lagsarlm {spdep} | R Documentation |
Maximum likelihood estimation of spatial simultaneous autoregressive lag and spatial Durbin (mixed) models of the form:
y = rho W y + X beta + e
where rho is found by optimize()
first, and beta and other parameters by generalized least squares subsequently (one-dimensional search using optim performs badly on some platforms). In the spatial Durbin (mixed) model, the spatially lagged independent variables are added to X. Note that interpretation of the fitted coefficients should use impact measures, because of the feedback loops induced by the data generation process for this model. 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.
lagsarlm(formula, data = list(), listw, na.action, type="lag", method="eigen", quiet=NULL, zero.policy=NULL, interval=NULL, tol.solve=1.0e-10, trs=NULL, control=list())
formula |
a symbolic description of the model to be fit. The details
of model specification are given for |
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 |
na.action |
a function (default |
type |
default "lag", may be set to "mixed"; when "mixed", 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 |
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 (default) assign NA - causing |
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 |
trs |
default NULL, if given, a vector of powered spatial weights matrix traces output by |
control |
list of extra control arguments - see section below |
The asymptotic standard error of rho 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, and with the elements involving rho 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.
Versions of the package from 0.4-38 include numerical Hessian values where asymptotic standard errors are not available. This change has been introduced to permit the simulation of distributions for impact measures. The warnings made above with regard to variable scaling also apply in this case.
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.
A list object of class sarlm
type |
"lag" or "mixed" |
rho |
simultaneous autoregressive lag coefficient |
coefficients |
GLS coefficient estimates |
rest.se |
asymptotic standard errors if ase=TRUE, otherwise approximate numeriacal Hessian-based values |
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 rho=0 |
AIC_lm.model |
AIC of the linear model for rho=0 |
method |
the method used to calculate the Jacobian |
call |
the call used to create this object |
residuals |
GLS residuals |
tarX |
model matrix of the GLS model |
tary |
response of the GLS model |
y |
response of the linear model for rho=0 |
X |
model matrix of the linear model for rho=0 |
opt |
object returned from numerical optimisation |
fitted.values |
Difference between residuals and response variable |
se.fit |
Not used yet |
ase |
TRUE if method=eigen |
rho.se |
if ase=TRUE, the asymptotic standard error of rho, otherwise approximate numeriacal Hessian-based value |
LMtest |
if ase=TRUE, the Lagrange Multiplier test for the absence of spatial autocorrelation in the lag model residuals |
resvar |
the asymptotic coefficient covariance matrix for (s2, rho, B) |
zero.policy |
zero.policy for this model |
aliased |
the aliased explanatory variables (if any) |
listw_style |
the style of the spatial weights used |
interval |
the line search interval used to find rho |
fdHess |
the numerical Hessian-based coefficient covariance matrix for (rho, B) if computed |
optimHess |
if TRUE and fdHess returned, |
insert |
if TRUE and fdHess returned, the asymptotic analytical values are inserted into the numerical Hessian instead of the approximated values, and its size increased to include the first row/column for sigma2 |
LLNullLlm |
Log-likelihood of the null linear model |
timings |
processing timings |
na.action |
(possibly) named vector of excluded or omitted observations if non-default na.action argument used |
The internal sar.lag.mixed.* functions return the value of the log likelihood function at rho.
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)
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
default FALSE, use fdHess
from nlme, if TRUE, use optim
to calculate Hessian at optimum
default FALSE; logical value used in the log likelihood function to choose compiled code for computing SSE
default 2; used for preparing the Cholesky decompositions for updating in the Jacobian function
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
default 5; highest power of the approximating polynomial for the Chebyshev approximation
default 16; number of random variates
default 30; number of products of random variates matrix and spatial weights matrix
default “MMD”, alternative “RCM”
default 0.1, coefficient value for initial Cholesky decomposition in “spam_update”
default “MC”, used with method “moments”; alternatives “mult” and “moments”, for use if trs
is missing, trW
default TRUE, used with method “moments” to compute the Smirnov/Anselin correction term
default TRUE, used with method “moments” to truncate the Smirnov/Anselin correction term
Roger Bivand Roger.Bivand@nhh.no, with thanks to Andrew Bernat for contributions to the asymptotic standard error code.
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; 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.
lm
, errorsarlm
,
eigenw
,
predict.sarlm
, impacts.sarlm
,
residuals.sarlm
, do_ldet
data(oldcol) COL.lag.eig <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, nb2listw(COL.nb, style="W"), method="eigen", quiet=FALSE) summary(COL.lag.eig, correlation=TRUE) COL.lag.eig$fdHess COL.lag.eig$resvar W <- as(as_dgRMatrix_listw(nb2listw(COL.nb)), "CsparseMatrix") trMatc <- trW(W, type="mult") COL.lag.eig1 <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, nb2listw(COL.nb, style="W"), control=list(fdHess=TRUE), trs=trMatc) COL.lag.eig1$fdHess system.time(COL.lag.M <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, nb2listw(COL.nb), method="Matrix", quiet=FALSE)) summary(COL.lag.M) impacts(COL.lag.M, listw=nb2listw(COL.nb)) ## Not run: system.time(COL.lag.sp <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, nb2listw(COL.nb), method="spam", quiet=FALSE)) summary(COL.lag.sp) ## End(Not run) COL.lag.B <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, nb2listw(COL.nb, style="B")) summary(COL.lag.B, correlation=TRUE) COL.mixed.B <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, nb2listw(COL.nb, style="B"), type="mixed", tol.solve=1e-9) summary(COL.mixed.B, correlation=TRUE) COL.mixed.W <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, nb2listw(COL.nb, style="W"), type="mixed") summary(COL.mixed.W, correlation=TRUE) NA.COL.OLD <- COL.OLD NA.COL.OLD$CRIME[20:25] <- NA COL.lag.NA <- lagsarlm(CRIME ~ INC + HOVAL, data=NA.COL.OLD, nb2listw(COL.nb), na.action=na.exclude, control=list(tol.opt=.Machine$double.eps^0.4)) COL.lag.NA$na.action COL.lag.NA resid(COL.lag.NA) data(boston) gp2mM <- lagsarlm(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, nb2listw(boston.soi), type="mixed", method="Matrix") summary(gp2mM) W <- as(as_dgRMatrix_listw(nb2listw(boston.soi)), "CsparseMatrix") trMatb <- trW(W, type="mult") gp2mMi <- lagsarlm(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, nb2listw(boston.soi), type="mixed", method="Matrix", trs=trMatb) summary(gp2mMi)