spatialprobit (version 0.9-1)

semprobit: 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)

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.

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 e

subset

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

...

additional arguments to be passed

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)

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 $\rho \sim Beta(a1,a2)$.

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)
# 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 <- sparseMatrix(i=rep(1:n, each=m), 
  j=replicate(n, sample(x=1:n, size=m, replace=FALSE)), 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)