Revisiting estimation methods for spatial econometric interaction models

Abstract

This article develops improved calculation techniques for estimating the spatial econometric interaction model of LeSage and Pace (2008) by maximum likelihood (MLE), Bayesian Markov Chain Monte Carlo (MCMC) and spatial two-stage least-squares (S2SLS). The refined estimation methods derive the parameter estimates and their standard errors exclusively from moment matrices with low dimensions. For the computation of these moments, we exploit efficiency gains linked to a matrix formulation of the model, which we generalize to make more flexible use of the exogenous variables. To improve the MLE we restructure the Hessian matrix and the quadratic term in the likelihood function. We also derive a moment based formulation of the Bayesian MCMC estimator from the same likelihood restructuring. Finally, the S2SLS estimator presented in this article is the first one to exploit the efficiency gains of the matrix formulation and also solves the problem of collinearity among spatial instruments. Several benchmarks show that these moment based estimators scale very well to large samples and can be used to estimate models with 100 million flows in just a few minutes. In addition to the improved estimation methods, this article presents a new way to define a feasible parameter space for the spatial econometric interaction model, which allows to verify the models consistency with a minimal computational burden. All of these developments indicate that the spatial econometric extension of the traditional gravity model has become an increasingly mature alternative and should eventually be considered a standard modeling approach for origin-destination flows.

Appendices

Appendix A Hessian matrix

This appendix develops the simplified analytical expression of the elements of the Hessian matrix that are presented in Sect. 4.2.2. These expressions are derived from the second order derivatives of the likelihood function given in (16). LeSage and Pace (2009) derive similar expressions for a spatial model of order one. Another treatment of the Hessian matrix for the spatial econometric models can be found in (Anselin 1988, ][see pages 74ff.), who provides many computation steps linked to derivatives of the spatial filter matrix that are similar to the ones shown below.

$$\begin{aligned} \begin{aligned} \frac{\partial A}{\partial \rho _d}&= -W_d\\ \frac{\partial A'A}{\partial \rho _d}&= -2A'W_d \end{aligned} \qquad \begin{aligned} \frac{\partial ln|A|}{\partial \rho _d} = {{\,\mathrm{tr}\,}}(A^{-1}\frac{\partial A}{\partial \rho _d}) = -{{\,\mathrm{tr}\,}}(A^{-1}W_d)\\ \frac{\partial A^{-1}}{\partial \rho _d} = -A^{-1}\frac{\partial A}{\partial \rho _d}A^{-1}=A^{-1}W_dA^{-1} \end{aligned} \end{aligned}$$

The following paragraphs show that the Hessian matrix of higher order spatial models, in particular the one of model (6), also admits very simple analytical solutions to most of its elements. Since the expression of the likelihood function is invariant by permutation of the terms $\rho _oW_o$, $\rho _dW_d$, $\rho _wW_w$ many elements of the Hessian matrix are up to the indexes (o, d, w) identical. This property allows shorten the following argumentation, since we only have to develop the expressions related to $\rho _d$, while the expressions related to $\rho _d$ and $\rho _w$ are inferred by exchanging the indexes.

1.1 Block $H_{11}$

The Hessian block $H_{11}$ given is the second order derivative of the likelihood function with respect to $\rho$

$$\begin{aligned} H_{11} = \frac{\partial ^{2} {\mathcal {L}}}{\partial \rho \partial \rho '}= \left[ \begin{array}{ccc} {\frac{\partial ^{2} {\mathcal {L}}}{\partial \rho _d^{2}}} &{} {\frac{\partial ^{2} {\mathcal {L}}}{\partial \rho _d \partial \rho _o}} &{} {\frac{\partial ^{2} {\mathcal {L}}}{\partial \rho _d \partial \rho _w}} \\ \bullet &{} {\frac{\partial ^{2} {\mathcal {L}}}{\partial \rho _o^2}} &{} {\frac{\partial ^{2} {\mathcal {L}}}{\partial \rho _o \partial \rho _w}} \\ \bullet &{} \bullet &{} {\frac{\partial ^{2} {\mathcal {L}}}{\partial \rho _w^2}} \end{array}\right] = \left[ \begin{array}{ccc} V_{dd} &{} V_{do} &{} V_{dw} \\ \bullet &{} V_{oo} &{} V_{ow} \\ \bullet &{} \bullet &{} V_{ww} \end{array}\right] , \end{aligned}$$

where for $j,k \in (d,o,w)$, $V_{jk} = -{{\,\mathrm{tr}\,}}(W_jA^{-1}W_kA^{-1}) -\frac{1}{\sigma ^2}y'W_j'W_ky$. The steps below derive the expression of $V_{dd}$, while all other elements can be found using the same arguments and only require to exchange the indexes.

$$\begin{aligned} \begin{aligned} \frac{\partial ^2 {\mathcal {L}}}{\partial \rho _d^2}&= \frac{\partial ^2}{\partial \rho _d^2}(\ln |A|) + \frac{\partial ^2}{\partial \rho _d^2}(-\frac{1}{2\sigma ^2}(Ay-Z\delta )'(Ay-Z\delta ))\\&= \frac{\partial }{\partial \rho _d}({{\,\mathrm{tr}\,}}(A^{-1}\frac{\partial }{\partial \rho _d}A)) + \frac{\partial ^2}{\partial \rho _d^2}(-\frac{1}{2\sigma ^2}y'A'Ay) \\&= \frac{\partial }{\partial \rho _d}(-{{\,\mathrm{tr}\,}}(A^{-1}W_d)) + \frac{\partial }{\partial \rho _d}(-\frac{1}{2\sigma ^2}2y'A'(-W_d)y) \\&= -{{\,\mathrm{tr}\,}}(-A^{-1}(-W_d)A^{-1}W_d)) -\frac{y'W_d'W_dy}{\sigma ^2} \\&= -{{\,\mathrm{tr}\,}}(W_dA^{-1}W_dA^{-1}) -\frac{y'W_d'W_dy}{\sigma ^2} := V_{dd} \end{aligned} \end{aligned}$$

1.2 Block $H_{21}$

The Hessian block $H_{21}$ given in (20) corresponds to the cross derivative of the likelihood function which we differentiate first with respect to $\theta = (\delta ' \sigma ^2)'$ and then with respect to $\rho$

$$\begin{aligned} H_{21}= \frac{\partial ^{2} {\mathcal {L}}}{\partial \theta \partial \rho '}= \begin{bmatrix} \frac{\partial ^{2} {\mathcal {L}}}{\partial \delta \partial \rho _d } &{} \frac{\partial ^{2} {\mathcal {L}}}{\partial \delta \partial \rho _o } &{} \frac{\partial ^{2} {\mathcal {L}}}{\partial \delta \partial \rho _w } \\ \frac{\partial ^{2} {\mathcal {L}}}{\partial \sigma ^2 \partial \rho _d } &{} \frac{\partial ^{2} {\mathcal {L}}}{\partial \sigma ^2 \partial \rho _o } &{} \frac{\partial ^{2} {\mathcal {L}}}{\partial \sigma ^2 \partial \rho _w } &{} \end{bmatrix} = \begin{bmatrix} -\frac{Z'L}{\sigma ^2} \\ \frac{\delta '(Z'L) - \tau (\rho )'(y_{\bullet }'L) }{\sigma ^4} \end{bmatrix}, \end{aligned}$$

where $L = (W_dy \; W_oy \; W_wy)$. The equations below derive analytical expressions for the elements in the first column of above matrix.

$$\begin{aligned} \begin{aligned} \frac{\partial ^{2}{\mathcal {L}}}{\partial \delta \partial \rho _d }&= \frac{\partial ^{2}}{\partial \delta \partial \rho _d }(-\frac{1}{2\sigma ^2}(Ay-Z\delta )'(Ay-Z\delta )) \\&= \frac{\partial ^{2}}{\partial \delta \partial \rho _d }(-\frac{1}{2\sigma ^2}(-2(Z\delta )'Ay) \\&= \frac{\partial }{\partial \delta }(\frac{(Z\delta )'(-W_d)y}{\sigma ^2}) \\&= \frac{-Z'W_dy}{\sigma ^2} \end{aligned} \quad \begin{aligned} \frac{\partial ^{2}{\mathcal {L}}}{\partial \sigma ^2 \partial \rho _d }&= \frac{\partial ^{2}}{\partial \sigma ^2 \partial \rho _d }(-\frac{1}{2\sigma ^2}(Ay-Z\delta )'(Ay-Z\delta )) \\&= \frac{\partial ^{2}}{\partial \sigma ^2 \partial \rho _d }(-\frac{1}{2\sigma ^2}(y'A'Ay-2(Z\delta )'Ay)) \\&= \frac{\partial }{\partial \rho _d}(\frac{y'A'Ay-2(Z\delta )'Ay}{2\sigma ^4}) \\&= \frac{-2(Ay)'W_dy+2\delta 'Z'W_dy}{2\sigma ^4} \end{aligned} \end{aligned}$$

The previously introduced permutation-invariance of the likelihood function then allows to apply the same steps to derive the other two columns. These expressions are simplified further using the multivariate formulation L of the spatial lags of the flow vector.

$$\begin{aligned} H_{21}&= \begin{bmatrix} -\frac{Z'W_dy}{\sigma ^2} &{} -\frac{Z'W_oy}{\sigma ^2} &{} -\frac{Z'W_wy}{\sigma ^2} &{} \\ \frac{\delta 'Z'W_dy-(Ay)'W_dy}{\sigma ^4} &{} \frac{\delta 'Z'W_oy-(Ay)'W_oy}{\sigma ^4} &{} \frac{\delta 'Z'W_wy-(Ay)'W_wy}{\sigma ^4} \end{bmatrix} \\&= \begin{bmatrix} -\frac{Z'L}{\sigma ^2} \\ \frac{\delta '(Z'L) - \tau (\rho )'(y_{\bullet }'L) }{\sigma ^4} \end{bmatrix} \end{aligned}$$

1.3 Block $H_{22}$

The Hessian block $H_{22}$ given in (20) contains the second order derivatives of the likelihood function with respect to $\theta$

$$\begin{aligned} H_{22} = {\frac{\partial ^{2} {\mathcal {L}}}{\partial \theta \partial \theta ^{\prime }}} =\left[ \begin{array}{ccc} {\frac{\partial ^{2} {\mathcal {L}}}{\partial \delta \partial \delta '}} &{} {\frac{\partial ^{2} {\mathcal {L}}}{\partial \delta \partial \sigma ^2}} \\ {\frac{\partial ^{2} {\mathcal {L}}}{\partial \sigma ^2 \partial \delta '}} &{} {\frac{\partial ^{2} {\mathcal {L}}}{\partial (\sigma ^2)^2}} \end{array}\right] =\left[ \begin{array}{ccc} -{\frac{Z'Z}{\sigma ^2}} &{} 0 \\ 0 &{} -{\frac{N}{2\sigma ^4}} \end{array}\right] \end{aligned}$$

The above block has the same expression as the Hessian matrix of a non-spatial linear model, when we evaluate at the maximum likelihood estimators in (17), in particular ${\hat{\sigma }}^2(\rho ) = (Ay - Z\delta (\rho ))'(Ay - Z\delta (\rho ))N^{-1}$ and ${\hat{\delta }}(\rho ) = (Z'Z)^{-1}Z'Ay$.

$$\begin{aligned} \begin{aligned} \frac{\partial ^{2}{\mathcal {L}} }{\partial \delta \partial \delta '}&= \frac{\partial ^{2}}{\partial \delta \partial \delta '} (-\frac{1}{2\sigma ^2}(Ay-Z\delta )'(Ay-Z\delta )) \\&= \frac{\partial ^{2}}{\partial \delta \partial \delta '} (\frac{-\delta 'Z'Z\delta }{2\sigma ^2}) \\&= \frac{-Z'Z}{\sigma ^2} \\~\\ \frac{\partial ^{2}{\mathcal {L}} }{\partial \delta \partial \sigma ^2}&= \frac{\partial ^{2}}{\partial \delta \partial \sigma ^2} (-\frac{1}{2\sigma ^2} (Ay - Z\delta )'(Ay - Z\delta )) \\&= \frac{\partial }{\partial \delta } (\frac{1}{2\sigma ^4} (Ay - Z\delta )'(Ay - Z\delta )) \\&= \frac{-Z'(Ay - Z\delta )}{\sigma ^4} \\&= \frac{-Z'Ay - Z'Z(Z'Z)^{-1}Z'Ay}{\sigma ^4} \\&= 0 \\~\\ \frac{\partial ^{2} {\mathcal {L}}}{\partial (\sigma ^2)^2}&= \frac{\partial ^{2}}{\partial (\sigma ^2)^2} (-\frac{N}{2}\ln (\sigma ^2)-\frac{1}{2\sigma ^2} (Ay - Z\delta )'(Ay - Z\delta )) \\&= \frac{\partial }{\partial \sigma ^2} (-\frac{N}{2\sigma ^2}+\frac{(Ay - Z\delta )'(Ay - Z\delta )}{2\sigma ^4})\\&= \frac{N}{2\sigma ^4} - \frac{(Ay - Z\delta )'(Ay - Z\delta )}{\sigma ^6} \\&= \frac{N}{2\sigma ^4} - \frac{N\sigma ^2}{\sigma ^6} \\&= -\frac{N}{2\sigma ^4} \end{aligned} \end{aligned}$$

Appendix B A matrix formulation of spatial instruments

This appendix shows how to derive the moments $U'y$, $U'L$ and $U'U$ of Sect. 4.4.2 that are required to perform S2SLS estimation based on the matrix formulation of the spatial interaction model. The first subsection presents the ten blocks of the moment $U'U$ that is linked to the empirical variance of the instrumental variables. The second subsection derives the moments $U'y$ and $U'L$, which are proportional to the empirical covariances of the instrumental variables with the original and lagged versions of the flow vector.

1.1 The variance moment $U'U$

The previously introduced groups of instrumental variables $U = (U_{\alpha } U_{\alpha _I} U_\beta U_\gamma )$ allow to express the moment matrix $U'U$ in terms of 16 blocks.

$$\begin{aligned} U'U = \begin{pmatrix} U_{\alpha \alpha } &{} U_{\alpha \alpha _I} &{} U_{\alpha \beta } &{} U_{\alpha \gamma } \\ \bullet &{} U_{\alpha _I\alpha _I} &{} U_{\alpha _I\beta } &{} U_{\alpha _I\gamma } \\ \bullet &{} \bullet &{} U_{\beta \beta } &{} U_{\beta \gamma } \\ \bullet &{} \bullet &{} \bullet &{} U_{\gamma \gamma } \end{pmatrix} \end{aligned}$$

The following four paragraphs derive the diagonal blocks of above matrix. For the six off-diagonal blocks we will start with those in the first row and move from top to bottom and left to right. Since $U'U$ is a symmetric matrix, we can focus on the upper triangle of the above matrix and infer all blocks on the lower triangle by symmetry.

1.1.1 Block 1 ($\alpha ^2$)

The first block of the $U'U$ moment contains only the inner product of intercept term $U_{\alpha } = \iota _{{N}}$.

$$\begin{aligned} U_{\alpha }'U_{\alpha } := U_{\alpha \alpha } = (\iota _{{n}}' \otimes \iota _{{n}}') (\iota _{{n}} \otimes \iota _{{n}}) = (\iota _{{n}}'\iota _{{n}}) (\iota _{{n}}'\iota _{{n}}) = n \times n = N \end{aligned}$$

1.1.2 Block 2 ($\alpha _I^2$)

The second diagonal block of the covariance moment $U_{\alpha _I}'U_{\alpha _I} := U_{\alpha _I\alpha _I}$ contains the inner product of the instruments derived from intra-regional constant. All columns in $U_{\alpha _I}$ are derived by applying ${{\,\mathrm{VEC}\,}}$-operator to the matrices $\mathrm {A}_i^H$, for $i = 1, ..., 9$, which are defined below.

$$\begin{aligned} U_{\alpha _I} = \begin{pmatrix} VEC(\mathrm {A}_1^U)&...&VEC(\mathrm {A}_9^U) \end{pmatrix} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \mathrm {A}_1^U&= {\mathbf {I}}_{n}\\ \mathrm {A}_2^U&= {W}\\ \mathrm {A}_3^U&= {W}' \end{aligned} \qquad \qquad \begin{aligned} \mathrm {A}_4^U&= {W^2}\\ \mathrm {A}_5^U&= {W^{2\prime }}\\ \mathrm {A}_6^U&= {W}{W}' \end{aligned} \qquad \qquad \begin{aligned} \mathrm {A}_7^U&= {W^2}{W^{2\prime }}\\ \mathrm {A}_8^U&= {W^2}{W}' \\ \mathrm {A}_9^U&= {W}{W^{2\prime }}\end{aligned} \end{aligned}$$

(47)

This structure allows to compute each element of $U_{\alpha _I\alpha _I}$ as $U_{\alpha _I\alpha _I,ij} = \iota _n'(\mathrm {A}_i^H \odot \mathrm {A}_i^U)\iota _n$, for $i,j = 1,...,9$. Since the block is symmetric we only need to compute the elements on the upper triangle. We can further reduce the number of elements to compute if we exploit the following properties: For two square matrices A and B with compatible dimension, we have $\iota _{{n}}'(A \odot B)\iota _{{n}}$ = $\iota _{{n}}'(A' \odot B')\iota _{{n}} = \iota _{{n}}'(B \odot A)\iota _{{n}}$. In addition, if either A or B is symmetric we have $\iota _{{n}}'(A \odot B)\iota _{{n}} = \iota _{{n}}'(A' \odot B)\iota _{{n}}$. Exploiting all symmetries of the problem we only need to compute 21 of the 81 elements that constitute the matrix $U_{\alpha _I\alpha _I}$ for the case when W itself is not symmetric. When W is symmetric we have to define $U_\alpha$ in terms of the five instruments ${{\,\mathrm{VEC}\,}}(W^m)$ for $m=0,1,2,3,4$ since four of the nine vectors in $\begin{pmatrix} VEC(\mathrm {A}_1^U)&...&VEC(\mathrm {A}_9^U) \end{pmatrix}$ would be redundant.

1.1.3 Block 3 ($\beta ^2$)

The next diagonal block contains the inner product of the instruments derived from the site attributes $U_{\beta } = ( \tilde{\tilde{X_d}} \; \tilde{\tilde{X_o}} \; \tilde{\tilde{X_I}} )$. We can exploit the structure of the three matrices $\tilde{\tilde{X_r}} (r = d,o,I)$ to derive this moment block from matrix products of much the much smaller matrices $\tilde{\tilde{DX}}, \tilde{\tilde{OX}}$ and $\tilde{\tilde{DX}}$.

$$\begin{aligned} U_{\beta }'U_{\beta } := U_{\beta \beta } = \begin{pmatrix} n \tilde{\tilde{DX}}' \tilde{\tilde{DX}} &{}&{} (\iota _{{n}}'\tilde{\tilde{DX}})'(\iota _{{n}}'\tilde{\tilde{OX}}) &{}&{} \tilde{\tilde{DX}}'\tilde{\tilde{IX}} \\ \bullet &{}&{} n \tilde{\tilde{OX}}' \tilde{\tilde{OX}} &{}&{} \tilde{\tilde{OX}}'\tilde{\tilde{IX}} \\ \bullet &{}&{} \bullet &{}&{} \tilde{\tilde{IX}}'\tilde{\tilde{IX}} \end{pmatrix} \end{aligned}$$

1.1.4 Block 4 ($\gamma ^2$)

The last diagonal block contains inner product of the instruments derived from the pair attributes $U_{\gamma } = ( {{\,\mathrm{VEC}\,}}(G) \; {{\,\mathrm{VEC}\,}}({\check{G}}) \; {{\,\mathrm{VEC}\,}}(\check{{\check{G}}}) )$, where ${\check{G}}= WGW'$ and $\check{{\check{G}}}= W{\check{G}}W'$. This structure allows to express the moment block in terms of the Hadamard product. When the matrix G represents the geographic distance, it is usually symmetric, which implies that ${\check{G}}$ and $\check{{\check{G}}}$ are also symmetric. In this case, we can exploit that for two symmetric matrices A and B with compatible dimensions, we only require the upper diagonal of both to compute $\iota _n'(A \odot B)\iota _n$. This reduces the memory and computational requirements by about 50%, which may be worthwhile because this block is responsible for a major part of the computational burden of the S2SLS estimator.

$$\begin{aligned} U_{\gamma }'U_{\gamma } := U_{\gamma \gamma } = \begin{pmatrix} \iota _n'(G \odot G)\iota _n &{} \iota _n'(G \odot {\check{G}})\iota _n &{} \iota _n'(G \odot \check{{\check{G}}})\iota _n \\ \bullet &{} \iota _n'({\check{G}}\odot {\check{G}})\iota _n &{} \iota _n'({\check{G}}\odot \check{{\check{G}}})\iota _n \\ \bullet &{} \bullet &{} \iota _n'(\check{{\check{G}}}\odot \check{{\check{G}}})\iota _n \end{pmatrix} \end{aligned}$$

1.1.5 Block 5 ($\alpha \alpha _I$)

The fifth block of $U'U$ is obtained as the inner product of the global constant $U_{\alpha }$ and the instruments derived from the intra-regional constant $U_{\alpha _I}$. Since all elements of $U_{\alpha }$ are one, computing this inner product corresponds to summing the elements in each of the nine vectors in $U_{\alpha _I}$. As W is assumed to be row-stochastic, we can directly conclude that the first five entries of $U_{\alpha \alpha _I}$ are equal to n. The remaining four entries are computed from the vectors of column sums of ${W}$ and ${W^2}$.

$$\begin{aligned} U_{\alpha }'U_{\alpha _I} := U_{\alpha \alpha _I} = \begin{pmatrix}n&n&n&n&n&(\iota _{{n}}'W)(\iota _{{n}}'W)'&(\iota _{{n}}'{W^2})(\iota _{{n}}'{W^2})'&(\iota _{{n}}'W)(\iota _{{n}}'{W^2})'&(\iota _{{n}}'W)(\iota _{{n}}'{W^2})' \end{pmatrix} \end{aligned}$$

1.1.6 Block 6 ($\alpha \beta$)

Block six is computed from the constant $U_{\alpha }$ and the instruments $U_{\beta }$ that are derived from the site attributes. Computing the elements of this block only requires the scalar n and the column sums of the matrices $\tilde{\tilde{DX}}, \tilde{\tilde{OX}}$ and $\tilde{\tilde{IX}}$.

$$\begin{aligned} U_{\alpha }'U_{\beta } := U_{\alpha \beta } = \begin{pmatrix} n \cdot \iota _{{n}}'\tilde{\tilde{DX}}&\,&n \cdot \iota _{{n}}'\tilde{\tilde{OX}}&\,&\iota _{{n}}'\tilde{\tilde{IX}} \end{pmatrix} \end{aligned}$$

1.1.7 Block 7 ($\alpha \gamma$)

This block contains the inner product of the constant $U_{\alpha }$ with the instruments derived from the exogenous attributes of the origin-destination pairs $U_{\gamma }$.

$$\begin{aligned} U_{\alpha }' U_{\gamma } := U_{\alpha \gamma } = \begin{pmatrix} \iota _n' G \iota _n &{}&{} \iota _n' {\check{G}}\iota _n &{}&{} \iota _n' \check{{\check{G}}}\iota _n \\ \end{pmatrix} \end{aligned}$$

1.1.8 Block 8 ($\alpha _I \beta$)

The eighth block of $U'U$ relates to the empirical covariance between $U_{\alpha _I}$ the instruments derived from the intra-regional constant and $U_{\beta }$ the instruments derived from the site attributes. The structure of $U_{\alpha _I}$ allows to express each of the nine rows of $U_{\alpha _I}'U_{\beta } := U_{\alpha _I\beta }$ using $\mathrm {A}_i^U$, for $i = 1, ... , 9$, as defined in (47).

$$\begin{aligned} U_{\alpha _I\beta ,i} = \begin{pmatrix} (\iota _n'\mathrm {A}_i^{H\prime }) \tilde{\tilde{DX}}&\,&(\iota _n'\mathrm {A}_i^H) \tilde{\tilde{OX}}&\,&{{\,\mathrm{diag}\,}}(\mathrm {A}_i^H) \tilde{\tilde{IX}} \end{pmatrix} \end{aligned}$$

1.1.9 Block 9 ($\alpha _I \gamma$)

This block is calculated as the inner product of the instruments derived from the intra-regional $U_{\alpha _I}$ constant and those derived from the exogenous origin-destination pair attributes $U_{\gamma }$. It is again possible to derive $U_{\alpha _I}'U_{\gamma } := U_{\alpha _I\gamma }$ for all rows ($i = 1,...,9$) using the notations in (47).

$$\begin{aligned} U_{\alpha _I\gamma ,i} = \begin{pmatrix} \iota _n'(\mathrm {A}_i^H \odot G)\iota _n &{}&{} \iota _n'(\mathrm {A}_i^H \odot {\check{G}})\iota _n &{}&{} \iota _n'(\mathrm {A}_i^H \odot \check{{\check{G}}})\iota _n \\ \end{pmatrix} \end{aligned}$$

We can reduce the computational burden of this block if we pay attention to symmetries that were already mentioned in the sections on the diagonal blocks $U_{\alpha _I\alpha _I}$ and $U_{\gamma \gamma }$.

1.1.10 Block 10 ($\beta \gamma$)

The last block of the $U'U$ moment matrix is computed as the inner product of $U_{\beta }$ the instruments derived from the site attributes with $U_{\gamma }$ the instruments derived from the pair attributes.

$$\begin{aligned} U_{\beta \gamma }'= \begin{pmatrix} (\iota _{{n}}'G') \tilde{\tilde{DX}} &{}&{} (\iota _{{n}}'G) \tilde{\tilde{OX}} &{}&{} {{\,\mathrm{diag}\,}}(G) \tilde{\tilde{IX}} \\ (\iota _{{n}}'{\check{G}}') \tilde{\tilde{DX}} &{}&{} (\iota _{{n}}'{\check{G}}) \tilde{\tilde{OX}} &{}&{} {{\,\mathrm{diag}\,}}({\check{G}}) \tilde{\tilde{IX}} \\ (\iota _{{n}}'\check{{\check{G}}}') \tilde{\tilde{DX}} &{}&{} (\iota _{{n}}'\check{{\check{G}}}) \tilde{\tilde{OX}} &{}&{} {{\,\mathrm{diag}\,}}(\check{{\check{G}}}) \tilde{\tilde{IX}} \end{pmatrix} \end{aligned}$$

1.2 The covariance moments $U'y_{\bullet }$

To construct the moments $U'y$ and $U'L$ we use the notations $Y^{(t)}$ and $y^{(t)}$ that are defined in (8). The elements of the four moments $U'y^{(t)}$, for $t = 1,2,3,4$, represent the columns of $U'y = U'y^{(1)}$ and $U'L = (U'y^{(2)}U'y^{(3)}U'y^{(4)})$.

$$\begin{aligned} U'y^{(t)} = \begin{pmatrix} (U_{\alpha }'y^{(t)})'&(U_{\alpha _I}'y^{(t)})'&(U_\beta 'y^{(t)})'&(U_\gamma 'y^{(t)})'\end{pmatrix}' \end{aligned}$$

The elements of the above moments are derived below, where the elementwise notation for the nine entries of $U_{\alpha _I}'y^{(t)}$ uses definition (47) of $\mathrm {A}_i^U$, for $i = 1, ... , 9$.

$$\begin{aligned} \begin{aligned} U_{\alpha }'y^{(t)}&= \begin{pmatrix} \iota _{{n}}'Y^{(t)}\iota _{{n}} \end{pmatrix} \\~\\ (U_{\alpha _I}'y^{(t)})_i&= \begin{pmatrix} \iota _{{n}}'(\mathrm {A}_i^H \odot Y^{(t)}) \iota _{{n}} \end{pmatrix}, \text { for } i = 1,...,9 \\~\\ U_{\beta }'y^{(t)}&= \begin{pmatrix} \tilde{\tilde{DX'}}(Y^{(t)} \iota _{{n}}) \\ \tilde{\tilde{OX'}}(Y^{(t)\prime } \iota _{{n}}) \\ \tilde{\tilde{IX'}} {{\,\mathrm{diag}\,}}(Y^{(t)}) \end{pmatrix} \\~\\ U_{\gamma }'y^{(t)}&= \begin{pmatrix} \iota _{{n}}'(G \odot Y^{(t)}) \iota _{{n}} \\ \iota _{{n}}'({\check{G}}\odot Y^{(t)}) \iota _{{n}} \\ \iota _{{n}}'(\check{{\check{G}}}\odot Y^{(t)}) \iota _{{n}} \end{pmatrix} \end{aligned} \end{aligned}$$

Appendix C Trace plots of the Bayesian MCMC estimates

Fig. 6

Trace plots of the parameter values during 5500 iterations of the MCMC sampling procedure