estimate

Fit Markov-switching dynamic regression model to data

Syntax

EstMdl = estimate(Mdl,Mdl0,Y)

EstMdl = estimate(Mdl,Mdl0,Y,Name,Value)

[EstMdl,SS,logL] = estimate(___)

[EstMdl,SS,logL] = estimate(ax,___)

[EstMdl,SS,logL,h] = estimate(ax,___)

Description

EstMdl = estimate(Mdl,Mdl0,Y) estimates parameters of the Markov-switching dynamic regression model Mdl. estimate fits the model to the response data Y, and initializes the estimation procedure by treating the parameter values of the fully specified Markov-switching dynamic regression model Mdl0 as initial values. estimate uses a version of the expectation-maximization (EM) algorithm described by Hamilton [3].

EstMdl = estimate(Mdl,Mdl0,Y,Name,Value) uses additional options specified by one or more name-value pair arguments. For example, 'IterationPlot',true displays a plot of the loglikelihood versus the iteration step, after the algorithm terminates.

[EstMdl,SS,logL] = estimate(___) also returns expected smoothed state probabilities SS and the estimated loglikelihood logL when the algorithm terminates, using any of the input argument combinations in the previous syntaxes.

[EstMdl,SS,logL] = estimate(ax,___) plots on the axes specified by ax instead of the current axes (gca). The option ax can precede any of the input argument combinations in the previous syntaxes.

[EstMdl,SS,logL,h] = estimate(ax,___) also returns the plot handle h. Use h to modify properties of the plot after you create it.

Examples

Estimate Markov-Switching Dynamic Regression Model

Consider a two-state Markov-switching dynamic regression model of the postwar US real GDP growth rate, as estimated in [1].

Create Partially Specified Model for Estimation

Create a Markov-switching dynamic regression model for the naive estimator by specifying a two-state discrete-time Markov chain with an unknown transition matrix and AR(0) (constant only) submodels for both regimes. Label the regimes.

P = NaN(2);
mc = dtmc(P,'StateNames',["Expansion" "Recession"]);
mdl = arima(0,0,0);
Mdl = msVAR(mc,[mdl; mdl]);

Mdl is a partially specified msVAR object. NaN-valued elements of the Switch and SubModels properties indicate estimable parameters.

Create Fully Specified Model Containing Initial Values

The estimation procedure requires initial values for all estimable parameters. Create a fully specified Markov-switching dynamic regression model that has the same structure as Mdl, but set all estimable parameters to initial values. This example uses arbitrary initial values.

P0 = 0.5*ones(2);
mc0 = dtmc(P0,'StateNames',Mdl.StateNames);
mdl01 = arima('Constant',1,'Variance',1);
mdl02 = arima('Constant',-1,'Variance',1);
Mdl0 = msVAR(mc0,[mdl01; mdl02]);

Mdl0 is a fully specified msVAR object.

Load and Preprocess Data

Load the US GDP data set.

load Data_GDP

Data contains quarterly measurements of the US real GDP in the period 1947:Q1–2005:Q2. The estimation period in [1] is 1947:Q2–2004:Q2. For more details on the data set, enter Description at the command line.

Transform the data to an annualized rate series by:

1. Converting the data to a quarterly rate within the estimation period

2. Annualizing the quarterly rates

qrate = diff(Data(2:230))./Data(2:229); % Quarterly rate
arate = 100*((1 + qrate).^4 - 1);       % Annualized rate

Estimate Model

Fit the model Mdl to the annualized rate series arate. Specify Mdl0 as the model containing the initial estimable parameter values.

EstMdl = estimate(Mdl,Mdl0,arate);

EstMdl is an estimated (fully specified) Markov-switching dynamic regression model. EstMdl.Switch is an estimated discrete-time Markov chain model (dtmc object), and EstMdl.Submodels is a vector of estimated univariate VAR(0) models (varm objects).

Display the estimated state-specific dynamic models.

EstMdlExp = EstMdl.Submodels(1)

EstMdlExp = 
  varm with properties:

     Description: "ARIMA(0,0,0) Model (Gaussian Distribution)"
     SeriesNames: "Y1" 
       NumSeries: 1
               P: 0
        Constant: 4.90146
              AR: {}
           Trend: 0
            Beta: [1×0 matrix]
      Covariance: 12.087
 
   1-Dimensional VAR(0) Model

EstMdlRec = EstMdl.Submodels(2)

EstMdlRec = 
  varm with properties:

     Description: "ARIMA(0,0,0) Model (Gaussian Distribution)"
     SeriesNames: "Y1" 
       NumSeries: 1
               P: 0
        Constant: 0.0084884
              AR: {}
           Trend: 0
            Beta: [1×0 matrix]
      Covariance: 12.6876
 
   1-Dimensional VAR(0) Model

Display the estimated state transition matrix.

EstP = EstMdl.Switch.P

EstP = 2×2

    0.9088    0.0912
    0.2303    0.7697

Display Behavior of EM Algorithm After Estimation

Consider the model and data in Estimate Markov-Switching Dynamic Regression Model.

Create the partially specified model for estimation.

P = NaN(2);
mc = dtmc(P,'StateNames',["Expansion" "Recession"]);
mdl = arima(0,0,0);
Mdl = msVAR(mc,[mdl; mdl]);

Create the fully specified model containing initial parameter values for the estimation procedure.

P0 = 0.5*ones(2);
mc0 = dtmc(P0,'StateNames',Mdl.StateNames);
mdl01 = arima('Constant',1,'Variance',1);
mdl02 = arima('Constant',-1,'Variance',1);
Mdl0 = msVAR(mc0,[mdl01; mdl02]);

Load and preprocess the data.

load Data_GDP
qrate = diff(Data(2:230))./Data(2:229); 
arate = 100*((1 + qrate).^4 - 1);       

Fit the model to the data. Plot the loglikelihood versus the iteration step when the estimation procedure terminates.

EstMdl = estimate(Mdl,Mdl0,arate,'IterationPlot',true);

Fit Model to Simulated Data

Assess estimation accuracy using simulated data from a known data-generating process (DGP). This example uses arbitrary parameter values.

Create Model for DGP

Create a fully specified, two-state discrete-time Markov chain model for the switching mechanism.

P = [0.7 0.3; 0.1 0.9];
mc = dtmc(P);

For each state, create a fully specified AR(1) model for the response process.

% Constants
C1 = 5;
C2 = -2;

% Autoregression coefficients
AR1 = 0.4;
AR2 = 0.2;

% Variances
V1 = 4;
V2 = 2;

% AR Submodels
dgp1 = arima('Constant',C1,'AR',AR1,'Variance',V1);
dgp2 = arima('Constant',C2,'AR',AR2,'Variance',V2);

Create a fully specified Markov-switching dynamic regression model for the DGP.

DGP = msVAR(mc,[dgp1,dgp2]);

Simulate Response Paths from DGP

Generate 10 random response paths of length 1000 from the DGP.

rng(1); % For reproducibility
N = 10;
n = 1000;
Data = simulate(DGP,n,'Numpaths',N);

Data is a 1000-by-10 matrix of simulated responses.

Create Model for Estimation

Create a partially specified Markov-switching dynamic regression model that has the same structure as the data-generating process, but specify an unknown transition matrix and unknown submodel coefficients.

PEst = NaN(2);
mcEst = dtmc(PEst);
mdl = arima(1,0,0);
Mdl = msVAR(mcEst,[mdl; mdl]);

Create Model Containing Initial Values

Create a fully specified Markov-switching dynamic regression model that has the same structure as Mdl, but set all estimable parameters to initial values.

P0 = 0.5*ones(2);
mc0 = dtmc(P0);
mdl01 = arima('Constant',1,'AR',0.5,'Variance',2);
mdl02 = arima('Constant',-1,'AR',0.5,'Variance',1);
Mdl0 = msVAR(mc0,[mdl01,mdl02]);

Estimate Models

Fit the model to each simulated path. For each path, plot the loglikelihood at each iteration of the EM algorithm.

c1 = zeros(N,1);
c2 = zeros(N,1);
v1 = zeros(N,1);
v2 = zeros(N,1);
ar1 = zeros(N,1);
ar2 = zeros(N,1);
PStack = zeros(2,2,N);

figure
hold on

for i = 1:N

    EstModel = estimate(Mdl,Mdl0,Data(:,i),'IterationPlot',true);
    
    c1(i) = EstModel.Submodels(1).Constant;
    c2(i) = EstModel.Submodels(2).Constant;
    v1(i) = EstModel.Submodels(1).Covariance;
    v2(i) = EstModel.Submodels(2).Covariance;
    ar1(i) = EstModel.Submodels(1).AR{1};
    ar2(i) = EstModel.Submodels(2).AR{1};
    PStack(:,:,i) = EstModel.Switch.P;
    
end

hold off

Assess Accuracy

Compute the Monte Carlo mean of each estimated parameter.

c1Mean = mean(c1);
c2Mean = mean(c2);
v1Mean = mean(v1);
v2Mean = mean(v2);
ar1Mean = mean(ar1);
ar2Mean = mean(ar2);
PMean = mean(PStack,3);

Compare population parameters to the corresponding Monte Carlo estimates.

DGPvsEstimate = [...
    C1  c1Mean
    C2  c2Mean
    V1  v1Mean
    V2  v2Mean
    AR1 ar1Mean
    AR2 ar2Mean]

DGPvsEstimate = 6×2

    5.0000    5.0260
   -2.0000   -1.9615
    4.0000    3.9710
    2.0000    1.9903
    0.4000    0.4061
    0.2000    0.2017

P

P = 2×2

    0.7000    0.3000
    0.1000    0.9000

PEstimate = PMean

PEstimate = 2×2

    0.7065    0.2935
    0.1023    0.8977

Specify Presample Data

Consider the data in Estimate Markov-Switching Dynamic Regression Model, but assume that the period of interest is 1960:Q1–2004:Q2. Also, consider adding an autoregressive term to each submodel.

Create a partially specified Markov-switching dynamic regression model for estimation. Specify AR(1) submodels.

P = NaN(2);
mc = dtmc(P,'StateNames',["Expansion" "Recession"]);
mdl = arima(1,0,0);
Mdl = msVAR(mc,[mdl; mdl]);

Because the submodels are AR(1), each requires one presample observation to initialize its dynamic component for estimation.

Create the model containing initial parameter values for the estimation procedure.

P0 = 0.5*ones(2);
mc0 = dtmc(P0,'StateNames',Mdl.StateNames);
mdl01 = arima('Constant',1,'Variance',1,'AR',0.001);
mdl02 = arima('Constant',-1,'Variance',1,'AR',0.001);
Mdl0 = msVAR(mc0,[mdl01; mdl02]);

Load the data. Transform the entire set to an annualized rate series.

load Data_GDP
qrate = diff(Data)./Data(1:(end - 1)); 
arate = 100*((1 + qrate).^4 - 1);

Identify the presample and estimation sample periods using the dates associated with the annualized rate series. Because the transformation applies the first difference, you must drop the first observation date from the original sample.

dates = datetime(dates(2:end),'ConvertFrom','datenum',...
    'Format','yyyy:QQQ','Locale','en_US');
estPrd = datetime(["1960:Q2" "2004:Q2"],'InputFormat','yyyy:QQQ',...
    'Format','yyyy:QQQ','Locale','en_US');
idxEst = isbetween(dates,estPrd(1),estPrd(2));
idxPre = dates < estPrd(1); % The presample is the previous quarter.

Fit the model to the estimation sample data. Specify the presample observation and plot the loglikelihood at each iteration when the estimation procedure terminates.

EstMdl = estimate(Mdl,Mdl0,arate(idxEst),'Y0',arate(idxPre),...
    'IterationPlot',true);

Access Expected Smoothed State Probabilities and Loglikelihood

Consider the model and data in Estimate Markov-Switching Dynamic Regression Model.

Create the partially specified model for estimation.

P = NaN(2);
mc = dtmc(P,'StateNames',["Expansion" "Recession"]);
mdl = arima(0,0,0);
Mdl = msVAR(mc,[mdl; mdl]);

Create the fully specified model containing initial parameter values for the estimation procedure.

P0 = 0.5*ones(2);
mc0 = dtmc(P0,'StateNames',Mdl.StateNames);
mdl01 = arima('Constant',1,'Variance',1);
mdl02 = arima('Constant',-1,'Variance',1);
Mdl0 = msVAR(mc0,[mdl01; mdl02]);

Load and preprocess the data.

load Data_GDP
qrate = diff(Data(2:230))./Data(2:229); 
arate = 100*((1 + qrate).^4 - 1);       

Fit the model to the data. Return the expected smoothed state probabilities and loglikelihood when the algorithm terminates.

[EstMdl,SS,logL] = estimate(Mdl,Mdl0,arate);

SS is a 228-by-2 matrix of expected smoothed state probabilities; rows correspond to periods in the estimation sample, and columns correspond to the regimes. logL is the final loglikelihood.

Display the expected smoothed state probabilities for the last period in the estimation sample, and display the final loglikelihood.

SS(end,:)

ans = 1×2

    0.8985    0.1015

logL

logL = -639.4962

Perform Constrained Estimation

Fit simulated data to a Markov-switching dynamic regression model with VARX submodels. Specify equality constraints for estimation. This example uses arbitrary parameter values.

Create Model for DGP

Create a fully specified, three-state discrete-time Markov chain model for the switching mechanism.

P = [0.8 0.1 0.1; 0.2 0.6 0.2; 0 0.1 0.9];
mc = dtmc(P);

For each state, create a fully specified VARX(1) model for the response process. Specify the same model constant and lag 1 AR coefficient matrix for all submodels. For each model, specify a different 2-by-1 regression coefficient for the one exogenous variable.

% Constants
C = [1;-1];

% Autoregression coefficients
AR = {[0.6 0.1; 0.4 0.2]};

% Regression coefficients
Beta1 = [0.2;-0.4];
Beta2 = [0.6;-1.0];
Beta3 = [0.9;-1.3];

% VAR Submodels
dgp = varm('Constant',C,'AR',AR,'Covariance',5*eye(2));
dgp1 = dgp;
dgp1.Beta = Beta1;
dgp2 = dgp;
dgp2.Beta = Beta2;
dgp3 = dgp;
dgp3.Beta = Beta3;

Create a fully specified Markov-switching dynamic regression model for the DGP.

DGP = msVAR(mc,[dgp1; dgp2; dgp3]);

Simulate Data from DGP

Simulate data for the exogenous series by generating 1000 observations from the Gaussian distribution with mean 0 and variance 100.

rng(1); % For reproducibility
X = 10*randn(1000,1);

Generate a random response path of length 1000 from the DGP. Specify the simulated exogenous data for the submodel regression components.

Data = simulate(DGP,1000,'X',X);

Data is a 1000-by-1 vector of simulated responses.

Create Model for Estimation

Create a partially specified Markov-switching dynamic regression model that has the same structure as the data-generating process, but specify an unknown transition matrix and unknown regression coefficients. Specify the true values of the constants and AR coefficient matrices.

PEst = NaN(3);
mcEst = dtmc(PEst);
mdl = varm(2,1);
mdl.Constant = C;
mdl.AR = AR;
mdl.Beta = NaN(2,1);
Mdl = msVAR(mcEst,[mdl; mdl; mdl]);

Because the values of the constants and AR coefficient matrices are specified in Mdl, estimate treats them as equality constraints for estimation.

Create Model Containing Initial Values

Create a fully specified Markov-switching dynamic regression model that has the same structure as Mdl, but set all estimable parameters to initial values and set parameters with equality constraints to their values specified in Mdl.

P0 = (1/3)*ones(3);
mc0 = dtmc(P0);
mdl0 = varm('Constant',C,'AR',AR,'Covariance',eye(2));
mdl01 = mdl0;
mdl01.Beta = [0.1;-0.1];
mdl02 = mdl0;
mdl02.Beta = [0.5;-0.5];
mdl03 = mdl0;
mdl03.Beta = [1;-1];
Mdl0 = msVAR(mc0,[mdl01; mdl02; mdl03]);

Estimate Model

Fit the model to the simulated data. Specify the exogenous data for the regression component. Plot the loglikelihood at each iteration of the EM algorithm.

figure
EstMdl = estimate(Mdl,Mdl0,Data,'X',X,'IterationPlot',true);

Assess Accuracy

Compare the estimated regression coefficient vectors and transition matrix to their true values.

Beta1

Beta1 = 2×1

    0.2000
   -0.4000

Beta1Estimate = EstMdl.Submodels(1).Beta

Beta1Estimate = 2×1

    0.1596
   -0.4040

Beta2

Beta2 = 2×1

    0.6000
   -1.0000

Beta2Estimate = EstMdl.Submodels(2).Beta

Beta2Estimate = 2×1

    0.5888
   -0.9771

Beta3

Beta3 = 2×1

    0.9000
   -1.3000

Beta3Estimate = EstMdl.Submodels(3).Beta

Beta3Estimate = 2×1

    0.8987
   -1.2991

P

P = 3×3

    0.8000    0.1000    0.1000
    0.2000    0.6000    0.2000
         0    0.1000    0.9000

PEstimate = EstMdl.Switch.P

PEstimate = 3×3

    0.7787    0.0856    0.1357
    0.1366    0.6906    0.1727
    0.0086    0.0787    0.9127

Input Arguments

Mdl — Partially specified Markov-switching dynamic regression model

msVAR model object

Partially specified Markov-switching dynamic regression model used to indicate constrained and estimable model parameters, specified as an msVAR model object returned by msVAR. Properties of Mdl describe the model structure and specify the parameters.

estimate treats specified parameters as equality constraints during estimation.

estimate fits unspecified (NaN-valued) parameters to the data Y.

Mdl0 — Fully specified Markov-switching dynamic regression model

msVAR model object

Fully specified Markov-switching dynamic regression model used to initialize the estimation procedure, specified as an msVAR model object returned by msVAR. Properties of a fully specified model object do not contain NaN values.

Mdl0 is a copy of Mdl with NaN values replaced by initial values.

Tip

For broad coverage of the parameter space, run the algorithm from multiple instances of Mdl0. For more details, see Algorithms.

Y — Observed response data

numeric matrix

Observed response data to which estimate fits the model, specified as a numObs-by-numSeries numeric matrix.

numObs is the sample size. numSeries is the number of response variables (Mdl.NumSeries).

Rows correspond to observations, and the last row contains the latest observation. Columns correspond to individual response variables.

Y represents the continuation of the presample response series in Y0.

Data Types: double

ax — Axes on which to plot loglikelihood for each iteration

Axes object

Axes on which to plot the loglikelihood for each iteration when IterationPlot is true, specified as an Axes object.

By default, estimate plots to the current axes (gca).

If IterationPlot is false, estimate ignores ax.

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'Y0',Y0,'X',X,'IterationPlot',true uses the matrix Y0 as presample responses required for estimation, includes a linear regression component in each submodel composed of the predictor data in X and the specified regression coefficients, and plots the loglikelihood for each iteration.

'Y0' — Presample response data

numeric matrix

Presample response data, specified as the comma-separated pair consisting of 'Y0' and a numPreSampleObs-by-numSeries numeric matrix.

The number of presample observations numPreSampleObs must be sufficient to initialize the AR terms of all submodels. If numPreSampleObs exceeds the AR order of any state, estimate uses the latest observations. By default, Y0 is the initial segment of Y, which reduces the effective sample size.

Data Types: double

'S0' — Initial state probabilities

nonnegative numeric vector

Initial state probabilities, specified as the comma-separated pair consisting of 'S0' and a nonnegative numeric vector of length numStates.

estimate normalizes S0 to produce a distribution.

By default, S0 is a steady-state distribution computed by asymptotics.

estimate updates S0 at each iteration using expected smoothed probabilities for the initial distribution.

Example: 'S0',[0.2 0.2 0.6]

Example: 'S0',[0 1] specifies state 2 as the initial state.

Data Types: double

'X' — Predictor data

numeric matrix | cell vector of numeric matrices

Predictor data used to evaluate regression components in all submodels of Mdl, specified as the comma-separated pair consisting of 'X' and a numeric matrix or a cell vector of numeric matrices.

To use a subset of the same predictors in each state, specify X as a matrix with numPreds columns and at least numObs rows. Columns correspond to distinct predictor variables. Submodels use initial columns of the associated matrix, in order, up to the number of submodel predictors. The number of columns in the Beta property of Mdl.SubModels(j) determines the number of exogenous variables in the regression component of submodel j. If the number of rows exceeds numObs, then estimate uses the latest observations.

To use different predictors in each state, specify a cell vector of such matrices with length numStates.

By default, estimate ignores regression components in Mdl.

Data Types: double

'Tolerance' — Convergence tolerance

0.001 (default) | positive numeric scalar

Convergence tolerance, specified as the comma-separated pair consisting of 'Tolerance' and a positive numeric scalar.

The EM algorithm terminates after an iteration in which the loglikelihood logL changes by less than Tolerance.

Example: 'Tolerance',1e-6

Data Types: double

'MaxIterations' — Maximum number of iterations

100 (default) | positive integer

Maximum number of iterations of the EM algorithm, specified as the comma-separated pair consisting of 'MaxIterations' and a positive integer.

Example: 'MaxIterations',1000

Data Types: double

'IterationPlot' — Flag for plotting loglikelihood at each iteration

false (default) | true

Flag for plotting the loglikelihood at each iteration, specified as the comma-separated pair consisting of 'IterationPlot' and a value in this table.

Value	Description
true	When the algorithm terminates, estimate shows a plot of the loglikelihood at each iteration step.
false	estimate does not show a plot.

Example: 'IterationPlot',true

Data Types: logical

Output Arguments

EstMdl — Estimated Markov-switching dynamic regression model

msVAR object

Estimated Markov-switching dynamic regression model, returned as an msVAR object. EstMdl is a copy of Mdl that has NaN values replaced with parameter estimates. EstMdl is fully specified.

Note

Row normalization can alter specified values in the transition matrix of the input model Mdl.Switch.P.

SS — Expected smoothed state probabilities

numeric matrix

Expected smoothed state probabilities when the algorithm terminates, returned as a numObs-by-numStates numeric matrix.

logL — Estimated loglikelihood

numeric scalar

Estimated loglikelihood of the response data Y with respect to the estimated model EstMdl when the algorithm terminates, returned as a numeric scalar.

h — Handle to iteration plot

graphics object

Handle to the iteration plot, returned as a graphics object when IterationPlot is true.

h contains a unique plot identifier, which you can use to query or modify properties of the plot.

Tips

• In [4], Hamilton cautions: "Although a researcher might be tempted to use the most general specification possible, with all the parameters changing across a large number of regimes...in practice this is usually asking more than the data can deliver." Hamilton recommends model parsimony and selective estimation, to "limit the focus to a few of the most important parameters that are likely to change."

• Data-generating processes with low variances can lead to difficulties in state inference and subsequent parameter estimation. In such cases, consider scaling the data. The variance scales quadratically.

Algorithms

• estimate implements a version of the EM algorithm described in [2], [3], and [4]. Given an initial estimate of model parameters Mdl0, estimate iterates the following process until convergence:

  1. Expectation step — estimate applies smooth to the data to obtain inferences of latent state probabilities at each time step and an estimate of the overall data loglikelihood.

  2. Maximization step — estimate uses the expected smoothed probabilities from the expectation step to weight the data before updating parameter estimates in each submodel.

• Likelihood surfaces for the mixture densities of switching models can contain local maxima and singularities [2]. If so, the largest local maximum with a nonzero model variance is considered the maximum likelihood estimate (MLE). If Mdl0 is in the neighborhood of the MLE, estimate typically converges to it, but this convergence is not guaranteed.

• estimate handles two types of constraints:

  • Constraints on submodel parameters, which the estimate object function of the varm object enforces at each maximization step

  • Constraints on transition probabilities, which estimate enforces by projecting a maximal estimate of the transition matrix onto the constrained parameter space after each iteration

  Constraints on submodel innovations variances and covariances are unsupported. estimate computes innovations covariances after estimation, regardless of their values.

References

[1] Chauvet, M., and J. D. Hamilton. "Dating Business Cycle Turning Points." In Nonlinear Analysis of Business Cycles (Contributions to Economic Analysis, Volume 276). (C. Milas, P. Rothman, and D. van Dijk, eds.). Amsterdam: Emerald Group Publishing Limited, 2006.

[2] Hamilton, J. D. "Analysis of Time Series Subject to Changes in Regime." Journal of Econometrics. Vol. 45, 1990, pp. 39–70.

[3] Hamilton, James D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.

[4] Hamilton, J. D. "Macroeconomic Regimes and Regime Shifts." In Handbook of Macroeconomics. (H. Uhlig and J. Taylor, eds.). Amsterdam: Elsevier, 2016.

[5] Kole, E. "Regime Switching Models: An Example for a Stock Market Index." Rotterdam, NL: Econometric Institute, Erasmus School of Economics, 2010.