spatialprobit (version 0.9-1)

sartobit: Bayesian estimation of the SAR Tobit model

Description

Bayesian estimation of the spatial autoregressive Tobit model (SAR Tobit model).

Usage

sartobit(formula, W, data, ...)

sar_tobit_mcmc(y, X, W, ndraw = 1000, burn.in = 100, thinning = 1, 
  prior=list(a1=1, a2=1, c=rep(0, ncol(X)), T=diag(ncol(X))*1e12, lflag = 0), 
  start = list(rho = 0.75, beta = rep(0, ncol(X)), sige = 1),
  m=10, computeMarginalEffects=FALSE, showProgress=FALSE)

Arguments

y

dependent variables. vector of zeros and ones

X

design matrix

W

spatial weight matrix

ndraw

number of MCMC iterations

burn.in

number of MCMC burn-in to be discarded

thinning

MCMC thinning factor, defaults to 1.

prior

A list of prior settings for $\rho \sim Beta(a1,a2)$ and $\beta \sim N(c,T)$. Defaults to diffuse prior for beta.

start

list of start values

m

Number of burn-in samples in innermost Gibbs sampler. Defaults to 10.

computeMarginalEffects

Flag if marginal effects are calculated. Defaults to FALSE. We recommend to enable it only when sample size is small.

showProgress

Flag if progress bar should be shown. Defaults to FALSE.

formula

an object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted.

data

an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the e

...

additional arguments to be passed

Value

• Returns a structure of class sartobit:
• beta
  posterior mean of bhat based on draws
• rho
  posterior mean of rho based on draws
• bdraw
  beta draws (ndraw-nomit x nvar)
• pdraw
  rho draws (ndraw-nomit x 1)
• sdraw
  sige draws (ndraw-nomit x 1)
• total
  a matrix (ndraw,nvars-1) total x-impacts
• direct
  a matrix (ndraw,nvars-1) direct x-impacts
• indirect
  a matrix (ndraw,nvars-1) indirect x-impacts
• rdraw
  r draws (ndraw-nomit x 1) (if m,k input)
• nobs
  # of observations
• nvar
  # of variables in x-matrix
• ndraw
  # of draws
• nomit
  # of initial draws omitted
• nsteps
  # of samples used by Gibbs sampler for TMVN
• y
  y-vector from input (nobs x 1)
• zip
  # of zero y-values
• a1
  a1 parameter for beta prior on rho from input, or default value
• a2
  a2 parameter for beta prior on rho from input, or default value
• time
  total time taken
• rmax
  1/max eigenvalue of W (or rmax if input)
• rmin
  1/min eigenvalue of W (or rmin if input)
• tflag
  'plevel' (default) for printing p-levels; 'tstat' for printing bogus t-statistics
• lflag
  lflag from input
• cflag
  1 for intercept term, 0 for no intercept term
• lndet
  a matrix containing log-determinant information (for use in later function calls to save time)

Details

Bayesian estimates of the spatial autoregressive Tobit model (SAR Tobit model) $$y = \rho W y + X \beta + \epsilon, \epsilon \sim N(0, \sigma^2_{e} I_n)$$ $$y = (I_n - \rho W)^{-1} X \beta + (I_n - \rho W)^{-1} \epsilon$$ where y $(n \times 1)$ is only observed for y >= 0 and censored to 0 otherwise. $\beta$ is a $(k \times 1)$ vector of parameters associated with the $(n \times k)$ data matrix X. The prior distributions are $\beta \sim N(c,T)$ and $\rho \sim Uni(rmin,rmax)$ or $\rho \sim Beta(a1,a2)$.

References

LeSage, J. and Pace, R. K. (2009), Introduction to Spatial Econometrics, CRC Press, chapter 10, section 10.3, 299--304

See Also

sar_lndet for computing log-determinants

Examples

# Example from LeSage/Pace (2009), section 10.3.1, p. 302-304
# Value of "a" is not stated in book! 
# Assuming a=-1 which gives approx. 50 library(spatialprobit)

a <- -1   # control degree of censored observation
n <- 1000
rho <- 0.7
beta <- c(0, 2)
sige <- 0.5
I_n <- sparseMatrix(i=1:n, j=1:n, x=1)
x <- runif(n, a, 1)
X <- cbind(1, x)
eps <- rnorm(n, sd=sqrt(sige))
param <- c(beta, sige, rho)

# random locational coordinates and 6 nearest neighbors
lat <- rnorm(n)
long <- rnorm(n)
W <- kNearestNeighbors(lat, long, k=6)

y <- as.double(solve(I_n - rho * W) %*% (X %*% beta + eps))
table(y > 0)

# full information
yfull <- y

# set negative values to zero to reflect sample truncation
ind <- which(y <=0)
y[ind] <- 0

# Fit SAR (with complete information)
fit_sar <- sartobit(yfull ~ X-1, W,ndraw=1000,burn.in=200, showProgress=FALSE)
summary(fit_sar)

# Fit SAR Tobit (with approx. 50% censored observations)
fit_sartobit <- sartobit(y ~ x,W,ndraw=1000,burn.in=200, showProgress=TRUE)

par(mfrow=c(2,2))
for (i in 1:4) {
 ylim1 <- range(fit_sar$B[,i], fit_sartobit$B[,i])
 plot(fit_sar$B[,i], type="l", ylim=ylim1, main=fit_sartobit$names[i], col="red")
 lines(fit_sartobit$B[,i], col="green")
 legend("topleft", legend=c("SAR", "SAR Tobit"), col=c("red", "green"), 
   lty=1, bty="n")
}