Spatial seemingly unrelated regression models. A comparison of spsur, spse and PySAL

Introduction

The main objective of this vignette is to present the results of the estimation of spatial SUR models using three alternative tools: Two R-packages, spsur (López, Mı́nguez, and Mur 2020; Minguez, Lopez, and Mur \noop{2022}forthcoming) and spse (Piras 2018) together with the Python spatial analysis library PySAL available at https://pysal.org/. Three SUR models are estimated in this vignette. The first one, is the baseline SUR-SIM model, a SUR model without spatial effects. This model will be estimated with spsur and PySAL. The second one is the SUR-SEM model; that is, a SUR model including a spatial lag in the errors. This model will be estimated by Maximum Likelihood (Anselin 1988), using the three alternative tools. The last one is the SUR-SLM model, including a spatial lag of the dependent variable, which will be estimated by the Three Stage Least Squares (3SLS) algorithm (López, Mı́nguez, and Mur 2020; Anselin 2016). This model will be estimated with spsur and PySAL.

The comparison is performed on the Baller et al. (2001) data set1. This is a well-known dataset downloaded from the GeoDa Data and Lab collection with information about homicide rates in 3,085 continental US counties for four years (1960, 1970, 1980, and 1990). The dataset includes a large number of socio-economic characteristics for these counties. This dataset is available from the spsur R-package with the name NCOVR and from PySAL with the name of NAT.dbf.

The model selected to illustrate the results of the estimation is the same as that included in the help area of PySAL.

The SUR-SIM model

This section considers the simplest case of the estimation of an SUR model without spatial effects: SUR-SIM. The model specification is the same as that which appears in the help of area of PySAL and we reproduce it in the equation @ref(eq:ncovr):

The R code to estimate equation @ref(eq:ncovr) with the spsur package is:

data("NCOVR", package = "spsur")
formula.spsur <- HR80 | HR90 ~ PS80 + UE80 | PS90 + UE90
control <- list(trace = FALSE)
spsur.sim <- spsurml(formula = formula.spsur, data = NCOVR.sf, 
                     type = "sim", control = control)

The Python code to estimate equation @ref(eq:ncovr) with PySAL is:

db = pysal.open(pysal.examples.get_path("NAT.dbf"),'r')
y_var = ['HR80','HR90']
x_var = [['PS80','UE80'],['PS90','UE90']]
bigy,bigX,bigyvars,bigXvars = pysal.spreg.sur_utils.sur_dictxy(db,y_var,x_var)
w = pysal.knnW_from_shapefile(pysal.examples.get_path("NAT.shp"), k = 10)
w.transform = 'r'
pysal_sim = SUR(bigy,bigX,w=w,iter=True,
                name_bigy=bigyvars,name_bigX=bigXvars,spat_diag=True,
                name_ds="nat")

Note that the Python code includes the definition of the W matrix. Following Baller et al. (2001) we choose a W matrix based on the k-nearest-neighbors, with k = 10.

Table 2.1 shows the values of the coefficients and the standard error (in parentheses). The full output of both codes is shown in the Appendix. The main result is that no relevant differences are founded. The results of the estimations are similar, both in terms of the parameters and the standard errors, and only extremely small numerical differences appears.

Table 2.1 Estimated coefficients and standard errors. SUR-SIM
β̂10 β̂11 β̂12 β̂12 β̂12 β̂12
spsur 5.1794 0.6775 0.2578 3.7811 1.0243 0.3614
(0.2595) (0.1219) (0.0338) (0.2531) (0.1133) (0.0340)
PySAL 5.1842 0.6776 0.2571 3.7973 1.0241 0.3590
(0.2594) (0.1219) (0.0338) (0.2531) (0.1133) (0.0340)

Estimation of SUR-SEM

This section presents the results of the estimation of the SUR-SEM model by Maximum Likelihood with spsur, spse and PySAL. The formal expression of SUR-SEM includes a spatial structure in the residuals of the model @ref(eq:ncovr),

where W is the N × N spatial weighting matrix and λi (i=1,2) are the parameters of spatial dependence. The W matrix has been previously defined in the PySAL code using neighborhood criteria based on the k-nearest-neighbors with k = 10. This W matrix can be imported to the R environment and transformed into an listw object. The W matrix has been standardized in PySAL.

listw <- mat2listw(as.matrix(py_to_r(py$w)$sparse))

The R code to estimate the SUR-SEM model @ref(eq:sur-sem) with spsur is:

spsur.sem <- spsurml(formula = formula.spsur, data = NCOVR.sf,
                     listw = listw , type = "sem", control = control)

In order to estimate the same model with spse, the sf object NCOVR.sf must be reordered to transform the data set from a data frame into another one with a structure of panel data.

data <- data.frame(
  index_indiv = factor(cbind(paste0("Indv_",rep(1:3085,
                                                each = 2)))),
  year = rep(c(1980,1990),3085),
  HR = c(rbind(NCOVR.sf$HR80,NCOVR.sf$HR90)),
  PS = c(rbind(NCOVR.sf$PS80,NCOVR.sf$PS90)),
  UE = c(rbind(NCOVR.sf$UE80,NCOVR.sf$UE90)))

With this data frame, model @ref(eq:sur-sem) can be estimated with spse:

eq <- HR ~ PS + UE
formula.spse <- list(tp1 = eq, tp2 = eq)
spse.sem <- spseml(formula.spse, data = data,
                       w = listw, model = "error", quiet = TRUE)

Finally, the PySAL code to estimate SUR-SEM model @ref(eq:sur-sem) is:

from spreg import SURerrorML
pysal_sem = SURerrorML(bigy,bigX,w=w,name_bigy=bigyvars,name_bigX=bigXvars,
                       name_ds="NAT",name_w="nat_queen")

Table 3.1 shows the coefficients and the standard errors obtained in the estimation of equation @ref(eq:sur-sem) with the three alternatives. The results are very similar, and only small differences appear when the results of spse are compared with spsur and PySAL. The use of different optimization routines is a possible source of the these small numerical differences. For example, the spsur optimizes the concentrated Log-Likelihood with the bobyqa() from minqa (Bates et al. 2014) while spse uses nlminb() from the stats package (R Core Team 2020). The full output of both codes is shown in the Appendix.

Table 3.1 Estimated coefficients and standard errors. SUR-SEM
β̂10 β̂11 β̂12 β̂20 β̂21 β̂22 λ̂1 λ̂2
spsur 3.9998 1.0185 0.4313 3.1256 1.1626 0.4532 0.6680 0.6252
(0.4256) (0.1414) (0.0436) (0.3753) (0.1354) (0.0408) (0.0216) (0.0233)
spse 4.0458 1.0090 0.4247 3.1730 1.1609 0.4462 0.6550 0.6245
(0.4179) (0.1414) (0.0435) (0.3754) (0.1357) (0.0408) (0.0187) (0.0197)
PySAL 3.9996 1.0186 0.4314 3.1253 1.1626 0.4533 0.6680 0.6252
(0.4256) (0.1414) (0.0436) (0.3753) (0.1354) (0.0408) (NA) (NA)

Estimation of SUR-SLM

The option to estimate an SUR-SLM model with the 3SLS algorithm (Anselin 2016; López, Mı́nguez, and Mur 2020) is available with spsur and PySAL. The specification of this model in our case is:

where ρi (i=1,2) are the parameters of spatial dependence. The R code to estimate SUR-SLM (@ref(eq:sur-slm)) using the 3SLS algorithm is:

spsur.slm.3sls <- spsur3sls(formula = formula.spsur, data = NCOVR.sf,
                            listw = listw , type = "slm", trace = FALSE)

The code to estimate the equation (@ref(eq:sur-slm)) with PySAL is,

from spreg import SURlagIV
pysal_iv = SURlagIV(bigy,bigX,w=w,w_lags=2,name_bigy=bigyvars,
                    name_bigX=bigXvars,name_ds="NAT",name_w="nat_queen")

Note that the instruments used by default with the function spsur3sls() are the first two spatial lags of the independent variables (see López, Mı́nguez, and Mur (2020)) while the function SURlagIV() in PySAL considers only the first one by default (see Anselin (2016)). Therefore, to obtain equivalent results, it is necessary to include the option w_lags = 2 in the PySAL code.

Table 4.1 shows the coefficients and standard error (in parentheses) of the estimation with spsur and PySAL. As in the case of the SUR-SEM estimation, minimal differences are founded. The results are practically identical. The full output of both codes is shown in the Appendix.

Table 4.1 Estimated coefficients and standard errors. SUR-SLM-IV
β̂10 β̂11 β̂12 β̂20 β̂21 β̂22 ρ̂1 ρ̂2
spsur 3.6000 0.5932 0.2913 2.3915 0.8871 0.3626 0.1957 0.2230
(1.7539) (0.1687) (0.0409) (0.3649) (0.1155) (0.0417) (0.2611) (0.0727)
PySAL 3.6000 0.5932 0.2913 2.3915 0.8871 0.3626 0.1957 0.2230
(1.7536) (0.1687) (0.0409) (0.3648) (0.1155) (0.0417) (0.2610) (0.0727)

Conclusion

In this vignette, a numerical check to compare the results of several spatial SUR models estimations is shown. Fortunately, some functionalities of spsur are also available in the spse package and also in PySAL so the user can choose. The well-known data set (Baller et al. 2001) is used with the objective of comparing the values of the estimated coefficients and standard errors. The results confirm that the three alternatives supply identical outputs with extremely small numerical differences.

Appendix

Full results SUR-SIM

spsur output

summary(spsur.sim)
## Call:
## spsurml(formula = formula.spsur, data = NCOVR.sf, type = "sim", 
##     control = control)
## 
##  
## Spatial SUR model type:  sim 
## 
## Equation  1 
##               Estimate Std. Error t value  Pr(>|t|)    
## (Intercept)_1 5.179417   0.259455 19.9627 < 2.2e-16 ***
## PS80_1        0.677534   0.121932  5.5567 2.865e-08 ***
## UE80_1        0.257775   0.033814  7.6233 2.846e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## R-squared: 0.02502 
##   Equation  2 
##               Estimate Std. Error t value  Pr(>|t|)    
## (Intercept)_2 3.781120   0.253129  14.938 < 2.2e-16 ***
## PS90_2        1.024287   0.113331   9.038 < 2.2e-16 ***
## UE90_2        0.361394   0.034047  10.614 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## R-squared: 0.1099 
##   
## Variance-Covariance Matrix of inter-equation residuals:                  
##  45.43619 21.56823
##  21.56823 39.72187
## Correlation Matrix of inter-equation residuals:                    
##  1.0000000 0.5076902
##  0.5076902 1.0000000
## 
##  R-sq. pooled: 0.06654 
##  Breusch-Pagan: 795.9  p-value: (4.3e-175)

PySal output

print(pysal_sim.summary)
## REGRESSION
## ----------
## SUMMARY OF OUTPUT: SEEMINGLY UNRELATED REGRESSIONS (SUR)
## --------------------------------------------------------
## Data set            :         nat
## Weights matrix      :     unknown
## Number of Equations :           2                Number of Observations:        3085
## Log likelihood (SUR):  -19860.068                Number of Iterations  :           3
## ----------
## 
## SUMMARY OF EQUATION 1
## ---------------------
## Dependent Variable  :        HR80                Number of Variables   :           3
## Mean dependent var  :      6.9276                Degrees of Freedom    :        3082
## S.D. dependent var  :      6.8251
## 
## ------------------------------------------------------------------------------------
##             Variable     Coefficient       Std.Error     z-Statistic     Probability
## ------------------------------------------------------------------------------------
##           Constant_1       5.1842323       0.2593924      19.9860602       0.0000000
##                 PS80       0.6775792       0.1219113       5.5579678       0.0000000
##                 UE80       0.2570650       0.0338051       7.6043173       0.0000000
## ------------------------------------------------------------------------------------
## 
## SUMMARY OF EQUATION 2
## ---------------------
## Dependent Variable  :        HR90                Number of Variables   :           3
## Mean dependent var  :      6.1829                Degrees of Freedom    :        3082
## S.D. dependent var  :      6.6403
## 
## ------------------------------------------------------------------------------------
##             Variable     Coefficient       Std.Error     z-Statistic     Probability
## ------------------------------------------------------------------------------------
##           Constant_2       3.7973181       0.2531089      15.0027035       0.0000000
##                 PS90       1.0241120       0.1133298       9.0365598       0.0000000
##                 UE90       0.3589567       0.0340440      10.5438928       0.0000000
## ------------------------------------------------------------------------------------
## 
## 
## REGRESSION DIAGNOSTICS
##                                      TEST         DF       VALUE           PROB
##                          LM test on Sigma         1      680.168           0.0000
##                          LR test on Sigma         1      854.181           0.0000
## 
## OTHER DIAGNOSTICS - CHOW TEST BETWEEN EQUATIONS
##                                 VARIABLES         DF       VALUE           PROB
##                    Constant_1, Constant_2         1       23.457           0.0000
##                                PS80, PS90         1        8.700           0.0032
##                                UE80, UE90         1        6.843           0.0089
## 
## DIAGNOSTICS FOR SPATIAL DEPENDENCE
## TEST                              DF       VALUE           PROB
## Lagrange Multiplier (error)       2        2278.632        0.0000
## Lagrange Multiplier (lag)         2        2153.976        0.0000
## 
## ERROR CORRELATION MATRIX
##   EQUATION 1  EQUATION 2
##     1.000000    0.507913
##     0.507913    1.000000
## ================================ END OF REPORT =====================================

Full results SUR-SEM

spsur output

summary(spsur.sem)
## Call:
## spsurml(formula = formula.spsur, data = NCOVR.sf, listw = listw, 
##     type = "sem", control = control)
## 
##  
## Spatial SUR model type:  sem 
## 
## Equation  1 
##               Estimate Std. Error t value  Pr(>|t|)    
## (Intercept)_1 3.999783   0.425587  9.3983 < 2.2e-16 ***
## PS80_1        1.018523   0.141382  7.2040 6.544e-13 ***
## UE80_1        0.431340   0.043645  9.8829 < 2.2e-16 ***
## lambda_1      0.667989   0.021592 30.9374 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## R-squared: 0.3561 
##   Equation  2 
##               Estimate Std. Error t value  Pr(>|t|)    
## (Intercept)_2 3.125570   0.375304  8.3281 < 2.2e-16 ***
## PS90_2        1.162587   0.135392  8.5868 < 2.2e-16 ***
## UE90_2        0.453225   0.040766 11.1178 < 2.2e-16 ***
## lambda_2      0.625186   0.023276 26.8594 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## R-squared: 0.3724 
##   
## Variance-Covariance Matrix of inter-equation residuals:                    
##  30.711725  8.751191
##   8.751191 28.408108
## Correlation Matrix of inter-equation residuals:                    
##  1.0000000 0.2962743
##  0.2962743 1.0000000
## 
##  R-sq. pooled: 0.3658 
##  Breusch-Pagan: 270.8  p-value: (7.63e-61) 
##  LMM: 267.61  p-value: (3.76e-60)

spse output

summary(spse.sem)
## 
## Simultaneous Equations Model:
## 
## Call:
## spseml(formula = formula.spse, data = data, w = listw, quiet = TRUE, 
##     model = "error")
##  
## Equation 1
##             Estimate Std.Error t value  Pr(>|t|)    
## (Intercept) 4.045752  0.417886  9.6815 < 2.2e-16 ***
## PS          1.009015  0.141429  7.1344 9.719e-13 ***
## UE          0.424679  0.043513  9.7598 < 2.2e-16 ***
##  
## Spatial autocorrelation coefficient: 0.655 Pr(>|t|) 0
## 
##  _______________________________________________________ 
##  
## Equation 2
##             Estimate Std.Error t value  Pr(>|t|)    
## (Intercept) 3.172955  0.375401  8.4522 < 2.2e-16 ***
## PS          1.160865  0.135689  8.5553 < 2.2e-16 ***
## UE          0.446204  0.040767 10.9452 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##  
## Spatial autocorrelation coefficient: 0.6245 Pr(>|t|) 0
## 
##  _______________________________________________________

PySal output

print(pysal_sem.summary)
## REGRESSION
## ----------
## SUMMARY OF OUTPUT: SEEMINGLY UNRELATED REGRESSIONS (SUR) - ML SPATIAL ERROR MODEL
## ---------------------------------------------------------------------------------
## Data set            :         NAT
## Weights matrix      :   nat_queen
## Number of Equations :           2                Number of Observations:        3085
## Log likelihood (SUR):  -19860.068
## Log likel. (error)  :  -19344.228                Log likel. (SUR error):  -19215.933
## ----------
## 
## SUMMARY OF EQUATION 1
## ---------------------
## Dependent Variable  :        HR80                Number of Variables   :           3
## Mean dependent var  :      6.9276                Degrees of Freedom    :        3082
## S.D. dependent var  :      6.8251
## 
## ------------------------------------------------------------------------------------
##             Variable     Coefficient       Std.Error     z-Statistic     Probability
## ------------------------------------------------------------------------------------
##           Constant_1       3.9995930       0.4255575       9.3984776       0.0000000
##                 PS80       1.0185647       0.1413612       7.2054039       0.0000000
##                 UE80       0.4313678       0.0436389       9.8849267       0.0000000
##             lambda_1       0.6680433    
## ------------------------------------------------------------------------------------
## 
## SUMMARY OF EQUATION 2
## ---------------------
## Dependent Variable  :        HR90                Number of Variables   :           3
## Mean dependent var  :      6.1829                Degrees of Freedom    :        3082
## S.D. dependent var  :      6.6403
## 
## ------------------------------------------------------------------------------------
##             Variable     Coefficient       Std.Error     z-Statistic     Probability
## ------------------------------------------------------------------------------------
##           Constant_2       3.1253307       0.3752706       8.3282066       0.0000000
##                 PS90       1.1625986       0.1353719       8.5881796       0.0000000
##                 UE90       0.4532595       0.0407597      11.1202836       0.0000000
##             lambda_2       0.6252406    
## ------------------------------------------------------------------------------------
## 
## 
## REGRESSION DIAGNOSTICS
##                                      TEST         DF       VALUE           PROB
##                          LR test on Sigma         1      256.591           0.0000
## 
## OTHER DIAGNOSTICS - CHOW TEST BETWEEN EQUATIONS
##                                 VARIABLES         DF       VALUE           PROB
##                    Constant_1, Constant_2         1        3.111           0.0778
##                                PS80, PS90         1        0.767           0.3812
##                                UE80, UE90         1        0.165           0.6847
## 
## ERROR CORRELATION MATRIX
##   EQUATION 1  EQUATION 2
##     1.000000    0.296257
##     0.296257    1.000000
## ================================ END OF REPORT =====================================

Full results SUR-SLM-IV

spsur output

summary(spsur.slm.3sls)
## Call:
## spsur3sls(formula = formula.spsur, data = NCOVR.sf, listw = listw, 
##     type = "slm", trace = FALSE)
## 
##  
## Spatial SUR model type:  slm 
## 
## Equation  1 
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)_1 3.600033   1.753859  2.0526 0.040150 *  
## PS80_1        0.593229   0.168744  3.5156 0.000442 ***
## UE80_1        0.291323   0.040887  7.1250 1.16e-12 ***
## rho_1         0.195653   0.261078  0.7494 0.453643    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## R-squared: 0.1995 
##   Equation  2 
##               Estimate Std. Error t value  Pr(>|t|)    
## (Intercept)_2 2.391472   0.364855  6.5546 6.033e-11 ***
## PS90_2        0.887118   0.115519  7.6794 1.848e-14 ***
## UE90_2        0.362642   0.041738  8.6885 < 2.2e-16 ***
## rho_2         0.222994   0.072671  3.0686   0.00216 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## R-squared: 0.2807 
##   
## Variance-Covariance Matrix of inter-equation residuals:                  
##  59.15973 21.01769
##  21.01769 34.68056
## Correlation Matrix of inter-equation residuals:                    
##  1.0000000 0.4397986
##  0.4397986 1.0000000
## 
##  R-sq. pooled: 0.2385 
##  Breusch-Pagan: 664.2  p-value: (1.8e-146)

PySal output

print(pysal_iv.summary)
## REGRESSION
## ----------
## SUMMARY OF OUTPUT: SEEMINGLY UNRELATED REGRESSIONS (SUR) - SPATIAL LAG MODEL
## ----------------------------------------------------------------------------
## Data set            :         NAT
## Weights matrix      :   nat_queen
## Number of Equations :           2                Number of Observations:        3085
## ----------
## 
## SUMMARY OF EQUATION 1
## ---------------------
## Dependent Variable  :        HR80                Number of Variables   :           4
## Mean dependent var  :      6.9276                Degrees of Freedom    :        3081
## S.D. dependent var  :      6.8251
## 
## ------------------------------------------------------------------------------------
##             Variable     Coefficient       Std.Error     z-Statistic     Probability
## ------------------------------------------------------------------------------------
##           Constant_1       3.6000333       1.7535743       2.0529688       0.0400756
##                 PS80       0.5932292       0.1687164       3.5161321       0.0004379
##                 UE80       0.2913228       0.0408809       7.1261406       0.0000000
##               W_HR80       0.1956528       0.2610360       0.7495244       0.4535412
## ------------------------------------------------------------------------------------
## Instrumented: W_HR80
## Instruments: WW_PS80, WW_UE80, W_PS80, W_UE80
## 
## SUMMARY OF EQUATION 2
## ---------------------
## Dependent Variable  :        HR90                Number of Variables   :           4
## Mean dependent var  :      6.1829                Degrees of Freedom    :        3081
## S.D. dependent var  :      6.6403
## 
## ------------------------------------------------------------------------------------
##             Variable     Coefficient       Std.Error     z-Statistic     Probability
## ------------------------------------------------------------------------------------
##           Constant_2       2.3914725       0.3647963       6.5556384       0.0000000
##                 PS90       0.8871183       0.1155004       7.6806495       0.0000000
##                 UE90       0.3626418       0.0417315       8.6898895       0.0000000
##               W_HR90       0.2229941       0.0726587       3.0690610       0.0021473
## ------------------------------------------------------------------------------------
## Instrumented: W_HR90
## Instruments: WW_PS90, WW_UE90, W_PS90, W_UE90
## 
## 
## OTHER DIAGNOSTICS - CHOW TEST BETWEEN EQUATIONS
##                                 VARIABLES         DF       VALUE           PROB
##                    Constant_1, Constant_2         1        0.496           0.4811
##                                PS80, PS90         1        3.288           0.0698
##                                UE80, UE90         1        1.945           0.1631
##                            W_HR80, W_HR90         1        0.011           0.9147
## 
## ERROR CORRELATION MATRIX
##   EQUATION 1  EQUATION 2
##     1.000000    0.464012
##     0.464012    1.000000
## ================================ END OF REPORT =====================================
Anselin, Luc. 1988. Spatial Econometrics: Methods and Models. Studies in Operational Regional Science. Dordrecht: Kluwer Academic Publishers.
———. 2016. “Estimation and Testing in the Spatial Seemingly Unrelated Regression (SUR).” Geoda Center for Geospatial Analysis; Computation, Arizona State University. Working Paper 2016-01. https://doi.org/10.13140/RG.2.2.15925.40163.
Baller, Robert D, Luc Anselin, Steven F Messner, Glenn Deane, and Darnell F Hawkins. 2001. “Structural Covariates of US County Homicide Rates: Incorporating Spatial Effects.” Criminology 39 (3): 561–88.
Bates, Douglas, Katharine M. Mullen, John C. Nash, and Ravi Varadhan. 2014. minqa: Derivative-Free Optimization Algorithms by Quadratic Approximation. https://CRAN.R-project.org/package=minqa.
López, Fernando A, Román Mı́nguez, and Jesús Mur. 2020. “ML Versus IV Estimates of Spatial SUR Models: Evidence from the Case of Airbnb in Madrid Urban Area.” The Annals of Regional Science 64 (2): 313–47. https://doi.org/10.1007/s00168-019-00914-1.
Minguez, Roman, Fernando Lopez, and Jesus Mur. \noop{2022}forthcoming. “spsur: an R package for dealing with Spatial Seemingly Unrelated Regression Models.” Journal of Statistical Software, \noop{2022}forthcoming. https://github.com/rominsal/spsur/tree/master/vignettes.
Piras, Gianfranco. 2018. “spse: Spatial Simultaneous Equations and Spatial SUR Model.” https://github.com/gpiras.
R Core Team. 2020. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/.

  1. see GeoDa for more details about the database↩︎