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.

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$double.eps,
 llprof=NULL, interval1=NULL, interval2=NULL, trs1=NULL, trs2=NULL,
 control = list())
# S3 method for class 'Sarlm'
summary(object, correlation = FALSE, Nagelkerke = FALSE,
 Hausman=FALSE, adj.se=FALSE, ...)
# S3 method for class 'Sarlm'
print(x, ...)
# S3 method for class 'summary.Sarlm'
print(x, digits = max(5, .Options$digits - 3),
 signif.stars = FALSE, ...)
# S3 method for class 'Sarlm'
residuals(object, ...)
# S3 method for class 'Sarlm'
deviance(object, ...)
# S3 method for class 'Sarlm'
coef(object, ...)
# S3 method for class 'Sarlm'
vcov(object, ...)
# S3 method for class '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
return_impacts: default TRUE; may be set FALSE to avoid problems calculating impacts with aliased variables
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
# \dontrun{
COL.lag.eig$fdHess
#> [1] FALSE
COL.lag.eig$resvar
#>                    sigma           rho (Intercept)         INC         HOVAL
#> sigma       379.77510023 -0.3236306420  16.3015085 -0.29590802 -0.0202478469
#> rho          -0.32363064  0.0138487528  -0.6975717  0.01266245  0.0008664428
#> (Intercept)  16.30150847 -0.6975716519  51.5143024 -1.32602702 -0.1616379845
#> INC          -0.29590802  0.0126624508  -1.3260270  0.09311223 -0.0117959714
#> HOVAL        -0.02024785  0.0008664428  -0.1616380 -0.01179597  0.0078320057
# 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)
#> 
#> Call:lagsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     method = "eigen", 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   9.617835  4.6870 2.772e-06
#> INC         -1.031616   0.326524 -3.1594  0.001581
#> HOVAL       -0.265926   0.088855 -2.9928  0.002764
#> 
#> Rho: 0.43102, LR test value: 9.9736, p-value: 0.001588
#> Asymptotic standard error: 0.17322
#>     z-value: 2.4884, p-value: 0.012833
#> Wald statistic: 6.1919, p-value: 0.012833
#> 
#> 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: -0.93825, p-value: 1
#> 
COL.lag.eigb$fdHess
#> [1] FALSE
COL.lag.eigb$resvar
#>                    sigma          rho (Intercept)         INC        HOVAL
#> sigma       388.59742708 -0.701154300  35.3176470 -0.64109252 -0.043867493
#> rho          -0.70115430  0.030003687  -1.5113073  0.02743353  0.001877171
#> (Intercept)  35.31764699 -1.511307337  92.5027548 -2.07005706 -0.212549096
#> INC          -0.64109252  0.027433533  -2.0700571  0.10661800 -0.010871823
#> HOVAL        -0.04386749  0.001877171  -0.2125491 -0.01087182  0.007895242
# 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)
#> 
#> Call:lagsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     method = "Matrix", control = list(small = 25))
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -37.68585  -5.35636   0.05421   6.02013  23.20555 
#> 
#> Type: lag 
#> Coefficients: (numerical Hessian approximate standard errors) 
#>             Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 45.07925    7.87142   5.727 1.022e-08
#> INC         -1.03162    0.32843  -3.141  0.001683
#> HOVAL       -0.26593    0.08823  -3.014  0.002578
#> 
#> Rho: 0.43102, LR test value: 9.9736, p-value: 0.001588
#> Approximate (numerical Hessian) standard error: 0.12363
#>     z-value: 3.4865, p-value: 0.00048934
#> Wald statistic: 12.156, p-value: 0.00048934
#> 
#> 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)
#> 
COL.lag.eig1$fdHess
#>                       rho (Intercept)         INC         HOVAL
#> rho          0.0152832589  -0.8346578  0.02005909  0.0002832041
#> (Intercept) -0.8346577895  61.9591934 -1.78361966 -0.1334688151
#> INC          0.0200590872  -1.7836197  0.10786711 -0.0122201879
#> HOVAL        0.0002832041  -0.1334688 -0.01222019  0.0077846116
# force LeSage & Pace (2008, p. 57) approximation 
COL.lag.eig1a <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw=listw, method="Matrix", control=list(small=25), trs=trMatc)
summary(COL.lag.eig1a)
#> 
#> Call:lagsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     method = "Matrix", trs = trMatc, control = list(small = 25))
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -37.68585  -5.35636   0.05421   6.02013  23.20555 
#> 
#> Type: lag 
#> Coefficients: (numerical Hessian approximate standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 45.079249   7.937572  5.6792 1.353e-08
#> INC         -1.031616   0.329337 -3.1324  0.001734
#> HOVAL       -0.265926   0.088222 -3.0143  0.002576
#> 
#> Rho: 0.43102, LR test value: 9.9736, p-value: 0.001588
#> Approximate (numerical Hessian) standard error: 0.12503
#>     z-value: 3.4473, p-value: 0.00056624
#> Wald statistic: 11.884, p-value: 0.00056624
#> 
#> Log likelihood: -182.3904 for lag model
#> ML residual variance (sigma squared): 95.494, (sigma: 9.7721)
#> Number of observations: 49 
#> Number of parameters estimated: 5 
#> AIC: 374.78, (AIC for lm: 382.75)
#> 
COL.lag.eig1a$fdHess
#>                    sigma2           rho (Intercept)         INC         HOVAL
#> sigma2      380.749548407 -0.3653290756  19.9519208 -0.47947507 -0.0067851124
#> rho          -0.365329076  0.0156331058  -0.8537795  0.02051762  0.0002903475
#> (Intercept)  19.951920812 -0.8537795385  63.0050547 -1.80875071 -0.1338515483
#> INC          -0.479475072  0.0205176238  -1.8087507  0.10846276 -0.0122071279
#> HOVAL        -0.006785112  0.0002903475  -0.1338515 -0.01220713  0.0077831895
COL.lag.eig$resvar[2,2]
#> [1] 0.01384875
# using the apparent sign in Ord (1975, equation B.1) 
COL.lag.eigb$resvar[2,2]
#> [1] 0.03000369
# force numerical Hessian
COL.lag.eig1$fdHess[1,1]
#> [1] 0.01528326
# force LeSage & Pace (2008, p. 57) approximation 
COL.lag.eig1a$fdHess[2,2]
#> [1] 0.01563311
# }
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
#> 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 
#>   0.187   0.000   0.188 
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
# \dontrun{
system.time(COL.lag.sp <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw=listw, method="spam", quiet=FALSE))
#> 
#> Spatial lag model
#> Jacobian calculated using sparse matrix Cholesky decomposition
#> 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 
#>   0.449   0.003   0.454 
summary(COL.lag.sp)
#> 
#> Call:lagsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     method = "spam", 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) 45.079250   7.177347  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
#> 
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)
#> 
#> Call:
#> lagsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = spdep::nb2listw(COL.nb, 
#>     style = "B"), control = list(pre_eig = ev))
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -33.6620  -4.8615  -1.3576   5.1567  25.7563 
#> 
#> Type: lag 
#> Coefficients: (asymptotic standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 51.604815   6.075285  8.4942 < 2.2e-16
#> INC         -1.154463   0.301808 -3.8252 0.0001307
#> HOVAL       -0.251633   0.087612 -2.8721 0.0040773
#> 
#> Rho: 0.054543, LR test value: 13.453, p-value: 0.00024461
#> Asymptotic standard error: 0.014836
#>     z-value: 3.6763, p-value: 0.00023662
#> Wald statistic: 13.515, p-value: 0.00023662
#> 
#> Log likelihood: -180.6507 for lag model
#> ML residual variance (sigma squared): 93.22, (sigma: 9.655)
#> Number of observations: 49 
#> Number of parameters estimated: 5 
#> AIC: 371.3, (AIC for lm: 382.75)
#> LM test for residual autocorrelation
#> test value: 0.006827, p-value: 0.93415
#> 
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)
#> 
#> Call:
#> lagsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = spdep::nb2listw(COL.nb, 
#>     style = "B"), type = "mixed", tol.solve = 1e-09, control = list(pre_eig = ev))
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -34.8460  -4.2057  -0.1195   4.6525  21.6112 
#> 
#> Type: mixed 
#> Coefficients: (asymptotic standard errors) 
#>                    Estimate  Std. Error z value  Pr(>|z|)
#> (Intercept)      5.4215e+01  5.6639e+00  9.5719 < 2.2e-16
#> INC             -8.2386e-01  3.1643e-01 -2.6036 0.0092262
#> HOVAL           -3.0085e-01  8.5629e-02 -3.5134 0.0004424
#> lag.(Intercept) -7.5493e+00  1.7307e+00 -4.3620 1.289e-05
#> lag.INC          2.1531e-05  1.2216e-01  0.0002 0.9998594
#> lag.HOVAL        7.2458e-02  3.9007e-02  1.8576 0.0632281
#> 
#> Rho: 0.15212, LR test value: 7.435, p-value: 0.0063967
#> Asymptotic standard error: 0.015565
#>     z-value: 9.7735, p-value: < 2.22e-16
#> Wald statistic: 95.522, p-value: < 2.22e-16
#> 
#> Log likelihood: -177.7722 for mixed model
#> ML residual variance (sigma squared): 82.502, (sigma: 9.0831)
#> Number of observations: 49 
#> Number of parameters estimated: 8 
#> AIC: 371.54, (AIC for lm: 376.98)
#> LM test for residual autocorrelation
#> test value: -26.79, p-value: 1
#> 
COL.mixed.W <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, type="mixed", control=list(pre_eig=ev))
summary(COL.mixed.W)
#> 
#> Call:lagsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     type = "mixed", control = list(pre_eig = ev))
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -37.47829  -6.46731  -0.33835   6.05200  22.62969 
#> 
#> Type: mixed 
#> Coefficients: (asymptotic standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 42.822414  12.667204  3.3806 0.0007233
#> INC         -0.914223   0.331094 -2.7612 0.0057586
#> HOVAL       -0.293738   0.089212 -3.2926 0.0009927
#> lag.INC     -0.520284   0.565129 -0.9206 0.3572355
#> lag.HOVAL    0.245640   0.178917  1.3729 0.1697756
#> 
#> Rho: 0.42634, LR test value: 5.3693, p-value: 0.020494
#> Asymptotic standard error: 0.15623
#>     z-value: 2.7288, p-value: 0.0063561
#> Wald statistic: 7.4465, p-value: 0.0063561
#> 
#> Log likelihood: -181.3935 for mixed model
#> ML residual variance (sigma squared): 91.791, (sigma: 9.5808)
#> Number of observations: 49 
#> Number of parameters estimated: 7 
#> AIC: 376.79, (AIC for lm: 380.16)
#> LM test for residual autocorrelation
#> test value: 0.28919, p-value: 0.59074
#> 
COL.mixed.D00 <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, Durbin=TRUE, control=list(pre_eig=ev))
summary(COL.mixed.D00)
#> 
#> Call:lagsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     Durbin = TRUE, control = list(pre_eig = ev))
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -37.47829  -6.46731  -0.33835   6.05200  22.62969 
#> 
#> Type: mixed 
#> Coefficients: (asymptotic standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 42.822414  12.667204  3.3806 0.0007233
#> INC         -0.914223   0.331094 -2.7612 0.0057586
#> HOVAL       -0.293738   0.089212 -3.2926 0.0009927
#> lag.INC     -0.520284   0.565129 -0.9206 0.3572355
#> lag.HOVAL    0.245640   0.178917  1.3729 0.1697756
#> 
#> Rho: 0.42634, LR test value: 5.3693, p-value: 0.020494
#> Asymptotic standard error: 0.15623
#>     z-value: 2.7288, p-value: 0.0063561
#> Wald statistic: 7.4465, p-value: 0.0063561
#> 
#> Log likelihood: -181.3935 for mixed model
#> ML residual variance (sigma squared): 91.791, (sigma: 9.5808)
#> Number of observations: 49 
#> Number of parameters estimated: 7 
#> AIC: 376.79, (AIC for lm: 380.16)
#> LM test for residual autocorrelation
#> test value: 0.28919, p-value: 0.59074
#> 
COL.mixed.D01 <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, Durbin=FALSE, control=list(pre_eig=ev))
summary(COL.mixed.D01)
#> 
#> Call:lagsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     Durbin = FALSE, control = list(pre_eig = ev))
#> 
#> 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
#> 
COL.mixed.D1 <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, Durbin= ~ INC + HOVAL, control=list(pre_eig=ev))
summary(COL.mixed.D1)
#> 
#> Call:lagsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     Durbin = ~INC + HOVAL, control = list(pre_eig = ev))
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -37.47829  -6.46731  -0.33835   6.05200  22.62969 
#> 
#> Type: mixed 
#> Coefficients: (asymptotic standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 42.822414  12.667204  3.3806 0.0007233
#> INC         -0.914223   0.331094 -2.7612 0.0057586
#> HOVAL       -0.293738   0.089212 -3.2926 0.0009927
#> lag.INC     -0.520284   0.565129 -0.9206 0.3572355
#> lag.HOVAL    0.245640   0.178917  1.3729 0.1697756
#> 
#> Rho: 0.42634, LR test value: 5.3693, p-value: 0.020494
#> Asymptotic standard error: 0.15623
#>     z-value: 2.7288, p-value: 0.0063561
#> Wald statistic: 7.4465, p-value: 0.0063561
#> 
#> Log likelihood: -181.3935 for mixed model
#> ML residual variance (sigma squared): 91.791, (sigma: 9.5808)
#> Number of observations: 49 
#> Number of parameters estimated: 7 
#> AIC: 376.79, (AIC for lm: 380.16)
#> LM test for residual autocorrelation
#> test value: 0.28919, p-value: 0.59074
#> 
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)
#> 
#> Call:
#> lagsarlm(formula = f, data = COL.OLD, listw = listw, Durbin = as.formula(delete.response(terms(f))), 
#>     control = list(pre_eig = ev))
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -37.47829  -6.46731  -0.33835   6.05200  22.62969 
#> 
#> Type: mixed 
#> Coefficients: (asymptotic standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 42.822414  12.667204  3.3806 0.0007233
#> INC         -0.914223   0.331094 -2.7612 0.0057586
#> HOVAL       -0.293738   0.089212 -3.2926 0.0009927
#> lag.INC     -0.520284   0.565129 -0.9206 0.3572355
#> lag.HOVAL    0.245640   0.178917  1.3729 0.1697756
#> 
#> Rho: 0.42634, LR test value: 5.3693, p-value: 0.020494
#> Asymptotic standard error: 0.15623
#>     z-value: 2.7288, p-value: 0.0063561
#> Wald statistic: 7.4465, p-value: 0.0063561
#> 
#> Log likelihood: -181.3935 for mixed model
#> ML residual variance (sigma squared): 91.791, (sigma: 9.5808)
#> Number of observations: 49 
#> Number of parameters estimated: 7 
#> AIC: 376.79, (AIC for lm: 380.16)
#> LM test for residual autocorrelation
#> test value: 0.28919, p-value: 0.59074
#> 
COL.mixed.D1a <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, Durbin= ~ INC, control=list(pre_eig=ev))
summary(COL.mixed.D1a)
#> 
#> Call:lagsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     Durbin = ~INC, control = list(pre_eig = ev))
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -37.82800  -5.85207   0.12047   6.00137  23.19963 
#> 
#> Type: mixed 
#> Coefficients: (asymptotic standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 48.814687  12.198232  4.0018 6.287e-05
#> INC         -1.006620   0.330641 -3.0445  0.002331
#> HOVAL       -0.265514   0.088768 -2.9911  0.002780
#> lag.INC     -0.186684   0.530550 -0.3519  0.724936
#> 
#> Rho: 0.39229, LR test value: 4.5007, p-value: 0.033881
#> Asymptotic standard error: 0.1561
#>     z-value: 2.513, p-value: 0.011971
#> Wald statistic: 6.3151, p-value: 0.011971
#> 
#> Log likelihood: -182.3328 for mixed model
#> ML residual variance (sigma squared): 96.122, (sigma: 9.8042)
#> Number of observations: 49 
#> Number of parameters estimated: 6 
#> AIC: 376.67, (AIC for lm: 379.17)
#> LM test for residual autocorrelation
#> test value: 2.6134, p-value: 0.10596
#> 
try(COL.mixed.D1 <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, Durbin= ~ inc + HOVAL, control=list(pre_eig=ev)))
#> Error in eval(predvars, data, env) : object 'inc' not found
try(COL.mixed.D1 <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, Durbin= ~ DISCBD + HOVAL, control=list(pre_eig=ev)))
#> Error in lagsarlm(CRIME ~ INC + HOVAL, data = COL.OLD, listw, Durbin = ~DISCBD +  : 
#>   WX variables not in X: DISCBD
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
#> 1020 1021 1022 1023 1024 1025 
#>   20   21   22   23   24   25 
#> attr(,"class")
#> [1] "exclude"
COL.lag.NA
#> 
#> Call:
#> lagsarlm(formula = CRIME ~ INC + HOVAL, data = NA.COL.OLD, listw = listw, 
#>     na.action = na.exclude)
#> Type: lag 
#> 
#> Coefficients:
#>         rho (Intercept)         INC       HOVAL 
#>   0.4537820  43.1054975  -0.9267352  -0.2715541 
#> 
#> Log likelihood: -160.8867 
resid(COL.lag.NA)
#>        1001        1002        1003        1004        1005        1006 
#>  -4.4352945 -13.2750948  -2.9233555 -37.3067542   1.3568011  -3.8828027 
#>        1007        1008        1009        1010        1011        1012 
#>   5.9980436 -12.8113318  -4.0482558  18.1813323   5.9714203   1.0035599 
#>        1013        1014        1015        1016        1017        1018 
#>  -1.7490666  -1.7490651   5.8456328  10.1074158  -4.3706895   3.3713771 
#>        1019        1020        1021        1022        1023        1024 
#>  -4.0823022          NA          NA          NA          NA          NA 
#>        1025        1026        1027        1028        1029        1030 
#>          NA  -6.0553296 -11.5813311  -8.0011389  -1.9047478   3.9744243 
#>        1031        1032        1033        1034        1035        1036 
#>   4.8148614   6.0673483  10.2059847  22.7419913  -2.0830998   0.1422777 
#>        1037        1038        1039        1040        1041        1042 
#>   8.6436388   8.8150864   6.4974685  15.4121789   9.4182148   5.6427603 
#>        1043        1044        1045        1046        1047        1048 
#>  -7.5727123  -7.5983733  -9.7792620 -10.0870764  -3.1242393   3.4921796 
#>        1049 
#>   0.7173251 
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$na.action
#> 1020 1021 1022 1023 1024 1025 
#>   20   21   22   23   24   25 
#> attr(,"class")
#> [1] "omit"
COL.lag.NA2 <- lagsarlm(CRIME ~ INC + HOVAL, data=NA.COL.OLD,
 listw, Durbin=~INC, na.action=na.exclude)
COL.lag.NA2$na.action
#> 1020 1021 1022 1023 1024 1025 
#>   20   21   22   23   24   25 
#> attr(,"class")
#> [1] "exclude"
# 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))
#> Warning: NAs found, precomputed eigenvalues ignored
COL.lag.NA3$na.action
#> 1020 1021 1022 1023 1024 1025 
#>   20   21   22   23   24   25 
#> attr(,"class")
#> [1] "omit"
# }

# \dontrun{
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)
#> 
#> Call:lagsarlm(formula = 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, listw = spdep::nb2listw(boston.soi), 
#>     type = "mixed", method = "Matrix")
#> 
#> Residuals:
#>        Min         1Q     Median         3Q        Max 
#> -0.6316833 -0.0629790 -0.0090776  0.0682421  0.6991072 
#> 
#> Type: mixed 
#> Coefficients: (asymptotic standard errors) 
#>                   Estimate  Std. Error  z value  Pr(>|z|)
#> (Intercept)     1.89816225  0.22759605   8.3400 < 2.2e-16
#> CRIM           -0.00571021  0.00093504  -6.1069 1.016e-09
#> ZN              0.00069091  0.00051869   1.3320  0.182851
#> INDUS          -0.00111343  0.00307354  -0.3623  0.717155
#> CHAS1          -0.04163225  0.02739364  -1.5198  0.128567
#> I(NOX^2)       -0.01034950  0.19360214  -0.0535  0.957367
#> I(RM^2)         0.00794979  0.00102063   7.7891 6.661e-15
#> AGE            -0.00128789  0.00048920  -2.6326  0.008473
#> log(DIS)       -0.12404108  0.09510940  -1.3042  0.192168
#> log(RAD)        0.05863502  0.02257078   2.5978  0.009382
#> TAX            -0.00049084  0.00012145  -4.0416 5.308e-05
#> PTRATIO        -0.01319853  0.00595352  -2.2169  0.026628
#> B               0.00056383  0.00011089   5.0847 3.682e-07
#> log(LSTAT)     -0.24724454  0.02262033 -10.9302 < 2.2e-16
#> lag.CRIM       -0.00464215  0.00172935  -2.6843  0.007267
#> lag.ZN         -0.00037937  0.00070584  -0.5375  0.590940
#> lag.INDUS       0.00025064  0.00385901   0.0649  0.948215
#> lag.CHAS1       0.12518252  0.04071559   3.0746  0.002108
#> lag.I(NOX^2)   -0.38640403  0.22157523  -1.7439  0.081177
#> lag.I(RM^2)    -0.00451252  0.00153180  -2.9459  0.003220
#> lag.AGE         0.00149678  0.00068418   2.1877  0.028693
#> lag.log(DIS)   -0.00453785  0.10046478  -0.0452  0.963973
#> lag.log(RAD)   -0.00940702  0.03104930  -0.3030  0.761912
#> lag.TAX         0.00041083  0.00017867   2.2994  0.021481
#> lag.PTRATIO     0.00060355  0.00788837   0.0765  0.939012
#> lag.B          -0.00050781  0.00014155  -3.5874  0.000334
#> lag.log(LSTAT)  0.09846781  0.03399423   2.8966  0.003772
#> 
#> Rho: 0.59578, LR test value: 181.68, p-value: < 2.22e-16
#> Asymptotic standard error: 0.038445
#>     z-value: 15.497, p-value: < 2.22e-16
#> Wald statistic: 240.16, p-value: < 2.22e-16
#> 
#> Log likelihood: 300.6131 for mixed model
#> ML residual variance (sigma squared): 0.016011, (sigma: 0.12654)
#> Number of observations: 506 
#> Number of parameters estimated: 29 
#> AIC: -543.23, (AIC for lm: -363.55)
#> LM test for residual autocorrelation
#> test value: 29.772, p-value: 4.8604e-08
#> 
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)
#> 
#> Call:lagsarlm(formula = 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, listw = spdep::nb2listw(boston.soi), 
#>     type = "mixed", method = "Matrix", trs = trMatb)
#> 
#> Residuals:
#>        Min         1Q     Median         3Q        Max 
#> -0.6316833 -0.0629790 -0.0090776  0.0682421  0.6991072 
#> 
#> Type: mixed 
#> Coefficients: (asymptotic standard errors) 
#>                   Estimate  Std. Error  z value  Pr(>|z|)
#> (Intercept)     1.89816225  0.22759605   8.3400 < 2.2e-16
#> CRIM           -0.00571021  0.00093504  -6.1069 1.016e-09
#> ZN              0.00069091  0.00051869   1.3320  0.182851
#> INDUS          -0.00111343  0.00307354  -0.3623  0.717155
#> CHAS1          -0.04163225  0.02739364  -1.5198  0.128567
#> I(NOX^2)       -0.01034950  0.19360214  -0.0535  0.957367
#> I(RM^2)         0.00794979  0.00102063   7.7891 6.661e-15
#> AGE            -0.00128789  0.00048920  -2.6326  0.008473
#> log(DIS)       -0.12404108  0.09510940  -1.3042  0.192168
#> log(RAD)        0.05863502  0.02257078   2.5978  0.009382
#> TAX            -0.00049084  0.00012145  -4.0416 5.308e-05
#> PTRATIO        -0.01319853  0.00595352  -2.2169  0.026628
#> B               0.00056383  0.00011089   5.0847 3.682e-07
#> log(LSTAT)     -0.24724454  0.02262033 -10.9302 < 2.2e-16
#> lag.CRIM       -0.00464215  0.00172935  -2.6843  0.007267
#> lag.ZN         -0.00037937  0.00070584  -0.5375  0.590940
#> lag.INDUS       0.00025064  0.00385901   0.0649  0.948215
#> lag.CHAS1       0.12518252  0.04071559   3.0746  0.002108
#> lag.I(NOX^2)   -0.38640403  0.22157523  -1.7439  0.081177
#> lag.I(RM^2)    -0.00451252  0.00153180  -2.9459  0.003220
#> lag.AGE         0.00149678  0.00068418   2.1877  0.028693
#> lag.log(DIS)   -0.00453785  0.10046478  -0.0452  0.963973
#> lag.log(RAD)   -0.00940702  0.03104930  -0.3030  0.761912
#> lag.TAX         0.00041083  0.00017867   2.2994  0.021481
#> lag.PTRATIO     0.00060355  0.00788837   0.0765  0.939012
#> lag.B          -0.00050781  0.00014155  -3.5874  0.000334
#> lag.log(LSTAT)  0.09846781  0.03399423   2.8966  0.003772
#> 
#> Rho: 0.59578, LR test value: 181.68, p-value: < 2.22e-16
#> Asymptotic standard error: 0.038445
#>     z-value: 15.497, p-value: < 2.22e-16
#> Wald statistic: 240.16, p-value: < 2.22e-16
#> 
#> Log likelihood: 300.6131 for mixed model
#> ML residual variance (sigma squared): 0.016011, (sigma: 0.12654)
#> Number of observations: 506 
#> Number of parameters estimated: 29 
#> AIC: -543.23, (AIC for lm: -363.55)
#> LM test for residual autocorrelation
#> test value: 29.772, p-value: 4.8604e-08
#> 
# }
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)
#> 
# \dontrun{
COL.SDEM.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, Durbin=TRUE, control=list(pre_eig=ev))
summary(COL.SDEM.eig)
#> 
#> Call:errorsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     Durbin = TRUE, 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)
#> 
COL.SDEM.eig <- errorsarlm(CRIME ~ DISCBD + INC + HOVAL, data=COL.OLD,
 listw, Durbin=~INC, control=list(pre_eig=ev))
summary(COL.SDEM.eig)
#> 
#> Call:errorsarlm(formula = CRIME ~ DISCBD + INC + HOVAL, data = COL.OLD, 
#>     listw = listw, Durbin = ~INC, control = list(pre_eig = ev))
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -34.61867  -7.31993   0.82879   5.92877  17.82211 
#> 
#> Type: error 
#> Coefficients: (asymptotic standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 68.961912   6.985784  9.8717 < 2.2e-16
#> DISCBD      -5.412936   2.009281 -2.6940  0.007061
#> INC         -0.899425   0.315174 -2.8537  0.004321
#> HOVAL       -0.202846   0.090125 -2.2507  0.024403
#> lag.INC      0.161858   0.603859  0.2680  0.788669
#> 
#> Lambda: 0.24524, LR test value: 1.2719, p-value: 0.25942
#> Asymptotic standard error: 0.18393
#>     z-value: 1.3334, p-value: 0.18241
#> Wald statistic: 1.7778, p-value: 0.18241
#> 
#> Log likelihood: -178.9895 for error model
#> ML residual variance (sigma squared): 85.935, (sigma: 9.2701)
#> Number of observations: 49 
#> Number of parameters estimated: 7 
#> AIC: 371.98, (AIC for lm: 371.25)
#> 
summary(impacts(COL.SDEM.eig))
#> Impact measures (SDEM, glht, n):
#>            Direct  Indirect      Total
#> DISCBD -5.4129365        NA -5.4129365
#> INC    -0.8994251 0.1618581 -0.7375670
#> HOVAL  -0.2028457        NA -0.2028457
#> ========================================================
#> Standard errors:
#>            Direct  Indirect      Total
#> DISCBD 2.00928140        NA 2.00928140
#> INC    0.31517363 0.6038588 0.67559401
#> HOVAL  0.09012493        NA 0.09012493
#> ========================================================
#> Z-values:
#>           Direct  Indirect     Total
#> DISCBD -2.693966        NA -2.693966
#> INC    -2.853745 0.2680396 -1.091731
#> HOVAL  -2.250717        NA -2.250717
#> 
#> p-values:
#>        Direct    Indirect Total    
#> DISCBD 0.0070607 NA       0.0070607
#> INC    0.0043207 0.78867  0.2749513
#> HOVAL  0.0244035 NA       0.0244035
#> 
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
#> 1020 1021 1022 1023 1024 1025 
#>   20   21   22   23   24   25 
#> attr(,"class")
#> [1] "exclude"
COL.err.NA
#> 
#> Call:
#> errorsarlm(formula = CRIME ~ INC + HOVAL, data = NA.COL.OLD, 
#>     listw = listw, na.action = na.exclude)
#> Type: error 
#> 
#> Coefficients:
#>      lambda (Intercept)         INC       HOVAL 
#>   0.5748430  58.2460528  -0.8473028  -0.3024909 
#> 
#> Log likelihood: -161.8763 
resid(COL.err.NA)
#>         1001         1002         1003         1004         1005         1006 
#>  -4.18270830 -11.44133843   0.31874928 -34.47163074   2.42244758  -4.32095072 
#>         1007         1008         1009         1010         1011         1012 
#>   8.66744165 -13.38669934  -1.92276585  17.85753950  -1.11484596  -2.30434792 
#>         1013         1014         1015         1016         1017         1018 
#>  -8.16935116  -5.80500231   0.14973721   5.93191445  -7.03028271   2.39112829 
#>         1019         1020         1021         1022         1023         1024 
#>  -8.95099917           NA           NA           NA           NA           NA 
#>         1025         1026         1027         1028         1029         1030 
#>           NA  -2.52940712  -9.60025384  -6.95635586  -0.43630579   5.98493664 
#>         1031         1032         1033         1034         1035         1036 
#>   6.25882669   7.75527032  10.83413236  23.23927250  -0.05594957   1.43808573 
#>         1037         1038         1039         1040         1041         1042 
#>   9.51995266  12.18295373   8.31031724  17.06834503   7.04418393   7.50088924 
#>         1043         1044         1045         1046         1047         1048 
#>  -7.78485976  -6.79207447  -7.94977534 -11.25362117  -5.68994630   5.04837399 
#>         1049 
#>   2.22497381 
print(system.time(ev <- eigenw(similar.listw(listw))))
#>    user  system elapsed 
#>   0.001   0.000   0.001 
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, method="eigen", control=list(pre_eig=ev))))
#>    user  system elapsed 
#>   0.165   0.000   0.166 
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))))
#>    user  system elapsed 
#>   0.161   0.000   0.161 
print(all.equal(ocoef, coefficients(COL.errW.eig)))
#> [1] TRUE
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, method="eigen", control=list(pre_eig=ev, compiled_sse=TRUE))))
#>    user  system elapsed 
#>   0.160   0.000   0.161 
print(all.equal(ocoef, coefficients(COL.errW.eig)))
#> [1] TRUE
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, method="Matrix_J", control=list(super=TRUE))))
#> 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
#>    user  system elapsed 
#>   0.185   0.000   0.186 
print(all.equal(ocoef, coefficients(COL.errW.eig)))
#> [1] TRUE
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, method="Matrix_J", control=list(super=FALSE))))
#>    user  system elapsed 
#>   0.177   0.000   0.178 
print(all.equal(ocoef, coefficients(COL.errW.eig)))
#> [1] TRUE
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, method="Matrix_J", control=list(super=as.logical(NA)))))
#>    user  system elapsed 
#>   0.178   0.000   0.178 
print(all.equal(ocoef, coefficients(COL.errW.eig)))
#> [1] TRUE
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, method="Matrix", control=list(super=TRUE))))
#>    user  system elapsed 
#>   0.169   0.000   0.169 
print(all.equal(ocoef, coefficients(COL.errW.eig)))
#> [1] TRUE
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, method="Matrix", control=list(super=FALSE))))
#>    user  system elapsed 
#>   0.164   0.000   0.165 
print(all.equal(ocoef, coefficients(COL.errW.eig)))
#> [1] TRUE
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, method="Matrix", control=list(super=as.logical(NA)))))
#>    user  system elapsed 
#>   0.165   0.000   0.166 
print(all.equal(ocoef, coefficients(COL.errW.eig)))
#> [1] TRUE
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, method="spam", control=list(spamPivot="MMD"))))
#>    user  system elapsed 
#>   0.175   0.001   0.176 
print(all.equal(ocoef, coefficients(COL.errW.eig)))
#> [1] TRUE
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, method="spam", control=list(spamPivot="RCM"))))
#>    user  system elapsed 
#>   0.184   0.000   0.184 
print(all.equal(ocoef, coefficients(COL.errW.eig)))
#> [1] TRUE
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, method="spam_update", control=list(spamPivot="MMD"))))
#>    user  system elapsed 
#>   0.216   0.000   0.216 
print(all.equal(ocoef, coefficients(COL.errW.eig)))
#> [1] TRUE
print(system.time(COL.errW.eig <- errorsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, method="spam_update", control=list(spamPivot="RCM"))))
#>    user  system elapsed 
#>   0.182   0.000   0.183 
print(all.equal(ocoef, coefficients(COL.errW.eig)))
#> [1] TRUE
# }
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) 47.783766   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 -0.5601501 -1.6234223
#> HOVAL -0.2919129 -0.1537847 -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 -0.8371024 -1.861677
#> HOVAL -3.159909 -0.7431009 -1.553156
#> 
#> Simulated p-values:
#>       Direct     Indirect Total   
#> INC   0.00073396 0.40254  0.062649
#> HOVAL 0.00157818 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   68.25722  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: 0.94581
#>     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   0.3799971 2.356727 2.5113549
#> HOVAL 0.1091241 0.761200 0.8220231
#> 
#> Simulated z-values:
#>          Direct   Indirect       Total
#> INC   -2.706969 -0.6434228 -1.01340192
#> HOVAL -2.478740  0.2512075 -0.09643407
#> 
#> Simulated p-values:
#>       Direct    Indirect Total  
#> INC   0.0067901 0.51995  0.31087
#> HOVAL 0.0131847 0.80165  0.92318
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   68.25722  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: 0.94581
#>     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   0.3799971 2.356727 2.5113549
#> HOVAL 0.1091241 0.761200 0.8220231
#> 
#> Simulated z-values:
#>          Direct   Indirect       Total
#> INC   -2.706969 -0.6434228 -1.01340192
#> HOVAL -2.478740  0.2512075 -0.09643407
#> 
#> Simulated p-values:
#>       Direct    Indirect Total  
#> INC   0.0067901 0.51995  0.31087
#> HOVAL 0.0131847 0.80165  0.92318
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 5.7201657 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
# \dontrun{
COL.mix.eig <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, type="mixed", method="eigen")
summary(COL.mix.eig, correlation=TRUE, Nagelkerke=TRUE)
#> 
#> Call:lagsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     type = "mixed", method = "eigen")
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -37.47829  -6.46731  -0.33835   6.05200  22.62969 
#> 
#> Type: mixed 
#> Coefficients: (asymptotic standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 42.822413  12.667204  3.3806 0.0007233
#> INC         -0.914223   0.331094 -2.7612 0.0057586
#> HOVAL       -0.293738   0.089212 -3.2926 0.0009927
#> lag.INC     -0.520283   0.565129 -0.9206 0.3572355
#> lag.HOVAL    0.245640   0.178917  1.3729 0.1697756
#> 
#> Rho: 0.42634, LR test value: 5.3693, p-value: 0.020494
#> Asymptotic standard error: 0.15623
#>     z-value: 2.7288, p-value: 0.0063561
#> Wald statistic: 7.4465, p-value: 0.0063561
#> 
#> Log likelihood: -181.3935 for mixed model
#> ML residual variance (sigma squared): 91.791, (sigma: 9.5808)
#> Nagelkerke pseudo-R-squared: 0.6494 
#> Number of observations: 49 
#> Number of parameters estimated: 7 
#> AIC: 376.79, (AIC for lm: 380.16)
#> LM test for residual autocorrelation
#> test value: 0.28919, p-value: 0.59074
#> 
#>  Correlation of coefficients 
#>             sigma rho   (Intercept) INC   HOVAL lag.INC
#> rho         -0.18                                      
#> (Intercept)  0.16 -0.89                                
#> INC         -0.03  0.14 -0.19                          
#> HOVAL        0.02 -0.09  0.03       -0.45              
#> lag.INC     -0.09  0.49 -0.53       -0.36  0.05        
#> lag.HOVAL   -0.04  0.19 -0.36        0.19 -0.24 -0.41  
#> 
COL.mix.M <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD,
 listw, type="mixed", method="Matrix")
summary(COL.mix.M, correlation=TRUE, Nagelkerke=TRUE)
#> 
#> Call:lagsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = listw, 
#>     type = "mixed", method = "Matrix")
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -37.47829  -6.46731  -0.33835   6.05200  22.62969 
#> 
#> Type: mixed 
#> Coefficients: (asymptotic standard errors) 
#>              Estimate Std. Error z value  Pr(>|z|)
#> (Intercept) 42.822416  12.667205  3.3806 0.0007233
#> INC         -0.914223   0.331094 -2.7612 0.0057586
#> HOVAL       -0.293738   0.089212 -3.2926 0.0009927
#> lag.INC     -0.520284   0.565129 -0.9206 0.3572354
#> lag.HOVAL    0.245640   0.178917  1.3729 0.1697756
#> 
#> Rho: 0.42634, LR test value: 5.3693, p-value: 0.020494
#> Asymptotic standard error: 0.15623
#>     z-value: 2.7288, p-value: 0.0063561
#> Wald statistic: 7.4465, p-value: 0.0063561
#> 
#> Log likelihood: -181.3935 for mixed model
#> ML residual variance (sigma squared): 91.791, (sigma: 9.5808)
#> Nagelkerke pseudo-R-squared: 0.6494 
#> Number of observations: 49 
#> Number of parameters estimated: 7 
#> AIC: 376.79, (AIC for lm: 380.16)
#> LM test for residual autocorrelation
#> test value: 0.28919, p-value: 0.59074
#> 
#>  Correlation of coefficients 
#>             sigma rho   (Intercept) INC   HOVAL lag.INC
#> rho         -0.18                                      
#> (Intercept)  0.16 -0.89                                
#> INC         -0.03  0.14 -0.19                          
#> HOVAL        0.02 -0.09  0.03       -0.45              
#> lag.INC     -0.09  0.49 -0.53       -0.36  0.05        
#> lag.HOVAL   -0.04  0.19 -0.36        0.19 -0.24 -0.41  
#> 
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)
#> 
#> Call:
#> errorsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = spdep::nb2listw(COL.nb, 
#>     style = "W"), method = "eigen")
#> 
#> 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)
#> Nagelkerke pseudo-R-squared: 0.61978 
#> Number of observations: 49 
#> Number of parameters estimated: 5 
#> AIC: 376.76, (AIC for lm: 382.75)
#> Hausman test: 4.902, df: 3, p-value: 0.17911
#> 
#>  Correlation of coefficients 
#>             sigma lambda (Intercept) INC  
#> lambda      -0.24                         
#> (Intercept)  0.00  0.00                   
#> INC          0.00  0.00  -0.56            
#> HOVAL        0.00  0.00  -0.26       -0.45
#> 
# }