Spatial simultaneous autoregressive model estimation by maximum likelihood

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.

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, ...)

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, listw2: a listw object created for example by nb2listw; if nb2listw 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 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.
weights: an optional vector of weights to be used in the fitting process. Non-NULL weights can be used to indicate that different observations have different variances (with the values in weights being inversely proportional to the variances); or equivalently, when the elements of weights are positive integers w_i, that each response y_i is the mean of w_i unit-weight observations (including the case that there are w_i observations equal to y_i and the data have been summarized) - lm
Durbin: default FALSE (spatial lag model); if TRUE, full spatial Durbin model; if a formula object, the subset of explanatory variables to lag
type: (use the ‘Durbin=’ argument - retained for backwards compatibility only) 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”
etype: (use the ‘Durbin=’ argument - retained for backwards compatibility only) default "error", may be set to "emixed" to include the spatially lagged independent variables added to X; when "emixed", 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 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
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
trs1, trs2: default NULL, if given, vectors for each weights object of powered spatial weights matrix traces output by trW; when given, used in some Jacobian methods
interval1, interval2: default is NULL, search intervals for each weights object for autoregressive parameters
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
object: Sarlm object from lagsarlm, errorsarlm or sacsarlm
correlation: logical; if 'TRUE', the correlation matrix of the estimated parameters including sigma is returned and printed (default=FALSE)
Nagelkerke: if TRUE, the Nagelkerke pseudo R-squared is reported
Hausman: if TRUE, the results of the Hausman test for error models are reported
adj.se: if TRUE, adjust the coefficient standard errors for the number of fitted coefficients
x: Sarlm object from lagsarlm, errorsarlm or sacsarlm in print.Sarlm, summary object from summary.Sarlm for print.summary.Sarlm
digits: the number of significant digits to use when printing
signif.stars: logical. If TRUE, "significance stars" are printed for each coefficient.
...: further arguments passed to or from other methods

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.

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

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. doi:10.18637/jss.v063.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.

Author

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

Examples

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))
#> 
#> Spatial lag model
#> Jacobian calculated using neighbourhood matrix eigenvalues
#> 
#> rho:	 -0.5674437 	function value:	 -202.2909 
#> rho:	 0.03126655 	function value:	 -186.749 
#> rho:	 0.4012898 	function value:	 -182.419 
#> rho:	 0.6138418 	function value:	 -183.5636 
#> rho:	 0.4157662 	function value:	 -182.398 
#> rho:	 0.4295565 	function value:	 -182.3905 
#> rho:	 0.4311288 	function value:	 -182.3904 
#> rho:	 0.4310273 	function value:	 -182.3904 
#> rho:	 0.4310232 	function value:	 -182.3904 
#> rho:	 0.4310232 	function value:	 -182.3904 
#> rho:	 0.4310232 	function value:	 -182.3904 
#> rho:	 0.4310232 	function value:	 -182.3904 
#> rho:	 0.4310232 	function value:	 -182.3904 
(x <- summary(COL.lag.eig, correlation=TRUE))
#> 
#> Call:lagsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     method = "eigen", quiet = FALSE, control = list(pre_eig = ev, 
#>         OrdVsign = 1))
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -37.68585  -5.35636   0.05421   6.02013  23.20555 
#> 
#> Type: lag 
#> Coefficients: (asymptotic standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 45.079249   7.177346  6.2808 3.369e-10
#> INC         -1.031616   0.305143 -3.3808 0.0007229
#> HOVAL       -0.265926   0.088499 -3.0049 0.0026570
#> 
#> Rho: 0.43102, LR test value: 9.9736, p-value: 0.001588
#> Asymptotic standard error: 0.11768
#>     z-value: 3.6626, p-value: 0.00024962
#> Wald statistic: 13.415, p-value: 0.00024962
#> 
#> Log likelihood: -182.3904 for lag model
#> ML residual variance (sigma squared): 95.494, (sigma: 9.7721)
#> Number of observations: 49 
#> Number of parameters estimated: 5 
#> AIC: 374.78, (AIC for lm: 382.75)
#> LM test for residual autocorrelation
#> test value: 0.31954, p-value: 0.57188
#> 
#>  Correlation of coefficients 
#>             sigma rho   (Intercept) INC  
#> rho         -0.14                        
#> (Intercept)  0.12 -0.83                  
#> INC         -0.05  0.35 -0.61            
#> HOVAL       -0.01  0.08 -0.25       -0.44
#> 
coef(x)
#>               Estimate Std. Error   z value     Pr(>|z|)
#> (Intercept) 45.0792493 7.17734647  6.280768 3.369043e-10
#> INC         -1.0316157 0.30514297 -3.380762 7.228519e-04
#> HOVAL       -0.2659263 0.08849862 -3.004863 2.657002e-03
if (FALSE) {
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]
}
system.time(COL.lag.M <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, method="Matrix", quiet=FALSE))
#> 
#> Spatial lag model
#> Jacobian calculated using sparse matrix Cholesky decomposition
#> Warning: the default value of argument 'sqrt' of method 'determinant(<CHMfactor>, <logical>)' may change from TRUE to FALSE as soon as the next release of Matrix; set 'sqrt' when programming
#> rho:	 -0.2364499 	function value:	 -192.9523 
#> rho:	 0.2354499 	function value:	 -183.542 
#> rho:	 0.5271001 	function value:	 -182.7039 
#> rho:	 0.4455543 	function value:	 -182.3974 
#> rho:	 0.4267907 	function value:	 -182.391 
#> rho:	 0.4311986 	function value:	 -182.3904 
#> rho:	 0.4310114 	function value:	 -182.3904 
#> rho:	 0.4310231 	function value:	 -182.3904 
#> rho:	 0.4310232 	function value:	 -182.3904 
#> rho:	 0.4310232 	function value:	 -182.3904 
#> rho:	 0.4310232 	function value:	 -182.3904 
#> rho:	 0.4310232 	function value:	 -182.3904 
#> Computing eigenvalues ...
#> 
#>    user  system elapsed 
#>   0.180   0.000   0.182 
summary(COL.lag.M)
#> 
#> Call:lagsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     method = "Matrix", quiet = FALSE)
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -37.68585  -5.35636   0.05421   6.02013  23.20555 
#> 
#> Type: lag 
#> Coefficients: (asymptotic standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 45.079249   7.177346  6.2808 3.369e-10
#> INC         -1.031616   0.305143 -3.3808 0.0007229
#> HOVAL       -0.265926   0.088499 -3.0049 0.0026570
#> 
#> Rho: 0.43102, LR test value: 9.9736, p-value: 0.001588
#> Asymptotic standard error: 0.11768
#>     z-value: 3.6626, p-value: 0.00024962
#> Wald statistic: 13.415, p-value: 0.00024962
#> 
#> Log likelihood: -182.3904 for lag model
#> ML residual variance (sigma squared): 95.494, (sigma: 9.7721)
#> Number of observations: 49 
#> Number of parameters estimated: 5 
#> AIC: 374.78, (AIC for lm: 382.75)
#> LM test for residual autocorrelation
#> test value: 0.31954, p-value: 0.57188
#> 
impacts(COL.lag.M, listw=listw)
#> Impact measures (lag, exact):
#>           Direct   Indirect      Total
#> INC   -1.0860220 -0.7270848 -1.8131068
#> HOVAL -0.2799509 -0.1874254 -0.4673763
if (FALSE) {
system.time(COL.lag.sp <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw=listw, method="spam", quiet=FALSE))
summary(COL.lag.sp)
COL.lag.B <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 spdep::nb2listw(COL.nb, style="B"), control=list(pre_eig=ev))
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
}

if (FALSE) {
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)
}
COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, quiet=FALSE, control=list(pre_eig=ev))
#> 
#> Spatial autoregressive error model
#> 
#> Jacobian calculated using neighbourhood matrix eigenvalues
#> 
#> lambda: -0.5674437  function: -195.8051  Jacobian: -1.636549  SSE: 7936.201 
#> lambda: 0.03126655  function: -187.0219  Jacobian: -0.005318373  SSE: 5927.009 
#> lambda: 0.4012898  function: -183.8422  Jacobian: -0.9953987  SSE: 4999.419 
#> lambda: 0.6299767  function: -183.4895  Jacobian: -2.818134  SSE: 4574.641 
#> lambda: 0.5811116  function: -183.3887  Jacobian: -2.314073  SSE: 4650.566 
#> lambda: 0.554104  function: -183.3817  Jacobian: -2.066354  SSE: 4696.49 
#> lambda: 0.5621834  function: -183.3805  Jacobian: -2.138326  SSE: 4682.474 
#> lambda: 0.5617028  function: -183.3805  Jacobian: -2.133995  SSE: 4683.301 
#> lambda: 0.5617888  function: -183.3805  Jacobian: -2.134769  SSE: 4683.153 
#> lambda: 0.5617902  function: -183.3805  Jacobian: -2.134782  SSE: 4683.151 
#> lambda: 0.5617903  function: -183.3805  Jacobian: -2.134782  SSE: 4683.151 
#> lambda: 0.5617902  function: -183.3805  Jacobian: -2.134782  SSE: 4683.151 
#> lambda: 0.5617902  function: -183.3805  Jacobian: -2.134782  SSE: 4683.151 
summary(COL.errW.eig)
#> 
#> Call:errorsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     quiet = FALSE, control = list(pre_eig = ev))
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -34.81174  -6.44031  -0.72142   7.61476  23.33626 
#> 
#> Type: error 
#> Coefficients: (asymptotic standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 59.893220   5.366162 11.1613 < 2.2e-16
#> INC         -0.941312   0.330569 -2.8476 0.0044057
#> HOVAL       -0.302250   0.090476 -3.3407 0.0008358
#> 
#> Lambda: 0.56179, LR test value: 7.9935, p-value: 0.0046945
#> Asymptotic standard error: 0.13387
#>     z-value: 4.1966, p-value: 2.7098e-05
#> Wald statistic: 17.611, p-value: 2.7098e-05
#> 
#> Log likelihood: -183.3805 for error model
#> ML residual variance (sigma squared): 95.575, (sigma: 9.7762)
#> Number of observations: 49 
#> Number of parameters estimated: 5 
#> AIC: 376.76, (AIC for lm: 382.75)
#> 
COL.errW.eig_ev <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, control=list(pre_eig=ev))
all.equal(coefficients(COL.errW.eig), coefficients(COL.errW.eig_ev))
#> [1] TRUE
COL.errB.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 spdep::nb2listw(COL.nb, style="B"))
summary(COL.errB.eig)
#> 
#> Call:
#> errorsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = spdep::nb2listw(COL.nb, 
#>     style = "B"))
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -32.19010  -5.22646  -0.69952   7.92588  24.23511 
#> 
#> Type: error 
#> Coefficients: (asymptotic standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 55.383119   5.449775 10.1625 < 2.2e-16
#> INC         -0.936595   0.319355 -2.9328 0.0033596
#> HOVAL       -0.299857   0.088678 -3.3814 0.0007212
#> 
#> Lambda: 0.12686, LR test value: 10.654, p-value: 0.0010983
#> Asymptotic standard error: 0.021745
#>     z-value: 5.8342, p-value: 5.4044e-09
#> Wald statistic: 34.038, p-value: 5.4044e-09
#> 
#> Log likelihood: -182.0502 for error model
#> ML residual variance (sigma squared): 88.744, (sigma: 9.4204)
#> Number of observations: 49 
#> Number of parameters estimated: 5 
#> AIC: 374.1, (AIC for lm: 382.75)
#> 
COL.errW.M <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, method="Matrix", quiet=FALSE, trs=trMatc)
#> 
#> Spatial autoregressive error model
#> 
#> Jacobian calculated using sparse matrix Cholesky decomposition
#> lambda: -0.2364499  function: -190.4287  Jacobian: -0.2899343  SSE: 6732.556 
#> lambda: 0.2354499  function: -185.0024  Jacobian: -0.3201148  SSE: 5388.346 
#> lambda: 0.5271001  function: -183.4053  Jacobian: -1.838306  SSE: 4744.975 
#> lambda: 0.728364  function: -184.1244  Jacobian: -4.106761  SSE: 4454.203 
#> lambda: 0.5304163  function: -183.4009  Jacobian: -1.865308  SSE: 4738.889 
#> lambda: 0.5557478  function: -183.3812  Jacobian: -2.080853  SSE: 4693.62 
#> lambda: 0.6216813  function: -183.4637  Jacobian: -2.727084  SSE: 4586.84 
#> lambda: 0.5627129  function: -183.3805  Jacobian: -2.143105  SSE: 4681.563 
#> lambda: 0.5618852  function: -183.3805  Jacobian: -2.135638  SSE: 4682.987 
#> lambda: 0.5617866  function: -183.3805  Jacobian: -2.134749  SSE: 4683.157 
#> lambda: 0.5617903  function: -183.3805  Jacobian: -2.134783  SSE: 4683.15 
#> lambda: 0.5617903  function: -183.3805  Jacobian: -2.134782  SSE: 4683.151 
#> lambda: 0.5617903  function: -183.3805  Jacobian: -2.134782  SSE: 4683.151 
#> lambda: 0.5617903  function: -183.3805  Jacobian: -2.134783  SSE: 4683.151 
#> lambda: 0.5617903  function: -183.3805  Jacobian: -2.134783  SSE: 4683.151 
#> lambda: 0.5617903  function: -183.3805  Jacobian: -2.134783  SSE: 4683.151 
summary(COL.errW.M)
#> 
#> Call:errorsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     method = "Matrix", quiet = FALSE, trs = trMatc)
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -34.81174  -6.44031  -0.72142   7.61476  23.33626 
#> 
#> Type: error 
#> Coefficients: (asymptotic standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 59.893219   5.366163 11.1613 < 2.2e-16
#> INC         -0.941312   0.330569 -2.8476 0.0044057
#> HOVAL       -0.302250   0.090476 -3.3407 0.0008358
#> 
#> Lambda: 0.56179, LR test value: 7.9935, p-value: 0.0046945
#> Asymptotic standard error: 0.13387
#>     z-value: 4.1966, p-value: 2.7098e-05
#> Wald statistic: 17.611, p-value: 2.7098e-05
#> 
#> Log likelihood: -183.3805 for error model
#> ML residual variance (sigma squared): 95.575, (sigma: 9.7762)
#> Number of observations: 49 
#> Number of parameters estimated: 5 
#> AIC: 376.76, (AIC for lm: 382.75)
#> 
COL.SDEM.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, etype="emixed", control=list(pre_eig=ev))
summary(COL.SDEM.eig)
#> 
#> Call:errorsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     etype = "emixed", control = list(pre_eig = ev))
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -37.31635  -6.54376  -0.22212   6.44591  23.15801 
#> 
#> Type: error 
#> Coefficients: (asymptotic standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 73.545133   8.783543  8.3731 < 2.2e-16
#> INC         -1.051673   0.319514 -3.2915 0.0009966
#> HOVAL       -0.275608   0.091151 -3.0236 0.0024976
#> lag.INC     -1.156711   0.578629 -1.9991 0.0456024
#> lag.HOVAL    0.111691   0.198993  0.5613 0.5746048
#> 
#> Lambda: 0.4254, LR test value: 4.9871, p-value: 0.025537
#> Asymptotic standard error: 0.15842
#>     z-value: 2.6852, p-value: 0.0072485
#> Wald statistic: 7.2103, p-value: 0.0072485
#> 
#> Log likelihood: -181.5846 for error model
#> ML residual variance (sigma squared): 92.531, (sigma: 9.6193)
#> Number of observations: 49 
#> Number of parameters estimated: 7 
#> AIC: 377.17, (AIC for lm: 380.16)
#> 
if (FALSE) {
COL.SDEM.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, Durbin=TRUE, control=list(pre_eig=ev))
summary(COL.SDEM.eig)
COL.SDEM.eig <- errorsarlm(CRIME ~ DISCBD + INC + HOVAL, data=COL.OLD,
 listw, Durbin=~INC, control=list(pre_eig=ev))
summary(COL.SDEM.eig)
summary(impacts(COL.SDEM.eig))
NA.COL.OLD <- COL.OLD
NA.COL.OLD$CRIME[20:25] <- NA
COL.err.NA <- errorsarlm(CRIME ~ INC + HOVAL, data=NA.COL.OLD,
 listw, na.action=na.exclude)
COL.err.NA$na.action
COL.err.NA
resid(COL.err.NA)
print(system.time(ev <- eigenw(similar.listw(listw))))
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, 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,
 listw, 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,
 listw, 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,
 listw, 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,
 listw, 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,
 listw, 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,
 listw, 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,
 listw, 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,
 listw, 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,
 listw, 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,
 listw, 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,
 listw, 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,
 listw, method="spam_update", control=list(spamPivot="RCM"))))
print(all.equal(ocoef, coefficients(COL.errW.eig)))
}
COL.sacW.eig <- sacsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, listw,
 control=list(pre_eig1=ev, pre_eig2=ev))
summary(COL.sacW.eig)
#> 
#> Call:sacsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     control = list(pre_eig1 = ev, pre_eig2 = ev))
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -37.32081  -5.33662  -0.20219   6.59672  23.25604 
#> 
#> Type: sac 
#> Coefficients: (asymptotic standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 47.783764   9.902659  4.8253 1.398e-06
#> INC         -1.025894   0.326326 -3.1438  0.001668
#> HOVAL       -0.281651   0.090033 -3.1283  0.001758
#> 
#> Rho: 0.36807
#> Asymptotic standard error: 0.19668
#>     z-value: 1.8714, p-value: 0.061285
#> Lambda: 0.16668
#> Asymptotic standard error: 0.29661
#>     z-value: 0.56196, p-value: 0.57415
#> 
#> LR test value: 10.285, p-value: 0.0058432
#> 
#> Log likelihood: -182.2348 for sac model
#> ML residual variance (sigma squared): 95.604, (sigma: 9.7777)
#> Number of observations: 49 
#> Number of parameters estimated: 6 
#> AIC: 376.47, (AIC for lm: 382.75)
#> 
set.seed(1)
summary(impacts(COL.sacW.eig, tr=trMatc, R=2000), zstats=TRUE, short=TRUE)
#> Impact measures (sac, trace):
#>           Direct   Indirect      Total
#> INC   -1.0632723 -0.5601502 -1.6234225
#> HOVAL -0.2919129 -0.1537848 -0.4456977
#> ========================================================
#> Simulation results ( variance matrix):
#> ========================================================
#> Simulated standard errors
#>          Direct  Indirect     Total
#> INC   0.3200735 0.8453130 0.9606199
#> HOVAL 0.0947660 0.2823186 0.3278763
#> 
#> Simulated z-values:
#>          Direct   Indirect     Total
#> INC   -3.376570 -0.8371025 -1.861677
#> HOVAL -3.159909 -0.7431010 -1.553156
#> 
#> Simulated p-values:
#>       Direct     Indirect Total   
#> INC   0.00073396 0.40253  0.062649
#> HOVAL 0.00157819 0.45742  0.120386
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)
#> 
#> Call:sacsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     type = "sacmixed", control = list(pre_eig1 = ev, pre_eig2 = ev))
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -37.8045  -6.5244  -0.2207   5.9944  22.8691 
#> 
#> Type: sacmixed 
#> Coefficients: (asymptotic standard errors) 
#>             Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 50.92026   68.25721  0.7460 0.455664
#> INC         -0.95072    0.44033 -2.1591 0.030841
#> HOVAL       -0.28650    0.09994 -2.8667 0.004148
#> lag.INC     -0.69261    1.69113 -0.4096 0.682132
#> lag.HOVAL    0.20852    0.28702  0.7265 0.467546
#> 
#> Rho: 0.31557
#> Asymptotic standard error: 0.9458
#>     z-value: 0.33365, p-value: 0.73864
#> Lambda: 0.15415
#> Asymptotic standard error: 1.0643
#>     z-value: 0.14484, p-value: 0.88484
#> 
#> LR test value: 12.07, p-value: 0.016837
#> 
#> Log likelihood: -181.3422 for sacmixed model
#> ML residual variance (sigma squared): 93.149, (sigma: 9.6514)
#> Number of observations: 49 
#> Number of parameters estimated: 8 
#> AIC: 378.68, (AIC for lm: 382.75)
#> 
set.seed(1)
summary(impacts(COL.msacW.eig, tr=trMatc, R=2000), zstats=TRUE, short=TRUE)
#> Impact measures (sacmixed, trace):
#>           Direct   Indirect      Total
#> INC   -1.0317003 -1.3693141 -2.4010144
#> HOVAL -0.2768608  0.1629265 -0.1139344
#> ========================================================
#> Simulation results ( variance matrix):
#> ========================================================
#> Simulated standard errors
#>          Direct  Indirect     Total
#> INC   0.3778064 2.1709289 2.3099448
#> HOVAL 0.1144714 0.7736216 0.8389226
#> 
#> Simulated z-values:
#>          Direct   Indirect       Total
#> INC   -2.737182 -0.7297008 -1.13347010
#> HOVAL -2.364129  0.2526689 -0.08958505
#> 
#> Simulated p-values:
#>       Direct    Indirect Total  
#> INC   0.0061968 0.46557  0.25702
#> HOVAL 0.0180725 0.80052  0.92862
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)
#> 
#> Call:sacsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     Durbin = TRUE, control = list(pre_eig1 = ev, pre_eig2 = ev))
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -37.8045  -6.5244  -0.2207   5.9944  22.8691 
#> 
#> Type: sacmixed 
#> Coefficients: (asymptotic standard errors) 
#>             Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 50.92026   68.25721  0.7460 0.455664
#> INC         -0.95072    0.44033 -2.1591 0.030841
#> HOVAL       -0.28650    0.09994 -2.8667 0.004148
#> lag.INC     -0.69261    1.69113 -0.4096 0.682132
#> lag.HOVAL    0.20852    0.28702  0.7265 0.467546
#> 
#> Rho: 0.31557
#> Asymptotic standard error: 0.9458
#>     z-value: 0.33365, p-value: 0.73864
#> Lambda: 0.15415
#> Asymptotic standard error: 1.0643
#>     z-value: 0.14484, p-value: 0.88484
#> 
#> LR test value: 12.07, p-value: 0.016837
#> 
#> Log likelihood: -181.3422 for sacmixed model
#> ML residual variance (sigma squared): 93.149, (sigma: 9.6514)
#> Number of observations: 49 
#> Number of parameters estimated: 8 
#> AIC: 378.68, (AIC for lm: 382.75)
#> 
set.seed(1)
summary(impacts(COL.msacW1.eig, tr=trMatc, R=2000), zstats=TRUE, short=TRUE)
#> Impact measures (sacmixed, trace):
#>           Direct   Indirect      Total
#> INC   -1.0317003 -1.3693141 -2.4010144
#> HOVAL -0.2768608  0.1629265 -0.1139344
#> ========================================================
#> Simulation results ( variance matrix):
#> ========================================================
#> Simulated standard errors
#>          Direct  Indirect     Total
#> INC   0.3778064 2.1709289 2.3099448
#> HOVAL 0.1144714 0.7736216 0.8389226
#> 
#> Simulated z-values:
#>          Direct   Indirect       Total
#> INC   -2.737182 -0.7297008 -1.13347010
#> HOVAL -2.364129  0.2526689 -0.08958505
#> 
#> Simulated p-values:
#>       Direct    Indirect Total  
#> INC   0.0061968 0.46557  0.25702
#> HOVAL 0.0180725 0.80052  0.92862
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)
#> 
#> Call:sacsarlm(formula = CRIME ~ DISCBD + INC + HOVAL, data = COL.OLD, 
#>     listw = listw, Durbin = ~INC, control = list(pre_eig1 = ev, 
#>         pre_eig2 = ev))
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -34.2794  -7.0786   1.0543   6.0019  17.8891 
#> 
#> Type: sacmixed 
#> Coefficients: (asymptotic standard errors) 
#>              Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 74.064502  37.738940  1.9625 0.049699
#> DISCBD      -5.707678   3.248629 -1.7569 0.078926
#> INC         -0.900975   0.314996 -2.8603 0.004233
#> HOVAL       -0.203399   0.092982 -2.1875 0.028705
#> lag.INC      0.061568   0.926947  0.0664 0.947043
#> 
#> Rho: -0.077599
#> Asymptotic standard error: 0.57955
#>     z-value: -0.1339, p-value: 0.89348
#> Lambda: 0.29646
#> Asymptotic standard error: 0.51021
#>     z-value: 0.58104, p-value: 0.56121
#> 
#> LR test value: 1.4982, p-value: 0.68269
#> 
#> Log likelihood: -178.963 for sacmixed model
#> ML residual variance (sigma squared): 85.135, (sigma: 9.2269)
#> Number of observations: 49 
#> Number of parameters estimated: 8 
#> AIC: 373.93, (AIC for lm: 369.42)
#> 
summary(impacts(COL.msacW2.eig, tr=trMatc, R=2000), zstats=TRUE, short=TRUE)
#> Impact measures (sacmixed, trace):
#>            Direct   Indirect      Total
#> DISCBD -5.7150799 0.41841671 -5.2966632
#> INC    -0.9031729 0.12421198 -0.7789609
#> HOVAL  -0.2036631 0.01491074 -0.1887524
#> ========================================================
#> Simulation results ( variance matrix):
#> ========================================================
#> Simulated standard errors
#>           Direct  Indirect    Total
#> DISCBD 3.1446963 5.7201656 5.757459
#> INC    0.3621137 1.7150155 1.876353
#> HOVAL  0.1090172 0.6360488 0.698836
#> 
#> Simulated z-values:
#>           Direct   Indirect      Total
#> DISCBD -1.852002  0.0280018 -0.9837341
#> INC    -2.461864  0.1861184 -0.3049951
#> HOVAL  -2.021743 -0.1841543 -0.4829973
#> 
#> Simulated p-values:
#>        Direct   Indirect Total  
#> DISCBD 0.064026 0.97766  0.32525
#> INC    0.013822 0.85235  0.76037
#> HOVAL  0.043203 0.85389  0.62910
if (FALSE) {
COL.mix.eig <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, type="mixed", method="eigen")
summary(COL.mix.eig, correlation=TRUE, Nagelkerke=TRUE)
COL.mix.M <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, 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)
}