summarize

Summarize Markov-switching dynamic regression model estimation results

Since R2021b

Syntax

summarize(Mdl)

summarize(Mdl,state)

results = summarize(___)

Description

summarize(Mdl) displays a summary of the Markov-switching dynamic regression model Mdl.

• If Mdl is an estimated model returned by estimate, then summarize displays estimation results to the MATLAB^® Command Window. The display includes:

  • A model description

  • Estimated transition probabilities

  • Fit statistics, which include the effective sample size, number of estimated submodel parameters and constraints, loglikelihood, and information criteria (AIC and BIC)

  • A table of submodel estimates and inferences, which includes coefficient estimates with standard errors, t-statistics, and p-values.

• If Mdl is an unestimated Markov-switching model returned by msVAR, summarize prints the standard object display (the same display that msVAR prints during model creation).

summarize(Mdl,state) displays only summary information for the submodel with name state.

results = summarize(___) returns one of the following variables and does not print to the Command Window.

• If Mdl is an estimated Markov-switching model, results is a table containing the submodel estimates and inferences.

• If Mdl is an unestimated model, results is an msVAR object that is equal to Mdl.

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:

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

2. Annualize 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: "1-Dimensional VAR(0) Model"
     SeriesNames: "Y1" 
       NumSeries: 1
               P: 0
        Constant: 4.90146
              AR: {}
           Trend: 0
            Beta: [1×0 matrix]
      Covariance: 12.087

EstMdlRec = EstMdl.Submodels(2)

EstMdlRec = 
  varm with properties:

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

Display the estimated state transition matrix.

EstP = EstMdl.Switch.P

EstP = 2×2

    0.9088    0.0912
    0.2303    0.7697

Display an estimation summary containing parameter estimates and inferences.

summarize(EstMdl)

Description
1-Dimensional msVAR Model with 2 Submodels

Switch
Estimated Transition Matrix:
0.909 0.091 
0.230 0.770 

Fit
Effective Sample Size: 228 
Number of Estimated Parameters: 2 
Number of Constrained Parameters: 0 
LogLikelihood: -639.496 
AIC: 1282.992 
BIC: 1289.851 

Submodels
                           Estimate     StandardError    TStatistic      PValue   
                           _________    _____________    __________    ___________

    State 1 Constant(1)       4.9015       0.23023          21.289     1.4301e-100
    State 2 Constant(1)    0.0084884        0.2359        0.035983          0.9713

Display Estimation Summary Separately for Each State

Create the following fully specified Markov-switching model the DGP.

• State transition matrix: $$P=\left[\begin{array}{ccc}0.5& 0.2& 0.3\\ 0.2& 0.6& 0.2\\ 0.2& 0.1& 0.7\end{array}\right]$$.

• State 1: $$\left[\begin{array}{c}{y}_{1,t}\\ y_{2,t}\end{array}\right]=\left[\begin{array}{c}-1\\ -1\end{array}\right]+\left[\begin{array}{cc}-0.5& 0.1\\ 0.2& -0.75\end{array}\right]\left[\begin{array}{c}{y}_{1,t-1}\\ y_{2,t-1}\end{array}\right]+{\epsilon }_{1,t}$$, where ${\epsilon }_{1,\mathit{t}}\sim {\mathit{N}}_2\left(\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{cc}0.5& 0\\ 0& 1\end{array}\right]\right)$.

• State 2: $$\left[\begin{array}{c}{y}_{1,t}\\ y_{2,t}\end{array}\right]=\left[\begin{array}{c}-1\\ 2\end{array}\right]+{\epsilon }_{2,t}$$, where ${\epsilon }_{2,\mathit{t}}\sim {\mathit{N}}_2\left(\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\right)$.

• State 3:$$\left[\begin{array}{c}{y}_{1,t}\\ y_{2,t}\end{array}\right]=\left[\begin{array}{c}1\\ 2\end{array}\right]+\left[\begin{array}{cc}0.5& 0.1\\ 0.2& 0.75\end{array}\right]\left[\begin{array}{c}{y}_{1,t-1}\\ y_{2,t-1}\end{array}\right]+{\epsilon }_{3,t}$$, where ${\epsilon }_{3,\mathit{t}}\sim {\mathit{N}}_2\left(\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{cc}1& -0.1\\ -0.1& 2\end{array}\right]\right)$.

PDGP = [0.5 0.2 0.3; 0.2 0.6 0.2; 0.2 0.1 0.7];
mcDGP = dtmc(PDGP);

constant1 = [-1; -1];
constant2 = [-1; 2];
constant3 = [1; 2];
AR1 = [-0.5 0.1; 0.2 -0.75];
AR3 = [0.5 0.1; 0.2 0.75];
Sigma1 = [0.5 0; 0 1];
Sigma2 = eye(2);
Sigma3 = [1 -0.1; -0.1 2];

mdl1DGP = varm(Constant=constant1,AR={AR1},Covariance=Sigma1);
mdl2DGP = varm(Constant=constant2,Covariance=Sigma2);
mdl3DGP = varm(Constant=constant3,AR={AR3},Covariance=Sigma3);
mdlDGP = [mdl1DGP; mdl2DGP; mdl3DGP];

MdlDGP = msVAR(mcDGP,mdlDGP);

Generate a random response path of length 1000 from the DGP.

rng(1) % For reproducibiliy
Y = simulate(MdlDGP,1000);

Create a partially specified Markov-switching model that has the same structure as the DGP, but the transition matrix, and all submodel coefficients and innovations covariance matrices are unknown and estimable.

mc = dtmc(nan(3));
mdlar = varm(2,1);
mdlc = varm(2,0);
Mdl = msVAR(mc,[mdlar; mdlc; mdlar]);

Initialize the estimation procedure by fully specifying a Markov-switching model that has the same structure as Mdl, but has the following parameter values:

• A randomly drawn transition matrix

• Randomly drawn contant vectors for each model

• AR self lags of 0.1 and cross lags of 0

• The identify matrix for the innovations covariance

P0 = randi(10,3,3);
mc0 = dtmc(P0);
constant01 = randn(2,1);
constant02 = randn(2,1);
constant03 = randn(2,1);
AR0 = 0.1*eye(2);
Sigma0 = eye(2);
mdl01 = varm(Constant=constant01,AR={AR0},Covariance=Sigma0);
mdl02 = varm(Constant=constant02,Covariance=Sigma0);
mdl03 = varm(Constant=constant03,AR={AR0},Covariance=Sigma0);
submdl0 = [mdl01; mdl02; mdl03];
Mdl0 = msVAR(mc0,submdl0);

Fit the Markov-switching model to the simulated series. Plot the loglikelihood after each iteration of the EM algorithm.

EstMdl = estimate(Mdl,Mdl0,Y,IterationPlot=true);

The plot displays the evolution of the loglikelihood with increasing iterations of the EM algorithm. The procedure terminates when one of the stopping criteria is satisfied.

Display an estimation summary of the model.

summarize(EstMdl)

Description
2-Dimensional msVAR Model with 3 Submodels

Switch
Estimated Transition Matrix:
0.501 0.245 0.254 
0.204 0.549 0.247 
0.188 0.102 0.710 

Fit
Effective Sample Size: 999 
Number of Estimated Parameters: 14 
Number of Constrained Parameters: 0 
LogLikelihood: -3634.005 
AIC: 7296.010 
BIC: 7364.704 

Submodels
                           Estimate    StandardError    TStatistic      PValue   
                           ________    _____________    __________    ___________

    State 1 Constant(1)    -0.98929       0.023779       -41.603                0
    State 1 Constant(2)     -1.0884       0.030164       -36.083      4.1957e-285
    State 1 AR{1}(1,1)     -0.48446        0.01547       -31.316      2.8121e-215
    State 1 AR{1}(2,1)       0.1835       0.019624        9.3509       8.6868e-21
    State 1 AR{1}(1,2)     0.083953      0.0070162        11.966       5.3839e-33
    State 1 AR{1}(2,2)     -0.72972      0.0089002       -81.989                0
    State 2 Constant(1)     -0.9082       0.030103        -30.17      5.9064e-200
    State 2 Constant(2)      1.9514       0.030483        64.016                0
    State 3 Constant(1)      1.1212       0.044427        25.237      1.5818e-140
    State 3 Constant(2)      1.9561         0.0593        32.986      1.2831e-238
    State 3 AR{1}(1,1)      0.48965       0.023149        21.152       2.6484e-99
    State 3 AR{1}(2,1)      0.22688       0.030899        7.3427       2.0936e-13
    State 3 AR{1}(1,2)     0.095847       0.012005        7.9838       1.4188e-15
    State 3 AR{1}(2,2)      0.72766       0.016024         45.41                0

Display an estimation summary separately for each state.

summarize(EstMdl,1)

Description
2-Dimensional VAR Submodel, State 1

Submodel
                           Estimate    StandardError    TStatistic      PValue   
                           ________    _____________    __________    ___________

    State 1 Constant(1)    -0.98929       0.023779       -41.603                0
    State 1 Constant(2)     -1.0884       0.030164       -36.083      4.1957e-285
    State 1 AR{1}(1,1)     -0.48446        0.01547       -31.316      2.8121e-215
    State 1 AR{1}(2,1)       0.1835       0.019624        9.3509       8.6868e-21
    State 1 AR{1}(1,2)     0.083953      0.0070162        11.966       5.3839e-33
    State 1 AR{1}(2,2)     -0.72972      0.0089002       -81.989                0

summarize(EstMdl,2)

Description
2-Dimensional VAR Submodel, State 2

Submodel
                           Estimate    StandardError    TStatistic      PValue   
                           ________    _____________    __________    ___________

    State 2 Constant(1)    -0.9082       0.030103         -30.17      5.9064e-200
    State 2 Constant(2)     1.9514       0.030483         64.016                0

summarize(EstMdl,3)

Description
2-Dimensional VAR Submodel, State 3

Submodel
                           Estimate    StandardError    TStatistic      PValue   
                           ________    _____________    __________    ___________

    State 3 Constant(1)      1.1212      0.044427         25.237      1.5818e-140
    State 3 Constant(2)      1.9561        0.0593         32.986      1.2831e-238
    State 3 AR{1}(1,1)      0.48965      0.023149         21.152       2.6484e-99
    State 3 AR{1}(2,1)      0.22688      0.030899         7.3427       2.0936e-13
    State 3 AR{1}(1,2)     0.095847      0.012005         7.9838       1.4188e-15
    State 3 AR{1}(2,2)      0.72766      0.016024          45.41                0

Return Estimation Summary Table

Consider the model for the US GDP growth rate in Estimate Markov-Switching Dynamic Regression Model.

Create a Markov-switching dynamic regression model for the naive estimator.

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

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,'StateNames',Mdl.StateNames);
mdl01 = arima('Constant',1,'Variance',1);
mdl02 = arima('Constant',-1,'Variance',1);
Mdl0 = msVAR(mc0,[mdl01; mdl02]);

Load the US GDP data set. Preprocess the data.

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

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);

Return an estimation summary table.

results = summarize(EstMdl)

results=2×4 table
                           Estimate     StandardError    TStatistic      PValue   
                           _________    _____________    __________    ___________

    State 1 Constant(1)       4.9015       0.23023          21.289     1.4301e-100
    State 2 Constant(1)    0.0084884        0.2359        0.035983          0.9713

results is a table containing estimates and inferences for all submodel coefficients.

Identify significant coefficient estimates.

results.Properties.RowNames(results.PValue < 0.05)

ans = 1×1 cell array
    {'State 1 Constant(1)'}

Input Arguments

Mdl — Markov-switching dynamic regression model

msVAR object

Markov-switching dynamic regression model, specified as an msVAR object returned by estimate or msVAR.

state — State to summarize

integer in 1:Mdl.NumStates (default) | state name in Mdl.StateNames

State to summarize, specified as an integer in 1:Mdl.NumStates or a state name in Mdl.StateNames.

The default summarizes all states.

Example: summarize(Mdl,3) summarizes the third state in Mdl.

Example: summarize(Mdl,"Recession") summarizes the state labeled "Recession" in Mdl.

Data Types: double | char | string

Output Arguments

results — Model summary

table | msVAR object

Model summary, returned as a table or an msVAR object.

• If Mdl is an estimated Markov-switching model returned by estimate, results is a table of summary information for the submodel parameter estimates. Each row corresponds to a submodel coefficient. Columns correspond to the estimate (Estimate), standard error (StandardError), t-statistic (TStatistic), and the p-value (PValue).

  When the summary includes all states (the default), results.Properties stores the following fit statistics:

  Field	Description
  Description	Model summary description (character vector)
  EffectiveSampleSize	Effective sample size (numeric scalar)
  NumEstimatedParameters	Number of estimated parameters (numeric scalar)
  NumConstraints	Number of equality constraints (numeric scalar)
  LogLikelihood	Optimized loglikelihood value (numeric scalar)
  AIC	Akaike information criterion (numeric scalar)
  BIC	Bayesian information criterion (numeric scalar)

• If Mdl is an unestimated model, results is an msVAR object that is equal to Mdl.

Note

When results is a table, it contains only submodel parameter estimates:

• Mdl.Switch contains estimated transition probabilities.

• Mdl.Submodels(j).Covariance contains the estimated residual covariance matrix of state j. For details, see msVAR.

Algorithms

estimate implements a version of Hamilton's Expectation-Maximization (EM) algorithm, as described in [3]. The standard errors, loglikelihood, and information criteria are conditional on optimal parameter values in the estimated transition matrix Mdl.Switch. In particular, standard errors do not account for variation in estimated transition probabilities.

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.

Version History

Introduced in R2021b