HMmdl function

Hidden Markov model

This function estimates a Hidden Markov model with k regimes.

HMmdl(Y, k, Z = NULL, control = list())

Arguments

• Y: a (T x q) matrix of observations.

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

• Z: an otpional (T x qz) matrix of exogenous regressors. Default is NULL.

• 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_variance_constraint: double used to determine the lower bound on the smallest eigenvalue for the covariance matrix of each regime. Default is 1e-3.
  • 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 HMmdl (S3 object) with model attributes including:

• y: a (T x q) matrix of observations.
• fitted: a (T x q) matrix of fitted values.
• resid: a (T x q) matrix of residuals.
• mu: a (k x q) matrix of estimated means of each process.
• beta: if q=1, this is a ((1 + qz) x k) matrix of estimated coefficients. If q\>1, this is a list containing k separate ((1 + qz) x q) matrix of estimated coefficients for each regime.
• betaZ: a (qz x q) matrix of estimated exogenous regressor coefficients.
• intercept: a (k x q) matrix of estimated intercept of each process. If Z is Null, this is the same as mu.
• stdev: If q=1, this is a (k x 1) matrix with estimated standard. If q\>1, this is a List with k (q x q) matrices with estimated standard deviation on the diagonal.
• sigma: If q=1, this is a (k x 1) matrix with variances. If q\>1, this is a List with k (q x q) 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 variance and covariances with 1 and 0 otherwise.
• theta_var_ind: vector indicating location of variances with 1 and 0 otherwise.
• theta_P_ind: vector indicating location of transition matrix elements with 1 and 0 otherwise.
• n: number of observations (same as T).
• q: number of series.
• 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.

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.

Krolzig, Hans-Martin. 1997. “The markov-switching vector autoregressive model.”. Springer.

See Also

Nmdl