The sar() function implements a standard spatial econometrics model (SAR) or a spatially lagged dependent
variable model using the Markov chain Monte Carlo (McMC) simulation approach.
Usage
sar(
formula,
data = NULL,
W,
burnin = 5000,
Nsim = 10000,
thinning = 1,
parameters.start = NULL
)Arguments
- formula
A symbolic description of the model to fit. A formula for the covariate part of the model using the syntax of the
stats::lm()function fitting standard linear regression models. Neither the response variable nor the explanatory variables are allowed to contain NA values.- data
A
data.framecontaining variables used in the formula object.- W
The N by N spatial weights matrix or neighbourhood matrix where N is the number of spatial units. The formulation of W could be based on geographical distances separating units or based on geographical contiguity. To ensure the maximum value of the spatial autoregressive parameter \(\rho\) less than 1, W is usually row-normalised before implementing the SAR model. As in most cases, spatial weights matrix is very sparse, therefore W here should be converted to a sparse matrix before imported into the
sar()function to save computational burden and reduce computing time. More specifically, W should be a column-oriented numeric sparse matrices of adgCMatrixclass defined in theMatrixpackage. The converion between a dense numeric matrix and a sparse numeric matrix is made quite convenient through theMatrixlibrary.- burnin
The number of McMC samples to discard as the burnin period.
- Nsim
The total number of McMC samples to generate.
- thinning
MCMC thinning factor.
- parameters.start
A list with names "rho", "sigma2e", and "beta" corresponding to initial values for the model parameters \(\rho, \sigma^2_e\) and the regression coefficients respectively.
Value
A list.
- cbetas
A matrix with the MCMC samples of the draws for the coefficients.
- Mbetas
A vector of estimated mean values of regression coefficients.
- SDbetas
The standard deviations of estimated regression coefficients.
- Mrho
The estimated mean of the lower-level spatial autoregressive parameter \(\rho\).
- SDrho
The standard deviation of the estimated lower-level spatial autoregressive parameter.
- Msigma2e
The estimated mean of the lower-level variance parameter \(\sigma^{2}_{e} \).
- SDsigma2e
The standard deviation of the estimated lower-level variance parameter \(\sigma^{2}_{e} \).
- DIC
The deviance information criterion (DIC) of the fitted model.
- pd
The effective number of parameters of the fitted model.
- Log_Likelihood
The log-likelihood of the fitted model.
- R_Squared
A pseudo R square model fit indicator.
- impact_direct
Summaries of the direct impact of a covariate effect on the outcome variable.
- impact_idirect
Summaries of the indirect impact of a covariate effect on the outcome variable.
- impact_total
Summaries of the total impact of a covariate effect on the outcome variable.
References
Anselin, L. (1988). Spatial Econometrics: Methods and Models. Dordrecht: Kluwer Academic Publishers.
LeSage, J. P., and R. K. Pace. (2009). Introduction to Spatial Econometrics. Boca Raton, FL: CRC Press/Taylor & Francis
Examples
data(landprice)
head(landprice)
#> obs lnprice lnarea lndcbd dsubway dpark dele popden crimerate
#> 1 190 7.16382 11.58780 9.93534 7.14334 6.78243 6.67827 0.548966 10.7511
#> 2 992 7.45757 9.20029 9.84785 7.78904 6.95662 7.05138 0.548966 10.7511
#> 3 189 5.57430 10.27820 9.94866 6.83023 7.06579 6.81916 0.548966 10.7511
#> 4 993 7.12569 7.97788 9.84388 7.81991 7.00792 7.11267 0.548966 10.7511
#> 5 969 6.81564 5.81928 9.91940 7.64640 6.88254 4.10025 0.548966 10.7511
#> 6 994 7.48522 7.78634 9.84203 7.83398 7.03089 7.13981 0.548966 10.7511
#> district.id year
#> 1 3 1
#> 2 3 0
#> 3 3 1
#> 4 3 0
#> 5 3 0
#> 6 3 0
data(land)
# extract the land parcel level spatial weights matrix
library(spdep)
library(Matrix)
nb.25 <- spdep::dnearneigh(land,0,2500)
#> Warning: neighbour object has 4 sub-graphs
# to a weights matrix
dist.25 <- spdep::nbdists(nb.25,land)
dist.25 <- lapply(dist.25,function(x) exp(-0.5 * (x / 2500)^2))
mat.25 <- spdep::nb2mat(nb.25,glist=dist.25,style="W")
W <- as(mat.25,"dgCMatrix")
## run the sar() function
res.formula <- lnprice ~ lnarea + lndcbd + dsubway + dpark + dele +
popden + crimerate + as.factor(year)
betas= coef(lm(formula=res.formula,data=landprice))
pars=list(rho = 0.5, sigma2e = 2.0, betas = betas)
# \donttest{
res <- sar(res.formula,data=landprice,W=W,
burnin=500, Nsim=1000, thinning=1,
parameters.start=pars)
#> Warning: style is M (missing); style should be set to a valid value
#> Warning: style is M (missing); style should be set to a valid value
#> Warning: neighbour object has 4 sub-graphs
#> Warning: neighbour object has 4 sub-graphs
summary(res)
#>
#> Call:
#> sar(formula = res.formula, data = landprice, W = W, burnin = 500,
#> Nsim = 1000, thinning = 1, parameters.start = pars)
#> Type: sar
#>
#> Coefficients:
#> Mean SD
#> (Intercept) 6.692659271 0.821582708
#> lnarea -0.005703754 0.017245283
#> lndcbd -0.093812794 0.048795925
#> dsubway -0.151066147 0.035977010
#> dpark -0.157082432 0.045501847
#> dele -0.029478755 0.031136321
#> popden 0.021820387 0.010123086
#> crimerate 0.006581084 0.004185754
#> as.factor(year)1 -0.188791418 0.055353624
#> as.factor(year)2 -0.028732593 0.117591116
#> as.factor(year)3 -0.141198327 0.106523962
#> as.factor(year)4 0.568283074 0.116757966
#> as.factor(year)5 0.398618141 0.127220897
#> as.factor(year)6 2.183383251 0.218838610
#>
#> Spatial Coefficients:
#> rho
#> 0.544068
#>
#> Diagnostics
#> Deviance information criterion (DIC): 4704.887
#> Effective number of parameters (pd): 956.0725
#> Log likelihood: -1396.371
#> Pseudo R squared: 0.3487775
#>
#> Impacts:
#> direct indirect total
#> (Intercept) 6.815286826 7.483055987 14.29834281
#> lnarea -0.005808262 -0.006377362 -0.01218562
#> lndcbd -0.095531697 -0.104891996 -0.20042369
#> dsubway -0.153834086 -0.168906916 -0.32274100
#> dpark -0.159960606 -0.175633718 -0.33559432
#> dele -0.030018885 -0.032960168 -0.06297905
#> popden 0.022220195 0.024397354 0.04661755
#> crimerate 0.006701667 0.007358304 0.01405997
#> as.factor(year)1 -0.192250585 -0.211087505 -0.40333809
#> as.factor(year)2 -0.029259052 -0.032125885 -0.06138494
#> as.factor(year)3 -0.143785461 -0.157873715 -0.30165918
#> as.factor(year)4 0.578695551 0.635396766 1.21409232
#> as.factor(year)5 0.405921899 0.445694565 0.85161646
#> as.factor(year)6 2.223388716 2.441238743 4.66462746
#>
#> Quantiles:
#> 5% 25% 50% 75%
#> (Intercept) 5.3931321636 6.102344739 6.667925879 7.261131168
#> lnarea -0.0328579787 -0.017805259 -0.005443984 0.005865107
#> lndcbd -0.1719583291 -0.126473435 -0.094248991 -0.061389837
#> dsubway -0.2086087870 -0.174525097 -0.151688345 -0.127059403
#> dpark -0.2366361542 -0.187072950 -0.157027536 -0.126171278
#> dele -0.0799836946 -0.051391759 -0.029007668 -0.008189104
#> popden 0.0054129196 0.014553396 0.021759805 0.028808175
#> crimerate -0.0004581391 0.003773817 0.006547580 0.009282714
#> as.factor(year)1 -0.2756505015 -0.227269916 -0.190331369 -0.150788147
#> as.factor(year)2 -0.2218306150 -0.106403948 -0.031509779 0.056263472
#> as.factor(year)3 -0.3146393375 -0.212914997 -0.142246504 -0.070137649
#> as.factor(year)4 0.3816845823 0.487045899 0.577141855 0.645571177
#> as.factor(year)5 0.1922468443 0.311307645 0.402879081 0.481189056
#> as.factor(year)6 1.8106568356 2.041326641 2.191568289 2.327773608
#> 95%
#> (Intercept) 8.03383299
#> lnarea 0.02201095
#> lndcbd -0.01651091
#> dsubway -0.09292392
#> dpark -0.08466489
#> dele 0.02294826
#> popden 0.03870791
#> crimerate 0.01327714
#> as.factor(year)1 -0.09626197
#> as.factor(year)2 0.15447486
#> as.factor(year)3 0.04359984
#> as.factor(year)4 0.76317569
#> as.factor(year)5 0.60496796
#> as.factor(year)6 2.55235508
# }