HSAR (version 0.6.0)

hsar: Hierarchical SAR model estimation

Description

The specification of a HSAR model is as follows: $$y_{i,j} = \rho *\mathbf{W}_i *\mathbf{y} + \mathbf{x}^\prime_{i,j} * \mathbf{\beta} + \mathbf{z}^\prime_j * \mathbf{\gamma} + \theta_j + \epsilon_{i,j} $$ $$\theta_j = \lambda * \mathbf{M}_j * \mathbf{\theta} + \mu_j $$ $$\epsilon_{i,j} \sim N(0,\sigma_e^2), \hspace{2cm} \mu_j \sim N(0,\sigma_u^2)$$ where \(i=1,2,...,n_j\) and \(j=1,2,...,J\) are indicators of lower- and higher-level spatial units. \(n_j\) is the number of lower-level units in the \(j-th\) higher level unit and \(\sum_{j=1}^J=\mathbf{N}\). \(\mathbf{x}^\prime_{i,j}\) and \(\mathbf{z}^\prime_j\) represent vectors of lower- and higher-level independent variables. \(\mathbf{\beta}\) and \(\mathbf{\gamma}\) are regression coefficients to estimate. \(\mathbf{\theta}\), a \(N \times J\) vector of higher-level random effects, also follows a simultaneous autoregressive process. \(\mathbf{W}\) and \(\mathbf{M}\) are two spatial weights matrices (or neighbourhood connection matrices) at the lower and higher levels, defining how spatial units at each level are connected. \(\rho\) and \(\lambda\) are two spatial autoregressive parameters measuring the strength of the dependencies/correlations at the two spatial scales. A succinct matrix formulation of the model is, $$\mathbf{y} = \rho * \mathbf{W} * \mathbf{y} + \mathbf{X} * \mathbf{\beta} + \mathbf{Z} * \mathbf{\gamma} + \Delta * \mathbf{\theta} + \mathbf{\epsilon} $$ $$\mathbf{\theta} = \lambda * \mathbf{M} * \mathbf{\theta} + \mathbf{\mu}$$ It is also useful to note that the HSAR model nests a standard (random intercept) multilevel model model when \(\rho\) and \(\lambda\) are both equal to zero and a standard spaital econometric model when \(\lambda\) and \(\sigma^2_u\) are both equal to zero.

Usage

hsar(
  formula,
  data = NULL,
  W = NULL,
  M = NULL,
  Delta,
  burnin = 5000,
  Nsim = 10000,
  thinning = 1,
  parameters.start = NULL
)

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.

Mlamda

The estimated mean of the higher-level spatial autoregressive parameter \(\lambda\).

SDlambda

The standard deviation of the estimated higher-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 \(\sigma^{2}_{e} \).

Msigma2u

The estimated mean of the higher-level variance parameter \(\sigma^2_u\).

SDsigma2u

The standard deviation of the estimated higher-level variance parameter \(\sigma^2_u\).

Mus

Mean values of \(\theta\)

SDus

Standard deviation of \(\theta\)

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.

Arguments

formula

A symbolic description of the model to fit. A formula for the covariate part of the model using the syntax of the lm() function fitting standard linear regression models. Neither the response variable nor the explanatory variables are allowed to contain NA values.

data

A data.frame containing variables used in the formula object.

W

The N by N lower-level spatial weights matrix or neighbourhood matrix where N is the total number of lower-level 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 should be row-normalised before running the HSAR 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 hsar() function to save computational burden and reduce computing time. More specifically, W should be a column-oriented numeric sparse matrices of a dgCMatrix class defined in the Matrix package. The converion between a dense numeric matrix and a sparse numeric matrix is made quite convenient through the Matrixlibrary.

M

The J by J higher-level spatial weights matrix or neighbourhood matrix where J is the total number of higher-level spatial units. Similar with W, the formulation of M could be based on geographical distances separating units or based on geographical contiguity. To ensure the maximum value of the spatial autoregressive parameter \(\lambda\) less than 1, M is also row-normalised before running the HSAR model. As with W, M should also be a column-oriented numeric sparse matrices.

Delta

The N by J random effect design matrix that links the J by 1 higher-level random effect vector back to the N by 1 response variable under investigation. It is simply how lower-level units are grouped into each high-level units with columns of the matrix being each higher-level units. As with W and M, \(\delta\) should also be a column-oriented numeric sparse matrices.

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", "lambda", "sigma2e", "sigma2u" and "beta" corresponding to initial values for the model parameters \(\rho, \lambda, \sigma^2_e, \sigma^2_u\) and the regression coefficients respectively.

References

Dong, G. and Harris, R. 2015. Spatial Autoregressive Models for Geographically Hierarchical Data Structures. Geographical Analysis, 47:173-191.

LeSage, J. P., and R. K. Pace. (2009). Introduction to Spatial Econometrics. Boca Raton, FL: CRC Press/Taylor & Francis.

Examples

library(spdep)

# Running the hsar() function using the Beijing land price data
data(landprice)

# load shapefiles of Beijing districts and land parcels
data(Beijingdistricts)
data(land)

plot(Beijingdistricts,border="green")
plot(land,add=TRUE,col="red",pch=16,cex=0.8)

# Define the random effect matrix
model.data <- landprice[order(landprice$district.id),]
head(model.data,50)

# the number of individuals within each neighbourhood
MM <- as.data.frame(table(model.data$district.id))
# the total number of neighbourhood, 100
Utotal <- dim(MM)[1]
Unum <- MM[,2]
Uid <- rep(c(1:Utotal),Unum)

n <- nrow(model.data)
Delta <- matrix(0,nrow=n,ncol=Utotal)
for(i in 1:Utotal) {
  Delta[Uid==i,i] <- 1
}
rm(i)
# Delta[1:50,1:10]
Delta <- as(Delta,"dgCMatrix")

# extract the district level spatial weights matrix using the queen's rule
nb.list <- spdep::poly2nb(Beijingdistricts)
mat.list <- spdep::nb2mat(nb.list,style="W")
M <- as(mat.list,"dgCMatrix")

# extract the land parcel level spatial weights matrix
nb.25 <- spdep::dnearneigh(land,0,2500)
# 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 hsar() 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,lambda = 0.5, sigma2e = 2.0, sigma2u = 2.0, betas = betas )

# \donttest{
  res <- hsar(res.formula, data=landprice, W=W, M=M, Delta=Delta,
              burnin=500, Nsim=1000, thinning = 1, parameters.start=pars)
  summary(res)

  # visualise the district level random effect
  groups <- sdsfun::discretize_vector(res$Mus,n = 4,method = "natural")
  palette <- RColorBrewer::brewer.pal(4, "Blues")
  plot(Beijingdistricts,col=palette[groups],border="grey")
# }