sartobit {spatialprobit}

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

# 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% censoring
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)