spdep (version 0.5-3)

sacsarlm: Spatial simultaneous autoregressive SAC model estimation

Description

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

sacsarlm(formula, data = list(), listw, listw2 = NULL, na.action, type="sac",
 method = "eigen", quiet = NULL, zero.policy = NULL, tol.solve = 1e-10,
 llprof=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

listw2

a listw object created for example by nb2listw, if not given, set to the same spatial weights as the listw argument

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

type

default "sac", may be set to "sacmixed" for the Manski model to include the spatially lagged independent variables added to X using listw; when "sacmixed", the lagged intercept is dropped for spatial weights style "W", that is row-standardise

method

"eigen" (default) - the Jacobian is computed as the product of (1 - rho*eigenvalue) using eigenw, and "spam" or "Matrix" for strictly symmetric weights lists of styles "B" and "C", or made symmetric by similarity (Ord, 1975, Appendix C) if p

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 sacsarlm() to terminate with an error

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

llprof

default NULL, can either be an integer, to divide the feasible ranges into a grid of points, or a two-column matrix of spatial coefficient values, at which to evaluate the likelihood function

control

list of extra control arguments - see section below

Value

• A list object of class sarlm
• type
  sac
• rho
  lag simultaneous autoregressive lag coefficient
• lambda
  error simultaneous autoregressive error coefficient
• coefficients
  GLS coefficient estimates
• rest.se
  asymptotic standard errors if ase=TRUE, otherwise approximate numeriacal Hessian-based values
• ase
  TRUE if method=eigen
• 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 $\rho=0$
• 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
• opt
  object returned from numerical optimisation
• pars
  starting parameter values for final optimization, either given or found by trial point evaluation
• mxs
  if default input pars, optimal objective function values at trial points
• fitted.values
  Difference between residuals and response variable
• se.fit
  Not used yet
• formula
  model formula
• rho.se
  if ase=TRUE, the asymptotic standard error of $\rho$, otherwise approximate numeriacal Hessian-based value
• lambda.se
  if ase=TRUE, the asymptotic standard error of $\lambda$
• resvar
  the asymptotic coefficient covariance matrix for (s2, rho, lambda, B)
• zero.policy
  zero.policy for this model
• aliased
  the aliased explanatory variables (if any)
• LLNullLlm
  Log-likelihood of the null linear model
• fdHess
  the numerical Hessian-based coefficient covariance matrix for (rho, B) if computed
• resvar
• optimHess
  if TRUE and fdHess returned, optim used to calculate Hessian at optimum
• timings
  processing timings
• na.action
  (possibly) named vector of excluded or omitted observations if non-default na.action argument used

Details

Because numerical optimisation is used to find the values of lambda and rho, 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

Anselin, L. 1988 Spatial econometrics: methods and models. (Dordrecht: Kluwer); LeSage J and RK Pace (2009) Introduction to Spatial Econometrics. CRC Press, Boca Raton

See Also

help,

Examples

data(oldcol)
COL.sacW.eig <- sacsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, 
 nb2listw(COL.nb, style="W"))
summary(COL.sacW.eig, correlation=TRUE)
W <- as(as_dgRMatrix_listw(nb2listw(COL.nb, style="W")), "CsparseMatrix")
trMatc <- trW(W, type="mult")
summary(impacts(COL.sacW.eig, tr=trMatc, R=2000), zstats=TRUE, short=TRUE)
COL.msacW.eig <- sacsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, 
 nb2listw(COL.nb, style="W"), type="sacmixed")
summary(COL.msacW.eig, correlation=TRUE)
summary(impacts(COL.msacW.eig, tr=trMatc, R=2000), zstats=TRUE, short=TRUE)