semprobit: Bayesian estimation of the SEM probit model

In spatialprobit: Spatial Probit Models

Bayesian estimation of the SEM probit model

Description

Bayesian estimation of the probit model with spatial errors (SEM probit model).

Usage

semprobit(formula, W, data, subset, ...)

sem_probit_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,
  nu = 0, d0 = 0, lflag = 0), 
  start = list(rho = 0.75, beta = rep(0, ncol(X)), sige = 1),
  m=10, showProgress=FALSE, univariateConditionals = TRUE)

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.
showProgress	Flag if progress bar should be shown. Defaults to FALSE.
univariateConditionals	Switch whether to draw from univariate or multivariate truncated normals. See notes. Defaults to TRUE.
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 semprobit is called.
subset	an optional vector specifying a subset of observations to be used in the fitting process.
...	additional arguments to be passed

Details

Bayesian estimates of the probit model with spatial errors (SEM probit model)

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

which leads to the data-generating process

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

where y is a binary 0,1 (n \times 1) vector of observations for z < 0 and z \ge 0. \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), \sigma^2_{\epsilon} \sim IG(a1, a2), and \rho \sim Uni(rmin,rmax) or \rho \sim Beta(a1,a2).

Value

Returns a structure of class semprobit:

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 code from James P. LeSage

References

LeSage, J. and Pace, R. K. (2009), Introduction to Spatial Econometrics, CRC Press, chapter 10

See Also

sar_lndet for computing log-determinants

Examples

## Not run: 
library(Matrix)
# number of observations
n <- 200

# true parameters
beta <- c(0, 1, -1)
sige <- 2
rho <- 0.75

# design matrix with two standard normal variates as "covariates"
X <- cbind(intercept=1, x=rnorm(n), y=rnorm(n))

# sparse identity matrix
I_n <- sparseMatrix(i=1:n, j=1:n, x=1)

# number of nearest neighbors in spatial weight matrix W
m <- 6

# spatial weight matrix with m=6 nearest neighbors
# W must not have non-zeros in the main diagonal!
i <- rep(1:n, each=m)
j <- rep(NA, n * m)
for (k in 1:n) {
  j[(((k-1)*m)+1):(k*m)] <- sample(x=(1:n)[-k], size=m, replace=FALSE)
}
W <- sparseMatrix(i, j, x=1/m, dims=c(n, n))

# innovations
eps <- sqrt(sige)*rnorm(n=n, mean=0, sd=1)

# generate data from model 
S <- I_n - rho * W
z <- X %*% beta + solve(qr(S), eps)
y <- as.double(z >= 0)  # 0 or 1, FALSE or TRUE

# estimate SEM probit model
semprobit.fit1 <- semprobit(y ~ X - 1, W, ndraw=500, burn.in=100, 
  thinning=1, prior=NULL)
summary(semprobit.fit1)

## End(Not run)

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