An Overview on Econometric Models for Linear Spatial Panel Data

Abstract

When spatial data are repeatedly collected from the same spatial locations over a short period of time, a spatial panel/longitudinal data set is generated. Thus, this type of spatial longitudinal data must exhibit both spatial and longitudinal correlations, which are not easy to model. This work is motivated by existing studies in statistics and econometrics literature but the proposed model and inference procedures should be applicable to the spatial panel data encountered in other fields as well such as environmental and/or ecological setups. Specifically, unlike the existing studies, we propose a new dynamic mixed model to accommodate both spatial and panel correlations. A complete theoretical analysis is given for the estimation of regression effects, and spatial and panel correlations by exploiting second and higher order moments based quasi-likelihood methods. Asymptotic properties are also studied in details. The step by step estimation results developed in the paper should be useful to the practitioners dealing with spatial panel data.

Appendix: Computation for the Derivatives and Asymptotic Properties

1.1 Formulas for the derivatives with respect to $\sigma ^{2}_{\gamma }$:

Formula for $\frac {\partial \lambda _{ii,t}(\sigma ^{2}_{\gamma },\phi ,\sigma ^{2}_{\epsilon },\boldsymbol {\xi }^{\prime })} {\partial \sigma ^{2}_{\gamma }}:$ Because by Eq. 5.27,

$$ \begin{array}{@{}rcl@{}} &&\lambda_{ii,t}(\sigma^{2}_{\gamma},\phi,\sigma^{2}_{\epsilon},\boldsymbol{\xi}^{\prime}) \\ &=&\left[\sigma^{2}_{\gamma}[\boldsymbol{a}^{\prime}_{i}\{\phi \boldsymbol{1}_{n_{i}}\boldsymbol{1}^{\prime}_{n_{i}}+(1-\phi)\boldsymbol{I}_{n_{i}}\}\boldsymbol{a}_{i}]+\sigma^{2}_{\epsilon}\right] +\left[\boldsymbol{z}^{\prime}_{i}\boldsymbol{\theta}+\boldsymbol{x}^{\prime}_{i}(t)\boldsymbol{\beta}\right]^{2}, \end{array} $$

one obtains

$$ \begin{array}{@{}rcl@{}} \frac{\partial \lambda_{ii,t}(\sigma^{2}_{\gamma},\phi,\sigma^{2}_{\epsilon},\boldsymbol{\xi}^{\prime})} {\partial \sigma^{2}_{\gamma}}&=&\boldsymbol{a}^{\prime}_{i}\{\phi \boldsymbol{1}_{n_{i}}\boldsymbol{1}^{\prime}_{n_{i}}+(1-\phi)\boldsymbol{I}_{n_{i}}\}\boldsymbol{a}_{i}. \end{array} $$

(a.1)

Formula for $\frac {\partial \lambda ^{*}_{ij,t}(\sigma ^{2}_{\gamma },\phi ,\sigma ^{2}_{\epsilon },\boldsymbol {\xi }^{\prime })} {\partial \sigma ^{2}_{\gamma }}$: By using Eqs. 3.27–3.28, one obtains this derivative as

$$ \begin{array}{@{}rcl@{}} &&\frac{\partial \lambda^{*}_{ij,t}(\sigma^{2}_{\gamma},\phi,\sigma^{2}_{\epsilon},\boldsymbol{\xi}^{\prime})} {\partial \sigma^{2}_{\gamma}} \\ & =&\left\{\begin{array}{ll} 0 & \{d_{s_{iu_{i}}s_{jv_{j}}}, \text{for all} u_{i}=1,\ldots,n_{i};v_{j}=1,\ldots,n_{j}\} >d^{*}\\ G_{ij}(\phi,\boldsymbol{a}_{i},\boldsymbol{a}_{j};n_{i},n_{j}) & \text{otherwise}. \end{array} \right.\\ \end{array} $$

(a.2)

Formula for the derivative with respect to ϕ:

Formula for $\frac {\partial \lambda ^{*}_{i(i+1),t}(\sigma ^{2}_{\gamma },\phi ,\sigma ^{2}_{\epsilon },\boldsymbol {\xi }^{\prime })} {\partial \phi }$: This derivative also follows from Eqs. 3.27–3.28. Specifically,

$$ \begin{array}{@{}rcl@{}} &&\!\!\!\frac{\partial \lambda^{*}_{i(i+1),t}(\sigma^{2}_{\gamma},\phi,\sigma^{2}_{\epsilon},\boldsymbol{\xi}^{\prime})} {\partial \phi} \\ \!\!\!& =&\!\!\!\left\{\begin{array}{ll} 0 & \{d_{s_{iu_{i}}s_{(i+1)v_{i+1}}},\! \text{for all} u_{i} = 1,\!\ldots\!,n_{i};v_{i+1} = 1,\!\ldots\!,n_{i+1}\} \!\!>\!\!d^{*}\\ \sigma^{2}_{\gamma} \frac{\partial G_{i,i+1}(\phi,\boldsymbol{a}_{i},\boldsymbol{a}_{i+1};n_{i},n_{i+1})}{\partial \phi} & \text{otherwise}, \end{array} \right.\\ \end{array} $$

(a.3)

where, by Eq. 3.27,

$$ \begin{array}{@{}rcl@{}} &&\frac{\partial G_{i,i+1}(\phi,\boldsymbol{a}_{i},\boldsymbol{a}_{i+1};n_{i},n_{i+1})}{\partial \phi} \\ &=& \left[\left\{{\boldsymbol{a}^{(1)}}^{\prime}_{i},[\boldsymbol{a}_{ij}+\tilde{\boldsymbol{a}}_{ij}]^{\prime},{\boldsymbol{a}^{(1)}}^{\prime}_{j}\right\} \frac{\partial \boldsymbol{C}^{\dagger}_{ij}(\phi;n_{i}+n^{(1)}_{j})}{\partial \phi}\right.\\ &&\left. \left\{{\boldsymbol{a}^{(1)}}^{\prime}_{i},[\boldsymbol{a}_{ij}+\tilde{\boldsymbol{a}}_{ij}]^{\prime},{\boldsymbol{a}^{(1)}}^{\prime}_{j}\right\}^{\prime} \right]_{|j=i+1}, \end{array} $$

(a.4)

with

$$ \begin{array}{@{}rcl@{}} &&\frac{\partial \boldsymbol{C}^{\dagger}_{ij}(\phi;n_{i}+n^{(1)}_{j})}{\partial \phi}, \end{array} $$

easily computed from Eq. 3.25.

Consistency of $\hat {\rho }_{\ell }$ (5.52)

Notice that as $K \rightarrow \infty ,$ and T is fixed and small, the first order approximation used in Eq. 5.52 is equivalent to write

$$ \begin{array}{@{}rcl@{}} E[\hat{\rho}_{\ell,N}]&=&E\left[\frac{\text{MSP}}{\text{MSS}}\right] \\ &=&\frac{E[\text{MSP}]}{E[\text{MSS}]}+o_{p}(1/\sqrt{K}). \end{array} $$

(a.5)

Consequently, by Eqs. 5.51 and 5.52, one writes

$$ \begin{array}{@{}rcl@{}} &&E[\hat{\rho}_{\ell}]=\rho_{\ell}+o_{p}(1/\sqrt{K}), \text{and} \\ &&[\hat{\rho}_{\ell}-\rho_{\ell}]=O(\sigma_{\hat{\rho}_{\ell}})+o_{p}(1/\sqrt{K}), \end{array} $$

(a.6)

where, taking B_K,5 as a K-dependent finite quantity, and assuming that

$$ \begin{array}{@{}rcl@{}} &&\left[\frac{\sigma^{2}_{\epsilon}}{K(T-\ell)}\sum\limits^{K}_{i=1}\sum\limits^{T-\ell}_{t=1} \frac{1}{\sqrt{\sigma_{tt|i}(\sigma^{2}_{\gamma},\phi,\sigma^{2}_{\epsilon}) \sigma_{(t+\ell,t+\ell)|i}(\sigma^{2}_{\gamma},\phi,\sigma^{2}_{\epsilon})}}\right]^{-1} \leq B_{K,5},\\ \end{array} $$

(a.7)

following Eq. 5.52, one obtains $\sigma ^{2}_{\hat {\rho }_{\ell }}$ in Eq. a.6 as

$$ \begin{array}{@{}rcl@{}} \sigma^{2}_{\hat{\rho}_{\ell}}&=&\text{var}[\hat{\rho}_{\ell}] =O(B_{K,5})\text{var}(\hat{\rho}_{\ell,N}). \end{array} $$

(a.8)

Next by Eq. 5.50, using the so-called first order approximation, we write

$$ \begin{array}{@{}rcl@{}} \text{var}(\hat{\rho}_{\ell,N})&=&\left[\text{var}\{\text{MSS}\}\right]^{-1}\text{var}\{\text{MSP}\}. \end{array} $$

(a.9)

To compute these variance components, we first recall that the full covariance matrix for the spatial panel data was computed by Eq. 3.42. We then assume that the fourth order moments for the spatial panel data also exist and they are bounded, specially, similar to spatial formula (5.46), assume that

$$\text{cov}[\{y_{s_{i}}(t)-\mu_{t|i}(\boldsymbol{\theta},\boldsymbol{\beta})\}^{2}, \{y_{s_{j}}(t^{*})-\mu_{t^{*}|j}(\boldsymbol{\theta},\boldsymbol{\beta})\}^{2}]=\omega^{*}_{(ii,jj)(tt,t^{*}t^{*})}(\cdot)$$

exist and bounded, so that for K-dependent finite quantities B_K,6 and B_K,7,

$$ \begin{array}{@{}rcl@{}} \text{var}\{(KT)\text{MSS}\}&=&\left[ \sum\limits^{K}_{i=1}\sum\limits^{T}_{t=1}[\sigma^{-2}_{tt|i}(\sigma^{2}_{\gamma},\phi,\sigma^{2}_{\epsilon})] \omega^{*}_{(ii,ii)(tt,tt)}(\cdot)\right.\\ &&+\left. \sum\limits^{K}_{i \neq j}\sum\limits^{T}_{t=1}\sum\limits^{T}_{t^{*} \neq t} [\sigma^{-2}_{tt|i}(\sigma^{2}_{\gamma},\phi,\sigma^{2}_{\epsilon})] [\omega^{*}_{(ii,jj)(tt,t^{*}t^{*})}(\cdot)]\right] \leq B_{K,6} \\ \end{array} $$

(a.10)

and

$$ \begin{array}{@{}rcl@{}} &&\text{var}\{[K(T-\ell)])\text{MSP}\} \\ &=&\left[ \sum\limits^{K}_{i=1}\sum\limits^{T}_{t=1}[\sigma^{-1}_{tt|i}(\sigma^{2}_{\gamma},\phi,\sigma^{2}_{\epsilon}) \sigma^{-1}_{(t+\ell,t+\ell)|i}(\sigma^{2}_{\gamma},\phi,\sigma^{2}_{\epsilon})] \omega^{*}_{(ii,ii)(t(t+\ell),t(t+\ell))}(\cdot)\right. \\ &&\quad+\left. \sum\limits^{K}_{i \neq j}\sum\limits^{T}_{t=1}\sum\limits^{T}_{t^{*} \neq t} [\sigma^{-1}_{tt|i}(\sigma^{2}_{\gamma},\phi,\sigma^{2}_{\epsilon}) \sigma^{-1}_{(t+\ell,t+\ell)|i}(\sigma^{2}_{\gamma},\phi,\sigma^{2}_{\epsilon})]\right. \\ &&\quad\times \left. [\omega^{*}_{(ii,jj)(t(t+\ell),t^{*}(t^{*}+\ell))}(\cdot)]\vphantom{\sum\limits^{K}_{i=1}}\right] \leq B_{K,7}. \end{array} $$

(a.11)

By applying Eqs. a.10 and a.11 to a.9, it then follows from Eq. a.8 that

$$ \begin{array}{@{}rcl@{}} \sigma^{2}_{\hat{\rho}_{\ell}} \equiv O(B_{K,5})O(B^{-1}_{K,6})O(B_{K,7}), \end{array} $$

(a.12)

implying by Eq. a.6 that

$$ \begin{array}{@{}rcl@{}} [\hat{\rho}_{\ell}-\rho_{\ell}]&=&O(\sigma_{\hat{\rho}_{\ell}})+o_{p}(1/\sqrt{K}) \\ &=&O(B^{\frac{1}{2}}_{K,5})O(B^{-\frac{1}{2}}_{K,6})O(B^{\frac{1}{2}}_{K,7})+o_{p}(1/\sqrt{K}) \\ &\equiv& o_{p}(1/\sqrt{K}), \end{array} $$

(a.13)

and hence

$$\lim\limits_{K \rightarrow \infty}|\hat{\rho}_{\ell}-\rho_{\ell}| \rightarrow_{p}0 \Rightarrow \hat{\rho}_{\ell} \rightarrow_{p} \rho_{\ell},$$

i.e., $\hat {\rho }_{\ell }$ defined by Eq. 5.52 is a consistent estimator for the panel correlation index parameter ρ_ℓ.