---
title: "Maximum Likelihood estimation of Spatial Seemingly Unrelated Regression models. A short Monte Carlo exercise with spsur and spse"
author:
- Fernando A. López, Technical University of Cartagena (Spain)
- Román Mínguez, University of Castilla-La Mancha (Spain)
- Jesús Mur, University of Zaragoza, (Spain) <br> <br> <br>
date: '2022-10-29 <br>  <br> <br>'
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---




# A short Monte Carlo exercise: spsur vs spse.

The goal of this vignette is to present the results obtained in a Monte Carlo exercise to evaluate the performance of the Maximum Likelihood (ML) estimation of three spatial SUR models using the *R*-package **spsur** [@MinguezJSS2022; @Lopez2020]. The results will be compared with the same estimation using the **spse** *R*-package [@Piras2010] when it is possible. We compare the two basic spatial SUR models, named SUR-SLM and SUR-SEM. In the case of SUR-SARAR, we only present the results obtained with **spsur** because the estimation of this model is not available with **spse**.

The design of the Monte Carlo is as follows: We simulate a spatial SUR model with two equations (G = 2), where each equation includes an intercept and two explanatory variables plus the corresponding spatial terms. For the general model the equation is:

```{=tex}
\begin{equation}
y_i = (I_N-\rho_iW)^{-1}(\beta_{i0} + X_{i1}\beta_{i1} + X_{i2}\beta_{i2} + (I_N-\lambda_iW)^{-1}\epsilon_i); \ cov(\epsilon_i,\epsilon_j)=\sigma_{ij} ; \ i=1,2
(\#eq:sur)
\end{equation}
```

During the experiment, the $\beta$ parameters are fixed for every model taking the values $\beta_{10}=\beta_{20}=1$; $\beta_{11}=\beta_{21}=2$ and $\beta_{12}=\beta_{22}=3$. The variance-covariance matrix $\Sigma=(\sigma_{ij})$ is defined by $\sigma_{ij}=0.5 \ (i \neq j)$  and $\sigma_{ii}=1 \ (i=1,2)$. Two sample sizes, small and medium, are choosen (N=52, 516). A regular hexagonal layout is selected, from which the **W** matrix is obtained, based on the border contiguity between the hexagons (rook neighborhood type). Figure \ref{Fig:geometry} shows the hexagonal lattices for the case of N = 516. The $X_{ij}$ (i,j=1,2) variables are drawn from an independent U(0,1), and the error terms from a bivariate normal distribution with a variance-covariance matrix $\Sigma$. For all the experiments, 1,000 replications are performed.

Several combinations of parameters are selected to evaluate the performance of the ML algorithm under different levels of spatial dependence.

     SUR-SLM: $(\rho_1,\rho_2)=(-0.4,0.6);(0.5,0.5);(0.2,0.8)$ and $(\lambda_1,\lambda_2)=(0,0)$

     SUR-SEM: $(\rho_1,\rho_2)=(0,0)$ and $(\lambda_1,\lambda_2)=(-0.4,0.6);(0.5,0.5);(0.2,0.8)$

     SUR-SARAR: $(\rho_1,\rho_2)=(\lambda_1,\lambda_2)=(-0.4,0.6);(0.5,0.5);(0.2,0.8)$

These spatial processes have been generated using the function  *dgp_spsur()*, available in the **spsur** package. To evaluate the performance of the Maximum Likelihood estimation, we report bias and root mean-squared errors (RMSE) for all the combinations of the spatial parameters.





If **spsur** and **spse** needed to be installed, the first one  is available in the CRAN repository and the second one can be installed from the following GitHub repository:


```r
# install_github("gpiras/spse",force = TRUE)
library(spse)
```

The package **sf** is used to generate hexagonal and regular lattices with the number of hexagons prefixed and **spdep** to obtain the **W** matrix based on a common border.


```r
library(sf)
library(spdep)
sfc <- st_sfc(st_polygon(list(rbind(c(0,0),c(1,0),c(1,1),c(0,1),c(0,0)))))
hexs.N52.sf <- st_sf(st_make_grid(sfc, cellsize = .19, square = FALSE))
hexs.N525.sf <- st_sf(st_make_grid(sfc, cellsize = .05, square = FALSE))
listw.N52 <- as(hexs.N52.sf, "Spatial") %>% 
             poly2nb(queen = FALSE) %>% nb2listw()
listw.N525 <- as(hexs.N525.sf, "Spatial") %>% 
              poly2nb(queen = FALSE) %>% nb2listw()
```



# Maximum Likelihood estimation of SUR-SLM models

This section presents the results of a Monte Carlo exercise for the ML estimation of SUR-SLM models. 

```{=tex}
\begin{equation}
y_i = (I_N-\rho_iW)^{-1}(\beta_{i0} + X_{i1}\beta_{i1} + X_{i2}\beta_{i2} + \epsilon_i) \ ; \ cov(\epsilon_i,\epsilon_j)=\sigma_{ij} \ ; \ \ i=1,2
(\#eq:sur-sem)
\end{equation}
```

Table 1 shows the mean of the bias and the RMSE of the $\beta's$ and $\rho's$ parameters for the 1,000 replications. In general, all the results are coherent. The estimations with both *R*-packages show similar results. The highest bias is observed in the estimates of the intercept of the second equation for both packages. When the model is estimated with **spsur** the maximum bias is reached for N = 52  and  when the model is estimated with **spse** the maximum bias corresponds to N = 516. In general, the results confirm that for both packages, the estimates of the parameters of spatial dependence present low biases. The RMSE values decrease when the sample size increases, as expected.















<table>
<caption>Table 1: SUR-SLM Bias and RMSE. Maximum Likelihood</caption>
 <thead>
  <tr>
   <th style="text-align:center;"> Pack. </th>
   <th style="text-align:center;"> N </th>
   <th style="text-align:center;"> $\rho_1$ </th>
   <th style="text-align:center;"> $\rho_2$ </th>
   <th style="text-align:center;"> $\hat\beta_{10}$ </th>
   <th style="text-align:center;"> $\hat\beta_{11}$ </th>
   <th style="text-align:center;"> $\hat\beta_{12}$ </th>
   <th style="text-align:center;"> $\hat\beta_{20}$ </th>
   <th style="text-align:center;"> $\hat\beta_{21}$ </th>
   <th style="text-align:center;"> $\hat\beta_{22}$ </th>
   <th style="text-align:center;"> $\hat\rho_1$ </th>
   <th style="text-align:center;"> $\hat\rho_2$ </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> spsur </td>
   <td style="text-align:center;"> 52 </td>
   <td style="text-align:center;"> -0.4 </td>
   <td style="text-align:center;"> 0.6 </td>
   <td style="text-align:center;"> 0.009 </td>
   <td style="text-align:center;"> 0.001 </td>
   <td style="text-align:center;"> 0.001 </td>
   <td style="text-align:center;"> 0.036 </td>
   <td style="text-align:center;"> 0.001 </td>
   <td style="text-align:center;"> -0.004 </td>
   <td style="text-align:center;"> -0.007 </td>
   <td style="text-align:center;"> -0.012 </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> -0.4 </td>
   <td style="text-align:center;"> 0.6 </td>
   <td style="text-align:center;"> (0.156) </td>
   <td style="text-align:center;"> (0.126) </td>
   <td style="text-align:center;"> (0.126) </td>
   <td style="text-align:center;"> (0.208) </td>
   <td style="text-align:center;"> (0.133) </td>
   <td style="text-align:center;"> (0.132) </td>
   <td style="text-align:center;"> (0.077) </td>
   <td style="text-align:center;"> (0.054) </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> 0.027 </td>
   <td style="text-align:center;"> -0.002 </td>
   <td style="text-align:center;"> 0.001 </td>
   <td style="text-align:center;"> 0.027 </td>
   <td style="text-align:center;"> 0.001 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> -0.012 </td>
   <td style="text-align:center;"> -0.011 </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> (0.201) </td>
   <td style="text-align:center;"> (0.129) </td>
   <td style="text-align:center;"> (0.129) </td>
   <td style="text-align:center;"> (0.199) </td>
   <td style="text-align:center;"> (0.132) </td>
   <td style="text-align:center;"> (0.126) </td>
   <td style="text-align:center;"> (0.061) </td>
   <td style="text-align:center;"> (0.059) </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 0.2 </td>
   <td style="text-align:center;"> 0.8 </td>
   <td style="text-align:center;"> 0.017 </td>
   <td style="text-align:center;"> -0.006 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> 0.065 </td>
   <td style="text-align:center;"> 0.006 </td>
   <td style="text-align:center;"> 0.002 </td>
   <td style="text-align:center;"> -0.011 </td>
   <td style="text-align:center;"> -0.012 </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 0.2 </td>
   <td style="text-align:center;"> 0.8 </td>
   <td style="text-align:center;"> (0.178) </td>
   <td style="text-align:center;"> (0.124) </td>
   <td style="text-align:center;"> (0.126) </td>
   <td style="text-align:center;"> (0.276) </td>
   <td style="text-align:center;"> (0.129) </td>
   <td style="text-align:center;"> (0.126) </td>
   <td style="text-align:center;"> (0.071) </td>
   <td style="text-align:center;"> (0.040) </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 516 </td>
   <td style="text-align:center;"> -0.4 </td>
   <td style="text-align:center;"> 0.6 </td>
   <td style="text-align:center;"> -0.002 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> -0.000 </td>
   <td style="text-align:center;"> 0.001 </td>
   <td style="text-align:center;"> 0.002 </td>
   <td style="text-align:center;"> 0.001 </td>
   <td style="text-align:center;"> -0.002 </td>
   <td style="text-align:center;"> -0.001 </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> -0.4 </td>
   <td style="text-align:center;"> 0.6 </td>
   <td style="text-align:center;"> (0.047) </td>
   <td style="text-align:center;"> (0.038) </td>
   <td style="text-align:center;"> (0.040) </td>
   <td style="text-align:center;"> (0.058) </td>
   <td style="text-align:center;"> (0.040) </td>
   <td style="text-align:center;"> (0.041) </td>
   <td style="text-align:center;"> (0.025) </td>
   <td style="text-align:center;"> (0.015) </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> 0.001 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> -0.001 </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> (0.057) </td>
   <td style="text-align:center;"> (0.039) </td>
   <td style="text-align:center;"> (0.041) </td>
   <td style="text-align:center;"> (0.058) </td>
   <td style="text-align:center;"> (0.039) </td>
   <td style="text-align:center;"> (0.041) </td>
   <td style="text-align:center;"> (0.017) </td>
   <td style="text-align:center;"> (0.017) </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 0.2 </td>
   <td style="text-align:center;"> 0.8 </td>
   <td style="text-align:center;"> 0.003 </td>
   <td style="text-align:center;"> 0.001 </td>
   <td style="text-align:center;"> -0.000 </td>
   <td style="text-align:center;"> 0.007 </td>
   <td style="text-align:center;"> 0.001 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> -0.001 </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 0.2 </td>
   <td style="text-align:center;"> 0.8 </td>
   <td style="text-align:center;"> (0.052) </td>
   <td style="text-align:center;"> (0.038) </td>
   <td style="text-align:center;"> (0.039) </td>
   <td style="text-align:center;"> (0.068) </td>
   <td style="text-align:center;"> (0.038) </td>
   <td style="text-align:center;"> (0.040) </td>
   <td style="text-align:center;"> (0.022) </td>
   <td style="text-align:center;"> (0.010) </td>
  </tr>
  <tr>
   <td style="text-align:center;"> spse </td>
   <td style="text-align:center;"> 52 </td>
   <td style="text-align:center;"> -0.4 </td>
   <td style="text-align:center;"> 0.6 </td>
   <td style="text-align:center;"> 0.018 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> -0.002 </td>
   <td style="text-align:center;"> 0.007 </td>
   <td style="text-align:center;"> -0.004 </td>
   <td style="text-align:center;"> -0.009 </td>
   <td style="text-align:center;"> -0.021 </td>
   <td style="text-align:center;"> -0.001 </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> -0.4 </td>
   <td style="text-align:center;"> 0.6 </td>
   <td style="text-align:center;"> (0.159) </td>
   <td style="text-align:center;"> (0.143) </td>
   <td style="text-align:center;"> (0.143) </td>
   <td style="text-align:center;"> (0.205) </td>
   <td style="text-align:center;"> (0.147) </td>
   <td style="text-align:center;"> (0.144) </td>
   <td style="text-align:center;"> (0.083) </td>
   <td style="text-align:center;"> (0.054) </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> 0.002 </td>
   <td style="text-align:center;"> -0.003 </td>
   <td style="text-align:center;"> -0.003 </td>
   <td style="text-align:center;"> 0.005 </td>
   <td style="text-align:center;"> -0.003 </td>
   <td style="text-align:center;"> -0.005 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> (0.201) </td>
   <td style="text-align:center;"> (0.149) </td>
   <td style="text-align:center;"> (0.146) </td>
   <td style="text-align:center;"> (0.200) </td>
   <td style="text-align:center;"> (0.150) </td>
   <td style="text-align:center;"> (0.141) </td>
   <td style="text-align:center;"> (0.063) </td>
   <td style="text-align:center;"> (0.061) </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 0.2 </td>
   <td style="text-align:center;"> 0.8 </td>
   <td style="text-align:center;"> 0.011 </td>
   <td style="text-align:center;"> -0.002 </td>
   <td style="text-align:center;"> -0.004 </td>
   <td style="text-align:center;"> 0.008 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> -0.009 </td>
   <td style="text-align:center;"> -0.006 </td>
   <td style="text-align:center;"> -0.001 </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 0.2 </td>
   <td style="text-align:center;"> 0.8 </td>
   <td style="text-align:center;"> (0.180) </td>
   <td style="text-align:center;"> (0.143) </td>
   <td style="text-align:center;"> (0.149) </td>
   <td style="text-align:center;"> (0.267) </td>
   <td style="text-align:center;"> (0.149) </td>
   <td style="text-align:center;"> (0.153) </td>
   <td style="text-align:center;"> (0.074) </td>
   <td style="text-align:center;"> (0.039) </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 516 </td>
   <td style="text-align:center;"> -0.4 </td>
   <td style="text-align:center;"> 0.6 </td>
   <td style="text-align:center;"> 0.007 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> -0.004 </td>
   <td style="text-align:center;"> -0.025 </td>
   <td style="text-align:center;"> -0.002 </td>
   <td style="text-align:center;"> -0.003 </td>
   <td style="text-align:center;"> -0.014 </td>
   <td style="text-align:center;"> 0.009 </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> -0.4 </td>
   <td style="text-align:center;"> 0.6 </td>
   <td style="text-align:center;"> (0.049) </td>
   <td style="text-align:center;"> (0.044) </td>
   <td style="text-align:center;"> (0.045) </td>
   <td style="text-align:center;"> (0.063) </td>
   <td style="text-align:center;"> (0.046) </td>
   <td style="text-align:center;"> (0.048) </td>
   <td style="text-align:center;"> (0.030) </td>
   <td style="text-align:center;"> (0.018) </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> -0.021 </td>
   <td style="text-align:center;"> -0.003 </td>
   <td style="text-align:center;"> -0.003 </td>
   <td style="text-align:center;"> -0.022 </td>
   <td style="text-align:center;"> -0.004 </td>
   <td style="text-align:center;"> -0.003 </td>
   <td style="text-align:center;"> 0.010 </td>
   <td style="text-align:center;"> 0.010 </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> (0.061) </td>
   <td style="text-align:center;"> (0.045) </td>
   <td style="text-align:center;"> (0.047) </td>
   <td style="text-align:center;"> (0.063) </td>
   <td style="text-align:center;"> (0.045) </td>
   <td style="text-align:center;"> (0.046) </td>
   <td style="text-align:center;"> (0.020) </td>
   <td style="text-align:center;"> (0.020) </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 0.2 </td>
   <td style="text-align:center;"> 0.8 </td>
   <td style="text-align:center;"> -0.004 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> -0.039 </td>
   <td style="text-align:center;"> -0.005 </td>
   <td style="text-align:center;"> -0.007 </td>
   <td style="text-align:center;"> 0.004 </td>
   <td style="text-align:center;"> 0.008 </td>
  </tr>
  <tr>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;"> 0.2 </td>
   <td style="text-align:center;"> 0.8 </td>
   <td style="text-align:center;"> (0.053) </td>
   <td style="text-align:center;"> (0.044) </td>
   <td style="text-align:center;"> (0.044) </td>
   <td style="text-align:center;"> (0.078) </td>
   <td style="text-align:center;"> (0.043) </td>
   <td style="text-align:center;"> (0.045) </td>
   <td style="text-align:center;"> (0.023) </td>
   <td style="text-align:center;"> (0.013) </td>
  </tr>
</tbody>
</table>


Figure 1 shows the boxplots of  $\gamma_{ij}=\hat\beta_{ij}^{spsur}-\hat\beta_{ij}^{spse}$ and $\delta_i=\hat\rho_{i}^{spsur}-\hat\rho_{i}^{spse}$, the difference between estimated parameters 'model to model' for N = 516 (the superscript indicates the package used to estimate the coefficient). These boxplots confirm that the main differences are founded in the intercept of the second equation.


![Figure 1: Difference between parameters 'model to model' SUR-SLM (N=516). (A) $\rho_1=-0.4; \rho_2=0.6$ ; (B) $\rho_1=0.5; \rho_2=0.5$ ; (C) $\rho_1 = 0.2; \rho_2 = 0.8$](boxplot_slm1.png)


# Maximum Likelihood estimation of SUR-SEM models

Table 1 shows the results of the bias and RMSE for the estimation of an SUR-SEM model with both R-packages. In general terms, the biases of the estimated parameters are lower than 0.01 in absolute values for all $\beta$ parameters. The estimation of the $\lambda's$ parameters for small sample (N = 52) has a bias higher than 0.01 with a tendency toward the underestimation in all the cases. For medium sample sizes (N = 516), the bias is lower than 0.01. The RMSE decreases when the sample size increase as expected.














\begin{table}

\caption{\label{tab:print-table-2}Table 2: Bias and RMSE. SUR-SEM Maximum Likelihood}
\centering
\begin{tabular}[t]{cccccccccccc}
\toprule
Pack & N & $\lambda_1$ & $\lambda_2$ & $\hat\beta_{10}$ & $\hat\beta_{11}$ & $\hat\beta_{12}$ & $\hat\beta_{20}$ & $\hat\beta_{21}$ & $\hat\beta_{22}$ & $\hat\lambda_1$ & $\hat\lambda_2$\\
\midrule
 &  &  &  & 0.004 & 0.005 & 0.003 & 0.008 & 0.002 & -0.006 & -0.071 & -0.080\\

 &  & \multirow[t]{-2}{*}{\centering\arraybackslash -0.4} & \multirow[t]{-2}{*}{\centering\arraybackslash 0.6} & (0.103) & (0.127) & (0.122) & (0.357) & (0.128) & (0.124) & (0.239) & (0.174)\\

 &  &  &  & 0.006 & -0.001 & 0.003 & 0.006 & -0.001 & 0.001 & -0.070 & -0.086\\

 &  & \multirow[t]{-2}{*}{\centering\arraybackslash 0.5} & \multirow[t]{-2}{*}{\centering\arraybackslash 0.5} & (0.289) & (0.125) & (0.126) & (0.286) & (0.126) & (0.123) & (0.186) & (0.190)\\

 &  &  &  & 0.003 & -0.003 & 0.000 & 0.021 & 0.002 & -0.003 & -0.082 & -0.074\\

 & \multirow[t]{-6}{*}{\centering\arraybackslash 52} & \multirow[t]{-2}{*}{\centering\arraybackslash 0.2} & \multirow[t]{-2}{*}{\centering\arraybackslash 0.8} & (0.174) & (0.123) & (0.127) & (0.688) & (0.117) & (0.115) & (0.228) & (0.138)\\
\cline{2-12}

 &  &  &  & -0.002 & 0.001 & -0.000 & -0.004 & 0.002 & 0.001 & -0.008 & -0.008\\

 &  & \multirow[t]{-2}{*}{\centering\arraybackslash -0.4} & \multirow[t]{-2}{*}{\centering\arraybackslash 0.6} & (0.030) & (0.037) & (0.039) & (0.110) & (0.038) & (0.039) & (0.073) & (0.046)\\

 &  &  &  & -0.004 & -0.001 & 0.001 & -0.007 & -0.001 & -0.001 & -0.005 & -0.009\\

 &  & \multirow[t]{-2}{*}{\centering\arraybackslash 0.5} & \multirow[t]{-2}{*}{\centering\arraybackslash 0.5} & (0.089) & (0.038) & (0.040) & (0.091) & (0.038) & (0.040) & (0.050) & (0.052)\\

 &  &  &  & 0.001 & 0.001 & 0.000 & 0.015 & 0.000 & 0.000 & -0.010 & -0.007\\

\multirow[t]{-12}{*}{\centering\arraybackslash spsur} & \multirow[t]{-6}{*}{\centering\arraybackslash 516} & \multirow[t]{-2}{*}{\centering\arraybackslash 0.2} & \multirow[t]{-2}{*}{\centering\arraybackslash 0.8} & (0.055) & (0.038) & (0.038) & (0.225) & (0.035) & (0.037) & (0.063) & (0.031)\\
\midrule
\cmidrule{1-12}
 &  &  &  & 0.004 & 0.005 & 0.004 & 0.008 & 0.002 & -0.007 & -0.033 & -0.107\\

 &  & \multirow[t]{-2}{*}{\centering\arraybackslash -0.4} & \multirow[t]{-2}{*}{\centering\arraybackslash 0.6} & (0.103) & (0.126) & (0.122) & (0.357) & (0.129) & (0.125) & (0.214) & (0.185)\\

 &  &  &  & 0.006 & -0.001 & 0.003 & 0.006 & -0.001 & 0.001 & -0.091 & -0.106\\

 &  & \multirow[t]{-2}{*}{\centering\arraybackslash 0.5} & \multirow[t]{-2}{*}{\centering\arraybackslash 0.5} & (0.289) & (0.126) & (0.126) & (0.286) & (0.127) & (0.123) & (0.190) & (0.195)\\

 &  &  &  & 0.003 & -0.003 & 0.001 & 0.022 & 0.001 & -0.003 & -0.087 & -0.103\\

 & \multirow[t]{-6}{*}{\centering\arraybackslash 52} & \multirow[t]{-2}{*}{\centering\arraybackslash 0.2} & \multirow[t]{-2}{*}{\centering\arraybackslash 0.8} & (0.174) & (0.122) & (0.126) & (0.687) & (0.118) & (0.117) & (0.218) & (0.158)\\
\cline{2-12}

 &  &  &  & -0.002 & 0.001 & -0.000 & -0.004 & 0.002 & 0.001 & -0.005 & -0.011\\

 &  & \multirow[t]{-2}{*}{\centering\arraybackslash -0.4} & \multirow[t]{-2}{*}{\centering\arraybackslash 0.6} & (0.030) & (0.037) & (0.039) & (0.110) & (0.038) & (0.039) & (0.072) & (0.046)\\

 &  &  &  & -0.004 & -0.001 & 0.001 & -0.007 & -0.001 & -0.001 & -0.008 & -0.012\\

 &  & \multirow[t]{-2}{*}{\centering\arraybackslash 0.5} & \multirow[t]{-2}{*}{\centering\arraybackslash 0.5} & (0.089) & (0.038) & (0.040) & (0.091) & (0.038) & (0.040) & (0.050) & (0.053)\\

 &  &  &  & 0.001 & 0.001 & 0.000 & 0.015 & 0.000 & 0.000 & -0.011 & -0.009\\

\multirow[t]{-12}{*}{\centering\arraybackslash spse} & \multirow[t]{-6}{*}{\centering\arraybackslash 516} & \multirow[t]{-2}{*}{\centering\arraybackslash 0.2} & \multirow[t]{-2}{*}{\centering\arraybackslash 0.8} & (0.055) & (0.038) & (0.038) & (0.225) & (0.035) & (0.037) & (0.063) & (0.032)\\
\bottomrule
\end{tabular}
\end{table}


As in the case of SUR-SLM, the Figure 2 shows the difference between the parameters estimated with **spsur** and **spse** for N = 516. These boxplots show that the biases in the SUR-SEM are lower than in the SUR-SLM for all the parameters.

![Figure 2: Difference between parameters 'model to model' SUR-SEM N=516.  (A) $\lambda_1=-0.4; \lambda_2=0.6$ ; (B) $\lambda_1=0.5; \lambda_2=0.5$ ; (C) $\lambda_1 = 0.2; \lambda_2 = 0.8$](boxplot_sem1.png)

# Maximum Likelihood estimation of SUR-SARAR models

Table 3 shows the results obtained for the bias and RMSE for the LM estimation of SUR-SARAR models. For this model, only the results obtained with the **spsur** package can be shown because this specification is not available for the **spse** package. As in the case of the estimations of the SUR-SLM and SUR-SEM models the worst results in terms of bias and RMSE are obtained when the sample size is small (N = 52). In the case of N = 52 the $\lambda's$ parameters are underestimated. This underestimation disappears when the sample size is medium (N = 516).









<table class="table table-striped" style="font-size: 7px; margin-left: auto; margin-right: auto;">
<caption style="font-size: initial !important;">Table 3: SUR-SARAR Bias and RMSE (in brackets). Maximum Likelihood with spsur</caption>
 <thead>
  <tr>
   <th style="text-align:center;"> N </th>
   <th style="text-align:center;"> $\rho_1;\lambda_1$ </th>
   <th style="text-align:center;"> $\rho_2;\lambda_2$ </th>
   <th style="text-align:center;"> $\hat\beta_{10}$ </th>
   <th style="text-align:center;"> $\hat\beta_{11}$ </th>
   <th style="text-align:center;"> $\hat\beta_{12}$ </th>
   <th style="text-align:center;"> $\hat\beta_{20}$ </th>
   <th style="text-align:center;"> $\hat\beta_{21}$ </th>
   <th style="text-align:center;"> $\hat\beta_{22}$ </th>
   <th style="text-align:center;"> $\hat\rho_1$ </th>
   <th style="text-align:center;"> $\hat\rho_2$ </th>
   <th style="text-align:center;"> $\hat\lambda_1$ </th>
   <th style="text-align:center;"> $\hat\lambda_2$ </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 52 </td>
   <td style="text-align:center;"> -0.4 </td>
   <td style="text-align:center;"> 0.6 </td>
   <td style="text-align:center;"> 0.005 </td>
   <td style="text-align:center;"> 0.003 </td>
   <td style="text-align:center;"> 0.002 </td>
   <td style="text-align:center;"> 0.032 </td>
   <td style="text-align:center;"> -0.002 </td>
   <td style="text-align:center;"> -0.012 </td>
   <td style="text-align:center;"> -0.003 </td>
   <td style="text-align:center;"> -0.009 </td>
   <td style="text-align:center;"> -0.077 </td>
   <td style="text-align:center;"> -0.104 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 52 </td>
   <td style="text-align:center;"> -0.4 </td>
   <td style="text-align:center;"> 0.6 </td>
   <td style="text-align:center;"> (0.121) </td>
   <td style="text-align:center;"> (0.127) </td>
   <td style="text-align:center;"> (0.124) </td>
   <td style="text-align:center;"> (0.439) </td>
   <td style="text-align:center;"> (0.133) </td>
   <td style="text-align:center;"> (0.131) </td>
   <td style="text-align:center;"> (0.080) </td>
   <td style="text-align:center;"> (0.083) </td>
   <td style="text-align:center;"> (0.252) </td>
   <td style="text-align:center;"> (0.210) </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 52 </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> 0.027 </td>
   <td style="text-align:center;"> -0.007 </td>
   <td style="text-align:center;"> -0.005 </td>
   <td style="text-align:center;"> 0.014 </td>
   <td style="text-align:center;"> -0.003 </td>
   <td style="text-align:center;"> -0.004 </td>
   <td style="text-align:center;"> -0.010 </td>
   <td style="text-align:center;"> -0.003 </td>
   <td style="text-align:center;"> -0.094 </td>
   <td style="text-align:center;"> -0.117 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 52 </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> (0.365) </td>
   <td style="text-align:center;"> (0.130) </td>
   <td style="text-align:center;"> (0.129) </td>
   <td style="text-align:center;"> (0.345) </td>
   <td style="text-align:center;"> (0.130) </td>
   <td style="text-align:center;"> (0.128) </td>
   <td style="text-align:center;"> (0.089) </td>
   <td style="text-align:center;"> (0.084) </td>
   <td style="text-align:center;"> (0.229) </td>
   <td style="text-align:center;"> (0.231) </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 52 </td>
   <td style="text-align:center;"> 0.2 </td>
   <td style="text-align:center;"> 0.8 </td>
   <td style="text-align:center;"> 0.010 </td>
   <td style="text-align:center;"> -0.007 </td>
   <td style="text-align:center;"> -0.004 </td>
   <td style="text-align:center;"> 0.098 </td>
   <td style="text-align:center;"> -0.002 </td>
   <td style="text-align:center;"> -0.009 </td>
   <td style="text-align:center;"> -0.006 </td>
   <td style="text-align:center;"> -0.013 </td>
   <td style="text-align:center;"> -0.103 </td>
   <td style="text-align:center;"> -0.093 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 52 </td>
   <td style="text-align:center;"> 0.2 </td>
   <td style="text-align:center;"> 0.8 </td>
   <td style="text-align:center;"> (0.214) </td>
   <td style="text-align:center;"> (0.126) </td>
   <td style="text-align:center;"> (0.127) </td>
   <td style="text-align:center;"> (0.931) </td>
   <td style="text-align:center;"> (0.122) </td>
   <td style="text-align:center;"> (0.123) </td>
   <td style="text-align:center;"> (0.083) </td>
   <td style="text-align:center;"> (0.080) </td>
   <td style="text-align:center;"> (0.253) </td>
   <td style="text-align:center;"> (0.177) </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 516 </td>
   <td style="text-align:center;"> -0.4 </td>
   <td style="text-align:center;"> 0.6 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> 0.001 </td>
   <td style="text-align:center;"> -0.000 </td>
   <td style="text-align:center;"> -0.002 </td>
   <td style="text-align:center;"> 0.002 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> -0.008 </td>
   <td style="text-align:center;"> -0.010 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 516 </td>
   <td style="text-align:center;"> -0.4 </td>
   <td style="text-align:center;"> 0.6 </td>
   <td style="text-align:center;"> (0.036) </td>
   <td style="text-align:center;"> (0.037) </td>
   <td style="text-align:center;"> (0.039) </td>
   <td style="text-align:center;"> (0.127) </td>
   <td style="text-align:center;"> (0.039) </td>
   <td style="text-align:center;"> (0.040) </td>
   <td style="text-align:center;"> (0.025) </td>
   <td style="text-align:center;"> (0.026) </td>
   <td style="text-align:center;"> (0.076) </td>
   <td style="text-align:center;"> (0.053) </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 516 </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> -0.002 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> -0.006 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> -0.002 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> -0.008 </td>
   <td style="text-align:center;"> -0.012 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 516 </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> 0.5 </td>
   <td style="text-align:center;"> (0.106) </td>
   <td style="text-align:center;"> (0.038) </td>
   <td style="text-align:center;"> (0.041) </td>
   <td style="text-align:center;"> (0.106) </td>
   <td style="text-align:center;"> (0.038) </td>
   <td style="text-align:center;"> (0.041) </td>
   <td style="text-align:center;"> (0.026) </td>
   <td style="text-align:center;"> (0.026) </td>
   <td style="text-align:center;"> (0.058) </td>
   <td style="text-align:center;"> (0.059) </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 516 </td>
   <td style="text-align:center;"> 0.2 </td>
   <td style="text-align:center;"> 0.8 </td>
   <td style="text-align:center;"> 0.002 </td>
   <td style="text-align:center;"> 0.001 </td>
   <td style="text-align:center;"> -0.000 </td>
   <td style="text-align:center;"> 0.021 </td>
   <td style="text-align:center;"> -0.000 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> -0.000 </td>
   <td style="text-align:center;"> -0.001 </td>
   <td style="text-align:center;"> -0.013 </td>
   <td style="text-align:center;"> -0.009 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 516 </td>
   <td style="text-align:center;"> 0.2 </td>
   <td style="text-align:center;"> 0.8 </td>
   <td style="text-align:center;"> (0.064) </td>
   <td style="text-align:center;"> (0.038) </td>
   <td style="text-align:center;"> (0.038) </td>
   <td style="text-align:center;"> (0.262) </td>
   <td style="text-align:center;"> (0.036) </td>
   <td style="text-align:center;"> (0.038) </td>
   <td style="text-align:center;"> (0.025) </td>
   <td style="text-align:center;"> (0.025) </td>
   <td style="text-align:center;"> (0.070) </td>
   <td style="text-align:center;"> (0.041) </td>
  </tr>
</tbody>
</table>

# Conclusion

This vignette shows the results of a sort Monte Carlo exercise to evaluate the ML estimation of three spatial SUR models, SUR-SLM, SUR-SEM, and SUR-SARAR. The first two models are estimated with the **spsur** and **spse** packages and the results are compared. In the case of the SUR-SARAR model only the results using the **spsur** are presented because the estimation of SUR-SARAR is no available.

In general, both packages present admissible results. When comparing the estimates of the coefficients for SUR-SLM some differences emerge, mainly in the estimation of the intercepts. In the case of SUR-SEM both *R*-packages give similar results for small and medium sample sizes.

A full Monte Carlo using irregular lattices, alternative **W** matrices, and non-ideal conditions would shed more light on the performance of the ML algorithm implemented in both *R*-packages. 

# References

