MCMCpack (version 1.4-4)

MCMCprobitChange: Markov Chain Monte Carlo for a linear Gaussian Multiple Changepoint Model

Description

This function generates a sample from the posterior distribution of a linear Gaussian model with multiple changepoints. 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

MCMCprobitChange(formula, data = parent.frame(), m = 1, burnin = 10000,
  mcmc = 10000, thin = 1, verbose = 0, seed = NA, beta.start = NA,
  P.start = NA, b0 = NULL, B0 = NULL, a = NULL, b = NULL,
  marginal.likelihood = c("none", "Chib95"), ...)

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.

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.

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.

...

further arguments to be passed

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).

Details

MCMCprobitChange simulates from the posterior distribution of a probit regression model with multiple parameter breaks. The simulation is based on Chib (1998) and Park (2011).

The model takes the following form:

$$\Pr(y_t = 1) = \Phi(x_i'\beta_m) \;\; m = 1, \ldots, M$$

Where \(M\) is the number of states, and \(\beta_m\) is a parameter 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.

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.

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. http://www.jstatsoft.org/v42/i09/.

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

Albert, J. H. and S. Chib. 1993. ``Bayesian Analysis of Binary and Polychotomous Response Data.'' J. Amer. Statist. Assoc. 88, 669-679

See Also

plotState, plotChangepoint

Examples

# NOT RUN {
# }
# NOT RUN {
set.seed(1973)
x1 <- rnorm(300, 0, 1)
true.beta <- c(-.5, .2, 1)
true.alpha <- c(.1, -1., .2)
X <- cbind(1, x1)

## set two true breaks at 100 and 200
true.phi1 <- pnorm(true.alpha[1] + x1[1:100]*true.beta[1])
true.phi2 <- pnorm(true.alpha[2] + x1[101:200]*true.beta[2])
true.phi3 <-  pnorm(true.alpha[3] + x1[201:300]*true.beta[3])

## generate y
y1 <- rbinom(100, 1, true.phi1)
y2 <- rbinom(100, 1, true.phi2)
y3 <- rbinom(100, 1, true.phi3)
Y <- as.ts(c(y1, y2, y3))

## fit multiple models with a varying number of breaks
out0 <- MCMCprobitChange(formula=Y~X-1, data=parent.frame(), m=0,
                         mcmc=1000, burnin=1000, thin=1, verbose=1000,
                         b0 = 0, B0 = 10, a = 1, b = 1,  marginal.likelihood = c("Chib95"))
out1 <- MCMCprobitChange(formula=Y~X-1, data=parent.frame(), m=1,
                         mcmc=1000, burnin=1000, thin=1, verbose=1000,
                         b0 = 0, B0 = 10, a = 1, b = 1,  marginal.likelihood = c("Chib95"))
out2 <- MCMCprobitChange(formula=Y~X-1, data=parent.frame(), m=2,
                         mcmc=1000, burnin=1000, thin=1, verbose=1000,
                         b0 = 0, B0 = 10, a = 1, b = 1,  marginal.likelihood = c("Chib95"))
out3 <- MCMCprobitChange(formula=Y~X-1, data=parent.frame(), m=3,
                         mcmc=1000, burnin=1000, thin=1, verbose=1000,
                         b0 = 0, B0 = 10, a = 1, b = 1,  marginal.likelihood = c("Chib95"))

## find the most reasonable one
BayesFactor(out0, out1, out2, out3)

## draw plots using the "right" model
plotState(out2)
plotChangepoint(out2)
# }
# NOT RUN {
# }