errorsarlm

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Spatial simultaneous autoregressive error model estimation

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

$$y = X \beta + u, u = \lambda W u + \varepsilon$$

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.

Keywords
spatial
Usage
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 sarlmThe 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

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

Aliases
  • errorsarlm
  • lmSLX
  • create_WX
Examples
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))

Documentation reproduced from package spdep, version 0.6-9, License: GPL (>= 2)

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