The function fits a spatial lag model by two stage least squares, with the option of adjusting the results for heteroskedasticity.

stsls(formula, data = list(), listw, zero.policy = NULL,
 na.action = na.fail, robust = FALSE, HC=NULL, legacy=FALSE, W2X = TRUE)
# S3 method for Stsls
impacts(obj, ..., tr, R = NULL, listw = NULL, evalues=NULL,
 tol = 1e-06, empirical = FALSE, Q=NULL)

Arguments

formula

a symbolic description of the model to be fit. The details of model specification are given for lm()

data

an optional data frame containing the variables in the model. By default the variables are taken from the environment which the function is called.

listw

a listw object created for example by nb2listw

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

na.action

a function (default na.fail), can also be na.omit or na.exclude with consequences for residuals and fitted values - in these cases the weights list will be subsetted to remove NAs in the data. It may be necessary to set zero.policy to TRUE because this subsetting may create no-neighbour observations. Note that only weights lists created without using the glist argument to nb2listw may be subsetted.

robust

default FALSE, if TRUE, apply a heteroskedasticity correction to the coefficients covariances

HC

default NULL, if robust is TRUE, assigned “HC0”, may take values “HC0” or “HC1” for White estimates or MacKinnon-White estimates respectively

legacy

the argument chooses between two implementations of the robustness correction: default FALSE - use the estimate of Omega only in the White consistent estimator of the variance-covariance matrix, if TRUE, use the original implementation which runs a GLS using the estimate of Omega, and yields different coefficient estimates as well - see example below

W2X

default TRUE, if FALSE only WX are used as instruments in the spatial two stage least squares; until release 0.4-60, only WX were used - see example below

obj

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

...

Arguments passed through to methods in the coda package

tr

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

evalues

vector of eigenvalues of spatial weights matrix for impacts calculations

R

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

tol

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

empirical

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

Q

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

Details

The fitting implementation fits a spatial lag model:

$$y = \rho W y + X \beta + \varepsilon$$

by using spatially lagged X variables as instruments for the spatially lagged dependent variable.

Value

an object of class "Stsls" containing:

coefficients

coefficient estimates

var

coefficient covariance matrix

sse

sum of squared errors

residuals

model residuals

df

degrees of freedom

References

Kelejian, H.H. and I.R. Prucha (1998). A generalized spatial two stage least squares procedure for estimating a spatial autoregressive model with autoregressive disturbances. Journal of Real Estate Finance and Economics 17, 99-121.

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

Author

Luc Anselin, Gianfranco Piras and Roger Bivand

See also

Examples

data(oldcol, package="spdep")
#require(spdep, quietly=TRUE)
lw <- spdep::nb2listw(COL.nb)
COL.lag.eig <- lagsarlm(CRIME ~ INC + HOVAL, data=COL.OLD, lw)
summary(COL.lag.eig, correlation=TRUE)
#> 
#> Call:lagsarlm(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = lw)
#> 
#> 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) 45.079251   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: 0.31955, 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
#> 
COL.lag.stsls <- stsls(CRIME ~ INC + HOVAL, data=COL.OLD, lw)
(x <- summary(COL.lag.stsls, correlation=TRUE))
#> 
#> Call:stsls(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = lw)
#> 
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -37.86437  -5.65096  -0.13669   6.23315  22.90823 
#> 
#> Coefficients: 
#>              Estimate Std. Error t value  Pr(>|t|)
#> Rho          0.454567   0.185118  2.4555  0.014067
#> (Intercept) 43.793442  10.952229  3.9986 6.372e-05
#> INC         -1.000716   0.383858 -2.6070  0.009134
#> HOVAL       -0.265489   0.091852 -2.8904  0.003847
#> 
#> Residual variance (sigma squared): 103.44, (sigma: 10.171)
#> 
#> Correlation of Coefficients:
#>      [,1]  [,2]  [,3] 
#> [1,] -0.92            
#> [2,]  0.63 -0.76      
#> [3,]  0.04 -0.16 -0.36
#> 
coef(x)
#>               Estimate  Std. Error   t value     Pr(>|t|)
#> Rho          0.4545669  0.18511845  2.455546 1.406706e-02
#> (Intercept) 43.7934425 10.95222944  3.998587 6.372175e-05
#> INC         -1.0007158  0.38385778 -2.606996 9.134037e-03
#> HOVAL       -0.2654890  0.09185167 -2.890410 3.847398e-03
W <- as(lw, "CsparseMatrix")
trMatc <- trW(W, type="mult")
loobj1 <- impacts(COL.lag.stsls, R=200, tr=trMatc)
summary(loobj1, zstats=TRUE, short=TRUE)
#> Impact measures (lag, trace):
#>           Direct   Indirect     Total
#> INC   -1.0607411 -0.7739768 -1.834718
#> HOVAL -0.2814136 -0.2053354 -0.486749
#> ========================================================
#> Simulation results ( variance matrix):
#> ========================================================
#> Simulated standard errors
#>          Direct  Indirect     Total
#> INC   0.4250637 0.7860257 0.9598058
#> HOVAL 0.1068486 0.6601511 0.7218709
#> 
#> Simulated z-values:
#>          Direct   Indirect      Total
#> INC   -2.474018 -1.0625140 -1.9657924
#> HOVAL -2.826222 -0.4999244 -0.8755077
#> 
#> Simulated p-values:
#>       Direct    Indirect Total   
#> INC   0.0133603 0.28800  0.049323
#> HOVAL 0.0047101 0.61713  0.381298
ev <- eigenw(lw)
loobj2 <- impacts(COL.lag.stsls, R=200, evalues=ev)
summary(loobj2, zstats=TRUE, short=TRUE)
#> Impact measures (lag, evalues):
#>           Direct   Indirect     Total
#> INC   -1.0607411 -0.7739768 -1.834718
#> HOVAL -0.2814136 -0.2053354 -0.486749
#> ========================================================
#> Simulation results ( variance matrix):
#> ========================================================
#> Simulated standard errors
#>          Direct  Indirect     Total
#> INC   0.3647280 1.2688013 1.3886153
#> HOVAL 0.1175608 0.8263993 0.9011554
#> 
#> Simulated z-values:
#>          Direct   Indirect      Total
#> INC   -2.955756 -0.7150236 -1.4296758
#> HOVAL -2.404673 -0.4145933 -0.6939034
#> 
#> Simulated p-values:
#>       Direct   Indirect Total  
#> INC   0.003119 0.47459  0.15281
#> HOVAL 0.016187 0.67844  0.48774
require(coda)
HPDinterval(loobj1)
#>            lower      upper
#> INC   -1.8854657 -0.2997080
#> HOVAL -0.4662763 -0.1121351
#> attr(,"Probability")
#> [1] 0.95
COL.lag.stslsW <- stsls(CRIME ~ INC + HOVAL, data=COL.OLD, lw, W2X=FALSE)
summary(COL.lag.stslsW, correlation=TRUE)
#> 
#> Call:stsls(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = lw, 
#>     W2X = FALSE)
#> 
#> Residuals:
#>        Min         1Q     Median         3Q        Max 
#> -37.785778  -5.442414  -0.052649   6.170104  23.039123 
#> 
#> Coefficients: 
#>              Estimate Std. Error t value  Pr(>|t|)
#> Rho          0.444202   0.189141  2.3485  0.018848
#> (Intercept) 44.359512  11.157079  3.9759 7.011e-05
#> INC         -1.014319   0.387469 -2.6178  0.008850
#> HOVAL       -0.265681   0.091954 -2.8893  0.003861
#> 
#> Residual variance (sigma squared): 103.67, (sigma: 10.182)
#> 
#> Correlation of Coefficients:
#>      [,1]  [,2]  [,3] 
#> [1,] -0.93            
#> [2,]  0.64 -0.77      
#> [3,]  0.04 -0.16 -0.36
#> 
COL.lag.stslsR <- stsls(CRIME ~ INC + HOVAL, data=COL.OLD, lw,
robust=TRUE, W2X=FALSE)
summary(COL.lag.stslsR, correlation=TRUE)
#> 
#> Call:stsls(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = lw, 
#>     robust = TRUE, W2X = FALSE)
#> 
#> Residuals:
#>        Min         1Q     Median         3Q        Max 
#> -37.785778  -5.442414  -0.052649   6.170104  23.039123 
#> 
#> Coefficients: 
#>             Estimate HC0 std. Error z value  Pr(>|z|)
#> Rho          0.44420        0.13748  3.2310  0.001234
#> (Intercept) 44.35951        7.67306  5.7812 7.417e-09
#> INC         -1.01432        0.44113 -2.2993  0.021486
#> HOVAL       -0.26568        0.17353 -1.5311  0.125752
#> 
#> Residual variance (sigma squared): 103.67, (sigma: 10.182)
#> 
#> Correlation of Coefficients:
#>      [,1]  [,2]  [,3] 
#> [1,] -0.90            
#> [2,]  0.15 -0.28      
#> [3,]  0.24 -0.24 -0.83
#> 
COL.lag.stslsRl <- stsls(CRIME ~ INC + HOVAL, data=COL.OLD, lw,
robust=TRUE, legacy=TRUE, W2X=FALSE)
summary(COL.lag.stslsRl, correlation=TRUE)
#> 
#> Call:stsls(formula = CRIME ~ INC + HOVAL, data = COL.OLD, listw = lw, 
#>     robust = TRUE, legacy = TRUE, W2X = FALSE)
#> 
#> Residuals:
#>        Min         1Q     Median         3Q        Max 
#> -38.654607  -5.141303  -0.065221   5.864384  23.671589 
#> 
#> Coefficients: 
#>             Estimate HC0 std. Error z value  Pr(>|z|)
#> Rho          0.40138        0.13554  2.9613  0.003064
#> (Intercept) 47.37696        7.49975  6.3171 2.664e-10
#> INC         -1.15183        0.43490 -2.6485  0.008085
#> HOVAL       -0.25047        0.17333 -1.4450  0.148461
#> 
#> Asymptotic robust residual variance: 96.446, (sigma: 9.8207)
#> 
#> Correlation of Coefficients:
#>             Rho   (Intercept) INC  
#> (Intercept) -0.89                  
#> INC          0.12 -0.26            
#> HOVAL        0.25 -0.26       -0.83
#> 
data(boston, package="spData")
gp2a <- stsls(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))
summary(gp2a)
#> 
#> Call:stsls(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))
#> 
#> Residuals:
#>        Min         1Q     Median         3Q        Max 
#> -0.5356002 -0.0758562 -0.0045074  0.0719613  0.7128012 
#> 
#> Coefficients: 
#>                Estimate  Std. Error  t value  Pr(>|t|)
#> Rho          4.5925e-01  3.8485e-02  11.9330 < 2.2e-16
#> (Intercept)  2.4025e+00  2.1710e-01  11.0661 < 2.2e-16
#> CRIM        -7.3557e-03  1.0345e-03  -7.1100 1.160e-12
#> ZN           3.6435e-04  3.9311e-04   0.9268 0.3540112
#> INDUS        1.1992e-03  1.8365e-03   0.6530 0.5137794
#> CHAS1        1.1929e-02  2.6632e-02   0.4479 0.6542202
#> I(NOX^2)    -2.8874e-01  9.2546e-02  -3.1199 0.0018091
#> I(RM^2)      6.6991e-03  1.0192e-03   6.5728 4.938e-11
#> AGE         -2.5810e-04  4.0940e-04  -0.6304 0.5284073
#> log(DIS)    -1.6043e-01  2.6107e-02  -6.1451 7.993e-10
#> log(RAD)     7.1704e-02  1.4926e-02   4.8038 1.557e-06
#> TAX         -3.6857e-04  9.5315e-05  -3.8668 0.0001103
#> PTRATIO     -1.2957e-02  4.1334e-03  -3.1347 0.0017203
#> B            2.8845e-04  8.0266e-05   3.5937 0.0003261
#> log(LSTAT)  -2.3984e-01  2.2470e-02 -10.6740 < 2.2e-16
#> 
#> Residual variance (sigma squared): 0.020054, (sigma: 0.14161)
#>