rsurGibbs {bayesm} R Documentation

## Gibbs Sampler for Seemingly Unrelated Regressions (SUR)

### Description

rsurGibbs implements a Gibbs Sampler to draw from the posterior of the Seemingly Unrelated Regression (SUR) Model of Zellner.

### Usage

rsurGibbs(Data, Prior, Mcmc)

### Arguments

 Data list(regdata) Prior list(betabar, A, nu, V) Mcmc list(R, keep)

### Details

#### Model and Priors

y_i = X_i\beta_i + e_i with i=1,\ldots,m for m regressions
(e(k,1), \ldots, e(k,m))' \sim N(0, \Sigma) with k=1, \ldots, n

Can be written as a stacked model:
y = X\beta + e where y is a nobs*m vector and p = length(beta) = sum(length(beta_i))

Note: must have the same number of observations (n) in each equation but can have a different number of X variables (p_i) for each equation where p = \sum p_i.

\beta \sim N(betabar, A^{-1})
\Sigma \sim IW(nu,V)

#### Argument Details

Data = list(regdata)

 regdata:  list of lists, regdata[[i]] = list(y=y_i, X=X_i), where y_i is n x 1 and X_i is n x p_i

Prior = list(betabar, A, nu, V) [optional]

 betabar:  p x 1 prior mean (def: 0) A:  p x p prior precision matrix (def: 0.01*I) nu:  d.f. parameter for Inverted Wishart prior (def: m+3) V:  m x m scale parameter for Inverted Wishart prior (def: nu*I)

Mcmc = list(R, keep) [only R required]

 R:  number of MCMC draws keep:  MCMC thinning parameter -- keep every keepth draw (def: 1) nprint:  print the estimated time remaining for every nprint'th draw (def: 100, set to 0 for no print)

### Value

A list containing:

 betadraw R x p matrix of betadraws Sigmadraw R x (m*m) array of Sigma draws

### Author(s)

Peter Rossi, Anderson School, UCLA, perossichi@gmail.com.

### References

For further discussion, see Chapter 3, Bayesian Statistics and Marketing by Rossi, Allenby, and McCulloch.

rmultireg

### Examples

if(nchar(Sys.getenv("LONG_TEST")) != 0) {R=1000} else {R=10}
set.seed(66)

## simulate data from SUR
beta1 = c(1,2)
beta2 = c(1,-1,-2)
nobs = 100
nreg = 2
iota = c(rep(1, nobs))
X1 = cbind(iota, runif(nobs))
X2 = cbind(iota, runif(nobs), runif(nobs))
Sigma = matrix(c(0.5, 0.2, 0.2, 0.5), ncol=2)
U = chol(Sigma)
E = matrix(rnorm(2*nobs),ncol=2)%*%U
y1 = X1%*%beta1 + E[,1]
y2 = X2%*%beta2 + E[,2]

## run Gibbs Sampler
regdata = NULL
regdata[] = list(y=y1, X=X1)
regdata[] = list(y=y2, X=X2)

out = rsurGibbs(Data=list(regdata=regdata), Mcmc=list(R=R))

cat("Summary of beta draws", fill=TRUE)
summary(out$betadraw, tvalues=c(beta1,beta2)) cat("Summary of Sigmadraws", fill=TRUE) summary(out$Sigmadraw, tvalues=as.vector(Sigma[upper.tri(Sigma,diag=TRUE)]))

## plotting examples