A Skew-Normal Spatial Simultaneous Autoregressive Model and its Implementation

Abstract

Abstract: We propose generalization of the spatial Simultaneous Autoregressive (SAR) model on a lattice towards modelling for asymmetry. Under the assumption of skew-normal error structure, expression for density and characteristic function for the induced distribution of response are obtained. Full-likelihood based implementation of the proposed model to a real data set is performed using Differential Evolution (DE). The relevant results are reported and compared with the results from existing models.

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Acknowledgements

We are thankful to the anonymous referees and the editor for their constructive comments and suggestions that help improve the article. We would like to thank the R Core Team (2018). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. The URL address is, http://www.R-project.org/. We extend our thanks to Lee (2013) and Bivand et al. (2015) and Ardia et al. (2016) for developing the useful R packages, ‘CARBayes’, ‘spdep’ and ‘DEoptim’ respectively.

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Correspondence to Sanjeeva Kumar Jha.

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Appendices

Appendix

We have \(\boldsymbol {Y} \sim Sn-SAR_{n} \left (X\boldsymbol {\beta }-\sqrt {\frac {2}{\pi } } \boldsymbol {\delta }, {\Delta } , D, \rho W\right ).\)

The characteristic function of response vector Y is

$$ \begin{array}{@{}rcl@{}} {\Psi}_{\boldsymbol{ Y}} \left( \boldsymbol{ t}\right)&=& {\int}_{R_{n} }\frac{e^{i\boldsymbol{t}^{\prime}\boldsymbol{y}} 2^{n} }{\left( 2\pi \right)^{\frac{n}{2} } \left|\left( I-\rho W\right)^{-1} Q^{-\frac{1}{2} } \right|} \\ && \times e^{-\frac{1}{2} \left\{\left( I-\rho W\right)\left( \boldsymbol{y}-X\boldsymbol{\beta }\right)+\sqrt{\frac{2}{\pi } } \boldsymbol{ \delta }\right\}^{\prime} Q\left\{\left( I-\rho W\right)\left( \boldsymbol{ y}-X\boldsymbol{\beta }\right)+\sqrt{\frac{2}{\pi } } \boldsymbol{\delta }\right\}} \\ && \times {\Phi} \left\{B^{-\frac{1}{2} } DQ\left( \left( I-\rho W\right)\left( \boldsymbol{ y}-X\boldsymbol{\beta }\right)+\sqrt{\frac{2}{\pi } } \boldsymbol{\delta }\right)\right\}d\boldsymbol{ y}. \end{array} $$
(A.1)

Letting \(\boldsymbol {u}=Q^{\frac {1}{2} } \left (\left (I-\rho W\right )\left (\boldsymbol { y}-X\boldsymbol { \beta }\right )+\sqrt {\frac {2}{\pi } } \boldsymbol {\delta }\right )\), we have

$$ \boldsymbol{y}=\left( I-\rho W\right)^{-1} \left( Q^{-\frac{1}{2} } \boldsymbol{ u}-\sqrt{\frac{2}{\pi } } \boldsymbol{\delta }\right)+X\boldsymbol{ \beta } $$

and differentiate with respect to u, we get

$$ \frac{d\boldsymbol{ y}}{d\boldsymbol{ u}} =\left|\left( I-\rho W\right)^{-1} Q^{-\frac{1}{2} } \right|. $$

Therefore, Eq. A.1 becomes

$$ \begin{array}{@{}rcl@{}} {\Psi}_{\boldsymbol{ Y}} \left( \boldsymbol{t}\right) &=& 2^{n} e^{i\boldsymbol{t}^{\prime}\left\{X\boldsymbol{\beta }-\sqrt{\frac{2}{\pi } } \left( I-\rho W\right)^{-1} \boldsymbol{\delta }\right\}} {\int}_{R_{n} }\frac{1}{\left( 2\pi \right)^{\frac{n}{2} } } e^{-\frac{1}{2} \left\{\boldsymbol{ u}^{\prime}-2i\boldsymbol{ t}^{\prime}\left( I-\rho W\right)^{-1} Q^{-\frac{1}{2} } \right\}\boldsymbol{u}} \\ && \times {\Phi} \left( B^{-\frac{1}{2} } DQ^{\frac{1}{2} } \boldsymbol{u}\right)d\boldsymbol{u}. \end{array} $$
(A.2)

Again letting, \(\boldsymbol {z}^{\prime }=\boldsymbol {u}^{\prime }-i\boldsymbol {t}^{\prime }\left (I-\rho W\right )^{-1} Q^{-\frac {1}{2} } \), we have

$$ \boldsymbol{ u}^{\prime}=\boldsymbol{z}^{\prime}+i\boldsymbol{t}^{\prime}\left( I-\rho W\right)^{-1} Q^{-\frac{1}{2} }. $$

And Eq. A.2 may be written as,

$$ \begin{array}{ll} {\Psi}_{\boldsymbol{ Y}} \left( \boldsymbol{t}\right)=& 2^{n} e^{i\boldsymbol{ t}^{\prime}\left\{X\boldsymbol{ \beta }-\sqrt{\frac{2}{\pi } } \left( I-\rho W\right)^{-1} \boldsymbol{\delta }\right\}-\frac{\boldsymbol{ t}^{\prime}\left( I-\rho W\right)^{-1} Q^{-1} \left( I-\rho W\right)^{-1} \boldsymbol{ t}}{2} } \\ & \times {\Phi} \left\{iD\left( I-\rho W\right)^{-1} \boldsymbol{t}\right\}. \end{array} $$
(A.3)

Equation A.3 provides the required expression for the characteristic function of the response vector Y.

\(E\left (\boldsymbol {Y}\right )\) may be obtained from Eq. A.3 as, Letting, \(S=i \left \{X\boldsymbol {\beta }-\sqrt {\frac {2}{\pi } }\right . (I-\) \(\left . \rho W)^{-1} \boldsymbol { \delta }\vphantom {\sqrt {\frac {2}{\pi } }}\right \}\), \(U=\left (I-\rho W\right )^{-1} Q^{-1} \left (I-\rho W\right )^{-1} \) and \(V=iD\left (I-\rho W\right )^{-1} \) in Eq. A.3, then the characteristic function is written as

$$ \begin{array}{@{}rcl@{}} {\Psi}_{\boldsymbol{Y}} \left( \boldsymbol{t}\right)=2^{n} e^{\boldsymbol{ t}^{\prime}S-\boldsymbol{t}^{\prime}U\boldsymbol{t}/2} {\Phi} \left\{V\boldsymbol{t}\right\}. \end{array} $$
(A.4)

Differentiating (A.4) with respect to t, we get

$$ \begin{array}{@{}rcl@{}} {\Psi}_{\boldsymbol{ Y}}^{\prime} \left( \boldsymbol{t}\right)=2^{n} \left\{\phi \left( V\boldsymbol{ t}\right)Ve^{\boldsymbol{ t}^{\prime}S-\boldsymbol{ t}^{\prime}U\boldsymbol{t}/2} +{\Phi} \left( V\boldsymbol{t}\right)e^{\boldsymbol{ t}S-\boldsymbol{ t}^{\prime}U\boldsymbol{ t}/2} \left( S-U\boldsymbol{t}\right)\right\}. \end{array} $$
(A.5)

And, second differentiation of Eq. A.4 with respect to t

$$ \begin{array}{ll} {\Psi}_{\boldsymbol{Y}}^{\prime\prime} \left( \boldsymbol{t}\right)\!= &\!\!\! 2^{n} \left\{\phi \right. {~}^{\prime}\left( V\boldsymbol{t}\right)VV^{\prime}e^{\boldsymbol{t}^{\prime}S-\boldsymbol{t}^{\prime}U\boldsymbol{t}/2} +\phi \left( V\boldsymbol{t}\right)Ve^{\boldsymbol{t}S-\boldsymbol{t}^{\prime}U\boldsymbol{t}/2} \left( S - U\boldsymbol{t}\right) \\ & \!\!\!+\phi \left( V\boldsymbol{t}\right)Ve^{\boldsymbol{t}^{\prime}S-\boldsymbol{t}^{\prime}U\boldsymbol{t}/2} S +{\Phi} \left( V\boldsymbol{t}\right)e^{\boldsymbol{t}S-\boldsymbol{t}^{\prime}U\boldsymbol{t}/2} \left( S-U\boldsymbol{t}\right)S^{\prime} \\ & \!\!\!-\phi \left( V\boldsymbol{t}\right)Ve^{\boldsymbol{t}^{\prime}S-\boldsymbol{t}^{\prime}U\boldsymbol{t}/2} U\boldsymbol{t} - {\Phi} \left( V\boldsymbol{t}\right)e^{\boldsymbol{t}S-\boldsymbol{t}^{\prime}U\boldsymbol{t}/2} \left( S - U\boldsymbol{t}\right)\left( U\boldsymbol{t}\right)^{\prime} \\ & \!\!\!-{\Phi} \left( V\boldsymbol{t}\right)e^{\boldsymbol{t}S-\boldsymbol{t}^{\prime}U\boldsymbol{t}/2} \left. U\right\}. \end{array} $$
(A.6)

On putting t = 0 in Eq. A.5 we obtain the first moment about the origin as,

$$ \begin{array}{@{}rcl@{}} E\left( \boldsymbol{Y}\right) &=& \left( -i\right)^{1} \left|{\Psi}_{\boldsymbol{Y}}^{\prime} \left( \boldsymbol{t}\right)\right|_{\boldsymbol{t}=\boldsymbol{0}} \\ & =&\left( -i\right)^{1} 2^{n} \left[\left\{\phi \left( V\boldsymbol{t}\right)Ve^{\boldsymbol{t}^{\prime}S-\boldsymbol{t}^{\prime}U\boldsymbol{t}/2} +{\Phi} \left( \boldsymbol{t}\right)e^{\boldsymbol{t}S-\boldsymbol{t}^{\prime}U\boldsymbol{t}/2} \left( S-U\boldsymbol{t}\right)\right\}\right]_{\boldsymbol{t}=\boldsymbol{0}} \\ &=&\left( -i\right)^{1} 2^{n} \left\{\phi \left( V\boldsymbol{0}\right)Ve^{\boldsymbol{0}^{\prime}S-\boldsymbol{0}^{\prime}U\boldsymbol{0}/2} +{\Phi} \left( V\boldsymbol{t}\right)e^{\boldsymbol{0}^{\prime}S-\boldsymbol{0}^{\prime}U\boldsymbol{0}/2} \left( S-U\boldsymbol{0}\right)\right\} \\ & =&\left( -i\right)^{1} 2^{n} \left\{\phi \left( \boldsymbol{0}\right)V+{\Phi} \left( \boldsymbol{0}\right)S\right\} \\ &=&\left( -i\right)^{1} 2^{n} \left\{\frac{i\left( I-\rho W\right)^{-1} \boldsymbol{\delta }}{\left( 2\pi \right)^{n/2} } +\frac{i}{2^{n} } \left( X\boldsymbol{\beta } -\sqrt{\frac{2}{\pi } } \left( I-\rho W\right)^{-1} \boldsymbol{\delta }\right)\right\} \\ &=&\left( -i\right)^{1} \left\{i\sqrt{\frac{2}{\pi } } (I-\rho W)^{-1} \boldsymbol{\delta }+iX\boldsymbol{\beta } -i\sqrt{\frac{2}{\pi } } (I-\rho W)^{-1} \boldsymbol{\delta }\right\} \\ &=&X\boldsymbol{\beta }. \end{array} $$
(A.7)

Similarly, the second moment about origin for the random response vector Y may be obtained on putting t = 0 in Eq. A.6.

$$ \begin{array}{@{}rcl@{}} E\left( \boldsymbol{Y}^{2} \right) \!\!\!\!& =&\!\!\!\!\left( -i\right)^{2} \left|{\Psi}_{\boldsymbol{Y}}^{\prime\prime} \left( \boldsymbol{t}\right)\right|_{\boldsymbol{t}=\boldsymbol{0}} \\ & =&\!\!\!\!\left( -i\right)^{2} 2^{n} \left\{2\phi \left( \boldsymbol{0}\right)VS+{\Phi} \left( \boldsymbol{0}\right)SS^{\prime}-{\Phi} \left( \boldsymbol{0}\right)U\right\} \\ & =&\!\!\!\!\left( -i\right)^{2} 2^{n} \left[\frac{2}{\left( 2\pi \right)^{n/2} } iD\left( I - \rho W\right)^{-1} \left\{i\left( X\boldsymbol{\beta } - \sqrt{\frac{2}{\pi } } \left( I-\rho W\right)^{-1} \boldsymbol{\delta }\right)\right\}\right. \\ && \!\!\!\!+\frac{1}{2^{n} } i^{2} \left( X\boldsymbol{\beta }-\sqrt{\frac{2}{\pi } } \left( I-\rho W\right)^{-1} \boldsymbol{\delta }\right)\left( X\boldsymbol{\beta} -\sqrt{\frac{2}{\pi } } \left( I-\rho W\right)^{-1} \boldsymbol{\delta }\right)^{\prime} \\ && {\left. \!\!\!\!-\frac{1}{2^{n} } \left( I-\rho W\right)^{-1} Q^{-1} \left( I-\rho W\right)^{-1} \right]} \\ & =&\!\!\!\!2\sqrt{\frac{2}{\pi } } D\left( I-\rho W\right)^{-1} X\boldsymbol{\beta }-2\left( \frac{2}{\pi } \right)\left( I-\rho W\right)^{-1} D^{2} \left( I-\rho W\right)^{-1} \\ && \!\!\!\!+X\boldsymbol{\beta \beta }^{\prime}X^{\prime} +\left( \frac{2}{\pi } \right)\left( I-\rho W\right)^{-1} D^{2} \left( I-\rho W\right)^{-1} \\ && \!\!\!\!- 2\sqrt{\frac{2}{\pi } } D\left( I-\rho W\right)^{-1} X\boldsymbol{\beta } \\ && \!\!\!\!+\left( I-\rho W\right)^{-1} Q^{-1} \left( I-\rho W\right)^{-1} \\ & =&\!\!\!\!\left( I-\rho W\right)^{-1} Q^{-1} \left( I-\rho W\right)^{-1} -\left( \frac{2}{\pi } \right)\left( I-\rho W\right)^{-1} D^{2} \left( I-\rho W\right)^{-1} \\ && \!\!\!\!+X\boldsymbol{\beta \beta }^{\prime}X^{\prime} \\ & =&\!\!\!\!\left( I-\rho W\right)^{-1} \left\{\Delta +\left( 1-\frac{2}{\pi } \right)D^{2} \right\}\left( I-\rho W\right)^{-1} +X\boldsymbol{\beta \beta }^{\prime}X^{\prime}. \end{array} $$
(A.8)

Using Eqs. A.7 and A.8, the covariance matrix of Y is

$$ \begin{array}{@{}rcl@{}} \text{cov}\left( \boldsymbol{Y}\right) =\left( I-\rho W\right)^{-1} \left\{\Delta +\left( 1-\frac{2}{\pi } \right)D^{2} \right\}\left( I-\rho W\right)^{-1}. \end{array} $$
(A.9)

Appendix

Figure 4
figure4

(a-d). Convergence plots of the maximum likelihood estimate of parameters and the corresponding bootstrap density plots for Property price data

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Jha, S.K., Singh, N.V. A Skew-Normal Spatial Simultaneous Autoregressive Model and its Implementation. Sankhya A (2021). https://doi.org/10.1007/s13171-021-00246-3

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Keywords

  • Spatial SAR models
  • Skew-normal distribution
  • Maximum likelihood
  • Differential evolution.

PACS Nos

  • Primary 62H11; Secondary 62M30