Testing for spatial group-wise heteroskedasticity in spatial autocorrelation regression models: Lagrange multiplier scan tests

Abstract

The aim of this paper is to develop a spatial group-wise heteroskedasticity test based on the scan approach, specifically developed for spatial autocorrelation regression models (spatial lag and spatial error models): the “scan-LM test.” Based on the Lagrange multiplier (LM) principle, its main advantage lies in its comparative ease of implementation as it is not necessary to obtain the maximum likelihood estimations for the alternative hypothesis. Moreover, when rejecting the null hypothesis, this test identifies the shape and size of the spatial clusters with different residual variance, a feature which proves very useful for specification search of the regression model. Another important benefit of the scan-LM test is that it does not require the specification of a spatial weights matrix. An extensive Monte Carlo simulation confirms the good properties of the scan-LM test in terms of size and power. This test is also robust in the presence of non-normality and other forms of a spatial heteroskedasticity. We finally propose an application on housing prices in the agglomeration of Madrid for a specific submarket: the attics.

Appendices

Appendix 1: Derivation of the scan-SAR LM test

In order to obtain the ${\text{LM}} - {\text{Scan}}_{\text{SAR}}^{\text{SGWH}}$ test, we evaluate the gradient of the log-likelihood function:

$$ \begin{aligned} \frac{\partial l}{\partial \beta } & = (Ay - X\beta )'\varOmega^{ - 1} X \\ \frac{\partial l}{\partial \rho } & = -\, tr\,A^{ - 1} W + (Ay - X\beta )'\varOmega^{ - 1} Wy \\ \frac{\partial l}{{\partial \sigma^{2} }} & = -\, \frac{1}{2}tr\,\varOmega^{ - 1} H_{{\sigma^{2} }} + \frac{1}{2}(Ay - X\beta )'\varOmega^{ - 2} H_{{\sigma^{2} }} (Ay - X\beta ) \\ \frac{\partial l}{\partial h} & = - \,\frac{1}{2}tr\,\varOmega^{ - 1} H_{h} + \frac{1}{2}(Ay - X\beta )'\varOmega^{ - 2} H_{h} (Ay - X\beta ) \\ \end{aligned} $$

where A = I − ρW; $H_{h} = \frac{\partial \varOmega }{\partial h}$ and $H_{{\sigma^{2} }} = \frac{\partial \varOmega }{{\partial \sigma^{2} }}$

Under the null hypothesis, we have:

$$ g(\theta )|_{{H_{0} }} = \left[ {\begin{array}{*{20}l} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {g_{\text{SAR}} (h_{\sigma } ) = - \frac{{n_{Z} }}{2} + \frac{1}{2}\frac{{u'_{Z} u_{Z} }}{{\sigma^{2} }}} \hfill \\ \end{array} } \right] $$

where n_Z = #(Z) and u = Ay − Xβ. By u_z, we denote the elements of the vector u that belong to zone Z. In order to obtain the elements of the information matrix $I = E\left[ {\frac{{\partial^{2} l}}{\partial \theta \partial \theta '}} \right]$, it is necessary to derive the following elements under the null:

$$ \begin{aligned} I_{\beta \beta } & = X'\varOmega^{ - 1} X \\ I_{\beta \rho } & = X'\varOmega^{ - 1} WA^{ - 1} X\beta \\ I_{{\beta \sigma^{2} }} & = 0 \\ I_{\beta h} & = 0 \\ I_{\rho \rho } & = tr\,WA^{ - 1} WA^{ - 1} + tr\,\varOmega A'^{ - 1} W'\varOmega^{ - 1} WA^{ - 1} + \beta 'X'A'^{ - 1} W'\varOmega^{ - 1} WA^{ - 1} X\beta \\ I_{{\rho \sigma^{2} }} & = tr\,\varOmega^{ - 1} H_{{\sigma^{2} }} WA^{ - 1} \\ I_{\rho h} & = tr\,\varOmega^{ - 1} H_{h} WA^{ - 1} \\ I_{{\sigma^{2} \sigma^{2} }} & = \frac{1}{2}tr\,\varOmega^{ - 2} H_{{\sigma^{2} }} H_{{\sigma^{2} }} \\ I_{{\sigma^{2} h}} & = \frac{1}{2}tr\,\varOmega^{ - 2} H_{{\sigma^{2} }} H_{h} \\ I_{hh} & = \frac{1}{2}tr\,\varOmega^{ - 2} H_{h} H_{h} \\ \end{aligned} $$

The information matrix under the null has the following expression:

$$ I_{\text{SAR}} (\theta )|_{{H_{0} }} = \frac{1}{{\sigma^{2} }}\left[ {\begin{array}{llll} {X'X} \hfill &\quad {X'WA^{ - 1} X\beta } \hfill &\quad 0 \hfill &\quad 0 \hfill \\ { - - } \hfill &\quad {\sigma^{2} tr\,WA^{ - 1} WA^{ - 1} + \sigma^{2} tr\,H + \beta 'X'HX\beta } \hfill &\quad {tr\,WA^{ - 1} } \hfill &\quad {tr\,HWA^{ - 1} } \hfill \\ { - - } \hfill &\quad { - - } \hfill &\quad {\frac{1}{2}\frac{n}{{\sigma^{2} }}} \hfill &\quad {\frac{{n_{Z} }}{2}} \hfill \\ { - - } \hfill &\quad { - - } \hfill &\quad { - - } \hfill &\quad {\frac{1}{2}\sigma^{2} n_{Z} } \hfill \\ \end{array} } \right] $$

with $H = A'^{ - 1} W'WA^{ - 1}$

The LM test has the following general expression:

$$ {\text{LM}}(Z)_{\text{SAR}}^{\text{SGWH}} = \left[ {g(\theta )|_{{H_{0} }} } \right]'\left[ {I_{\text{SAR}} (\theta )|_{{H_{0} }} } \right]^{ - 1} \left[ {g(\theta )|_{{H_{0} }} } \right] $$

After, some computations, it can be expressed as:

$$ {\text{LM}} - {\text{Scan}}_{\text{SAR}}^{\text{SGWH}} = \mathop {sup}\limits_{Z \in \varTheta } \frac{{g_{\text{SAR}} (h_{\sigma } )^{2} a^{44} }}{{det(I_{\text{SAR}} (\theta )|_{{H_{0} }} )}} $$

The expression of the determinant of the information matrix can be written in compact form as:

$$ \begin{aligned} \det (I(\theta )|_{{H_{0} }} ) & = \left( {\frac{1}{{\sigma^{2} }}} \right)^{k + 3} \left[ {\frac{{\sigma^{2} n_{Z} }}{2}\det (A_{33} ) - \frac{{n_{Z} }}{2}\left( {\frac{{n_{Z} }}{2}\det (A_{22} ) - trWA^{ - 1} \det (X^{\prime }X)trHWA^{ - 1} } \right)} \right. \\ & \quad + \;\left. {trHWA^{ - 1} \det (X^{\prime }X)\left( {\frac{{n_{Z} }}{2}trWA^{ - 1} - \frac{n}{{2\sigma^{2} }}trHWA^{ - 1} } \right)} \right] \\ \end{aligned} $$

where A₂₂ is the following matrix that does not depend upon Z,

$$ A_{22} = \left[ {\begin{array}{ll} {X^{\prime } X} \hfill &\quad {X^{\prime } WA^{ - 1} X\beta } \hfill \\ { - - } \hfill &\quad {\sigma^{2} trWA^{ - 1} WA^{ - 1} + \sigma^{2} trH + \beta^{\prime } X^{\prime } HX\beta } \hfill \\ \end{array} } \right] $$

and

$$ A_{33} = \left[ {\begin{array}{*{20}l} {X^{\prime } X} \hfill &\quad {X^{\prime } WA^{ - 1} X\beta } \hfill &\quad 0 \hfill \\ { - - } \hfill &\quad {\sigma^{2} trWA^{ - 1} WA^{ - 1} + \sigma^{2} trH + \beta^{\prime } X^{\prime} HX\beta } \hfill & \quad{trWA^{ - 1} } \hfill \\ { - - } \hfill &\quad { - - } \hfill &\quad {\frac{1}{2}\frac{n}{{\sigma^{2} }}} \hfill \\ \end{array} } \right] $$

finally, $a^{44}$ is a term that does not depend on the set Z

$$ a^{44} = \left( {\frac{1}{{\sigma^{2} }}} \right)^{k + 2} det\left( {\left[ {\begin{array}{lll} {X'X} \hfill &\quad {X'WA^{ - 1} X\beta } \hfill &\quad 0 \hfill \\ {} \hfill &\quad {\sigma^{2} tr\,WA^{ - 1} WA^{ - 1} + \sigma^{2} tr\,H + \beta 'X'HX\beta } \hfill &\quad {tr\,WA^{ - 1} } \hfill \\ {} \hfill &\quad {} \hfill &\quad {\frac{1}{2}\frac{n}{{\sigma^{2} }}} \hfill \\ \end{array} } \right]} \right) $$

Appendix 2: Derivation of the scan-SEM LM test

In order to obtain the ${\text{LM}} - {\text{Scan}}_{\text{SEM}}^{\text{SGWH}}$ test, we evaluate the gradient of the log-likelihood function:

$$ \begin{aligned} \frac{\partial l}{\partial \beta } & = (y - X\beta )'B'\varOmega^{ - 1} BX \\ \frac{\partial l}{\partial \rho } & = - tr\,B^{ - 1} W + (y - X\beta )'B'\varOmega^{ - 1} W(y - X\beta ) \\ \frac{\partial l}{{\partial \sigma^{2} }} & = - \frac{1}{2}tr\,\varOmega^{ - 1} H_{{\sigma^{2} }} + \frac{1}{2}(y - X\beta )'B'\varOmega^{ - 2} H_{{\sigma^{2} }} B(y - X\beta ) \\ \frac{\partial l}{\partial h} & = - \frac{1}{2}tr\,\varOmega^{ - 1} H_{h} + \frac{1}{2}(y - X\beta )'B'\varOmega^{ - 2} H_{h} B(y - X\beta ) \\ \end{aligned} $$

where B = I − λW; $H_{h} = \frac{\partial \varOmega }{\partial h}$ and $H_{{\sigma^{2} }} = \frac{\partial \varOmega }{{\partial \sigma^{2} }}$

The gradient under the null hypothesis is:

$$ g(\theta )|_{{H_{0} }} = \left[ {\begin{array}{*{20}l} 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {g_{\text{SEM}} (h_{\sigma } ) = - \frac{{n_{Z} }}{2} + \frac{1}{2}\frac{{v'_{Z} v_{Z} }}{{\sigma^{2} }}} \hfill \\ \end{array} } \right] $$

where v = Bu and by v_Z, we denote the elements of the vector v that belong to zone Z.

In order to obtain the elements of the information matrix $I = E\left[ {\frac{{\partial^{2} l}}{\partial \theta \partial \theta '}} \right]$ matrix, it is necessary to derive the following elements under the null:

$$ \begin{aligned} I_{\beta \beta } & = X'B'\varOmega^{ - 1} BX \\ I_{\beta \lambda } & = 0 \\ I_{{\beta \sigma^{2} }} & = 0 \\ I_{\beta h} & = 0 \\ I_{\lambda \lambda } & = tr\,WB^{ - 1} WB^{ - 1} + tr\,\varOmega B'^{ - 1} W'\varOmega^{ - 1} WB^{ - 1} \\ I_{{\lambda \sigma^{2} }} & = tr\,\varOmega^{ - 1} H_{{\sigma^{2} }} WB^{ - 1} \\ I_{\lambda h} & = tr\,\varOmega^{ - 1} H_{h} WB^{ - 1} \\ I_{{\sigma^{2} \sigma^{2} }} & = \frac{1}{2}tr\,\varOmega^{ - 2} H_{{\sigma^{2} }} H_{{\sigma^{2} }} \\ I_{{\sigma^{2} h}} & = \frac{1}{2}tr\,\varOmega^{ - 2} H_{{\sigma^{2} }} H_{h} \\ I_{hh} & = \frac{1}{2}tr\,\varOmega^{ - 2} H_{h} H_{h} \\ \end{aligned} $$

The information matrix under the null has the following expression:

$$ I_{\text{SEM}} (\theta )|_{{H_{0} }} = \frac{1}{{\sigma^{2} }}\left[ {\begin{array}{llll} {X'B'BX} \hfill &\quad 0 \hfill &\quad 0 \hfill &\quad 0 \hfill \\ { - - } \hfill &\quad {\sigma^{2} tr\,WB^{ - 1} WB^{ - 1} + \sigma^{2} tr\,B'^{ - 1} W'WB^{ - 1} } \hfill &\quad {tr\,WB^{ - 1} } \hfill &\quad {tr\,HWB^{ - 1} } \hfill \\ { - - } \hfill &\quad { - - } \hfill &\quad {\frac{1}{2}\frac{n}{{\sigma^{2} }}} \hfill &\quad {\frac{{n_{Z} }}{2}} \hfill \\ { - - } \hfill &\quad { - - } \hfill &\quad { - - } \hfill &\quad {\frac{1}{2}\sigma^{2} n_{Z} } \hfill \\ \end{array} } \right] $$

After some calculations, we obtain:

$$ {\text{LM}}_{\text{SEM}}^{\text{SGWH}} = \frac{{g_{\text{SEM}} (h_{\sigma } )^{2} b^{44} }}{{det(I_{\text{SEM}} (\theta )|_{{H_{0} }} )}} \sim \chi_{1}^{2} $$

where the term b⁴⁴ is not dependent on the set Z

$$ \begin{aligned} b^{44} & = \left( {\frac{1}{{\sigma^{2} }}} \right)^{k + 2} det\left( {\left[ {\begin{array}{lll} {X'X} \hfill &\quad 0 \hfill &\quad 0 \hfill \\ { - - } \hfill &\quad {\sigma^{2} tr\,WB^{ - 1} \,WB^{ - 1} + \sigma^{2} tr\,B'^{ - 1} \,W'WB^{ - 1} } \hfill &\quad {tr\,WB^{ - 1} } \hfill \\ { - - } \hfill &\quad { - - } \hfill &\quad {\frac{1}{2}\frac{n}{{\sigma^{2} }}} \hfill \\ \end{array} } \right]} \right) \\ & = \left( {\frac{1}{{\sigma^{2} }}} \right)^{k + 2} det(X'X)\left( {\frac{1}{2}\frac{n}{{\sigma^{2} }}(\sigma^{2} tr\,WB^{ - 1} \,WB^{ - 1} + \sigma^{2} tr\,B'^{ - 1} \,W'WB^{ - 1} ) - (tr\,WB^{ - 1} )^{2} } \right) \\ \end{aligned} $$

The determinant of the information matrix can be expressed as a linear combination of terms that depend on Z:

$$ \det (I_{SEM} (\theta )|_{{H_{0} }} ) = \left( {\frac{1}{{\sigma^{2} }}} \right)^{k + 3} \det \left( {X^{\prime }{\rm B}^{\prime }BX} \right)\left[ \begin{aligned} I_{22} \frac{{nn_{Z} }}{4} + R_{Z} trWB^{ - 1} trH_{h} WB^{ - 1} - \frac{n}{{2\sigma^{2} }}\left( {trH_{h} WB^{ - 1} } \right)^{2} \hfill \\ - \frac{{\sigma^{2} n_{Z} }}{2}\left( {trWB^{ - 1} } \right)^{2} - \frac{{n_{Z}^{2} }}{4}I_{22} \hfill \\ \end{aligned} \right] $$

with

$$ I_{22} = \sigma^{2} trWB^{ - 1} WB^{ - 1} + \sigma^{2} trB^{ - 1} W^{\prime }WB^{. - 1} $$