Spatial simultaneous autoregressive lag model estimation

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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 + 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.

The spBreg_lag function is an early-release version of the Matlab Spatial Econometrics Toolbox function sar_g.m, using drawing by inversion, and not accommodating heteroskedastic disturbances.

Usage

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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())
spBreg_lag(formula, data = list(), listw, na.action, type="lag",
    zero.policy=NULL, control=list())

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.

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; “Durbin” may be used instead of “mixed”

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 or Matrix packages 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 (default) assign NA - causing lagsarlm() 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; the use of trs is recommended

control

list of extra control arguments - see section below

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 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.

Value

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, optim used to calculate Hessian at optimum

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

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

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.

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)

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

Extra Bayesian control arguments

ldet_method

default “SE_classic”; equivalent to the method argument in lagsarlm

interval

default c(-1, 1); used unmodified or set internally by jacobianSetup

ndraw

default 2500L; integer total number of draws

nomit

default 500L; integer total number of omitted burn-in draws

thin

default 1L; integer thinning proportion

verbose

default FALSE; inverse of quiet argument in lagsarlm

detval

default NULL; not yet in use, precomputed matrix of log determinants

prior

a list with the following components:

Tbeta

default NULL; values of the betas variance-covariance matrix, set to diag(k)*1e+12 if NULL

c_beta

default NULL; values of the betas set to 0 if NULL

rho

default 0.5; value of the autoregressive coefficient

sige

default 1; value of the residual variance

nu

default 0; informative Gamma(nu,d0) prior on sige

d0

default 0; informative Gamma(nu,d0) prior on sige

a1

default 1.01; parameter for beta(a1,a2) prior on rho

a2

default 1.01; parameter for beta(a1,a2) prior on rho

Author(s)

Roger Bivand Roger.Bivand@nhh.no, with thanks to Andrew Bernat for contributions to the asymptotic standard error code.

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, errorsarlm, summary.sarlm, eigenw, predict.sarlm, impacts.sarlm, residuals.sarlm, do_ldet

Examples

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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(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)
## Not run: 
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(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)

## End(Not run)
summary(COL.lag.eig)
COL.lag.Bayes <- spBreg_lag(CRIME ~ INC + HOVAL, data=COL.OLD,
 nb2listw(COL.nb, style="W"))
summary(COL.lag.Bayes)
set.seed(1)
summary(impacts(COL.lag.Bayes, tr=trMatc), short=TRUE, zstats=TRUE)
## Not run: 
data(elect80)
lw <- nb2listw(e80_queen, zero.policy=TRUE)
el_ml <- lagsarlm(log(pc_turnout) ~ log(pc_college) + log(pc_homeownership)
 + log(pc_income), data=elect80, listw=lw, zero.policy=TRUE, method="LU")
summary(el_ml)
set.seed(1)
el_B <- spBreg_lag(log(pc_turnout) ~ log(pc_college) + log(pc_homeownership)
 + log(pc_income), data=elect80, listw=lw, zero.policy=TRUE)
summary(el_B)
el_ml$timings
attr(el_B, "timings")

## End(Not run)

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