Spatial Durbin linear (SLX, spatially lagged X) model

lmSLX fits an lm model augmented with the spatially lagged RHS variables, including the lagged intercept when the spatial weights are not row-standardised. create_WX creates spatially lagged RHS variables, and is exposed for use in model fitting functions.

Usage

lmSLX(formula, data = list(), listw, na.action, weights=NULL, Durbin=TRUE,
 zero.policy=NULL, return_impacts=TRUE)
# S3 method for class 'SlX'
print(x, digits = max(3L, getOption("digits") - 3L), ...)
# S3 method for class 'SlX'
summary(object, correlation = FALSE, symbolic.cor = FALSE, ...)
# S3 method for class 'summary.SlX'
print(x, digits = max(3L, getOption("digits") - 3L),
 symbolic.cor = x$symbolic.cor, signif.stars = getOption("show.signif.stars"), ...)
# S3 method for class 'SlX'
impacts(obj, ...)
# S3 method for class 'WXimpact'
print(x, ...)
# S3 method for class 'WXimpact'
summary(object, ..., adjust_k=(attr(object, "type") == "SDEM"))
# S3 method for class 'SlX'
predict(object, newdata, listw, zero.policy=NULL, ...)
create_WX(x, listw, zero.policy=NULL, prefix="")

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
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 spatial 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 TRUE for lmSLX (Durbin model including WX); if TRUE, full spatial Durbin model; if a formula object, the subset of explanatory variables to lag
zero.policy: default NULL, use global option value; if TRUE assign zero to the lagged value of zones without neighbours, if FALSE assign NA
return_impacts: default TRUE; may be set FALSE to avoid problems calculating impacts with aliased variables
digits: the number of significant digits to use when printing
correlation: logical; if TRUE, the correlation matrix of the estimated parameters is returned and printed
symbolic.cor: logical. If TRUE, print the correlations in a symbolic form (see 'symnum') rather than as numbers
signif.stars: logical. If TRUE, 'significance stars' are printed for each coefficient
obj: A spatial regression object created by lmSLX
...: Arguments passed through
prefix: default empty string, may be “lag” in some cases
x, object: model matrix to be lagged; lagImpact objects created by impacts methods
adjust_k: default TRUE if SDEM else FALSE, adjust internal OLS SDEM standard errors by dividing by n rather than (n-k) (default changed and bug fixed after 0.7-8; standard errors now ML in SDEM summary and impacts summary and identical - for SLX use FALSE)
newdata: data frame in which to predict — if NULL, predictions are for the data on which the model was fitted. Should have row names corresponding to region.id. If row names are exactly the same than the ones used for training, it uses in-sample predictors for forecast.

Value

The lmSLX function returns an “lm” object with a “mixedImps” list of three impact matrixes (impacts and standard errors) for direct, indirect and total impacts; total impacts calculated using a simplified local copy of the estimable function from the gmodels package.

Author

Roger Bivand Roger.Bivand@nhh.no

Examples

data(oldcol, package="spdep")
lw <- spdep::nb2listw(COL.nb, style="W")
COL.SLX <- lmSLX(CRIME ~ INC + HOVAL, data=COL.OLD, listw=lw)
summary(COL.SLX)
#> 
#> Call:
#> lm(formula = formula(paste("y ~ ", paste(colnames(x)[-1], collapse = "+"))), 
#>     data = as.data.frame(x), weights = weights)
#> 
#> Coefficients:
#>              Estimate    Std. Error  t value     Pr(>|t|)  
#> (Intercept)   7.503e+01   6.626e+00   1.132e+01   1.261e-14
#> INC          -1.109e+00   3.738e-01  -2.967e+00   4.854e-03
#> HOVAL        -2.897e-01   1.014e-01  -2.858e+00   6.486e-03
#> lag.INC      -1.371e+00   5.613e-01  -2.443e+00   1.867e-02
#> lag.HOVAL     1.918e-01   2.003e-01   9.572e-01   3.437e-01
#> 
summary(impacts(COL.SLX))
#> Impact measures (SlX, glht, n-k):
#>           Direct   Indirect       Total
#> INC   -1.1089293 -1.3709725 -2.47990173
#> HOVAL -0.2897283  0.1917608 -0.09796753
#> ========================================================
#> Standard errors:
#>          Direct  Indirect     Total
#> INC   0.3738129 0.5612771 0.4965456
#> HOVAL 0.1013673 0.2003335 0.2028016
#> ========================================================
#> Z-values:
#>          Direct   Indirect      Total
#> INC   -2.966535 -2.4425945 -4.9943086
#> HOVAL -2.858202  0.9572079 -0.4830709
#> 
#> p-values:
#>       Direct    Indirect Total     
#> INC   0.0030118 0.014582 5.9047e-07
#> HOVAL 0.0042605 0.338462 0.62905   
#> 
COL.SLX <- lmSLX(CRIME ~ INC + HOVAL + I(HOVAL^2), data=COL.OLD, listw=lw, Durbin=TRUE)
summary(impacts(COL.SLX))
#> Impact measures (SlX, glht, n-k):
#>                  Direct     Indirect       Total
#> INC        -0.947594274 -1.275338647 -2.22293292
#> HOVAL      -0.777427839 -0.355048446 -1.13247628
#> I(HOVAL^2)  0.004639919  0.005608104  0.01024802
#> ========================================================
#> Standard errors:
#>                 Direct   Indirect      Total
#> INC        0.398832844 0.59687399 0.56325440
#> HOVAL      0.464456540 0.98213200 1.07348981
#> I(HOVAL^2) 0.004226259 0.00908385 0.01025212
#> ========================================================
#> Z-values:
#>               Direct   Indirect      Total
#> INC        -2.375918 -2.1366966 -3.9465877
#> HOVAL      -1.673844 -0.3615079 -1.0549483
#> I(HOVAL^2)  1.097879  0.6173709  0.9996008
#> 
#> p-values:
#>            Direct   Indirect Total     
#> INC        0.017505 0.032623 7.9273e-05
#> HOVAL      0.094161 0.717720 0.29145   
#> I(HOVAL^2) 0.272258 0.536990 0.31750   
#> 
summary(COL.SLX)
#> 
#> Call:
#> lm(formula = formula(paste("y ~ ", paste(colnames(x)[-1], collapse = "+"))), 
#>     data = as.data.frame(x), weights = weights)
#> 
#> Coefficients:
#>                 Estimate    Std. Error  t value     Pr(>|t|)  
#> (Intercept)      9.246e+01   1.928e+01   4.796e+00   2.058e-05
#> INC             -9.476e-01   3.988e-01  -2.376e+00   2.214e-02
#> HOVAL           -7.774e-01   4.645e-01  -1.674e+00   1.016e-01
#> I.HOVAL.2.       4.640e-03   4.226e-03   1.098e+00   2.785e-01
#> lag.INC         -1.275e+00   5.969e-01  -2.137e+00   3.849e-02
#> lag.HOVAL       -3.550e-01   9.821e-01  -3.615e-01   7.195e-01
#> lag.I.HOVAL.2.   5.608e-03   9.084e-03   6.174e-01   5.403e-01
#> 
COL.SLX <- lmSLX(CRIME ~ INC + HOVAL + I(HOVAL^2), data=COL.OLD, listw=lw, Durbin=~INC)
summary(impacts(COL.SLX))
#> Impact measures (SlX, glht, n-k):
#>                  Direct  Indirect        Total
#> INC        -1.079064628 -1.010896 -2.089960575
#> HOVAL      -0.634518755        NA -0.634518755
#> I(HOVAL^2)  0.003455273        NA  0.003455273
#> ========================================================
#> Standard errors:
#>                Direct  Indirect      Total
#> INC        0.38471071 0.4552667 0.44650193
#> HOVAL      0.44760078        NA 0.44760078
#> I(HOVAL^2) 0.00411036        NA 0.00411036
#> ========================================================
#> Z-values:
#>                Direct  Indirect      Total
#> INC        -2.8048728 -2.220448 -4.6807425
#> HOVAL      -1.4175997        NA -1.4175997
#> I(HOVAL^2)  0.8406254        NA  0.8406254
#> 
#> p-values:
#>            Direct    Indirect Total     
#> INC        0.0050336 0.026388 2.8584e-06
#> HOVAL      0.1563077 NA       0.15631   
#> I(HOVAL^2) 0.4005579 NA       0.40056   
#> 
summary(COL.SLX)
#> 
#> Call:
#> lm(formula = formula(paste("y ~ ", paste(colnames(x)[-1], collapse = "+"))), 
#>     data = as.data.frame(x), weights = weights)
#> 
#> Coefficients:
#>              Estimate    Std. Error  t value     Pr(>|t|)  
#> (Intercept)   8.368e+01   9.265e+00   9.032e+00   1.401e-11
#> INC          -1.079e+00   3.847e-01  -2.805e+00   7.466e-03
#> HOVAL        -6.345e-01   4.476e-01  -1.418e+00   1.634e-01
#> I.HOVAL.2.    3.455e-03   4.110e-03   8.406e-01   4.051e-01
#> lag.INC      -1.011e+00   4.553e-01  -2.220e+00   3.159e-02
#> 
COL.SLX <- lmSLX(CRIME ~ INC, data=COL.OLD, listw=lw)
summary(COL.SLX)
#> 
#> Call:
#> lm(formula = formula(paste("y ~ ", paste(colnames(x)[-1], collapse = "+"))), 
#>     data = as.data.frame(x), weights = weights)
#> 
#> Coefficients:
#>              Estimate    Std. Error  t value     Pr(>|t|)  
#> (Intercept)   7.398e+01   6.208e+00   1.192e+01   1.155e-15
#> INC          -1.589e+00   3.564e-01  -4.458e+00   5.276e-05
#> lag.INC      -1.086e+00   4.812e-01  -2.257e+00   2.882e-02
#> 
summary(impacts(COL.SLX))
#> Impact measures (SlX, glht, n-k):
#>        Direct  Indirect     Total
#> INC -1.588901 -1.085867 -2.674768
#> ========================================================
#> Standard errors:
#>        Direct  Indirect    Total
#> INC 0.3564039 0.4811809 0.407313
#> ========================================================
#> Z-values:
#>        Direct  Indirect     Total
#> INC -4.458147 -2.256671 -6.566861
#> 
#> p-values:
#>     Direct     Indirect Total     
#> INC 8.2671e-06 0.024029 5.1387e-11
#> 
if (FALSE) { # \dontrun{
crds <- cbind(COL.OLD$X, COL.OLD$Y)
mdist <- sqrt(sum(diff(apply(crds, 2, range))^2))
dnb <- spdep::dnearneigh(crds, 0, mdist)
dists <- spdep::nbdists(dnb, crds)
f <- function(x, form, data, dnb, dists, verbose) {
  glst <- lapply(dists, function(d) 1/(d^x))
  lw <- spdep::nb2listw(dnb, glist=glst, style="B")
  res <- logLik(lmSLX(form=form, data=data, listw=lw))
  if (verbose) cat("power:", x, "logLik:", res, "\n")
  res
}
opt <- optimize(f, interval=c(0.1, 4), form=CRIME ~ INC + HOVAL,
 data=COL.OLD, dnb=dnb, dists=dists, verbose=TRUE, maximum=TRUE)
glst <- lapply(dists, function(d) 1/(d^opt$maximum))
lw <- spdep::nb2listw(dnb, glist=glst, style="B")
SLX <- lmSLX(CRIME ~ INC + HOVAL, data=COL.OLD, listw=lw)
summary(SLX)
summary(impacts(SLX))
} # }
COL.SLX <- lmSLX(CRIME ~ INC + HOVAL, data=COL.OLD, listw=lw)
pslx0 <- predict(COL.SLX)
pslx1 <- predict(COL.SLX, newdata=COL.OLD, listw=lw)
all.equal(pslx0, pslx1)
#> [1] TRUE
COL.OLD1 <- COL.OLD
COL.OLD1$INC <- COL.OLD1$INC + 1
pslx2 <- predict(COL.SLX, newdata=COL.OLD1, listw=lw)
sum(coef(COL.SLX)[c(2,4)])
#> [1] -2.479902
mean(pslx2-pslx1)
#> [1] -2.479902

Spatial Durbin linear (SLX, spatially lagged X) model

Usage

Arguments

Value

See also

Author

Examples