MMCLRTest function

Maximized Monte Carlo Likelihood Ratio Test

This function performs the Maximized Monte Carlo likelihood ratio test (MMC-LRT) proposed in Rodriguez-Rondon & Dufour (2024).

MMCLRTest(Y, p, k0, k1, Z = NULL, control = list())

Arguments

• Y: Series to be tested. Must be a (T x q) matrix where T is the number of time observations and q is the number of variables.

• p: Number of autoregressive lags. Must be greater than or equal to 0.

• k0: Number of regimes under null hypothesis. Must be greater than or equal to 1.

• k1: Number of regimes under alternative hypothesis. Must be greater than k0.

• Z: Exogenous regressors. Optional input and default is NULL. When used, it should be a (T x qz) matrix where T is the number of time observations and q is the number of exogenous variables.

• control: List with test procedure options including:

  • N: Integer determining the number of Monte Carlo simulations. Default is set to 99 as in paper.
  • burnin: Number of simulated observations to remove from beginning. Default is 100.
  • converge_check: String of NULL determining if convergence of model(s) should be verified. Allowed inputs are: "null", "alt", "both", or NULL. If NULL (default) no model convergence is verified.
  • workers: Integer determining the number of workers to use for parallel computing version of test. Note that parallel pool must already be open. Default is 0.
  • type: String that determines the type of optimization algorithm used. Arguments allowed are: "pso", "GenSA", and "GA". Default is "pso".
  • eps: Double determining the constant value that defines a consistent set for search. Default is 0.1.
  • CI_union: Boolean determining if union of set determined by eps and confidence set should be used to define consistent set for search. Default is TRUE.
  • lambda: Double determining penalty on nonlinear constraint. Default is 100.
  • stationary_constraint: Boolean determining if only stationary solutions are considered (if TRUE) or not (if FALSE). Default is TRUE.
  • phi_low: Vector with lower bound for autoregressive parameters when optimizing. Default is NULL.
  • phi_upp: Vector with upper bound for autoregressive parameters when optimizing. Default is NULL.
  • P_low: Value with lower bound for transition probabilities when optimizing. Default is 0.
  • P_upp: Value with upper bound for transition probabilities when optimizing. Default is 1.
  • variance_constraint: Double used to determine the lower bound for variance in parameter set for search. Value should be between 0 and 1 as it is multiplied by consistent point estimates of variances. Default is 0.01 (i.e., 1% of consistent point estimates.
  • silence: Boolean determining if optimization steps should be silenced (if TRUE) or not (if FALSE). Default is FALSE.
  • threshold_stop: Double determining the global optimum of function. Default is 1.
  • mdl_h0_control: List with restricted model options. See Nmdl, ARmdl, VARmdl, HMmdl, MSARmdl, or MSVARmdl documentation for available and default values.
  • mdl_h1_control: List with unrestricted model options. See HMmdl, MSARmdl, or MSVARmdl documentation for available and default values.
  • use_diff_init_sim: Value which determines the number of initial values to use when estimating models for null distribution. Default is set to use the same as specified in mdl_h0_control and mdl_h1_control.
  • optim_control: List with optimization algorithm options. See psoptim, GenSA, ga. Default is to set list(maxit = 200) so that maximum number of iterations is 200.

Returns

List of class LMCLRTest (S3 object) with attributes including:

• mdl_h0: List with restricted model attributes.
• mdl_h1: List with unrestricted model attributes.
• LRT_0: Value of test statistic from observed data.
• LRN: A (N x 1) vector of test statistics from data simulated under the null hypothesis.
• pval: P-value of Local Monte Carlo Likelihood Ratio Test.
• LRN_cv: Vector with 90%, 95%, and 99% Monte Carlo simulated critical values (from vector LRN). These are not asymptotic critical values.
• control: List with test procedure options used.

Examples

set.seed(1234)
# Define DGP of MS AR process
mdl_ms2 <- list(n     = 200, 
                mu    = c(5,10),
                sigma = c(1,4),
                phi   = c(0.5),
                k     = 2,
                P     = rbind(c(0.90, 0.10),
                              c(0.10, 0.90)))

# Simulate process using simuMSAR() function
y_ms_simu <- simuMSAR(mdl_ms2)

# Set test procedure options
mmc_control = list(N = 19,
                   burnin = 100,
                   converge_check = NULL,
                   eps = 0.1,
                   CI_union = TRUE,
                   silence = FALSE,
                   threshold_stop = 0.05 + 1e-6,
                   type = "pso",
                   maxit = 100,
                   mdl_h0_control = list(const  = TRUE, 
                                         getSE  = TRUE),
                   mdl_h1_control = list(msmu   = TRUE, 
                                         msvar  = TRUE,
                                         getSE  = TRUE,
                                         method = "EM",
                                         use_diff_init = 1))

  MMCtest <- MMCLRTest(y_ms_simu$y, p = 1 , k0 = 1 , k1 = 2, control = mmc_control)
  summary(MMCtest)

References

Rodriguez-Rondon, Gabriel and Jean-Marie Dufour. 2022. "Simulation-Based Inference for Markov Switching Models” JSM Proceedings, Business and Economic Statistics Section: American Statistical Association.

Rodriguez-Rondon, Gabriel and Jean-Marie Dufour. 2024. “Monte Carlo Likelihood Ratio Tests for Markov Switching Models.” Unpublished manuscript.