spsurml: Maximum likelihood estimation of spatial SUR model.
In rominsal/spsur: Spatial Seemingly Unrelated Regression Models
spsurml R Documentation
Maximum likelihood estimation of spatial SUR model.
Description
This function estimates spatial SUR models using
maximum-likelihood methods.The number of equations, time periods
and cross-sectional units is not restricted.The user can choose
between different spatial specifications as described below.
The estimation procedure allows for the introduction of linear
restrictions on the β parameters associated to the
regressors.
Usage
spsurml(formula = NULL, data = NULL, na.action,
listw = NULL, type = "sim", Durbin = NULL,
method = "eigen", zero.policy = NULL, interval = NULL,
trs = NULL, R = NULL, b = NULL, X = NULL, Y = NULL,
G = NULL, N = NULL, Tm = NULL,p = NULL,
control = list() )
Arguments
formula
An object type Formula
similar to objects created with the package Formula
describing the equations to be estimated in the model.
This model may contain several responses (explained
variables) and a varying number of regressors in each equation.
data
An object of class data.frame or a matrix.
na.action
A function (default options("na.action")),
can also be na.omit or na.exclude with consequences
for residuals and fitted values. It may be necessary to set
zero.policy to TRUE because this subsetting may
create no-neighbour observations.
listw
A listw object created for example by
nb2listw from spatialreg package; if
nb2listw not given, set to
the same spatial weights as the listw argument. It can
also be a spatial weighting matrix of order (NxN) instead of
a listw object. Default = NULL.
type
Type of spatial model specification: "sim",
"slx", "slm", "sem", "sdm",
"sdem", "sarar" or "gnm". Default = "sim".
Durbin
If a formula object and model is type "sdm", "sdem"
or "slx" the subset of explanatory variables to lag for each equation.
method
Similar to the corresponding parameter of
lagsarlm function in spatialreg package.
"eigen" (default) - the Jacobian is computed as the product of
(1 - rho*eigenvalue) using eigenw, and
"spam" or "Matrix_J" for strictly symmetric weights lists of
styles "B" and "C", or made symmetric by similarity
(Ord, 1975, Appendix C) if possible for styles "W" and "S",
using code from the spam or Matrix packages to calculate the
determinant; "Matrix" and "spam_update" provide updating Cholesky
decomposition methods; "LU" provides an alternative sparse matrix
decomposition approach. In addition, there are "Chebyshev" and
Monte Carlo "MC" approximate log-determinant methods;
the Smirnov/Anselin (2009) trace approximation is available
as "moments". Three methods: "SE_classic", "SE_whichMin",
and "SE_interp" are provided experimentally, the first to
attempt to emulate the behaviour of Spatial Econometrics
toolbox ML fitting functions. All use grids of log determinant
values, and the latter two attempt to ameliorate some features
of "SE_classic".
zero.policy
Similar to the corresponding parameter of
lagsarlm function in spatialreg package.
If TRUE assign zero to the lagged value of zones without
neighbours, if FALSE assign NA - causing
spsurml() to terminate with an error. Default = NULL.
interval
Search interval for autoregressive parameter.
Default = NULL.
trs
Similar to the corresponding parameter of
lagsarlm function in spatialreg package.
Default NULL, if given, a vector of powered spatial weights
matrix traces output by trW.
R
A row vector of order (1xpr) with the set of
r linear constraints on the beta parameters. The
first restriction appears in the first p terms,
the second restriction in the next p terms and so on.
Default = NULL.
b
A column vector of order (rx1) with the values of
the linear restrictions on the beta parameters.
Default = NULL.
X
A data matrix of order (NTmGxp) with the observations
of the regressors. The number of covariates in the SUR model is
p = sum(p_{g}) where p_{g} is the number
of regressors (including the intercept) in the g-th equation,
g = 1,...,G). The specification of "X" is only
necessary if not available a Formula and a
data frame. Default = NULL.
Y
A column vector of order (NTmGx1), with the
observations of the explained variables. The ordering of the data
must be (first) equation, (second) time dimension and (third)
cross-sectional/spatial units. The specification of Y is
only necessary if not available a Formula
and a data frame. Default = NULL.
G
Number of equations.
N
Number of cross-section or spatial units
Tm
Number of time periods.
p
Number of regressors by equation, including the intercept.
p can be a row vector of order (1xG), if the number
of regressors is not the same for all the equations, or a scalar,
if the G equations have the same number of regressors. The
specification of p is only necessary if not available a
Formula and a data frame.
control
List of additional control arguments.
Details
The list of (spatial) models that can be estimated with the spsurml function are:
"sim": SUR model with no spatial effects
y_{tg} = X_{tg} β_{g} + ε_{tg}
"slx": SUR model with spatial lags of the regressors
y_{tg} = X_{tg} β_{g} + WX_{tg} θ_{g} + ε_{tg}
"slm": SUR model with spatial lags of the explained variables
y_{tg} = ρ_{g} Wy_{tg} + X_{tg} β_{g} + ε_{tg}
"sem": SUR model with spatial errors
y_{tg} = X_{tg} β_{g} + u_{tg}
u_{tg} = λ_{g} Wu_{tg} + ε_{tg}
"sdm": SUR model of the Spatial Durbin type
y_{tg} = ρ_{g} Wy_{tg} + X_{tt} β_{g} + WX_{tg} θ_{g} + ε_{tg}
"sdem": SUR model with spatial lags of the regressors and spatial errors
y_{tg} = X_{tg} β_{g} + WX_{tg} θ_{g} + u_{tg}
u_{tg} = λ_{g} W u_{tg} + ε_{tg}
"sarar": SUR model with spatial lags of the explained variables and spatial
errors
y_{tg} = ρ_{g} Wy_{tg} + X_{tg} β_{g} + u_{tg}
u_{tg} = λ_{g} W u_{tg} + ε_{tg}
"gnm": SUR model with spatial lags of the explained variables,
regressors and spatial errors
y_{tg} = ρ_{g} Wy_{tg} + X_{tg} β_{g} +
WX_{tg} θ_{g} + u_{tg}
u_{tg} = λ_{g} W u_{tg} + ε_{tg}
Value
Object of spsur class with the output of the
maximum-likelihood estimation of the specified spatial SUR model.
A list with:
call Matched call.
type Type of model specified.
method Value of method argument to compute the
Jacobian
Durbin Value of Durbin argument.
coefficients Estimated coefficients for the regressors.
deltas Estimated spatial coefficients.
rest.se Estimated standard errors for the
estimates of beta.
deltas.se Estimated standard errors for the estimates of
the spatial coefficients (deltas).
resvar Estimated covariance matrix for the estimates of
beta's and spatial coefficients (deltas).
LL Value of the likelihood function at the
maximum-likelihood estimates.
R2 Coefficient of determination for each equation,
obtained as the squared of the correlation coefficient between the
corresponding explained variable and its estimate.
spsurml also shows a global coefficient of
determination obtained, in the same manner, for the set of
the G equations.
Sigma Estimated covariance matrix for the residuals of
the G equations.
fdHess Logical value of fdHess argument when
computing numerical covariances.
residuals Residuals of the model.
df.residuals Degrees of freedom for the residuals.
fitted.values Estimated values for the dependent
variables.
BP Value of the Breusch-Pagan statistic to test the
null hypothesis of diagonality among the errors of the G
equations.
LMM Marginal Lagrange Multipliers,
LM(ρ|λ) and
LM(λ|ρ), to test for omitted spatial effects
in the specification.
G Number of equations.
N Number of cross-sections or spatial units.
Tm Number of time periods.
p Number of regressors by equation (including intercepts).
Y If data is NULL, vector Y of the explained variables of the
SUR model.
X If data is NULL, matrix X of the regressors of the SUR model.
W Spatial weighting matrix.
zero.policy Logical value of zero.policy .
interval Search interval for spatial parameter.
listw_style Style of neighborhood matrix W.
trs Either NULL or vector of powered spatial weights
matrix traces output by trW.
insert Logical value to check if is.null(trs).
Control arguments
tol Numerical value for the tolerance for the estimation
algorithm until convergence. Default = 1e-3.
maxit Maximum number of iterations until convergence;
it must be an integer value. Default = 200.
trace A logical value to show intermediate results during
the estimation process. Default = TRUE.
fdHess Compute variance-covariance matrix using the numerical
hessian. Suited for large samples. Default = FALSE
Imult default 2; used for preparing the Cholesky
decompositions for updating in the Jacobian function
super if NULL (default), set to FALSE to use
a simplicial decomposition for the sparse Cholesky decomposition and
method "Matrix_J", set to as.logical(NA) for method "Matrix", if
TRUE, use a supernodal decomposition
cheb_q default 5; highest power of the approximating
polynomial for the Chebyshev approximation
MC_p default 16; number of random variates
MC_m default 30; number of products of random variates
matrix and spatial weights matrix
spamPivot default "MMD", alternative "RCM"
in_coef default 0.1, coefficient value for initial Cholesky
decomposition in "spam_update"
type default "MC", used with method "moments"; alternatives
"mult" and "moments", for use if trs is missing
correct default TRUE, used with method "moments" to
compute the Smirnov/Anselin correction term
trunc default TRUE, used with method "moments" to
truncate the Smirnov/Anselin correction term
SE_method default "LU", may be "MC"
nrho default 200, as in SE toolbox; the size of the first
stage lndet grid; it may be reduced to for example 40
interpn default 2000, as in SE toolbox; the size of the
second stage lndet grid
SElndet default NULL, may be used to pass a
pre-computed SE toolbox style matrix of coefficients and their lndet
values to the "SE_classic" and "SE_whichMin" methods
LU_order default FALSE; used in "LU_prepermutate",
note warnings given for lu method
pre_eig default NULL; may be used to pass a
pre-computed vector of eigenvalues
Author(s)
Fernando Lopez fernando.lopez@upct.es
Roman Minguez roman.minguez@uclm.es
Jesus Mur jmur@unizar.es
References
Anselin, L. (1988). Spatial econometrics: methods and models.
Dordrecht: Kluwer
Bivand, R.S. and Piras G. (2015). Comparing Implementations of
Estimation Methods for Spatial Econometrics. Journal of
Statistical Software, 63(18), 1-36.
<doi: 10.18637/jss.v063.i18>
Bivand, R. S., Hauke, J., and Kossowski, T. (2013).
Computing the Jacobian in Gaussian spatial autoregressive models: An
illustrated comparison of available methods. Geographical
Analysis, 45(2), 150-179. <doi:10.1111/gean.12008>
Breusch T., Pagan A. (1980). The Lagrange multiplier test and its
applications to model specification in econometrics.
Rev Econ Stud 47: 239-254
Cliff, A.D. and Ord, J.K. (1981). Spatial processes: Models
and applications, Pion.
LeSage J and Pace, R.K. (2009). Introduction to Spatial
Econometrics. CRC Press, Boca Raton.
Lopez, F.A., Mur, J., and Angulo, A. (2014). Spatial model
selection strategies in a SUR framework. The case of regional
productivity in EU. Annals of Regional Science, 53(1), 197-220.
<doi:10.1007/s00168-014-0624-2>
Mur, J., Lopez, F., and Herrera, M. (2010). Testing for spatial
effects in seemingly unrelated regressions.
Spatial Economic Analysis, 5(4), 399-440.
<doi:10.1080/17421772.2010.516443>
Ord, J.K. (1975). Estimation methods for models of spatial
interaction, Journal of the American Statistical Association,
70, 120-126.
See Also
spsur3sls, lagsarlm,
lmtestspsur, wald_betas,
lrtest
Examples
#################################################
######## CROSS SECTION DATA (G>1; Tm=1) ########
#################################################
#### Example 1: Spatial Phillips-Curve. Anselin (1988, p. 203)
rm(list = ls()) # Clean memory
data(spc)
Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA
spcsur.sim <- spsurml(formula = Tformula, data = spc, type = "sim")
summary(spcsur.sim)
# All the coefficients in a single table.
print(spcsur.sim)
# Plot of the coefficients of each equation in different graphs
plot(spcsur.sim)
## A SUR-SLX model
## (listw argument can be either a matrix or a listw object )
spcsur.slx <- spsurml(formula = Tformula, data = spc, type = "slx",
listw = Wspc, Durbin = TRUE)
summary(spcsur.slx)
# All the coefficients in a single table.
print(spcsur.slx)
# Plot of the coefficients in a single graph
if (require(gridExtra)) {
pl <- plot(spcsur.slx, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
nrow = 2)
}
## VIP: The output of the whole set of the examples can be examined
## by executing demo(demo_spsurml, package="spsur")
### A SUR-SLM model
spcsur.slm <- spsurml(formula = Tformula, data = spc, type = "slm",
listw = Wspc)
summary(spcsur.slm)
if (require(gridExtra)) {
pl <- plot(spcsur.slm, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
#' ### A SUR-SEM model
spcsur.sem <- spsurml(formula = Tformula, data = spc, type = "sem",
listw = Wspc)
summary(spcsur.sem)
print(spcsur.sem)
if (require(gridExtra)) {
pl <- plot(spcsur.sem, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
### A SUR-SDM model
spcsur.sdm <- spsurml(formula = Tformula, data = spc, type = "sdm",
listw = Wspc)
summary(spcsur.sdm)
print(spcsur.sdm)
if (require(gridExtra)) {
pl <- plot(spcsur.sdm, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
## A SUR-SDM model with different spatial lags in each equation
TformulaD <- ~ UN83 + NMR83 + SMSA | UN80
spcsur.sdm2 <- spsurml(formula = Tformula, data = spc, type = "sdm",
listw = Wspc, Durbin = TformulaD)
summary(spcsur.sdm2)
if (require(gridExtra)) {
pl <- plot(spcsur.sdm2, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
### A SUR-SDEM model
spcsur.sdem <- spsurml(formula = Tformula, data = spc, type = "sdem",
listw = Wspc)
print(spcsur.sdem)
if (require(gridExtra)) {
pl <- plot(spcsur.sdem, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
### A SUR-SARAR model
spcsur.sarar <- spsurml(formula = Tformula, data = spc, type = "sarar",
listw = Wspc, control = list(tol = 0.1))
print(spcsur.sarar)
if (require(gridExtra)) {
pl <- plot(spcsur.sarar, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
### A SUR-GNM model
spcsur.gnm <- spsurml(formula = Tformula, data = spc, type = "gnm",
listw = Wspc, control = list(tol = 0.1))
print(spcsur.gnm)
if (require(gridExtra)) {
pl <- plot(spcsur.gnm, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
## A A SUR-GNM model model with different spatial lags in each equation
TformulaD <- ~ UN83 + NMR83 + SMSA | UN80
spcsur.gnm2 <-spsurml(formula = Tformula, data = spc, type = "gnm",
listw = Wspc, Durbin = TformulaD,
control = list(tol = 0.1))
print(spcsur.gnm2)
if (require(gridExtra)) {
pl <- plot(spcsur.gnm2, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
##################################################
######### CLASSIC PANEL DATA G=1; Tm>1 ########
##################################################
#
##### Example 2: Homicides + Socio-Economics (1960-90)
## Homicides and selected socio-economic characteristics for continental
## U.S. counties.
## Data for four decennial census years: 1960, 1970, 1980 and 1990.
## \url{https://geodacenter.github.io/data-and-lab/ncovr/}
### It usually requires 1-2 minutes maximum...
rm(list = ls()) # Clean memory
### Read NCOVR.sf object
data(NCOVR, package = "spsur")
nbncovr <- spdep::poly2nb(NCOVR.sf, queen = TRUE)
### Some regions with no links...
lwncovr <- spdep::nb2listw(nbncovr, style = "W", zero.policy = TRUE)
Tformula <- HR80 | HR90 ~ PS80 + UE80 | PS90 + UE90
### A SUR-SIM model
NCOVRSUR.sim <- spsurml(formula = Tformula, data = NCOVR.sf, type = "sim")
summary(NCOVRSUR.sim)
if (require(gridExtra)) {
pl <- plot(NCOVRSUR.sim, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]], nrow = 3)
}
### A SUR-SLX model
NCOVRSUR.slx <- spsurml(formula = Tformula, data = NCOVR.sf, type = "slx",
listw = lwncovr, zero.policy = TRUE)
print(NCOVRSUR.slx)
if (require(gridExtra)) {
pl <- plot(NCOVRSUR.slx, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]], nrow = 2)
}
### A SUR-SLM model
### method = "Matrix" (Cholesky) instead of "eigen"
### (fdHess = TRUE to compute numerical covariances )
NCOVRSUR.slm <- spsurml(formula = Tformula, data = NCOVR.sf,
type = "slm", listw = lwncovr, method = "Matrix",
zero.policy = TRUE, control = list(fdHess = TRUE))
summary(NCOVRSUR.slm)
if (require(gridExtra)) {
pl <- plot(NCOVRSUR.slm, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
# LR test for nested models
anova(NCOVRSUR.sim, NCOVRSUR.slm)
### A SUR-SDM model with different spatial lags in each equation
### Analytical covariances (default)
TformulaD <- ~ PS80 + UE80 | PS90
NCOVRSUR.sdm <- spsurml(formula = Tformula, data = NCOVR.sf,
type = "sdm", listw = lwncovr, method = "Matrix",
Durbin = TformulaD, zero.policy = TRUE)
print(NCOVRSUR.sdm)
if (require(gridExtra)) {
pl <- plot(NCOVRSUR.sdm, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
### A SUR-SEM model
NCOVRSUR.sem <- spsurml(formula = Tformula, data = NCOVR.sf,
type = "sem", listw = lwncovr, method = "Matrix",
zero.policy = TRUE, control = list(fdHess = TRUE))
print(NCOVRSUR.sem)
if (require(gridExtra)) {
pl <- plot(NCOVRSUR.sem, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
### A SUR-SDEM model
NCOVRSUR.sdem <-spsurml(formula = Tformula, data = NCOVR.sf,
type = "sdem", listw = lwncovr, method = "Matrix",
zero.policy = TRUE, control = list(fdHess = TRUE))
print(NCOVRSUR.sdem)
if (require(gridExtra)) {
pl <- plot(NCOVRSUR.sdem, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
###############################################
## MULTI-DIMENSIONAL SUR PANEL G>1; Tm>1 ###
###############################################
##### Reshape NCOVR in panel format
N <- nrow(NCOVR.sf)
Tm <- 4
index_time <- rep(1:Tm, each = N)
index_indiv <- rep(1:N, Tm)
pHR <- c(NCOVR.sf$HR60, NCOVR.sf$HR70, NCOVR.sf$HR80, NCOVR.sf$HR90)
pPS <- c(NCOVR.sf$PS60, NCOVR.sf$PS70, NCOVR.sf$PS80, NCOVR.sf$PS90)
pUE <- c(NCOVR.sf$UE60, NCOVR.sf$UE70, NCOVR.sf$UE80, NCOVR.sf$UE90)
pDV <- c(NCOVR.sf$DV60, NCOVR.sf$DV70, NCOVR.sf$DV80, NCOVR.sf$DV90)
pFP <- c(NCOVR.sf$FP59, NCOVR.sf$FP70, NCOVR.sf$FP80, NCOVR.sf$FP90)
pSOUTH <- rep(NCOVR.sf$SOUTH, Tm)
pNCOVR <- data.frame(indiv = index_indiv, time = index_time,
HR = pHR, PS = pPS, UE = pUE, DV = pDV,
FP = pFP, SOUTH = pSOUTH)
pform <- HR | DV | FP ~ PS + UE | PS + UE + SOUTH | PS
### SIM
### Remark: It is necessary to provide Tm value as argument
### when G>1 && Tm>1
pNCOVRSUR.sim <- spsurml(formula = pform, data = pNCOVR,
type = "sim", Tm = Tm)
print(pNCOVRSUR.sim)
if (require(gridExtra)) {
pl <- plot(pNCOVRSUR.sim, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$lplbetas[[3]], nrow = 3)
}
# SLM
pNCOVRSUR.slm <- spsurml(formula = pform, data = pNCOVR,
listw = lwncovr, type = "slm", method = "Matrix", Tm = Tm,
zero.policy = TRUE, control= list(fdHess = TRUE))
print(pNCOVRSUR.slm)
if (require(gridExtra)) {
pl <- plot(pNCOVRSUR.slm, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$lplbetas[[3]], pl$pldeltas, nrow = 4)
}
pNCOVRSUR.sem <- spsurml(formula = pform, data = pNCOVR,
listw = lwncovr, type = "sem", method = "Matrix", Tm = Tm,
zero.policy = TRUE, control= list(fdHess = TRUE))
print(pNCOVRSUR.sem)
if (require(gridExtra)) {
pl <- plot(pNCOVRSUR.sem, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$lplbetas[[3]], pl$pldeltas, nrow = 4)
}
rominsal/spsur documentation built on April 27, 2022, 2:31 a.m.
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| spsurml | R Documentation |
Maximum likelihood estimation of spatial SUR model.
Description
This function estimates spatial SUR models using maximum-likelihood methods.The number of equations, time periods and cross-sectional units is not restricted.The user can choose between different spatial specifications as described below. The estimation procedure allows for the introduction of linear restrictions on the β parameters associated to the regressors.
Usage
spsurml(formula = NULL, data = NULL, na.action,
listw = NULL, type = "sim", Durbin = NULL,
method = "eigen", zero.policy = NULL, interval = NULL,
trs = NULL, R = NULL, b = NULL, X = NULL, Y = NULL,
G = NULL, N = NULL, Tm = NULL,p = NULL,
control = list() )
Arguments
formula |
An object type |
data |
An object of class data.frame or a matrix. |
na.action |
A function (default |
listw |
A |
type |
Type of spatial model specification: "sim", "slx", "slm", "sem", "sdm", "sdem", "sarar" or "gnm". Default = "sim". |
Durbin |
If a formula object and model is type "sdm", "sdem" or "slx" the subset of explanatory variables to lag for each equation. |
method |
Similar to the corresponding parameter of
|
zero.policy |
Similar to the corresponding parameter of
|
interval |
Search interval for autoregressive parameter.
Default = |
trs |
Similar to the corresponding parameter of
|
R |
A row vector of order (1xpr) with the set of
r linear constraints on the beta parameters. The
first restriction appears in the first p terms,
the second restriction in the next p terms and so on.
Default = |
b |
A column vector of order (rx1) with the values of
the linear restrictions on the beta parameters.
Default = |
X |
A data matrix of order (NTmGxp) with the observations
of the regressors. The number of covariates in the SUR model is
p = sum(p_{g}) where p_{g} is the number
of regressors (including the intercept) in the g-th equation,
g = 1,...,G). The specification of "X" is only
necessary if not available a |
Y |
A column vector of order (NTmGx1), with the
observations of the explained variables. The ordering of the data
must be (first) equation, (second) time dimension and (third)
cross-sectional/spatial units. The specification of Y is
only necessary if not available a |
G |
Number of equations. |
N |
Number of cross-section or spatial units |
Tm |
Number of time periods. |
p |
Number of regressors by equation, including the intercept.
p can be a row vector of order (1xG), if the number
of regressors is not the same for all the equations, or a scalar,
if the G equations have the same number of regressors. The
specification of p is only necessary if not available a
|
control |
List of additional control arguments. |
Details
The list of (spatial) models that can be estimated with the spsurml function are:
"sim": SUR model with no spatial effects
y_{tg} = X_{tg} β_{g} + ε_{tg}
"slx": SUR model with spatial lags of the regressors
y_{tg} = X_{tg} β_{g} + WX_{tg} θ_{g} + ε_{tg}
"slm": SUR model with spatial lags of the explained variables
y_{tg} = ρ_{g} Wy_{tg} + X_{tg} β_{g} + ε_{tg}
"sem": SUR model with spatial errors
y_{tg} = X_{tg} β_{g} + u_{tg}
u_{tg} = λ_{g} Wu_{tg} + ε_{tg}
"sdm": SUR model of the Spatial Durbin type
y_{tg} = ρ_{g} Wy_{tg} + X_{tt} β_{g} + WX_{tg} θ_{g} + ε_{tg}
"sdem": SUR model with spatial lags of the regressors and spatial errors
y_{tg} = X_{tg} β_{g} + WX_{tg} θ_{g} + u_{tg}
u_{tg} = λ_{g} W u_{tg} + ε_{tg}
"sarar": SUR model with spatial lags of the explained variables and spatial errors
y_{tg} = ρ_{g} Wy_{tg} + X_{tg} β_{g} + u_{tg}
u_{tg} = λ_{g} W u_{tg} + ε_{tg}
"gnm": SUR model with spatial lags of the explained variables, regressors and spatial errors
y_{tg} = ρ_{g} Wy_{tg} + X_{tg} β_{g} + WX_{tg} θ_{g} + u_{tg}
u_{tg} = λ_{g} W u_{tg} + ε_{tg}
Value
Object of spsur class with the output of the
maximum-likelihood estimation of the specified spatial SUR model.
A list with:
call | Matched call. |
type | Type of model specified. |
method | Value of method argument to compute the
Jacobian |
Durbin | Value of Durbin argument. |
coefficients | Estimated coefficients for the regressors. |
deltas | Estimated spatial coefficients. |
rest.se | Estimated standard errors for the estimates of beta. |
deltas.se | Estimated standard errors for the estimates of
the spatial coefficients (deltas). |
resvar | Estimated covariance matrix for the estimates of
beta's and spatial coefficients (deltas). |
LL | Value of the likelihood function at the maximum-likelihood estimates. |
R2 | Coefficient of determination for each equation,
obtained as the squared of the correlation coefficient between the
corresponding explained variable and its estimate.
spsurml also shows a global coefficient of
determination obtained, in the same manner, for the set of
the G equations. |
Sigma | Estimated covariance matrix for the residuals of the G equations. |
fdHess | Logical value of fdHess argument when
computing numerical covariances. |
residuals | Residuals of the model. |
df.residuals | Degrees of freedom for the residuals. |
fitted.values | Estimated values for the dependent variables. |
BP | Value of the Breusch-Pagan statistic to test the null hypothesis of diagonality among the errors of the G equations. |
LMM | Marginal Lagrange Multipliers, LM(ρ|λ) and LM(λ|ρ), to test for omitted spatial effects in the specification. |
G | Number of equations. |
N | Number of cross-sections or spatial units. |
Tm | Number of time periods. |
p | Number of regressors by equation (including intercepts). |
Y | If data is NULL, vector Y of the explained variables of the SUR model. |
X | If data is NULL, matrix X of the regressors of the SUR model. |
W | Spatial weighting matrix. |
zero.policy | Logical value of zero.policy . |
interval | Search interval for spatial parameter. |
listw_style | Style of neighborhood matrix W. |
trs | Either NULL or vector of powered spatial weights
matrix traces output by trW. |
insert | Logical value to check if is.null(trs). |
Control arguments
tol | Numerical value for the tolerance for the estimation algorithm until convergence. Default = 1e-3. |
maxit | Maximum number of iterations until convergence; it must be an integer value. Default = 200. |
trace | A logical value to show intermediate results during
the estimation process. Default = TRUE. |
fdHess | Compute variance-covariance matrix using the numerical
hessian. Suited for large samples. Default = FALSE |
Imult | default 2; used for preparing the Cholesky decompositions for updating in the Jacobian function |
super | if NULL (default), set to FALSE to use
a simplicial decomposition for the sparse Cholesky decomposition and
method "Matrix_J", set to as.logical(NA) for method "Matrix", if
TRUE, use a supernodal decomposition |
cheb_q | default 5; highest power of the approximating polynomial for the Chebyshev approximation |
MC_p | default 16; number of random variates |
MC_m | default 30; number of products of random variates matrix and spatial weights matrix |
spamPivot | default "MMD", alternative "RCM" |
in_coef | default 0.1, coefficient value for initial Cholesky decomposition in "spam_update" |
type | default "MC", used with method "moments"; alternatives "mult" and "moments", for use if trs is missing |
correct | default TRUE, used with method "moments" to
compute the Smirnov/Anselin correction term |
trunc | default TRUE, used with method "moments" to
truncate the Smirnov/Anselin correction term |
SE_method | default "LU", may be "MC" |
nrho | default 200, as in SE toolbox; the size of the first stage lndet grid; it may be reduced to for example 40 |
interpn | default 2000, as in SE toolbox; the size of the second stage lndet grid |
SElndet | default NULL, may be used to pass a
pre-computed SE toolbox style matrix of coefficients and their lndet
values to the "SE_classic" and "SE_whichMin" methods |
LU_order | default FALSE; used in "LU_prepermutate",
note warnings given for lu method |
pre_eig | default NULL; may be used to pass a
pre-computed vector of eigenvalues |
Author(s)
| Fernando Lopez | fernando.lopez@upct.es |
| Roman Minguez | roman.minguez@uclm.es |
| Jesus Mur | jmur@unizar.es |
References
Anselin, L. (1988). Spatial econometrics: methods and models. Dordrecht: Kluwer
Bivand, R.S. and Piras G. (2015). Comparing Implementations of Estimation Methods for Spatial Econometrics. Journal of Statistical Software, 63(18), 1-36. <doi: 10.18637/jss.v063.i18>
Bivand, R. S., Hauke, J., and Kossowski, T. (2013). Computing the Jacobian in Gaussian spatial autoregressive models: An illustrated comparison of available methods. Geographical Analysis, 45(2), 150-179. <doi:10.1111/gean.12008>
Breusch T., Pagan A. (1980). The Lagrange multiplier test and its applications to model specification in econometrics. Rev Econ Stud 47: 239-254
Cliff, A.D. and Ord, J.K. (1981). Spatial processes: Models and applications, Pion.
LeSage J and Pace, R.K. (2009). Introduction to Spatial Econometrics. CRC Press, Boca Raton.
Lopez, F.A., Mur, J., and Angulo, A. (2014). Spatial model selection strategies in a SUR framework. The case of regional productivity in EU. Annals of Regional Science, 53(1), 197-220. <doi:10.1007/s00168-014-0624-2>
Mur, J., Lopez, F., and Herrera, M. (2010). Testing for spatial effects in seemingly unrelated regressions. Spatial Economic Analysis, 5(4), 399-440. <doi:10.1080/17421772.2010.516443>
Ord, J.K. (1975). Estimation methods for models of spatial interaction, Journal of the American Statistical Association, 70, 120-126.
See Also
spsur3sls, lagsarlm,
lmtestspsur, wald_betas,
lrtest
Examples
#################################################
######## CROSS SECTION DATA (G>1; Tm=1) ########
#################################################
#### Example 1: Spatial Phillips-Curve. Anselin (1988, p. 203)
rm(list = ls()) # Clean memory
data(spc)
Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA
spcsur.sim <- spsurml(formula = Tformula, data = spc, type = "sim")
summary(spcsur.sim)
# All the coefficients in a single table.
print(spcsur.sim)
# Plot of the coefficients of each equation in different graphs
plot(spcsur.sim)
## A SUR-SLX model
## (listw argument can be either a matrix or a listw object )
spcsur.slx <- spsurml(formula = Tformula, data = spc, type = "slx",
listw = Wspc, Durbin = TRUE)
summary(spcsur.slx)
# All the coefficients in a single table.
print(spcsur.slx)
# Plot of the coefficients in a single graph
if (require(gridExtra)) {
pl <- plot(spcsur.slx, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
nrow = 2)
}
## VIP: The output of the whole set of the examples can be examined
## by executing demo(demo_spsurml, package="spsur")
### A SUR-SLM model
spcsur.slm <- spsurml(formula = Tformula, data = spc, type = "slm",
listw = Wspc)
summary(spcsur.slm)
if (require(gridExtra)) {
pl <- plot(spcsur.slm, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
#' ### A SUR-SEM model
spcsur.sem <- spsurml(formula = Tformula, data = spc, type = "sem",
listw = Wspc)
summary(spcsur.sem)
print(spcsur.sem)
if (require(gridExtra)) {
pl <- plot(spcsur.sem, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
### A SUR-SDM model
spcsur.sdm <- spsurml(formula = Tformula, data = spc, type = "sdm",
listw = Wspc)
summary(spcsur.sdm)
print(spcsur.sdm)
if (require(gridExtra)) {
pl <- plot(spcsur.sdm, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
## A SUR-SDM model with different spatial lags in each equation
TformulaD <- ~ UN83 + NMR83 + SMSA | UN80
spcsur.sdm2 <- spsurml(formula = Tformula, data = spc, type = "sdm",
listw = Wspc, Durbin = TformulaD)
summary(spcsur.sdm2)
if (require(gridExtra)) {
pl <- plot(spcsur.sdm2, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
### A SUR-SDEM model
spcsur.sdem <- spsurml(formula = Tformula, data = spc, type = "sdem",
listw = Wspc)
print(spcsur.sdem)
if (require(gridExtra)) {
pl <- plot(spcsur.sdem, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
### A SUR-SARAR model
spcsur.sarar <- spsurml(formula = Tformula, data = spc, type = "sarar",
listw = Wspc, control = list(tol = 0.1))
print(spcsur.sarar)
if (require(gridExtra)) {
pl <- plot(spcsur.sarar, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
### A SUR-GNM model
spcsur.gnm <- spsurml(formula = Tformula, data = spc, type = "gnm",
listw = Wspc, control = list(tol = 0.1))
print(spcsur.gnm)
if (require(gridExtra)) {
pl <- plot(spcsur.gnm, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
## A A SUR-GNM model model with different spatial lags in each equation
TformulaD <- ~ UN83 + NMR83 + SMSA | UN80
spcsur.gnm2 <-spsurml(formula = Tformula, data = spc, type = "gnm",
listw = Wspc, Durbin = TformulaD,
control = list(tol = 0.1))
print(spcsur.gnm2)
if (require(gridExtra)) {
pl <- plot(spcsur.gnm2, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
##################################################
######### CLASSIC PANEL DATA G=1; Tm>1 ########
##################################################
#
##### Example 2: Homicides + Socio-Economics (1960-90)
## Homicides and selected socio-economic characteristics for continental
## U.S. counties.
## Data for four decennial census years: 1960, 1970, 1980 and 1990.
## \url{https://geodacenter.github.io/data-and-lab/ncovr/}
### It usually requires 1-2 minutes maximum...
rm(list = ls()) # Clean memory
### Read NCOVR.sf object
data(NCOVR, package = "spsur")
nbncovr <- spdep::poly2nb(NCOVR.sf, queen = TRUE)
### Some regions with no links...
lwncovr <- spdep::nb2listw(nbncovr, style = "W", zero.policy = TRUE)
Tformula <- HR80 | HR90 ~ PS80 + UE80 | PS90 + UE90
### A SUR-SIM model
NCOVRSUR.sim <- spsurml(formula = Tformula, data = NCOVR.sf, type = "sim")
summary(NCOVRSUR.sim)
if (require(gridExtra)) {
pl <- plot(NCOVRSUR.sim, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]], nrow = 3)
}
### A SUR-SLX model
NCOVRSUR.slx <- spsurml(formula = Tformula, data = NCOVR.sf, type = "slx",
listw = lwncovr, zero.policy = TRUE)
print(NCOVRSUR.slx)
if (require(gridExtra)) {
pl <- plot(NCOVRSUR.slx, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]], nrow = 2)
}
### A SUR-SLM model
### method = "Matrix" (Cholesky) instead of "eigen"
### (fdHess = TRUE to compute numerical covariances )
NCOVRSUR.slm <- spsurml(formula = Tformula, data = NCOVR.sf,
type = "slm", listw = lwncovr, method = "Matrix",
zero.policy = TRUE, control = list(fdHess = TRUE))
summary(NCOVRSUR.slm)
if (require(gridExtra)) {
pl <- plot(NCOVRSUR.slm, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
# LR test for nested models
anova(NCOVRSUR.sim, NCOVRSUR.slm)
### A SUR-SDM model with different spatial lags in each equation
### Analytical covariances (default)
TformulaD <- ~ PS80 + UE80 | PS90
NCOVRSUR.sdm <- spsurml(formula = Tformula, data = NCOVR.sf,
type = "sdm", listw = lwncovr, method = "Matrix",
Durbin = TformulaD, zero.policy = TRUE)
print(NCOVRSUR.sdm)
if (require(gridExtra)) {
pl <- plot(NCOVRSUR.sdm, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
### A SUR-SEM model
NCOVRSUR.sem <- spsurml(formula = Tformula, data = NCOVR.sf,
type = "sem", listw = lwncovr, method = "Matrix",
zero.policy = TRUE, control = list(fdHess = TRUE))
print(NCOVRSUR.sem)
if (require(gridExtra)) {
pl <- plot(NCOVRSUR.sem, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
### A SUR-SDEM model
NCOVRSUR.sdem <-spsurml(formula = Tformula, data = NCOVR.sf,
type = "sdem", listw = lwncovr, method = "Matrix",
zero.policy = TRUE, control = list(fdHess = TRUE))
print(NCOVRSUR.sdem)
if (require(gridExtra)) {
pl <- plot(NCOVRSUR.sdem, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
###############################################
## MULTI-DIMENSIONAL SUR PANEL G>1; Tm>1 ###
###############################################
##### Reshape NCOVR in panel format
N <- nrow(NCOVR.sf)
Tm <- 4
index_time <- rep(1:Tm, each = N)
index_indiv <- rep(1:N, Tm)
pHR <- c(NCOVR.sf$HR60, NCOVR.sf$HR70, NCOVR.sf$HR80, NCOVR.sf$HR90)
pPS <- c(NCOVR.sf$PS60, NCOVR.sf$PS70, NCOVR.sf$PS80, NCOVR.sf$PS90)
pUE <- c(NCOVR.sf$UE60, NCOVR.sf$UE70, NCOVR.sf$UE80, NCOVR.sf$UE90)
pDV <- c(NCOVR.sf$DV60, NCOVR.sf$DV70, NCOVR.sf$DV80, NCOVR.sf$DV90)
pFP <- c(NCOVR.sf$FP59, NCOVR.sf$FP70, NCOVR.sf$FP80, NCOVR.sf$FP90)
pSOUTH <- rep(NCOVR.sf$SOUTH, Tm)
pNCOVR <- data.frame(indiv = index_indiv, time = index_time,
HR = pHR, PS = pPS, UE = pUE, DV = pDV,
FP = pFP, SOUTH = pSOUTH)
pform <- HR | DV | FP ~ PS + UE | PS + UE + SOUTH | PS
### SIM
### Remark: It is necessary to provide Tm value as argument
### when G>1 && Tm>1
pNCOVRSUR.sim <- spsurml(formula = pform, data = pNCOVR,
type = "sim", Tm = Tm)
print(pNCOVRSUR.sim)
if (require(gridExtra)) {
pl <- plot(pNCOVRSUR.sim, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$lplbetas[[3]], nrow = 3)
}
# SLM
pNCOVRSUR.slm <- spsurml(formula = pform, data = pNCOVR,
listw = lwncovr, type = "slm", method = "Matrix", Tm = Tm,
zero.policy = TRUE, control= list(fdHess = TRUE))
print(pNCOVRSUR.slm)
if (require(gridExtra)) {
pl <- plot(pNCOVRSUR.slm, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$lplbetas[[3]], pl$pldeltas, nrow = 4)
}
pNCOVRSUR.sem <- spsurml(formula = pform, data = pNCOVR,
listw = lwncovr, type = "sem", method = "Matrix", Tm = Tm,
zero.policy = TRUE, control= list(fdHess = TRUE))
print(pNCOVRSUR.sem)
if (require(gridExtra)) {
pl <- plot(pNCOVRSUR.sem, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$lplbetas[[3]], pl$pldeltas, nrow = 4)
}
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