Spatial simultaneous autoregressive error model estimation

Description

Maximum likelihood estimation of spatial simultaneous autoregressive error models of the form:

y = X beta + u, u = lambda W u + e

where lambda is found by optimize() first, and beta and other parameters by generalized least squares subsequently. With one of the sparse matrix methods, larger numbers of observations can be handled, but the interval= argument may need be set when the weights are not row-standardised. When etype is “emixed”, a so-called spatial Durbin error model is fitted, while 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

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errorsarlm(formula, data=list(), listw, na.action, weights=NULL,
 etype="error", method="eigen", quiet=NULL, zero.policy=NULL,
 interval = NULL, tol.solve=1.0e-10, trs=NULL, control=list())
lmSLX(formula, data = list(), listw, na.action, weights=NULL, 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 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

etype

default "error", may be set to "emixed" to include the spatially lagged independent variables added to X; when "emixed", the lagged intercept is dropped for spatial weights style "W", that is row-standardised weights, but otherwise included

method

"eigen" (default) - the Jacobian is computed as the product of (1 - rho*eigenvalue) using eigenw, and "spam" or "Matrix_J" for strictly symmetric weights lists of styles "B" and "C", or made symmetric by similarity (Ord, 1975, Appendix C) if possible for styles "W" and "S", using code from the spam package or Matrix package to calculate the determinant; “Matrix” and “spam_update” provide updating Cholesky decomposition methods; "LU" provides an alternative sparse matrix decomposition approach. In addition, there are "Chebyshev" and Monte Carlo "MC" approximate log-determinant methods; the Smirnov/Anselin (2009) trace approximation is available as "moments". Three methods: "SE_classic", "SE_whichMin", and "SE_interp" are provided experimentally, the first to attempt to emulate the behaviour of Spatial Econometrics toolbox ML fitting functions. All use grids of log determinant values, and the latter two attempt to ameliorate some features of "SE_classic".

quiet

default NULL, use !verbose global option value; if FALSE, reports function values during optimization.

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 errorsarlm() to terminate with an error

interval

default is NULL, search interval for autoregressive parameter

tol.solve

the tolerance for detecting linear dependencies in the columns of matrices to be inverted - passed to solve() (default=1.0e-10). This may be used if necessary to extract coefficient standard errors (for instance lowering to 1e-12), but 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

trs

default NULL, if given, a vector of powered spatial weights matrix traces output by trW; when given, insert the asymptotic analytical values into the numerical Hessian instead of the approximated values; may be used to get around some problems raised when the numerical Hessian is poorly conditioned, generating NaNs in subsequent operations. When using the numerical Hessian to get the standard error of lambda, it is very strongly advised that trs be given, as the parts of fdHess corresponding to the regression coefficients are badly approximated, affecting the standard error of lambda; the coefficient correlation matrix is unusable

control

list of extra control arguments - see section below

x

model matrix to be lagged

prefix

default empty string, may be “lag” in some cases

Details

The asymptotic standard error of lambda is only computed when method=eigen, because the full matrix operations involved would be costly for large n typically associated with the choice of method="spam" or "Matrix". The same applies to the coefficient covariance matrix. Taken as the asymptotic matrix from the literature, it is typically badly scaled, being block-diagonal, and with the elements involving lambda being very small, while other parts of the matrix can be very large (often many orders of magnitude in difference). It often happens that the tol.solve argument needs to be set to a smaller value than the default, or the RHS variables can be centred or reduced in range.

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.

Value

A list object of class sarlm

type

"error"

lambda

simultaneous autoregressive error coefficient

coefficients

GLS coefficient estimates

rest.se

GLS coefficient standard errors (are equal to asymptotic standard errors)

LL

log likelihood value at computed optimum

s2

GLS residual variance

SSE

sum of squared GLS errors

parameters

number of parameters estimated

logLik_lm.model

Log likelihood of the linear model for lambda=0

AIC_lm.model

AIC of the linear model for lambda=0

coef_lm.model

coefficients of the linear model for lambda=0

tarX

model matrix of the GLS model

tary

response of the GLS model

y

response of the linear model for lambda=0

X

model matrix of the linear model for lambda=0

method

the method used to calculate the Jacobian

call

the call used to create this object

residuals

GLS residuals

opt

object returned from numerical optimisation

fitted.values

Difference between residuals and response variable

ase

TRUE if method=eigen

se.fit

Not used yet

lambda.se

if ase=TRUE, the asymptotic standard error of lambda

LMtest

NULL for this model

aliased

if not NULL, details of aliased variables

LLNullLlm

Log-likelihood of the null linear model

Hcov

Spatial DGP covariance matrix for Hausman test if available

interval

line search interval

fdHess

finite difference Hessian

optimHess

optim or fdHess used

insert

logical; is TRUE, asymptotic values inserted in fdHess where feasible

timings

processing timings

f_calls

number of calls to the log likelihood function during optimization

hf_calls

number of calls to the log likelihood function during numerical Hessian computation

intern_classic

a data frame of detval matrix row choices used by the SE toolbox classic method

zero.policy

zero.policy for this model

na.action

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

weights

weights used in model fitting

emixedImps

for “emixed” models, a list of three impact matrixes (impacts and standard errors) for direct, indirect and total impacts; total impacts calculated using gmodels::estimable

The internal sar.error.* functions return the value of the log likelihood function at lambda.

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 gmodels::estimable.

Control arguments

tol.opt:

the desired accuracy of the optimization - passed to optimize() (default=square root of double precision machine tolerance, a larger root may be used needed, see help(boston) for an example)

returnHcov:

default TRUE, return the Vo matrix for a spatial Hausman test

pWOrder:

default 250, if returnHcov=TRUE and the method is not “eigen”, pass this order to powerWeights as the power series maximum limit

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

LAPACK:

default FALSE; logical value passed to qr in the SSE log likelihood function

compiled_sse:

default FALSE; logical value used in the log likelihood function to choose compiled code for computing SSE

Imult:

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

super:

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

cheb_q:

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

MC_p:

default 16; number of random variates

MC_m:

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

spamPivot:

default “MMD”, alternative “RCM”

in_coef

default 0.1, coefficient value for initial Cholesky decomposition in “spam_update”

type

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

Author(s)

Roger Bivand Roger.Bivand@nhh.no

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; Anselin, L. 1988 Spatial econometrics: methods and models. (Dordrecht: Kluwer); Anselin, L. 1995 SpaceStat, a software program for the analysis of spatial data, version 1.80. Regional Research Institute, West Virginia University, Morgantown, WV; Anselin L, Bera AK (1998) Spatial dependence in linear regression models with an introduction to spatial econometrics. In: Ullah A, Giles DEA (eds) Handbook of applied economic statistics. Marcel Dekker, New York, pp. 237-289; Cressie, N. A. C. 1993 Statistics for spatial data, Wiley, New York; LeSage J and RK Pace (2009) Introduction to Spatial Econometrics. CRC Press, Boca Raton.

Roger Bivand, Gianfranco Piras (2015). Comparing Implementations of Estimation Methods for Spatial Econometrics. Journal of Statistical Software, 63(18), 1-36. http://www.jstatsoft.org/v63/i18/.

Bivand, R. S., Hauke, J., and Kossowski, T. (2013). Computing the Jacobian in Gaussian spatial autoregressive models: An illustrated comparison of available methods. Geographical Analysis, 45(2), 150-179.

See Also

lm, lagsarlm, similar.listw, summary.sarlm, predict.sarlm, residuals.sarlm, do_ldet, estimable

Examples

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data(oldcol)
lw <- nb2listw(COL.nb, style="W")
COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="eigen", quiet=FALSE)
summary(COL.errW.eig, correlation=TRUE)
ev <- eigenw(similar.listw(lw))
COL.errW.eig_ev <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="eigen", control=list(pre_eig=ev))
all.equal(coefficients(COL.errW.eig), coefficients(COL.errW.eig_ev))
COL.errB.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 nb2listw(COL.nb, style="B"), method="eigen", quiet=FALSE)
summary(COL.errB.eig, correlation=TRUE)
W <- as(nb2listw(COL.nb), "CsparseMatrix")
trMatc <- trW(W, type="mult")
COL.errW.M <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="Matrix", quiet=FALSE, trs=trMatc)
summary(COL.errW.M, correlation=TRUE)
COL.SDEM.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="eigen", etype="emixed")
summary(COL.SDEM.eig, correlation=TRUE)
summary(impacts(COL.SDEM.eig))
summary(impacts(COL.SDEM.eig), adjust_k=TRUE)
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)
summary(COL.SLX)
COL.SLX <- lmSLX(CRIME ~ INC, data=COL.OLD, listw=lw)

crds <- cbind(COL.OLD$X, COL.OLD$Y)
mdist <- sqrt(sum(diff(apply(crds, 2, range))^2))
dnb <- dnearneigh(crds, 0, mdist)
dists <- nbdists(dnb, crds)
f <- function(x, form, data, dnb, dists, verbose) {
  glst <- lapply(dists, function(d) 1/(d^x))
  lw <- 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 <- nb2listw(dnb, glist=glst, style="B")
SLX <- lmSLX(CRIME ~ INC + HOVAL, data=COL.OLD, listw=lw)
summary(SLX)
summary(impacts(SLX))

NA.COL.OLD <- COL.OLD
NA.COL.OLD$CRIME[20:25] <- NA
COL.err.NA <- errorsarlm(CRIME ~ INC + HOVAL, data=NA.COL.OLD,
 nb2listw(COL.nb), na.action=na.exclude)
COL.err.NA$na.action
COL.err.NA
resid(COL.err.NA)

lw <- nb2listw(COL.nb, style="W")
system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="eigen"))
ocoef <- coefficients(COL.errW.eig)
system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="eigen", control=list(LAPACK=FALSE)))
all.equal(ocoef, coefficients(COL.errW.eig))
system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="eigen", control=list(compiled_sse=TRUE)))
all.equal(ocoef, coefficients(COL.errW.eig))
system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="Matrix_J", control=list(super=TRUE)))
all.equal(ocoef, coefficients(COL.errW.eig))
system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="Matrix_J", control=list(super=FALSE)))
all.equal(ocoef, coefficients(COL.errW.eig))
system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="Matrix_J", control=list(super=as.logical(NA))))
all.equal(ocoef, coefficients(COL.errW.eig))
system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="Matrix", control=list(super=TRUE)))
all.equal(ocoef, coefficients(COL.errW.eig))
system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="Matrix", control=list(super=FALSE)))
all.equal(ocoef, coefficients(COL.errW.eig))
system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="Matrix", control=list(super=as.logical(NA))))
all.equal(ocoef, coefficients(COL.errW.eig))
system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="spam", control=list(spamPivot="MMD")))
all.equal(ocoef, coefficients(COL.errW.eig))
system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="spam", control=list(spamPivot="RCM")))
all.equal(ocoef, coefficients(COL.errW.eig))
system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="spam_update", control=list(spamPivot="MMD")))
all.equal(ocoef, coefficients(COL.errW.eig))
system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="spam_update", control=list(spamPivot="RCM")))
all.equal(ocoef, coefficients(COL.errW.eig))

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