msVAR

Create Markov-switching dynamic regression model

Description

The msVAR function returns an msVAR object that specifies the functional form of a Markov-switching dynamic regression model for the univariate or multivariate response process y_t. The msVAR object also stores the parameter values of the model.

An msVAR object has two key components:

• Switching mechanism among states, represented by a discrete-time Markov chain (dtmc object)

• State-specific submodels, either autoregressive (ARX) or vector autoregression (VARX) models (arima or varm objects), which can contain exogenous regression components

The components completely specify the model structure. The Markov chain transition matrix and submodel parameters, such as the AR coefficients and innovation-distribution variance, are unknown and estimable unless you specify their values.

To estimate a model containing unknown parameter values, pass the model and data to estimate. To work with an estimated or fully specified msVAR object, pass it to an object function.

Alternatively, to create a threshold-switching dynamic regression model, which has a switching mechanism governed by threshold transitions and observations of a threshold variable, see threshold and tsVAR.

Creation

Syntax

Mdl = msVAR(mc,mdl)

Mdl = msVAR(mc,mdl,'SeriesNames',seriesNames)

Description

Mdl = msVAR(mc,mdl) creates a Markov-switching dynamic regression model Mdl (an msVAR object) that has the discrete-time Markov chain, switching mechanism among states mc and the state-specific, stable dynamic regression submodels mdl.

Mdl = msVAR(mc,mdl,'SeriesNames',seriesNames) optionally sets the SeriesNames property, which associates the names seriesNames to the time series of the model.

Input Arguments

mc — Discrete-time Markov chain for switching mechanism among states

dtmc object

Discrete-time Markov chain for the switching mechanism among states, specified as a dtmc object.

The states represented in the rows and columns of the transition matrix mc.P correspond to the states represented in the submodel vector mdl.

msVAR processes and stores mc in the Switch property.

mdl — State-specific dynamic regression submodels

vector of arima objects | vector of varm objects

State-specific dynamic regression submodels, specified as a length mc.NumStates vector of model objects individually constructed by arima or varm. All submodels must be of the same type (arima or varm) and have the same number of series.

Unlike other model estimation tools, estimate does not infer the size of submodel regression coefficient arrays during estimation. Therefore, you must specify the Beta property of each submodel appropriately. For example, to include and estimate three predictors of the regression component of univariate submodel j, set mdl(j).Beta = NaN(3,1).

msVAR processes and stores mdl in the property Submodels.

Properties

You can set only the SeriesNames property when you create a model by using name-value argument syntax or by using dot notation. MATLAB^® derives the values of all other properties from inputs mc and mdl.

For example, create a Markov-switching model for a 2-D response series, and then label the first and second series "GDP" and "CPI", respectively.

Mdl = msVAR(mc,mdl);
Mdl.SeriesNames = ["GDP" "CPI"];

NumStates — Number of states

Read-only: positive scalar

This property is read-only.

Number of states, specified as a positive scalar.

Data Types: double

NumSeries — Number of time series

Read-only: positive integer

This property is read-only.

Number of time series, specified as a positive integer. NumSeries specifies the dimensionality of the response variable and innovation in all submodels.

Data Types: double

StateNames — State labels

Read-only: string vector

This property is read-only.

State labels, specified as a string vector of length NumStates.

Data Types: string

SeriesNames — Unique Series labels

string(1:numSeries) (default) | string vector | cell array of character vectors | numeric vector

Unique series labels, specified as a string vector, cell array of character vectors, or a numeric vector of length numSeries. msVAR stores the series names as a string vector.

Data Types: string

Switch — Discrete-time Markov chain for switching mechanism among states

Read-only: dtmc object

This property is read-only.

Discrete-time Markov chain for the switching mechanism among states, specified as a dtmc object.

Submodels — State-specific vector autoregression submodels

Read-only: vector of varm objects

This property is read-only.

State-specific vector autoregression submodels, specified as a vector of varm objects of length NumStates.

msVAR removes unsupported submodel components.

• For arima submodels, msVAR does not support the moving average (MA), differencing, and seasonal components. If any submodel is a composite conditional mean and variance model (for example, its Variance property is a garch object), msVAR issues an error.

• For varm submodels, msVAR does not support the trend component.

msVAR converts submodels specified as arima objects to 1-D varm objects.

Notes

• NaN-valued elements in either the properties of Switch or the submodels of Submodels indicate unknown, estimable parameters. Specified elements, except submodel innovation variances, indicate equality constraints on parameters in model estimation.

• All unknown submodel parameters are state dependent.

Object Functions

estimate	Fit Markov-switching dynamic regression model to data
filter	Filtered inference of operative latent states in Markov-switching dynamic regression data
forecast	Forecast sample paths from Markov-switching dynamic regression model
simulate	Simulate sample paths of Markov-switching dynamic regression model
smooth	Smoothed inference of operative latent states in Markov-switching dynamic regression data
summarize	Summarize Markov-switching dynamic regression model estimation results

Examples

Create Fully Specified Univariate Model

Create a two-state Markov-switching dynamic regression model for a 1-D response process. Specify all parameter values (this example uses arbitrary values).

Create a two-state discrete-time Markov chain model that describes the regime switching mechanism. Label the regimes.

P = [0.9 0.1; 0.3 0.7];
mc = dtmc(P,'StateNames',["Expansion" "Recession"])

mc = 
  dtmc with properties:

             P: [2×2 double]
    StateNames: ["Expansion"    "Recession"]
     NumStates: 2

mc is a dtmc object.

For each regime, use arima to create an AR model that describes the response process within the regime.

% Constants
C1 = 5;
C2 = -5;

% AR coefficients
AR1 = [0.3 0.2]; % 2 lags
AR2 = 0.1;       % 1 lag

% Innovations variances
v1 = 2;
v2 = 1;

% AR Submodels
mdl1 = arima('Constant',C1,'AR',AR1,...
    'Variance',v1,'Description','Expansion State')

mdl1 = 
  arima with properties:

     Description: "Expansion State"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
               P: 2
               D: 0
               Q: 0
        Constant: 5
              AR: {0.3 0.2} at lags [1 2]
             SAR: {}
              MA: {}
             SMA: {}
     Seasonality: 0
            Beta: [1×0]
        Variance: 2
 
   ARIMA(2,0,0) Model (Gaussian Distribution)

mdl2 = arima('Constant',C2,'AR',AR2,...
    'Variance',v2,'Description','Recession State')

mdl2 = 
  arima with properties:

     Description: "Recession State"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
               P: 1
               D: 0
               Q: 0
        Constant: -5
              AR: {0.1} at lag [1]
             SAR: {}
              MA: {}
             SMA: {}
     Seasonality: 0
            Beta: [1×0]
        Variance: 1
 
   ARIMA(1,0,0) Model (Gaussian Distribution)

mdl1 and mdl2 are fully specified arima objects.

Store the submodels in a vector with order corresponding to the regimes in mc.StateNames.

mdl = [mdl1; mdl2];

Use msVAR to create a Markov-switching dynamic regression model from the switching mechanism mc and the state-specific submodels mdl.

Mdl = msVAR(mc,mdl)

Mdl = 
  msVAR with properties:

      NumStates: 2
      NumSeries: 1
     StateNames: ["Expansion"    "Recession"]
    SeriesNames: "1"
         Switch: [1×1 dtmc]
      Submodels: [2×1 varm]

Mdl.Submodels(1)

ans = 
  varm with properties:

     Description: "AR-Stationary 1-Dimensional VAR(2) Model"
     SeriesNames: "Y1" 
       NumSeries: 1
               P: 2
        Constant: 5
              AR: {0.3 0.2} at lags [1 2]
           Trend: 0
            Beta: [1×0 matrix]
      Covariance: 2

Mdl.Submodels(2)

ans = 
  varm with properties:

     Description: "AR-Stationary 1-Dimensional VAR(1) Model"
     SeriesNames: "Y1" 
       NumSeries: 1
               P: 1
        Constant: -5
              AR: {0.1} at lag [1]
           Trend: 0
            Beta: [1×0 matrix]
      Covariance: 1

Mdl is a fully specified msVAR object representing a univariate two-state Markov-switching dynamic regression model. msVAR stores specified arima submodels as varm objects.

Because Mdl is fully specified, you can pass it to any msVAR object function for further analysis (see Object Functions). Or, you can specify that the parameters of Mdl are initial values for the estimation procedure (see estimate).

Create Fully Specified Model for US GDP Rate

Consider a two-state Markov-switching dynamic regression model of the postwar US real GDP growth rate. The model has the parameter estimates presented in [1].

Create a discrete-time Markov chain model that describes the regime switching mechanism. Label the regimes.

P = [0.92 0.08; 0.26 0.74];
mc = dtmc(P,'StateNames',["Expansion" "Recession"]);

mc is a fully specified dtmc object.

Create separate AR(0) models (constant only) for the two regimes.

sigma = 3.34; % Homoscedastic models across states
mdl1 = arima('Constant',4.62,'Variance',sigma^2);
mdl2 = arima('Constant',-0.48,'Variance',sigma^2);
mdl = [mdl1 mdl2];

Create the Markov-switching dynamic regression model that describes the behavior of the US GDP growth rate.

Mdl = msVAR(mc,mdl)

Mdl = 
  msVAR with properties:

      NumStates: 2
      NumSeries: 1
     StateNames: ["Expansion"    "Recession"]
    SeriesNames: "1"
         Switch: [1×1 dtmc]
      Submodels: [2×1 varm]

Mdl is a fully specified msVAR object.

Create Partially Specified Univariate Model for Estimation

Consider fitting to data a two-state Markov-switching model for a 1-D response process.

Create a discrete-time Markov chain model for the switching mechanism. Specify a 2-by-2 matrix of NaN values for the transition matrix. This setting indicates that you want to estimate all transition probabilities. Label the states.

P = NaN(2);
mc = dtmc(P,'StateNames',["Expansion" "Recession"])

mc = 
  dtmc with properties:

             P: [2×2 double]
    StateNames: ["Expansion"    "Recession"]
     NumStates: 2

mc.P

ans = 2×2

   NaN   NaN
   NaN   NaN

mc is a partially specified dtmc object. The transition matrix mc.P is completely unknown and estimable.

Create AR(1) and AR(2) models by using the shorthand syntax of arima. After you create each model, specify the model description by using dot notation.

mdl1 = arima(1,0,0);
mdl1.Description = "Expansion State"

mdl1 = 
  arima with properties:

     Description: "Expansion State"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
               P: 1
               D: 0
               Q: 0
        Constant: NaN
              AR: {NaN} at lag [1]
             SAR: {}
              MA: {}
             SMA: {}
     Seasonality: 0
            Beta: [1×0]
        Variance: NaN
 
   ARIMA(1,0,0) Model (Gaussian Distribution)

mdl2 = arima(2,0,0);
mdl2.Description = "Recession State"

mdl2 = 
  arima with properties:

     Description: "Recession State"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
               P: 2
               D: 0
               Q: 0
        Constant: NaN
              AR: {NaN NaN} at lags [1 2]
             SAR: {}
              MA: {}
             SMA: {}
     Seasonality: 0
            Beta: [1×0]
        Variance: NaN
 
   ARIMA(2,0,0) Model (Gaussian Distribution)

mdl1 and mdl2 are partially specified arima objects. NaN-valued properties correspond to unknown, estimable parameters.

Store the submodels in a vector with order corresponding to the regimes in mc.StateNames.

mdl = [mdl1; mdl2];

Create a Markov-switching model template from the switching mechanism mc and the state-specific submodels mdl.

Mdl = msVAR(mc,mdl)

Mdl = 
  msVAR with properties:

      NumStates: 2
      NumSeries: 1
     StateNames: ["Expansion"    "Recession"]
    SeriesNames: "1"
         Switch: [1×1 dtmc]
      Submodels: [2×1 varm]

Mdl is a partially specified msVAR object representing a univariate two-state Markov-switching dynamic regression model.

Mdl.Submodels(1)

ans = 
  varm with properties:

     Description: "1-Dimensional VAR(1) Model"
     SeriesNames: "Y1" 
       NumSeries: 1
               P: 1
        Constant: NaN
              AR: {NaN} at lag [1]
           Trend: 0
            Beta: [1×0 matrix]
      Covariance: NaN

Mdl.Submodels(2)

ans = 
  varm with properties:

     Description: "1-Dimensional VAR(2) Model"
     SeriesNames: "Y1" 
       NumSeries: 1
               P: 2
        Constant: NaN
              AR: {NaN NaN} at lags [1 2]
           Trend: 0
            Beta: [1×0 matrix]
      Covariance: NaN

msVAR converts the arima object submodels to 1-D varm object equivalents.

Mdl is prepared for estimation. You can pass Mdl, along with data and a fully specified model containing initial values for optimization, to estimate.

Create Fully Specified Multivariate Model

Create a three-state Markov-switching dynamic regression model for a 2-D response process. Specify all parameter values (this example uses arbitrary values).

Create a three-state discrete-time Markov chain model that describes the regime switching mechanism.

P = [10 1 1; 1 10 1; 1 1 10];
mc = dtmc(P);
mc.P

ans = 3×3

    0.8333    0.0833    0.0833
    0.0833    0.8333    0.0833
    0.0833    0.0833    0.8333

mc is a dtmc object. dtmc normalizes P so that each row sums to 1.

For each regime, use varm to create a VAR model that describes the response process within the regime. Specify all parameter values.

% Constants (numSeries x 1 vectors)
C1 = [1;-1];
C2 = [2;-2];
C3 = [3;-3];

% Autoregression coefficients (numSeries x numSeries matrices)
AR1 = {};                            % 0 lags
AR2 = {[0.5 0.1; 0.5 0.5]};          % 1 lag
AR3 = {[0.25 0; 0 0] [0 0; 0.25 0]}; % 2 lags

% Innovations covariances (numSeries x numSeries matrices)
Sigma1 = [1 -0.1; -0.1 1];
Sigma2 = [2 -0.2; -0.2 2];
Sigma3 = [3 -0.3; -0.3 3];

% VAR Submodels
mdl1 = varm('Constant',C1,'AR',AR1,'Covariance',Sigma1);
mdl2 = varm('Constant',C2,'AR',AR2,'Covariance',Sigma2);
mdl3 = varm('Constant',C3,'AR',AR3,'Covariance',Sigma3);

mdl1, mdl2, and mdl3 are fully specified varm objects.

Store the submodels in a vector with order corresponding to the regimes in mc.StateNames.

mdl = [mdl1; mdl2; mdl3];

Use msVAR to create a Markov-switching dynamic regression model from the switching mechanism mc and the state-specific submodels mdl.

Mdl = msVAR(mc,mdl)

Mdl = 
  msVAR with properties:

      NumStates: 3
      NumSeries: 2
     StateNames: ["1"    "2"    "3"]
    SeriesNames: ["1"    "2"]
         Switch: [1×1 dtmc]
      Submodels: [3×1 varm]

Mdl.Submodels(1)

ans = 
  varm with properties:

     Description: "2-Dimensional VAR(0) Model"
     SeriesNames: "Y1"  "Y2" 
       NumSeries: 2
               P: 0
        Constant: [1 -1]'
              AR: {}
           Trend: [2×1 vector of zeros]
            Beta: [2×0 matrix]
      Covariance: [2×2 matrix]

Mdl.Submodels(2)

ans = 
  varm with properties:

     Description: "AR-Stationary 2-Dimensional VAR(1) Model"
     SeriesNames: "Y1"  "Y2" 
       NumSeries: 2
               P: 1
        Constant: [2 -2]'
              AR: {2×2 matrix} at lag [1]
           Trend: [2×1 vector of zeros]
            Beta: [2×0 matrix]
      Covariance: [2×2 matrix]

Mdl.Submodels(3)

ans = 
  varm with properties:

     Description: "AR-Stationary 2-Dimensional VAR(2) Model"
     SeriesNames: "Y1"  "Y2" 
       NumSeries: 2
               P: 2
        Constant: [3 -3]'
              AR: {2×2 matrices} at lags [1 2]
           Trend: [2×1 vector of zeros]
            Beta: [2×0 matrix]
      Covariance: [2×2 matrix]

Mdl is a fully specified msVAR object representing a multivariate three-state Markov-switching dynamic regression model.

Create Fully Specified Model Containing Regression Component

Consider including regression components for exogenous variables in each submodel of the Markov-switching dynamic regression model in Create Fully Specified Multivariate Model.

Create a three-state discrete-time Markov chain model that describes the regime switching mechanism.

P = [10 1 1; 1 10 1; 1 1 10];
mc = dtmc(P);

For each regime, use varm to create a VARX model that describes the response process within the regime. Specify all parameter values.

% Constants (numSeries x 1 vectors)
C1 = [1;-1];
C2 = [2;-2];
C3 = [3;-3];

% Autoregression coefficients (numSeries x numSeries matrices)
AR1 = {};                            % 0 lags
AR2 = {[0.5 0.1; 0.5 0.5]};          % 1 lag
AR3 = {[0.25 0; 0 0] [0 0; 0.25 0]}; % 2 lags

% Regression coefficients (numSeries x numRegressors matrices)
Beta1 = [1;-1];           % 1 regressor
Beta2 = [2 2;-2 -2];      % 2 regressors
Beta3 = [3 3 3;-3 -3 -3]; % 3 regressors

% Innovations covariances (numSeries x numSeries matrices)
Sigma1 = [1 -0.1; -0.1 1];
Sigma2 = [2 -0.2; -0.2 2];
Sigma3 = [3 -0.3; -0.3 3];

% VARX Submodels
mdl1 = varm('Constant',C1,'AR',AR1,'Beta',Beta1,...
    'Covariance',Sigma1);
mdl2 = varm('Constant',C2,'AR',AR2,'Beta',Beta2,...
    'Covariance',Sigma2);
mdl3 = varm('Constant',C3,'AR',AR3,'Beta',Beta3,...
    'Covariance',Sigma3);

mdl1, mdl2, and mdl3 are fully specified varm objects representing the state-specified submodels.

Store the submodels in a vector with order corresponding to the regimes in mc.StateNames.

mdl = [mdl1; mdl2; mdl3];

Use msVAR to create a Markov-switching dynamic regression model from the switching mechanism mc and the state-specific submodels mdl.

Mdl = msVAR(mc,mdl)

Mdl = 
  msVAR with properties:

      NumStates: 3
      NumSeries: 2
     StateNames: ["1"    "2"    "3"]
    SeriesNames: ["1"    "2"]
         Switch: [1×1 dtmc]
      Submodels: [3×1 varm]

Mdl.Submodels(1)

ans = 
  varm with properties:

     Description: "2-Dimensional VARX(0) Model with 1 Predictor"
     SeriesNames: "Y1"  "Y2" 
       NumSeries: 2
               P: 0
        Constant: [1 -1]'
              AR: {}
           Trend: [2×1 vector of zeros]
            Beta: [2×1 matrix]
      Covariance: [2×2 matrix]

Mdl.Submodels(2)

ans = 
  varm with properties:

     Description: "AR-Stationary 2-Dimensional VARX(1) Model with 2 Predictors"
     SeriesNames: "Y1"  "Y2" 
       NumSeries: 2
               P: 1
        Constant: [2 -2]'
              AR: {2×2 matrix} at lag [1]
           Trend: [2×1 vector of zeros]
            Beta: [2×2 matrix]
      Covariance: [2×2 matrix]

Mdl.Submodels(3)

ans = 
  varm with properties:

     Description: "AR-Stationary 2-Dimensional VARX(2) Model with 3 Predictors"
     SeriesNames: "Y1"  "Y2" 
       NumSeries: 2
               P: 2
        Constant: [3 -3]'
              AR: {2×2 matrices} at lags [1 2]
           Trend: [2×1 vector of zeros]
            Beta: [2×3 matrix]
      Covariance: [2×2 matrix]

Create Partially Specified Multivariate Model for Estimation

Consider fitting to data a three-state Markov-switching model for a 2-D response process.

Create a discrete-time Markov chain model for the switching mechanism. Specify a 3-by-3 matrix of NaN values for the transition matrix. This setting indicates that you want to estimate all transition probabilities.

P = nan(3);
mc = dtmc(P);

mc is a partially specified dtmc object. The transition matrix mc.P is completely unknown and estimable.

Create 2-D VAR(0), VAR(1), and VAR(2) models by using the shorthand syntax of varm. Store the models in a vector.

mdl1 = varm(2,0);
mdl2 = varm(2,1);
mdl3 = varm(2,2);

mdl = [mdl1 mdl2 mdl3];
mdl(1)

ans = 
  varm with properties:

     Description: "2-Dimensional VAR(0) Model"
     SeriesNames: "Y1"  "Y2" 
       NumSeries: 2
               P: 0
        Constant: [2×1 vector of NaNs]
              AR: {}
           Trend: [2×1 vector of zeros]
            Beta: [2×0 matrix]
      Covariance: [2×2 matrix of NaNs]

mdl contains three state-specific varm model templates for estimation. NaN values in the properties indicate estimable parameters.

Create a Markov-switching model template from the switching mechanism mc and the state-specific submodels mdl.

Mdl = msVAR(mc,mdl)

Mdl = 
  msVAR with properties:

      NumStates: 3
      NumSeries: 2
     StateNames: ["1"    "2"    "3"]
    SeriesNames: ["1"    "2"]
         Switch: [1×1 dtmc]
      Submodels: [3×1 varm]

Mdl.Submodels(1)

ans = 
  varm with properties:

     Description: "2-Dimensional VAR(0) Model"
     SeriesNames: "Y1"  "Y2" 
       NumSeries: 2
               P: 0
        Constant: [2×1 vector of NaNs]
              AR: {}
           Trend: [2×1 vector of zeros]
            Beta: [2×0 matrix]
      Covariance: [2×2 matrix of NaNs]

Mdl.Submodels(2)

ans = 
  varm with properties:

     Description: "2-Dimensional VAR(1) Model"
     SeriesNames: "Y1"  "Y2" 
       NumSeries: 2
               P: 1
        Constant: [2×1 vector of NaNs]
              AR: {2×2 matrix of NaNs} at lag [1]
           Trend: [2×1 vector of zeros]
            Beta: [2×0 matrix]
      Covariance: [2×2 matrix of NaNs]

Mdl.Submodels(3)

ans = 
  varm with properties:

     Description: "2-Dimensional VAR(2) Model"
     SeriesNames: "Y1"  "Y2" 
       NumSeries: 2
               P: 2
        Constant: [2×1 vector of NaNs]
              AR: {2×2 matrices of NaNs} at lags [1 2]
           Trend: [2×1 vector of zeros]
            Beta: [2×0 matrix]
      Covariance: [2×2 matrix of NaNs]

Mdl is a partially specified msVAR model for estimation.

Specify Model Regression Component for Estimation

Consider including regression components for exogenous variables in the submodels of the Markov-switching dynamic regression model in Create Partially Specified Multivariate Model for Estimation. Assume that the VAR(0) model includes the regressor ${\mathit{x}}_{1\mathit{t}}$, the VAR(1) model includes the regressors ${\mathit{x}}_{1\mathit{t}}$ and ${\mathit{x}}_{2\mathit{t}}$, and the VAR(2) model includes the regressors ${\mathit{x}}_{1\mathit{t}}$, ${\mathit{x}}_{2\mathit{t}}$, and ${\mathit{x}}_{3\mathit{t}}$.

Create the discrete-time Markov chain.

P = nan(3);
mc = dtmc(P);

Create 2-D VARX(0), VARX(1), and VARX(2) models by using the shorthand syntax of varm. For each model, set the Beta property to a numSeries-by-numRegressors matrix of NaN values by using dot notation. Store all models in a vector.

numSeries = 2;

mdl1 = varm(numSeries,0);
mdl1.Beta = NaN(numSeries,1);

mdl2 = varm(numSeries,1);
mdl2.Beta = NaN(numSeries,2);

mdl3 = varm(numSeries,2);
mdl3.Beta = nan(numSeries,3);

mdl = [mdl1; mdl2; mdl3];

Create a Markov-switching dynamic regression model from the switching mechanism mc and the state-specific submodels mdl.

Mdl = msVAR(mc,mdl); 
Mdl.Submodels(2)

ans = 
  varm with properties:

     Description: "2-Dimensional VARX(1) Model with 2 Predictors"
     SeriesNames: "Y1"  "Y2" 
       NumSeries: 2
               P: 1
        Constant: [2×1 vector of NaNs]
              AR: {2×2 matrix of NaNs} at lag [1]
           Trend: [2×1 vector of zeros]
            Beta: [2×2 matrix of NaNs]
      Covariance: [2×2 matrix of NaNs]

Create Model Specifying Equality Constraints for Estimation

Consider the model in Create Partially Specified Multivariate Model for Estimation. Suppose theory dictates that states do not persist.

Create a discrete-time Markov chain model for the switching mechanism. Specify a 3-by-3 matrix of NaN values for the transition matrix. Indicate that states do not persist by setting the diagonal elements of the matrix to 0.

P = nan(3);
P(logical(eye(3))) = 0;
mc = dtmc(P);

mc is a partially specified dtmc object.

Create the submodels and store them in a vector.

mdl1 = varm(2,0);
mdl2 = varm(2,1);
mdl3 = varm(2,2);
submdl = [mdl1; mdl2; mdl3];

Create a Markov-switching dynamic regression model from the switching mechanism mc and the state-specific submodels mdl.

Mdl = msVAR(mc,submdl);
Mdl.Switch.P

ans = 3×3

     0   NaN   NaN
   NaN     0   NaN
   NaN   NaN     0

estimate treats the known diagonal elements of the transition matrix as equality constraints during estimation. For more details, see estimate.

More About

Markov-Switching Dynamic Regression Model

A Markov-switching dynamic regression model of a univariate or multivariate response series y_t describes the dynamic behavior of the series in the presence of structural breaks or regime changes. A collection of state-specific dynamic regression submodels describes the dynamic behavior of y_t within the regimes.

$$y_t~\{\begin{array}{cc}{f}_1\left(y_t;x_t,{\theta }_1\right)& ,s_t=1\\ f_2\left(y_t;x_t,{\theta }_2\right)& ,s_t=2\\ ⋮& ⋮\\ f_n\left(y_t;x_t,{\theta }_n\right)& ,s_t=n\end{array},$$

where:

• s_t is a discrete-time Markov chain representing the switching mechanism among regimes (Switch).

• n is the number of regimes (NumStates).

• $$f_i\left(y_t;x_t,{\theta }_i\right)$$ is the regime i dynamic regression model of y_t (Submodels(i)). Submodels are either univariate (ARX) or multivariate (VARX).

• x_t is a vector of observed exogenous variables at time t.

• θ_i is the regime i collection of parameters of the dynamic regression model, such as AR coefficients and the innovation variances.

Hamilton [2] proposes a general model, known as Markov-switching autoregression (MSAR), allowing for lagged values of the switching state s. Hamilton [3] shows how to convert an MSAR model into a dynamic regression model with a higher-dimensional state space, supported by msVAR.

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. "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle." Econometrica. Vol. 57, 1989, pp. 357–384.

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

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

[5] Krolzig, H.-M. Markov-Switching Vector Autoregressions. Berlin: Springer, 1997.

Version History

Introduced in R2019b