This function performs a break analysis for univariate time series data
using a linear Gaussian changepoint model. The code is written mainly for an
internal use in testpanelSubjectBreak.
MCMCresidualBreakAnalysis(
resid,
m = 1,
b0 = 0,
B0 = 0.001,
c0 = 0.1,
d0 = 0.1,
a = NULL,
b = NULL,
mcmc = 1000,
burnin = 1000,
thin = 1,
verbose = 0,
seed = NA,
beta.start = NA,
P.start = NA,
random.perturb = FALSE,
WAIC = FALSE,
marginal.likelihood = c("none", "Chib95"),
...
)An mcmc object that contains the posterior sample. This object can be summarized by functions provided by the coda package.
Univariate time series
The number of breaks.
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.
\(c_0/2\) is the shape parameter for the inverse Gamma prior on \(\sigma^2\) (the variance of the disturbances). The amount of information in the inverse Gamma prior is something like that from \(c_0\) pseudo-observations.
\(d_0/2\) is the scale parameter for the inverse Gamma prior on \(\sigma^2\) (the variance of the disturbances). In constructing the inverse Gamma prior, \(d_0\) acts like the sum of squared errors from the \(c_0\) pseudo-observations.
\(a\) 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.
The number of MCMC iterations after burnin.
The number of burn-in iterations for the sampler.
The thinning interval used in the simulation. The number of MCMC iterations must be divisible by this value.
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 OLS 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 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.
If TRUE, randomly sample hidden states whenever regularly sampled hidden states have at least one single observation state. It's one method to avoid overfitting in a non-ergodic hidden Markov models. See Park and Sohn (2017).
Compute the Widely Applicable Information Criterion (Watanabe 2010).
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
MCMCresidualBreakAnalysis simulates from the posterior distribution
using standard Gibbs sampling (a multivariate Normal draw for the betas, and
an inverse Gamma draw for the conditional error variance). The simulation
proper is done in compiled C++ code to maximize efficiency. Please consult
the coda documentation for a comprehensive list of functions that can be
used to analyze the posterior sample.
The model takes the following form:
$$y_{i} \sim \mathcal{N}(\beta_{m}, \sigma^2_{m}) \;\; m = 1, \ldots, M$$
We assume standard, semi-conjugate priors:
$$\beta \sim \mathcal{N}(b_0,B_0^{-1})$$
And: $$\sigma^{-2} \sim \mathcal{G}amma(c_0/2, d_0/2)$$
Where \(\beta\) and \(\sigma^{-2}\) are assumed a priori independent.
And:
$$p_{mm} \sim \mathcal{B}eta(a, b),\;\; m = 1, \ldots, M$$
Where \(M\) is the number of states.
Jong Hee Park and Yunkyu Sohn. 2017. "Detecting Structural Changes in Network Data: An Application to Changes in Military Alliance Networks, 1816-2012". Working Paper.
Jong Hee Park, 2012. ``Unified Method for Dynamic and Cross-Sectional Heterogeneity: Introducing Hidden Markov Panel Models.'' American Journal of Political Science.56: 1040-1054. <doi: 10.1111/j.1540-5907.2012.00590.x>
Sumio Watanabe. 2010. "Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory" Journal of Machine Learning Research. 11: 3571-3594.
Siddhartha Chib. 1995. "Marginal Likelihood from the Gibbs Output." Journal of the American Statistical Association. 90: 1313-1321. <doi: 10.1016/S0304-4076(97)00115-2>
Siddhartha Chib. 1998. "Estimation and comparison of multiple change-point models." Journal of Econometrics. 86: 221-241. <doi: 10.1080/01621459.1995.10476635>
plot.mcmc, summary.mcmc,
lm
if (FALSE) {
line <- list(X = c(-2,-1,0,1,2), Y = c(1,3,3,3,5))
ols <- lm(Y~X)
residual <- rstandard(ols)
posterior <- MCMCresidualBreakAnalysis(residual, m = 1, data=line, mcmc=1000, verbose=200)
plotState(posterior)
summary(posterior)
}
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