Spatial simultaneous autoregressive SAC model estimation by GMM

An implementation of Kelejian and Prucha's generalised moments estimator for the autoregressive parameter in a spatial model with a spatially lagged dependent variable.

gstsls(formula, data = list(), listw, listw2 = NULL, na.action = na.fail, 
    zero.policy = NULL, pars=NULL, scaleU=FALSE, control = list(), 
    verbose=NULL, method="nlminb", robust=FALSE, legacy=FALSE, W2X=TRUE) 
# S3 method for Gmsar
impacts(obj, ..., n = NULL, tr = NULL, R = NULL,
 listw = NULL, evalues=NULL, tol = 1e-06, empirical = FALSE, Q=NULL)

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 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.
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 GMerrorsar() to terminate with an error
pars

starting values for \(\lambda\) and \(\sigma^2\) for GMM optimisation, if missing (default), approximated from initial 2sls model as the autocorrelation coefficient corrected for weights style and model sigma squared
scaleU

Default FALSE: scale the OLS residuals before computing the moment matrices; only used if the pars argument is missing
control

A list of control parameters. See details in optim or nlminb
verbose

default NULL, use global option value; if TRUE, reports function values during optimization.
method

default nlminb, or optionally a method passed to optim to use an alternative optimizer
robust

see stsls
legacy

see stsls
W2X

see stsls
obj

A spatial regression object created by lagsarlm, lagmess or by lmSLX; in HPDinterval.LagImpact, a LagImpact object
...

Arguments passed through to methods in the coda package
tr

A vector of traces of powers of the spatial weights matrix created using trW, for approximate impact measures; if not given, listw must be given for exact measures (for small to moderate spatial weights matrices); the traces must be for the same spatial weights as were used in fitting the spatial regression, and must be row-standardised
evalues

vector of eigenvalues of spatial weights matrix for impacts calculations
R

If given, simulations are used to compute distributions for the impact measures, returned as mcmc objects; the objects are used for convenience but are not output by an MCMC process
tol

Argument passed to mvrnorm: tolerance (relative to largest variance) for numerical lack of positive-definiteness in the coefficient covariance matrix
empirical

Argument passed to mvrnorm (default FALSE): if true, the coefficients and their covariance matrix specify the empirical not population mean and covariance matrix
Q

default NULL, else an integer number of cumulative power series impacts to calculate if tr is given
n

defaults to length(obj$residuals); in the method for Gmsar objects it may be used in panel settings to compute the impacts for cross-sectional weights only, suggested by Angela Parenti

Details

When the control list is set with care, the function will converge to values close to the ML estimator without requiring computation of the Jacobian, the most resource-intensive part of ML estimation.

Value

A list object of class Gmsar

lambda

simultaneous autoregressive error coefficient

coefficients

GMM coefficient estimates (including the spatial autocorrelation coefficient)

rest.se

GMM coefficient standard errors

s2

GMM residual variance

SSE

sum of squared GMM errors

parameters

number of parameters estimated

lm.model

NULL

call

the call used to create this object

residuals

GMM residuals

lm.target

NULL

fitted.values

Difference between residuals and response variable

formula

model formula

aliased

NULL

zero.policy

zero.policy for this model

LL

NULL

vv

list of internal bigG and litg components for testing optimisation surface

optres

object returned by optimizer

pars

start parameter values for optimisation

Hcov

NULL

na.action

(possibly) named vector of excluded or omitted observations if non-default na.action argument used

References

Kelejian, H. H., and Prucha, I. R., 1999. A Generalized Moments Estimator for the Autoregressive Parameter in a Spatial Model. International Economic Review, 40, pp. 509--533; Cressie, N. A. C. 1993 Statistics for spatial data, Wiley, New York.

Roger Bivand, Gianfranco Piras (2015). Comparing Implementations of Estimation Methods for Spatial Econometrics. Journal of Statistical Software, 63(18), 1-36. doi: 10.18637/jss.v063.i18 .

Author

Gianfranco Piras and Roger Bivand

See also

optim, nlminb, GMerrorsar, GMargminImage

Examples

#require("spdep", quietly=TRUE) 
data(oldcol, package="spdep")
COL.errW.GM <- gstsls(CRIME ~ INC + HOVAL, data=COL.OLD, spdep::nb2listw(COL.nb, style="W"))
summary(COL.errW.GM)
#> 
#> Call:
#> gstsls(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = spdep::nb2listw(COL.nb, 
#>     style = "W"))
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -37.87710  -5.74981  -0.21447   6.19490  22.85527 
#> 
#> Type: GM SARAR estimator
#> Coefficients: (GM standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> Rho_Wy       0.454171   0.185680  2.4460   0.01445
#> (Intercept) 43.782818  10.923192  4.0082 6.117e-05
#> INC         -0.994831   0.382774 -2.5990   0.00935
#> HOVAL       -0.267076   0.091984 -2.9035   0.00369
#> 
#> Lambda: 0.016647
#> Residual variance (sigma squared): 103.36, (sigma: 10.167)
#> GM argmin sigma squared: 93.729
#> Number of observations: 49 
#> Number of parameters estimated: 6 
#> 
aa <- GMargminImage(COL.errW.GM)
levs <- quantile(aa$z, seq(0, 1, 1/12))
image(aa, breaks=levs, xlab="lambda", ylab="s2")
points(COL.errW.GM$lambda, COL.errW.GM$s2, pch=3, lwd=2)
contour(aa, levels=signif(levs, 4), add=TRUE)

COL.errW.GM <- gstsls(CRIME ~ INC + HOVAL, data=COL.OLD,
 spdep::nb2listw(COL.nb, style="W"), scaleU=TRUE)
summary(COL.errW.GM)
#> 
#> Call:
#> gstsls(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = spdep::nb2listw(COL.nb, 
#>     style = "W"), scaleU = TRUE)
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -37.87710  -5.74981  -0.21447   6.19490  22.85527 
#> 
#> Type: GM SARAR estimator
#> Coefficients: (GM standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> Rho_Wy       0.454171   0.185680  2.4460   0.01445
#> (Intercept) 43.782818  10.923192  4.0082 6.117e-05
#> INC         -0.994831   0.382774 -2.5990   0.00935
#> HOVAL       -0.267076   0.091984 -2.9035   0.00369
#> 
#> Lambda: 0.016647
#> Residual variance (sigma squared): 103.36, (sigma: 10.167)
#> (scaled) GM argmin sigma squared: 0.96653
#> Number of observations: 49 
#> Number of parameters estimated: 6 
#> 
listw <- spdep::nb2listw(COL.nb)
W <- as(listw, "CsparseMatrix")
trMat <- trW(W, type="mult")
impacts(COL.errW.GM, tr=trMat)
#> Impact measures (lag, trace):
#>           Direct   Indirect      Total
#> INC   -1.0543769 -0.7682309 -1.8226078
#> HOVAL -0.2830618 -0.2062420 -0.4893038