Spatial panel simultaneous equations models with error components

Abstract

This paper develops limited and full information estimators for a simultaneous panel data model with spatial lags on the dependent variables and spatially autocorrelated error processes in the form of spatial autoregressive or spatial moving average processes. The spatial error components are estimated with various generalized moment procedures. Monte Carlo experiments show that the proposed estimators outperform traditional estimators and also provide results on the impact of misspecifying the error process. We illustrate the various estimators on an empirical example pertaining to competition in current and capital expenditure between French municipalities in the capital region of Ile-de-France.

A Appendix

1.1 A.1 Notations

The following list summarizes some frequently used notations in either the text or the Appendices:

• $A^\top$ is the transpose operator for any $m\times n$ matrix A.

• $\text {E}(A)$ is the expected value of any input $m\times n$ random matrix A.

• $\text {tr}(A)=\sum _{i=1}^ma_{ii}$ is the sum of diagonal elements for any $m\times m$ matrix A.

• For a $n\times m$ matrix B the vectorization is denoted $\text {vec}(B)=\big (b_{11},\ldots ,b_{n1},\ldots ,b_{1m},\ldots ,b_{nm}\big )$

• If A, B and C are conformable matrices, then $\text {vec}(ABC)= (C^\top \otimes A)\text {vec}(B)$.

• For any vector $\textbf{a}$ of order n and $\forall k\in {\mathbb {N}}$, let $\overset{k}{\bar{\textbf{a}}}=M_n^k\textbf{a}$. This means that: a) for $k=0$, $\overset{0}{\bar{\textbf{a}}}={\textbf{a}}$; b) for $k=1$, $\overset{1}{\bar{\textbf{a}}}=\bar{\textbf{a}}$; and for $k=2$, $\overset{2}{\bar{{\textbf{a}}}}=M\bar{{\textbf{a}}}=\bar{\bar{{\textbf{a}}}}$.

• Let $\upsilon _l$ and $\upsilon _q$ be $n\times 1$ vectors of spatial processes; $\forall a,b\in {\mathbb {N}}$, $\gamma ^{lq,h}_{ab} = {\overset{a}{{\bar{\upsilon }}}^{\top }_lQ_{nT,h}\overset{b}{{\bar{\upsilon }}}_q}/{\text {tr}\left( Q_{nT,h}\right) }.$

• For any spatial matrix $M_n$ of order n and $\forall a,b\in {\mathbb {N}}$, let $\phi _{ab}=\phi _{ba}=\frac{1}{n}\text {tr}\left( M^{\top {a}}_{n}M^b_{n}\right)$. This means that $\phi _{00}=1$ and $\phi _{01}=\phi _{10}=0$.

• Let $A=[a_{ij}]$ and $B=[b_{ij}]$ be $n\times n$ nonstochastic matrices with zero diagonals. Let $\epsilon _1$, $\epsilon _2$, $\epsilon _3$ and $\epsilon _4$ be $n\times 1$ vectors of independent, centered random variables. Let $\Sigma _{lq}=\text {E}(\epsilon _l\epsilon _q^\top )$ for $l,q=1,2,3,4$. Then, $\text {E}(\epsilon _1^\top A\epsilon _2\epsilon _3^\top B\epsilon _4)=\text {tr}(\Sigma _{13}A\Sigma _{24}B^\top ) + \text {tr}(\Sigma _{14}A\Sigma _{23}B)$. See Liu and Saraiva (2019)

• Let $A=[a_{ij}]$ be a $n\times n$ nonstochastic matrix with zero diagonals and $\textbf{c}=[c_{1},\ldots ,c_L]$ be a $n\times 1$ nonstochastic vector. Let $\epsilon _1$, $\epsilon _2$ and $\epsilon _3$ be $n\times 1$ vectors of independents, centered random variables. Then, $\text {E}(\epsilon _1^\top A\epsilon _2\epsilon _3^\top \textbf{c}) = 0$

• $\text {diag}(A_n)=\text {diag}(a_{11},a_{22},\ldots ,a_{nn})$ is a diagonal $n\times n$ matrix from a square matrix A of order n.

• $\text {diag}_{h=1}^p(A_{n,h})\equiv \text {diag}_{h}(A_{n,h})= \text {diag}(A_{n,1},\ldots ,A_{n,p})$ is a block diagonal $np\times np$ matrix.

• $I_n$ is an identity matrix of order n.

• $\iota _n$ is a $n\times 1$ vector of ones with $n\in \mathbb {N}^*$.

• $\Lambda _\lambda =\text {diag}_l(\Lambda _{nT}(\lambda _l))$, where $\Lambda _{nT}(\lambda _l)=I_{T}\otimes \Lambda _n(\lambda _l)$; with $\Lambda _n(\lambda _l)=(I_{n}-\lambda _l W_{n})^{-1}$ for SAR case or $\Lambda _n(\lambda _l)=(I_{n}+\lambda _l W_{n})$ for SMA case.

1.2 A.2 Simultaneous moment functions

1.2.1 A.2.1 Proof of Equation (14)

Such that $\overset{a}{{\bar{\epsilon }}} = \underset{a\text { times}}{\underbrace{M_{nT}\dots M_{nT}}}\epsilon =M_{nT}^a\epsilon$, $M_{nT} Q_{nT,h}=(I_T\otimes M_n)(Q_{T,h}\otimes I_n)=Q_{nT,h} M_{nT}$ and $Q_{nT,h}\Sigma _{lq}=\sigma _{h_{lq}}Q_{nT,h}$ for $h=0,1$, one can write

$$\begin{aligned}{} & {} \text {E}\left( \overset{a}{{\bar{\epsilon }}_l^{\top }}Q_{nT,h}\overset{b}{{\bar{\epsilon }}_q}\right) = \text{ E }\left[ \text {tr}(\epsilon ^{\top }_lM_{nT}^{\top a}Q_{nT,h}M_{nT}^b\epsilon _q)\right] = \sigma _{h_{lq}}{\text {tr}\left( M_n^{\top a}M_n^b\right) }\text {tr}\left( Q_{T,h}\right) \\{} & {} \quad = \sigma _{h_{lq}}\phi _{ab}\text {tr}\left( Q_{nT,h}\right) \end{aligned}$$

1.2.2 A.2.2 Proof of Equation (15)

Since, $\overset{a}{{\overline{\epsilon }}}_l=\overset{a}{{\overline{\upsilon }}_l}-\lambda _l\overset{a+1}{\overline{\upsilon }_l}$,

$$\begin{aligned}{} & {} \overset{a}{{\overline{\epsilon }}_l^{\top }}Q_{nT,h}\overset{b}{{\overline{\epsilon }}_q}= \overset{a}{{\overline{\upsilon }}_l}^{\top } Q_{nT,h} \overset{b}{{\overline{\upsilon }}_q} - \lambda _l\overset{a+1}{\overline{\upsilon }_l}^{\top } Q_{nT,h} \overset{b}{{\overline{\upsilon }}_q} - \lambda _q\overset{a}{{\overline{\upsilon }}_l}^{\top } Q_{nT,h} \overset{b}{\overline{{\bar{\upsilon }}}_q} \\{} & {} \quad +\, \lambda _l\lambda _q\overset{a+1}{\overline{\upsilon }_l}^{\top } Q_{nT,h} \overset{b+1}{\overline{\upsilon }_q} \end{aligned}$$

Such that $\overset{a}{{\overline{\upsilon }}}^{\top }_l Q_{nT,h} \overset{b}{{\overline{\upsilon }}}_q=\text {tr}(Q_{nT,h})\gamma ^{lq,h}_{ab}$ we finally get,

$$\begin{aligned} \overset{a}{{\bar{\epsilon }}_l^{\top }}Q_{nT,h}\overset{b}{{\bar{\epsilon }}_q} =\text {tr}(Q_{h})(\gamma _{ab}^{lq,h} - \lambda _l\gamma _{a+1,b}^{lq,h} - \lambda _q\gamma _{a,b+1}^{lq,h} + \lambda _l\lambda _q\gamma _{a+1,b+1}^{lq,h}) \end{aligned}$$

1.2.3 A.2.3 Proof of Equation (20)

Since, for all $a\in \mathbb {N}$, $\overset{a}{{\overline{\upsilon }}}_l=\overset{a}{{\overline{\epsilon }}_l}+\lambda _l\overset{a+1}{\overline{\epsilon }_l}$

$$\begin{aligned}{} & {} \overset{a}{{\overline{\upsilon }}_l^{\top }}Q_{nT,h}\overset{b}{{\overline{\upsilon }}_q} = \overset{a}{{\overline{\epsilon }}_l}^{\top }Q_{nT,h}\overset{b}{{\overline{\epsilon }}_q} + \lambda _l\overset{a+1}{{{\overline{\epsilon }}}_l}^{\top }Q_{nT,h}\overset{b}{{\overline{\epsilon }}_q} + \lambda _q\overset{a}{{\overline{\epsilon }}_l}^{\top }Q_{nT,h}\overset{b+1}{{{\overline{\epsilon }}}_q} \\{} & {} \quad +\, \lambda _l\lambda _q\overset{a+1}{{{\overline{\epsilon }}}_l}^{\top }Q_{nT,h}\overset{b+1}{\overline{\epsilon }_q} \end{aligned}$$

Using (14), the expectation gives,

$$\begin{aligned} \text {E}\left[ \frac{\overset{a}{{\bar{\upsilon }}_l^{\top }}Q_{nT,h}\overset{b}{{\bar{\upsilon }}_q}}{\text {tr}\left( Q_{nT,h}\right) }\right] =\text {E}\left[ \gamma _{ab}^{lq,h}\right] = \sigma _{h_{lq}}\left[ \phi _{ab} + \lambda _l\phi _{a+1,b} + \lambda _q\phi _{a,b+1} + \lambda _l\lambda _q\phi _{a+1,b+1} \right] \end{aligned}$$

1.3 A.3 Derivation of $\Omega _{\textbf{g}}$ in (52)

Since $\Omega _{\textbf{g},11}=\text {E}\left( \textbf{g}_{1}\textbf{g}^\top _{1}\right)$ has a typical element $\begin{bmatrix}\text {E}\left( \textbf{g}_{1,l}\textbf{g}^\top _{1,q}\right) \end{bmatrix}$ and such that

$$\begin{aligned}\text {E}\left( \textbf{g}_{1,l}\textbf{g}^\top _{1,q}\right) =\sum _h\sigma _{h_{lq}}{\textbf{H}}^\top \Lambda _{nT}(\lambda _l) Q_{nT,h}\Lambda ^\top _{nT}(\lambda _q){\textbf{H}}\end{aligned}$$

, then,

$$\begin{aligned}\begin{aligned}&\Omega _{\textbf{g},11} =\sum _{h}\left[ \sigma _{h_{lq}}\textbf{H}^\top \Lambda _{nT}(\lambda _l) Q_{nT,h}\Lambda _{nT}^\top (\lambda _q)\textbf{H}\right] \\&\quad =\sum _h\Sigma _h\otimes (\textbf{H}^\top \Lambda _{nT}(\lambda _l) Q_{nT,h}\Lambda _{nT}^\top (\lambda _q)\textbf{H}) \end{aligned} \end{aligned}$$

$\Omega _{\textbf{g},22}=\text {E}\left( \textbf{g}_{2}\textbf{g}^\top _{2}\right)$ has a typical element $\begin{bmatrix}\text {E}\left( \textbf{g}_{2,lq}\textbf{g}^\top _{2,kt}\right) \end{bmatrix}$ for $l,q,k,t=1,\ldots ,L$; since

$$\begin{aligned}{} & {} \text {E}\left( \textbf{g}_{2,lq}\textbf{g}^\top _{2,kt}\right) =\text {E}\left( \epsilon _l^\top \Xi _{nT,r}\epsilon _q\epsilon _k^\top \Xi _{nT,s}\epsilon _t\right) \\{} & {} \quad =\sum _h\sum _{h'}\begin{bmatrix} \sigma _{h_{lk}}\sigma _{h_{qt}}\text {tr} \left( Q_{h}\Xi _{nT,r}Q_{h'}\Xi _{nT,s}^\top \right) + \sigma _{h_{lt}}\sigma _{h_{qk}}\text {tr} \left( Q_{h}\Xi _{nT,r}Q_{h'}\Xi _{nT,s}\right) \end{bmatrix} \end{aligned}$$

and such that, $Q_{nT,h}=Q_{T,h}\otimes I_n$ for $h=0,1$ with $Q_{T,h}Q_{T,h'}=0$ if $h\ne h'$ and $\Xi _{nT,r}=I_T\otimes \Xi _{n,r}$ for $r,s=1,\ldots ,p$, we have,

$$\begin{aligned} \begin{aligned} \text {E}\left( \textbf{g}_{2,lq}\textbf{g}^\top _{2,kt}\right)&=\sum _h\text {tr}(Q_{T,h}) \left[ \sigma _{h_{lk}}\sigma _{h_{qt}} \text {tr}(\Xi _{n,r}\Xi _{n,s}^\top ) + \sigma _{h_{lt}}\sigma _{h_{qk}}\text {tr}(\Xi _{n,r}\Xi _{n,s})\right] \\&=\sum _h\text {tr}\left( Q_{T,h}\right) \left( \sigma _{h_{lk}}\sigma _{h_{qt}}\Upsilon _1 + \sigma _{h_{lt}}\sigma _{h_{qk}}\Upsilon _2\right) \end{aligned}\end{aligned}$$

where,

$$\begin{aligned} \Upsilon _1= \begin{bmatrix}\text {tr}(\Xi _{n,1}\Xi _{n,1}^\top )&{}\quad \cdots &{}\quad \text {tr}(\Xi _{n,1}\Xi _{n,p}^\top )\\ \vdots &{}\quad \ddots &{}\quad \vdots \\ \text {tr}(\Xi _{n,p}\Xi _{n,1}^\top )&{}\quad \cdots &{}\quad \text {tr}(\Xi _{n,p}\Xi _{n,p}^\top )\end{bmatrix} \text { and } \Upsilon _2= \begin{bmatrix}\text {tr}\left( \Xi _{n,1}\Xi _{n,1}\right) &{}\quad \cdots &{}\quad \text {tr}\left( \Xi _{n,1}\Xi _{n,p}\right) \\ \vdots &{}\quad \ddots &{}\quad \vdots \\ \text {tr}\left( \Xi _{n,p}\Xi _{n,1}\right) &{}\quad \cdots &{}\quad \text {tr}\left( \Xi _{n,p}\Xi _{n,p}\right) \end{bmatrix} \end{aligned}$$

are $p\times p$ matrix of constants. Hence,

$$\begin{aligned}\begin{aligned} \Omega _{\textbf{g},22}=\begin{bmatrix}\text {E}\left( \textbf{g}_{2,lq}\textbf{g}^\top _{2,kt}\right) \end{bmatrix}&=\sum _h\text {tr}\left( Q_{T,h}\right) \left[ \sigma _{h_{lk}}\sigma _{h_{qt}}\Upsilon _1 + \sigma _{h_{lt}}\sigma _{h_{qk}}\Upsilon _2\right] \\&=\sum _h\text {tr}\left( Q_{T,h}\right) \left\{ \left[ \sigma _{h_{lk}}\sigma _{h_{qt}}\Upsilon _1\right] + \left[ \sigma _{h_{lt}}\sigma _{h_{qk}}\Upsilon _2\right] \right\} \\&=\sum _h\text {tr}\left( Q_{T,h}\right) \left( \Psi _{1,h}\otimes \Upsilon _1 + \Psi _{2,h}\otimes \Upsilon _2\right) \end{aligned} \end{aligned}$$

where, with the following matrix construction plan: $\Psi _{row}({t,q})=[\circ \overset{t}{\rightarrow }\circ \; \circ \overset{t,q}{\rightarrow \rightarrow }\ ]$ and $\Psi _{column}({l,k})=[\circ \overset{k}{\rightarrow }\circ \; \circ \overset{l,k}{\rightarrow \rightarrow }]$, the matrices $\Psi _{1,h}$ $\Psi _{2,h}$ have the following forms:

$$\begin{aligned} \Psi _{1,h}=\begin{bmatrix}\sigma _{h_{lk}}\sigma _{h_{qt}}\end{bmatrix} =\begin{bmatrix} \sigma _{h_{11}}\sigma _{h_{11}}&{}\quad \cdots &{}\quad \sigma _{h_{11}}\sigma _{h_{1L}} &{}\quad \sigma _{h_{11}}\sigma _{h_{21}}&{}\quad \cdots \cdots &{}\quad \sigma _{h_{11}}\sigma _{h_{LL}} \\ \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ \sigma _{h_{1L}}\sigma _{h_{11}}&{}\quad \cdots &{}\quad \sigma _{h_{1L}}\sigma _{h_{1L}} &{}\quad \sigma _{h_{1L}}\sigma _{h_{21}}&{}\quad \cdots \cdots &{}\quad \sigma _{h_{1L}}\sigma _{h_{LL}}\\ \sigma _{h_{21}}\sigma _{h_{11}}&{}\quad \cdots &{}\quad \sigma _{h_{21}}\sigma _{h_{1L}} &{}\quad \sigma _{h_{21}}\sigma _{h_{21}}&{}\quad \cdots \cdots &{}\quad \sigma _{h_{21}}\sigma _{h_{LL}} \\ \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ \\ \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ \sigma _{h_{LL}}\sigma _{h_{11}}&{}\quad \cdots &{}\quad \sigma _{h_{LL}}\sigma _{h_{1L}}&{}\quad \sigma _{h_{LL}}\sigma _{h_{21}}&{}\quad \cdots \cdots &{}\quad \sigma _{h_{LL}}\sigma _{h_{LL}} \end{bmatrix} \end{aligned}$$

and

$$\begin{aligned} \Psi _{2,h}=\begin{bmatrix}\sigma _{h_{lt}}\sigma _{h_{qk}}\end{bmatrix} =\begin{bmatrix} \sigma _{h_{11}}\sigma _{h_{11}}&{}\quad \cdots &{}\quad \sigma _{h_{1L}}\sigma _{h_{11}} &{}\quad \sigma _{h_{11}}\sigma _{h_{21}}&{}\quad \cdots \cdots &{}\quad \sigma _{h_{1L}}\sigma _{h_{L1}} \\ \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ \sigma _{h_{11}}\sigma _{h_{1L}}&{}\quad \cdots &{}\quad \sigma _{h_{1L}}\sigma _{h_{1L}} &{}\quad \sigma _{h_{11}}\sigma _{h_{2L}}&{}\quad \cdots \cdots &{}\quad \sigma _{h_{1L}}\sigma _{h_{LL}}\\ \sigma _{h_{21}}\sigma _{h_{11}}&{}\quad \cdots &{}\quad \sigma _{h_{2L}}\sigma _{h_{11}} &{}\quad \sigma _{h_{21}}\sigma _{h_{21}}&{}\quad \cdots \cdots &{}\quad \sigma _{h_{2L}}\sigma _{h_{L1}} \\ \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ \\ \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ \sigma _{h_{L1}}\sigma _{h_{1L}}&{}\quad \cdots &{}\quad \sigma _{h_{LL}}\sigma _{h_{1L}} &{}\quad \sigma _{h_{L1}}\sigma _{h_{2L}}&{}\quad \cdots \cdots &{}\quad \sigma _{h_{LL}}\sigma _{h_{LL}} \end{bmatrix} \end{aligned}$$

1.4 A.4 Identification

The identification of the spatial panel simultaneous equations model (1) relies on the two-step strategy initially adopted in Yang and Lee (2017), and Liu and Saraiva (2019). The first-step consists in the identification of the “pseudo” reduced form parameters and the second-step recovers the structural parameters from the pseudo” reduced form parameters.

1.4.1 A.4.1 Identification of the pseudo reduced form parameters

Let $\Theta _0$, $\Delta _0$, ${\textbf{B}}_0$ and ${\textbf{A}}_0$ be the matrices of the true parameters in the DGP. When $\Theta _0$ is non singular, the pseudo reduced form of model (62) is

$$\begin{aligned} \textbf{Y}= \overline{\textbf{Y}}\Phi _0+ \textbf{X}\Pi _0+ \textbf{V} \end{aligned}$$

(62)

where $\Phi _0=\Delta _0(I_{L}-\Theta _0)^{-1}$, $\Pi _0={\textbf{B}}_0(I_{L}-\Theta _0)^{-1}$ and $\textbf{V}=\textbf{U}(I_{L}-\Theta _0)^{-1}$. With $\Phi _0=[\varphi _{lq,0}]$ and $\Pi _0=[\pi _{1,0},...,\pi _{L,0}]$, under the GMM framework, the q-th equation in model (62) is given by

$$\begin{aligned} y_q = \sum _{l=1}^L\varphi _{lq,0}W_{nT}y_l+ {\textbf{X}}\pi _{q,0}+ v_q \end{aligned}$$

(63)

where

$$\begin{aligned} W_{nT}y_l = {\textbf{R}}_l(\Pi _0^\top \otimes I_{nT}){\textbf{x}} +{\textbf{R}}_l{\textbf{v}} \end{aligned}$$

(64)

with ${\textbf{R}}_l=(i_{L,l}^\top \otimes W_{nT})(I_{nTL}-\Phi _0^\top \otimes W_{nT})^{-1}$, ${\textbf{x}}=\text {vec}({\textbf{X}})$, ${\textbf{v}}=\text {vec}({\textbf{V}})$ and $i_{L,l}$ be the l-th vector of an identity matrix $I_L$. We note that $\text {E}(W_{nT}y_l)={\textbf{R}}_l(\Pi _0^\top \otimes I_{nT}){\textbf{x}}$ since $\text {E}({\textbf{R}}_l{\textbf{v}})=0$. Now, the residual function

$$\begin{aligned} \text {v}_q(\rho _q) = y_q - \sum _{l=1}^L\varphi _{lq}W_{nT}y_l- {\textbf{X}}\pi _{q} \end{aligned}$$

(65)

for the q-th equation can be rewritten as

$$\begin{aligned} \text {v}_q(\rho _q) = d_q(\rho _q)+ \text {v}_q + \sum _{l=1}^L(\varphi _{lq,0}-\varphi _{lq}){\textbf{R}}_q{\textbf{v}} \end{aligned}$$

(66)

where $\rho _{q}=(\varphi _{1q},\ldots ,\varphi _{Lq},\pi _{q}^\top )^\top$ and $d_q(\rho _q)=[\text {E}(W_{nT}y_1),\ldots ,\text {E}(W_{nT}y_L),{\textbf{X}}](\rho _{q,0}-\rho _{q})$. The pseudo reduced form parameters in model (62) can be identified by the moment conditions proposed in the Sect. (4.3.1) on the main text. Then, the linear and quadratic moment functions can be written as

$$\begin{aligned} {\textbf{f}}_{1,l}\equiv & {} {\textbf{f}}_{1,l}(\rho _l) = \textbf{H}^\top \text {v}_l(\rho _l), \\ \textbf{f}_{2,lq}\equiv \textbf{f}_{2,lq}(\rho _l,\rho _q)= & {} \text {v}_q^\top (\rho _q) \Xi ^{lq}_{nT,r} \text {v}_l(\rho _l) \quad \text { for }\quad r=1,\ldots ,p; l,q=1,\ldots ,L\\= & {} \text {v}_q^\top (\rho _q) (\Lambda _{nT}^{-1}({\hat{\lambda }}_q))^{\top }\Xi _{nT,r}\Lambda _{nT}^{-1}({\hat{\lambda }}_l) \text {v}_l(\rho _l) \end{aligned}$$

Let $\textbf{f}({\rho })=\begin{bmatrix}\textbf{f}^\top _{1}({\rho })&\textbf{f}^\top _{2}({\rho })\end{bmatrix}^\top$ where $\rho =(\rho _1,\ldots ,\rho _L)$, $\textbf{f}_{1}({\rho })=\begin{bmatrix}\textbf{f}_{1,1}^\top&\quad \cdots&\quad \textbf{f}_{1,L}^\top \end{bmatrix}^\top$ and $\textbf{f}_{2}({\rho })=\begin{bmatrix}\textbf{f}_{2,11}^\top&\quad \cdots&\quad \textbf{f}_{2,1L}^\top&\quad \textbf{f}_{2,21}^\top&\quad \cdots&\quad \cdots&\quad \textbf{f}_{2,LL}^\top \end{bmatrix}^\top$. The identification of $\rho _0$ by means of the moment conditions $\text {E}[{\textbf{f}}(\rho _0)] =0$, requires that the moment equations $\lim _{n\rightarrow \infty }(nT)^{-1}\text {E}[{\textbf{f}}(\rho _0)] =0$ have a unique solution at $\rho =\rho _0$ (see Hansen 1982). From (66), it follows that

$$\begin{aligned} \begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{nT}\text {E}[{\textbf{f}}_{1,l}(\rho _l)]&= \lim _{n\rightarrow \infty }\frac{1}{nT}\textbf{H}^\top \text {v}_l(\rho _l)\\&= \lim _{n\rightarrow \infty }\frac{1}{nT}\textbf{H}^\top d_l(\rho _l) \\&= \lim _{n\rightarrow \infty }\frac{1}{nT}\textbf{H}^\top [\text {E}(W_{nT}y_1),\ldots ,\text {E}(W_{nT}y_L),{\textbf{X}}](\rho _{l,0}-\rho _{l}) \\ \end{aligned} \end{aligned}$$

for $l=1,\ldots ,L$. The linear moment equations, $\lim _{n\rightarrow \infty }\frac{1}{nT}\text{ E }[{\textbf{f}}_{1,l}(\rho _l)]=0$, have a unique solution at $\rho _l=\rho _{l,0}$ if $\textbf{H}^\top [\text {E}(W_{nT}y_1),\ldots ,\text {E}(W_{nT}y_L),{\textbf{X}}]$ has full column rank for n sufficiently large. A necessary condition for this rank condition is that $\text {rank}[\text {E}(W_{nT}y_1),\ldots ,\text {E}(W_{nT}y_L),{\textbf{X}}]=k+L$ and $\text {rank}(\textbf{H})\ge k+L$ for n sufficiently large.

However, if the rank condition fails (i.e., $[\text {E}(W_{nT}y_1),\ldots ,\text {E}(W_{nT}y_L),{\textbf{X}}]$ does not have full column rank), identification of the model may still be feasible through the quadratic moment conditions. Suppose for some $m=\{0,1,\ldots ,L-1\}$, $\text {E}(W_{nT}y_l)$ and $[\text {E}(W_{nT}y_1),\ldots ,\text {E}(W_{nT}y_{m}),{\textbf{X}}]$ are linearly dependent¹² such that $\text {E}(W_{nT}y_l)=\sum _{q=1}^{m}c_{1,ql}\text {E}(W_{nT}y_q)+{\textbf{X}} c_{2,l}$ for $l=m+1,\ldots ,L$, where $(c_{1,1l},\ldots ,c_{1,m l},c_{2,l}^\top )\in {\mathbb {R}}^{k+m}$ is a vector of constants. This allows to write

$$\begin{aligned}{} & {} d_q(\rho _q)=\sum _{j=1}^{m}\text {E}(W_{nT}y_j)\left[ \varphi _{jq,0}-\varphi _{jq}+\sum _{l=m+1}^Lc_{1,jl}(\varphi _{lq,0}-\varphi _{lq})\right] \\{} & {} \quad +\,{\textbf{X}}\left[ \pi _{q,0}-\pi _{q} + \sum _{l=m+1}^Lc_{2,l}(\varphi _{lq,0}-\varphi _{lq})\right] \end{aligned}$$

and thus, $\lim _{n\rightarrow \infty }\frac{1}{nT}\text{ E }[{\textbf{f}}_{1,l}(\rho _l)]=0$ entails that:

$$\begin{aligned} \begin{aligned} \varphi _{jq}&= \varphi _{jq,0}+\sum _{l=m+1}^Lc_{1,jl}(\varphi _{lq,0}-\varphi _{lq})\\ \pi _{q}&= \pi _{q,0} + \sum _{l=m+1}^Lc_{2,l}(\varphi _{lq,0}-\varphi _{lq}) \end{aligned} \end{aligned}$$

(67)

for $j=1,\ldots ,m$ and $q=1,\ldots ,L$, provided that $\textbf{H}^\top [\text {E}(W_{nT}y_1),\ldots ,\text {E}(W_{nT}y_m),{\textbf{X}}]$ has full column rank for large enough n. Therefore, $(\varphi _{1q,0},\ldots ,\varphi _{mq,0},\pi _{q,0}^\top )$ can be identified if $\varphi _{lq,0}$ (for $l=m+1,\ldots ,L$) can be identified from the quadratic moment conditions. With $\rho _q$ given by (67), we have

$$\begin{aligned} \begin{aligned} \text {E}(\textbf{f}_{2,lq})&= \sum _{s=1}^L(\varphi _{sl,0}-\varphi _{sl})\text {tr}\left[ (\Lambda _{nT}^{-1}({\hat{\lambda }}_q))\top \Xi _{nT,r}\Lambda _{nT}^{-1}({\hat{\lambda }}_l){\textbf{R}}_l\text{ E }({\textbf{v}}{\textbf{v}}_q^\top ) \right] \\&\quad + \sum _{j=1}^L(\varphi _{jl,0}-\varphi _{jl})\text {tr}\left[ (\Lambda _{nT}^{-1}({\hat{\lambda }}_q))^{\top }\Xi _{nT,r}\Lambda _{nT}^{-1}({\hat{\lambda }}_l)\text {E}({\textbf{v}}_l{\textbf{v}}^\top ){\textbf{R}}_q^\top \right] \\&\quad +\sum _{s=1}^L\sum _{j=1}^L (\varphi _{sl,0}-\varphi _{sl})(\varphi _{jl,0}-\varphi _{jl})\text {tr}\\&\quad \left[ (\Lambda _{nT}^{-1}({\hat{\lambda }}_q))^{\top }\Xi _{nT,r}\Lambda _{nT}^{-1}({\hat{\lambda }}_l){\textbf{R}}_l\text{ E }({\textbf{v}}{\textbf{v}}^\top ){\textbf{R}}_q^\top \right] \end{aligned} \end{aligned}$$

where $\text{ E }({\textbf{v}}{\textbf{v}}^\top )=\sum _{h=0}^1(I_L-\Theta _0)^{-1}({\hat{\Lambda }}_{\lambda }^{-1})^{\top }(\Sigma _h\otimes Q_{nT,h}){\hat{\Lambda }}^{-1}_{\lambda }(I_L-\Theta _0^\top )^{-1}$ and $\text {E}({\textbf{v}}{\textbf{v}}_q^\top )= \text{ E }({\textbf{v}}{\textbf{v}}^\top )(i_{L,q}\otimes I_{nT})$. Thus, at $\Phi _0$, the quadratic moment equations, $\lim _{n\rightarrow \infty }\frac{1}{nT}\text{ E }[{\textbf{f}}_{2,lq}(\rho _l,\rho _q)]=0$ for $l,q=1,\ldots ,L$, have a unique solution, if the equations

$$\begin{aligned} \begin{aligned}&\lim _{n\rightarrow \infty }\frac{1}{nT} \left[ \sum _{s=1}^L(\varphi _{sl,0}-\varphi _{sl})\text {tr}\left[ (\Lambda _{nT}^{-1}({\hat{\lambda }}_q))^{\top }\Xi _{nT,r}\Lambda _{nT}^{-1}({\hat{\lambda }}_l){\textbf{R}}_l\text {E}({\textbf{v}}{\textbf{v}}_q^\top ) \right] \right] \\&\quad +\lim _{n\rightarrow \infty }\frac{1}{nT} \left[ \sum _{j=1}^L(\varphi _{jl,0}-\varphi _{jl})\text {tr}\left[ (\Lambda _{nT}^{-1}({\hat{\lambda }}_q))^{\top }\Xi _{nT,r}\Lambda _{nT}^{-1}({\hat{\lambda }}_l)\text {E}({\textbf{v}}_l{\textbf{v}}^\top ){\textbf{R}}_q^\top \right] \right] \\&\quad +\lim _{n\rightarrow \infty }\frac{1}{nT} \left[ \sum _{s=1}^L\sum _{j=1}^L (\varphi _{sl,0}-\varphi _{sl})(\varphi _{jl,0}-\varphi _{jl})\text {tr}\right. \\&\quad \left. \left[ (\Lambda _{nT}^{-1}({\hat{\lambda }}_q))^{\top }\Xi _{nT,r}\Lambda _{nT}^{-1}({\hat{\lambda }}_l){\textbf{R}}_l\text{ E }({\textbf{v}}{\textbf{v}}^\top ){\textbf{R}}_q^\top \right] \right] =0 \end{aligned}\qquad \quad \end{aligned}$$

(68)

for $r=1,\ldots ,p$ and $l,q=1,\ldots ,L$. The following assumption provides sufficient conditions for the identification of the pseudo reduced form parameters.

Assumption 9

At least one of the following conditions holds:

(a) $\lim _{n\rightarrow \infty }\frac{1}{nT}\textbf{H}^\top [\text {E}(W_{nT}y_1),\ldots ,\text {E}(W_{nT}y_L),{\textbf{X}}]$ exists and has full column rank.

(b) $\lim _{n\rightarrow \infty }\frac{1}{nT}\textbf{H}^\top [\text {E}(W_{nT}y_1),\ldots ,\text {E}(W_{nT}y_m),{\textbf{X}}]$ exists and has full column rank for some $0 \le m \le L-1$. Equation (67) for $r=1,\ldots ,p$ and $l,q=1,\ldots ,L$, have a unique solution at $\Phi _0$.

1.4.2 A.4.2 Identification of the structural parameters

Having identified the “pseudo” reduced form parameters $\Phi _0$ and $\Pi _0$ from the linear and quadratic moment conditions, we can proceed on the identification of the structural parameters ${\textbf{C}}_0=[(I_L-\Theta _0)^\top ,-\Delta _0^\top ,-{\textbf{B}}_0^\top ]^\top$ through the linear restrictions $\Phi _0=\Delta _0(I_{L}-\Theta _0)^{-1}$ and $\Pi _0={\textbf{B}}_0(I_{L}-\Theta _0)^{-1}$ in a similar way to the classical linear simultaneous equations model. Given $D_l$ restrictions on ${\textbf{c}}_{l,0}$ such that ${\textbf{D}}_l{\textbf{c}}_{l,0}$=0, where ${\textbf{D}}_l$ and ${\textbf{c}}_{l,0}$ are, respectively, $D_l\times (k+2L)$ matrix of known constants and the l-th column of ${\textbf{C}}_0$, the rank and order conditions for ${\textbf{c}}_{l,0}$ to be identified are as follow: (i) - the sufficient and necessary rank condition is $rank({\textbf{D}}_l{\textbf{C}}) = L-1$; (ii) - the necessary order condition is $D_l\ge L-1$, for $l=1,\ldots ,L$.

Assumption 10

The following conditions holds for $l=1,\ldots ,L$, ${\textbf{D}}_l{\textbf{c}}_{l,0}=0$ for some $D_l\times (k+2L)$ constant matrix $D_l$ with $rank({\textbf{D}}_l{\textbf{C}}) = L-1$.

1.5 A.5 Tables

See Tables 1, 2, 3, 4, 5, 6, 7, 8, 9.

Table 1 SEC estimation: change in spatial coefficient $\lambda _2$, for $T=8$, $n=30$ in the SAR case

Table 2 SEC estimation: change in the number of neighbors, for $T=8$, $n=30$ in the SAR case

Table 3 SEC estimation: change in covariance design $\Sigma _h$, for $T=8$, $n=30$ in the SAR case

Table 4 Estimation results for the structural parameters for $T=8$, $n=30$, $\lambda _1=3\lambda _2=0.6$, $d_1$, $J=2$ and $\Lambda _{sar}$

Table 5 Estimation results for the structural parameters for $T=8$, $n=30$, $\lambda =0.2$, $d_1$, $J=2$ and $\Lambda _{sar}$

Table 6 Estimation results for the structural parameters for $T=8$, $n=30$, $\lambda _1=3\lambda _2=0.6$, $d_2$, $J=2$ and $\Lambda _{sar}$

Table 7 Estimation results for the structural parameters for $T=8$, $n=30$, $\lambda _1=3\lambda _2=0.6$, $d_2$, $J=6$ and $\Lambda _{sar}$

Table 8 Estimation results for the simultaneous panel data model (Current expenditure)

Table 9 Estimates for the simultaneous panel data model (Capital expenditure)