spatialprobit (version 1.0.4)

sarprobit: Bayesian estimation of the SAR probit model

Description

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

Usage

sarprobit(formula, W, data, subset, ...)
sar_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, lflag = 0), 
  start = list(rho = 0.75, beta = rep(0, ncol(X))),
  m=10, computeMarginalEffects=TRUE, showProgress=FALSE, verbose=FALSE)

Value

Returns a structure of class sarprobit:

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)

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)

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 TRUE

showProgress

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

verbose

Flag for more verbose output. Default 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.

subset

an optional vector specifying a subset of observations to be used in the fitting process.

...

additional arguments to be passed

Author

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

Details

Bayesian estimates of the spatial autoregressive probit model (SAR probit model) $$ z = \rho W z + X \beta + \epsilon, \epsilon \sim N(0, I_n) $$ $$ z = (I_n - \rho W)^{-1} X \beta + (I_n - \rho W)^{-1} \epsilon $$ where y is a binary 0,1 \((n \times 1)\) vector of observations for z < 0 and z >= 0. \(\beta\) is a \((k \times 1)\) vector of parameters associated with the \((n \times k)\) data matrix X. The error variance \(\sigma_e\) is set to 1 for identification.

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

See Also

sar_lndet for computing log-determinants

Examples

library(Matrix)
set.seed(2)

# number of observations
n <- 100

# true parameters
beta <- c(0, 1, -1)
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 <- rnorm(n=n, mean=0, sd=1)

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

# estimate SAR probit model
sarprobit.fit1 <- sar_probit_mcmc(y, X, W, ndraw=100, thinning=1, prior=NULL)
summary(sarprobit.fit1)