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R-CMD-check [CRAN status [CRAN downloads-last-month [CRAN downloads-grand-total Lifecycle: stable

spsur

The spsur package allows for the estimation of the most popular Spatial Seemingly Unrelated Regression models by maximum likelihood or instrumental variable procedures (SUR-SLX; SUR-SLM; SUR-SEM;SUR-SDM; SUR-SDEM and SUR-SARAR) and non spatial SUR model (SUR-SIM). Moreover, spsur implements a collection of Lagrange Multipliers and Likelihood Ratios to test for misspecifications in the SUR. Additional functions allow for the estimation of the so-called spatial impacts (direct, indirect and total effects) and also obtains random data sets, of a SUR nature, with the features decided by the user. An important aspect of spsur is that it operates both in a pure cross-sectional setting or in panel data sets.

Installation

You can install the released version of spsur from CRAN with:

install.packages("spsur")

Main functionalities of spsur

A few functions are necessary to test spatial autocorrelation in SUR and estimate the spatial SUR models. Figure show the main functionalities of the spsur package. The spsur package has one function to test spatial autocorrelation on the residuals of a SUR model (lmtestspsur()); two functions to estimate SUR models with several spatial structures, by maximum likelihood (spsurml()) and instrumental variables (spsur3sls()); three functions to help to the user to select the correct espeficication (lrtestspsur(); wald_betas() and wald_deltas(); one function to get the impacts (impacts()). Finally another function has been included in this package to help to the user to develop Monte Carlo exercices (dgp_spsur()).

\label{Fig1} Main functionatities of spsur package

Data sets in spSUR

The spSUR package include two data sets:

The spc (Spatial Phillips-Curve). A classical data set from Anselin (1988, p.203)

A total of N=25 observations and Tm=2 time periods

COUNTY WAGE83 UN83 NMR83 SMSA WAGE82 WAGE81 UN80 NMR80 WAGE80
UNION 1.003127 0.080500 -0.002217 1 1.108662 1.146178 0.130375 -0.010875 1.084886
DELAWARE 1.039972 0.122174 0.018268 1 1.071271 1.104241 0.189603 0.041886 1.110426
LICKING 1.050196 0.095821 -0.013681 1 1.058375 1.094732 0.124125 -0.004158 1.069776

Homicides + Socio-Economics characteristics for U.S. counties (1960-90)

from [https://geodacenter.github.io/data-and-lab/ncovr/]

Homicides and selected socio-economic characteristics for continental U.S. counties. Data for four decennial census years: 1960, 1970, 1980 and 1990.
A total of N=3,085 US counties

0

NAME STATE_NAME STATE_FIPS CNTY_FIPS FIPS STFIPS COFIPS FIPSNO SOUTH
Lake of the Woods Minnesota 27 077 27077 27 77 27077 0
Ferry Washington 53 019 53019 53 19 53019 0
Stevens Washington 53 065 53065 53 65 53065 0

How to specify multiple equations: The Formula package

By example: two equations with different number of

regressors

(Y_{1} = \beta_{10} + \beta_{11} \ X_{11}+\beta_{12}X_{12}+\epsilon_{1})
(Y_{2} = \beta_{20} + \beta_{21} \ X_{21}+\epsilon_{2})

formula <- (Y_{1}) | (Y_{2}) ~ (X_{11}) + (X_{12})   |   (X_{21})

Note that in the left side of the formula, two dependent variables has been included separated by the symbol |. In right side, the independent variables for each equation are included separated newly by a vertical bar |, keeping the same order that in the left side.

The spSUR package step by step

Step 1: Testing for spatial effects

Step 2: Estimation of the Spatial SUR models

Step 3: Looking for the correct especification

Step 4: Impacts: Directs, Indirects and Total effects

Step 5: spSUR in a panel data framework

Step 6: Additional functionalities

Step 7: Conclusion and work to do

Step 1: Testing for: lmtestspsur

The function lmtestspsur() obtain five LM statistis for testing spatial dependence in Seemingly Unrelated Regression models
(Mur J, López FA, Herrera M, 2010: Testing for spatial effect in Seemingly Unrelated Regressions. Spatial Economic Analysis 5(4) 399-440).

(H_{0}:) No spatial autocorrelation
(H_{A}:) SUR-SAR or
(H_{A}:) SUR-SEM or
(H_{A}:) SUR-SARAR

  • LM-SUR-SAR
  • LM-SUR-SEM
  • LM-SUR-SARAR

and two robust LM tests

  • LM*-SUR-SAR
  • LM*-SUR-SEM

Example 1: with Anselin’s data we can test spatial effects in the SUR model:

(WAGE_{83} = \beta_{10} + \beta_{11} \ UN_{83} + \beta_{12} \ NMR_{83} + \beta_{13} \ SMSA + \epsilon_{83})
(WAGE_{81} = \beta_{20} + \beta_{21} \ UN_{80} + \beta_{22} \ NMR_{80} + \beta_{23} \ SMSA+ \epsilon_{81})
(Corr(\epsilon_{83},\epsilon_{81}) \neq 0)

library("spsur")
data("spc")
Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA
LMs <- lmtestspsur(Form=Tformula,data=spc,W=Wspc)
#>              LM-Stat. DF p-value  
#> LM-SUR-SLM     5.2472  2  0.0725 .
#> LM-SUR-SEM     3.3050  2  0.1916  
#> LM*-SUR-SLM    2.1050  2  0.3491  
#> LM*-SUR-SEM    0.1628  2  0.9218  
#> LM-SUR-SARAR   5.7703  4  0.2170  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

In this example no spatial autocorrelation is identify! (25 observations).

Step 2: Estimation of a Spatial SUR

Two alternative estimation methods are implemented:

2.1. Maximum likelihood estimation: spsurml

Maximum likelihood estimation for differents spatial SUR models using spsurml. The models are:

  • SUR-SIM: with out spatial autocorrelation
  • SUR-SLX: Spatial Lag of X SUR model
  • SUR-SLM: Spatial Autorregresive SUR model
  • SUR-SEM: Spatial Error SUR model
  • SUR-SDM: Spatial Durbin SUR model
  • SUR-SDEM: Spatial Durbin Error SUR model
  • SUR-SARAR: Spatial Autorregresive with Spatial Error SUR model

2.1.1 Anselin data set

SUR-SAR: Spatial autorregresive model:
((y_{t} = \lambda_{t} Wy_{t} + X_{t} \beta_{t} + \epsilon_{t}; \ t=1,...,T) )

(WAGE_{83} = \lambda_{83} WWAGE_{83} + \beta_{10} + \beta_{11} UN_{83} + \beta_{12} NMR_{83} + \beta_{13} SMSA + \epsilon_{83})
(WAGE_{81} = \lambda_{81} WWAGE_{81} + \beta_{20} + \beta_{21} UN_{80} + \beta_{22} NMR_{80} + \beta_{23} SMSA+ \epsilon_{81})
(Corr(\epsilon_{83},\epsilon_{81}) \neq 0)

spcSUR.slm <-spsurml(Form=Tformula,data=spc,type="slm",W=Wspc)
#> Initial point:   log_lik:  113.197  lambdas:  -0.472 -0.446 
#> Iteration:  1   log_lik:  114.085  lambdas:  -0.506 -0.482 
#> Iteration:  2   log_lik:  114.096  lambdas:  -0.506 -0.482 
#> Time to fit the model:  3.11  seconds
#> Computing marginal test... 
#> Time to compute covariances:  0.4  seconds
summary(spcSUR.slm)
#> Call:
#> spsurml(Form = Tformula, data = spc, W = Wspc, type = "slm")
#> 
#>  
#> Spatial SUR model type:  slm 
#> 
#> Equation  1 
#>                 Estimate Std. Error t value  Pr(>|t|)    
#> (Intercept)_1  1.4955217  0.2467240  6.0615 5.183e-07 ***
#> UN83_1         0.8070029  0.2557439  3.1555  0.003179 ** 
#> NMR83_1       -0.5194114  0.2590550 -2.0050  0.052318 .  
#> SMSA_1        -0.0073247  0.0118519 -0.6180  0.540347    
#> lambda_1      -0.5057334  0.2405734 -2.1022  0.042401 *  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.6224 
#>   Equation  2 
#>                 Estimate Std. Error t value  Pr(>|t|)    
#> (Intercept)_2  1.7094414  0.2925620  5.8430 1.024e-06 ***
#> UN80_2        -0.6745562  0.3870737 -1.7427   0.08969 .  
#> NMR80_2        0.7502934  0.3842670  1.9525   0.05847 .  
#> SMSA_2         0.0014181  0.0241859  0.0586   0.95356    
#> lambda_2      -0.4821428  0.2557758 -1.8850   0.06730 .  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.4743 
#>   Variance-Covariance Matrix of inter-equation residuals:                            
#>   0.0003085954 -0.0003561928
#>  -0.0003561928  0.0015864976
#> Correlation Matrix of inter-equation residuals:                    
#>   1.000000 -0.509062
#>  -0.509062  1.000000
#> 
#>  R-sq. pooled: 0.6603 
#>  Log-Likelihood:  114.096
#>  Breusch-Pagan: 6.516  p-value: (0.0107) 
#>  LMM: 0.50489  p-value: (0.477)

Only change the ‘type’ argument in spsurml function it is possible to estimate several spatial model

SUR-SEM: Spatial error model:
((y_{t} = X_{t}\ \beta_{t} + u_{t}\ ; u_{t}=\rho u_{t}+ \epsilon_{t}\ t=1,...,T))

(WAGE_{83} = \beta_{10} + \beta_{11} UN_{83} + \beta_{12} NMR_{83} + \beta_{13} SMSA + u_{83}; \ u_{83}=\rho W \ u_{83} + \epsilon_{83})
(WAGE_{81} =\beta_{20} + \beta_{21} UN_{80} + \beta_{22} NMR_{80} + \beta_{23} SMSA+ u_{81}; \ u_{81}=\rho W \ u_{81} + \epsilon_{81})
(Corr(\epsilon_{83},\epsilon_{81}) \neq 0)

spcSUR.sem <-spsurml(Form=Tformula,data=spc,type="sem",W=Wspc)
#> Initial point:   log_lik:  112.821  deltas:  -0.556 -0.477 
#> Iteration:  1  log_lik:  113.695  rhos:  -0.618 -0.537 
#> Iteration:  2  log_lik:  113.719  rhos:  -0.628 -0.548 
#> Time to fit the model:  3.99  seconds
#> Computing marginal test... 
#> Time to compute covariances:  0.36  seconds
summary(spcSUR.sem)
#> Call:
#> spsurml(Form = Tformula, data = spc, W = Wspc, type = "sem")
#> 
#>  
#> Spatial SUR model type:  sem 
#> 
#> Equation  1 
#>                 Estimate Std. Error t value  Pr(>|t|)    
#> (Intercept)_1  0.9805218  0.0151899 64.5511 < 2.2e-16 ***
#> UN83_1         0.7383349  0.2277247  3.2422  0.002513 ** 
#> NMR83_1       -0.4859228  0.2550377 -1.9053  0.064535 .  
#> SMSA_1        -0.0132403  0.0099122 -1.3358  0.189790    
#> rho_1         -0.6280610  0.2774391 -2.2638  0.029541 *  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.6607 
#>   Equation  2 
#>                 Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)_2  1.1479484  0.0386961 29.6657  < 2e-16 ***
#> UN80_2        -0.4406330  0.3614882 -1.2189  0.23058    
#> NMR80_2        0.8223976  0.4062173  2.0245  0.05018 .  
#> SMSA_2         0.0041942  0.0204639  0.2050  0.83873    
#> rho_2         -0.5480668  0.2817155 -1.9455  0.05935 .  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.5092 
#>   Variance-Covariance Matrix of inter-equation residuals:                            
#>   0.0002970481 -0.0003217158
#>  -0.0003217158  0.0015512097
#> Correlation Matrix of inter-equation residuals:                      
#>   1.0000000 -0.4739403
#>  -0.4739403  1.0000000
#> 
#>  R-sq. pooled: 0.673 
#>  Log-Likelihood:  113.719
#>  Breusch-Pagan: 5.512  p-value: (0.0189) 
#>  LMM: 1.4742  p-value: (0.225)

Step 3: Testing for misspecification in spatial SUR

3.1 Testing for the diagonality of (\Sigma)

The Breush-Pagan test of diagonality of (\Sigma)

(H_{0}: \Sigma = \sigma^2 I_{R})
(H_{A}: \Sigma \neq \sigma^2 I_{R})

3.2 Marginal tests: (LM(\rho|\lambda)) & (LM(\lambda|\rho))

The Marginal Multiplier tests (LMM) are used to test for no correlation in one part of the model allowing for spatial correlation in the other.

  • The (LM(\rho|\lambda)) is the test for spatial error correlation in a model with subtantive spatial correlation (SUR-SAR; SUR-SDM).

(H_{0}: SUR-SAR)
(H_{A}: SUR-SARAR)

  • The (LM(\lambda|\rho)) is the test for subtantive spatial autocorrelation in a model with spatial autocorrelation in error term (SUR-SEM; SUR-SDEM).

(H_{0}: SUR-SEM)
(H_{A}: SUR-SARAR)

summary(spcSUR.sem)
#> Call:
#> spsurml(Form = Tformula, data = spc, W = Wspc, type = "sem")
#> 
#>  
#> Spatial SUR model type:  sem 
#> 
#> Equation  1 
#>                 Estimate Std. Error t value  Pr(>|t|)    
#> (Intercept)_1  0.9805218  0.0151899 64.5511 < 2.2e-16 ***
#> UN83_1         0.7383349  0.2277247  3.2422  0.002513 ** 
#> NMR83_1       -0.4859228  0.2550377 -1.9053  0.064535 .  
#> SMSA_1        -0.0132403  0.0099122 -1.3358  0.189790    
#> rho_1         -0.6280610  0.2774391 -2.2638  0.029541 *  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.6607 
#>   Equation  2 
#>                 Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)_2  1.1479484  0.0386961 29.6657  < 2e-16 ***
#> UN80_2        -0.4406330  0.3614882 -1.2189  0.23058    
#> NMR80_2        0.8223976  0.4062173  2.0245  0.05018 .  
#> SMSA_2         0.0041942  0.0204639  0.2050  0.83873    
#> rho_2         -0.5480668  0.2817155 -1.9455  0.05935 .  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.5092 
#>   Variance-Covariance Matrix of inter-equation residuals:                            
#>   0.0002970481 -0.0003217158
#>  -0.0003217158  0.0015512097
#> Correlation Matrix of inter-equation residuals:                      
#>   1.0000000 -0.4739403
#>  -0.4739403  1.0000000
#> 
#>  R-sq. pooled: 0.673 
#>  Log-Likelihood:  113.719
#>  Breusch-Pagan: 5.512  p-value: (0.0189) 
#>  LMM: 1.4742  p-value: (0.225)

3.3 Coefficient stability/homogeneity

3.3.1 Wald tests for beta coefficients: wald_betas

In a SUR-SAR the

model:

(WAGE_{83} = \lambda_{83} W \ WAGE_{83} + \beta_{10} + \beta_{11} UN_{83} + \beta_{12} NMR_{83} + \boldsymbol{\beta_{13}} SMSA + \epsilon_{83})
(WAGE_{81} = \lambda_{81} W \ WAGE_{81} + \beta_{20} + \beta_{21} UN_{80} + \beta_{22} NMR_{80} + \boldsymbol{\beta_{23}} SMSA + \epsilon_{81})

It’s possible to test equality between SMSA coefficients in both equations:

(H_{0}: \beta_{13} = \beta_{23})
(H_{A}: \beta_{13} \neq \beta_{23})

Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA
spcSUR.slm <-spsurml(Form=Tformula,data=spc,type="slm",W=Wspc)
#> Initial point:   log_lik:  113.197  lambdas:  -0.472 -0.446 
#> Iteration:  1   log_lik:  114.085  lambdas:  -0.506 -0.482 
#> Iteration:  2   log_lik:  114.096  lambdas:  -0.506 -0.482 
#> Time to fit the model:  2.89  seconds
#> Computing marginal test... 
#> Time to compute covariances:  0.2  seconds
R1 <- matrix(c(0,0,0,1,0,0,0,-1),nrow=1)
b1 <- matrix(0,ncol=1)
Wald_beta <- wald_betas(results=spcSUR.slm,R=R1,b=b1)
#> Wald stat.: 0.079 p-value: (0.779)

More complex hypothesis about (\beta) coefficients could be tested using R1 vector

(H_{0}: \beta_{13} = \beta_{23}) and (\beta_{12} = \beta_{22})
(H_{A}: \beta_{13} \neq \beta_{23}) or (\beta_{12} \neq \beta_{22})

# Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA
# spcSUR.slm <-spsurml(Form=Tformula,data=spc,type="slm",W=Wspc)
R1 <- t(matrix(c(0,0,0,1,0,0,0,-1,0,0,1,0,0,0,-1,0),ncol=2))
b1 <- matrix(0,ncol=2)
print(R1)
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
#> [1,]    0    0    0    1    0    0    0   -1
#> [2,]    0    0    1    0    0    0   -1    0
Wald_beta <- wald_betas(results=spcSUR.slm,R=R1,b=b1)
#> Wald stat.: 6.179 p-value: (0.046)

Estimate the restricted model

In case don’t reject the null, it’s possible to estimate the model with equal coefficient in both equations:

(WAGE_{83} = \lambda_{83} W \ WAGE_{83} + \beta_{10} + \beta_{11} UN_{83} + \beta_{12} NMR_{83} + \boldsymbol{\beta_{13}} SMSA + \epsilon_{83})
(WAGE_{81} = \lambda_{81} W \ WAGE_{81} + \beta_{20} + \beta_{21} UN_{80} + \beta_{22} NMR_{80} + \boldsymbol{\beta_{13}} SMSA + \epsilon_{81})

Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA
R1 <- matrix(c(0,0,0,1,0,0,0,-1),nrow=1)
b1 <- matrix(0,ncol=1)
spcSUR.sar.restring <- spsurml(Form=Tformula, data=spc, type="slm", W=Wspc,R=R1,b=b1)
#> Initial point:   log_lik:  113.161  lambdas:  -0.428 -0.421 
#> Iteration:  1   log_lik:  114.01  lambdas:  -0.482 -0.465 
#> Iteration:  2   log_lik:  114.049  lambdas:  -0.495 -0.476 
#> Time to fit the model:  2.8  seconds
#> Computing marginal test... 
#> Time to compute covariances:  0.21  seconds
summary(spcSUR.sar.restring)
#> Call:
#> spsurml(Form = Tformula, data = spc, R = R1, b = b1, W = Wspc, 
#>     type = "slm")
#> 
#>  
#> Spatial SUR model type:  slm 
#> 
#> Equation  1 
#>                 Estimate Std. Error t value  Pr(>|t|)    
#> (Intercept)_1  1.4876833  0.2444552  6.0857 4.343e-07 ***
#> UN83_1         0.7486660  0.2290706  3.2683  0.002301 ** 
#> NMR83_1       -0.4565917  0.2559136 -1.7842  0.082384 .  
#> SMSA_1        -0.0041128  0.0081857 -0.5024  0.618257    
#> lambda_1      -0.4950681  0.2392546 -2.0692  0.045377 *  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.6115 
#>   Equation  2 
#>               Estimate Std. Error t value  Pr(>|t|)    
#> (Intercept)_2  1.69869    0.28885  5.8809 8.294e-07 ***
#> UN80_2        -0.61498    0.32522 -1.8910   0.06627 .  
#> NMR80_2        0.68793    0.37391  1.8398   0.07362 .  
#> NA                  NA         NA      NA        NA    
#> lambda_2      -0.47556    0.25016 -1.9010   0.06490 .  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.4691 
#>   Variance-Covariance Matrix of inter-equation residuals:                           
#>   0.0003220152 -0.000394289
#>  -0.0003942890  0.001616933
#> Correlation Matrix of inter-equation residuals:                      
#>   1.0000000 -0.5464249
#>  -0.5464249  1.0000000
#> 
#>  R-sq. pooled: 0.6531 
#>  Log-Likelihood:  114.049
#>  Breusch-Pagan: 7.318  p-value: (0.00683) 
#>  LMM: 0.36616  p-value: (0.545)

3.3.2 Wald test for ‘spatial’ coefficients homogeneity: wald_deltas

In same way a test for equal spatial autocorrelation coefficients can be obtain with wald_deltas function:
In the

model:

(WAGE_{83} = \boldsymbol{\lambda_{83}} W \ WAGE_{83} + \beta_{10} + \beta_{11} UN_{83} + \beta_{12} NMR_{83} + \beta_{13} SMSA + \epsilon_{83})
(WAGE_{81} = \boldsymbol{\lambda_{81}} W \ WAGE_{81} + \beta_{20} + \beta_{21} UN_{80} + \beta_{22} NMR_{80} + \beta_{23} SMSA + \epsilon_{81})

In this case the null is:

(H_{0}: \lambda_{83} = \lambda_{81})
(H_{A}: \lambda_{83} \neq \lambda_{81})

# Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA
# spcSUR.slm <-spsurml(Form=Tformula,data=spc,type="slm",W=Wspc,trace=F)
R1 <- matrix(c(1,-1),nrow=1)
b1 <- matrix(0,ncol=1)
res1 <- wald_deltas(results=spcSUR.slm,R=R1,b=b1)
#> 
#>  Wald stat.: 0.006 (0.939)

3.3.3 Likekihood ratio tests lr_betas_spsur

Alternatively to wald test, the Likelihoo Ration (LR) tests can be obtain using the lr_betas_spsur function.

Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA
R1 <- matrix(c(0,0,0,1,0,0,0,-1),nrow=1)
b1 <- matrix(0,ncol=1)
LR_SMSA <-  lr_betas_spsur(Form=Tformula,data=spc,W=Wspc,type="slm",R=R1,b=b1,trace=F,printmodels=F)
#> 
#>  Fitting unrestricted model ... 
#> 
#>  Time to fit unrestricted model:  2.88  seconds
#> 
#>  Fitting restricted model ... 
#> Time to fit restricted model:  2.83  seconds
#> 
#>  LR-Test 
#> 
#>  Log-likelihood unrestricted model:  114.096
#>  Log-likelihood restricted model:  114.049
#>  LR statistic:  0.095  degrees of freedom:  1  p-value: ( 0.7576548 )

4. Step 4: Marginal Effects: impacts

The marginal effects impacts of spatial autoregressive models (SUR-SAR; SUR-SDM; SUR-SARAR) has been calculated following the propose of LeSage and Pace (2009).

eff.spcSUR.sar <-impacts(spcSUR.slm,nsim=299)
#> 
#> Spatial SUR model type:  slm 
#> 
#>  Direct effects 
#> 
#>                mean          sd  t-stat     p-val    
#> UN83_1   8.6479e-01  2.5919e-01  3.3365 0.0008484 ***
#> NMR83_1 -5.7166e-01  2.9096e-01 -1.9647 0.0494447 *  
#> SMSA_1  -7.8388e-03  1.2291e-02 -0.6378 0.5236346    
#> UN80_2  -6.8648e-01  4.0118e-01 -1.7112 0.0870514 .  
#> NMR80_2  7.9222e-01  4.2757e-01  1.8528 0.0639067 .  
#> SMSA_2   1.7235e-05  2.4841e-02  0.0007 0.9994464    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#>  Indirect effects 
#> 
#>                mean          sd  t-stat   p-val  
#> UN83_1  -0.31238169  0.16813634 -1.8579 0.06318 .
#> NMR83_1  0.20488538  0.14047116  1.4586 0.14469  
#> SMSA_1   0.00224931  0.00480408  0.4682 0.63964  
#> UN80_2   0.22953581  0.19511769  1.1764 0.23944  
#> NMR80_2 -0.25803529  0.21139368 -1.2206 0.22222  
#> SMSA_2   0.00093014  0.00929599  0.1001 0.92030  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#>  Total effects 
#> 
#>                mean          sd  t-stat    p-val   
#> UN83_1   0.55240913  0.19422962  2.8441 0.004454 **
#> NMR83_1 -0.36677246  0.20738882 -1.7685 0.076973 . 
#> SMSA_1  -0.00558949  0.00835723 -0.6688 0.503609   
#> UN80_2  -0.45694388  0.28367671 -1.6108 0.107225   
#> NMR80_2  0.53418244  0.30849200  1.7316 0.083346 . 
#> SMSA_2   0.00094737  0.01705941  0.0555 0.955713   
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

5. Step 5: The spSUR in a panel data framework

5.1 The spSUR with G equation and T periods

Case of T temporal cross-sections and G equations

By example with NAT data set:

  • T = 4 (Four temporal periods)
  • G = 2 (Two equations with different numbers of independent variables)
  • R = 3085 (Spatial observations)

A SUR-SLM-PANEL model
(y_{gt} = \lambda_{g} Wy_{gt} + X_{gt} \beta_{g} + \epsilon_{gt};)
(\ g=1,...,G; \ t=1,...,T)
(Corr(\epsilon_{gt},\epsilon_{g't})=Corr(\epsilon_{g},\epsilon_{g'}) \neq 0) for ((\forall t))

Tformula <- HR80 | HC80 ~ UE80 + RD80 | UE80
spSUR.sar.panel <- spsurml(Form = Tformula,data=NCOVR,W=W,type="slm",N=3085,G=2,Tm=4)
#> Initial point:   log_lik:  -26267.63  lambdas:  0.376 0.145 
#> Iteration:  1   log_lik:  -26253.87  lambdas:  0.389 0.146 
#> Iteration:  2   log_lik:  -26253.87  lambdas:  0.389 0.146 
#> Time to fit the model:  6.3  seconds
#> Computing marginal test... 
#> Time to compute covariances:  37.11  seconds
summary(spSUR.sar.panel)
#> Call:
#> spsurml(Form = Tformula, data = NCOVR, W = W, G = 2, N = 3085, 
#>     Tm = 4, type = "slm")
#> 
#>  
#> Spatial SUR model type:  slm 
#> 
#> Equation  1 
#>               Estimate Std. Error t value  Pr(>|t|)    
#> (Intercept)_1 3.664135   0.267777 13.6835 < 2.2e-16 ***
#> UE80_1        0.081705   0.029855  2.7367  0.006224 ** 
#> RD80_1        2.537307   0.115395 21.9881 < 2.2e-16 ***
#> lambda_1      0.389281   0.021784 17.8697 < 2.2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.3779 
#>   Equation  2 
#>               Estimate Std. Error t value  Pr(>|t|)    
#> (Intercept)_2 5.602462   2.232275  2.5098   0.01211 *  
#> UE80_2        0.110066   0.294648  0.3736   0.70875    
#> lambda_2      0.145692   0.028112  5.1825 2.259e-07 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.01993 
#>   Variance-Covariance Matrix of inter-equation residuals:                    
#>  29.07190   60.48062
#>  60.48062 2940.02430
#> Correlation Matrix of inter-equation residuals:                    
#>  1.0000000 0.2068731
#>  0.2068731 1.0000000
#> 
#>  R-sq. pooled: 0.02445 
#>  Log-Likelihood:  -26253.9
#>  Breusch-Pagan:   132  p-value: (1.48e-30) 
#>  LMM: 371.03  p-value: (1.12e-82)

5.1.1 Unobserved effects: The demaining option

spsur package offers the possibility to transform the original data in order to remove potential unobserved effects.
The most popular transformation is demeaning the data: To subtract the sampling averages of each individual, in every equation, from the corresponding observation.

Tformula <- HR80 | HC80 ~ UE80 + RD80 | UE80
spSUR.sar.panel <- spsurml(Form = Tformula,data=NCOVR,W=W,type="slm",N=3085,G=2,Tm=4,demean = T)
#> Initial point:   log_lik:  -26267.63  lambdas:  0.376 0.145 
#> Iteration:  1   log_lik:  -26253.87  lambdas:  0.389 0.146 
#> Iteration:  2   log_lik:  -26253.87  lambdas:  0.389 0.146 
#> Time to fit the model:  6.51  seconds
#> Computing marginal test... 
#> Time to compute covariances:  35.82  seconds
summary(spSUR.sar.panel)
#> Call:
#> spsurml(Form = Tformula, data = NCOVR, W = W, G = 2, N = 3085, 
#>     Tm = 4, demean = T, type = "slm")
#> 
#>  
#> Spatial SUR model type:  slm 
#> 
#> Equation  1 
#>               Estimate Std. Error t value  Pr(>|t|)    
#> (Intercept)_1 3.664135   0.267777 13.6835 < 2.2e-16 ***
#> UE80_1        0.081705   0.029855  2.7367  0.006224 ** 
#> RD80_1        2.537307   0.115395 21.9881 < 2.2e-16 ***
#> lambda_1      0.389281   0.021784 17.8697 < 2.2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.3779 
#>   Equation  2 
#>               Estimate Std. Error t value  Pr(>|t|)    
#> (Intercept)_2 5.602462   2.232275  2.5098   0.01211 *  
#> UE80_2        0.110066   0.294648  0.3736   0.70875    
#> lambda_2      0.145692   0.028112  5.1825 2.259e-07 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.01993 
#>   Variance-Covariance Matrix of inter-equation residuals:                    
#>  29.07190   60.48062
#>  60.48062 2940.02430
#> Correlation Matrix of inter-equation residuals:                    
#>  1.0000000 0.2068731
#>  0.2068731 1.0000000
#> 
#>  R-sq. pooled: 0.02445 
#>  Log-Likelihood:  -26253.9
#>  Breusch-Pagan:   132  p-value: (1.48e-30) 
#>  LMM: 371.03  p-value: (1.12e-82)

6. Aditional functionalities: dgp_spSUR

A Data Generating Process of a spatial SUR models is avalible using the function dgp_spSUR

nT <- 1 # Number of time periods
nG <- 3 # Number of equations
nR <- 500 # Number of spatial elements
p <- 3 # Number of independent variables
Sigma <- matrix(0.3,ncol=nG,nrow=nG)
diag(Sigma)<-1
Coeff <- c(2,3)
rho <- 0.5 # level of spatial dependence
lambda <- 0.0 # spatial autocorrelation error term = 0
# ramdom coordinates
# co <- cbind(runif(nR,0,1),runif(nR,0,1))
# W <- spdep::nb2mat(spdep::knn2nb(spdep::knearneigh(co,k=5,longlat=F)))
# DGP <- dgp_spSUR(Sigma=Sigma,Coeff=Coeff,rho=rho,lambda=lambda,nT=nT,nG=nG,nR=nR,p=p,W=W)

7. Conclusion & work to do

  • spSUR is a powerful R-package to test, estimate and looking for the correct specification
  • More functionalities and estimation algorithm coming soon
    • GMM estimation
    • ML estimation with equal level of spatial dependence ((\lambda)/(\rho)=constant)
    • Orthogonal deamining for space-time SUR models
    • …..
  • The spSUR is avaliable in GitHub [https://github.com/rominsal/spSUR/]

References

  • López, F.A., P. J. Martínez-Ortiz, and J.G. Cegarra-Navarro (2017). Spatial spillovers in public expenditure on a municipal level in spain. The Annals of Regional Science 58 (1), 39–65.
  • López, F.A., J. Mur, and A. Angulo (2014). Spatial model selection strategies in a sur framework. the case of regional productivity in eu. The Annals of Regional Science 53 (1), 197–220.
  • Mur, J., F. López, and M. Herrera (2010). Testing for spatial effects in seemingly unrelated regressions. Spatial Economic Analysis 5 (4), 399–440.

Example

This is a basic example which shows you how test spatial structure in the residual of a SUR model:

## Testing for spatial effects in SUR model
library("spsur")
data("spc")
Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA
LMs <- lmtestspsur(Form = Tformula, data = spc, W = Wspc)
#>              LM-Stat. DF p-value  
#> LM-SUR-SLM     5.2472  2  0.0725 .
#> LM-SUR-SEM     3.3050  2  0.1916  
#> LM*-SUR-SLM    2.1050  2  0.3491  
#> LM*-SUR-SEM    0.1628  2  0.9218  
#> LM-SUR-SARAR   5.7703  4  0.2170  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Several Spatial SUR model can be estimated by maximun likelihhod:

## A SUR-SLM model
spcsur.slm <-spsurml(Form = Tformula, data = spc, type = "slm", W = Wspc)
#> Initial point:   log_lik:  113.197  lambdas:  -0.472 -0.446 
#> Iteration:  1   log_lik:  114.085  lambdas:  -0.506 -0.482 
#> Iteration:  2   log_lik:  114.096  lambdas:  -0.506 -0.482 
#> Time to fit the model:  3.04  seconds
#> Computing marginal test... 
#> Time to compute covariances:  0.21  seconds
summary(spcsur.slm)
#> Call:
#> spsurml(Form = Tformula, data = spc, W = Wspc, type = "slm")
#> 
#>  
#> Spatial SUR model type:  slm 
#> 
#> Equation  1 
#>                 Estimate Std. Error t value  Pr(>|t|)    
#> (Intercept)_1  1.4955217  0.2467240  6.0615 5.183e-07 ***
#> UN83_1         0.8070029  0.2557439  3.1555  0.003179 ** 
#> NMR83_1       -0.5194114  0.2590550 -2.0050  0.052318 .  
#> SMSA_1        -0.0073247  0.0118519 -0.6180  0.540347    
#> lambda_1      -0.5057334  0.2405734 -2.1022  0.042401 *  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.6224 
#>   Equation  2 
#>                 Estimate Std. Error t value  Pr(>|t|)    
#> (Intercept)_2  1.7094414  0.2925620  5.8430 1.024e-06 ***
#> UN80_2        -0.6745562  0.3870737 -1.7427   0.08969 .  
#> NMR80_2        0.7502934  0.3842670  1.9525   0.05847 .  
#> SMSA_2         0.0014181  0.0241859  0.0586   0.95356    
#> lambda_2      -0.4821428  0.2557758 -1.8850   0.06730 .  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> R-squared: 0.4743 
#>   Variance-Covariance Matrix of inter-equation residuals:                            
#>   0.0003085954 -0.0003561928
#>  -0.0003561928  0.0015864976
#> Correlation Matrix of inter-equation residuals:                    
#>   1.000000 -0.509062
#>  -0.509062  1.000000
#> 
#>  R-sq. pooled: 0.6603 
#>  Log-Likelihood:  114.096
#>  Breusch-Pagan: 6.516  p-value: (0.0107) 
#>  LMM: 0.50489  p-value: (0.477)