ML_models: Spatial simultaneous autoregressive model estimation by maximum likelihood

Description

The lagsarlm function provides Maximum likelihood estimation of spatial simultaneous autoregressive lag and spatial Durbin (mixed) models of the form:

$$y = \rho W y + X \beta + \varepsilon$$

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.

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.

Maximum likelihood estimation of spatial simultaneous autoregressive “SAC/SARAR” models of the form:

$$y = \rho W1 y + X \beta + u, u = \lambda W2 u + \varepsilon$$

where $\rho$ and $\lambda$ are found by nlminb or optim() first, and $\beta$ and other parameters by generalized least squares subsequently.

Usage

lagsarlm(formula, data = list(), listw, na.action, Durbin, type,
 method="eigen", quiet=NULL, zero.policy=NULL, interval=NULL,
 tol.solve=.Machine$double.eps, trs=NULL, control=list())
errorsarlm(formula, data=list(), listw, na.action, weights=NULL,
 Durbin, etype, method="eigen", quiet=NULL, zero.policy=NULL,
 interval = NULL, tol.solve=.Machine$double.eps, trs=NULL, control=list())
sacsarlm(formula, data = list(), listw, listw2 = NULL, na.action, Durbin, type,
 method="eigen", quiet=NULL, zero.policy=NULL, tol.solve=.Machine$double.eps,
 llprof=NULL, interval1=NULL, interval2=NULL, trs1=NULL, trs2=NULL,
 control = list())
# S3 method for sarlm
summary(object, correlation = FALSE, Nagelkerke = FALSE,
 Hausman=FALSE, adj.se=FALSE, ...)
# S3 method for sarlm
print(x, ...)
# S3 method for summary.sarlm
print(x, digits = max(5, .Options$digits - 3),
 signif.stars = FALSE, ...)
# S3 method for sarlm
residuals(object, ...)
# S3 method for sarlm
deviance(object, ...)
# S3 method for sarlm
coef(object, ...)
# S3 method for sarlm
vcov(object, ...)
# S3 method for sarlm
fitted(object, ...)

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:: (error model) default TRUE, return the Vo matrix for a spatial Hausman test
pWOrder:: (error model) 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
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
OrdVsign: default 1; used to set the sign of the final component to negative if -1 (alpha times ((sigma squared) squared) in Ord (1975) equation B.1).
opt_method:: default “nlminb”, may be set to “L-BFGS-B” to use box-constrained optimisation in optim
opt_control:: default list(), a control list to pass to nlminb or optim
pars:: default NULL, for which five trial starting values spanning the lower/upper range are tried and the best selected, starting values of $\rho$ and $\lambda$
npars: default integer 4L, four trial points; if not default value, nine trial points
pre_eig1, pre_eig2: default NULL; may be used to pass pre-computed vectors of eigenvalues

Details

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 $\rho$ (lag model) or $\lambda$ (error model) 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. Refer to the help page of predict.sarlm for discussions and references.

Because numerical optimisation is used to find the values of lambda and rho in sacsarlm, care needs to be shown. It has been found that the surface of the 2D likelihood function often forms a “banana trench” from (low rho, high lambda) through (high rho, high lambda) to (high rho, low lambda) values. In addition, sometimes the banana has optima towards both ends, one local, the other global, and conseqently the choice of the starting point for the final optimization becomes crucial. The default approach is not to use just (0, 0) as a starting point, nor the (rho, lambda) values from gstsls, which lie in a central part of the “trench”, but either four values at (low rho, high lambda), (0, 0), (high rho, high lambda), and (high rho, low lambda), and to use the best of these start points for the final optimization. Optionally, nine points can be used spanning the whole (lower, upper) space.

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; Nagelkerke NJD (1991) A note on a general definition of the coefficient of determination. Biometrika 78: 691-692; 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. https://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.

Examples

Run this code

# NOT RUN {
data(oldcol, package="spdep")
listw <- spdep::nb2listw(COL.nb, style="W")
ev <- eigenw(listw)
W <- as(listw, "CsparseMatrix")
trMatc <- trW(W, type="mult")
COL.lag.eig <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw=listw,
 method="eigen", quiet=FALSE, control=list(pre_eig=ev, OrdVsign=1))
summary(COL.lag.eig, correlation=TRUE)
# }
# NOT RUN {
COL.lag.eig$fdHess
COL.lag.eig$resvar
# using the apparent sign in Ord (1975, equation B.1) 
COL.lag.eigb <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw=listw,
 method="eigen", control=list(pre_eig=ev, OrdVsign=-1))
summary(COL.lag.eigb)
COL.lag.eigb$fdHess
COL.lag.eigb$resvar
# force numerical Hessian
COL.lag.eig1 <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw=listw, method="Matrix", control=list(small=25))
summary(COL.lag.eig1)
COL.lag.eig1$fdHess
# force LeSage & Pace (2008, p. 57) approximation 
COL.lag.eig1a <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw=listw, method="Matrix", control=list(small=25), trs=trMatc)
summary(COL.lag.eig1a)
COL.lag.eig1a$fdHess
COL.lag.eig$resvar[2,2]
# using the apparent sign in Ord (1975, equation B.1) 
COL.lag.eigb$resvar[2,2]
# force numerical Hessian
COL.lag.eig1$fdHess[1,1]
# force LeSage & Pace (2008, p. 57) approximation 
COL.lag.eig1a$fdHess[2,2]
# }
# NOT RUN {
system.time(COL.lag.M <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 spdep::nb2listw(COL.nb), method="Matrix", quiet=FALSE))
summary(COL.lag.M)
impacts(COL.lag.M, listw=spdep::nb2listw(COL.nb))
# }
# NOT RUN {
system.time(COL.lag.sp <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw=listw, method="spam", quiet=FALSE))
summary(COL.lag.sp)
# }
# NOT RUN {
COL.lag.B <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 spdep::nb2listw(COL.nb, style="B"))
summary(COL.lag.B)
COL.mixed.B <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 spdep::nb2listw(COL.nb, style="B"), type="mixed", tol.solve=1e-9,
 control=list(pre_eig=ev))
summary(COL.mixed.B)
COL.mixed.W <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, type="mixed", control=list(pre_eig=ev))
summary(COL.mixed.W)
COL.mixed.D00 <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, Durbin=TRUE, control=list(pre_eig=ev))
summary(COL.mixed.D00)
COL.mixed.D01 <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, Durbin=FALSE, control=list(pre_eig=ev))
summary(COL.mixed.D01)
COL.mixed.D1 <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, Durbin= ~ INC + HOVAL, control=list(pre_eig=ev))
summary(COL.mixed.D1)
f <- CRIME ~ INC + HOVAL
COL.mixed.D2 <- lagsarlm(f, data=COL.OLD, listw,
 Durbin=as.formula(delete.response(terms(f))),
 control=list(pre_eig=ev))
summary(COL.mixed.D2)
COL.mixed.D1a <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, Durbin= ~ INC, control=list(pre_eig=ev))
summary(COL.mixed.D1a)
try(COL.mixed.D1 <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, Durbin= ~ inc + HOVAL, control=list(pre_eig=ev)))
try(COL.mixed.D1 <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, Durbin= ~ DISCBD + HOVAL, control=list(pre_eig=ev)))
NA.COL.OLD <- COL.OLD
NA.COL.OLD$CRIME[20:25] <- NA
COL.lag.NA <- lagsarlm(CRIME ~ INC + HOVAL, data=NA.COL.OLD,
 listw, na.action=na.exclude)
COL.lag.NA$na.action
COL.lag.NA
resid(COL.lag.NA)
COL.lag.NA1 <- lagsarlm(CRIME ~ INC + HOVAL, data=NA.COL.OLD,
 listw, Durbin=~INC) # https://github.com/r-spatial/spatialreg/issues/10
COL.lag.NA1$na.action
COL.lag.NA2 <- lagsarlm(CRIME ~ INC + HOVAL, data=NA.COL.OLD,
 listw, Durbin=~INC, na.action=na.exclude)
COL.lag.NA2$na.action
# https://github.com/r-spatial/spatialreg/issues/11
COL.lag.NA3 <- lagsarlm(CRIME ~ INC + HOVAL, data=NA.COL.OLD,
 listw, control=list(pre_eig=ev))
COL.lag.NA3$na.action

# }
# NOT RUN {
data(boston, package="spData")
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, spdep::nb2listw(boston.soi), type="mixed", method="Matrix")
summary(gp2mM)
W <- as(spdep::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, spdep::nb2listw(boston.soi), type="mixed", method="Matrix", 
trs=trMatb)
summary(gp2mMi)
# }
# NOT RUN {
lw <- spdep::nb2listw(COL.nb, style="W")
COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, quiet=FALSE, control=list(pre_eig=ev))
summary(COL.errW.eig)
COL.errW.eig_ev <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, 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,
 spdep::nb2listw(COL.nb, style="B"))
summary(COL.errB.eig)
W <- as(spdep::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)
COL.SDEM.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, etype="emixed", control=list(pre_eig=ev))
summary(COL.SDEM.eig)
COL.SDEM.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, Durbin=TRUE, control=list(pre_eig=ev))
summary(COL.SDEM.eig)
COL.SDEM.eig <- errorsarlm(CRIME ~ DISCBD + INC + HOVAL, data=COL.OLD,
 lw, Durbin=~INC, control=list(pre_eig=ev))
summary(COL.SDEM.eig)
summary(impacts(COL.SDEM.eig))
NA.COL.OLD <- COL.OLD
NA.COL.OLD$CRIME[20:25] <- NA
COL.err.NA <- errorsarlm(CRIME ~ INC + HOVAL, data=NA.COL.OLD,
 spdep::nb2listw(COL.nb), na.action=na.exclude)
COL.err.NA$na.action
COL.err.NA
resid(COL.err.NA)
# }
# NOT RUN {
lw <- spdep::nb2listw(COL.nb, style="W")
print(system.time(ev <- eigenw(similar.listw(lw))))
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="eigen", control=list(pre_eig=ev))))
ocoef <- coefficients(COL.errW.eig)
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="eigen", control=list(pre_eig=ev, LAPACK=FALSE))))
print(all.equal(ocoef, coefficients(COL.errW.eig)))
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="eigen", control=list(pre_eig=ev, compiled_sse=TRUE))))
print(all.equal(ocoef, coefficients(COL.errW.eig)))
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="Matrix_J", control=list(super=TRUE))))
print(all.equal(ocoef, coefficients(COL.errW.eig)))
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="Matrix_J", control=list(super=FALSE))))
print(all.equal(ocoef, coefficients(COL.errW.eig)))
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="Matrix_J", control=list(super=as.logical(NA)))))
print(all.equal(ocoef, coefficients(COL.errW.eig)))
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="Matrix", control=list(super=TRUE))))
print(all.equal(ocoef, coefficients(COL.errW.eig)))
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="Matrix", control=list(super=FALSE))))
print(all.equal(ocoef, coefficients(COL.errW.eig)))
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="Matrix", control=list(super=as.logical(NA)))))
print(all.equal(ocoef, coefficients(COL.errW.eig)))
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="spam", control=list(spamPivot="MMD"))))
print(all.equal(ocoef, coefficients(COL.errW.eig)))
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="spam", control=list(spamPivot="RCM"))))
print(all.equal(ocoef, coefficients(COL.errW.eig)))
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="spam_update", control=list(spamPivot="MMD"))))
print(all.equal(ocoef, coefficients(COL.errW.eig)))
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 lw, method="spam_update", control=list(spamPivot="RCM"))))
print(all.equal(ocoef, coefficients(COL.errW.eig)))
# }
# NOT RUN {
listw <- spdep::nb2listw(COL.nb, style="W")
COL.sacW.eig <- sacsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw,
 control=list(pre_eig1=ev, pre_eig2=ev))
summary(COL.sacW.eig)
W <- as(listw, "CsparseMatrix")
trMatc <- trW(W, type="mult")
set.seed(1)
summary(impacts(COL.sacW.eig, tr=trMatc, R=2000), zstats=TRUE, short=TRUE)
COL.msacW.eig <- sacsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw,
 type="sacmixed", control=list(pre_eig1=ev, pre_eig2=ev))
summary(COL.msacW.eig)
set.seed(1)
summary(impacts(COL.msacW.eig, tr=trMatc, R=2000), zstats=TRUE, short=TRUE)
COL.msacW1.eig <- sacsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw,
 Durbin=TRUE, control=list(pre_eig1=ev, pre_eig2=ev))
summary(COL.msacW1.eig)
set.seed(1)
summary(impacts(COL.msacW1.eig, tr=trMatc, R=2000), zstats=TRUE, short=TRUE)
COL.msacW2.eig <- sacsarlm(CRIME ~ DISCBD + INC + HOVAL, data=COL.OLD, 
 listw, Durbin= ~ INC, control=list(pre_eig1=ev, pre_eig2=ev))
summary(COL.msacW2.eig)
summary(impacts(COL.msacW2.eig, tr=trMatc, R=2000), zstats=TRUE, short=TRUE)
# }
# NOT RUN {
COL.mix.eig <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 spdep::nb2listw(COL.nb), type="mixed", method="eigen")
summary(COL.mix.eig, correlation=TRUE, Nagelkerke=TRUE)
COL.mix.M <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 spdep::nb2listw(COL.nb), type="mixed", method="Matrix")
summary(COL.mix.M, correlation=TRUE, Nagelkerke=TRUE)
COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
  spdep::nb2listw(COL.nb, style="W"), method="eigen")
summary(COL.errW.eig, correlation=TRUE, Nagelkerke=TRUE, Hausman=TRUE)
# }