Function taking family and weights arguments for spatial autoregression model estimation by Maximum Likelihood, using dense matrix methods, not suited to large data sets with thousands of observations. With one of the sparse matrix methods, larger numbers of observations can be handled, but the interval= argument should be set. The implementation is GLS using the single spatial coefficient value, here termed lambda, found by line search using optimize to maximise the log likelihood.

spautolm(formula, data = list(), listw, weights,
 na.action, family = "SAR", method="eigen", verbose = NULL, trs=NULL,
 interval=NULL, zero.policy = NULL, tol.solve=.Machine$double.eps,
 llprof=NULL, control=list())
# S3 method for Spautolm
summary(object, correlation = FALSE, adj.se=FALSE,
 Nagelkerke=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 nb2listw

weights

an optional vector of weights to be used in the fitting process

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. Note that only weights lists created without using the glist argument to nb2listw may be subsetted.

family

character string: either "SAR" or "CAR" for simultaneous or conditional autoregressions; "SMA" for spatial moving average added thanks to Jielai Ma - "SMA" is only implemented for method="eigen" because it necessarily involves dense matrices

method

character string: default "eigen" for use of dense matrices, "Matrix_J" for sparse matrices (restricted to spatial weights symmetric or similar to symmetric) using methods in the Matrix package; “Matrix” provides updating Cholesky decomposition methods. Values of method may also include "LU", which provides an alternative sparse matrix decomposition approach, and the "Chebyshev" and Monte Carlo "MC" approximate log-determinant methods.

verbose

default NULL, use global option value; if TRUE, reports function values during optimization.

trs

default NULL, if given, a vector of powered spatial weights matrix traces output by trW; when given, used in some Jacobian methods

interval

search interval for autoregressive parameter when not using method="eigen"; default is c(-1,0.999), optimize will reset NA/NaN to a bound and gives a warning when the interval is poorly set; method="Matrix" will attempt to search for an appropriate interval, if find\_interval=TRUE (fails on some platforms)

zero.policy

default NULL, use global option value; Include list of no-neighbour observations in output if TRUE --- otherwise zero.policy is handled within the listw argument

tol.solve

the tolerance for detecting linear dependencies in the columns of matrices to be inverted - passed to solve() (default=double precision machine tolerance). Errors in solve() may constitute indications of poorly scaled variables: if the variables have scales differing much from the autoregressive coefficient, the values in this matrix may be very different in scale, and inverting such a matrix is analytically possible by definition, but numerically unstable; rescaling the RHS variables alleviates this better than setting tol.solve to a very small value

llprof

default NULL, can either be an integer, to divide the feasible range into llprof points, or a sequence of spatial coefficient values, at which to evaluate the likelihood function

control

list of extra control arguments - see section below

object

Spautolm object from spautolm

correlation

logical; if 'TRUE', the correlation matrix of the estimated parameters is returned and printed (default=FALSE)

adj.se

if TRUE, adjust the coefficient standard errors for the number of fitted coefficients

Nagelkerke

if TRUE, the Nagelkerke pseudo R-squared is reported

...

further arguments passed to or from other methods

Details

This implementation is based on lm.gls and errorsarlm. In particular, the function does not (yet) prevent asymmetric spatial weights being used with "CAR" family models. It appears that both numerical issues (convergence in particular) and uncertainties about the exact spatial weights matrix used make it difficult to reproduce Cressie and Chan's 1989 results, also given in Cressie 1993.

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.

Control arguments

tol.opt:

the desired accuracy of the optimization - passed to optimize() (default=.Machine$double.eps^(2/3))

fdHess:

default NULL, then set to (method != "eigen") internally; use fdHess to compute an approximate Hessian using finite differences when using sparse matrix methods; used to make a coefficient covariance matrix when the number of observations is large; may be turned off to save resources if need be

optimHess:

default FALSE, use fdHess from nlme, if TRUE, use optim to calculate Hessian at optimum

optimHessMethod:

default “optimHess”, may be “nlm” or one of the optim methods

Imult:

default 2; used for preparing the Cholesky decompositions for updating in the Jacobian function

super:

if NULL (default), set to FALSE to use a simplicial decomposition for the sparse Cholesky decomposition and method “Matrix_J”, set to as.logical(NA) for method “Matrix”, if TRUE, use a supernodal decomposition

cheb_q:

default 5; highest power of the approximating polynomial for the Chebyshev approximation

MC_p:

default 16; number of random variates

MC_m:

default 30; number of products of random variates matrix and spatial weights matrix

type

default “MC”, used with method “moments”; alternatives “mult” and “moments”, for use if trs is missing, trW

correct

default TRUE, used with method “moments” to compute the Smirnov/Anselin correction term

trunc

default TRUE, used with method “moments” to truncate the Smirnov/Anselin correction term

SE_method

default “LU”, may be “MC”

nrho

default 200, as in SE toolbox; the size of the first stage lndet grid; it may be reduced to for example 40

interpn

default 2000, as in SE toolbox; the size of the second stage lndet grid

small_asy

default TRUE; if the method is not “eigen”, use asymmetric covariances rather than numerical Hessian ones if n <= small

small

default 1500; threshold number of observations for asymmetric covariances when the method is not “eigen”

SElndet

default NULL, may be used to pass a pre-computed SE toolbox style matrix of coefficients and their lndet values to the "SE_classic" and "SE_whichMin" methods

LU_order

default FALSE; used in “LU_prepermutate”, note warnings given for lu method

pre_eig

default NULL; may be used to pass a pre-computed vector of eigenvalues

Value

A list object of class Spautolm:

fit

a list, with items:

coefficients

ML coefficient estimates

SSE

ML sum of squared errors

s2

ML residual variance

imat

ML coefficient covariance matrix (before multiplying by s2)

signal\_trend

non-spatial component of fitted.values

signal\_stochastic

spatial component of fitted.values

fitted.values

sum of non-spatial and spatial components of fitted.values

residuals

difference between observed and fitted values

lambda

ML autoregressive coefficient

LL

log likelihood for fitted model

LL0

log likelihood for model with lambda=0

call

the call used to create this object

parameters

number of parameters estimated

aliased

if not NULL, details of aliased variables

method

Jacobian method chosen

family

family chosen

zero.policy

zero.policy used

weights

case weights used

interval

the line search interval used

timings

processing timings

na.action

(possibly) named vector of excluded or omitted observations if non-default na.action argument used

llprof

if not NULL, a list with components lambda and ll of equal length

lambda.se

Numerical Hessian-based standard error of lambda

fdHess

Numerical Hessian-based variance-covariance matrix

X

covariates used in model fitting

Y

response used in model fitting

weights

weights used in model fitting

References

Cliff, A. D., Ord, J. K. 1981 Spatial processes, Pion; Ord, J. K. 1975 Estimation methods for models of spatial interaction, Journal of the American Statistical Association, 70, 120-126; Waller, L. A., Gotway, C. A. 2004 Applied spatial statistics for public health, Wiley, Hoboken, NJ, 325-380; Cressie, N. A. C. 1993 Statistics for spatial data, Wiley, New York, 548-568; Ripley, B. D. 1981 Spatial statistics, Wiley, New York, 88-95; LeSage J and RK Pace (2009) Introduction to Spatial Econometrics. CRC Press, Boca Raton.

Author

Roger Bivand Roger.Bivand@nhh.no

Note

The standard errors given in Waller and Gotway (2004) are adjusted for the numbers of parameters estimated, and may be reproduced by using the additional argument adj.se=TRUE in the summary method. In addition, the function returns fitted values and residuals as given by Cressie (1993) p. 564.

See also

Examples

require("sf", quietly=TRUE)
nydata <- st_read(system.file("shapes/NY8_bna_utm18.gpkg", package="spData")[1], quiet=TRUE)
if (FALSE) {
lm0 <- lm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata)
summary(lm0)
lm0w <- lm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata, weights=POP8)
summary(lm0w)
}
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$AREAKEY), 2, 10))
#> [1] TRUE
#require("spdep", quietly=TRUE)
nyadjlw <- spdep::mat2listw(nyadjmat, as.character(nydata$AREAKEY))
listw_NY <- spdep::nb2listw(nyadjlw$neighbours, style="B")
eigs <- eigenw(listw_NY)
if (FALSE) {
esar0 <- errorsarlm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY)
summary(esar0)
}
system.time(esar1f <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY, family="SAR", method="eigen",
 control=list(pre_eig=eigs)))
#>    user  system elapsed 
#>   0.291   0.000   0.292 
res <- summary(esar1f)
print(res)
#> 
#> Call: 
#> spautolm(formula = Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data = nydata, 
#>     listw = listw_NY, family = "SAR", method = "eigen", control = list(pre_eig = eigs))
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -1.56754 -0.38239 -0.02643  0.33109  4.01219 
#> 
#> Coefficients: 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) -0.618193   0.176784 -3.4969 0.0004707
#> PEXPOSURE    0.071014   0.042051  1.6888 0.0912635
#> PCTAGE65P    3.754200   0.624722  6.0094 1.862e-09
#> PCTOWNHOME  -0.419890   0.191329 -2.1946 0.0281930
#> 
#> Lambda: 0.040487 LR test value: 5.2438 p-value: 0.022026 
#> Numerical Hessian standard error of lambda: 0.017209 
#> 
#> Log likelihood: -276.1069 
#> ML residual variance (sigma squared): 0.41388, (sigma: 0.64333)
#> Number of observations: 281 
#> Number of parameters estimated: 6 
#> AIC: 564.21
#> 
coef(res)
#>                Estimate Std. Error   z value     Pr(>|z|)
#> (Intercept) -0.61819272 0.17678351 -3.496891 4.707136e-04
#> PEXPOSURE    0.07101384 0.04205063  1.688770 9.126351e-02
#> PCTAGE65P    3.75419997 0.62472153  6.009397 1.862141e-09
#> PCTOWNHOME  -0.41988961 0.19132936 -2.194591 2.819298e-02
if (FALSE) {
sqrt(diag(res$resvar))
sqrt(diag(esar1f$fit$imat)*esar1f$fit$s2)
sqrt(diag(esar1f$fdHess))
system.time(esar1M <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY, family="SAR", method="Matrix"))
summary(esar1M)
system.time(esar1M <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY, family="SAR", method="Matrix",
 control=list(super=TRUE)))
summary(esar1M)
}
esar1wf <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, weights=POP8, family="SAR", method="eigen",
 control=list(pre_eig=eigs))
summary(esar1wf)
#> 
#> Call: 
#> spautolm(formula = Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data = nydata, 
#>     listw = listw_NY, weights = POP8, family = "SAR", method = "eigen", 
#>     control = list(pre_eig = eigs))
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -1.48488 -0.26823  0.09489  0.46552  4.28343 
#> 
#> Coefficients: 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) -0.797063   0.144054 -5.5331 3.146e-08
#> PEXPOSURE    0.080545   0.028334  2.8428  0.004473
#> PCTAGE65P    3.816731   0.576037  6.6258 3.453e-11
#> PCTOWNHOME  -0.380778   0.156507 -2.4330  0.014975
#> 
#> Lambda: 0.0095636 LR test value: 0.32665 p-value: 0.56764 
#> Numerical Hessian standard error of lambda: 0.016466 
#> 
#> Log likelihood: -251.6017 
#> ML residual variance (sigma squared): 1104.1, (sigma: 33.229)
#> Number of observations: 281 
#> Number of parameters estimated: 6 
#> AIC: 515.2
#> 
if (FALSE) {
system.time(esar1wM <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY, weights=POP8, family="SAR", method="Matrix"))
summary(esar1wM)
esar1wlu <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, weights=POP8, family="SAR", method="LU")
summary(esar1wlu)
esar1wch <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, weights=POP8, family="SAR", method="Chebyshev")
summary(esar1wch)
}
ecar1f <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, family="CAR", method="eigen",
 control=list(pre_eig=eigs))
summary(ecar1f)
#> 
#> Call: 
#> spautolm(formula = Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data = nydata, 
#>     listw = listw_NY, family = "CAR", method = "eigen", control = list(pre_eig = eigs))
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -1.539732 -0.384311 -0.030646  0.335126  3.808848 
#> 
#> Coefficients: 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) -0.648362   0.181129 -3.5796 0.0003442
#> PEXPOSURE    0.077899   0.043692  1.7829 0.0745986
#> PCTAGE65P    3.703830   0.627185  5.9055 3.516e-09
#> PCTOWNHOME  -0.382789   0.195564 -1.9574 0.0503053
#> 
#> Lambda: 0.084123 LR test value: 5.8009 p-value: 0.016018 
#> Numerical Hessian standard error of lambda: 0.030868 
#> 
#> Log likelihood: -275.8283 
#> ML residual variance (sigma squared): 0.40758, (sigma: 0.63842)
#> Number of observations: 281 
#> Number of parameters estimated: 6 
#> AIC: 563.66
#> 
if (FALSE) {
system.time(ecar1M <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY, family="CAR", method="Matrix"))
summary(ecar1M)
}
ecar1wf <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, weights=POP8, family="CAR", method="eigen",
 control=list(pre_eig=eigs))
summary(ecar1wf)
#> 
#> Call: 
#> spautolm(formula = Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data = nydata, 
#>     listw = listw_NY, weights = POP8, family = "CAR", method = "eigen", 
#>     control = list(pre_eig = eigs))
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -1.491042 -0.270906  0.081435  0.451556  4.198134 
#> 
#> Coefficients: 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) -0.790154   0.144862 -5.4545 4.910e-08
#> PEXPOSURE    0.081922   0.028593  2.8651  0.004169
#> PCTAGE65P    3.825858   0.577720  6.6223 3.536e-11
#> PCTOWNHOME  -0.386820   0.157436 -2.4570  0.014010
#> 
#> Lambda: 0.022419 LR test value: 0.38785 p-value: 0.53343 
#> Numerical Hessian standard error of lambda: 0.038543 
#> 
#> Log likelihood: -251.5711 
#> ML residual variance (sigma squared): 1102.9, (sigma: 33.21)
#> Number of observations: 281 
#> Number of parameters estimated: 6 
#> AIC: 515.14
#> 
if (FALSE) {
system.time(ecar1wM <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY, weights=POP8, family="CAR", method="Matrix"))
summary(ecar1wM)
}
require("sf", quietly=TRUE)
nc.sids <- st_read(system.file("shapes/sids.shp", package="spData")[1], quiet=TRUE)
ft.SID74 <- sqrt(1000)*(sqrt(nc.sids$SID74/nc.sids$BIR74) +
 sqrt((nc.sids$SID74+1)/nc.sids$BIR74))
lm_nc <- lm(ft.SID74 ~ 1)
sids.nhbr30 <- spdep::dnearneigh(cbind(nc.sids$east, nc.sids$north), 0, 30,
 row.names=row.names(nc.sids))
sids.nhbr30.dist <- spdep::nbdists(sids.nhbr30, cbind(nc.sids$east, nc.sids$north))
sids.nhbr <- spdep::listw2sn(spdep::nb2listw(sids.nhbr30,
 glist=sids.nhbr30.dist, style="B", zero.policy=TRUE))
#> Warning: zero sum general weights
dij <- sids.nhbr[,3]
n <- nc.sids$BIR74
el1 <- min(dij)/dij
el2 <- sqrt(n[sids.nhbr$to]/n[sids.nhbr$from])
sids.nhbr$weights <- el1*el2
sids.nhbr.listw <- spdep::sn2listw(sids.nhbr)
#> Warning: 56, 87 are not origins
both <- factor(paste(nc.sids$L_id, nc.sids$M_id, sep=":"))
ft.NWBIR74 <- sqrt(1000)*(sqrt(nc.sids$NWBIR74/nc.sids$BIR74) +
 sqrt((nc.sids$NWBIR74+1)/nc.sids$BIR74))
mdata <- data.frame(both, ft.NWBIR74, ft.SID74, BIR74=nc.sids$BIR74)
outl <- which.max(rstandard(lm_nc))
as.character(nc.sids$NAME[outl])
#> [1] "Anson"
mdata.4 <- mdata[-outl,]
W <- spdep::listw2mat(sids.nhbr.listw)
W.4 <- W[-outl, -outl]
sids.nhbr.listw.4 <- spdep::mat2listw(W.4)
esarI <- errorsarlm(ft.SID74 ~ 1, data=mdata, listw=sids.nhbr.listw,
 zero.policy=TRUE)
summary(esarI)
#> 
#> Call:errorsarlm(formula = ft.SID74 ~ 1, data = mdata, listw = sids.nhbr.listw, 
#>     zero.policy = TRUE)
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -1.887117 -0.636573 -0.043429  0.448767  3.406724 
#> 
#> Type: error 
#> Regions with no neighbours included:
#>  56 87 
#> Coefficients: (asymptotic standard errors) 
#>             Estimate Std. Error z value  Pr(>|z|)
#> (Intercept)  2.97463    0.13011  22.862 < 2.2e-16
#> 
#> Lambda: 0.66864, LR test value: 10.214, p-value: 0.0013939
#> Asymptotic standard error: 0.11473
#>     z-value: 5.8278, p-value: 5.6146e-09
#> Wald statistic: 33.964, p-value: 5.6146e-09
#> 
#> Log likelihood: -133.8616 for error model
#> ML residual variance (sigma squared): 0.81932, (sigma: 0.90516)
#> Number of observations: 100 
#> Number of parameters estimated: 3 
#> AIC: 273.72, (AIC for lm: 281.94)
#> 
esarIa <- spautolm(ft.SID74 ~ 1, data=mdata, listw=sids.nhbr.listw,
 family="SAR")
summary(esarIa)
#> 
#> Call: spautolm(formula = ft.SID74 ~ 1, data = mdata, listw = sids.nhbr.listw, 
#>     family = "SAR")
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -1.887117 -0.636573 -0.043429  0.448767  3.406724 
#> 
#> Coefficients: 
#>             Estimate Std. Error z value  Pr(>|z|)
#> (Intercept)  2.97463    0.13011  22.862 < 2.2e-16
#> 
#> Lambda: 0.66864 LR test value: 10.214 p-value: 0.0013939 
#> Numerical Hessian standard error of lambda: 0.16506 
#> 
#> Log likelihood: -133.8616 
#> ML residual variance (sigma squared): 0.81932, (sigma: 0.90516)
#> Number of observations: 100 
#> Number of parameters estimated: 3 
#> AIC: 273.72
#> 
esarIV <- errorsarlm(ft.SID74 ~ ft.NWBIR74, data=mdata, listw=sids.nhbr.listw,
 zero.policy=TRUE)
summary(esarIV)
#> 
#> Call:
#> errorsarlm(formula = ft.SID74 ~ ft.NWBIR74, data = mdata, listw = sids.nhbr.listw, 
#>     zero.policy = TRUE)
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -2.123648 -0.573163  0.017859  0.468022  2.693604 
#> 
#> Type: error 
#> Regions with no neighbours included:
#>  56 87 
#> Coefficients: (asymptotic standard errors) 
#>             Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 1.549443   0.219230  7.0677 1.576e-12
#> ft.NWBIR74  0.041974   0.006171  6.8018 1.033e-11
#> 
#> Lambda: 0.18465, LR test value: 0.50496, p-value: 0.47733
#> Asymptotic standard error: 0.20648
#>     z-value: 0.89424, p-value: 0.37119
#> Wald statistic: 0.79967, p-value: 0.37119
#> 
#> Log likelihood: -117.7464 for error model
#> ML residual variance (sigma squared): 0.61546, (sigma: 0.78451)
#> Number of observations: 100 
#> Number of parameters estimated: 4 
#> AIC: 243.49, (AIC for lm: 242)
#> 
esarIVa <- spautolm(ft.SID74 ~ ft.NWBIR74, data=mdata, listw=sids.nhbr.listw,
 family="SAR")
summary(esarIVa)
#> 
#> Call: 
#> spautolm(formula = ft.SID74 ~ ft.NWBIR74, data = mdata, listw = sids.nhbr.listw, 
#>     family = "SAR")
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -2.123648 -0.573163  0.017859  0.468022  2.693604 
#> 
#> Coefficients: 
#>             Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 1.549443   0.219230  7.0677 1.576e-12
#> ft.NWBIR74  0.041974   0.006171  6.8018 1.033e-11
#> 
#> Lambda: 0.18465 LR test value: 0.50496 p-value: 0.47733 
#> Numerical Hessian standard error of lambda: 0.25591 
#> 
#> Log likelihood: -117.7464 
#> ML residual variance (sigma squared): 0.61546, (sigma: 0.78451)
#> Number of observations: 100 
#> Number of parameters estimated: 4 
#> AIC: 243.49
#> 
esarIaw <- spautolm(ft.SID74 ~ 1, data=mdata, listw=sids.nhbr.listw,
 weights=BIR74, family="SAR")
summary(esarIaw)
#> 
#> Call: spautolm(formula = ft.SID74 ~ 1, data = mdata, listw = sids.nhbr.listw, 
#>     weights = BIR74, family = "SAR")
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -1.867485 -0.568644  0.019717  0.502197  3.498013 
#> 
#> Coefficients: 
#>             Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 2.852052   0.090271  31.594 < 2.2e-16
#> 
#> Lambda: 0.7338 LR test value: 12.917 p-value: 0.00032554 
#> Numerical Hessian standard error of lambda: 0.13886 
#> 
#> Log likelihood: -130.0975 
#> ML residual variance (sigma squared): 1539.4, (sigma: 39.236)
#> Number of observations: 100 
#> Number of parameters estimated: 3 
#> AIC: 266.19
#> 
esarIIaw <- spautolm(ft.SID74 ~ both - 1, data=mdata, listw=sids.nhbr.listw,
 weights=BIR74, family="SAR")
summary(esarIIaw)
#> 
#> Call: 
#> spautolm(formula = ft.SID74 ~ both - 1, data = mdata, listw = sids.nhbr.listw, 
#>     weights = BIR74, family = "SAR")
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -2.590809 -0.432976  0.016736  0.357284  3.536718 
#> 
#> Coefficients: 
#>         Estimate Std. Error z value  Pr(>|z|)
#> both1:2  2.05545    0.22184  9.2654 < 2.2e-16
#> both1:3  2.87260    0.16181 17.7531 < 2.2e-16
#> both1:4  4.16365    0.34330 12.1283 < 2.2e-16
#> both2:1  2.47255    0.29757  8.3090 < 2.2e-16
#> both2:2  2.15307    0.21172 10.1692 < 2.2e-16
#> both2:3  2.64235    0.17296 15.2770 < 2.2e-16
#> both2:4  3.26604    0.28287 11.5459 < 2.2e-16
#> both3:1  3.11277    0.34166  9.1107 < 2.2e-16
#> both3:2  2.76541    0.15667 17.6508 < 2.2e-16
#> both3:3  2.86582    0.18593 15.4134 < 2.2e-16
#> both3:4  3.18142    0.21617 14.7169 < 2.2e-16
#> both4:3  3.69333    0.23348 15.8188 < 2.2e-16
#> 
#> Lambda: 0.32136 LR test value: 1.4004 p-value: 0.23666 
#> Numerical Hessian standard error of lambda: 0.25503 
#> 
#> Log likelihood: -109.8922 
#> ML residual variance (sigma squared): 1071.6, (sigma: 32.735)
#> Number of observations: 100 
#> Number of parameters estimated: 14 
#> AIC: 247.78
#> 
esarIVaw <- spautolm(ft.SID74 ~ ft.NWBIR74, data=mdata,
 listw=sids.nhbr.listw, weights=BIR74, family="SAR")
summary(esarIVaw)
#> 
#> Call: 
#> spautolm(formula = ft.SID74 ~ ft.NWBIR74, data = mdata, listw = sids.nhbr.listw, 
#>     weights = BIR74, family = "SAR")
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -2.00956 -0.45229  0.12547  0.55952  2.92223 
#> 
#> Coefficients: 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 1.5769279  0.2501334  6.3043 2.894e-10
#> ft.NWBIR74  0.0368573  0.0069413  5.3099 1.097e-07
#> 
#> Lambda: 0.3839 LR test value: 1.9983 p-value: 0.15747 
#> Numerical Hessian standard error of lambda: 0.25778 
#> 
#> Log likelihood: -119.5648 
#> ML residual variance (sigma squared): 1295.8, (sigma: 35.997)
#> Number of observations: 100 
#> Number of parameters estimated: 4 
#> AIC: 247.13
#> 
ecarIaw <- spautolm(ft.SID74 ~ 1, data=mdata.4, listw=sids.nhbr.listw.4,
 weights=BIR74, family="CAR")
#> Warning: Non-symmetric spatial weights in CAR model
summary(ecarIaw)
#> 
#> Call: 
#> spautolm(formula = ft.SID74 ~ 1, data = mdata.4, listw = sids.nhbr.listw.4, 
#>     weights = BIR74, family = "CAR")
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -2.009350 -0.638915 -0.060761  0.428526  2.019409 
#> 
#> Coefficients: 
#>             Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 2.942864   0.095304  30.879 < 2.2e-16
#> 
#> Lambda: 0.86832 LR test value: 23.003 p-value: 1.6172e-06 
#> Numerical Hessian standard error of lambda: 0.048102 
#> 
#> Log likelihood: -118.7564 
#> ML residual variance (sigma squared): 1264, (sigma: 35.553)
#> Number of observations: 99 
#> Number of parameters estimated: 3 
#> AIC: 243.51
#> 
ecarIIaw <- spautolm(ft.SID74 ~ both - 1, data=mdata.4,
 listw=sids.nhbr.listw.4, weights=BIR74, family="CAR")
#> Warning: Non-symmetric spatial weights in CAR model
#> Warning: NaNs produced
summary(ecarIIaw)
#> 
#> Call: 
#> spautolm(formula = ft.SID74 ~ both - 1, data = mdata.4, listw = sids.nhbr.listw.4, 
#>     weights = BIR74, family = "CAR")
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -2.564067 -0.461531 -0.020982  0.384458  2.054255 
#> 
#> Coefficients: 
#>         Estimate Std. Error z value  Pr(>|z|)
#> both1:2  2.06282    0.20065 10.2806 < 2.2e-16
#> both1:3  2.91982    0.14171 20.6048 < 2.2e-16
#> both1:4  4.12159    0.30076 13.7037 < 2.2e-16
#> both2:1  2.58281    0.27014  9.5611 < 2.2e-16
#> both2:2  2.17549    0.18265 11.9104 < 2.2e-16
#> both2:3  2.67030    0.15355 17.3910 < 2.2e-16
#> both2:4  3.10806    0.24748 12.5588 < 2.2e-16
#> both3:1  2.93237    0.30007  9.7724 < 2.2e-16
#> both3:2  2.65317    0.14139 18.7646 < 2.2e-16
#> both3:3  2.91685    0.17134 17.0234 < 2.2e-16
#> both3:4  3.20447    0.20402 15.7063 < 2.2e-16
#> both4:3  3.80672    0.20831 18.2742 < 2.2e-16
#> 
#> Lambda: 0.22163 LR test value: 1.3827 p-value: 0.23964 
#> Numerical Hessian standard error of lambda: NaN 
#> 
#> Log likelihood: -99.2181 
#> ML residual variance (sigma squared): 890.66, (sigma: 29.844)
#> Number of observations: 99 
#> Number of parameters estimated: 14 
#> AIC: 226.44
#> 
ecarIVaw <- spautolm(ft.SID74 ~ ft.NWBIR74, data=mdata.4,
 listw=sids.nhbr.listw.4, weights=BIR74, family="CAR")
#> Warning: Non-symmetric spatial weights in CAR model
summary(ecarIVaw)
#> 
#> Call: 
#> spautolm(formula = ft.SID74 ~ ft.NWBIR74, data = mdata.4, listw = sids.nhbr.listw.4, 
#>     weights = BIR74, family = "CAR")
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -1.99259 -0.44794  0.15464  0.60748  1.95751 
#> 
#> Coefficients: 
#>             Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 1.434705   0.225521  6.3618 1.995e-10
#> ft.NWBIR74  0.040903   0.006299  6.4936 8.382e-11
#> 
#> Lambda: 0.22724 LR test value: 1.1936 p-value: 0.2746 
#> Numerical Hessian standard error of lambda: 0.5473 
#> 
#> Log likelihood: -114.0196 
#> ML residual variance (sigma squared): 1201, (sigma: 34.655)
#> Number of observations: 99 
#> Number of parameters estimated: 4 
#> AIC: 236.04
#> 
nc.sids$fitIV <- append(fitted.values(ecarIVaw), NA, outl-1)
plot(nc.sids[,"fitIV"], nbreaks=12) # Cressie 1993, p. 565

if (FALSE) {
data(oldcol, package="spdep")
COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 spdep::nb2listw(COL.nb, style="W"))
summary(COL.errW.eig)
COL.errW.sar <- spautolm(CRIME ~ INC + HOVAL, data=COL.OLD,
 spdep::nb2listw(COL.nb, style="W"))
summary(COL.errW.sar)
data(boston, package="spData")
gp1 <- spautolm(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), family="SMA")
summary(gp1)
}