Package 'spatialreg'

Approximate profile-likelihood estimator (APLE)

Description

The Approximate profile-likelihood estimator (APLE) of the simultaneous autoregressive model's spatial dependence parameter was introduced in Li et al. (2007). It employs a correction term using the eigenvalues of the spatial weights matrix, and consequently should not be used for large numbers of observations. It also requires that the variable has a mean of zero, and it is assumed that it has been detrended. The spatial weights object is assumed to be row-standardised, that is using default style="W" in nb2listw.

Usage

aple(x, listw, override_similarity_check=FALSE, useTrace=TRUE)

Arguments

x	a zero-mean detrended continuous variable
listw	a listw object from for example spdep::nb2listw
override_similarity_check	default FALSE, if TRUE - typically for row-standardised weights with asymmetric underlying general weights - similarity is not checked
useTrace	default TRUE, use trace of sparse matrix W %*% W (Li et al. (2010)), if FALSE, use crossproduct of eigenvalues of W as in Li et al. (2007)

Details

This implementation has been checked with Hongfei Li's own implementation using her data; her help was very valuable.

Value

A scalar APLE value.

Author(s)

Roger Bivand [email protected]

References

Li, H, Calder, C. A. and Cressie N. A. C. (2007) Beyond Moran's I: testing for spatial dependence based on the spatial autoregressive model. Geographical Analysis 39, 357-375; Li, H, Calder, C. A. and Cressie N. A. C. (2012) One-step estimation of spatial dependence parameters: Properties and extensions of the APLE statistic, Journal of Multivariate Analysis 105, 68-84.

See Also

nb2listw, aple.mc, aple.plot

Examples

wheat <- st_read(system.file("shapes/wheat.gpkg", package="spData")[1], quiet=TRUE)
library(spdep)
nbr1 <- spdep::poly2nb(wheat, queen=FALSE)
nbrl <- spdep::nblag(nbr1, 2)
nbr12 <- spdep::nblag_cumul(nbrl)
cms0 <- with(as.data.frame(wheat), tapply(yield, c, median))
cms1 <- c(model.matrix(~ factor(c) -1, data=wheat) %*% cms0)
wheat$yield_detrend <- wheat$yield - cms1
isTRUE(all.equal(c(with(as.data.frame(wheat),
 tapply(yield_detrend, c, median))), rep(0.0, 25),
 check.attributes=FALSE))
spdep::moran.test(wheat$yield_detrend, spdep::nb2listw(nbr12, style="W"))
aple(as.vector(scale(wheat$yield_detrend, scale=FALSE)), spdep::nb2listw(nbr12, style="W"))
## Not run: 
errorsarlm(yield_detrend ~ 1, wheat, spdep::nb2listw(nbr12, style="W"))

## End(Not run)

---

Approximate profile-likelihood estimator (APLE) permutation test

Description

A permutation bootstrap test for the approximate profile-likelihood estimator (APLE).

Usage

aple.mc(x, listw, nsim, override_similarity_check=FALSE, useTrace=TRUE)

Arguments

x	a zero-mean detrended continuous variable
listw	a listw object from for example spdep::nb2listw
nsim	number of simulations
override_similarity_check	default FALSE, if TRUE - typically for row-standardised weights with asymmetric underlying general weights - similarity is not checked
useTrace	default TRUE, use trace of sparse matrix W %*% W (Li et al. (2010)), if FALSE, use crossproduct of eigenvalues of W as in Li et al. (2007)

Value

A boot object as returned by the boot function.

Author(s)

Roger Bivand [email protected]

References

Li, H, Calder, C. A. and Cressie N. A. C. (2007) Beyond Moran's I: testing for spatial dependence based on the spatial autoregressive model. Geographical Analysis 39, 357-375; Li, H, Calder, C. A. and Cressie N. A. C. (2012) One-step estimation of spatial dependence parameters: Properties and extensions of the APLE statistic, Journal of Multivariate Analysis 105, 68-84.

See Also

aple, boot

Examples

## Not run: 
wheat <- st_read(system.file("shapes/wheat.gpkg", package="spData")[1], quiet=TRUE)
nbr1 <- spdep::poly2nb(wheat, queen=FALSE)
nbrl <- spdep::nblag(nbr1, 2)
nbr12 <- spdep::nblag_cumul(nbrl)
wheat_g <- wheat
st_geometry(wheat_g) <- NULL
cms0 <- with(wheat_g, tapply(yield, c, median))
cms1 <- c(model.matrix(~ factor(c) -1, data=wheat) %*% cms0)
wheat$yield_detrend <- wheat$yield - cms1
oldRNG <- RNGkind()
RNGkind("L'Ecuyer-CMRG")
set.seed(1L)
boot_out_ser <- aple.mc(as.vector(scale(wheat$yield_detrend, scale=FALSE)),
 spdep::nb2listw(nbr12, style="W"), nsim=500)
plot(boot_out_ser)
boot_out_ser
library(parallel)
oldCores <- set.coresOption(NULL)
nc <- max(2L, detectCores(logical=FALSE), na.rm = TRUE)-1L
# set nc to 1L here
if (nc > 1L) nc <- 1L
invisible(set.coresOption(nc))
set.seed(1L)
if (!get.mcOption()) {
  cl <- makeCluster(nc)
  set.ClusterOption(cl)
} else{
  mc.reset.stream()
}
boot_out_par <- aple.mc(as.vector(scale(wheat$yield_detrend, scale=FALSE)),
    spdep::nb2listw(nbr12, style="W"), nsim=500)
if (!get.mcOption()) {
  set.ClusterOption(NULL)
  stopCluster(cl)
}
boot_out_par
invisible(set.coresOption(oldCores))
RNGkind(oldRNG[1], oldRNG[2])

## End(Not run)

---

Approximate profile-likelihood estimator (APLE) scatterplot

Description

A scatterplot decomposition of the approximate profile-likelihood estimator, and a local APLE based on the list of vectors returned by the scatterplot function.

Usage

aple.plot(x, listw, override_similarity_check=FALSE, useTrace=TRUE, do.plot=TRUE, ...)
localAple(x, listw, override_similarity_check=FALSE, useTrace=TRUE)

Arguments

x	a zero-mean detrended continuous variable
listw	a listw object from for example spdep::nb2listw
override_similarity_check	default FALSE, if TRUE - typically for row-standardised weights with asymmetric underlying general weights - similarity is not checked
useTrace	default TRUE, use trace of sparse matrix W %*% W (Li et al. (2010)), if FALSE, use crossproduct of eigenvalues of W as in Li et al. (2007)
do.plot	default TRUE: should a scatterplot be drawn
...	other arguments to be passed to plot

Details

The function solves a secondary eigenproblem of size n internally, so constructing the values for the scatterplot is quite compute and memory intensive, and is not suitable for very large n.

Value

aple.plot returns list with components:

X	A vector as described in Li et al. (2007), p. 366.
Y	A vector as described in Li et al. (2007), p. 367.

localAple returns a vector of local APLE values.

Author(s)

Roger Bivand [email protected]

References

Li, H, Calder, C. A. and Cressie N. A. C. (2007) Beyond Moran's I: testing for spatial dependence based on the spatial autoregressive model. Geographical Analysis 39, pp. 357-375; Li, H, Calder, C. A. and Cressie N. A. C. (2012) One-step estimation of spatial dependence parameters: Properties and extensions of the APLE statistic, Journal of Multivariate Analysis 105, 68-84.

See Also

aple

Examples

## Not run: 
wheat <- st_read(system.file("shapes/wheat.gpkg", package="spData")[1], quiet=TRUE)
nbr1 <- spdep::poly2nb(wheat, queen=FALSE)
nbrl <- spdep::nblag(nbr1, 2)
nbr12 <- spdep::nblag_cumul(nbrl)
cms0 <- with(as.data.frame(wheat), tapply(yield, c, median))
cms1 <- c(model.matrix(~ factor(c) -1, data=wheat) %*% cms0)
wheat$yield_detrend <- wheat$yield - cms1
plt_out <- aple.plot(as.vector(scale(wheat$yield_detrend, scale=FALSE)),
 spdep::nb2listw(nbr12, style="W"), cex=0.6)
lm_obj <- lm(Y ~ X, plt_out)
abline(lm_obj)
abline(v=0, h=0, lty=2)
zz <- summary(influence.measures(lm_obj))
infl <- as.integer(rownames(zz))
points(plt_out$X[infl], plt_out$Y[infl], pch=3, cex=0.6, col="red")
crossprod(plt_out$Y, plt_out$X)/crossprod(plt_out$X)
wheat$localAple <- localAple(as.vector(scale(wheat$yield_detrend, scale=FALSE)),
 spdep::nb2listw(nbr12, style="W"))
mean(wheat$localAple)
hist(wheat$localAple)
opar <- par(no.readonly=TRUE)
plot(wheat[,"localAple"], reset=FALSE)
text(st_coordinates(st_centroid(st_geometry(wheat)))[infl,], labels=rep("*", length(infl)))
par(opar)

## End(Not run)

---

Spatial neighbour sparse representation

Description

Interface between Matrix class objects and weights lists. The as.spam.listw method converts a "listw" object to a sparse matrix as defined in the spam package.

Usage

as.spam.listw(listw)
listw2U_spam(lw)
listw2U_Matrix(lw)
as_dgRMatrix_listw(listw)
as_dsTMatrix_listw(listw)
as_dsCMatrix_I(n)
as_dsCMatrix_IrW(W, rho)
Jacobian_W(W, rho)
powerWeights(W, rho, order=250, X, tol=.Machine$double.eps^(3/5))

Arguments

listw, lw	a listw object from for example nb2listw
W	a dsTMatrix object created using as_dsTMatrix_listw from a symmetric listw object
rho	spatial regression coefficient
n	length of diagonal for identity matrix
order	Power series maximum limit
X	A numerical matrix
tol	Tolerance for convergence of power series

Author(s)

Roger Bivand [email protected]

See Also

nb2listw

Examples

## Not run: 
require(sf, quietly=TRUE)
columbus <- st_read(system.file("shapes/columbus.gpkg", package="spData")[1], quiet=TRUE)
#require(spdep, quietly=TRUE)
col.gal.nb <- spdep::read.gal(system.file("weights/columbus.gal", package="spData")[1])
col.listw <- spdep::nb2listw(col.gal.nb)
if (require("spam", quietly=TRUE)) {
  col.sp <- as.spam.listw(col.listw)
  str(col.sp)
}
suppressMessages(nyadjmat <- as.matrix(foreign::read.dbf(system.file(
 "misc/nyadjwts.dbf", package="spData")[1])[-1]))
nyadjlw <- spdep::mat2listw(nyadjmat)
listw_NY <- spdep::nb2listw(nyadjlw$neighbours, style="B")
W_C <- as(listw_NY, "CsparseMatrix")
W_R <- as(listw_NY, "RsparseMatrix")
W_S <- as(listw_NY, "symmetricMatrix")
n <- nrow(W_S)
I <- Diagonal(n)
rho <- 0.1
c(determinant(I - rho * W_S, logarithm=TRUE)$modulus)
sum(log(1 - rho * eigenw(listw_NY)))
nW <- - W_S
nChol <- Cholesky(nW, Imult=8)
n * log(rho) + (2 * c(determinant(update(nChol, nW, 1/rho))$modulus))

## End(Not run)
nb7rt <- spdep::cell2nb(7, 7, torus=TRUE)
x <- matrix(sample(rnorm(500*length(nb7rt))), nrow=length(nb7rt))
lw <- spdep::nb2listw(nb7rt)
if (FALSE) {
# Only needed in some simulation settings where the input and
# output distributions must agree in all but autocorrelation
e <- eigenw(lw)
x <- apply(x, 2, scale)
st <- apply(x, 2, function(x) shapiro.test(x)$p.value)
x <- x[, (st > 0.2 & st < 0.8)]
x <- apply(x, 2, function(v) residuals(spautolm(v ~ 1, listw=lw,
 method="eigen", control=list(pre_eig=e, fdHess=FALSE))))
x <- apply(x, 2, scale)
}
W <- as(lw, "CsparseMatrix")
system.time(e <- invIrM(nb7rt, rho=0.98, method="solve", feasible=NULL) %*% x)
system.time(ee <- powerWeights(W, rho=0.98, X=x))
str(attr(ee, "internal"))
all.equal(e, as(ee, "matrix"), check.attributes=FALSE)
## Not run: 
system.time(ee <- powerWeights(W, rho=0.9, X=x))
system.time(ee <- powerWeights(W, rho=0.98, order=1000, X=x))
all.equal(e, as(ee, "matrix"), check.attributes=FALSE)
nb60rt <- spdep::cell2nb(60, 60, torus=TRUE)
W <- as(spdep::nb2listw(nb60rt), "CsparseMatrix")
set.seed(1)
x <- matrix(rnorm(dim(W)[1]), ncol=1)
system.time(ee <- powerWeights(W, rho=0.3, X=x))
str(as(ee, "matrix"))
obj <- errorsarlm(as(ee, "matrix")[,1] ~ 1, listw=spdep::nb2listw(nb60rt), method="Matrix")
coefficients(obj)

## End(Not run)

---

Spatial regression model Jacobian computations

Description

These functions are made available in the package namespace for other developers, and are not intended for users. They provide a shared infrastructure for setting up data for Jacobian computation, and then for caclulating the Jacobian, either exactly or approximately, in maximum likelihood fitting of spatial regression models. The techniques used are the exact eigenvalue, Cholesky decompositions (Matrix, spam), and LU ones, with Chebyshev and Monte Carlo approximations; moments use the methods due to Martin and Smirnov/Anselin.

Usage

do_ldet(coef, env, which=1)
jacobianSetup(method, env, con, pre_eig=NULL, trs=NULL, interval=NULL, which=1)
cheb_setup(env, q=5, which=1)
mcdet_setup(env, p=16, m=30, which=1)
eigen_setup(env, which=1)
eigen_pre_setup(env, pre_eig, which=1)
spam_setup(env, pivot="MMD", which=1)
spam_update_setup(env, in_coef=0.1, pivot="MMD", which=1)
Matrix_setup(env, Imult, super=as.logical(NA), which=1)
Matrix_J_setup(env, super=FALSE, which=1)
LU_setup(env, which=1)
LU_prepermutate_setup(env, coef=0.1, order=FALSE, which=1)
moments_setup(env, trs=NULL, m, p, type="MC", correct=TRUE, trunc=TRUE, eq7=TRUE, which=1)
SE_classic_setup(env, SE_method="LU", p=16, m=30, nrho=200, interpn=2000,
 interval=c(-1,0.999), SElndet=NULL, which=1)
SE_whichMin_setup(env, SE_method="LU", p=16, m=30, nrho=200, interpn=2000,
 interval=c(-1,0.999), SElndet=NULL, which=1)
SE_interp_setup(env, SE_method="LU", p=16, m=30, nrho=200,
 interval=c(-1,0.999), which=1)
can.be.simmed(listw)

Arguments

coef	spatial coefficient value
env	environment containing pre-computed objects, fixed after assignment in setup functions
which	default 1; if 2, use second listw object
method	string value, used by jacobianSetup to choose method
con	control list passed from model fitting function and parsed in jacobianSetup to set environment variables for method-specific setup
pre_eig	pre-computed eigenvalues of length n
q	Chebyshev approximation order; default in calling spdep functions is 5, here it cannot be missing and does not have a default
p	Monte Carlo approximation number of random normal variables; default calling spdep functions is 16, here it cannot be missing and does not have a default
m	Monte Carlo approximation number of series terms; default in calling spdep functions is 30, here it cannot be missing and does not have a default; m serves the same purpose in the moments method
pivot	default “MMD”, may also be “RCM” for Cholesky decompisition using spam
in_coef	fill-in initiation coefficient value, default 0.1
Imult	see Cholesky; numeric scalar which defaults to zero. The matrix that is decomposed is A+m*I where m is the value of Imult and I is the identity matrix of order ncol(A). Default in calling spdep functions is 2, here it cannot be missing and does not have a default, but is rescaled for binary weights matrices in proportion to the maximim row sum in those calling functions
super	see Cholesky; logical scalar indicating is a supernodal decomposition should be created. The alternative is a simplicial decomposition. Default in calling spdep functions is FALSE for “Matrix_J” and as.logical(NA) for “Matrix”. Setting it to NA leaves the choice to a CHOLMOD-internal heuristic
order	default FALSE; used in LU_prepermutate, note warnings given for lu method
trs	A numeric vector of m traces, as from trW
type	moments trace type, see trW
correct	default TRUE: use Smirnov correction term, see trW
trunc	default TRUE: truncate Smirnov correction term, see trW
eq7	default TRUE; use equation 7 in Smirnov and Anselin (2009), if FALSE no unit root correction
SE_method	default “LU”, alternatively “MC”; underlying lndet method to use for generating SE toolbox emulation grid
nrho	default 200, number of lndet values in first stage SE toolbox emulation grid
interval	default c(-1,0.999) if interval argument NULL, bounds for SE toolbox emulation grid
interpn	default 2000, number of lndet values to interpolate in second stage SE toolbox emulation grid
SElndet	default NULL, used to pass a pre-computed two-column matrix of coefficient values and corresponding interpolated lndet values
listw	a spatial weights object

Details

Since environments are containers in the R workspace passed by reference rather than by value, they are useful for passing objects to functions called in numerical optimisation, here for the maximum likelihood estimation of spatial regression models. This technique can save a little time on each function call, balanced against the need to access the objects in the environment inside the function. The environment should contain a family string object either “SAR”, “CAR” or “SMA” (used in do_ldet to choose spatial moving average in spautolm, and these specific objects before calling the set-up functions:

eigen

Classical Ord eigenvalue computations - either:

listw

A listw spatial weights object

can.sim

logical scalar: can the spatial weights be made symmetric by similarity

verbose

logical scalar: legacy report print control, for historical reasons only

or:

pre_eig

pre-computed eigenvalues

and assigns to the environment:

eig

a vector of eigenvalues

eig.range

the search interval for the spatial coefficient

method

string: “eigen”

Matrix

Sparse matrix pre-computed Cholesky decomposition with fast updating:

listw

A listw spatial weights object

can.sim

logical scalar: can the spatial weights be made symmetric by similarity

and assigns to the environment:

csrw

sparse spatial weights matrix

nW

negative sparse spatial weights matrix

pChol

a “CHMfactor” from factorising csrw with Cholesky

nChol

a “CHMfactor” from factorising nW with Cholesky

method

string: “Matrix”

Matrix_J

Standard Cholesky decomposition without updating:

listw

A listw spatial weights object

can.sim

logical scalar: can the spatial weights be made symmetric by similarity

n

number of spatial objects

and assigns to the environment:

csrw

sparse spatial weights matrix

I

sparse identity matrix

super

the value of the super argument

method

string: “Matrix_J”

spam

Standard Cholesky decomposition without updating:

listw

A listw spatial weights object

can.sim

logical scalar: can the spatial weights be made symmetric by similarity

n

number of spatial objects

and assigns to the environment:

csrw

sparse spatial weights matrix

I

sparse identity matrix

pivot

string — pivot method

method

string: “spam”

spam_update

Pre-computed Cholesky decomposition with updating:

listw

A listw spatial weights object

can.sim

logical scalar: can the spatial weights be made symmetric by similarity

n

number of spatial objects

and assigns to the environment:

csrw

sparse spatial weights matrix

I

sparse identity matrix

csrwchol

A Cholesky decomposition for updating

method

string: “spam”

LU

Standard LU decomposition without updating:

listw

A listw spatial weights object

n

number of spatial objects

and assigns to the environment:

W

sparse spatial weights matrix

I

sparse identity matrix

method

string: “LU”

LU_prepermutate

Standard LU decomposition with updating (pre-computed fill-reducing permutation):

listw

A listw spatial weights object

n

number of spatial objects

and assigns to the environment:

W

sparse spatial weights matrix

lu_order

order argument to lu

pq

2-column matrix for row and column permutation for fill-reduction

I

sparse identity matrix

method

string: “LU”

MC

Monte Carlo approximation:

listw

A listw spatial weights object

and assigns to the environment:

clx

list of Monte Carlo approximation terms (the first two simulated traces are replaced by their analytical equivalents)

W

sparse spatial weights matrix

method

string: “MC”

cheb

Chebyshev approximation:

listw

A listw spatial weights object

and assigns to the environment:

trT

vector of Chebyshev approximation terms

W

sparse spatial weights matrix

method

string: “Chebyshev”

moments

moments approximation:

listw

A listw spatial weights object

can.sim

logical scalar: can the spatial weights be made symmetric by similarity

and assigns to the environment:

trs

vector of traces, possibly approximated

q12

integer vector of length 2, unit roots terms, ignored until 0.5-52

eq7

logical scalar: use equation 7

correct

logical scalar: use Smirnov correction term

trunc

logical scalar: truncate Smirnov correction term

method

string: “moments”

SE_classic

:

listw

A listw spatial weights object

n

number of spatial objects

and assigns to the environment:

detval

two column matrix of lndet grid values

method

string: “SE_classic”

SE_method

string: “LU” or “MC”

SE_whichMin

:

listw

A listw spatial weights object

n

number of spatial objects

and assigns to the environment:

detval

two column matrix of lndet grid values

method

string: “SE_whichMin”

SE_method

string: “LU” or “MC”

SE_interp

:

listw

A listw spatial weights object

n

number of spatial objects

and assigns to the environment:

fit

fitted spline object from which to predict lndet values

method

string: “SE_interp”

SE_method

string: “LU” or “MC”

Some set-up functions may also assign similar to the environment if the weights were made symmetric by similarity.

Three set-up functions emulate the behaviour of the Spatial Econometrics toolbox (March 2010) maximum likelihood lndet grid performance. The toolbox lndet functions compute a smaller number of lndet values for a grid of coefficient values (spacing 0.01), and then interpolate to a finer grid of values (spacing 0.001). “SE_classic”, which is an implementation of the SE toolbox code, for example in f_sar.m, appears to have selected a row in the grid matrix one below the correct row when the candidate coefficient value was between 0.005 and 0.01-fuzz, always rounding the row index down. A possible alternative is to choose the index that is closest to the candidate coefficient value (“SE_whichMin”). Another alternative is to fit a spline model to the first stage coarser grid, and pass this fitted model to the log likelihood function to make a point prediction using the candidate coefficient value, rather than finding the grid index (“SE_interp”).

Value

do_ldet returns the value of the Jacobian for the calculation method recorded in the environment argument, and for the Monte Carlo approximation, returns a measure of the spread of the approximation as an “sd” attribute; the remaining functions modify the environment in place as a side effect and return nothing.

Author(s)

Roger Bivand [email protected]

References

LeSage J and RK Pace (2009) Introduction to Spatial Econometrics. CRC Press, Boca Raton, pp. 77–110.

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.

See Also

spautolm, lagsarlm, errorsarlm, Cholesky

Examples

data(boston, package="spData")
#require("spdep", quietly=TRUE)
lw <- spdep::nb2listw(boston.soi)
can.sim <- can.be.simmed(lw)
env <- new.env(parent=globalenv())
assign("listw", lw, envir=env)
assign("can.sim", can.sim, envir=env)
assign("similar", FALSE, envir=env)
assign("verbose", FALSE, envir=env)
assign("family", "SAR", envir=env)
eigen_setup(env)
get("similar", envir=env)
do_ldet(0.5, env)
rm(env)
env <- new.env(parent=globalenv())
assign("listw", lw, envir=env)
assign("can.sim", can.sim, envir=env)
assign("similar", FALSE, envir=env)
assign("verbose", FALSE, envir=env)
assign("family", "SAR", envir=env)
assign("n", length(boston.soi), envir=env)
eigen_pre_setup(env, pre_eig=eigenw(similar.listw(lw)))
do_ldet(0.5, env)
rm(env)
env <- new.env(parent=globalenv())
assign("listw", lw, envir=env)
assign("can.sim", can.sim, envir=env)
assign("similar", FALSE, envir=env)
assign("family", "SAR", envir=env)
assign("n", length(boston.soi), envir=env)
Matrix_setup(env, Imult=2, super=FALSE)
get("similar", envir=env)
do_ldet(0.5, env)
rm(env)
env <- new.env(parent=globalenv())
assign("listw", lw, envir=env)
assign("n", length(boston.soi), envir=env)
assign("can.sim", can.sim, envir=env)
assign("similar", FALSE, envir=env)
assign("family", "SAR", envir=env)
spam_setup(env)
get("similar", envir=env)
do_ldet(0.5, env)
rm(env)
env <- new.env(parent=globalenv())
assign("listw", lw, envir=env)
assign("n", length(boston.soi), envir=env)
assign("similar", FALSE, envir=env)
assign("family", "SAR", envir=env)
LU_setup(env)
get("similar", envir=env)
do_ldet(0.5, env)
rm(env)
env <- new.env(parent=globalenv())
assign("listw", lw, envir=env)
assign("n", length(boston.soi), envir=env)
assign("similar", FALSE, envir=env)
assign("family", "SAR", envir=env)
LU_prepermutate_setup(env)
get("similar", envir=env)
do_ldet(0.5, env)
rm(env)
env <- new.env(parent=globalenv())
assign("listw", lw, envir=env)
assign("similar", FALSE, envir=env)
assign("family", "SAR", envir=env)
cheb_setup(env, q=5)
get("similar", envir=env)
do_ldet(0.5, env)
rm(env)
env <- new.env(parent=globalenv())
assign("listw", lw, envir=env)
assign("n", length(boston.soi), envir=env)
assign("similar", FALSE, envir=env)
assign("family", "SAR", envir=env)
set.seed(12345)
mcdet_setup(env, p=16, m=30)
get("similar", envir=env)
do_ldet(0.5, env)
rm(env)

---

Spatial simultaneous autoregressive error model estimation by GMM

Description

An implementation of Kelejian and Prucha's generalised moments estimator for the autoregressive parameter in a spatial model.

Usage

GMerrorsar(formula, data = list(), listw, na.action = na.fail,
 zero.policy = attr(listw, "zero.policy"), method="nlminb", arnoldWied=FALSE, 
 control = list(), pars, scaleU=FALSE, verbose=NULL, legacy=FALSE,
 se.lambda=TRUE, returnHcov=FALSE, pWOrder=250, tol.Hcov=1.0e-10)
## S3 method for class 'Gmsar'
summary(object, correlation = FALSE, Hausman=FALSE, ...)
GMargminImage(obj, lambdaseq, s2seq)

Arguments

formula	a symbolic description of the model to be fit. The details of model specification are given for lm()
data	an optional data frame containing the variables in the model. By default the variables are taken from the environment which the function is called.
listw	a listw object created for example by nb2listw
na.action	a function (default na.fail), can also be na.omit or na.exclude with consequences for residuals and fitted values - in these cases the weights list will be subsetted to remove NAs in the data. It may be necessary to set zero.policy to TRUE because this subsetting may create no-neighbour observations. Note that only weights lists created without using the glist argument to nb2listw may be subsetted.
zero.policy	default NULL, use global option value; if TRUE assign zero to the lagged value of zones without neighbours, if FALSE (default) assign NA - causing GMerrorsar() to terminate with an error
method	default "nlminb", or optionally a method passed to optim to use an alternative optimizer
arnoldWied	default FALSE
control	A list of control parameters. See details in optim or nlminb.
pars	starting values for $\lambda$ and $\sigma^2$ for GMM optimisation, if missing (default), approximated from initial OLS model as the autocorrelation coefficient corrected for weights style and model sigma squared
scaleU	Default FALSE: scale the OLS residuals before computing the moment matrices; only used if the pars argument is missing
verbose	default NULL, use global option value; if TRUE, reports function values during optimization.
legacy	default FALSE - compute using the unfiltered values of the response and right hand side variables; if TRUE - compute the fitted value and residuals from the spatially filtered model using the spatial error parameter
se.lambda	default TRUE, use the analytical method described in http://econweb.umd.edu/~prucha/STATPROG/OLS/desols.pdf
returnHcov	default FALSE, return the Vo matrix for a spatial Hausman test
tol.Hcov	the tolerance for computing the Vo matrix (default=1.0e-10)
pWOrder	default 250, if returnHcov=TRUE, pass this order to powerWeights as the power series maximum limit
object, obj	Gmsar object from GMerrorsar
correlation	logical; (default=FALSE), TRUE not available
Hausman	if TRUE, the results of the Hausman test for error models are reported
...	summary arguments passed through
lambdaseq	if given, an increasing sequence of lambda values for gridding
s2seq	if given, an increasing sequence of sigma squared values for gridding

Details

When the control list is set with care, the function will converge to values close to the ML estimator without requiring computation of the Jacobian, the most resource-intensive part of ML estimation.

Note that the fitted() function for the output object assumes that the response variable may be reconstructed as the sum of the trend, the signal, and the noise (residuals). Since the values of the response variable are known, their spatial lags are used to calculate signal components (Cressie 1993, p. 564). This differs from other software, including GeoDa, which does not use knowledge of the response variable in making predictions for the fitting data.

The GMargminImage may be used to visualize the shape of the surface of the argmin function used to find lambda.

Value

A list object of class Gmsar

type	"ERROR"
lambda	simultaneous autoregressive error coefficient
coefficients	GMM coefficient estimates
rest.se	GMM coefficient standard errors
s2	GMM residual variance
SSE	sum of squared GMM errors
parameters	number of parameters estimated
lm.model	the lm object returned when estimating for $\lambda=0$
call	the call used to create this object
residuals	GMM residuals
lm.target	the lm object returned for the GMM fit
fitted.values	Difference between residuals and response variable
formula	model formula
aliased	if not NULL, details of aliased variables
zero.policy	zero.policy for this model
vv	list of internal bigG and litg components for testing optimisation surface
optres	object returned by optimizer
pars	start parameter values for optimisation
Hcov	Spatial DGP covariance matrix for Hausman test if available
legacy	input choice of unfiltered or filtered values
lambda.se	value computed if input argument TRUE
arnoldWied	were Arnold-Wied moments used
GMs2	GM argmin sigma squared
scaleU	input choice of scaled OLS residuals
vcov	variance-covariance matrix of regression coefficients
na.action	(possibly) named vector of excluded or omitted observations if non-default na.action argument used

Author(s)

Luc Anselin and Roger Bivand

References

Kelejian, H. H., and Prucha, I. R., 1999. A Generalized Moments Estimator for the Autoregressive Parameter in a Spatial Model. International Economic Review, 40, pp. 509–533; Cressie, N. A. C. 1993 Statistics for spatial data, Wiley, New York.

Roger Bivand, Gianfranco Piras (2015). Comparing Implementations of Estimation Methods for Spatial Econometrics. Journal of Statistical Software, 63(18), 1-36. doi:10.18637/jss.v063.i18.

See Also

optim, nlminb, errorsarlm

Examples

#require("spdep", quietly=TRUE)
data(oldcol, package="spdep")
COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 spdep::nb2listw(COL.nb, style="W"), method="eigen")
(x <- summary(COL.errW.eig, Hausman=TRUE))
coef(x)
COL.errW.GM <- GMerrorsar(CRIME ~ INC + HOVAL, data=COL.OLD,
 spdep::nb2listw(COL.nb, style="W"), returnHcov=TRUE)
(x <- summary(COL.errW.GM, Hausman=TRUE))
coef(x)
aa <- GMargminImage(COL.errW.GM)
levs <- quantile(aa$z, seq(0, 1, 1/12))
image(aa, breaks=levs, xlab="lambda", ylab="s2")
points(COL.errW.GM$lambda, COL.errW.GM$s2, pch=3, lwd=2)
contour(aa, levels=signif(levs, 4), add=TRUE)
COL.errW.GM1 <- GMerrorsar(CRIME ~ INC + HOVAL, data=COL.OLD,
 spdep::nb2listw(COL.nb, style="W"))
summary(COL.errW.GM1)
require("sf", quietly=TRUE)
nydata <- st_read(system.file("shapes/NY8_bna_utm18.gpkg", package="spData")[1], quiet=TRUE)
suppressMessages(nyadjmat <- as.matrix(foreign::read.dbf(system.file(
 "misc/nyadjwts.dbf", package="spData")[1])[-1]))
suppressMessages(ID <- as.character(names(foreign::read.dbf(system.file(
 "misc/nyadjwts.dbf", package="spData")[1]))[-1]))
identical(substring(ID, 2, 10), substring(as.character(nydata$AREAKEY), 2, 10))
listw_NY <- spdep::mat2listw(nyadjmat, as.character(nydata$AREAKEY), style="B")
esar1f <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, family="SAR", method="eigen")
summary(esar1f)
esar1gm <- GMerrorsar(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY)
summary(esar1gm)
esar1gm1 <- GMerrorsar(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY, method="Nelder-Mead")
summary(esar1gm1)

---

Spatial weights matrix eigenvalues

Description

The eigenw function returns a numeric vector of eigenvalues of the weights matrix generated from the spatial weights object listw. The eigenvalues are used to speed the computation of the Jacobian in spatial model estimation:

$\log(\det[I - \rho W]) = \sum_{i=1}^{n}\log(1 - \rho \lambda_i)$

where $W$ is the n by n spatial weights matrix, and $\lambda_i$ are the eigenvalues of $W$.

Usage

eigenw(listw, quiet=NULL)
griffith_sone(P, Q, type="rook")
subgraph_eigenw(nb, glist=NULL, style="W", zero.policy=NULL, quiet=NULL)

Arguments

listw	a listw object created for example by nb2listw
quiet	default NULL, use global !verbose option value; set to FALSE for short summary
P	number of columns in the grid (number of units in a horizontal axis direction)
Q	number of rows in the grid (number of units in a vertical axis direction.)
type	“rook” or “queen”
nb	an object of class nb
glist	list of general weights corresponding to neighbours
style	style can take values “W”, “B”, “C”, “U”, “minmax” and “S”
zero.policy	default NULL, use global option value; if FALSE stop with error for any empty neighbour sets, if TRUE permit the weights list to be formed with zero-length weights vectors

Details

The griffith_sone function function may be used, following Ord and Gasim (for references see Griffith and Sone (1995)), to calculate analytical eigenvalues for binary rook or queen contiguous neighbours where the data are arranged as a regular P times Q grid. The subgraph_eigenw function may be used when there are multiple graph components, of which the largest may be handled as a dense matrix. Here the eigenvalues are computed for each subgraph in turn, and catenated to reconstruct the complete set. The functions may be used to provide pre-computed eigenvalues for spatial regression functions.

Value

a numeric or complex vector of eigenvalues of the weights matrix generated from the spatial weights object.

Author(s)

Roger Bivand [email protected]

References

Cliff, A. D., Ord, J. K. 1981 Spatial processes, Pion, p. 155; Ord, J. K. 1975 Estimation methods for models of spatial interaction, Journal of the American Statistical Association, 70, 120-126.; Griffith, D. A. and Sone, A. (1995). Trade-offs associated with normalizing constant computational simplifications for estimating spatial statistical models. Journal of Statistical Computation and Simulation, 51, 165-183.

See Also

eigen

Examples

#require(spdep)
data(oldcol, package="spdep")
W.eig <- eigenw(spdep::nb2listw(COL.nb, style="W"))
1/range(W.eig)
S.eig <- eigenw(spdep::nb2listw(COL.nb, style="S"))
1/range(S.eig)
B.eig <- eigenw(spdep::nb2listw(COL.nb, style="B"))
1/range(B.eig)
# cases for intrinsically asymmetric weights
crds <- cbind(COL.OLD$X, COL.OLD$Y)
k3 <- spdep::knn2nb(spdep::knearneigh(crds, k=3))
spdep::is.symmetric.nb(k3)
k3eig <- eigenw(spdep::nb2listw(k3, style="W"))
is.complex(k3eig)
rho <- 0.5
Jc <- sum(log(1 - rho * k3eig))
# complex eigenvalue Jacobian
Jc
# subgraphs
nc <- attr(k3, "ncomp")
if (is.null(nc)) nc <- spdep::n.comp.nb(k3)
nc$nc
table(nc$comp.id)
k3eigSG <- subgraph_eigenw(k3, style="W")
all.equal(sort(k3eig), k3eigSG)
W <- as(spdep::nb2listw(k3, style="W"), "CsparseMatrix")
I <- diag(length(k3))
Jl <- sum(log(abs(diag(slot(lu(I - rho * W), "U")))))
# LU Jacobian equals complex eigenvalue Jacobian
Jl
all.equal(Re(Jc), Jl)
# wrong value if only real part used
Jr <- sum(log(1 - rho * Re(k3eig)))
Jr
all.equal(Jr, Jl)
# construction of Jacobian from complex conjugate pairs (Jan Hauke)
Rev <- Re(k3eig)[which(Im(k3eig) == 0)]
# real eigenvalues
Cev <- k3eig[which(Im(k3eig) != 0)]
pCev <- Cev[Im(Cev) > 0]
# separate complex conjugate pairs
RpCev <- Re(pCev)
IpCev <- Im(pCev)
# reassemble Jacobian
Jc1 <- sum(log(1 - rho*Rev)) + sum(log((1 - rho * RpCev)^2 + (rho^2)*(IpCev^2)))
all.equal(Re(Jc), Jc1)
# impact of omitted complex part term in real part only Jacobian
Jc2 <- sum(log(1 - rho*Rev)) + sum(log((1 - rho * RpCev)^2))
all.equal(Jr, Jc2)
# trace of asymmetric (WW) and crossprod of complex eigenvalues for APLE
sum(diag(W %*% W))
crossprod(k3eig)
# analytical regular grid eigenvalues
rg <- spdep::cell2nb(ncol=7, nrow=7, type="rook")
rg_eig <- eigenw(spdep::nb2listw(rg, style="B"))
rg_GS <- griffith_sone(P=7, Q=7, type="rook")
all.equal(rg_eig, rg_GS)
## Not run: 
run <- FALSE
if (require("RSpectra", quietly=TRUE)) run <- TRUE
if (run) {
B <- as(spdep::nb2listw(rg, style="B"), "CsparseMatrix")
res1 <- eigs(B, k=1, which="LR")$values
resn <- eigs(B, k=1, which="SR")$values
print(Re(c(resn, res1)))
}
if (run) {
print(all.equal(range(Re(rg_eig)), c(resn, res1))) 
}
if (run) {
lw <- spdep::nb2listw(rg, style="W")
rg_eig <- eigenw(similar.listw(lw))
print(range(Re(rg_eig)))
}
if (run) {
W  <- as(lw, "CsparseMatrix")
print(Re(c(eigs(W, k=1, which="SR")$values, eigs(W, k=1, which="LR")$values)))
}
## End(Not run)

---

Spatial simultaneous autoregressive SAC model estimation by GMM

Description

An implementation of Kelejian and Prucha's generalised moments estimator for the autoregressive parameter in a spatial model with a spatially lagged dependent variable.

Usage

gstsls(formula, data = list(), listw, listw2 = NULL, na.action = na.fail, 
    zero.policy = attr(listw, "zero.policy"), pars=NULL, scaleU=FALSE, control = list(), 
    verbose=NULL, method="nlminb", robust=FALSE, legacy=FALSE, W2X=TRUE, sig2n_k=FALSE) 
## S3 method for class 'Gmsar'
impacts(obj, ..., n = NULL, tr = NULL, R = NULL,
 listw = NULL, evalues=NULL, tol = 1e-06, empirical = FALSE, Q=NULL)

Arguments

formula	a symbolic description of the model to be fit. The details of model specification are given for lm()
data	an optional data frame containing the variables in the model. By default the variables are taken from the environment which the function is called.
listw	a listw object created for example by nb2listw
listw2	a listw object created for example by nb2listw, if not given, set to the same spatial weights as the listw argument
na.action	a function (default na.fail), can also be na.omit or na.exclude with consequences for residuals and fitted values - in these cases the weights list will be subsetted to remove NAs in the data. It may be necessary to set zero.policy to TRUE because this subsetting may create no-neighbour observations. Note that only weights lists created without using the glist argument to nb2listw may be subsetted.
zero.policy	default NULL, use global option value; if TRUE assign zero to the lagged value of zones without neighbours, if FALSE (default) assign NA - causing GMerrorsar() to terminate with an error
pars	starting values for $\lambda$ and $\sigma^2$ for GMM optimisation, if missing (default), approximated from initial 2sls model as the autocorrelation coefficient corrected for weights style and model sigma squared
scaleU	Default FALSE: scale the OLS residuals before computing the moment matrices; only used if the pars argument is missing
control	A list of control parameters. See details in optim or nlminb
verbose	default NULL, use global option value; if TRUE, reports function values during optimization.
method	default nlminb, or optionally a method passed to optim to use an alternative optimizer
robust	see stsls
legacy	see stsls
W2X	see stsls
sig2n_k	see stsls
obj	A spatial regression object created by lagsarlm, lagmess or by lmSLX; in HPDinterval.LagImpact, a LagImpact object
...	Arguments passed through to methods in the coda package
tr	A vector of traces of powers of the spatial weights matrix created using trW, for approximate impact measures; if not given, listw must be given for exact measures (for small to moderate spatial weights matrices); the traces must be for the same spatial weights as were used in fitting the spatial regression, and must be row-standardised
evalues	vector of eigenvalues of spatial weights matrix for impacts calculations
R	If given, simulations are used to compute distributions for the impact measures, returned as mcmc objects; the objects are used for convenience but are not output by an MCMC process
tol	Argument passed to mvrnorm: tolerance (relative to largest variance) for numerical lack of positive-definiteness in the coefficient covariance matrix
empirical	Argument passed to mvrnorm (default FALSE): if true, the coefficients and their covariance matrix specify the empirical not population mean and covariance matrix
Q	default NULL, else an integer number of cumulative power series impacts to calculate if tr is given
n	defaults to length(obj$residuals); in the method for Gmsar objects it may be used in panel settings to compute the impacts for cross-sectional weights only, suggested by Angela Parenti

Details

When the control list is set with care, the function will converge to values close to the ML estimator without requiring computation of the Jacobian, the most resource-intensive part of ML estimation.

Value

A list object of class Gmsar

lambda	simultaneous autoregressive error coefficient
coefficients	GMM coefficient estimates (including the spatial autocorrelation coefficient)
rest.se	GMM coefficient standard errors
s2	GMM residual variance
SSE	sum of squared GMM errors
parameters	number of parameters estimated
lm.model	NULL
call	the call used to create this object
residuals	GMM residuals
lm.target	NULL
fitted.values	Difference between residuals and response variable
formula	model formula
aliased	NULL
zero.policy	zero.policy for this model
LL	NULL
vv	list of internal bigG and litg components for testing optimisation surface
optres	object returned by optimizer
pars	start parameter values for optimisation
Hcov	NULL
na.action	(possibly) named vector of excluded or omitted observations if non-default na.action argument used

Author(s)

Gianfranco Piras and Roger Bivand

References

Kelejian, H. H., and Prucha, I. R., 1999. A Generalized Moments Estimator for the Autoregressive Parameter in a Spatial Model. International Economic Review, 40, pp. 509–533; Cressie, N. A. C. 1993 Statistics for spatial data, Wiley, New York.

Roger Bivand, Gianfranco Piras (2015). Comparing Implementations of Estimation Methods for Spatial Econometrics. Journal of Statistical Software, 63(18), 1-36. doi:10.18637/jss.v063.i18.

See Also

optim, nlminb, GMerrorsar, GMargminImage

Examples

#require("spdep", quietly=TRUE) 
data(oldcol, package="spdep")
COL.errW.GM <- gstsls(CRIME ~ INC + HOVAL, data=COL.OLD, spdep::nb2listw(COL.nb, style="W"))
summary(COL.errW.GM)
aa <- GMargminImage(COL.errW.GM)
levs <- quantile(aa$z, seq(0, 1, 1/12))
image(aa, breaks=levs, xlab="lambda", ylab="s2")
points(COL.errW.GM$lambda, COL.errW.GM$s2, pch=3, lwd=2)
contour(aa, levels=signif(levs, 4), add=TRUE)
COL.errW.GM <- gstsls(CRIME ~ INC + HOVAL, data=COL.OLD,
 spdep::nb2listw(COL.nb, style="W"), scaleU=TRUE)
summary(COL.errW.GM)
listw <- spdep::nb2listw(COL.nb)
W <- as(listw, "CsparseMatrix")
trMat <- trW(W, type="mult")
impacts(COL.errW.GM, tr=trMat)

---

Impacts in spatial lag models

Description

The calculation of impacts for spatial lag and spatial Durbin models is needed in order to interpret the regression coefficients correctly, because of the spillovers between the terms in these data generation processes (unlike the spatial error model). Methods for “SLX” and Bayesian fitted models are also provided, the former do not need MC simulations, while the latter pass through MCMC draws.

Usage

#\method{impacts}{sarlm}(obj, \dots, tr, R = NULL, listw = NULL, evalues=NULL,
# useHESS = NULL, tol = 1e-06, empirical = FALSE, Q=NULL)
#\method{impacts}{lagmess}(obj, ..., R=NULL, listw=NULL, tol=1e-6,
# empirical=FALSE)
#\method{impacts}{SLX}(obj, ...)
#\method{impacts}{MCMC_sar_g}(obj, ..., tr=NULL, listw=NULL, evalues=NULL, Q=NULL)
#\method{impacts}{MCMC_sem_g}(obj, ..., tr=NULL, listw=NULL, evalues=NULL, Q=NULL)
#\method{impacts}{MCMC_sac_g}(obj, ..., tr=NULL, listw=NULL, evalues=NULL, Q=NULL)
## S3 method for class 'LagImpact'
plot(x, ..., choice="direct", trace=FALSE, density=TRUE)
## S3 method for class 'LagImpact'
print(x, ..., reportQ=NULL)
## S3 method for class 'LagImpact'
summary(object, ..., zstats=FALSE, short=FALSE, reportQ=NULL)
#\method{print}{WXImpact}(x, ...)
#\method{summary}{WXImpact}(object, ..., adjust_k=(attr(object, "type") == "SDEM"))
## S3 method for class 'LagImpact'
HPDinterval(obj, prob = 0.95, ..., choice="direct")
intImpacts(rho, beta, P, n, mu, Sigma, irho, drop2beta, bnames, interval,
 type, tr, R, listw, evalues, tol, empirical, Q, icept, iicept, p, mess=FALSE,
 samples=NULL, zero_fill = NULL, dvars = NULL)

Arguments

obj	A spatial regression object created by lagsarlm or by lmSLX; in HPDinterval.LagImpact, a LagImpact object
...	Arguments passed through to methods in the coda package
tr	A vector of traces of powers of the spatial weights matrix created using trW, for approximate impact measures; if not given, listw must be given for exact measures (for small to moderate spatial weights matrices); the traces must be for the same spatial weights as were used in fitting the spatial regression, and must be row-standardised
listw	If tr is not given, a spatial weights object as created by nb2listw; they must be the same spatial weights as were used in fitting the spatial regression, but do not have to be row-standardised
evalues	vector of eigenvalues of spatial weights matrix for impacts calculations
n	defaults to length(obj$residuals); in the method for gmsar objects it may be used in panel settings to compute the impacts for cross-sectional weights only, suggested by Angela Parenti
R	If given, simulations are used to compute distributions for the impact measures, returned as mcmc objects; the objects are used for convenience but are not output by an MCMC process
useHESS	Use the Hessian approximation (if available) even if the asymptotic coefficient covariance matrix is available; used for comparing methods
tol	Argument passed to mvrnorm: tolerance (relative to largest variance) for numerical lack of positive-definiteness in the coefficient covariance matrix
empirical	Argument passed to mvrnorm (default FALSE): if true, the coefficients and their covariance matrix specify the empirical not population mean and covariance matrix
Q	default NULL, else an integer number of cumulative power series impacts to calculate if tr is given
reportQ	default NULL; if TRUE and Q given as an argument to impacts, report impact components
x, object	LagImpact objects created by impacts methods
zstats	default FALSE, if TRUE, also return z-values and p-values for the impacts based on the simulations
short	default FALSE, if TRUE passed to the print summary method to omit printing of the mcmc summaries
choice	One of three impacts: direct, indirect, or total
trace	Argument passed to plot.mcmc: plot trace plots
density	Argument passed to plot.mcmc: plot density plots
prob	Argument passed to HPDinterval.mcmc: a numeric scalar in the interval (0,1) giving the target probability content of the intervals
adjust_k	default TRUE if SDEM else FALSE, adjust internal OLS SDEM standard errors by dividing by n rather than (n-k) (default changed and bug fixed after 0.7-8; standard errors now ML in SDEM summary and impacts summary and identical - for SLX use FALSE)
rho, beta, P, mu, Sigma, irho, drop2beta, bnames, interval, type, icept, iicept, p, mess, samples, zero_fill, dvars	internal arguments shared inside impacts methods

Details

If called without R being set, the method returns the direct, indirect and total impacts for the variables in the model, for the variables themselves in tha spatial lag model case, for the variables and their spatial lags in the spatial Durbin (mixed) model case. The spatial lag impact measures are computed using eq. 2.46 (LeSage and Pace, 2009, p. 38), either using the exact dense matrix (when listw is given), or traces of powers of the weights matrix (when tr is given). When the traces are created by powering sparse matrices, the exact and the trace methods should give very similar results, unless the number of powers used is very small, or the spatial coefficient is close to its bounds.

If R is given, simulations will be used to create distributions for the impact measures, provided that the fitted model object contains a coefficient covariance matrix. The simulations are made using mvrnorm with the coefficients and their covariance matrix from the fitted model.

The simulations are stored as mcmc objects as defined in the coda package; the objects are used for convenience but are not output by an MCMC process. The simulated values of the coefficients are checked to see that the spatial coefficient remains within its valid interval — draws outside the interval are discarded.

If a model is fitted with the “Durbin=” set to a formula subsetting the explanatory variables, the impacts object returned reports Durbin impacts for variables included in the formula and lag impacts for the other variables.

When Q and tr are given, addition impact component results are provided for each step in the traces of powers of the weights matrix up to and including the Q'th power. This increases computing time because the output object is substantially increased in size in proportion to the size of Q.

The method for gmsar objects is only for those of type SARAR output by gstsls, and assume that the spatial error coefficient is fixed, and thus omitted from the coefficients and covariance matrix used for simulation.

Value

An object of class LagImpact.

If no simulation is carried out, the object returned is a list with:

direct	numeric vector
indirect	numeric vector
total	numeric vector

and a matching Qres list attribute if Q was given.

If simulation is carried out, the object returned is a list with:

res	a list with three components as for the non-simulation case, with a matching Qres list attribute if Q was given
sres	a list with three mcmc matrices, for the direct, indirect and total impacts with a matching Qmcmc list attribute if Q was given

Author(s)

Roger Bivand [email protected]

References

LeSage J and RK Pace (2009) Introduction to Spatial Econometrics. CRC Press, Boca Raton, pp. 33–42, 114–115; LeSage J and MM Fischer (2008) Spatial growth regressions: model specification, estimation and interpretation. Spatial Economic Analysis 3 (3), pp. 275–304.

Roger Bivand, Gianfranco Piras (2015). Comparing Implementations of Estimation Methods for Spatial Econometrics. Journal of Statistical Software, 63(18), 1-36. doi:10.18637/jss.v063.i18.

See Also

trW, lagsarlm, nb2listw, mvrnorm, plot.mcmc, summary.mcmc, HPDinterval

Examples

require("sf", quietly=TRUE)
columbus <- st_read(system.file("shapes/columbus.gpkg", package="spData")[1], quiet=TRUE)
#require("spdep", quietly=TRUE)
col.gal.nb <- spdep::read.gal(system.file("weights/columbus.gal", package="spData")[1])
listw <- spdep::nb2listw(col.gal.nb)
ev <- eigenw(listw)
lobj <- lagsarlm(CRIME ~ INC + HOVAL, columbus, listw,
 control=list(pre_eig=ev))
summary(lobj)
mobj <- lagsarlm(CRIME ~ INC + HOVAL, columbus, listw, Durbin=TRUE,
 control=list(pre_eig=ev))
summary(mobj)
mobj1 <- lagsarlm(CRIME ~ INC + HOVAL, columbus, listw, Durbin= ~ INC,
 control=list(pre_eig=ev))
summary(mobj1)
W <- as(listw, "CsparseMatrix")
trMatc <- trW(W, type="mult")
trMC <- trW(W, type="MC")
set.seed(1)
impacts(lobj, listw=listw)
impacts(lobj, tr=trMatc)
impacts(lobj, tr=trMC)
impacts(lobj, evalues=ev)
library(coda)
lobjIQ5 <- impacts(lobj, tr=trMatc, R=200, Q=5)
summary(lobjIQ5, zstats=TRUE, short=TRUE)
summary(lobjIQ5, zstats=TRUE, short=TRUE, reportQ=TRUE)
impacts(mobj, listw=listw)
impacts(mobj, tr=trMatc)
impacts(mobj, tr=trMC)
impacts(mobj1, tr=trMatc)
impacts(mobj1, listw=listw)
## Not run: 
try(impacts(mobj, evalues=ev), silent=TRUE)

## End(Not run)
summary(impacts(mobj, tr=trMatc, R=200), short=TRUE, zstats=TRUE)
summary(impacts(mobj1, tr=trMatc, R=200), short=TRUE, zstats=TRUE)
xobj <- lmSLX(CRIME ~ INC + HOVAL, columbus, listw)
summary(impacts(xobj))
eobj <- errorsarlm(CRIME ~ INC + HOVAL, columbus, listw, etype="emixed")
summary(impacts(eobj), adjust_k=TRUE)
## Not run: 
mobj1 <- lagsarlm(CRIME ~ INC + HOVAL, columbus, listw, type="mixed", 
method="Matrix", control=list(fdHess=TRUE))
summary(mobj1)
set.seed(1)
summary(impacts(mobj1, tr=trMatc, R=1000), zstats=TRUE, short=TRUE)
summary(impacts(mobj, tr=trMatc, R=1000), zstats=TRUE, short=TRUE)
mobj2 <- lagsarlm(CRIME ~ INC + HOVAL, columbus, listw, type="mixed", 
method="Matrix", control=list(fdHess=TRUE, optimHess=TRUE))
summary(impacts(mobj2, tr=trMatc, R=1000), zstats=TRUE, short=TRUE)
mobj3 <- lagsarlm(CRIME ~ INC + HOVAL, columbus, listw, type="mixed", 
method="spam", control=list(fdHess=TRUE))
summary(impacts(mobj3, tr=trMatc, R=1000), zstats=TRUE, short=TRUE)

## End(Not run)
## Not run: 
data(boston, package="spData")
Wb <- as(spdep::nb2listw(boston.soi), "CsparseMatrix")
trMatb <- trW(Wb, type="mult")
gp2mMi <- lagsarlm(log(CMEDV) ~ CRIM + ZN + INDUS + CHAS + I(NOX^2) + 
I(RM^2) +  AGE + log(DIS) + log(RAD) + TAX + PTRATIO + B + log(LSTAT), 
data=boston.c, spdep::nb2listw(boston.soi), type="mixed", method="Matrix", 
control=list(fdHess=TRUE), trs=trMatb)
summary(gp2mMi)
summary(impacts(gp2mMi, tr=trMatb, R=1000), zstats=TRUE, short=TRUE)
#data(house, package="spData")
#lw <- spdep::nb2listw(LO_nb)
#form <- formula(log(price) ~ age + I(age^2) + I(age^3) + log(lotsize) +
#   rooms + log(TLA) + beds + syear)
#lobj <- lagsarlm(form, house, lw, method="Matrix",
# control=list(fdHess=TRUE), trs=trMat)
#summary(lobj)
#loobj <- impacts(lobj, tr=trMat, R=1000)
#summary(loobj, zstats=TRUE, short=TRUE)
#lobj1 <- stsls(form, house, lw)
#loobj1 <- impacts(lobj1, tr=trMat, R=1000)
#summary(loobj1, zstats=TRUE, short=TRUE)
#mobj <- lagsarlm(form, house, lw, type="mixed",
# method="Matrix", control=list(fdHess=TRUE), trs=trMat)
#summary(mobj)
#moobj <- impacts(mobj, tr=trMat, R=1000)
#summary(moobj, zstats=TRUE, short=TRUE)

## End(Not run)

---

Compute SAR generating operator

Description

Computes the matrix used for generating simultaneous autoregressive random variables, for a given value of rho, a neighbours list object or a matrix, a chosen coding scheme style, and optionally a list of general weights corresponding to neighbours.

Usage

invIrM(neighbours, rho, glist=NULL, style="W", method="solve",
 feasible=NULL)
invIrW(x, rho, method="solve", feasible=NULL)

Arguments

neighbours	an object of class nb
rho	autoregressive parameter
glist	list of general weights corresponding to neighbours
style	style can take values W, B, C, and S
method	default solve, can also take value chol
feasible	if NULL, the given value of rho is checked to see if it lies within its feasible range, if TRUE, the test is not conducted
x	either a listw object from for example nb2listw or a square spatial weights matrix, optionally a sparse matrix

Details

The invIrW function generates the full weights matrix V, checks that rho lies in its feasible range between 1/min(eigen(V)) and 1/max(eigen(V)), and returns the nxn inverted matrix

$(I - \rho V)^{-1}$

. With method=“chol” (only for a listw object), Cholesky decomposition is used, thanks to contributed code by Markus Reder and Werner Mueller.

Note that, in some situations in simulation, it may matter that the random vector from rnorm or similar will not be exactly N(0, 1), and it will also contain random amounts of spatial autocorrelection itself, which will mix with the spatial autocorrelection injected by the process operator

$(I - \rho V)^{-1}$

. In addition, it will not follow the stipulated distribution exactly either, so that several steps may be needed to scale the random vector, to remove artefacts coming from its deviance from distributional parameters, and to remove random spatial autocorrelation - see the examples below. Thanks to Rune Østergaard Pedersen for bring up this question.

The powerWeights function uses power series summation to cumulate the product

$(I - \rho V)^{-1} \%*\% X$

from

$(I + \rho V + (\rho V)^2 + \dots) \%*\% X$

, which can be done by storing only sparse V and several matrices of the same dimensions as X. This makes it possible to handle larger spatial weights matrices, but is sensitive to the power weights order and the tolerance arguments when the spatial coefficient is close to its bounds, leading to incorrect estimates of the implied inverse matrix.

Value

An nxn matrix with a "call" attribute; the powerWeights function returns a matrix of the same dimensions as X which has been multipled by the power series equivalent of the dense matrix

$(I - \rho V)^{-1}$

.

Note

Before version 0.6-10, powerWeights only worked correctly for positive rho, with differences from true values increasing as rho approached -1, and exploding between -1 and the true negative bound.

Author(s)

Roger Bivand [email protected]

References

Tiefelsdorf, M., Griffith, D. A., Boots, B. 1999 A variance-stabilizing coding scheme for spatial link matrices, Environment and Planning A, 31, pp. 165-180; Tiefelsdorf, M. 2000 Modelling spatial processes, Lecture notes in earth sciences, Springer, p. 76; Haining, R. 1990 Spatial data analysis in the social and environmental sciences, Cambridge University Press, p. 117; Cliff, A. D., Ord, J. K. 1981 Spatial processes, Pion, p. 152; Reder, M. and Mueller, W. (2007) An Improvement of the invIrM Routine of the Geostatistical R-package spdep by Cholesky Inversion, Statistical Projects, LV No: 238.205, SS 2006, Department of Applied Statistics, Johannes Kepler University, Linz

See Also

nb2listw

Examples

library(spdep)
nb7rt <- cell2nb(7, 7, torus=TRUE)
lw <- nb2listw(nb7rt, style="W")
set.seed(1)
x <- matrix(sample(rnorm(500*length(nb7rt))), nrow=length(nb7rt))
if (requireNamespace("spatialreg", quietly=TRUE)) {
# Only needed in some simulation settings where the input and
# output distributions must agree in all but autocorrelation
if (FALSE) {
e <- spatialreg::eigenw(lw)
x <- apply(x, 2, scale)
st <- apply(x, 2, function(x) shapiro.test(x)$p.value)
x <- x[, (st > 0.2 & st < 0.8)]
x <- apply(x, 2, function(v) spatialreg::residuals.spautolm(
 spatialreg::spautolm(v ~ 1, listw=lw, method="eigen",
 control=list(pre_eig=e, fdHess=FALSE))))
x <- apply(x, 2, scale)
}
res0 <- apply(invIrM(nb7rt, rho=0.0, method="chol",
 feasible=TRUE) %*% x, 2, function(x) var(x)/length(x))
res2 <- apply(invIrM(nb7rt, rho=0.2, method="chol",
 feasible=TRUE) %*% x, 2, function(x) var(x)/length(x))
res4 <- apply(invIrM(nb7rt, rho=0.4, method="chol",
 feasible=TRUE) %*% x, 2, function(x) var(x)/length(x))
res6 <- apply(invIrM(nb7rt, rho=0.6, method="chol",
 feasible=TRUE) %*% x, 2, function(x) var(x)/length(x))
res8 <- apply(invIrM(nb7rt, rho=0.8, method="chol",
 feasible=TRUE) %*% x, 2, function(x) var(x)/length(x))
res9 <- apply(invIrM(nb7rt, rho=0.9, method="chol",
 feasible=TRUE) %*% x, 2, function(x) var(x)/length(x))
plot(density(res9), col="red", xlim=c(-0.01, max(density(res9)$x)),
  ylim=range(density(res0)$y),
  xlab="estimated variance of the mean",
  main=expression(paste("Effects of spatial autocorrelation for different ",
    rho, " values")))
lines(density(res0), col="black")
lines(density(res2), col="brown")
lines(density(res4), col="green")
lines(density(res6), col="orange")
lines(density(res8), col="pink")
legend(c(-0.02, 0.01), c(7, 25),
 legend=c("0.0", "0.2", "0.4", "0.6", "0.8", "0.9"),
 col=c("black", "brown", "green", "orange", "pink", "red"), lty=1, bty="n")
}
## Not run: 
x <- matrix(rnorm(length(nb7rt)), ncol=1)
system.time(e <- invIrM(nb7rt, rho=0.9, method="chol", feasible=TRUE) %*% x)
system.time(e <- invIrM(nb7rt, rho=0.9, method="chol", feasible=NULL) %*% x)
system.time(e <- invIrM(nb7rt, rho=0.9, method="solve", feasible=TRUE) %*% x)
system.time(e <- invIrM(nb7rt, rho=0.9, method="solve", feasible=NULL) %*% x)

## End(Not run)

---

Matrix exponential spatial lag model

Description

The function fits a matrix exponential spatial lag model, using optim to find the value of alpha, the spatial coefficient.

Usage

lagmess(formula, data = list(), listw, zero.policy = NULL, na.action = na.fail,
 q = 10, start = -2.5, control=list(), method="BFGS", verbose=NULL,
 use_expm=FALSE)

Arguments

formula	a symbolic description of the model to be fit. The details of model specification are given for lm()
data	an optional data frame containing the variables in the model. By default the variables are taken from the environment which the function is called.
listw	a listw object created for example by spdep::nb2listw()
zero.policy	default NULL, use global option value; if TRUE assign zero to the lagged value of zones without neighbours, if FALSE assign NA - causing lagmess() to terminate with an error
na.action	a function (default options("na.action")), can also be na.omit or na.exclude with consequences for residuals and fitted values - in these cases the weights list will be subsetted to remove NAs in the data. It may be necessary to set zero.policy to TRUE because this subsetting may create no-neighbour observations. Note that only weights lists created without using the glist argument to nb2listw may be subsetted.
q	default 10; number of powers of the spatial weights to use
start	starting value for numerical optimization, should be a small negative number
control	control parameters passed to optim
method	default BFGS, method passed to optim
verbose	default NULL, use global option value; if TRUE report function values during optimization
use_expm	default FALSE; if TRUE use expm::expAtv instead of a truncated power series of W

Details

The underlying spatial lag model:

$y = \rho W y + X \beta + \varepsilon$

where $\rho$ is the spatial parameter may be fitted by maximum likelihood. In that case, the log likelihood function includes the logarithm of cumbersome Jacobian term $|I - \rho W|$. If we rewrite the model as:

$S y = X \beta + \varepsilon$

we see that in the ML case $S y = (I - \rho W) y$. If W is row-stochastic, S may be expressed as a linear combination of row-stochastic matrices. By pre-computing the matrix $[y, Wy, W^2y, ..., W^{q-1}y]$, the term $S y (\alpha)$ can readily be found by numerical optimization using the matrix exponential approach. $\alpha$ and $\rho$ are related as $\rho = 1 - \exp{\alpha}$, conditional on the number of matrix power terms taken q.

Value

The function returns an object of class Lagmess with components:

lmobj	the lm object returned after fitting alpha
alpha	the spatial coefficient
alphase	the standard error of the spatial coefficient using the numerical Hessian
rho	the value of rho implied by alpha
bestmess	the object returned by optim
q	the number of powers of the spatial weights used
start	the starting value for numerical optimization used
na.action	(possibly) named vector of excluded or omitted observations if non-default na.action argument used
nullLL	the log likelihood of the aspatial model for the same data

Author(s)

Roger Bivand [email protected] and Eric Blankmeyer

References

J. P. LeSage and R. K. Pace (2007) A matrix exponential specification. Journal of Econometrics, 140, 190-214; J. P. LeSage and R. K. Pace (2009) Introduction to Spatial Econometrics. CRC Press, Chapter 9.

See Also

lagsarlm, optim

Examples

#require(spdep, quietly=TRUE)
data(baltimore, package="spData")
baltimore$AGE <- ifelse(baltimore$AGE < 1, 1, baltimore$AGE)
lw <- spdep::nb2listw(spdep::knn2nb(spdep::knearneigh(cbind(baltimore$X, baltimore$Y), k=7)))
obj1 <- lm(log(PRICE) ~ PATIO + log(AGE) + log(SQFT),
 data=baltimore)
spdep::lm.morantest(obj1, lw)
spdep::lm.LMtests(obj1, lw, test="all")
system.time(obj2 <- lagmess(log(PRICE) ~ PATIO + log(AGE) + log(SQFT), data=baltimore, listw=lw))
(x <- summary(obj2))
coef(x)
has_expm <- require("expm", quietly=TRUE)
if (has_expm) {
system.time(
obj2a <- lagmess(log(PRICE) ~ PATIO + log(AGE) + log(SQFT), data=baltimore, listw=lw, use_expm=TRUE)
)
summary(obj2a)
}
obj3 <- lagsarlm(log(PRICE) ~ PATIO + log(AGE) + log(SQFT), data=baltimore, listw=lw)
summary(obj3)

data(boston, package="spData")
lw <- spdep::nb2listw(boston.soi)
gp2 <- lagsarlm(log(CMEDV) ~ CRIM + ZN + INDUS + CHAS + I(NOX^2) + I(RM^2)
 +  AGE + log(DIS) + log(RAD) + TAX + PTRATIO + B + log(LSTAT),
 data=boston.c, lw, method="Matrix")
summary(gp2)
gp2a <- lagmess(CMEDV ~ CRIM + ZN + INDUS + CHAS + I(NOX^2) + I(RM^2)
 +  AGE + log(DIS) + log(RAD) + TAX + PTRATIO + B + log(LSTAT),
 data=boston.c, lw)
summary(gp2a)

---

Find extreme eigenvalues of binary symmetric spatial weights

Description

The functions find extreme eigenvalues of binary symmetric spatial weights, when these form planar graphs; general weights are not permiited. l_max finds the largest eigenvalue using Rayleigh quotient methods of any “listw” object. lextrB first calls l_max, and uses its output to find the smallest eigenvalue in addition for binary symmetric spatial weights. lextrW extends these to find the smallest eigenvalue for intrinsically symmetric row-standardized binary weights matrices (transformed to symmetric through similarity internally). lextrS does the same for variance-stabilized (“S” style) intrinsically symmetric binary weights matrices (transformed to symmetric through similarity internally).

Usage

lextrB(lw, zero.policy = TRUE, control = list())
lextrW(lw, zero.policy=TRUE, control=list())
lextrS(lw, zero.policy=TRUE, control=list())
l_max(lw, zero.policy=TRUE, control=list())

Arguments

lw	a binary symmetric listw object from, for example, nb2listw with style “B” for lextrB, style “W” for lextrW and style “S” for lextrS; for l_max, the object may be asymmetric and does not have to be binary
zero.policy	default NULL, use global option value; if TRUE assign zero to the lagged value of zones without neighbours, if FALSE assign NA
control	a list of control arguments

Value

The functions return approximations to the extreme eigenvalues with the eigenvectors returned as attributes of this object.

Control arguments

trace

report values in while loops, default NULL assuming FALSE; logical

tol

tolerance for breaking while loops, default .Machine$double.eps^(1/2); numeric

maxiter

maximum number of iterations in while loops, default 6 * (length(lw$neighbours) - 2; integer

useC

use C code, default TRUE, logical (not in l_max)

Note

It may be necessary to modify control arguments if warnings about lack of convergence are seen.

Author(s)

Roger Bivand, Yongwan Chun, Daniel Griffith

References

Griffith, D. A. (2004). Extreme eigenfunctions of adjacency matrices for planar graphs employed in spatial analyses. Linear Algebra and its Applications, 388:201–219.

Examples

data(boston, package="spData")
#require(spdep, quietly=TRUE)
ab.listb <- spdep::nb2listw(boston.soi, style="B")
er <- range(eigenw(ab.listb))
er
res_1 <- lextrB(ab.listb)
c(res_1)
run <- FALSE
if (require("RSpectra", quietly=TRUE)) run <- TRUE
if (run) {
B <- as(ab.listb, "CsparseMatrix")
eigs(B, k=1, which="SR")$values
}
if (run) {
eigs(B, k=1, which="LR")$values
}
k5 <- spdep::knn2nb(spdep::knearneigh(boston.utm, k=5))
c(l_max(spdep::nb2listw(k5, style="B")))
max(Re(eigenw(spdep::nb2listw(k5, style="B"))))
c(l_max(spdep::nb2listw(k5, style="C")))
max(Re(eigenw(spdep::nb2listw(k5, style="C"))))
ab.listw <- spdep::nb2listw(boston.soi, style="W")
er <- range(eigenw(similar.listw(ab.listw)))
er
res_1 <- lextrW(ab.listw)
c(res_1)
if (run) {
B <- as(similar.listw(ab.listw), "CsparseMatrix")
eigs(B, k=1, which="SR")$values
}
if (run) {
eigs(B, k=1, which="LR")$values
}
## Not run: 
ab.listw <- spdep::nb2listw(boston.soi, style="S")
er <- range(eigenw(similar.listw(ab.listw)))
er
res_1 <- lextrS(ab.listw)
c(res_1)

## End(Not run)
if (run) {
B <- as(similar.listw(ab.listw), "CsparseMatrix")
eigs(B, k=1, which="SR")$values
}
if (run) {
eigs(B, k=1, which="LR")$values
}

---

Spatial Durbin linear (SLX, spatially lagged X) model

Description

lmSLX fits an lm model augmented with the spatially lagged RHS variables, including the lagged intercept when the spatial weights are not row-standardised. create_WX creates spatially lagged RHS variables, and is exposed for use in model fitting functions.

Usage

lmSLX(formula, data = list(), listw, na.action, weights=NULL, Durbin=TRUE,
 zero.policy=NULL, return_impacts=TRUE)
## S3 method for class 'SlX'
print(x, digits = max(3L, getOption("digits") - 3L), ...)
## S3 method for class 'SlX'
summary(object, correlation = FALSE, symbolic.cor = FALSE, ...)
## S3 method for class 'summary.SlX'
print(x, digits = max(3L, getOption("digits") - 3L),
 symbolic.cor = x$symbolic.cor, signif.stars = getOption("show.signif.stars"), ...)
## S3 method for class 'SlX'
impacts(obj, ...)
## S3 method for class 'WXimpact'
print(x, ...)
## S3 method for class 'WXimpact'
summary(object, ..., adjust_k=(attr(object, "type") == "SDEM"))
## S3 method for class 'SlX'
predict(object, newdata, listw, zero.policy=NULL, ...)
create_WX(x, listw, zero.policy=NULL, prefix="")

Arguments

formula	a symbolic description of the model to be fit. The details of model specification are given for lm()
data	an optional data frame containing the variables in the model. By default the variables are taken from the environment which the function is called.
listw	a listw object created for example by nb2listw
na.action	a function (default options("na.action")), can also be na.omit or na.exclude with consequences for residuals and fitted values - in these cases the spatial weights list will be subsetted to remove NAs in the data. It may be necessary to set zero.policy to TRUE because this subsetting may create no-neighbour observations. Note that only weights lists created without using the glist argument to nb2listw may be subsetted.
weights	an optional vector of weights to be used in the fitting process. Non-NULL weights can be used to indicate that different observations have different variances (with the values in weights being inversely proportional to the variances); or equivalently, when the elements of weights are positive integers w_i, that each response y_i is the mean of w_i unit-weight observations (including the case that there are w_i observations equal to y_i and the data have been summarized) - lm
Durbin	default TRUE for lmSLX (Durbin model including WX); if TRUE, full spatial Durbin model; if a formula object, the subset of explanatory variables to lag
zero.policy	default NULL, use global option value; if TRUE assign zero to the lagged value of zones without neighbours, if FALSE assign NA
return_impacts	default TRUE; may be set FALSE to avoid problems calculating impacts with aliased variables
digits	the number of significant digits to use when printing
correlation	logical; if TRUE, the correlation matrix of the estimated parameters is returned and printed
symbolic.cor	logical. If TRUE, print the correlations in a symbolic form (see 'symnum') rather than as numbers
signif.stars	logical. If TRUE, 'significance stars' are printed for each coefficient
obj	A spatial regression object created by lmSLX
...	Arguments passed through
prefix	default empty string, may be “lag” in some cases
x, object	model matrix to be lagged; lagImpact objects created by impacts methods
adjust_k	default TRUE if SDEM else FALSE, adjust internal OLS SDEM standard errors by dividing by n rather than (n-k) (default changed and bug fixed after 0.7-8; standard errors now ML in SDEM summary and impacts summary and identical - for SLX use FALSE)
newdata	data frame in which to predict — if NULL, predictions are for the data on which the model was fitted. Should have row names corresponding to region.id. If row names are exactly the same than the ones used for training, it uses in-sample predictors for forecast.

Value

The lmSLX function returns an “lm” object with a “mixedImps” list of three impact matrixes (impacts and standard errors) for direct, indirect and total impacts; total impacts calculated using a simplified local copy of the estimable function from the gmodels package.

Author(s)

Roger Bivand [email protected]

See Also

lm

Examples

data(oldcol, package="spdep")
lw <- spdep::nb2listw(COL.nb, style="W")
COL.SLX <- lmSLX(CRIME ~ INC + HOVAL, data=COL.OLD, listw=lw)
summary(COL.SLX)
summary(impacts(COL.SLX))
COL.SLX <- lmSLX(CRIME ~ INC + HOVAL + I(HOVAL^2), data=COL.OLD, listw=lw, Durbin=TRUE)
summary(impacts(COL.SLX))
summary(COL.SLX)
COL.SLX <- lmSLX(CRIME ~ INC + HOVAL + I(HOVAL^2), data=COL.OLD, listw=lw, Durbin=~INC)
summary(impacts(COL.SLX))
summary(COL.SLX)
COL.SLX <- lmSLX(CRIME ~ INC, data=COL.OLD, listw=lw)
summary(COL.SLX)
summary(impacts(COL.SLX))
## Not run: 
crds <- cbind(COL.OLD$X, COL.OLD$Y)
mdist <- sqrt(sum(diff(apply(crds, 2, range))^2))
dnb <- spdep::dnearneigh(crds, 0, mdist)
dists <- spdep::nbdists(dnb, crds)
f <- function(x, form, data, dnb, dists, verbose) {
  glst <- lapply(dists, function(d) 1/(d^x))
  lw <- spdep::nb2listw(dnb, glist=glst, style="B")
  res <- logLik(lmSLX(form=form, data=data, listw=lw))
  if (verbose) cat("power:", x, "logLik:", res, "\n")
  res
}
opt <- optimize(f, interval=c(0.1, 4), form=CRIME ~ INC + HOVAL,
 data=COL.OLD, dnb=dnb, dists=dists, verbose=TRUE, maximum=TRUE)
glst <- lapply(dists, function(d) 1/(d^opt$maximum))
lw <- spdep::nb2listw(dnb, glist=glst, style="B")
SLX <- lmSLX(CRIME ~ INC + HOVAL, data=COL.OLD, listw=lw)
summary(SLX)
summary(impacts(SLX))

## End(Not run)
COL.SLX <- lmSLX(CRIME ~ INC + HOVAL, data=COL.OLD, listw=lw)
pslx0 <- predict(COL.SLX)
pslx1 <- predict(COL.SLX, newdata=COL.OLD, listw=lw)
all.equal(pslx0, pslx1)
COL.OLD1 <- COL.OLD
COL.OLD1$INC <- COL.OLD1$INC + 1
pslx2 <- predict(COL.SLX, newdata=COL.OLD1, listw=lw)
sum(coef(COL.SLX)[c(2,4)])
mean(pslx2-pslx1)

---

Likelihood ratio test

Description

The LR.Sarlm() function provides a likelihood ratio test for objects for which a logLik() function exists for their class, or for objects of class logLik. LR1.Sarlm() and Wald1.Sarlm() are used internally in summary.Sarlm(), but may be accessed directly; they report the values respectively of LR and Wald tests for the absence of spatial dependence in spatial lag or error models. The spatial Hausman test is available for models fitted with errorSarlm and GMerrorsar.

Usage

LR.Sarlm(x, y)
## S3 method for class 'Sarlm'
logLik(object, ...)
LR1.Sarlm(object)
Wald1.Sarlm(object)
## S3 method for class 'Sarlm'
Hausman.test(object, ..., tol=NULL)
## S3 method for class 'Sarlm'
anova(object, ...)
bptest.Sarlm(object, varformula=NULL, studentize = TRUE, data=list())
## S3 method for class 'Sarlm'
impacts(obj, ..., tr, R = NULL, listw = NULL, evalues=NULL,
 useHESS = NULL, tol = 1e-06, empirical = FALSE, Q=NULL)

Arguments

x	a logLik object or an object for which a logLik() function exists
y	a logLik object or an object for which a logLik() function exists
object, obj	a Sarlm object
...	further arguments passed to or from other methods
varformula	a formula describing only the potential explanatory variables for the variance (no dependent variable needed). By default the same explanatory variables are taken as in the main regression model
studentize	logical. If set to TRUE Koenker's studentized version of the test statistic will be used.
data	an optional data frame containing the variables in the varformula
tr	A vector of traces of powers of the spatial weights matrix created using trW, for approximate impact measures; if not given, listw must be given for exact measures (for small to moderate spatial weights matrices); the traces must be for the same spatial weights as were used in fitting the spatial regression, and must be row-standardised
listw	If tr is not given, a spatial weights object as created by nb2listw; they must be the same spatial weights as were used in fitting the spatial regression, but do not have to be row-standardised
evalues	vector of eigenvalues of spatial weights matrix for impacts calculations
R	If given, simulations are used to compute distributions for the impact measures, returned as mcmc objects; the objects are used for convenience but are not output by an MCMC process
useHESS	Use the Hessian approximation (if available) even if the asymptotic coefficient covariance matrix is available; used for comparing methods
tol	Argument passed to mvrnorm and solve: tolerance (relative to largest variance) for numerical lack of positive-definiteness in the coefficient covariance matrix
empirical	Argument passed to mvrnorm (default FALSE): if true, the coefficients and their covariance matrix specify the empirical not population mean and covariance matrix
Q	default NULL, else an integer number of cumulative power series impacts to calculate if tr is given

Value

The tests return objects of class htest with:

statistic	value of statistic
parameter	degrees of freedom
p.value	Probability value
estimate	varies with test
method	description of test method

logLik.Sarlm() returns an object of class logLik LR1.Sarlm, Hausman.Sarlm and Wald1.Sarlm returm objects of class htest

Note

The numbers of degrees of freedom returned by logLik.Sarlm() include nuisance parameters, that is the number of regression coefficients, plus sigma, plus spatial parameter esitmate(s).

Author(s)

Roger Bivand [email protected], bptest: Torsten Hothorn and Achim Zeileis, modified by Roger Bivand

References

LeSage J and RK Pace (2009) Introduction to Spatial Econometrics. CRC Press, Boca Raton, pp. 61–63; Pace RK and LeSage J (2008) A spatial Hausman test. Economics Letters 101, 282–284. T.S. Breusch & A.R. Pagan (1979), A Simple Test for Heteroscedasticity and Random Coefficient Variation. Econometrica 47, 1287–1294

W. Krämer & H. Sonnberger (1986), The Linear Regression Model under Test. Heidelberg: Physica.

L. Anselin (1988) Spatial econometrics: methods and models. Dordrecht: Kluwer, pp. 121–122.

See Also

logLik.lm, anova.Sarlm

Examples

require("sf", quietly=TRUE)
columbus <- st_read(system.file("shapes/columbus.gpkg", package="spData")[1], quiet=TRUE)
#require("spdep", quietly=TRUE)
col.gal.nb <- spdep::read.gal(system.file("weights/columbus.gal", package="spData")[1])
lm.mod <- lm(CRIME ~ HOVAL + INC, data=columbus)
lag <- lagsarlm(CRIME ~ HOVAL + INC, data=columbus, spdep::nb2listw(col.gal.nb))
mixed <- lagsarlm(CRIME ~ HOVAL + INC, data=columbus, spdep::nb2listw(col.gal.nb), type="mixed")
error <- errorsarlm(CRIME ~ HOVAL + INC, data=columbus, spdep::nb2listw(col.gal.nb))
Hausman.test(error)
LR.Sarlm(mixed, error)
anova(lag, lm.mod)
anova(lag, error, mixed)
AIC(lag, error, mixed)
bptest.Sarlm(error)
bptest.Sarlm(error, studentize=FALSE)

---

MCMC sample from fitted spatial regression

Description

The MCMCsamp method uses rwmetrop, a random walk Metropolis algorithm, from LearnBayes to make MCMC samples from fitted maximum likelihood spatial regression models.

Usage

MCMCsamp(object, mcmc = 1L, verbose = NULL, ...)
## S3 method for class 'Spautolm'
MCMCsamp(object, mcmc = 1L, verbose = NULL, ...,
 burnin = 0L, scale=1, listw, control = list())
## S3 method for class 'Sarlm'
MCMCsamp(object, mcmc = 1L, verbose = NULL, ...,
    burnin=0L, scale=1, listw, listw2=NULL, control=list())

Arguments

object	A spatial regression model object fitted by maximum likelihood with spautolm
mcmc	The number of MCMC iterations after burnin
verbose	default NULL, use global option value; if TRUE, reports progress
...	Arguments passed through
burnin	The number of burn-in iterations for the sampler
scale	a positive scale parameter
listw, listw2	listw objects created for example by nb2listw; should be the same object(s) used for fitting the model
control	list of extra control arguments - see spautolm

Value

An object of class “mcmc” suited to coda, with attributes: “accept” acceptance rate; “type” input ML fitted model type “SAR”, “CAR”, “SMA”, “lag”, “mixed”, “error”, “sac”, “sacmixed”; “timings” run times

Note

If the acceptance rate is below 0.05, a warning will be issued; consider increasing mcmc.

Author(s)

Roger Bivand [email protected]

References

Jim Albert (2007) Bayesian Computation with R, Springer, New York, pp. 104-105.

See Also

rwmetrop, spautolm, lagsarlm, errorsarlm, sacsarlm

Examples

require("sf", quietly=TRUE)
nydata <- st_read(system.file("shapes/NY8_bna_utm18.gpkg", package="spData")[1], quiet=TRUE)
suppressMessages(nyadjmat <- as.matrix(foreign::read.dbf(system.file(
 "misc/nyadjwts.dbf", package="spData")[1])[-1]))
suppressMessages(ID <- as.character(names(foreign::read.dbf(system.file(
 "misc/nyadjwts.dbf", package="spData")[1]))[-1]))
identical(substring(ID, 2, 10), substring(as.character(nydata$AREAKEY), 2, 10))
#require("spdep", quietly=TRUE)
listw_NY <- spdep::mat2listw(nyadjmat, as.character(nydata$AREAKEY), style="B")
esar1f <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, family="SAR", method="eigen")
summary(esar1f)
res <- MCMCsamp(esar1f, mcmc=1000, burnin=200, listw=listw_NY)
summary(res)
## Not run: 
esar1fw <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, weights=POP8, family="SAR", method="eigen")
summary(esar1fw)
res <- MCMCsamp(esar1fw, mcmc=5000, burnin=500, listw=listw_NY)
summary(res)
ecar1f <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, family="CAR", method="eigen")
summary(ecar1f)
res <- MCMCsamp(ecar1f, mcmc=5000, burnin=500, listw=listw_NY)
summary(res)
esar1fw <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, weights=POP8, family="SAR", method="eigen")
summary(esar1fw)
res <- MCMCsamp(esar1fw, mcmc=5000, burnin=500, listw=listw_NY)
summary(res)
ecar1fw <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, weights=POP8, family="CAR", method="eigen")
summary(ecar1fw)
res <- MCMCsamp(ecar1fw, mcmc=5000, burnin=500, listw=listw_NY)
summary(res)

## End(Not run)
esar0 <- errorsarlm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY)
summary(esar0)
res <- MCMCsamp(esar0, mcmc=1000, burnin=200, listw=listw_NY)
summary(res)
## Not run: 
esar0w <- errorsarlm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, weights=POP8)
summary(esar0)
res <- MCMCsamp(esar0w, mcmc=5000, burnin=500, listw=listw_NY)
summary(res)
esar1 <- errorsarlm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, etype="emixed")
summary(esar1)
res <- MCMCsamp(esar1, mcmc=5000, burnin=500, listw=listw_NY)
summary(res)
lsar0 <- lagsarlm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY)
summary(lsar0)
res <- MCMCsamp(lsar0, mcmc=5000, burnin=500, listw=listw_NY)
summary(res)
lsar1 <- lagsarlm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, type="mixed")
summary(lsar1)
res <- MCMCsamp(lsar1, mcmc=5000, burnin=500, listw=listw_NY)
summary(res)
ssar0 <- sacsarlm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY)
summary(ssar0)
res <- MCMCsamp(ssar0, mcmc=5000, burnin=500, listw=listw_NY)
summary(res)
ssar1 <- sacsarlm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, type="sacmixed")
summary(ssar1)
res <- MCMCsamp(ssar1, mcmc=5000, burnin=500, listw=listw_NY)
summary(res)

## End(Not run)

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