MSARmdl function

Markov-switching autoregressive model

Markov-switching autoregressive model

This function estimates a Markov-switching autoregressive model

MSARmdl(Y, p, k, control = list())

Arguments

  • Y: (T x 1) vector with observational data.

  • p: integer for the number of lags to use in estimation. Must be greater than or equal to 1.

  • k: integer for the number of regimes to use in estimation. Must be greater than or equal to 2.

  • control: List with model options including:

    • getSE: Boolean. If TRUE standard errors are computed and returned. If FALSE standard errors are not computed. Default is TRUE.
    • msmu: Boolean. If TRUE model is estimated with switch in mean. If FALSE model is estimated with constant mean. Default is TRUE.
    • msvar: Boolean. If TRUE model is estimated with switch in variance. If FALSE model is estimated with constant variance. Default is TRUE.
    • init_theta: vector of initial values. vector must contain (1 x q) vector mu, vech(sigma), and vec(P) where sigma is a (q x q) covariance matrix.This is optional. Default is NULL, in which case initVals_MSARmdl is used to generate initial values.
    • method: string determining which method to use. Options are 'EM' for EM algorithm or 'MLE' for Maximum Likelihood Estimation. Default is 'EM'.
    • maxit: integer determining the maximum number of EM iterations.
    • thtol: double determining the convergence criterion for the absolute difference in parameter estimates theta between iterations. Default is 1e-6.
    • maxit_converge: integer determining the maximum number of initial values attempted until solution is finite. For example, if parameters in theta or logLike are NaN another set of initial values (up to maxit_converge) is attempted until finite values are returned. This does not occur frequently for most types of data but may be useful in some cases. Once finite values are obtained, this counts as one iteration towards use_diff_init. Default is 500.
    • use_diff_init: integer determining how many different initial values to try (that do not return NaN; see maxit_converge). Default is 1.
    • mle_stationary_constraint: Boolean determining if only stationary solutions are considered (if TRUE) or not (if FALSE). Default is TRUE.
    • mle_variance_constraint: Double used to determine the lower bound for variance in each regime. Value should be between 0 and 1 as it is multiplied by single regime variance. Default is 0.01 (i.e., 1% of single regime variance.
    • mle_theta_low: Vector with lower bounds on parameters (Used only if method = "MLE"). Default is NULL.
    • mle_theta_upp: Vector with upper bounds on parameters (Used only if method = "MLE"). Default is NULL.

Returns

List of class MSARmdl (S3 object) with model attributes including:

  • y: a (T x 1) matrix of observations.
  • X: a (T-p x p + const) matrix of lagged observations with a leading column of 1s.
  • x: a (T-p x p) matrix of lagged observations.
  • fitted: a (T x 1) matrix of fitted values.
  • resid: a (T x 1) matrix of residuals.
  • intercept: a (k x 1) vector of estimated intercepts of each process.
  • mu: a (k x 1) vector of estimated means of each process.
  • beta: a ((1 + p) x k) matrix of estimated coefficients.
  • phi: estimates of autoregressive coefficients.
  • stdev: a (k x 1) vector of estimated standard deviation of each process.
  • sigma: a (k x 1) estimated covariance matrix.
  • theta: vector containing: mu and vech(sigma).
  • theta_mu_ind: vector indicating location of mean with 1 and 0 otherwise.
  • theta_sig_ind: vector indicating location of variances with 1 and 0 otherwise.
  • theta_var_ind: vector indicating location of variances with 1 and 0 otherwise. This is the same as theta_sig_ind in MSARmdl.
  • theta_P_ind: vector indicating location of transition matrix elements with 1 and 0 otherwise.
  • stationary: Boolean indicating if process is stationary if TRUE or non-stationary if FALSE.
  • n: number of observations (same as T).
  • p: number of autoregressive lags.
  • q: number of series. This is always 1 in MSARmdl.
  • k: number of regimes in estimated model.
  • P: a (k x k) transition matrix.
  • pinf: a (k x 1) vector with limiting probabilities of each regime.
  • St: a (T x k) vector with smoothed probabilities of each regime at each time t.
  • deltath: double with maximum absolute difference in vector theta between last iteration.
  • iterations: number of EM iterations performed to achieve convergence (if less than maxit).
  • theta_0: vector of initial values used.
  • init_used: number of different initial values used to get a finite solution. See description of input maxit_converge.
  • msmu: Boolean. If TRUE model was estimated with switch in mean. If FALSE model was estimated with constant mean.
  • msvar: Boolean. If TRUE model was estimated with switch in variance. If FALSE model was estimated with constant variance.
  • control: List with model options used.
  • logLike: log-likelihood.
  • AIC: Akaike information criterion.
  • BIC: Bayesian (Schwarz) information criterion.
  • Hess: Hessian matrix. Approximated using hessian and only returned if getSE=TRUE.
  • info_mat: Information matrix. Computed as the inverse of -Hess. If matrix is not PD then nearest PD matrix is obtained using nearest_spd. Only returned if getSE=TRUE.
  • nearPD_used: Boolean determining whether nearPD function was used on info_mat if TRUE or not if FALSE. Only returned if getSE=TRUE.
  • theta_se: standard errors of parameters in theta. Only returned if getSE=TRUE.
  • trace: List with Lists of estimation output for each initial value used due to use_diff_init \> 1.

Examples

# --------------------------- Use simulated process ----------------------------
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 options for model estimation
control <- list(msmu   = TRUE, 
                msvar  = TRUE, 
                method = "EM",
                use_diff_init = 1)

# Estimate model

  ms_mdl <- MSARmdl(y_ms_simu$y, p = 1, k = 2, control = control)
  summary(ms_mdl)

References

Dempster, A. P., N. M. Laird, and D. B. Rubin. 1977. “Maximum Likelihood from Incomplete Data via the EM Algorithm.” Journal of the Royal Statistical Society. Series B 39 (1): 1–38..

Hamilton, James D. 1990. “Analysis of time series subject to changes in regime.” Journal of econometrics, 45 (1-2): 39–70.

See Also

ARmdl

MSTest packageRead PDF manual
  • Maintainer: Gabriel Rodriguez Rondon
  • License: GPL (>= 2)
  • Last published: 2025-02-24

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