# MCMCresidualBreakAnalysis: Break Analysis of Univariate Time Series using Markov Chain... In MCMCpack: Markov Chain Monte Carlo (MCMC) Package

## Description

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.

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 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"), ... ) 

## Arguments

 resid Univariate time series m The number of breaks. b0 The prior mean of β. 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 β. 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 c_0/2 is the shape parameter for the inverse Gamma prior on σ^2 (the variance of the disturbances). The amount of information in the inverse Gamma prior is something like that from c_0 pseudo-observations. d0 d_0/2 is the scale parameter for the inverse Gamma prior on σ^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 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. mcmc The number of MCMC iterations after burnin. burnin The number of burn-in iterations for the sampler. 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 β 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 β 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 β 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. random.perturb 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). WAIC Compute the Widely Applicable Information Criterion (Watanabe 2010). 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

## Details

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}(β_{m}, σ^2_{m}) \;\; m = 1, …, M

We assume standard, semi-conjugate priors:

β \sim \mathcal{N}(b_0,B_0^{-1})

And:

σ^{-2} \sim \mathcal{G}amma(c_0/2, d_0/2)

Where β and σ^{-2} are assumed a priori independent.

And:

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

Where M is the number of states.

## Value

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

## References

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
 1 2 3 4 5 6 7 8 9 ## Not run: 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) ## End(Not run)