Maximum likelihood estimation of spatial simultaneous autoregressive
error models of the form:$$y = X \beta + u, u = \lambda W u + \varepsilon$$
where $$ is found by optim() using method "L-BFGS-B" first and $$ and other parameters by generalized least squares subsequently.
errorsarlm(formula, data=list(), listw, na.action=na.fail, method="eigen",
quiet=TRUE, zero.policy=FALSE, interval = c(-1, 0.999), tol.solve=1.0e-10,
tol.opt=.Machine$double.eps^0.5, control, optim=FALSE, sparsedebug=FALSE)- 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.}
- method
{"eigen" (default) - the Jacobian is computed as the product
of (1 - rho*eigenvalue) using eigenw, "SparseM" for strictly symmetric weights lists of styles "B", "C" and "U", or made symmetric by similarity (Ord, 1975, Appendix C) if possible for styles "W" and "S", using code from the SparseM package to calculate the determinant, and "sparse" - (deprecated from this release) computes the determinant of the sparse matrix (I - rho*W) directly using logSpwdet.
}
- quiet
{default=TRUE; if FALSE, reports function values during optimization.}
- zero.policy
{if TRUE assign zero to the lagged value of zones without
neighbours, if FALSE (default) assign NA - causing errorsarlm() to terminate with an error}
- interval
{search interval for autoregressive parameter when not using method="eigen"; default is c(-1,1)}
- 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}
- tol.opt
{the desired accuracy of the optimization - passed to optim() (default=square root of double precision machine tolerance, a larger root may be used if the warning: ERROR: ABNORMAL_TERMINATION_IN_LNSRCH is seen, see help(boston) for an example)}
- control
{A list of control parameters passed to optim, se details in optim}
- optim
{If TRUE use experimental optim branch and control argument}
- sparsedebug
{if TRUE, writes a log file on sparse matrix operations
(name sparsestats) in the current directory. To be used if sparse estimation
fails!}
The asymptotic standard error of $$ 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=sparse. 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.
When using the sparse method, the user takes (unfortunately) full
responsibility for possible failures, including R terminating with
a core dump! Safeguards have been put in place to try to trap errant
behaviour in the sparse functions' memory allocation, but they may
not always help. When sparsedebug is TRUE, a log file (sparsestats)
is written in the working directory - the figure of interest is the
number of allocated blocks. At present, spwdet will fail when
this increases over the number initially allocated, but will not release
memory allocated by the sparse functions. In the event of problems,
save your workspace and quit R. Problems seem to be related to larger
n, and to an unknown trigger precipitating uncontrolled fillin, in the
course of which the sparse routines lose track of their memory pointers,
and then provoke a segmentation fault trying to free unallocated memory.
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}
- lm.model
{the lm object returned when estimating for $=0$
- method
{the method used to calculate the Jacobian}
- call
{the call used to create this object}
- residuals
{GLS residuals}
- lm.target
{the lm object returned for the GLS fit}
- fitted.values
{Difference between residuals and response variable}
- ase
{TRUE if method=eigen}
- formula
{model formula}
- se.fit
{Not used yet}
- lambda.se
{if ase=TRUE, the asymptotic standard error of $$
- LMtest
{NULL for this model}
- zero.policy
{zero.policy for this model}
- na.action
{(possibly) named vector of excluded or omitted observations if non-default na.action argument used}
The internal sar.error.* functions return the value of the log likelihood function at $$.
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 (www.spacestat.com);
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.
[object Object],[object Object],[object Object]
lm, lagsarlm,
eigenw, logSpwdet, asMatrixCsrListw, similar.listw, predict.sarlm,
residuals.sarlm
data(oldcol)
COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
nb2listw(COL.nb, style="W"), method="eigen", quiet=FALSE)
COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
nb2listw(COL.nb, style="W"), method="eigen", control=list(trace=3, fnscale=-1,
factr=.Machine$double.eps^0.5, pgtol=.Machine$double.eps^0.5))
COL.errB.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
nb2listw(COL.nb, style="B"), method="eigen", quiet=FALSE)
COL.errW.SM <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
nb2listw(COL.nb, style="W"), method="SparseM", quiet=FALSE)
summary(COL.errW.eig, correlation=TRUE)
summary(COL.errB.eig, correlation=TRUE)
summary(COL.errW.SM, correlation=TRUE)
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)
spatial