sarorderedprobit: Bayesian estimation of the SAR ordered probit model

Description

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

Usage

sarorderedprobit(formula, W, data, subset, ...)
sar_ordered_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)), 
  phi = c(-Inf, 0:(max(y)-1), Inf)),
  m=10, computeMarginalEffects=TRUE, showProgress=FALSE)

Value

Returns a structure of class sarprobit:

beta: posterior mean of bhat based on draws
rho: posterior mean of rho based on draws
phi: posterior mean of phi based on draws
coefficients: posterior mean of coefficients
fitted.values: fitted values
fitted.reponse: fitted reponse
ndraw: # of draws
bdraw: beta draws (ndraw-nomit x nvar)
pdraw: rho draws (ndraw-nomit x 1)
phidraw: phi draws (ndraw-nomit x 1)
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)
W: spatial weights matrix
X: regressor matrix

Arguments

y: dependent variables. vector of discrete choices from 1 to J ({1,2,...,J})
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. Currently without effect.
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.
subset: an optional vector specifying a subset of observations to be used in the fitting process.
...: additional arguments to be passed

Author

Stefan Wilhelm <wilhelm@financial.com>

Details

Bayesian estimates of the spatial autoregressive ordered probit model (SAR ordered 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 $(n \times 1)$ vector of discrete choices from 1 to J, $y \in \{1,2,...,J\}$, where

$y = 1$ for $-\infty \le z \le \phi_1 = 0$
$y = 2$ for $\phi_1 \le z \le \phi_2$
...
$y = j$ for $\phi_{j-1} \le z \le \phi_j$
...
$y = J$ for $\phi_{J-1} \le z \le \infty$

The vector $\phi=(\phi_1,...,\phi_{J-1})'$ $(J-1 \times 1)$ represents the cut points (threshold values) that need to be estimated. $\phi_1 = 0$ is set to zero by default.

$\beta$ is a $(k \times 1)$ vector of parameters associated with the $(n \times k)$ data matrix X.

$\rho$ is the spatial dependence parameter.

The error variance $\sigma_e$ is set to 1 for identification.

Computation of marginal effects is currently not implemented.

References

LeSage, J. and Pace, R. K. (2009), Introduction to Spatial Econometrics, CRC Press, chapter 10, section 10.2

Examples

Run this code

# \donttest{
library(spatialprobit)
set.seed(1)

################################################################################
#
# Example with J = 4 alternatives
#
################################################################################

# set up a model like in SAR probit
J <- 4   
# ordered alternatives j=1, 2, 3, 4 
# --> (J-2)=2 cutoff-points to be estimated phi_2, phi_3
phi <- c(-Inf, 0,  +1, +2, Inf)    # phi_0,...,phi_j, vector of length (J+1)
# phi_1 = 0 is a identification restriction

# generate random samples from true model
n <- 400             # number of items
k <- 3               # 3 beta parameters
beta <- c(0, -1, 1)  # true model parameters k=3 beta=(beta1,beta2,beta3)
rho <- 0.75
# design matrix with two standard normal variates as "coordinates"
X <- cbind(intercept=1, x=rnorm(n), y=rnorm(n))

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

# build spatial weight matrix W from coordinates in X
W <- kNearestNeighbors(x=rnorm(n), y=rnorm(n), k=6)

# create samples from epsilon using independence of distributions (rnorm()) 
# to avoid dense matrix I_n
eps <- rnorm(n=n, mean=0, sd=1)
z   <- solve(qr(I_n - rho * W), X %*% beta + eps)

# ordered variable y:
# y_i = 1 for phi_0 < z <= phi_1; -Inf < z <= 0
# y_i = 2 for phi_1 < z <= phi_2
# y_i = 3 for phi_2 < z <= phi_3
# y_i = 4 for phi_3 < z <= phi_4

# y in {1, 2, 3} 
y   <- cut(as.double(z), breaks=phi, labels=FALSE, ordered_result = TRUE)
table(y)

#y
#  1   2   3   4 
#246  55  44  55

# estimate SAR Ordered Probit
res <- sar_ordered_probit_mcmc(y=y, X=X, W=W, showProgress=TRUE)
summary(res)

#----MCMC spatial autoregressive ordered probit----
#Execution time  = 12.152 secs
#
#N draws         =   1000, N omit (burn-in)=    100
#N observations  =    400, K covariates    =      3
#Min rho         = -1.000, Max rho         =  1.000
#--------------------------------------------------
#
#y
#  1   2   3   4 
#246  55  44  55 
#          Estimate Std. Dev  p-level t-value Pr(>|z|)    
#intercept -0.10459  0.05813  0.03300  -1.799   0.0727 .  
#x         -0.78238  0.07609  0.00000 -10.283   <2e-16 ***
#y          0.83102  0.07256  0.00000  11.452   <2e-16 ***
#rho        0.72289  0.04045  0.00000  17.872   <2e-16 ***
#y>=2       0.00000  0.00000  1.00000      NA       NA    
#y>=3       0.74415  0.07927  0.00000   9.387   <2e-16 ***
#y>=4       1.53705  0.10104  0.00000  15.212   <2e-16 ***
#---

addmargins(table(y=res$y, fitted=res$fitted.response))

#     fitted
#y       1   2   3   4 Sum
#  1   218  26   2   0 246
#  2    31  19   5   0  55
#  3    11  19  12   2  44
#  4     3  14  15  23  55
#  Sum 263  78  34  25 400
# }

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