MSBVAR (version 0.9-2)

msbvar: Markov-switching Bayesian reduced form vector autoregression model setup and posterior mode estimation

Description

Sets up and estimates the posterior mode of a reduced form Markov-switching Bayesian vector autoregression model with a Sims-Zha prior. This is the setup and input function for the Gibbs sampler for this model.

Usage

msbvar(Y, z=NULL, p, h, lambda0, lambda1, lambda3, lambda4, lambda5, mu5, mu6, qm, alpha.prior=100*diag(h) + matrix(2, h, h), prior=0, max.iter=40, initialize.opt=NULL)

Arguments

Y

A $T x m$ multiple time series object created with ts().

z

NOT IMPLEMENTED AT PRESENT: THIS SHOULD BE A $T x k$ matrix of exogenous variables. Can be z = NULL if there are none (the default).

p

Lag length, aninteger

h

Number of regimes / states, an integer

lambda0

Value in $[0,1]$, Overall tightness of the prior (discounting of prior scale).

lambda1

Value in $[0,1]$, Standard deviation or tightness of the prior around the AR(1) parameters.

lambda3

Lag decay ($>0$, with 1=harmonic)

lambda4

Standard deviation or tightness around the intercept $>0$

lambda5

Standard deviation or tightness around the exogneous variable coefficients $>0$

mu5

Sum of coefficients prior weight $\ge0$. Larger values imply difference stationarity.

mu6

Dummy initial observations or drift prior $\ge0$. Larger values allow for common trends.

qm

Frequency of the data for lag decay equivalence. Default is 4, and a value of 12 will match the lag decay of monthly to quarterly data. Other values have the same effect as "4"

alpha.prior

Prior for the Dirichlet process for the MS process. Default is 100 * diag(h) + matrix(2, h, h), but the model will be sensitive to this.

prior

One of three values: 0= Normal-Wishart prior, 1 = Normal-flat prior, 2 = flat-flat prior (i.e., akin to MLE). The conjugate prior is the first one, which is the default.

max.iter

Maximum number of iterations for the block EM algorithm used to fit an initial guess of the model posterior. Default value is 40 iterations. Larger problems will need more iterations.

initialize.opt

Initial values for the block optimization algorithm. If default=NULL initialize.msbvar is called to provide values. User can specify values as long as they conform to the structure produced by initialize.msbvar.

Value

A list describing the posterior mode of the MSBVAR model and the inputs necessary for the subsequent Gibbs sampler.

init.model

An object of the class BVAR that describes the setup of the model. See szbvar for details.

hreg

A list containing the regime-specific moment matrices, VAR coefficients, and error covariances

Q

The $h x h$ Markov transition matrix.

fp

The $T x h$ matrix of the filtered regime probabilities. First column is the first regime, etc.

m

Integer, the number of endogenous variables in the system.

p

Integer, the lag length of the VAR.

h

Integer, the number of regimes in the MS process.

alpha.prior

The $h x h$ matrix for the prior for the Dirichlet density for the MS process.

Details

This function estimates the posterior mode of a reduced form Bayesian Markov-switching VAR model. The MSBVAR mode is estimated using block EM algorithm where the blocks are 1) the BVAR regression coefficients for each regime (separating optimands for intercepts, AR coefficients, and error covariances) and 2) the transition matrix. Starting values are randomly drawn, so a random number seed should be set prior to calling the function in order to make the results replicable.

The steps of the blockwise optimization follow the suggestions of Sims, Waggoner, and Zha (2008). The joint optimization problem is partitioned into the following separate blocks. For each block, a separate call to optim is made, holding all of the other blocks constant:

1. Maximize over the intercepts
2. Maximize over the AR(p) coefficients
3. Maximize over the error covariances $Sigma$
4. Maximize over the transition matrix $Q$

These four blocks are iterated a total of max.iter times. Internal to each block, the state-space filtering algorithm for the regime classifications is computing using compiled Fortran code for speed. Despite the use of compiled code, this algorithm can take several minutes to compute.

The user should try multiple starting values and number of iterations to ensure convergence. The algorithm will improve with each step of the optimization, although sometimes this can be very incremental improvement.

The results for posterior sampling via gibbs.msbvar will be sensitive to the choice of alpha.prior. This is the prior for the independent Dirichlet process for the MS process. Note that the prior is roughly proportionate to the number of time periods spent in each regime, since the estimated MS probabilities map to the duration of the regime via $1/(1-p)$ where $p$ is the probability of staying in the regime. This function should NOT be used for inference, since it only finds the posterior mode of the model. This function is intended to generate starting values for the Gibbs sampling of the model. See gibbs.msbvar for further details of the Gibbs sampling.

References

Brandt, Patrick T. 2009. "Empirical, Regime-Specific Models of International, Inter-group Conflict, and Politics" Fruhwirth-Schnatter, Sylvia. 2001. "Markov Chain Monte Carlo Estimation of Classical and Dynamic Switching and Mixture Models". Journal of the American Statistical Association. 96(153):194--209.

Fruhwirth-Schnatter, Sylvia. 2006. Finite Mixture and Markov Switching Models. Springer Series in Statistics New York: Springer.

Kim, Chang-Jin and Charles R. Nelson. 1999. State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications. Cambridge: MIT Press. Sims, Christopher A. and Daniel F. Waggoner and Tao Zha. 2008. "Methods for inference in large multiple-equation Markov-switching models" Journal of Econometrics 146(2):255--274. Sims, Christopher A. and Tao A. Zha. 1998. "Bayesian Methods for Dynamic Multivariate Models" International Economic Review 39(4):949-968. Sims, Christopher A. and Tao A. Zha. 2006. "Were There Regime Switches in U.S. Monetary Policy?" American Economic Review. 96(1):54--81.

See Also

gibbs.msbvar for the MCMC sampler after using this function, szbvar for a non-switching, Bayesian VAR and more details.

Examples

## Not run: 
# # Simple replication of Hamilton (1989) as in
# # Kim and Nelson (1999: 79, 220)
# 
# data(HamiltonGDP)
# set.seed(214)
# 
# m2 <- msbvar(HamiltonGDP, p=1, h=2,
#              lambda0=0.8, lambda1=0.15, lambda3=1, lambda4=0.25,
#              lambda5=1, mu5=0, mu6=0, qm=12,
#              alpha.prior=c(100, 30)*diag(2) +
#              matrix(12, 2, 2), prior=0, max.iter=30,
#              initialize.opt=NULL)
# 
# # Now plot the filtered probabilities of a recession
# # Compare to Kim and Nelson (1999: 79, 220)
# 
# fp.rec <- ts(m2$fp[,1], start=tsp(HamiltonGDP)[1],
#              freq=tsp(HamiltonGDP)[3])
# plot(fp.rec)
# 
# ## End(Not run)