sartobit: Bayesian estimation of the SAR Tobit model

In spatialprobit: Spatial Probit Models

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 environment from which sarprobit is called.
...	additional arguments to be passed

Details

Bayesian estimates of the spatial autoregressive Tobit model (SAR Tobit model)

z = \rho W y + X \beta + \epsilon, \epsilon \sim N(0, \sigma^2_{e} I_n)

z = (I_n - \rho W)^{-1} X \beta + (I_n - \rho W)^{-1} \epsilon

y = max(z, 0)

where y (n \times 1) is only observed for z \ge 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).

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)

Author(s)

adapted to and optimized for R by Stefan Wilhelm <[e-mail]> based on Matlab code from James P. LeSage

References

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

See Also

sarprobit, sarorderedprobit or semprobit for SAR probit/SAR Ordered Probit/ SEM probit model fitting

Examples

## Not run: 
# 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")
}

# Fit SAR Tobit (with approx. 50% censored observations)
fit_sartobit <- sartobit(y ~ x,W,ndraw=1000,burn.in=0, showProgress=TRUE, 
    computeMarginalEffects=TRUE)
# Print SAR Tobit marginal effects
impacts(fit_sartobit)
#--------Marginal Effects--------
#
#(a) Direct effects
#  lower_005 posterior_mean upper_095
#x     1.013          1.092     1.176
#
#(b) Indirect effects
#  lower_005 posterior_mean upper_095
#x     2.583          2.800     3.011
#
#(c) Total effects
#  lower_005 posterior_mean upper_095
#x     3.597          3.892     4.183
#mfx <- marginal.effects(fit_sartobit)

## End(Not run)

spatialprobit documentation built on Aug. 22, 2023, 9:09 a.m.