Skip to contents

Hidden Markov Model (HMM) Bootstrap for Multivariate Time Series

hmm_bootstrap.Rd

Fits a Hidden Markov Model (HMM) to a multivariate time series and generates bootstrap replicates by resampling regime-specific blocks. Supports both Gaussian emissions (via depmixS4) and non-Gaussian emissions including skew Student-t distributions (via MSGARCH or standalone EM).

Usage

hmm_bootstrap(
  x,
  n_boot = NULL,
  num_states = 2,
  num_blocks = NULL,
  num_boots = 100,
  distribution = c("gaussian", "sstd", "std", "snorm", "sged", "norm", "ged", "sstd_raw",
    "std_raw", "norm_raw"),
  variance_model = c("sGARCH", "eGARCH", "gjrGARCH", "tGARCH"),
  micro_block_length = 1L,
  regime_basis = "market",
  parallel = FALSE,
  num_cores = 1L,
  return_fit = FALSE,
  collect_diagnostics = FALSE,
  verbose = FALSE,
  seed = NULL,
  ...
)

Arguments

x

Numeric vector or matrix representing the time series.

n_boot

Integer, length of each bootstrap series. If NULL, defaults to the length of the original series.

num_states

Integer, number of hidden states for the HMM. Default is 2.

num_blocks

Integer, number of blocks to sample for each bootstrap replicate. Only used when distribution = "gaussian".

num_boots

Integer, number of bootstrap replicates to generate. Default is 100.

distribution

Character, emission distribution for the HMM. One of:

"gaussian"

(Default) Gaussian emissions via depmixS4. This is the original behavior.

"sstd"

Skew Student-t via MSGARCH (with GARCH dynamics).

"std"

Student-t via MSGARCH (with GARCH dynamics).

"snorm"

Skew normal via MSGARCH (with GARCH dynamics).

"sged"

Skew GED via MSGARCH (with GARCH dynamics).

"norm"

Normal via MSGARCH (with GARCH dynamics).

"ged"

GED via MSGARCH (with GARCH dynamics).

"sstd_raw"

Skew Student-t without GARCH (standalone HMM-EM).

"std_raw"

Student-t without GARCH (standalone HMM-EM).

"norm_raw"

Normal without GARCH (standalone HMM-EM, equivalent to depmixS4 but using our EM implementation).

variance_model

Character, GARCH specification when using MSGARCH-based distributions. One of "sGARCH" (default), "eGARCH", "gjrGARCH", or "tGARCH". Ignored for Gaussian and raw distributions.

micro_block_length

Integer, block length for within-state sampling when using MSGARCH-based bootstrap. Use 1 (default) for iid sampling within states, or >1 to preserve some local dependence. Ignored for Gaussian distribution which uses the traditional block sampling.

regime_basis

For multivariate data with MSGARCH: how to identify regimes. One of "market" (equal-weighted average, default), "first_pc" (first principal component), or an integer column index. Ignored for Gaussian distribution.

parallel

Logical, parallelize computation? Default is FALSE.

num_cores

Integer, number of cores for parallel processing.

return_fit

Logical. If TRUE, returns the fitted model along with bootstrap samples. Default is FALSE.

collect_diagnostics

Logical. If TRUE, collects detailed diagnostic information including regime composition. Default is FALSE.

verbose

Logical. If TRUE, prints fitting information. Default is FALSE.

seed

Integer, random seed for reproducibility. Default is NULL.

...

Additional arguments passed to the underlying fitting functions.

Value

If return_fit = FALSE and collect_diagnostics = FALSE: A list of bootstrap replicate matrices.

If return_fit = TRUE or collect_diagnostics = TRUE: A list containing:

bootstrap_series

List of bootstrap replicate matrices

fit

(if return_fit = TRUE) The fitted model object

states

The Viterbi state sequence for the original data

smoothed_probabilities

State probabilities matrix

diagnostics

(if collect_diagnostics = TRUE) A tsbs_diagnostics object

method

Character indicating which method was used

Details

Method Selection

When distribution = "gaussian", uses depmixS4 for HMM fitting with Gaussian emissions. This is the original behavior and uses block resampling within identified regimes.

When distribution is one of the MSGARCH distributions ("sstd", "std", etc.), uses the MSGARCH package for Markov-switching GARCH estimation. The bootstrap is then semi-parametric: state sequences are simulated from the fitted Markov chain, while observations are resampled from state-specific empirical pools.

When distribution ends with "_raw" (e.g., "sstd_raw"), fits an HMM directly to the returns without the GARCH volatility layer, using a standalone EM algorithm with the specified emission distribution.

Literature Background

The MSGARCH-based implementation combines several established techniques:

  • Markov-switching GARCH: Haas, Mittnik & Paolella (2004), implemented via the MSGARCH package (Ardia et al., 2019)

  • Skew Student-t distribution: Fernández & Steel (1998) transformation

  • Semi-parametric bootstrap: State sequences are simulated from the fitted Markov chain (parametric), while innovations are resampled from empirical state-specific pools (nonparametric)

Multivariate Handling

For multivariate data:

  • With Gaussian: fits a joint HMM where all variables depend on the same hidden state sequence.

  • With MSGARCH: fits the regime model to an aggregate series (see regime_basis), then samples full cross-sections synchronously to preserve empirical dependence structure.

References

Ardia, D., Bluteau, K., Boudt, K., Catania, L., & Trottier, D.-A. (2019). Markov-Switching GARCH Models in R: The MSGARCH Package. Journal of Statistical Software, 91(4), 1-38. doi:10.18637/jss.v091.i04

Fernández, C., & Steel, M. F. (1998). On Bayesian modeling of fat tails and skewness. Journal of the American Statistical Association, 93(441), 359-371.

Haas, M., Mittnik, S., & Paolella, M. S. (2004). A New Approach to Markov- Switching GARCH Models. Journal of Financial Econometrics, 2, 493-530.

Hamilton, J. D. (1989). A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle. Econometrica, 57(2), 357-384.

Holst, U., Lindgren, G., Holst, J. and Thuvesholmen, M. (1994), Recursive Estimation In Switching Autoregressions With A Markov Regime. Journal of Time Series Analysis, 15: 489-506.

See also

msvar_bootstrap, ms_varma_garch_bs

Examples

# \donttest{
## Example 1: Traditional Gaussian HMM bootstrap (original behavior)
set.seed(123)
x <- matrix(rnorm(500), ncol = 2)
boot_gaussian <- hmm_bootstrap(x, n_boot = 400, num_states = 2, num_boots = 50)
#> converged at iteration 184 with logLik: -687.2476 

## Example 2: Skew Student-t with MSGARCH (requires MSGARCH package)
if (requireNamespace("MSGARCH", quietly = TRUE)) {
  # Univariate
  y <- rnorm(300)
  boot_sstd <- hmm_bootstrap(y, num_states = 2, num_boots = 50,
                              distribution = "sstd")

  # With diagnostics
  result <- hmm_bootstrap(y, num_states = 2, num_boots = 20,
                           distribution = "sstd",
                           return_fit = TRUE,
                           collect_diagnostics = TRUE)
}
#> Error: Viterbi decoding assigned no observations to state(s): 2. The model effectively has fewer states than specified. Consider reducing 'num_states' or using different data.

## Example 3: Raw returns HMM without GARCH (requires fGarch)
if (requireNamespace("fGarch", quietly = TRUE)) {
  boot_raw <- hmm_bootstrap(y, num_states = 2, num_boots = 50,
                             distribution = "sstd_raw")
}
# }