spsur3sls: Three Stages Least Squares estimation,3sls, of spatial SUR...
In spsur: Spatial Seemingly Unrelated Regression Models
spsur3sls R Documentation
Three Stages Least Squares estimation,3sls, of spatial SUR models.
Description
The function estimates spatial SUR models using three stages
least squares, where the instruments are obtained from the spatial lags
of the X variables, assumed to be exogenous. The number of
equations, time periods and spatial units is not restricted. The user can
choose between a Spatial Durbin Model or a Spatial Lag Model, as described
below. The estimation procedure allows for the introduction of linear
restrictions on the β parameters associated to the regressors.
Usage
spsur3sls (formula = NULL, data = NULL, na.action,
R = NULL, b = NULL, listw = NULL,
zero.policy = NULL, X= NULL, Y = NULL, G = NULL,
N = NULL, Tm = NULL, p = NULL,
type = "slm", Durbin = NULL, maxlagW = NULL,
trace = TRUE)
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.
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.
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.
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.
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.
type
Type of spatial model, restricted to cases where lags of the
explained variable appear in the rigth hand side of the equations. There
are two possibilities: "slm" or "sdm". Default = "slm".
Durbin
If a formula object and model is type "sdm" the subset
of explanatory variables to lag for each equation.
maxlagW
Maximum spatial lag order of the regressors employed to
produce spatial instruments for the spatial lags of the explained variables.
Default = 2. Note that in case of type="sdm", the default value for
maxlagW is set to 3 because the first lag of the regressors,
WX_{tg}, can not be used as spatial instruments.
trace
A logical value to show intermediate results during
the estimation process. Default = TRUE.
Details
spsur3sls can be used to estimate two groups of spatial models:
"slm": SUR model with spatial lags of the endogenous in
the right hand side of the equations
y_{tg} = ρ_{g} Wy_{tg} + X_{tg} β_{g} + ε_{tg}
"sdm": SUR model of the Spatial Durbin type
y_{tg} = ρ_{g} Wy_{tg} + X_{tg} β_{g} +
WX_{tg} θ_{g} + ε_{tg}
where y_{tg} and ε_{tg} are (Nx1) vectors,
corresponding to the g-th equation and time period t; X_{tg} is
the matrix of regressors, of order (Nxp_g). Moreover,
ρ_{g} is a spatial coefficient and W is a
(NxN) spatial weighting matrix.
By default, the input of this function is an object created with
Formula and a data frame. However,
spsur3sls also allows for the direct specification of vector
Y and matrix X, with the explained variables and regressors
respectively, as inputs (these terms may be the result, for example,
of dgp_spsur).
spsur3sls is a Least-Squares procedure in three-stages designed
to circumvent the endogeneity problems due to the presence of spatial
lags of the explained variable in the right hand side of the equations
do the SUR. The instruments are produced internally by spsur3sls
using a sequence of spatial lags of the X variables, which are
assumed to be exogenous. The user must define the number of (spatial)
instruments to be used in the procedure, through the argument
maxlagW (i.e. maxlagW = 3). Then, the collection of instruments
generated is [WX_{tg}; W*WX_{tg}; W*W*WX_{tg}]. In the case of
a SDM, the first lag of the X matrix already is in the
equation and cannot be used as instrument. In the example above, the
list of instruments for a SDM model would be
[W^{2}X_{tg}; W^{3}X_{tg}].
The first stage of the procedure consists in the least squares
of the Y variables on the set of instruments. From this
estimation, the procedure retains the estimates of Y
in the so-called Yls variables. In the second stage,
the Y variables that appear in the right hand side of the equation
are substituted by Yls and the SUR model
is estimated by Least Squares. The third stage improves the
estimates of the second stage through a Feasible Generalized Least Squares
estimation of the parameters of the model,
using the residuals of the second stage to estimate the
Sigma matrix.
The arguments R and b allows to introduce linear
restrictions on the beta coefficients of the G equations.
spsur3sls, first, introduces the linear restrictions in
the SUR model and builds, internally, the corresponding constrained
SUR model. Then, the function estimates the restricted model which is
shown in the output. The function does not compute the unconstrained
model nor test for the linear restrictions. The user may ask for the
unconstrained estimation using another spsurml
estimation. Moreover, the function wald_betas obtains
the Wald test of a set of linear restrictions for an object created
previously by spsurml or spsur3sls.
Value
Object of spsur class with the output of the three-stages
least-squares estimation of the specified spatial model.
A list with:
call Matched call.
type Type of model specified.
Durbin Value of Durbin argument.
coefficients Estimated coefficients for the regressors.
deltas Estimated spatial coefficients.
rest.se Estimated standard errors for the estimates of
β coefficients.
deltas.se Estimated standard errors for the estimates of
the spatial coefficients.
resvar Estimated covariance matrix for the estimates of
beta's and spatial coefficients.
R2 Coefficient of determination for each equation,
obtained as the squared of the correlation coefficient between the
corresponding explained variable and fitted values.
R2 pooled Global coefficient of determination
obtained for the set of the G equations.
It is computed in the same way than uniequational R2 but
joining the dependent variable and fitted values in single vectors
instead of one vector for each equation.
Sigma Estimated covariance matrix for the residuals of the
G equations.
residuals Residuals of the model.
df.residuals Degrees of freedom for the residuals.
fitted.values Estimated values for the dependent variables.
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 .
listw_style Style of neighborhood matrix W.
Author(s)
Fernando Lopez fernando.lopez@upct.es
Roman Minguez roman.minguez@uclm.es
Jesus Mur jmur@unizar.es
References
Anselin, L. (2016) Estimation and Testing in the Spatial Seemingly
Unrelated Regression (SUR). Geoda Center for Geospatial Analysis
and Computation, Arizona State University. Working Paper 2016-01.
<doi:10.13140/RG.2.2.15925.40163>
, Anselin, L. (1988). Spatial Econometrics: Methods and Models.
Kluwer Academic Publishers, Dordrecht, The Netherlands (p. 146).
Anselin, L., Le Gallo, J., Hubert J. (2008) Spatial Panel Econometrics.
In The econometrics of panel data.
Fundamentals and recent developments in theory and practice. (Chap 19, p. 653)
Minguez, R., Lopez, F.A. and Mur, J. (2022).
spsur: An R Package for Dealing with Spatial
Seemingly Unrelated Regression Models.
Journal of Statistical Software,
104(11), 1–43.
<doi:10.18637/jss.v104.i11>
Lopez, F. A., Minguez, R., Mur, J. (2020). ML versus IV estimates
of spatial SUR models: evidence from the case of Airbnb in Madrid urban
area. The Annals of Regional Science, 64(2), 313-347.
<doi:10.1007/s00168-019-00914-1>
See Also
spsurml, stsls,
wald_betas
Examples
#################################################
######## CLASSIC PANEL DATA (G=1; Tm>1) ########
#################################################
#### Example 1: Spatial Phillips-Curve. Anselin (1988, p. 203)
## A SUR model without spatial effects
rm(list = ls()) # Clean memory
data(spc)
lwspc <- spdep::mat2listw(Wspc, style = "W")
Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA
## A SUR-SLM model (3SLS Estimation)
spcsur_slm_3sls <-spsur3sls(formula = Tformula, data = spc,
type = "slm", listw = lwspc)
summary(spcsur_slm_3sls)
print(spcsur_slm_3sls)
if (require(gridExtra)) {
pl <- plot(spcsur_slm_3sls, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
## VIP: The output of the whole set of the examples can be examined
## by executing demo(demo_spsur3sls, package="spsur")
## A SUR-SDM model (3SLS Estimation)
spcsur_sdm_3sls <- spsur3sls(formula = Tformula, data = spc,
type = "sdm", listw = lwspc)
summary(spcsur_sdm_3sls)
if (require(gridExtra)) {
pl <- plot(spcsur_sdm_3sls, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
rm(spcsur_sdm_3sls)
## A SUR-SDM model with different spatial lags in each equation
TformulaD <- ~ UN83 + NMR83 + SMSA | UN80 + NMR80
spcsur_sdm2_3sls <-spsur3sls(formula = Tformula, data = spc,
type = "sdm", listw = lwspc,
Durbin = TformulaD)
summary(spcsur_sdm2_3sls)
if (require(gridExtra)) {
pl <- plot(spcsur_sdm2_3sls, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
rm(spcsur_sdm2_3sls)
#################################################
### MULTI-DIMENSIONAL PANEL DATA (G>1; Tm>1) ###
#################################################
#### Example 3: Homicides + Socio-Economics (1960-90)
# Homicides and selected socio-economic characteristics for continental
# U.S. counties.
# Data for four decennial census years: 1960, 1970, 1980 and 1990.
# https://geodacenter.github.io/data-and-lab/ncovr/
rm(list = ls()) # Clean memory
data(NCOVR, package = "spsur")
nbncovr <- spdep::poly2nb(NCOVR.sf, queen = TRUE)
## Some regions with no links...
lwncovr <- spdep::nb2listw(nbncovr, style = "W", zero.policy = TRUE)
Tformula <- HR80 | HR90 ~ PS80 + UE80 | PS90 + UE90
## A SUR-SLM model
NCOVRSUR_slm_3sls <- spsur3sls(formula = Tformula, data = NCOVR.sf,
type = "slm", zero.policy = TRUE,
listw = lwncovr, trace = FALSE)
summary(NCOVRSUR_slm_3sls)
if (require(gridExtra)) {
pl <- plot(NCOVRSUR_slm_3sls, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
rm(NCOVRSUR_slm_3sls)
spsur documentation built on Oct. 30, 2022, 1:06 a.m.
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| spsur3sls | R Documentation |
Three Stages Least Squares estimation,3sls, of spatial SUR models.
Description
The function estimates spatial SUR models using three stages least squares, where the instruments are obtained from the spatial lags of the X variables, assumed to be exogenous. The number of equations, time periods and spatial units is not restricted. The user can choose between a Spatial Durbin Model or a Spatial Lag Model, as described below. The estimation procedure allows for the introduction of linear restrictions on the β parameters associated to the regressors.
Usage
spsur3sls (formula = NULL, data = NULL, na.action,
R = NULL, b = NULL, listw = NULL,
zero.policy = NULL, X= NULL, Y = NULL, G = NULL,
N = NULL, Tm = NULL, p = NULL,
type = "slm", Durbin = NULL, maxlagW = NULL,
trace = TRUE)
Arguments
formula |
An object type |
data |
An object of class data.frame or a matrix. |
na.action |
A function (default |
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 = |
listw |
A |
zero.policy |
Similar to the corresponding parameter of
|
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
|
type |
Type of spatial model, restricted to cases where lags of the explained variable appear in the rigth hand side of the equations. There are two possibilities: "slm" or "sdm". Default = "slm". |
Durbin |
If a formula object and model is type "sdm" the subset of explanatory variables to lag for each equation. |
maxlagW |
Maximum spatial lag order of the regressors employed to
produce spatial instruments for the spatial lags of the explained variables.
Default = 2. Note that in case of |
trace |
A logical value to show intermediate results during
the estimation process. Default = |
Details
spsur3sls can be used to estimate two groups of spatial models:
"slm": SUR model with spatial lags of the endogenous in the right hand side of the equations
y_{tg} = ρ_{g} Wy_{tg} + X_{tg} β_{g} + ε_{tg}
"sdm": SUR model of the Spatial Durbin type
y_{tg} = ρ_{g} Wy_{tg} + X_{tg} β_{g} + WX_{tg} θ_{g} + ε_{tg}
where y_{tg} and ε_{tg} are (Nx1) vectors, corresponding to the g-th equation and time period t; X_{tg} is the matrix of regressors, of order (Nxp_g). Moreover, ρ_{g} is a spatial coefficient and W is a (NxN) spatial weighting matrix.
By default, the input of this function is an object created with
Formula and a data frame. However,
spsur3sls also allows for the direct specification of vector
Y and matrix X, with the explained variables and regressors
respectively, as inputs (these terms may be the result, for example,
of dgp_spsur).
spsur3sls is a Least-Squares procedure in three-stages designed
to circumvent the endogeneity problems due to the presence of spatial
lags of the explained variable in the right hand side of the equations
do the SUR. The instruments are produced internally by spsur3sls
using a sequence of spatial lags of the X variables, which are
assumed to be exogenous. The user must define the number of (spatial)
instruments to be used in the procedure, through the argument
maxlagW (i.e. maxlagW = 3). Then, the collection of instruments
generated is [WX_{tg}; W*WX_{tg}; W*W*WX_{tg}]. In the case of
a SDM, the first lag of the X matrix already is in the
equation and cannot be used as instrument. In the example above, the
list of instruments for a SDM model would be
[W^{2}X_{tg}; W^{3}X_{tg}].
The first stage of the procedure consists in the least squares of the Y variables on the set of instruments. From this estimation, the procedure retains the estimates of Y in the so-called Yls variables. In the second stage, the Y variables that appear in the right hand side of the equation are substituted by Yls and the SUR model is estimated by Least Squares. The third stage improves the estimates of the second stage through a Feasible Generalized Least Squares estimation of the parameters of the model, using the residuals of the second stage to estimate the Sigma matrix.
The arguments R and b allows to introduce linear
restrictions on the beta coefficients of the G equations.
spsur3sls, first, introduces the linear restrictions in
the SUR model and builds, internally, the corresponding constrained
SUR model. Then, the function estimates the restricted model which is
shown in the output. The function does not compute the unconstrained
model nor test for the linear restrictions. The user may ask for the
unconstrained estimation using another spsurml
estimation. Moreover, the function wald_betas obtains
the Wald test of a set of linear restrictions for an object created
previously by spsurml or spsur3sls.
Value
Object of spsur class with the output of the three-stages
least-squares estimation of the specified spatial model.
A list with:
call | Matched call. |
type | Type of model specified. |
Durbin | Value of Durbin argument. |
coefficients | Estimated coefficients for the regressors. |
deltas | Estimated spatial coefficients. |
rest.se | Estimated standard errors for the estimates of β coefficients. |
deltas.se | Estimated standard errors for the estimates of the spatial coefficients. |
resvar | Estimated covariance matrix for the estimates of beta's and spatial coefficients. |
R2 | Coefficient of determination for each equation, obtained as the squared of the correlation coefficient between the corresponding explained variable and fitted values. |
R2 pooled | Global coefficient of determination
obtained for the set of the G equations.
It is computed in the same way than uniequational R2 but
joining the dependent variable and fitted values in single vectors
instead of one vector for each equation. |
Sigma | Estimated covariance matrix for the residuals of the G equations. |
residuals | Residuals of the model. |
df.residuals | Degrees of freedom for the residuals. |
fitted.values | Estimated values for the dependent variables. |
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 . |
listw_style | Style of neighborhood matrix W. |
Author(s)
| Fernando Lopez | fernando.lopez@upct.es |
| Roman Minguez | roman.minguez@uclm.es |
| Jesus Mur | jmur@unizar.es |
References
Anselin, L. (2016) Estimation and Testing in the Spatial Seemingly Unrelated Regression (SUR). Geoda Center for Geospatial Analysis and Computation, Arizona State University. Working Paper 2016-01. <doi:10.13140/RG.2.2.15925.40163>
, Anselin, L. (1988). Spatial Econometrics: Methods and Models. Kluwer Academic Publishers, Dordrecht, The Netherlands (p. 146).
Anselin, L., Le Gallo, J., Hubert J. (2008) Spatial Panel Econometrics. In The econometrics of panel data. Fundamentals and recent developments in theory and practice. (Chap 19, p. 653)
Minguez, R., Lopez, F.A. and Mur, J. (2022). spsur: An R Package for Dealing with Spatial Seemingly Unrelated Regression Models. Journal of Statistical Software, 104(11), 1–43. <doi:10.18637/jss.v104.i11>
Lopez, F. A., Minguez, R., Mur, J. (2020). ML versus IV estimates of spatial SUR models: evidence from the case of Airbnb in Madrid urban area. The Annals of Regional Science, 64(2), 313-347. <doi:10.1007/s00168-019-00914-1>
See Also
spsurml, stsls,
wald_betas
Examples
#################################################
######## CLASSIC PANEL DATA (G=1; Tm>1) ########
#################################################
#### Example 1: Spatial Phillips-Curve. Anselin (1988, p. 203)
## A SUR model without spatial effects
rm(list = ls()) # Clean memory
data(spc)
lwspc <- spdep::mat2listw(Wspc, style = "W")
Tformula <- WAGE83 | WAGE81 ~ UN83 + NMR83 + SMSA | UN80 + NMR80 + SMSA
## A SUR-SLM model (3SLS Estimation)
spcsur_slm_3sls <-spsur3sls(formula = Tformula, data = spc,
type = "slm", listw = lwspc)
summary(spcsur_slm_3sls)
print(spcsur_slm_3sls)
if (require(gridExtra)) {
pl <- plot(spcsur_slm_3sls, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
## VIP: The output of the whole set of the examples can be examined
## by executing demo(demo_spsur3sls, package="spsur")
## A SUR-SDM model (3SLS Estimation)
spcsur_sdm_3sls <- spsur3sls(formula = Tformula, data = spc,
type = "sdm", listw = lwspc)
summary(spcsur_sdm_3sls)
if (require(gridExtra)) {
pl <- plot(spcsur_sdm_3sls, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
rm(spcsur_sdm_3sls)
## A SUR-SDM model with different spatial lags in each equation
TformulaD <- ~ UN83 + NMR83 + SMSA | UN80 + NMR80
spcsur_sdm2_3sls <-spsur3sls(formula = Tformula, data = spc,
type = "sdm", listw = lwspc,
Durbin = TformulaD)
summary(spcsur_sdm2_3sls)
if (require(gridExtra)) {
pl <- plot(spcsur_sdm2_3sls, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
rm(spcsur_sdm2_3sls)
#################################################
### MULTI-DIMENSIONAL PANEL DATA (G>1; Tm>1) ###
#################################################
#### Example 3: Homicides + Socio-Economics (1960-90)
# Homicides and selected socio-economic characteristics for continental
# U.S. counties.
# Data for four decennial census years: 1960, 1970, 1980 and 1990.
# https://geodacenter.github.io/data-and-lab/ncovr/
rm(list = ls()) # Clean memory
data(NCOVR, package = "spsur")
nbncovr <- spdep::poly2nb(NCOVR.sf, queen = TRUE)
## Some regions with no links...
lwncovr <- spdep::nb2listw(nbncovr, style = "W", zero.policy = TRUE)
Tformula <- HR80 | HR90 ~ PS80 + UE80 | PS90 + UE90
## A SUR-SLM model
NCOVRSUR_slm_3sls <- spsur3sls(formula = Tformula, data = NCOVR.sf,
type = "slm", zero.policy = TRUE,
listw = lwncovr, trace = FALSE)
summary(NCOVRSUR_slm_3sls)
if (require(gridExtra)) {
pl <- plot(NCOVRSUR_slm_3sls, viewplot = FALSE)
grid.arrange(pl$lplbetas[[1]], pl$lplbetas[[2]],
pl$pldeltas, nrow = 3)
}
rm(NCOVRSUR_slm_3sls)
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