MCMCoprobitChange: Markov Chain Monte Carlo for Ordered Probit Changepoint...

In MCMCpack: Markov Chain Monte Carlo (MCMC) Package

Markov Chain Monte Carlo for Ordered Probit Changepoint Regression Model

Description

This function generates a sample from the posterior distribution of an ordered probit regression model with multiple parameter breaks. The function uses the Markov chain Monte Carlo method of Chib (1998). The user supplies data and priors, and a sample from the posterior distribution is returned as an mcmc object, which can be subsequently analyzed with functions provided in the coda package.

Usage

MCMCoprobitChange(
  formula,
  data = parent.frame(),
  m = 1,
  burnin = 1000,
  mcmc = 1000,
  thin = 1,
  tune = NA,
  verbose = 0,
  seed = NA,
  beta.start = NA,
  gamma.start = NA,
  P.start = NA,
  b0 = NULL,
  B0 = NULL,
  a = NULL,
  b = NULL,
  marginal.likelihood = c("none", "Chib95"),
  gamma.fixed = 0,
  ...
)

Arguments

formula	Model formula.
data	Data frame.
m	The number of changepoints.
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.
tune	The tuning parameter for the Metropolis-Hastings step. Default of NA corresponds to a choice of 0.05 divided by the number of categories in the response variable.
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, the \beta vector, and the error variance 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 the \beta vector. This can either be a scalar or a column vector with dimension equal to the number of betas. The default value of of NA will use the MLE estimate of \beta as the starting value. If this is a scalar, that value will serve as the starting value mean for all of the betas.
gamma.start	The starting values for the \gamma vector. This can either be a scalar or a column vector with dimension equal to the number of gammas. The default value of of NA will use the MLE estimate of \gamma as the starting value. If this is a scalar, that value will serve as the starting value mean for all of the gammas.
P.start	The starting values for the transition matrix. A user should provide a square matrix with dimension equal to the number of states. By default, draws from the Beta(0.9, 0.1) are used to construct a proper transition matrix for each raw except the last raw.
b0	The prior mean of \beta. This can either be a scalar or a column vector with dimension equal to the number of betas. If this takes a scalar value, then that value will serve as the prior mean for all of the betas.
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.
a	a is the shape1 beta prior for transition probabilities. By default, the expected duration is computed and corresponding a and b values are assigned. The expected duration is the sample period divided by the number of states.
b	b is the shape2 beta prior for transition probabilities. By default, the expected duration is computed and corresponding a and b values are assigned. The expected duration is the sample period divided by the number of states.
marginal.likelihood	How should the marginal likelihood be calculated? Options are: none in which case the marginal likelihood will not be calculated, and Chib95 in which case the method of Chib (1995) is used.
gamma.fixed	1 if users want to constrain \gamma values to be constant. By default, \gamma values are allowed to vary across regimes.
...	further arguments to be passed

Details

MCMCoprobitChange simulates from the posterior distribution of an ordinal probit regression model with multiple parameter breaks. The simulation of latent states is based on the linear approximation method discussed in Park (2011).

The model takes the following form:

\Pr(y_t = 1) = \Phi(\gamma_{c, m} - x_i'\beta_m) - \Phi(\gamma_{c-1, m} - x_i'\beta_m)\;\; m = 1, \ldots, M

Where M is the number of states, and \gamma_{c, m} and \beta_m are paramters when a state is m at t.

We assume Gaussian distribution for prior of \beta:

\beta_m \sim \mathcal{N}(b_0,B_0^{-1}),\;\; m = 1, \ldots, M

And:

p_{mm} \sim \mathcal{B}eta(a, b),\;\; m = 1, \ldots, M

Where M is the number of states.

Note that when the fitted changepoint model has very few observations in any of states, the marginal likelihood outcome can be “nan," which indicates that too many breaks are assumed given the model and data.

Value

An mcmc object that contains the posterior sample. This object can be summarized by functions provided by the coda package. The object contains an attribute prob.state storage matrix that contains the probability of state_i for each period, the log-likelihood of the model (loglike), and the log-marginal likelihood of the model (logmarglike).

References

Jong Hee Park. 2011. “Changepoint Analysis of Binary and Ordinal Probit Models: An Application to Bank Rate Policy Under the Interwar Gold Standard." Political Analysis. 19: 188-204. <doi:10.1093/pan/mpr007>

Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park. 2011. “MCMCpack: Markov Chain Monte Carlo in R.”, Journal of Statistical Software. 42(9): 1-21. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v042.i09")}.

Siddhartha Chib. 1998. “Estimation and comparison of multiple change-point models.” Journal of Econometrics. 86: 221-241.

See Also

plotState, plotChangepoint

Examples

set.seed(1909)
N <- 200
x1 <- rnorm(N, 1, .5);

## set a true break at 100
z1 <- 1 + x1[1:100] + rnorm(100);
z2 <- 1 -0.2*x1[101:200] + rnorm(100);
z <- c(z1,  z2);
y <- z

## generate y
y[z < 1] <- 1;
y[z >= 1 & z < 2] <- 2;
y[z >= 2] <- 3;

## inputs
formula <- y ~ x1

## fit multiple models with a varying number of breaks
out1 <- MCMCoprobitChange(formula, m=1,
      	mcmc=100, burnin=100, thin=1, tune=c(.5, .5), verbose=100,
     	b0=0, B0=0.1, marginal.likelihood = "Chib95")
out2 <- MCMCoprobitChange(formula, m=2,
      	mcmc=100, burnin=100, thin=1, tune=c(.5, .5, .5), verbose=100,
     	b0=0, B0=0.1, marginal.likelihood = "Chib95")

## Do model comparison
## NOTE: the chain should be run longer than this example!
BayesFactor(out1, out2)

## draw plots using the "right" model
plotState(out1)
plotChangepoint(out1)

MCMCpack documentation built on Sept. 11, 2024, 8:13 p.m.