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_iβ_i + e_i with i=1,…,m for m regressions
(e(k,1), …, e(k,m))' ~ N(0, Σ) with k=1, …, n

Can be written as a stacked model:
y = Xβ + 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 = ∑ p_i.

β ~ N(betabar, A^{-1})
Σ ~ 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 `keep`th 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.
http://www.perossi.org/home/bsm-1

`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)