HSAR (version 0.3.6)

hsar: Hierarchical SAR model estimation

Description

A Bayesian MCMC approach for estimation of the Hierarchical SAR model of the form: $$Y = \rho W Y + X \beta + Z \gamma +\delta \theta + \epsilon, \theta= \lambda M \theta + u$$ where Y is an Nx1 vector of the outcome variable, X is an NxK matrix of the independent variables, W and M are the weight matrices at the lower and upper level respectively,$\rho$ and $\lambda$ are the autoregressive parameters indicating the strength of spatial interaction at the lower and higher level respectively, $\beta$ is a kx1 vector of regression coefficients to estimate, $\delta$ is a NxJ block diagonal matrix and $\theta$ is a Jx1 vector of the regional effects. Two special cases of HSAR are included as well. In the first one, only the regional effects are considered independent (higher level) so the autoregressive parameter $\lambda$ is zero. In the second only the lower level effects are considered independent and so the autoregressive parameter $\rho$ is zero.

Usage

hsar( X, y, W=NULL, M=NULL, Z, Unum, burnin=5000, Nsim=10000)

Arguments

X

matrix of independent variables

y

vector of outcome variable values

W

spatial weight matrix

M

spatial weight matrix at the higher level

Z

matrix of the higher level variables

Unum

Relation between higher level regions and lower level areas

burnin

Number of samples before start collecting points

Nsim

Total number of samples in MC

Value

• a list with
• Mbetas
  a vector with the mean values of the vector of the regression coefficients estimated
• SDbetas
  a vector with the standard deviation of the vector of the regression coefficients estimated
• Mrho
  Mean values of the strength of the spatial interaction rho
• SDrho
  Standard deviation of rho
• Mlamda
  Mean values of the strength of the spatial interaction at the higher level
• SDlambda
  Standard deviation of lambda
• Msigma2e
  Mean value of $\sigma^{2}_{e}$
• SDsigma2e
  Standard deviation of $\sigma^{2}_{e}$
• Msigma2u
  Mean value of $\sigma^{2}_{u}$
• SDsigma2u
  Standard deviation of $\sigma^{2}_{u}$
• Mus
  Mean values of $\theta$
• SDus
  Standard deviation of $\theta$
• DIC
  deviance information criterion (DIC)
• pd
  effective number of parameters
• Log_Likelihood
  Log likelihood
• R_Squared
  pseudo R squared
• impact_direct
  Direct effect
• impact_idirect
  Indirect effect
• impact_total
  Total effect

Details

In order to run the full HSAR model both W and M matrices need to be defined.In case W is set to NULL only the regional effects are considered independent (higher level) so the autoregressive parameter $\lambda$ is zero. In case M is set to NULL only the lower level effects are considered independent and so the autoregressive parameter $\rho$ is zero. Both W and M can not be set to NULL.

References

Dong, D. and Harris, R. 2014. Spatial Autoregressive Models for Geographically Hierarchical Data Structures. Geographical Analysis, 1-19.

See Also

sar

Examples

data(land_prices)
result <- hsar( X, y, W, M, Z, Unum, 50, 100)
result$Mrho