summarize

Display estimation results of vector error-correction (VEC) model

Syntax

summarize(Mdl)

results = summarize(Mdl)

Description

summarize(Mdl) displays a summary of the VEC(p – 1) model Mdl.

• If Mdl is an estimated VEC model returned by estimate, then summarize prints estimation results to the MATLAB^® Command Window. The display includes an estimation summary and a table of parameter estimates with corresponding standard errors, t statistics, and p-values. The estimation summary includes fit statistics, such as the Akaike Information Criterion (AIC), and the estimated innovations covariance and correlation matrices.

• If Mdl is an unestimated VEC model returned by vecm, then summarize prints the standard object display (the same display that vecm prints during model creation).

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

• If Mdl is an estimated VEC model, then results is a structure containing estimation results.

• If Mdl is an unestimated VEC model, then results is a vecm model object that is equal to Mdl.

Examples

Fit VEC(1) Model to Matrix of Response Data

Fit a VEC(1) model to seven macroeconomic series. Supply the response data as a numeric matrix.

Consider a VEC model for the following macroeconomic series:

• Gross domestic product (GDP)

• GDP implicit price deflator

• Paid compensation of employees

• Nonfarm business sector hours of all persons

• Effective federal funds rate

• Personal consumption expenditures

• Gross private domestic investment

Suppose that a cointegrating rank of 4 and one short-run term are appropriate, that is, consider a VEC(1) model.

Load the Data_USEconVECModel data set.

load Data_USEconVECModel

For more information on the data set and variables, enter Description at the command line.

Determine whether the data needs to be preprocessed by plotting the series on separate plots.

figure
tiledlayout(2,2)
nexttile
plot(FRED.Time,FRED.GDP);
title("Gross Domestic Product");
ylabel("Index");
xlabel("Date");
nexttile
plot(FRED.Time,FRED.GDPDEF);
title("GDP Deflator");
ylabel("Index");
xlabel("Date");
nexttile
plot(FRED.Time,FRED.COE);
title("Paid Compensation of Employees");
ylabel("Billions of $");
xlabel("Date");
nexttile
plot(FRED.Time,FRED.HOANBS);
title("Nonfarm Business Sector Hours");
ylabel("Index");
xlabel("Date");

figure
tiledlayout(2,2)
nexttile
plot(FRED.Time,FRED.FEDFUNDS)
title("Federal Funds Rate")
ylabel("Percent")
xlabel("Date")
nexttile
plot(FRED.Time,FRED.PCEC)
title("Consumption Expenditures")
ylabel("Billions of $")
xlabel("Date")
nexttile
plot(FRED.Time,FRED.GPDI)
title("Gross Private Domestic Investment")
ylabel("Billions of $")
xlabel("Date")

Stabilize all series, except the federal funds rate, by applying the log transform. Scale the resulting series by 100 so that all series are on the same scale.

FRED.GDP = 100*log(FRED.GDP);      
FRED.GDPDEF = 100*log(FRED.GDPDEF);
FRED.COE = 100*log(FRED.COE);       
FRED.HOANBS = 100*log(FRED.HOANBS); 
FRED.PCEC = 100*log(FRED.PCEC);     
FRED.GPDI = 100*log(FRED.GPDI);

Create a VEC(1) model using the shorthand syntax. Specify the variable names.

Mdl = vecm(7,4,1);
Mdl.SeriesNames = FRED.Properties.VariableNames

Mdl = 
  vecm with properties:

             Description: "7-Dimensional Rank = 4 VEC(1) Model with Linear Time Trend"
             SeriesNames: "GDP"  "GDPDEF"  "COE"  ... and 4 more
               NumSeries: 7
                    Rank: 4
                       P: 2
                Constant: [7×1 vector of NaNs]
              Adjustment: [7×4 matrix of NaNs]
           Cointegration: [7×4 matrix of NaNs]
                  Impact: [7×7 matrix of NaNs]
   CointegrationConstant: [4×1 vector of NaNs]
      CointegrationTrend: [4×1 vector of NaNs]
                ShortRun: {7×7 matrix of NaNs} at lag [1]
                   Trend: [7×1 vector of NaNs]
                    Beta: [7×0 matrix]
              Covariance: [7×7 matrix of NaNs]

Mdl is a vecm model object. All properties containing NaN values correspond to parameters to be estimated given data.

Estimate the model using the entire data set and the default options.

EstMdl = estimate(Mdl,FRED.Variables)

EstMdl = 
  vecm with properties:

             Description: "7-Dimensional Rank = 4 VEC(1) Model"
             SeriesNames: "GDP"  "GDPDEF"  "COE"  ... and 4 more
               NumSeries: 7
                    Rank: 4
                       P: 2
                Constant: [14.1329 8.77841 -7.20359 ... and 4 more]'
              Adjustment: [7×4 matrix]
           Cointegration: [7×4 matrix]
                  Impact: [7×7 matrix]
   CointegrationConstant: [-28.6082 -109.555 77.0912 ... and 1 more]'
      CointegrationTrend: [4×1 vector of zeros]
                ShortRun: {7×7 matrix} at lag [1]
                   Trend: [7×1 vector of zeros]
                    Beta: [7×0 matrix]
              Covariance: [7×7 matrix]

EstMdl is an estimated vecm model object. It is fully specified because all parameters have known values. By default, estimate imposes the constraints of the H1 Johansen VEC model form by removing the cointegrating trend and linear trend terms from the model. Parameter exclusion from estimation is equivalent to imposing equality constraints to zero.

Display a short summary from the estimation.

results = summarize(EstMdl)

results = struct with fields:
               Description: "7-Dimensional Rank = 4 VEC(1) Model"
                     Model: "H1"
                SampleSize: 238
    NumEstimatedParameters: 112
             LogLikelihood: -1.4939e+03
                       AIC: 3.2118e+03
                       BIC: 3.6007e+03
                     Table: [133x4 table]
                Covariance: [7x7 double]
               Correlation: [7x7 double]

The Table field of results is a table of parameter estimates and corresponding statistics.

Compare Several VEC Model Fits

Consider the model and data in Fit VEC(1) Model to Matrix of Response Data and these four alternative VEC models: VEC(0), VEC(1), VEC(3), and VEC(7). Using historical data, estimate each of the four models, and then compare the model fits using the resulting Bayesian Information Criterion (BIC).

Load the Data_USEconVECModel data set and preprocess the data.

load Data_USEconVECModel
FRED.GDP = 100*log(FRED.GDP);      
FRED.GDPDEF = 100*log(FRED.GDPDEF);
FRED.COE = 100*log(FRED.COE);       
FRED.HOANBS = 100*log(FRED.HOANBS); 
FRED.PCEC = 100*log(FRED.PCEC);     
FRED.GPDI = 100*log(FRED.GPDI);

Within a loop:

• Create a VEC model using the shorthand syntax.

• Estimate the VEC Model. Reserve the maximum value of p as presample observations.

• Store the estimation results.

numlags = [0 1 3 7];
p = numlags + 1;
Y0 = FRED{1:max(p),:};
Y = FRED{((max(p) + 1):end),:};

for j = 1:numel(p)
    Mdl = vecm(7,4,numlags(j));
    EstMdl = estimate(Mdl,Y,'Y0',Y);
    results(j) = summarize(EstMdl);
end

results is a 4-by-1 structure array containing the estimation results of each model.

Extract the BIC from each set of results.

BIC = [results.BIC]

BIC = 1×4
103 ×

    5.3948    5.4372    5.8254    6.5536

The model corresponding to the lowest BIC has the best fit among the models considered. Therefore, the VEC(0) model is the best fitting model.

Input Arguments

Mdl — VEC model

vecm model object

VEC model, specified as a vecm model object returned by estimate or vecm.

Output Arguments

results — Model summary

structure array | vecm model object

Model summary, returned as a structure array or a vecm model object.

• If Mdl is an estimated VEC model, then results is a structure array containing the fields in this table.

  Field	Description
  Description	Model summary description (string)
  Model	Johansen model of deterministic terms ("H2", "H1*", "H1", "H*", "H") [1]
  SampleSize	Effective sample size (numeric scalar)
  NumEstimatedParameters	Number of estimated parameters (numeric scalar)
  LogLikelihood	Optimized loglikelihood value (numeric scalar)
  AIC	Akaike Information Criterion (numeric scalar)
  BIC	Bayesian Information Criterion (numeric scalar)
  Table	Parameter estimates with corresponding standard errors, t statistics (estimate divided by standard error), and p-values (assuming normality); a table with rows corresponding to model parameters
  Covariance	Estimated residual covariance matrix (the maximum likelihood estimate), an Mdl.NumSeries-by-Mdl.NumSeries numeric matrix with rows and columns corresponding to the innovations in the response equations ordered by the columns of Y
  Correlation	Estimated residual correlation matrix whose dimensions correspond to the dimensions of Covariance

  summarize uses mvregress to implement multivariate normal, maximum likelihood estimation. For more details on estimates and standard errors, see Estimation of Multivariate Regression Models.

• If Mdl is an unestimated VEC model, then results is a vecm model object that is equal to Mdl.

References

[1] Johansen, S. Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford: Oxford University Press, 1995.

Version History

Introduced in R2017b