summarize
Summarize Markov-switching dynamic regression model estimation results
Since R2021b
Description
summarize(
displays a summary of the
Markov-switching dynamic regression model Mdl
)Mdl
.
If
Mdl
is an estimated model returned byestimate
, thensummarize
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 bymsVAR
,summarize
prints the standard object display (the same display thatmsVAR
prints during model creation).
returns
one of the following variables and does not print to the Command Window.results
= summarize(___)
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 anmsVAR
object that is equal toMdl
.
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:
Convert the data to a quarterly rate within the estimation period.
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: .
State 1: , where .
State 2: , where .
State 3:, where .
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 = 1x1 cell array
{'State 1 Constant(1)'}
Input Arguments
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 byestimate
,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 anmsVAR
object that is equal toMdl
.
Note
When results
is a table, it contains only submodel parameter estimates:
Mdl.Switch
contains estimated transition probabilities.Mdl.Submodels(
contains the estimated residual covariance matrix of statej
).Covariance
. For details, seej
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
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