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 (Mínguez, López, and Mur
2022; López, Mı́nguez,
and Mur 2020). The results will be compared with the same
estimation using the spse R-package (Piras et
al. 2010) 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:
\[\begin{equation}
y_i = (I_N-\rho_iW)^{-1}(\beta_{i0} + X_{i1}\beta_{i1} +
X_{i2}\beta_{i2} + (I_N-\lambda_iW)^{-1}\epsilon_i); \
cov(\epsilon_i,\epsilon_j)=\sigma_{ij} ; \ i=1,2
(\#eq:sur)
\end{equation}\]
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:
# 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.
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.
\[\begin{equation}
y_i = (I_N-\rho_iW)^{-1}(\beta_{i0} + X_{i1}\beta_{i1} +
X_{i2}\beta_{i2} + \epsilon_i) \ ; \
cov(\epsilon_i,\epsilon_j)=\sigma_{ij} \ ; \ \ i=1,2
(\#eq:sur-sem)
\end{equation}\]
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 1: SUR-SLM Bias and RMSE. Maximum Likelihood
|
Pack.
|
N
|
\(\rho_1\)
|
\(\rho_2\)
|
\(\hat\beta_{10}\)
|
\(\hat\beta_{11}\)
|
\(\hat\beta_{12}\)
|
\(\hat\beta_{20}\)
|
\(\hat\beta_{21}\)
|
\(\hat\beta_{22}\)
|
\(\hat\rho_1\)
|
\(\hat\rho_2\)
|
|
spsur
|
52
|
-0.4
|
0.6
|
0.009
|
0.001
|
0.001
|
0.036
|
0.001
|
-0.004
|
-0.007
|
-0.012
|
|
|
|
-0.4
|
0.6
|
(0.156)
|
(0.126)
|
(0.126)
|
(0.208)
|
(0.133)
|
(0.132)
|
(0.077)
|
(0.054)
|
|
|
|
0.5
|
0.5
|
0.027
|
-0.002
|
0.001
|
0.027
|
0.001
|
0.000
|
-0.012
|
-0.011
|
|
|
|
0.5
|
0.5
|
(0.201)
|
(0.129)
|
(0.129)
|
(0.199)
|
(0.132)
|
(0.126)
|
(0.061)
|
(0.059)
|
|
|
|
0.2
|
0.8
|
0.017
|
-0.006
|
-0.001
|
0.065
|
0.006
|
0.002
|
-0.011
|
-0.012
|
|
|
|
0.2
|
0.8
|
(0.178)
|
(0.124)
|
(0.126)
|
(0.276)
|
(0.129)
|
(0.126)
|
(0.071)
|
(0.040)
|
|
|
516
|
-0.4
|
0.6
|
-0.002
|
0.000
|
-0.000
|
0.001
|
0.002
|
0.001
|
-0.002
|
-0.001
|
|
|
|
-0.4
|
0.6
|
(0.047)
|
(0.038)
|
(0.040)
|
(0.058)
|
(0.040)
|
(0.041)
|
(0.025)
|
(0.015)
|
|
|
|
0.5
|
0.5
|
0.000
|
-0.001
|
0.001
|
-0.001
|
-0.001
|
-0.001
|
-0.001
|
-0.001
|
|
|
|
0.5
|
0.5
|
(0.057)
|
(0.039)
|
(0.041)
|
(0.058)
|
(0.039)
|
(0.041)
|
(0.017)
|
(0.017)
|
|
|
|
0.2
|
0.8
|
0.003
|
0.001
|
-0.000
|
0.007
|
0.001
|
0.000
|
-0.001
|
-0.001
|
|
|
|
0.2
|
0.8
|
(0.052)
|
(0.038)
|
(0.039)
|
(0.068)
|
(0.038)
|
(0.040)
|
(0.022)
|
(0.010)
|
|
spse
|
52
|
-0.4
|
0.6
|
0.018
|
-0.001
|
-0.002
|
0.007
|
-0.004
|
-0.009
|
-0.021
|
-0.001
|
|
|
|
-0.4
|
0.6
|
(0.159)
|
(0.143)
|
(0.143)
|
(0.205)
|
(0.147)
|
(0.144)
|
(0.083)
|
(0.054)
|
|
|
|
0.5
|
0.5
|
0.002
|
-0.003
|
-0.003
|
0.005
|
-0.003
|
-0.005
|
0.000
|
0.000
|
|
|
|
0.5
|
0.5
|
(0.201)
|
(0.149)
|
(0.146)
|
(0.200)
|
(0.150)
|
(0.141)
|
(0.063)
|
(0.061)
|
|
|
|
0.2
|
0.8
|
0.011
|
-0.002
|
-0.004
|
0.008
|
-0.001
|
-0.009
|
-0.006
|
-0.001
|
|
|
|
0.2
|
0.8
|
(0.180)
|
(0.143)
|
(0.149)
|
(0.267)
|
(0.149)
|
(0.153)
|
(0.074)
|
(0.039)
|
|
|
516
|
-0.4
|
0.6
|
0.007
|
-0.001
|
-0.004
|
-0.025
|
-0.002
|
-0.003
|
-0.014
|
0.009
|
|
|
|
-0.4
|
0.6
|
(0.049)
|
(0.044)
|
(0.045)
|
(0.063)
|
(0.046)
|
(0.048)
|
(0.030)
|
(0.018)
|
|
|
|
0.5
|
0.5
|
-0.021
|
-0.003
|
-0.003
|
-0.022
|
-0.004
|
-0.003
|
0.010
|
0.010
|
|
|
|
0.5
|
0.5
|
(0.061)
|
(0.045)
|
(0.047)
|
(0.063)
|
(0.045)
|
(0.046)
|
(0.020)
|
(0.020)
|
|
|
|
0.2
|
0.8
|
-0.004
|
0.000
|
-0.001
|
-0.039
|
-0.005
|
-0.007
|
0.004
|
0.008
|
|
|
|
0.2
|
0.8
|
(0.053)
|
(0.044)
|
(0.044)
|
(0.078)
|
(0.043)
|
(0.045)
|
(0.023)
|
(0.013)
|
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.
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.
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.
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.
