sartobit function

Bayesian estimation of the SAR Tobit model

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

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 $or rho ~ Beta(a1,a2)$

  and $beta ~ N(c,T)$. 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)

where $y$ $(n x 1)$ is only observed for $z >= 0$ and censored to 0 otherwise. $beta$ is a $(k x 1)$

vector of parameters associated with the $(n x k)$ data matrix X.

The prior distributions are $beta ~ N(c,T)$

and $rho ~ Uni(rmin,rmax)$

or $or rho ~ Beta(a1,a2)$.

Returns

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)

References

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

Author(s)

adapted to and optimized for R by Stefan Wilhelm wilhelm@financial.com based on Matlab code from James P. LeSage

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)