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rhierNegbinRw

MCMC Algorithm for Hierarchical Negative Binomial Regression

Description

rhierNegbinRw implements an MCMC algorithm 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.

Usage

rhierNegbinRw(Data, Prior, Mcmc)

Arguments

`Data`	list(regdata, Z)
`Prior`	list(Deltabar, Adelta, nu, V, a, b)
`Mcmc`	list(R, keep, nprint, s_beta, s_alpha, alpha, c, Vbeta0, Delta0)

Details

Model and Priors

y_i ~ NBD(mean=λ, over-dispersion=alpha)
λ = exp(X_iβ_i)

β_i ~ N(Δ'z_i,Vbeta)

vec(Δ|Vbeta) ~ N(vec(Deltabar), Vbeta (x) Adelta)
Vbeta ~ IW(nu, V)
alpha ~ Gamma(a, b) (unless Mcmc$alpha specified)
Note: prior mean of alpha = a/b, variance = a/(b^2)

Argument Details

Data = list(regdata, Z) [Z optional]

`regdata:`	list of lists with data on each of `nreg` units
`regdata[[i]]$X:`	nobs_i x nvar matrix of X variables
`regdata[[i]]$y:`	nobs_i x 1 vector of count responses
`Z:`	nreg x nz matrix of unit characteristics (def: vector of ones)

Prior = list(Deltabar, Adelta, nu, V, a, b) [optional]

`Deltabar:`	nz x nvar prior mean matrix (def: 0)
`Adelta:`	nz x nz PDS prior precision matrix (def: 0.01*I)
`nu:`	d.f. parameter for Inverted Wishart prior (def: nvar+3)
`V:`	location matrix of Inverted Wishart prior (def: nu*I)
`a:`	Gamma prior parameter (def: 0.5)
`b:`	Gamma prior parameter (def: 0.1)

Mcmc = list(R, keep, nprint, s_beta, s_alpha, alpha, c, Vbeta0, Delta0) [only R required]

`R:`	number of MCMC draws
`keep:`	MCMC thinning parameter -- keep every `keep`th draw (def: 1)
`nprint:`	print the estimated time remaining for every `nprint`'th draw (def: 100, set to 0 for no print)
`s_beta:`	scaling for beta \| alpha RW inc cov (def: 2.93/`sqrt(nvar)`)
`s_alpha:`	scaling for alpha \| beta RW inc cov (def: 2.93)
`alpha:`	over-dispersion parameter (def: alpha ~ Gamma(a,b))
`c:`	fractional likelihood weighting parm (def: 2)
`Vbeta0:`	starting value for Vbeta (def: I)
`Delta0:`	starting value for Delta (def: 0)

Value

A list containing:

`llike`	R/keep x 1 vector of values of log-likelihood
`betadraw`	nreg x nvar x R/keep array of beta draws
`alphadraw`	R/keep x 1 vector of alpha draws
`acceptrbeta`	acceptance rate of the beta draws
`acceptralpha`	acceptance rate of the alpha draws

Note

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.

Author(s)

Sridhar Narayanan (Stanford GSB) and Peter Rossi (Anderson School, UCLA), perossichi@gmail.com.

References

For further discussion, see Chapter 5, Bayesian Statistics and Marketing by Rossi, Allenby, and McCulloch.
http://www.perossi.org/home/bsm-1

Examples

if(nchar(Sys.getenv("LONG_TEST")) != 0) {R=2000} else {R=10}
set.seed(66)

# Simulate from the Negative Binomial Regression
simnegbin = function(X, beta, alpha) {
  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(4,2), c(0.1,-1))
alpha = 5
Vbeta = rbind(c(2,1), c(1,2))

## 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)

## plotting examples
if(0){
  plot(out$betadraw)
  plot(out$alpha,tvalues=alpha)
  plot(out$Deltadraw,tvalues=as.vector(Delta))
}

bayesm

Bayesian Inference for Marketing/Micro-Econometrics

v3.1-4

GPL (>= 2)

Authors

Peter Rossi <perossichi@gmail.com>

Initial release

2019-10-14