Testing Markov Switching Models

Description

This package implements hypothesis testing procedures that can be used to identify the number of regimes in a Markov-Switching model.

Author(s)

Gabriel Rodriguez Rondon, gabriel.rodriguezrondon@mail.mcgill.ca (Maintainer)

Jean-Marie Dufour, jean-marie.dufour@mcgill.ca

References

Carrasco, Marine, Liang Hu, and Werner Ploberger. 2014. “Optimal test for Markov switching parameters.” Econometrica 82 (2): 765–784.

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..

Dufour, Jean-Marie, and Richard Luger. 2017. “Identification-robust moment-based tests for Markov switching in autoregressive models.” Econometric Reviews 36 (6-9): 713–727.

Kasahara, Hiroyuk, and Katsum Shimotsu. 2018. “Testing the number of regimes in Markov regime switching models.” arXiv preprint arXiv:1801.06862.

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

Hamilton, James D. 1989. “A new approach to the economic analysis of nonstationary time series and the business cycle.” Econometrica 57 (2): 357–384.

Hamilton, James D. 1994. "Time series analysis". Princeton university press.

Hansen, Bruce E. 1992. “The likelihood ratio test under nonstandard conditions: testing the Markov switching model of GNP.” Journal of applied Econometrics 7 (S1): S61–S82.

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. 2022. “Monte Carlo Likelihood Ratio Tests for Markov Switching Models.” Unpublished manuscript.

Rodriguez Rondon, Gabriel and Jean-Marie Dufour. 2022. “MSTest: An R-package for Testing Markov-Switching Models.” Unpublished manuscript.

Qu, Zhongjun, and Fan Zhuo. 2021. “Likelihood Ratio-Based Tests for Markov Regime Switching.” The Review of Economic Studies 88 (2): 937–968.

AIC of a ARmdl object

Description

This is a method for the function AIC() for objects of the class ARmdl.

Usage

## S3 method for class 'ARmdl'
AIC(object, ..., k = 2)

Arguments

Value

AIC value.

AIC of a HMmdl object

Description

This is a method for the function AIC() for objects of the class HMmdl.

Usage

## S3 method for class 'HMmdl'
AIC(object, ..., k = 2)

Arguments

Value

AIC value.

AIC of a MSARmdl object

Description

This is a method for the function AIC() for objects of the class MSARmdl.

Usage

## S3 method for class 'MSARmdl'
AIC(object, ..., k = 2)

Arguments

Value

AIC value.

AIC of a MSVARmdl object

Description

This is a method for the function AIC() for objects of the class MSVARmdl.

Usage

## S3 method for class 'MSVARmdl'
AIC(object, ..., k = 2)

Arguments

Value

AIC value.

AIC of a Nmdl object

Description

This is a method for the function AIC() for objects of the class Nmdl.

Usage

## S3 method for class 'Nmdl'
AIC(object, ..., k = 2)

Arguments

Value

AIC value.

AIC of a VARmdl object

Description

This is a method for the function AIC() for objects of the class VARmdl.

Usage

## S3 method for class 'VARmdl'
AIC(object, ..., k = 2)

Arguments

Value

AIC value.

Autoregressive X Model

Description

This function estimates an ARX model with p lags. This can be used for the null hypothesis of a linear model against an alternative hypothesis of a Markov switching autoregressive model with k regimes.

Usage

ARXmdl(Y, p, Z, control = list())

Arguments

Value

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

• y: a (T-p x 1) matrix of observations.

• X: a (T-p x p + const) matrix of lagged observations with a leading column of 1s if const=TRUE or not if const=FALSE.

• x: a (T-p x p) matrix of lagged observations.

• fitted: a (T-p x 1) matrix of fitted values.

• resid: a (T-p x 1) matrix of residuals.

• mu: estimated mean of the process.

• beta: a ((1 + p + qz) x 1) matrix of estimated coefficients.

• betaZ: a (qz x q) matrix of estimated exogenous regressor coefficients.

• intercept: estimate of intercept.

• phi: estimates of autoregressive coefficients.

• stdev: estimated standard deviation of the process.

• sigma: estimated variance of the process.

• theta: vector containing: mu, sigma, and phi.

• theta_mu_ind: vector indicating location of mean with 1 and 0 otherwise.

• theta_sig_ind: vector indicating location of variance with 1 and 0 otherwise.

• theta_var_ind: vector indicating location of variance with 1 and 0 otherwise. This is the same as theta_sig_ind in ARmdl.

• theta_phi_ind: vector indicating location of autoregressive coefficients with 1 and 0 otherwise.

• stationary: Boolean indicating if process is stationary if TRUE or non-stationary if FALSE.

• n: number of observations after lag transformation (i.e., n = T-p).

• p: number of autoregressive lags.

• q: number of series. This is always 1 in ARmdl.

• k: number of regimes. This is always 1 in ARmdl.

• 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.

See Also

MSARmdl

Examples

set.seed(1234)
# Define DGP of AR process
mdl_ar <- list(n     = 500, 
               mu    = 5,
               sigma = 2,
               phi   = c(0.5,0.2))

# Simulate process using simuAR() function
y_simu <- simuAR(mdl_ar)

# Set options for model estimation
control <- list(const  = TRUE, 
                getSE  = TRUE)

# Estimate model
y_ar_mdl <- ARmdl(y_simu$y, p = 2, control)
y_ar_mdl

Autoregressive Model

Description

This function estimates an autoregresive model with p lags. This can be used for the null hypothesis of a linear model against an alternative hypothesis of a Markov switching autoregressive model with k regimes.

Usage

ARmdl(Y, p, control = list())

Arguments

Value

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

• y: a (T-p x 1) matrix of observations.

• X: a (T-p x p + const) matrix of lagged observations with a leading column of 1s if const=TRUE or not if const=FALSE.

• x: a (T-p x p) matrix of lagged observations.

• fitted: a (T-p x 1) matrix of fitted values.

• resid: a (T-p x 1) matrix of residuals.

• mu: estimated mean of the process.

• beta: a ((1 + p) x 1) matrix of estimated coefficients.

• intercept: estimate of intercept.

• phi: estimates of autoregressive coefficients.

• stdev: estimated standard deviation of the process.

• sigma: estimated variance of the process.

• theta: vector containing: mu, sigma, and phi.

• theta_mu_ind: vector indicating location of mean with 1 and 0 otherwise.

• theta_sig_ind: vector indicating location of variance with 1 and 0 otherwise.

• theta_var_ind: vector indicating location of variance with 1 and 0 otherwise. This is the same as theta_sig_ind in ARmdl.

• theta_phi_ind: vector indicating location of autoregressive coefficients with 1 and 0 otherwise.

• stationary: Boolean indicating if process is stationary if TRUE or non-stationary if FALSE.

• n: number of observations after lag transformation (i.e., n = T-p).

• p: number of autoregressive lags.

• q: number of series. This is always 1 in ARmdl.

• k: number of regimes. This is always 1 in ARmdl.

• 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.

See Also

MSARmdl

Examples

set.seed(1234)
# Define DGP of AR process
mdl_ar <- list(n     = 500, 
               mu    = 5,
               sigma = 2,
               phi   = c(0.5,0.2))

# Simulate process using simuAR() function
y_simu <- simuAR(mdl_ar)

# Set options for model estimation
control <- list(const  = TRUE, 
                getSE  = TRUE)

# Estimate model
y_ar_mdl <- ARmdl(y_simu$y, p = 2, control)
y_ar_mdl

BIC of a ARmdl object

Description

This is a method for the function BIC() for objects of the class ARmdl.

Usage

## S3 method for class 'ARmdl'
BIC(object, ...)

Arguments

Value

BIC value.

BIC of a HMmdl object

Description

This is a method for the function BIC() for objects of the class HMmdl.

Usage

## S3 method for class 'HMmdl'
BIC(object, ...)

Arguments

Value

BIC value.

BIC of a MSARmdl object

Description

This is a method for the function BIC() for objects of the class MSARmdl.

Usage

## S3 method for class 'MSARmdl'
BIC(object, ...)

Arguments

Value

BIC value.

BIC of a MSVARmdl object

Description

This is a method for the function BIC() for objects of the class MSVARmdl.

Usage

## S3 method for class 'MSVARmdl'
BIC(object, ...)

Arguments

Value

BIC value.

BIC of a Nmdl object

Description

This is a method for the function BIC() for objects of the class Nmdl.

Usage

## S3 method for class 'Nmdl'
BIC(object, ...)

Arguments

Value

BIC value.

BIC of a VARmdl object

Description

This is a method for the function BIC() for objects of the class VARmdl.

Usage

## S3 method for class 'VARmdl'
BIC(object, ...)

Arguments

Value

BIC value.

Carrasco, Hu, and Ploberger (2014) parameter stability test

Description

This function performs the CHP (2014) parameter stability test as outline in Carrasco, M., Hu, L. and Ploberger, W. (2014). Original source code can be found here.

Usage

CHPTest(Y, p, control = list())

Arguments

Value

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

• mdl_h0: List with restricted model attributes. This will be of class ARmdl (S3 object). See ARmdl.

• supTS: supTS test statistic value.

• expTS: expTS test statistic value.

• supTS_N: A (N x 1) vector with simulated supTS test statistics under null hypothesis.

• expTS_N: A (N x 1) vector with simulated expTS test statistics under null hypothesis.

• pval_supTS: P-value for supTS version of parameter stability test.

• pval_expTS: P-value for expTS version of parameter stability test.

• supTS_cv: Vector with 90%, 95%, and 99% bootstrap critical values for supTS version of parameter stability test.

• expTS_cv: Vector with 90%, 95%, and 99% bootstrap critical values for expTS version of parameter stability test.

• control: List with test procedure options used.

References

Carrasco, Marine, Liang Hu, and Werner Ploberger. 2014. “Optimal test for Markov switching parameters.” Econometrica 82 (2): 765–784.

Examples

# load data used in Hamilton 1989 
y84 <- as.matrix(hamilton84GNP$GNP_gr)


# Set test procedure options
control = list(N = 1000, 
               rho_b = 0.7, 
               msvar = FALSE)

# perform test with switch in mean only on Hamilton 1989 data

  mdl_84_msmu <- CHPTest(y84, p = 4, control = control)
  summary(mdl_84_msmu)

Bootstrap critical values for CHP 2014 parameter stability test

Description

This bootstrap procedure is described on pg. 771 of CHP 2014.

Usage

CHPbootCV(mdl, rho_b, N, msvar)

Arguments

Value

Bootstrap critical values

References

Carrasco, Marine, Liang Hu, and Werner Ploberger. 2014. “Optimal test for Markov switching parameters.” Econometrica 82 (2): 765–784.

Monte Carlo moment-based test for Markov switching model

Description

This function performs the Local Monte Carlo moment-based test for Markov switching autoregressive models proposed in Dufour & Luger (2017).

Usage

DLMCTest(Y, p, control = list())

Arguments

Value

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

• mdl_h0: List with restricted model attributes. This will be of class ARmdl if p>0 or Nmdl otherwise (S3 objects). See ARmdl or Nmdl.

• theta: Value of nuisance parameters. Specifically, these are the consistent estimates of nuisance parameters as discussed in Dufour & Luger (2017) LMC procedure.

• S0: A (1 x 4)) matrix containing the four moment-based test statistics defined in (11) - (14) in Dufour & Luger (2017).

• F0_min: Test statistic value for min version of Local Monte Carlo moment-based test.

• F0_prod: Test statistic value for prod version of Local Monte Carlo moment-based test.

• FN_min: A (N x 1) vector with simulated test statistics for min version of Local Monte Carlo moment-based test under null hypothesis.

• FN_prod: A (N x 1) vector with simulated test statistics for prod version of Local Monte Carlo moment-based test under null hypothesis.

• pval_min: P-value for min version of Local Monte Carlo moment-based test.

• pval_prod: P-value for prod version of Local Monte Carlo moment-based test.

• FN_min_cv: Vector with 90%, 95%, and 99% Monte Carlo critical values for min version of Local Monte Carlo moment-based test.

• FN_prod_cv: Vector with 90%, 95%, and 99% Monte Carlo critical values for prod version of Local Monte Carlo moment-based test.

• control: List with test procedure options used.

References

Dufour, J. M., & Luger, R. 2017. "Identification-robust moment-based tests for Markov switching in autoregressive models." Econometric Reviews, 36(6-9), 713-727.

Examples

set.seed(1234)
# load data used in Hamilton 1989 and extended data used in CHP 2014 
y84 <- as.matrix(hamilton84GNP$GNP_gr)
y10 <- as.matrix(chp10GNP$GNP_gr)

# Set test procedure options
lmc_control = list(N = 99,
                   simdist_N = 10000,
                   getSE = TRUE)

# perform test on Hamilton 1989 data
lmc_gnp84 <- DLMCTest(y84, p = 4, control = lmc_control)
summary(lmc_gnp84)

Maximized Monte Carlo moment-based test for Markov switching model

Description

This function performs the maximized Monte Carlo moment-based test for Markov switching autoregressive models proposed in Dufour & Luger (2017).

Usage

DLMMCTest(Y, p, control = list())

Arguments

Value

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

• mdl_h0: List with restricted model attributes. This will be of class ARmdl if p>0 or Nmdl otherwise (S3 objects). See ARmdl or Nmdl.

• theta_max_min: Value of nuisance parameters when min version of p-value is maximized as discussed in Dufour & Luger (2017) MMC procedure.

• theta_max_prod: Value of nuisance parameters when prod version of p-value is maximized as discussed in Dufour & Luger (2017) MMC procedure.

• theta_low: Lower bound on nuisance parameter values used when searching for maximum p-value.

• theta_upp: Upper bound on nuisance parameter values used when searching for maximum p-value.

• S0_min: A (1 x 4)) matrix containing the four moment-based test statistics defined in (11) - (14) in Dufour & Luger (2017) when theta_min is used.

• S0_prod: A (1 x 4)) matrix containing the four moment-based test statistics defined in (11) - (14) in Dufour & Luger (2017) when theta_prod is used.

• F0_min: Test statistic value for min version of Maximized Monte Carlo moment-based test.

• F0_prod: Test statistic value for prod version of Maximized Monte Carlo moment-based test.

• FN_min: A (N x 1) vector with simulated test statistics for min version of Maximized Monte Carlo moment-based test under null hypothesis.

• FN_prod: A (N x 1) vector with simulated test statistics for prod version of Maximized Monte Carlo moment-based test under null hypothesis.

• pval_min: Maximum p-value for min version of Maximized Monte Carlo moment-based test.

• pval_prod: Maximum p-value for prod version of Local Monte Carlo moment-based test.

• FN_min_cv: Vector with 90%, 95%, and 99% Monte Carlo critical values for min version of Local Monte Carlo moment-based test.

• FN_prod_cv: Vector with 90%, 95%, and 99% Monte Carlo critical values for prod version of Local Monte Carlo moment-based test.

• control: List with test procedure options used.

• optim_min_output: List with optimization output for min version of Maximized Monte Carlo moment-based test.

• optim_prod_output: List with optimization output for prod version of Maximized Monte Carlo moment-based test.

References

Dufour, J. M., & Luger, R. 2017. "Identification-robust moment-based tests for Markov switching in autoregressive models." Econometric Reviews, 36(6-9), 713-727.

Examples

set.seed(1234)
# load data used in Hamilton 1989 and extended data used in CHP 2014 
y84 <- as.matrix(hamilton84GNP$GNP_gr)
y10 <- as.matrix(chp10GNP$GNP_gr)

# Set test procedure options
mmc_control <- list(N = 99,
                    simdist_N = 10000,
                    getSE = TRUE,
                    eps = 0, 
                    CI_union = TRUE,
                    lambda = 100,
                    stationary_ind = TRUE,
                    optim_type = "GenSA",
                    silence = FALSE,
                    threshold_stop = 1,
                    maxit = 200)


# perform test on Hamilton 1989 data

  mmc_gnp84 <- DLMMCTest(y84, p = 4, control = mmc_control)
  summary(mmc_gnp84)

MMC nuisance parameter bounds for Moment-based test

Description

This function is used to determine the lower and upper bounds for the MMC LRT parameter search.

Usage

DLMMC_bounds(mdl_h0, con)

Arguments

Value

List with theta_low, vector of parameter lower bounds, and theta_upp, vector of parameter upper bounds.

Moment-based MMC test p-value

Description

This functions is used by numerical optimization algorithms for find maximum p-value given parameter vector theta.

Usage

DLMMCpval_fun(
  theta,
  y,
  x,
  params,
  sim_stats,
  pval_type,
  stationary_ind,
  lambda
)

Arguments

Value

Maximized Monte Carlo p-value.

References

Dufour, J. M., & Luger, R. 2017. "Identification-robust moment-based tests for Markov switching in autoregressive models." Econometric Reviews, 36(6-9), 713-727.

Moment-based MMC test (negative) p-value

Description

This functions is used by numerical optimization algorithms for find negative of maximum p-value given parameter vector theta.

Usage

DLMMCpval_fun_min(
  theta,
  y,
  x,
  params,
  sim_stats,
  pval_type,
  stationary_ind,
  lambda
)

Arguments

Value

Negative Maximized Monte Carlo p-value.

References

Dufour, J. M., & Luger, R. 2017. "Identification-robust moment-based tests for Markov switching in autoregressive models." Econometric Reviews, 36(6-9), 713-727.

Maximization step of EM algorithm for Hidden Markov model

Description

This function performs the maximization step of the Expectation Maximization algorithm for Hidden Markov models.

Usage

EMaximization_HMmdl(theta, mdl, MSloglik_output, k)

Arguments

Value

List with new maximized parameters.

Maximization step of EM algorithm for Markov-switching ARX model

Description

This function performs the maximization step of the Expectation Maximization algorithm for Markov-switching ARX model.

Usage

EMaximization_MSARXmdl(theta, mdl, MSloglik_output, k)

Arguments

Value

List with new maximized parameters.

Maximization step of EM algorithm for Markov-switching autoregressive model

Description

This function performs the maximization step of the Expectation Maximization algorithm for Markov-switching autoregressive model.

Usage

EMaximization_MSARmdl(theta, mdl, MSloglik_output, k)

Arguments

Value

List with new maximized parameters.

Maximization step of EM algorithm for Markov-switching VARX model

Description

This function performs the maximization step of the Expectation Maximization algorithm for Markov-switching VARX model.

Usage

EMaximization_MSVARXmdl(theta, mdl, MSloglik_output, k)

Arguments

Value

List with new maximized parameters.

Maximization step of EM algorithm for Markov-switching vector autoregressive model

Description

This function performs the maximization step of the Expectation Maximization algorithm for Markov-switching vector autoregressive model.

Usage

EMaximization_MSVARmdl(theta, mdl, MSloglik_output, k)

Arguments

Value

List with new maximized parameters.

EM algorithm iteration for Hidden Markov model

Description

This function performs the one iteration (E-step and M-step) of the Expectation Maximization algorithm for a Hidden Markov model.

Usage

EMiter_HMmdl(mdl, EMest_output, k)

Arguments

Value

List with attributes from new iteration.

EM algorithm iteration for Markov-switching ARX model

Description

This function performs the one iteration (E-step and M-step) of the Expectation Maximization algorithm for Markov-switching ARX model.

Usage

EMiter_MSARXmdl(mdl, EMest_output, k)

Arguments

Value

List with attributes from new iteration.

EM algorithm iteration for Markov-switching autoregressive model

Description

This function performs the one iteration (E-step and M-step) of the Expectation Maximization algorithm for Markov-switching autoregressive model.

Usage

EMiter_MSARmdl(mdl, EMest_output, k)

Arguments

Value

List with attributes from new iteration.

EM algorithm iteration for Markov-switching VARX model

Description

This function performs the one iteration (E-step and M-step) of the Expectation Maximization algorithm for Markov-switching VARX model.

Usage

EMiter_MSVARXmdl(mdl, EMest_output, k)

Arguments

Value

List with attributes from new iteration.

EM algorithm iteration for Markov-switching vector autoregressive model

Description

This function performs the one iteration (E-step and M-step) of the Expectation Maximization algorithm for Markov-switching vector autoregressive model.

Usage

EMiter_MSVARmdl(mdl, EMest_output, k)

Arguments

Value

List with attributes from new iteration.

Hidden Markov model log-likelihood function

Description

This function computes the log-likelihood for a Hidden Markov model and uses the Hamilton smoother to obtain smoothed probabilities of each state. This is also the expectation step in the Expectation Maximization algorithm for a Markov-switching autoregressive model.

Usage

ExpectationM_HMmdl(theta, mdl, k)

Arguments

Value

List which includes log-likelihood value and smoothed probabilities of each regime.

Markov-switching ARX log-likelihood function

Description

This function computes the log-likelihood for a markov-switching autoregressive model and uses the Hamilton smoother to obtain smoothed probabilities of each state. This is also the expectation step in the Expectation Maximization algorithm for a Markov-switching ARX model.

Usage

ExpectationM_MSARXmdl(theta, mdl, k)

Arguments

Value

List which includes log-likelihood and smoothed probabilities of each regime.

Markov-switching autoregressive log-likelihood function

Description

This function computes the log-likelihood for a markov-switching autoregressive model and uses the Hamilton smoother to obtain smoothed probabilities of each state. This is also the expectation step in the Expectation Maximization algorithm for a Markov-switching autoregressive model.

Usage

ExpectationM_MSARmdl(theta, mdl, k)

Arguments

Value

List which includes log-likelihood and smoothed probabilities of each regime.

Markov-switching VARX log-likelihood function

Description

This function computes the log-likelihood for a markov-switching VARX model and uses the Hamilton smoother to obtain smoothed probabilities of each state. This is also the expectation step in the Expectation Maximization algorithm for a Markov-switching autoregressive model.

Usage

ExpectationM_MSVARXmdl(theta, mdl, k)

Arguments

Value

List which includes log-likelihood and smoothed probabilities of each regime.

Markov-switching vector autoregressive log-likelihood function

Description

This function computes the log-likelihood for a markov-switching vector autoregressive model and uses the Hamilton smoother to obtain smoothed probabilities of each state. This is also the expectation step in the Expectation Maximization algorithm for a Markov-switching autoregressive model.

Usage

ExpectationM_MSVARmdl(theta, mdl, k)

Arguments

Value

List which includes log-likelihood and smoothed probabilities of each regime.

Hansen (1992) likelihood ratio test

Description

This function performs Hansen's likelihood ratio test as described in Hansen (1992). Original source code can be found here.

Usage

HLRTest(Y, p, control = list())

Arguments

Value

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

• mdl_h0: List with restricted model attributes. This will be of class ARmdl (S3 object). See ARmdl.

• LR0: Likelihood ratio test statistic value.

• LRN: A (N x 1) vector with simulated LRT statistics under null hypothesis.

• pval: P-value.

• LR_cv: A (nwband x 3) matrix with 90%, 95%, and 99% critical values in each column respectively.

• coef: Vector of coefficients from restricted model and grid search that maximized standardized LRT.

• control: List with test procedure options used.

References

Hansen, Bruce E. 1992. “The likelihood ratio test under nonstandard conditions: testing the Markov switching model of GNP.” Journal of applied Econometrics 7 (S1): S61–S82.

Hansen, Bruce E. 1996. “Erratum: The likelihood ratio test under nonstandard conditions: testing the Markov switching model of GNP.” Journal of applied Econometrics 7 (S1): S61–S82.

Examples

# --------------------------- Use simulated process ----------------------------
set.seed(1234)
# Define DGP of MS AR process
mdl_ms2 <- list(n     = 200, 
                mu    = c(5,1),
                sigma = c(1,1),
                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)

hlrt_control  <- list(ix          = 1, 
                      gridsize    = 7,
                      pgrid_from  = 0.05,
                      pgrid_by    = 0.05,
                      pgrid_to    = 0.95,
                      mugrid_from = 0,
                      mugrid_by   = 1)


  hlrt <- HLRTest(y_ms_simu$y, p = 1, control = hlrt_control)
  summary(hlrt)

HLR param search

Description

This function performs the parameter grid search needed for the likelihood ratio test described in Hansen (1992).

Usage

HLRparamSearch(gx, gp, gq, b, null, HLR_opt_ls)

Arguments

Value

List which contains:

• cs: Vector with standardized LRT statistic for each grid point.

• draws: List with a (nwband+1 x N matrix for each grid point. Each row of these matrices is a vector of simulated test statistics under the null hypothesis for a value of bandwidth .

• coefficients: A matrix with coefficients for each grid point.

Hidden Markov model

Description

This function estimates a Hidden Markov model with k regimes.

Usage

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

Arguments

Value

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

Estimation of Hidden Markov model by EM Algorithm

Description

Estimate Hidden Markov model by EM algorithm. This function is used by HMmdl which organizes the output and takes raw data as input.

Usage

HMmdl_em(theta_0, mdl, k, optim_options)

Arguments

Value

List with model results.

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.

Hidden Markov model maximum likelihood estimation

Description

This function computes estimate a Hidden Markov model using MLE.

Usage

HMmdl_mle(theta_0, mdl_in, k, optim_options)

Arguments

Value

List with model attributes

Monte Carlo Likelihood Ratio Test

Description

This function performs the Local Monte Carlo likelihood ratio test (LMC-LRT) proposed in Rodriguez-Rondon & Dufour (2024). As discussed in their work, this test can be applied in very general settings and can be used to compare varioous regimes under the null and under the alternative.

Usage

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

Arguments

Value

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.

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.

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)


# ------ MS-AR example ----- #
# Set test procedure options
lmc_control = list(N = 19,
                   burnin = 100,
                   converge_check = NULL,
                   mdl_h0_control = list(const  = TRUE, 
                                         getSE  = TRUE),
                   mdl_h1_control = list(msmu   = TRUE, 
                                         msvar  = TRUE,
                                         getSE  = TRUE,
                                         method = "EM",
                                         use_diff_init = 1))


  lmctest <- LMCLRTest(y_ms_simu$y, p = 1, k0 = 1 , k1 = 2, control = lmc_control)
  summary(lmctest)

Likelihood Ratio Test Statistic Sample Distribution

Description

This function simulates the sample distribution under the null hypothesis.

Usage

LR_samp_dist(mdl_h0, k1, N, burnin, Z, mdl_h0_control, mdl_h1_control)

Arguments

Value

vector of simulated LRT statistics

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. 2022. “Monte Carlo Likelihood Ratio Tests for Markov Switching Models.” Unpublished manuscript.

Monte Carlo Likelihood Ratio Test sample distribution (parallel version)

Description

This function simulates the sample distribution under the null hypothesis using a parallel pool.

Usage

LR_samp_dist_par(
  mdl_h0,
  k1,
  N,
  burnin,
  Z,
  mdl_h0_control,
  mdl_h1_control,
  workers
)

Arguments

Value

vector of simulated LRT statistics

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. 2023. “Monte Carlo Likelihood Ratio Tests for Markov Switching Models.” Unpublished manuscript.

Monte Carlo P-value

Description

This function computes the Monte Carlo P-value.

Usage

MCpval(test_stat, null_vec, type = "geq")

Arguments

Value

MC p-value of test

References

Dufour, Jean-Marie 2006. "Monte Carlo tests with nuisance parameters: A general approach to finite-sample inference and nonstandard asymptotics". Journal of Econometrics, 133(2), 443-477.

Dufour, Jean-Marie, and Richard Luger. 2017. "Identification-robust moment-based tests for Markov switching in autoregressive models". Econometric Reviews, 36(6-9), 713-727.

Maximized Monte Carlo Likelihood Ratio Test

Description

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

Usage

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

Arguments

Value

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.

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.

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)

Monte Carlo Likelihood Ratio Test P-value Function

Description

This function computes the Maximum Monte Carlo P-value.

Usage

MMCLRpval_fun(
  theta_h0,
  mdl_h0,
  k1,
  LT_h1,
  N,
  burnin,
  workers,
  lambda,
  stationary_constraint,
  thtol,
  Z,
  exog,
  mdl_h0_control,
  mdl_h1_control
)

Arguments

Value

MMC p-value

Monte Carlo Likelihood Ratio Test P-value Function

Description

This function computes the (negative) Maximum Monte Carlo P-value.

Usage

MMCLRpval_fun_min(
  theta,
  mdl_h0,
  k1,
  LT_h1,
  N,
  burnin,
  workers,
  lambda,
  stationary_constraint,
  thtol,
  Z,
  exog,
  mdl_h0_control,
  mdl_h1_control
)

Arguments

Value

negative MMC p-value

MMC nuisance parameter bounds

Description

This function is used to determine the lower and upper bounds for the MMC LRT parameter search.

Usage

MMC_bounds(mdl_h0, con)

Arguments

Value

List with theta_low, vector of parameter lower bounds, and theta_upp, vector of parameter upper bounds.

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.

Markov-switching autoregressive model

Description

This function estimates a Markov-switching autoregressive model

Usage

MSARXmdl(Y, p, k, Z, control = list())

Arguments

Value

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 + qz) x k) matrix of estimated coefficients.

• betaZ: a (qz x q) matrix of estimated exogenous regressor 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.

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

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)

Estimation of Markov-switching ARX model by EM Algorithm

Description

Estimate Markov-switching ARX model by EM algorithm. This function is used by MSARmdl which organizes the output and takes raw data as input.

Usage

MSARXmdl_em(theta_0, mdl, k, optim_options)

Arguments

Value

List with model results.

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.

Markov-switching autoregressive model

Description

This function estimates a Markov-switching autoregressive model

Usage

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

Arguments

Value

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.

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

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)

Estimation of Markov-switching autoregressive model by EM Algorithm

Description

Estimate Markov-switching autoregressive model by EM algorithm. This function is used by MSARmdl which organizes the output and takes raw data as input.

Usage

MSARmdl_em(theta_0, mdl, k, optim_options)

Arguments

Value

List with model results.

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.

Markov-switching autoregressive maximum likelihood estimation

Description

This function computes estimate a Markov-switching autoregressive model using MLE.

Usage

MSARmdl_mle(theta_0, mdl_in, k, optim_options)

Arguments

Value

List with model attributes

Markov-switching vector autoregressive model

Description

This function estimates a Markov-switching vector autoregressive model

Usage

MSVARXmdl(Y, p, k, Z, control = list())

Arguments

Value

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

• y: a (T-p x q) matrix of observations.

• X: a (T-p x p*q + const) matrix of lagged observations with a leading column of 1s.

• x: a (T-p x p*q) matrix of lagged observations.

• resid: a (T-p x q) matrix of residuals.

• fitted: a (T x q) matrix of fitted values.

• intercept: a (k x q) matrix of estimated intercepts of each process.

• mu: a (k x q) matrix of estimated means of each process.

• beta: a list containing k separate ((1 + p + qz) x q) matrix of estimated coefficients for each regime.

• betaZ: a (qz x q) matrix of estimated exogenous regressor coefficients.

• phi: estimates of autoregressive coefficients.

• Fmat: Companion matrix containing autoregressive coefficients.

• stdev: List with k (q x q) matrices with estimated standard deviation on the diagonal.

• sigma: List with k (q x q) matrices with 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.

• 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.

• 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.

List with model characteristics

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..

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

See Also

VARmdl

Examples

set.seed(123)
# Define DGP of MS VAR process
mdl_msvar2 <- list(n     = 200, 
                   p     = 1,
                   q     = 2,
                   mu    = rbind(c(5, -2),
                                 c(10, 2)),
                   sigma = list(rbind(c(5.0, 1.5),
                                      c(1.5, 1.0)),
                                rbind(c(7.0, 3.0),
                                      c(3.0, 2.0))),
                   phi   = rbind(c(0.50, 0.30),
                                 c(0.20, 0.70)),
                   k     = 2,
                   P     = rbind(c(0.90, 0.10),
                                 c(0.10, 0.90)))

# Simulate process using simuMSVAR() function
y_msvar_simu <- simuMSVAR(mdl_msvar2)

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

  y_msvar_mdl <- MSVARmdl(y_msvar_simu$y, p = 1, k = 2, control = control)
  summary(y_msvar_mdl)

Estimation of Markov-switching VARX model by EM Algorithm

Description

Estimate Markov-switching VARX model by EM algorithm. This function is used by MSVARmdl which organizes the output and takes raw data as input.

Usage

MSVARXmdl_em(theta_0, mdl, k, optim_options)

Arguments

Value

List with model results.

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.

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

Markov-switching vector autoregressive model

Description

This function estimates a Markov-switching vector autoregressive model

Usage

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

Arguments

Value

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

• y: a (T-p x q) matrix of observations.

• X: a (T-p x p*q + const) matrix of lagged observations with a leading column of 1s.

• x: a (T-p x p*q) matrix of lagged observations.

• fitted: a (T x q) matrix of fitted values.

• resid: a (T-p x q) matrix of residuals.

• intercept: a (k x q) matrix of estimated intercepts of each process.

• mu: a (k x q) matrix of estimated means of each process.

• beta: a list containing k separate ((1 + p) x q) matrix of estimated coefficients for each regime.

• phi: estimates of autoregressive coefficients.

• Fmat: Companion matrix containing autoregressive coefficients.

• stdev: List with k (q x q) matrices with estimated standard deviation on the diagonal.

• sigma: List with k (q x q) matrices with 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.

• 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.

• 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.

List with model characteristics

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..

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

See Also

VARmdl

Examples

set.seed(123)
# Define DGP of MS VAR process
mdl_msvar2 <- list(n     = 200, 
                   p     = 1,
                   q     = 2,
                   mu    = rbind(c(5, -2),
                                 c(10, 2)),
                   sigma = list(rbind(c(5.0, 1.5),
                                      c(1.5, 1.0)),
                                rbind(c(7.0, 3.0),
                                      c(3.0, 2.0))),
                   phi   = rbind(c(0.50, 0.30),
                                 c(0.20, 0.70)),
                   k     = 2,
                   P     = rbind(c(0.90, 0.10),
                                 c(0.10, 0.90)))

# Simulate process using simuMSVAR() function
y_msvar_simu <- simuMSVAR(mdl_msvar2)

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

  y_msvar_mdl <- MSVARmdl(y_msvar_simu$y, p = 1, k = 2, control = control)
  summary(y_msvar_mdl)

Estimation of Markov-switching vector autoregressive model by EM Algorithm

Description

Estimate Markov-switching vector autoregressive model by EM algorithm. This function is used by MSVARmdl which organizes the output and takes raw data as input.

Usage

MSVARmdl_em(theta_0, mdl, k, optim_options)

Arguments

Value

List with model results.

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.

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

Markov-switching vector autoregressive maximum likelihood estimation

Description

This function computes estimate a Markov-switching vector autoregressive model using MLE.

Usage

MSVARmdl_mle(theta_0, mdl_in, k, optim_options)

Arguments

Value

List with model attributes

Normal distribution model

Description

This function estimates a univariate or multivariate normally distributed model. This can be used for the null hypothesis of a linear model against an alternative hypothesis of a HMM with k regimes.

Usage

Nmdl(Y, Z = NULL, control = list())

Arguments

Value

List of class Nmdl (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 (1 x q) vector of estimated means of each process.

• intercept: a (1 x q) vector of estimated intercept of each process. If Z is NULL, it is the same as mu.

• beta: a ((1 + p + qz) x q) matrix of estimated coefficients.

• betaZ: a (qz x q) matrix of estimated exogenous regressor coefficients (If Z is provided).

• stdev: a (q x 1) vector of estimated standard deviation of each process.

• sigma: a (q x q) estimated covariance matrix.

• theta: vector containing: mu, betaZ (if matrix Z is provided), 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.

• n: number of observations (same as T).

• q: number of series.

• k: number of regimes. This is always 1 in Nmdl.

• 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.

Examples

set.seed(1234)

# ----- Univariate ----- # 
# Define DGP 
mdl_norm <- list(n     = 1000, 
                 q     = 1,
                 mu    = as.matrix(5),
                 sigma = as.matrix(5.0))

# Simulate process using simuNorm() function
y_norm_simu <- simuNorm(mdl_norm)

# estimate parameters
y_norm_mdl <- Nmdl(y_norm_simu$y)
summary(y_norm_mdl)


# ----- Multivariate ----- # 
# Define DGP 
mdl_norm <- list(n     = 1000, 
                 q     = 2,
                 mu    = c(5, -2),
                 sigma = rbind(c(5.0, 1.5),
                               c(1.5, 1.0)))

# Simulate process using simuNorm() function
y_norm_simu <- simuNorm(mdl_norm)

# estimate parameters
y_norm_mdl <- Nmdl(y_norm_simu$y)
summary(y_norm_mdl)

US GNP data 1947Q2 - 2024Q2

Description

US GNP data 1947Q2 - 2024Q2

Usage

USGNP

Format

This data is used in Rodriguez-Rondon & Dufour (2024). The series ranges from 1947Q2 to 2024Q2.

Date

Vector of dates

GNP

US GNP series

GNP_gr

log difference of US GNP series

Source

https://fred.stlouisfed.org/graph/?g=UKDQ

References

U.S. Bureau of Economic Analysis, Gross National Product [GNP], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/GNP, September, 2024.

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

US Real GDP data 1947Q2 - 2024Q2

Description

US Real GDP data 1947Q2 - 2024Q2

Usage

USRGDP

Format

This data is used in Rodriguez-Rondon & Dufour (2024). The series ranges from 1947Q2 to 2024Q2.

Date

Vector of dates

RGDP

US Real GDP series

RGDP_gr

log difference of US Real GDP series

Source

https://fred.stlouisfed.org/series/GDPC1

References

U.S. Bureau of Economic Analysis, Real Gross Domestic Product [GDPC1], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/GDPC1, September, 2024.

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

Vector X autoregressive model

Description

This function estimates a vector autoregresive model with p lags. This can be used for the null hypothesis of a linear model against an alternative hypothesis of a Markov switching vector autoregressive model with k regimes.

Usage

VARXmdl(Y, p, Z, control = list())

Arguments

Value

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

• y: a (T-p x q) matrix of observations.

• X: a (T-p x p*q + const) matrix of lagged observations with a leading column of 1s if const=TRUE or not if const=FALSE.

• x: a (T-p x p*q) matrix of lagged observations.

• fitted: a (T-p x q) matrix of fitted values.

• resid: a (T-p x q) matrix of residuals.

• mu: a (1 x q) vector of estimated means of each process.

• beta: a ((1 + p + qz) x q) matrix of estimated coefficients.

• betaZ: a (qz x q) matrix of estimated exogenous regressor coefficients.

• intercept: estimate of intercepts.

• phi: a (q x p*q) matrix of estimated autoregressive coefficients.

• Fmat: Companion matrix containing autoregressive coefficients.

• stdev: a (q x 1) vector of estimated standard deviation of each process.

• sigma: a (q x q) estimated covariance matrix.

• theta: vector containing: mu, vech(sigma), and vec(t(phi)).

• 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_phi_ind: vector indicating location of autoregressive coefficients with 1 and 0 otherwise.

• stationary: Boolean indicating if process is stationary if TRUE or non-stationary if FALSE.

• n: number of observations after lag transformation (i.e., n = T-p).

• p: number of autoregressive lags.

• q: number of series.

• k: number of regimes. This is always 1 in VARmdl.

• Fmat: matrix from companion form. Used to determine is process is stationary.

• 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.

See Also

MSVARmdl

Examples

# ----- Bivariate VAR(1) process ----- #
set.seed(1234)
# Define DGP of VAR process
mdl_var <- list(n     = 1000, 
                p     = 1,
                q     = 2,
                mu    = c(5,-2),
                sigma = rbind(c(5.0, 1.5),
                              c(1.5, 1.0)),
                phi   = rbind(c(0.50, 0.30),
                              c(0.20, 0.70)))

# Simulate process using simuVAR() function
y_simu <- simuVAR(mdl_var)

# Set options for model estimation
control <- list(const  = TRUE, 
                getSE  = TRUE)

# Estimate model
y_var_mdl <- VARmdl(y_simu$y, p = 2, control = control)
summary(y_var_mdl)

Vector autoregressive model

Description

This function estimates a vector autoregresive model with p lags. This can be used for the null hypothesis of a linear model against an alternative hypothesis of a Markov switching vector autoregressive model with k regimes.

Usage

VARmdl(Y, p, control = list())

Arguments

Value

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

• y: a (T-p x q) matrix of observations.

• X: a (T-p x p*q + const) matrix of lagged observations with a leading column of 1s if const=TRUE or not if const=FALSE.

• x: a (T-p x p*q) matrix of lagged observations.

• fitted: a (T-p x q) matrix of fitted values.

• resid: a (T-p x q) matrix of residuals.

• mu: a (1 x q) vector of estimated means of each process.

• beta: a ((1 + p) x q) matrix of estimated coefficients. .

• intercept: estimate of intercepts.

• phi: a (q x p*q) matrix of estimated autoregressive coefficients.

• Fmat: Companion matrix containing autoregressive coefficients.

• stdev: a (q x 1) vector of estimated standard deviation of each process.

• sigma: a (q x q) estimated covariance matrix.

• theta: vector containing: mu, vech(sigma), and vec(t(phi)).

• 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_phi_ind: vector indicating location of autoregressive coefficients with 1 and 0 otherwise.

• stationary: Boolean indicating if process is stationary if TRUE or non-stationary if FALSE.

• n: number of observations after lag transformation (i.e., n = T-p).

• p: number of autoregressive lags.

• q: number of series.

• k: number of regimes. This is always 1 in VARmdl.

• Fmat: matrix from companion form. Used to determine is process is stationary.

• 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.

See Also

MSVARmdl

Examples

# ----- Bivariate VAR(1) process ----- #
set.seed(1234)
# Define DGP of VAR process
mdl_var <- list(n     = 1000, 
                p     = 1,
                q     = 2,
                mu    = c(5,-2),
                sigma = rbind(c(5.0, 1.5),
                              c(1.5, 1.0)),
                phi   = rbind(c(0.50, 0.30),
                              c(0.20, 0.70)))

# Simulate process using simuVAR() function
y_simu <- simuVAR(mdl_var)

# Set options for model estimation
control <- list(const  = TRUE, 
                getSE  = TRUE)

# Estimate model
y_var_mdl <- VARmdl(y_simu$y, p = 2, control = control)
summary(y_var_mdl)

Approximate CDF distribution

Description

This function obtains the parameters in eq. 16 of the CDF distribution needed for combining moment-based test statistics.

Usage

approxDistDL(Tsize, simdist_N)

Arguments

Value

A (2 x 4) matrix with parameters of CDF distributions. The first row contains \gamma_0 for each moment and the second row contains \gamma_1 for each moment.

References

Dufour, J. M., & Luger, R. 2017. "Identification-robust moment-based tests for Markov switching in autoregressive models." Econometric Reviews, 36(6-9), 713-727.

Loop for approxDistDL

Description

This function performs the loop in required in approxDistDL.

Usage

approx_dist_loop(SN2)

Arguments

Value

The test statistics from simulated data. Used for NLS to get params needed to combine p-values.

References

Dufour, J. M., & Luger, R. 2017. "Identification-robust moment-based tests for Markov switching in autoregressive models." Econometric Reviews, 36(6-9), 713-727.

Autoregressive transition matrix

Description

This function converts a transition matrix to the transition matrix consistent with a Markov-switching autoregressive model.

Usage

arP(P, k, ar)

Arguments

Value

transformed transition matrix.

Autoregressive moment grid

Description

This function creates a grid of mean and variance consistent with a Markov-switching autoregressive model.

Usage

argrid_MSARmdl(mu, sig, k, ar, msmu, msvar)

Arguments

Value

List with (M x ar+1) matrix of means for each regime M (where M = k^(ar+1)) and each time t,... t-ar, vector with variance for each regime M, and vector indicating the corresponded 1,..., k regime.

Vector autoregressive moment grid

Description

Creates grid of means and covariance matrices consistent with a Markov-switching vector autoregressive model.

Usage

argrid_MSVARmdl(mu, sigma, k, ar, msmu, msvar)

Arguments

Value

List with M regime specific (q x k) matrices of means, List with M regime specific covariance matrices, and vector indicating the corresponded 1,..., k regime.

Markov-switching autoregressive model residuals

Description

This function computes residuals of a Markov-switching autoregressive model.

Usage

calcResid_MSARXmdl(mdl, mu, k)

Arguments

Value

A (TxM) matrix of residuals in each regime M where M=k^(ar+1).

Markov-switching autoregressive model residuals

Description

This function computes residuals of a Markov-switching autoregressive model.

Usage

calcResid_MSARmdl(mdl, mu, k)

Arguments

Value

A (TxM) matrix of residuals in each regime M where M=k^(ar+1).

Markov-switching VARX model residuals

Description

This function computes residuals of a Markov-switching VARX model.

Usage

calcResid_MSVARXmdl(mdl, mu, k)

Arguments

Value

List with M (Txq) matrices of residuals in each regime M where M=k^(ar+1).

Markov-switching vector autoregressive model residuals

Description

This function computes residuals of a Markov-switching vector autoregressive model.

Usage

calcResid_MSVARmdl(mdl, mu, k)

Arguments

Value

List with M (Txq) matrices of residuals in each regime M where M=k^(ar+1).

Moment-based test statistics

Description

This function computes the four moment-based test statistics (eq. 11 - 14) discussed in Dufour & Luger 2017.

Usage

calc_DLmoments(ehat)

Arguments

Value

Vector containing the four test statistics.

References

Dufour, J. M., & Luger, R. 2017. "Identification-robust moment-based tests for Markov switching in autoregressive models." Econometric Reviews, 36(6-9), 713-727.

Test statistic for switch in mean only

Description

This function computes part of the test statistic given by eq. 2.5 of CHP 2014 when the alternative has switching mean only. The output is used in chpStat which computes the full test statistics.

Usage

calc_mu2t(mdl, rho, ltmt)

Arguments

Value

Part of test statistic given rho and hv value.

References

Carrasco, Marine, Liang Hu, and Werner Ploberger. 2014. “Optimal test for Markov switching parameters.” Econometrica 82 (2): 765–784.

Test statistic for switch in mean and variance

Description

This function computes part of the test statistic given by eq. 2.5 of CHP 2014 when the alternative has switching mean and variance. The output is used in chpStat which computes the full test statistics.

Usage

calc_mu2t_mv(mdl, rho, ltmt, hv)

Arguments

Value

Part of test statistic given rho and hv value.

References

Carrasco, Marine, Liang Hu, and Werner Ploberger. 2014. “Optimal test for Markov switching parameters.” Econometrica 82 (2): 765–784.

Carrasco, Hu, & Ploberger 2010 GNP data

Description

Carrasco, Hu, & Ploberger 2010 GNP data

Usage

chp10GNP

Format

This data is the extension of the GNP series used in CHP (2014), Econometrica. This series ranges from 1951Q2 to 2010Q4.

Date

Vector of dates

GNP

US GNP series

GNP_gr

log difference of US GNP series

Source

https://www.econometricsociety.org/content/supplement-optimal-test-markov-switching-parameters

References

Carrasco, Marine, Liang Hu, and Werner Ploberger. 2014. “Optimal test for Markov switching parameters.” Econometrica 82 (2): 765–784.

Derivative matrix

Description

This function organizes the first and second derivatives of the log-likelihood.

Usage

chpDmat(mdl, msvar)

Arguments

Value

List containing relevant first and second derivatives of log-likelihood function.

References

Carrasco, Marine, Liang Hu, and Werner Ploberger. 2014. “Optimal test for Markov switching parameters.” Econometrica 82 (2): 765–784.

Test statistic for CHP 2014 parameter stability test

Description

This function computes the supTS and expTS test-statistics proposed in CHP 2014.

Usage

chpStat(mdl, rho_b, ltmt, msvar)

Arguments

Value

A (2 x 1) vector with supTS test statistic as first element and expTS test-statistics as second element.

References

Carrasco, Marine, Liang Hu, and Werner Ploberger. 2014. “Optimal test for Markov switching parameters.” Econometrica 82 (2): 765–784.

Parameter vector & likelihood function used by HLRTest()

Description

This function combines parameters of restricted model with parameters of unrestricted model and then computes the likelihood using marklike().

Usage

clike(b, HLR_opt_ls)

Arguments

Value

Value of likelihood at each period t.

coef of a ARmdl object

Description

This is a method for the function coef() for objects of the class ARmdl.

Usage

## S3 method for class 'ARmdl'
coef(object, ...)

Arguments

Value

vector of coefficients.

coef of a HMmdl object

Description

This is a method for the function coef() for objects of the class HMmdl.

Usage

## S3 method for class 'HMmdl'
coef(object, ...)

Arguments

Value

vector of coefficients.

coef of a MSARmdl object

Description

This is a method for the function coef() for objects of the class MSARmdl.

Usage

## S3 method for class 'MSARmdl'
coef(object, ...)

Arguments

Value

vector of coefficients.

coef of a MSVARmdl object

Description

This is a method for the function coef() for objects of the class MSVARmdl.

Usage

## S3 method for class 'MSVARmdl'
coef(object, ...)

Arguments

Value

vector of coefficients.

coef of a Nmdl object

Description

This is a method for the function coef() for objects of the class Nmdl.

Usage

## S3 method for class 'Nmdl'
coef(object, ...)

Arguments

Value

vector of coefficients.

coef of a VARmdl object

Description

This is a method for the function coef() for objects of the class VARmdl.

Usage

## S3 method for class 'VARmdl'
coef(object, ...)

Arguments

Value

vector of coefficients.

Combine p-values

Description

This function is used to combine the four moment-based p-values as in eq. 17 and 18 of Dufour & Luger 2017.

Usage

combine_stat(stats, params, type)

Arguments

Value

A (N x 1) vector with test statistics. The last element is the test statistic from observed data.

References

Dufour, J. M., & Luger, R. 2017. "Identification-robust moment-based tests for Markov switching in autoregressive models." Econometric Reviews, 36(6-9), 713-727.

Tippett, L. 1931. "The Method of Statistics". London: Williams & Norgate.

Wilkinson, B. 1951. "A statistical consideration in psychological research." Psychology Bulletin 48:156–158.

Pearson, K. 1933. "On a method of determining whether a sample of size n supposed to have been drawn from a parent population having a known probability integral has probably been drawn at random". Biometrika 25:379–410.

Fisher, R. 1932. "Statistical Methods for Research Workers." Edinburgh: Oliver and Boyd.

Companion Matrix

Description

This function converts the (q x 1) vector of constants and (q x qp) matrix of autoregressive coefficients into (qp x qp) matrix belonging to the companion form

Usage

companionMat(phi, p, q)

Arguments

Value

matrix of dimension (qp x qp) of companion form.

Computes test stat using new parameter vectors

Description

This function computes the LRT statistic.

Usage

compu_tstat(theta_h0, mdl_h0, LT_h1, p, q, k0, exog)

Arguments

Value

LRT statistic

Covariance to correlation matrix

Description

This function takes an (n x n) covariance matrix and returns the associated (n x n) correlation matrix.

Usage

cov2corr(cov_mat)

Arguments

Value

A (n x n) correlation matrix.

Covariance vech to matrix

Description

This function undoes the half-vectorization of a covariance matrix.

Usage

covar_unvech(sig, n)

Arguments

Value

A (n x n) covariance matrix.

Covariance vech function

Description

This function returns the half-vectorization of an input matrix as a column vector.

Usage

covar_vech(mat)

Arguments

Value

A (n+1)*n/2 column vector.

Gradient of likelihood function.

Description

this function computes the score vector.

Usage

dmclike(th, HLR_opt_ls)

Arguments

Value

vector with gradient of likelihood function for each parameter. Used in HLRpramSearch().

Estimate model for likelihood ratio test

Description

This function is used by the Monte Carlo testing procedures to estimate restricted and unrestricted models.

Usage

estimMdl(Y, p, q, k, Z = NULL, control = list())

Arguments

Value

List with estimated model properties.

fitted values of a ARmdl object

Description

This is a method for the function fitted() for objects of the class ARmdl.

Usage

## S3 method for class 'ARmdl'
fitted(object, ...)

Arguments

Value

matrix with fitted values

fitted values of a HMmdl object

Description

This is a method for the function fitted() for objects of the class HMmdl.

Usage

## S3 method for class 'HMmdl'
fitted(object, ...)

Arguments

Value

matrix with fitted values

fitted values of a MSARmdl object

Description

This is a method for the function fitted() for objects of the class MSARmdl.

Usage

## S3 method for class 'MSARmdl'
fitted(object, ...)

Arguments

Value

matrix with fitted values

fitted values of a MSVARmdl object

Description

This is a method for the function fitted() for objects of the class MSVARmdl.

Usage

## S3 method for class 'MSVARmdl'
fitted(object, ...)

Arguments

Value

matrix with fitted values

fitted values of a Nmdl object

Description

This is a method for the function fitted() for objects of the class Nmdl.

Usage

## S3 method for class 'Nmdl'
fitted(object, ...)

Arguments

Value

matrix with fitted values

fitted values of a VARmdl object

Description

This is a method for the function fitted() for objects of the class VARmdl.

Usage

## S3 method for class 'VARmdl'
fitted(object, ...)

Arguments

Value

matrix with fitted values

Hessian matrix

Description

This function is used to obtain a numerical approximation of a Hessian matrix.

Usage

getHessian(mdl)

Arguments

Value

Hessian matrix.

Hessian matrix of autoregressive model

Description

This function is used to obtain a numerical approximation of a Hessian matrix for an autoregressive model.

Usage

## S3 method for class 'ARmdl'
getHessian(mdl)

Arguments

Value

Hessian matrix.

Hessian matrix of Hidden Markov model

Description

This function is used to obtain a numerical approximation of a Hessian matrix for a Hidden Markov model.

Usage

## S3 method for class 'HMmdl'
getHessian(mdl)

Arguments

Value

Hessian matrix.

Hessian matrix of Markov-switching autoregressive model

Description

This function is used to obtain a numerical approximation of a Hessian matrix for a Markov-switching autoregressive model.

Usage

## S3 method for class 'MSARmdl'
getHessian(mdl)

Arguments

Value

Hessian matrix.

Hessian matrix of Markov-switching vector autoregressive model

Description

This function is used to obtain a numerical approximation of a Hessian matrix for a Markov-switching vector autoregressive model.

Usage

## S3 method for class 'MSVARmdl'
getHessian(mdl)

Arguments

Value

Hessian matrix.

Hessian matrix of normal model

Description

This function is used to obtain a numerical approximation of a Hessian matrix for a normally distributed model.

Usage

## S3 method for class 'Nmdl'
getHessian(mdl)

Arguments

Value

Hessian matrix.

Hessian matrix of vector autoregressive model

Description

This function is used to obtain a numerical approximation of a Hessian matrix for a vector autoregressive model.

Usage

## S3 method for class 'VARmdl'
getHessian(mdl)

Arguments

Value

Hessian matrix.

Hamilton 1984 & Hansen 1992 GNP data

Description

Hamilton 1984 & Hansen 1992 GNP data

Usage

hamilton84GNP

Format

This data set is used in Hansen (1992) to test the US GNP model proposed by Hamilton (1989). This series ranges from 1951Q2 to 1984Q4.

Date

Vector of dates

GNP

US GNP series

GNP_gr

US GNP log difference

Source

https://www.ssc.wisc.edu/~bhansen/progs/jae_92.html

References

Hansen, Bruce E. 1992. “The likelihood ratio test under nonstandard conditions: testing the Markov switching model of GNP.” Journal of applied Econometrics 7 (S1): S61–S82.

Hamilton, James D. 1989. “A new approach to the economic analysis of nonstationary time series and the business cycle.” Econometrica 57 (2): 357–384.

Initial values for Hidden Markov model

Description

This function generates a random parameter vector to be used as initial values for a Hidden Markov model.

Usage

initVals_HMmdl(mdl, k)

Arguments

Value

Vector of initial parameter values.

Initial values for Markov-switching ARX model

Description

This function generates a random parameter vector to be used as initial values for a Markov-switching ARX model.

Usage

initVals_MSARXmdl(mdl, k)

Arguments

Value

Vector of initial parameter values.

Initial values for Markov-switching autoregressive model

Description

This function generates a random parameter vector to be used as initial values for a Markov-switching autoregressive model.

Usage

initVals_MSARmdl(mdl, k)

Arguments

Value

Vector of initial parameter values.

Initial values for Markov-switching VARX model

Description

This function generates a random parameter vector to be used as initial values for a Markov-switching VARX model.

Usage

initVals_MSVARXmdl(mdl, k)

Arguments

Value

Vector of initial parameter values.

Initial values for Markov-switching vector autoregressive model

Description

This function generates a random parameter vector to be used as initial values for a Markov-switching vector autoregressive model.

Usage

initVals_MSVARmdl(mdl, k)

Arguments

Value

Vector of initial parameter values.

Intercept from mu for MSARmdl

Description

This function computes the intercept for each regime k for an Markov switching AR model

Usage

interMSARmdl(mdl)

Arguments

Value

a (k x 1) vector of intercepts

Intercept from mu for MSVARmdl

Description

This function computes the intercept for each regime k for an Markov switching VAR model

Usage

interMSVARmdl(mdl)

Arguments

Value

a (k x q) vector of intercepts

Ergodic (limiting) probabilities of states

Description

Takes a transition matrix and returns the limiting probabilities.

Usage

limP(P)

Arguments

Value

A (k x 1) vector of limiting probabilities.

Log likelihood for autoregressive model

Description

This function is used to compute the log-likelihood for an autoregressive model.

Usage

## S3 method for class 'ARmdl'
logLik(object, ...)

Arguments

Value

Log-likelihood value.

Log likelihood for Hidden Markov model

Description

This function is used to compute the log-likelihood for a Hidden Markov model.

Usage

## S3 method for class 'HMmdl'
logLik(object, ...)

Arguments

Value

Log-likelihood value.

Log likelihood for Markov-switching autoregressive model

Description

This function is used to compute the log-likelihood for a Markov-switching autoregressive model.

Usage

## S3 method for class 'MSARmdl'
logLik(object, ...)

Arguments

Value

Log-likelihood value.

Log likelihood for Markov-switching vector autoregressive model

Description

This function is used to compute the log-likelihood for a Markov-switching vector autoregressive model.

Usage

## S3 method for class 'MSVARmdl'
logLik(object, ...)

Arguments

Value

Log-likelihood value.

Log likelihood for Normal model

Description

This function is used to compute the log-likelihood for a normally distributed model.

Usage

## S3 method for class 'Nmdl'
logLik(object, ...)

Arguments

Value

Log-likelihood value.

Log likelihood for vector autoregressive model

Description

This function is used to compute the log-likelihood for a vector autoregressive model.

Usage

## S3 method for class 'VARmdl'
logLik(object, ...)

Arguments

Value

Log-likelihood value.

ARX log-likelihood objective function

Description

This function computes the log-likelihood for an autoregressive model with exogenous regressors.

Usage

logLike_ARXmdl(theta, mdl)

Arguments

Value

Log-likelihood value.

Autoregressive log-likelihood objective function

Description

This function computes the log-likelihood for an autoregressive model.

Usage

logLike_ARmdl(theta, mdl)

Arguments

Value

Log-likelihood value.

Hidden Markov model log-likelihood function

Description

This function computes the log-likelihood for a markov-switching autoregressive model.

Usage

logLike_HMmdl(theta, mdl, k)

Arguments

Value

Log-likelihood value.

Hidden Markov model log-likelihood function (minimization version)

Description

This function computes the (negative) log-likelihood for a markov-switching autoregressive model.

Usage

logLike_HMmdl_min(theta, mdl, k)

Arguments

Value

Negative log-likelihood value.

Markov-switching ARX log-likelihood objective function

Description

This function computes the log-likelihood for a markov-switching ARX model.

Usage

logLike_MSARXmdl(theta, mdl, k)

Arguments

Value

Log-likelihood value.

Markov-switching ARX log-likelihood objective function (minimization version)

Description

This function computes the (negative) log-likelihood for a markov-switching ARX model.

Usage

logLike_MSARXmdl_min(theta, mdl, k)

Arguments

Value

Negative log-likelihood value.

Markov-switching autoregressive log-likelihood objective function

Description

This function computes the log-likelihood for a markov-switching autoregressive model.

Usage

logLike_MSARmdl(theta, mdl, k)

Arguments

Value

Log-likelihood value.

Markov-switching autoregressive log-likelihood objective function (minimization version)

Description

This function computes the (negative) log-likelihood for a markov-switching autoregressive model.

Usage

logLike_MSARmdl_min(theta, mdl, k)

Arguments

Value

Negative log-likelihood value.

Markov-switching VARX log-likelihood objective function

Description

This function computes the log-likelihood for a markov-switching VARX model.

Usage

logLike_MSVARXmdl(theta, mdl, k)

Arguments

Value

Log-likelihood value.

Markov-switching VARX log-likelihood objective function (minimization version)

Description

This function computes the (negative) log-likelihood for a markov-switching VARX model

Usage

logLike_MSVARXmdl_min(theta, mdl, k)

Arguments

Value

Negative log-likelihood value.

Markov-switching vector autoregressive log-likelihood objective function

Description

This function computes the log-likelihood for a markov-switching vector autoregressive model.

Usage

logLike_MSVARmdl(theta, mdl, k)

Arguments

Value

Log-likelihood value.

Markov-switching vector autoregressive log-likelihood objective function (minimization version)

Description

This function computes the (negative) log-likelihood for a markov-switching vector autoregressive model

Usage

logLike_MSVARmdl_min(theta, mdl, k)

Arguments

Value

Negative log-likelihood value.

Normal log-likelihood objective function

Description

This function computes the log-likelihood for a normally distributed model.

Usage

logLike_Nmdl(theta, mdl)

Arguments

Value

Log-likelihood value.

VARX log-likelihood objective function

Description

This function computes the log-likelihood for a VARX model.

Usage

logLike_VARXmdl(theta, mdl)

Arguments

Value

Log-likelihood value.

Vector autoregressive log-likelihood objective function

Description

This function computes the log-likelihood for a vector autoregressive model.

Usage

logLike_VARmdl(theta, mdl)

Arguments

Value

Log-likelihood value.

Likelihood function used by HLRTest()

Description

this function computes the sum Markov likelihood

Usage

marklike(ths, HLR_opt_ls)

Arguments

Value

Vector of likelihood values.

Sum of likelihood used by HLRTest()

Description

This function computes the sum of the likelihood.

Usage

mclike(th, HLR_opt_ls)

Arguments

Value

Sum of likelihood.

Change model List with new parameters

Description

This function is used by the MMC LRT procedure. It makes a copy of the restricted model under the null hypothesis and changes the parameters used to simulate the process under the null with values being considered in the search/optimization.

Usage

mdledit(mdl_h0, theta_h0, p, q, k0, exog)

Arguments

Value

List with model properties

Nobs of a ARmdl object

Description

This is a method for the function nobs() for objects of the class ARmdl.

Usage

## S3 method for class 'ARmdl'
nobs(object, ...)

Arguments

Value

Number of time series observations.

Nobs of a HMmdl object

Description

This is a method for the function nobs() for objects of the class HMmdl.

Usage

## S3 method for class 'HMmdl'
nobs(object, ...)

Arguments

Value

Number of time series observations.

Nobs of a MSARmdl object

Description

This is a method for the function nobs() for objects of the class MSARmdl.

Usage

## S3 method for class 'MSARmdl'
nobs(object, ...)

Arguments

Value

Number of time series observations.

Nobs of a MSVARmdl object

Description

This is a method for the function nobs() for objects of the class MSVARmdl.

Usage

## S3 method for class 'MSVARmdl'
nobs(object, ...)

Arguments

Value

Number of time series observations.

Nobs of a Nmdl object

Description

This is a method for the function nobs() for objects of the class Nmdl.

Usage

## S3 method for class 'Nmdl'
nobs(object, ...)

Arguments

Value

Number of time series observations.

Nobs of a VARmdl object

Description

This is a method for the function nobs() for objects of the class VARmdl.

Usage

## S3 method for class 'VARmdl'
nobs(object, ...)

Arguments

Value

Number of time series observations.

Parameter list for Markov-switching ARX model

Description

This function takes the parameter vector of interest and converts it to a list with specific parameter vectors needed for univariate Markov-switching functions.

Usage

paramList_MSARXmdl(theta, p, k, qz, msmu, msvar)

Arguments

Value

List with the mean, variance, transition matrix, limiting probabilities, and a vector of state indicators.

Parameter list for Markov-switching autoregressive model

Description

This function takes the parameter vector of interest and converts it to a list with specific parameter vectors needed for univariate Markov-switching functions.

Usage

paramList_MSARmdl(theta, p, k, msmu, msvar)

Arguments

Value

List with the mean, variance, transition matrix, limiting probabilities, and a vector of state indicators.

Parameter list for Markov-switching VARX model

Description

This function takes the parameter vector of interest and converts it to a list with specific parameter vectors needed for multivariate Markov-switching functions.

Usage

paramList_MSVARXmdl(theta, q, p, k, qz, msmu, msvar)

Arguments

Value

List with the mean, variance, transition matrix, limiting probabilities, and a vector of state indicators.

Parameter list for Markov-switching vector autoregressive model

Description

This function takes the parameter vector of interest and converts it to a list with specific parameter vectors needed for multivariate Markov-switching functions.

Usage

paramList_MSVARmdl(theta, q, p, k, msmu, msvar)

Arguments

Value

List with the mean, variance, transition matrix, limiting probabilities, and a vector of state indicators.

Plot of a ARmdl object

Description

This is a method for the function plot() for objects of the class ARmdl.

Usage

## S3 method for class 'ARmdl'
plot(x, ...)

Arguments

Value

The ARmdl object is returned invisibly.

Plot of a HMmdl object

Description

This is a method for the function plot() for objects of the class HMmdl.

Usage

## S3 method for class 'Hmmdl'
plot(x, ...)

Arguments

Value

The Hmmdl object is returned invisibly.

Plot of a MSARmdl object

Description

This is a method for the function plot() for objects of the class MSARmdl.

Usage

## S3 method for class 'MSARmdl'
plot(x, ...)

Arguments

Value

The MSARmdl object is returned invisibly.

Plot of a MSVARmdl object

Description

This is a method for the function plot() for objects of the class MSVARmdl.

Usage

## S3 method for class 'MSVARmdl'
plot(x, ...)

Arguments

Value

The MSVARmdl object is returned invisibly.

Plot of a Nmdl object

Description

This is a method for the function plot() for objects of the class Nmdl.

Usage

## S3 method for class 'Nmdl'
plot(x, ...)

Arguments

Value

The Nmdl object is returned invisibly.

Plot of a VARmdl object

Description

This is a method for the function plot() for objects of the class VARmdl.

Usage

## S3 method for class 'VARmdl'
plot(x, ...)

Arguments

Value

The VARmdl object is returned invisibly.

Plot of a simuAR object

Description

This is a method for the function plot() for objects of the class simuAR.

Usage

## S3 method for class 'simuAR'
plot(x, ...)

Arguments

Value

The simuAR object is returned invisibly.

Plot of a simuARX object

Description

This is a method for the function plot() for objects of the class simuAR.

Usage

## S3 method for class 'simuARX'
plot(x, ...)

Arguments