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.
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,
...
)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).
Model formula.
Data frame.
The number of changepoints.
The number of burn-in iterations for the sampler.
The number of MCMC iterations after burnin.
The thinning interval used in the simulation. The number of MCMC iterations must be divisible by this value.
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.
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.
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.
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.
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.
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.
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.
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.
\(a\) 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.
\(b\) 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.
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.
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
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.
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. 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.
plotState, plotChangepoint
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)
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