In rominsal/spsur: Spatial Seemingly Unrelated Regression Models
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>",
fig.path = "man/figures/README-",
out.width = "100%"
)
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 (impactspsur()). Another function has been included in this package to help to the user to develop Monte Carlo exercices (dgp_spsur()). Finally, the package include the usual methods for objects of class spsur like anova, coef, fitted, logLik, print, residuals, summary and vcov.
```r Main functionatities of spsur package"}
setwd("C:/Users/roman/Dropbox/spsurdev")
knitr::include_graphics('/notes/Journal of Statisitical Software/Figure1_JSS.png')
## 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
```r
data("spc")
knitr::kable(head(spc,3))
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
data("NCOVR")
knitr::kable(head(NCOVR[,1:9],3))
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(formula = Tformula, data = spc, listw = Wspc)
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(formula = Tformula, data=spc,
type="slm", listw = Wspc)
summary(spcSUR.slm)
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(formula = Tformula, data = spc,
type = "sem", listw=Wspc)
summary(spcSUR.sem)
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)
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)
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)
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)
Wald_beta <- wald_betas(results=spcSUR.slm,R=R1,b=b1)
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)
summary(spcSUR.sar.restring)
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)
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)
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)
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)
summary(spSUR.sar.panel)
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)
summary(spSUR.sar.panel)
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)
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)
summary(spcsur.slm)
rominsal/spsur documentation built on April 27, 2022, 2:31 a.m.
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knitr::opts_chunk$set( collapse = TRUE, comment = "#>", fig.path = "man/figures/README-", out.width = "100%" )
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 (impactspsur()). Another function has been included in this package to help to the user to develop Monte Carlo exercices (dgp_spsur()). Finally, the package include the usual methods for objects of class spsur like anova, coef, fitted, logLik, print, residuals, summary and vcov.
```r Main functionatities of spsur package"} setwd("C:/Users/roman/Dropbox/spsurdev") knitr::include_graphics('/notes/Journal of Statisitical Software/Figure1_JSS.png')
## 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
```r
data("spc")
knitr::kable(head(spc,3))
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
data("NCOVR") knitr::kable(head(NCOVR[,1:9],3))
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(formula = Tformula, data = spc, listw = Wspc)
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(formula = Tformula, data=spc, type="slm", listw = Wspc) summary(spcSUR.slm)
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(formula = Tformula, data = spc, type = "sem", listw=Wspc) summary(spcSUR.sem)
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)
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) 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)
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) Wald_beta <- wald_betas(results=spcSUR.slm,R=R1,b=b1)
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) summary(spcSUR.sar.restring)
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)
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)
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)
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) summary(spSUR.sar.panel)
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) summary(spSUR.sar.panel)
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
spSURis 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)
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) summary(spcsur.slm)
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