MCMCquantreg {MCMCpack}R Documentation

Bayesian quantile regression using Gibbs sampling

Description

This function fits quantile regression models under Bayesian inference. The function samples from the posterior distribution using Gibbs sampling with data augmentation. A multivariate normal prior is assumed for beta and an inverse Gamma prior is assumed for sigma. The user supplies the prior parameters. A sample of the posterior distribution is returned as an mcmc object, which can then be analysed by functions in the coda package.

Usage

MCMCquantreg(formula, tau=0.5, data = NULL, burnin = 1000,
   mcmc = 10000, thin = 1, verbose = 0, seed = NA,
   beta.start = NA, b0 = 0, B0 = 0, c0 = 0.001, d0 = 0.001, ...)

Arguments

formula Model formula.
tau The quantile of interest. Must be between 0 and 1. The default value of 0.5 corresponds to median regression.
data Data frame.
burnin The number of burn-in iterations for the sampler.
mcmc The number of MCMC iterations after burnin.
thin The thinning interval used in the simulation. The number of MCMC iterations must be divisible by this value.
verbose A switch which determines whether or not the progress of the sampler is printed to the screen. If verbose is greater than 0 the iteration number and the most recently sampled values of beta and sigma are printed to the screen every verboseth iteration.
seed The seed for the random number generator. If NA, the Mersenne Twister generator is used with default seed 12345; if an integer is passed it is used to seed the Mersenne twister. The user can also pass a list of length two to use the L'Ecuyer random number generator, which is suitable for parallel computation. The first element of the list is the L'Ecuyer seed, which is a vector of length six or NA (if NA a default seed of rep(12345,6) is used). The second element of list is a positive substream number. See the MCMCpack specification for more details.
beta.start The starting values for beta. This can either be a scalar or a column vector with dimension equal to the dimension of beta. The default value of NA will use the OLS estimate beta^hat with sigma^hat*Phi^(-1)(tau) added on to the first element of beta^hat as the starting value. ((sigma^hat)^2 denotes the usual unbiased estimator of sigma^2 under ordinary mean regression and Phi^(-1)(tau) denotes the inverse of the cumulative density function of the standard normal distribution.) Note that the default value assume that an intercept is included in the model. If a scalar is given, that value will serve as the starting value for all beta.
b0 The prior mean of beta. This can either be a scalar or a column vector with dimension equal to the dimension of beta. If this takes a scalar value, then that value will serve as the prior mean for all beta.
B0 The prior precision of beta. This can either be a scalar or a square matrix with dimensions equal to the number of betas. If this takes a scalar value, then that value times an identity matrix serves as the prior precision of beta. Default value of 0 is equivalent to an improper uniform prior for beta.
c0 c0/2 is the shape parameter for the inverse Gamma prior on sigma.
d0 d0/2 is the scale parameter for the inverse Gamma prior on sigma.
... further arguments to be passed

Details

MCMCquantreg simulates from the posterior distribution using a partially collapsed Gibbs sampler with data augmentation (see http://people.brunel.ac.uk/~mastkky/). This involves a multivariate Normal draw for beta, and an inverse Gamma draw for sigma. The augmented data are drawn conditionally from the inverse Gaussian distribution. The simulation is carried out in compiled C++ code to maximise efficiency. Please consult the coda documentation for a comprehensive list of functions that can be used to analyse the posterior sample.

The model takes the following form:

y_i = x_i'beta + epsilon_i

The errors are assumed to have an Asymmetric Laplace distribution with tauth quantile equal to zero and shape parameter equal to sigma:

epsilon_i(tau) ~ AL(0, sigma, tau),

where beta and sigma depend on tau. We assume standard, semi-conjugate priors:

beta ~ N(b0,B0^(-1)),

and

sigma^(-1) ~ Gamma(c0/2, d0/2),

where beta and sigma^(-1) are assumed a priori independent of each other and the parameters associated with the other quantiles. Only starting values for beta are allowed for this sampler.

Value

An mcmc object that contains the posterior sample. This object can be summarised by functions provided by the coda package.

Author(s)

Craig Reed

References

Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. Scythe Statistical Library 1.2. http://scythe.wustl.edu.

Craig Reed and Keming Yu. 2009. "A Partially Collapsed Gibbs Sampler for Bayesian Quantile Regression." Technical Report.

Keming Yu and Jin Zhang. 2005. "A Three Parameter Asymmetric Laplace Distribution and it's extensions." Communications in Statistics - Theory and Methods, 34, 1867-1879.

Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2002. Output Analysis and Diagnostics for MCMC (CODA). http://www-fis.iarc.fr/coda/.

See Also

MCMCregress, plot.mcmc, summary.mcmc, lm, rq

Examples

## Not run: 

x<-rep(1:10,5)
y<-rnorm(50,mean=x)
posterior_50 <- MCMCquantreg(y~x)
posterior_95 <- MCMCquantreg(y~x, tau=0.95, verbose=10000,
    mcmc=50000, thin=10, seed=2)
plot(posterior_50)
plot(posterior_95)
raftery.diag(posterior_50)
autocorr.plot(posterior_95)
summary(posterior_50)
summary(posterior_95)
## End(Not run)

[Package MCMCpack version 1.0-4 Index]