Spatial simultaneous autoregressive error model estimation by GMM

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

GMerrorsar(formula, data = list(), listw, na.action = na.fail,
 zero.policy = NULL, 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 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

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 .

Author

Luc Anselin and Roger Bivand

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))
#> 
#> Call:
#> errorsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = spdep::nb2listw(COL.nb, 
#>     style = "W"), method = "eigen")
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -34.81174  -6.44031  -0.72142   7.61476  23.33626 
#> 
#> Type: error 
#> Coefficients: (asymptotic standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 59.893219   5.366163 11.1613 < 2.2e-16
#> INC         -0.941312   0.330569 -2.8476 0.0044057
#> HOVAL       -0.302250   0.090476 -3.3407 0.0008358
#> 
#> Lambda: 0.56179, LR test value: 7.9935, p-value: 0.0046945
#> Asymptotic standard error: 0.13387
#>     z-value: 4.1966, p-value: 2.7098e-05
#> Wald statistic: 17.611, p-value: 2.7098e-05
#> 
#> Log likelihood: -183.3805 for error model
#> ML residual variance (sigma squared): 95.575, (sigma: 9.7762)
#> Number of observations: 49 
#> Number of parameters estimated: 5 
#> AIC: 376.76, (AIC for lm: 382.75)
#> Hausman test: 4.902, df: 3, p-value: 0.17911
#> 
coef(x)
#>               Estimate Std. Error   z value     Pr(>|z|)
#> (Intercept) 59.8932192 5.36616252 11.161276 0.0000000000
#> INC         -0.9413120 0.33056857 -2.847554 0.0044056569
#> HOVAL       -0.3022502 0.09047605 -3.340665 0.0008357788
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))
#> 
#> Call:
#> GMerrorsar(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = spdep::nb2listw(COL.nb, 
#>     style = "W"), returnHcov = TRUE)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -30.5432  -6.5553  -2.1921  10.0553  28.7497 
#> 
#> Type: GM SAR estimator
#> Coefficients: (GM standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 62.513752   5.121339  12.207 < 2.2e-16
#> INC         -1.128283   0.339745  -3.321  0.000897
#> HOVAL       -0.296957   0.095699  -3.103  0.001915
#> 
#> Lambda: 0.40196 (standard error): 0.42334 (z-value): 0.94948
#> Residual variance (sigma squared): 106.64, (sigma: 10.327)
#> GM argmin sigma squared: 106.36
#> Number of observations: 49 
#> Number of parameters estimated: 5 
#> Hausman test: 6.6406, df: 3, p-value: 0.08428
#> 
coef(x)
#>               Estimate Std. Error   z value     Pr(>|z|)
#> (Intercept) 62.5137525 5.12133863 12.206526 0.0000000000
#> INC         -1.1282834 0.33974492 -3.320972 0.0008970455
#> HOVAL       -0.2969573 0.09569889 -3.103038 0.0019154486
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)
#> 
#> Call:
#> GMerrorsar(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = spdep::nb2listw(COL.nb, 
#>     style = "W"))
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -30.5432  -6.5553  -2.1921  10.0553  28.7497 
#> 
#> Type: GM SAR estimator
#> Coefficients: (GM standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 62.513752   5.121339  12.207 < 2.2e-16
#> INC         -1.128283   0.339745  -3.321  0.000897
#> HOVAL       -0.296957   0.095699  -3.103  0.001915
#> 
#> Lambda: 0.40196 (standard error): 0.42334 (z-value): 0.94948
#> Residual variance (sigma squared): 106.64, (sigma: 10.327)
#> GM argmin sigma squared: 106.36
#> Number of observations: 49 
#> Number of parameters estimated: 5 
#> 
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$AREAKEY), 2, 10))
#> [1] TRUE
nyadjlw <- spdep::mat2listw(nyadjmat, as.character(nydata$AREAKEY))
listw_NY <- spdep::nb2listw(nyadjlw$neighbours, style="B")
esar1f <- spautolm(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data=nydata,
 listw=listw_NY, family="SAR", method="eigen")
summary(esar1f)
#> 
#> Call: 
#> spautolm(formula = Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, data = nydata, 
#>     listw = listw_NY, family = "SAR", method = "eigen")
#> 
#> 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
#> 
esar1gm <- GMerrorsar(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY)
summary(esar1gm)
#> 
#> Call:GMerrorsar(formula = Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, 
#>     data = nydata, listw = listw_NY)
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -1.641411 -0.396370 -0.026618  0.341740  4.204264 
#> 
#> Type: GM SAR estimator
#> Coefficients: (GM standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) -0.604364   0.174576 -3.4619 0.0005364
#> PEXPOSURE    0.067902   0.041164  1.6496 0.0990337
#> PCTAGE65P    3.775308   0.622965  6.0602 1.359e-09
#> PCTOWNHOME  -0.437545   0.188906 -2.3162 0.0205468
#> 
#> Lambda: 0.03605 (standard error): 0.2022 (z-value): 0.17829
#> Residual variance (sigma squared): 0.41585, (sigma: 0.64487)
#> GM argmin sigma squared: 0.45141
#> Number of observations: 281 
#> Number of parameters estimated: 6 
#> 
esar1gm1 <- GMerrorsar(Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME,
 data=nydata, listw=listw_NY, method="Nelder-Mead")
summary(esar1gm1)
#> 
#> Call:GMerrorsar(formula = Z ~ PEXPOSURE + PCTAGE65P + PCTOWNHOME, 
#>     data = nydata, listw = listw_NY, method = "Nelder-Mead")
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -1.641390 -0.396374 -0.026616  0.341745  4.204277 
#> 
#> Type: GM SAR estimator
#> Coefficients: (GM standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) -0.604384   0.174580 -3.4619 0.0005363
#> PEXPOSURE    0.067907   0.041165  1.6496 0.0990225
#> PCTAGE65P    3.775275   0.622969  6.0601  1.36e-09
#> PCTOWNHOME  -0.437518   0.188910 -2.3160 0.0205572
#> 
#> Lambda: 0.036057 (standard error): 0.20221 (z-value): 0.17832
#> Residual variance (sigma squared): 0.41585, (sigma: 0.64487)
#> GM argmin sigma squared: 0.45139
#> Number of observations: 281 
#> Number of parameters estimated: 6 
#>