| rhierNegbinRw {bayesm} | R Documentation |
rhierNegbinRw implements an MCMC strategy for the hierarchical Negative
Binomial (NBD) regression model. Metropolis steps for each unit level set of
regression parameters are automatically tuned by optimization. Over-dispersion
parameter (alpha) is common across units.
rhierNegbinRw(Data, Prior, Mcmc)
Data |
list(regdata,Z) |
Prior |
list(Deltabar,Adelta,nu,V,a,b) |
Mcmc |
list(R,keep,s_beta,s_alpha,c,Vbeta0,Delta0) |
Model: y_i ~ NBD(mean=lambda, over-dispersion=alpha).
lambda=exp(X_ibeta_i)
Prior: beta_i ~ N(Delta'z_i,Vbeta).
vec(Delta|Vbeta) ~ N(vec(Deltabar),Vbeta (x) Adelta).
Vbeta ~ IW(nu,V).
alpha ~ Gamma(a,b).
note: prior mean of alpha = a/b, variance = a/(b^2)
list arguments contain:
regdataregdata[[i]]$Xregdata[[i]]$yZDeltabarAdeltanuVabRkeeps\_betas\_alphacVbeta0Delta0a list containing:
llike |
R/keep vector of values of log-likelihood |
betadraw |
nreg x nvar x R/keep array of beta draws |
alphadraw |
R/keep vector of alpha draws |
acceptrbeta |
acceptance rate of the beta draws |
acceptralpha |
acceptance rate of the alpha draws |
The NBD regression encompasses Poisson regression in the sense that as alpha goes to
infinity the NBD distribution tends to the Poisson.
For "small" values of alpha, the dependent variable can be extremely variable so that
a large number of observations may be required to obtain precise inferences.
For ease of interpretation, we recommend demeaning Z variables.
Sridhar Narayanam & Peter Rossi, Graduate School of Business, University of Chicago, Peter.Rossi@ChicagoGsb.edu.
For further discussion, see Bayesian Statistics and Marketing
by Rossi, Allenby and McCulloch, Chapter 5.
http://faculty.chicagogsb.edu/peter.rossi/research/bsm.html
##
if(nchar(Sys.getenv("LONG_TEST")) != 0) {R=2000} else {R=10}
##
set.seed(66)
simnegbin =
function(X, beta, alpha) {
# Simulate from the Negative Binomial Regression
lambda = exp(X %*% beta)
y=NULL
for (j in 1:length(lambda))
y = c(y,rnbinom(1,mu = lambda[j],size = alpha))
return(y)
}
nreg = 100 # Number of cross sectional units
T = 50 # Number of observations per unit
nobs = nreg*T
nvar=2 # Number of X variables
nz=2 # Number of Z variables
# Construct the Z matrix
Z = cbind(rep(1,nreg),rnorm(nreg,mean=1,sd=0.125))
Delta = cbind(c(0.4,0.2), c(0.1,0.05))
alpha = 5
Vbeta = rbind(c(0.1,0),c(0,0.1))
# Construct the regdata (containing X)
simnegbindata = NULL
for (i in 1:nreg) {
betai = as.vector(Z[i,]%*%Delta) + chol(Vbeta)%*%rnorm(nvar)
X = cbind(rep(1,T),rnorm(T,mean=2,sd=0.25))
simnegbindata[[i]] = list(y=simnegbin(X,betai,alpha), X=X,beta=betai)
}
Beta = NULL
for (i in 1:nreg) {Beta=rbind(Beta,matrix(simnegbindata[[i]]$beta,nrow=1))}
Data1 = list(regdata=simnegbindata, Z=Z)
Mcmc1 = list(R=R)
out = rhierNegbinRw(Data=Data1, Mcmc=Mcmc1)
cat("Summary of Delta draws",fill=TRUE)
summary(out$Deltadraw,tvalues=as.vector(Delta))
cat("Summary of Vbeta draws",fill=TRUE)
summary(out$Vbetadraw,tvalues=as.vector(Vbeta[upper.tri(Vbeta,diag=TRUE)]))
cat("Summary of alpha draws",fill=TRUE)
summary(out$alpha,tvalues=alpha)
if(0){
## plotting examples
plot(out$betadraw)
plot(out$alpha,tvalues=alpha)
plot(out$Deltadraw,tvalues=as.vector(Delta))
}