bayesian approach

Abstract

In this paper, we extend the Bayesian methodology introduced by Beamonte et al. (Stat Modelling 8:285–311, 2008) for the estimation and comparison of spatio-temporal autoregressive models (STAR) with neighbourhood effects, providing a more general treatment that uses larger and denser nets for the number of spatial and temporal influential neighbours and continuous distributions for their smoothing weights. This new treatment also reduces the computational time and the RAM necessities of the estimation algorithm in Beamonte et al. (Stat Modelling 8:285–311, 2008). The procedure is illustrated by an application to the Zaragoza (Spain) real estate market, improving the goodness of fit and the outsampling behaviour of the model thanks to a more flexible estimation of the neighbourhood parameters.

Appendix

In this Appendix, the posterior distribution of the parameters of model given in Eq. (1) and the algorithm used to calculate it are briefly described. From now on, we use the following notation. Given two random variables, X and Y, we will designate the marginal and conditioned densities of X and Y|X by [X] and [Y|X], respectively.

As in Pace et al. (1998, 2000) and Beamonte et al. (2008), we specify n ₀ (max{m _Tnet,t , m _Snet,s } < n ₀ < n), the number of observations to which we condition the likelihood function of model given in Eq. (1) in order to diminish the influence of the oldest transactions in the estimation of the parameters.

Let ${\varvec{\uptheta}} = ({\varvec{\upalpha}}^{'} ,{\varvec{\upbeta}}^{'} ,{\varvec{\upphi}}^{'} ,\tau ,m_{T} ,\gamma ,m_{s} ,\lambda )^{'}$ be the model parameter vector and ${\varvec{\upphi}} = (\phi_{T} ,\phi_{S} ,\phi_{ST} ,\phi_{TS} )^{'}$. Let ${\mathbf{Y}}_{0} = (y_{1} , \ldots ,y_{{n_{0} }} )^{\prime }$and ${\mathbf{Y}}_{1} = (y_{{n_{0 + 1} }} , \ldots ,y_{n} )^{\prime }$

Applying the Bayes Theorem, the posterior distribution of the parameters in Eq. (1) is given by:

$$ \begin{aligned} [{\varvec{\uptheta}}|{\mathbf{Y}},{\mathbf{Z}},{\mathbf{X}}] \\ & \propto [{\mathbf{Y}}_{1} |{\varvec{\upalpha}},{\varvec{\upbeta}},{\varvec{\upphi}} ,\tau ,m_{T} ,\gamma ,m_{s} ,\lambda ,{\mathbf{Y}}_{0}, {\mathbf{Z}},{\mathbf{X}}][{\varvec{\upalpha}}][{\varvec{\upbeta}}][{\varvec{\upphi}} ][\tau ][m_{T} ][\gamma ][m_{s} ][\lambda ] \\ \propto \tau^{{{\frac{{(n - n_{0} )}}{2}}}} \exp \left[ { - {\frac{\tau }{2}}\user2{\varepsilon }_{1}^{\prime } ({\varvec{\uptheta}})\user2{\varepsilon }_{1} ({\varvec{\uptheta}})} \right] \\ & \times \exp \left[ { - \frac{1}{2}{\varvec{\upalpha}}^{\prime } \sum\nolimits_{\alpha }^{ - 1} {\varvec{\upalpha}} } \right]\exp \left[ { - \frac{1}{2}{\varvec{\upbeta}}^{\prime } \sum\nolimits_{\beta }^{ - 1} {\varvec{\upbeta}} } \right]\prod\limits_{{i \in \{ T,S,ST,TS\} }} {{\rm I}_{( - 1,1)} } (\phi_{i} ) \\ & \times \tau^{{{\frac{{d_{0} }}{2}} - 1 }}\exp \left[ { - {\frac{{d_{0} s_{0} }}{2}}\tau } \right]{\rm I}_{(0,\infty )} (\tau ){\rm I}_{{{\mathbf{m}}_{{\bf{Tnet}}} }} (m_{\text{T}} ){\rm I}_{{(\gamma_{\min } ,\gamma_{\max } )}} (\gamma ){\rm I}_{{{\mathbf{m}}_{\bf{Snet}} }} (m_{\text{S}} ){\rm I}_{{(\lambda_{\min } ,\lambda_{\max } )}} (\lambda ) \\ \end{aligned} $$

(4)

where I _A (x) = 1 if x ∈ A and 0 otherwise and ε ₁(θ) is the (n−n ₀)×1 vector that contains the last n−n ₀ components of the errors vector ε(θ), where:

$$ \begin{aligned} {\varvec{\varepsilon }}({\varvec{\uptheta}})\,=\,& {\varvec{\varepsilon }}({\varvec{\upalpha}},{\varvec{\upbeta}},{\varvec{\upphi}} ,\tau ,m_{T} ,\gamma ,m_{S} ,\lambda ) \\ = & {\mathbf{Y}} - {\mathbf{Z{\varvec{\upalpha}} }} - {\mathbf{X{\varvec{\upbeta}} }} - \phi_{T} {\mathbf{T}}({\mathbf{Y}} - {\mathbf{X{\varvec{\upbeta}} }}) - \phi_{S} {\mathbf{S}}({\mathbf{Y}} - {\mathbf{X{\varvec{\upbeta}} }}) - \phi_{ST} {\mathbf{ST}}({\mathbf{Y}} - {\mathbf{X{\varvec{\upbeta}} }}) - \phi_{TS} {\mathbf{TS}}({\mathbf{Y}} - {\mathbf{X{\varvec{\upbeta}} }}) \\ \end{aligned} $$

The algorithm used to draw a sample from Eq. (4) is based on Gibbs sampling and is similar to that described in Beamonte et al. (2008) with only two minor differences which affect the sampling of the neighbourhood parameters (m _T , γ, m _S , λ). To be precise,

• (m_T, m_S)|α, β, ϕ, τ, γ, λ, Y, Z, X is a discrete distribution with support m_Tnet × m_Snet and a probability function given by:

$$ \begin{gathered} P(m_{T} = m_{{{\text{Tnet}},a}} ,\, m_{S} = m_{{{\text{Snet}},c}} |{\varvec{\upalpha}} ,{\varvec{\upbeta}} ,{\varvec{\upphi}} ,\tau ,\gamma ,\lambda ,{\mathbf{Y}},{\mathbf{Z}},{\mathbf{X}}] \hfill \\ \propto \exp \left[ { - {\frac{\tau }{2}}{\varvec{\varepsilon }}_{1}^{'} \left( {{\varvec{\upalpha}},{\varvec{\upbeta}},{\varvec{\upphi}} ,m_{{{\text{Tnet}},a}} ,\gamma ,m_{{{\text{Snet}},c}} ,\lambda } \right){\varvec{\varepsilon }}_{{\mathbf{1}}} \left( {{\varvec{\upalpha}},{\varvec{\upbeta}},{\varvec{\upphi}} ,m_{{{\text{Tnet}},a}} ,\gamma ,m_{{{\text{Snet}},c}} ,\lambda } \right)} \right]\quad a = 1, \ldots ,t \quad c = 1, \ldots ,s \hfill \\ \end{gathered} $$

(5)

A Hasting-Metropolis step is used to draw a sample from this distribution, taking U _D(m _Tnet × m _snet) as the transition density.

• (γ, λ)|α, β, ϕ, τ, m _T , m _S , Y, Z, X is a continuous distribution with support (γ_min,γ_max)×(λ_min,λ_max) and a density function that is proportional to:

$$ \exp \left[ { - {\frac{\tau }{2}}\varepsilon_{1}^{'} \left( {{\varvec{\upalpha}},{\varvec{\upbeta}},{\varvec{\upphi }},m_{T} ,\gamma ,m_{S} ,\lambda } \right)\varepsilon_{1} \left( {{\varvec{\upalpha}},{\varvec{\upbeta}},{\varvec{\upphi}} ,m_{T} ,\gamma ,m_{S} ,\lambda } \right)} \right]I_{{\left( {\gamma_{\min } ,\gamma_{\max } } \right) \times \left( {\lambda_{\min } ,\lambda_{\max } } \right)}} \left( {\gamma ,\lambda } \right) $$

(6)

This distribution is not standard and we use a Hasting-Metropolis step to draw a sample from it, taking the uniform density in (γ _min, γ _max)×(λ _min, λ _max) as the transition density.