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varm

Convert vector error-correction (VEC) model to vector autoregression (VAR) model

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Syntax

VARMdl = varm(Mdl)

Description

VARMdl = varm(Mdl) converts the VEC(p – 1) model Mdl to its equivalent VAR(p) model representation VARMdl.

example

Examples

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Open Live Script

Consider a VEC model for the following seven 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;
subplot(2,2,1)
plot(FRED.Time,FRED.GDP);
title('Gross Domestic Product');
ylabel('Index');
xlabel('Date');
subplot(2,2,2)
plot(FRED.Time,FRED.GDPDEF);
title('GDP Deflator'); 
ylabel('Index');
xlabel('Date');
subplot(2,2,3)
plot(FRED.Time,FRED.COE);
title('Paid Compensation of Employees');
ylabel('Billions of $');
xlabel('Date');
subplot(2,2,4)
plot(FRED.Time,FRED.HOANBS);
title('Nonfarm Business Sector Hours');
ylabel('Index');
xlabel('Date');

Figure contains 4 axes objects. Axes object 1 with title Gross Domestic Product, xlabel Date, ylabel Index contains an object of type line. Axes object 2 with title GDP Deflator, xlabel Date, ylabel Index contains an object of type line. Axes object 3 with title Paid Compensation of Employees, xlabel Date, ylabel Billions of $ contains an object of type line. Axes object 4 with title Nonfarm Business Sector Hours, xlabel Date, ylabel Index contains an object of type line.

figure;
subplot(2,2,1)
plot(FRED.Time,FRED.FEDFUNDS);
title('Federal Funds Rate');
ylabel('Percent');
xlabel('Date');
subplot(2,2,2)
plot(FRED.Time,FRED.PCEC);
title('Consumption Expenditures');
ylabel('Billions of $');
xlabel('Date');
subplot(2,2,3)
plot(FRED.Time,FRED.GPDI);
title('Gross Private Domestic Investment');
ylabel('Billions of $');
xlabel('Date');

Figure contains 3 axes objects. Axes object 1 with title Federal Funds Rate, xlabel Date, ylabel Percent contains an object of type line. Axes object 2 with title Consumption Expenditures, xlabel Date, ylabel Billions of $ contains an object of type line. Axes object 3 with title Gross Private Domestic Investment, xlabel Date, ylabel Billions of $ contains an object of type line.

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.

Convert the estimated VEC(1) model to its equivalent VAR(2) model representation.

VARMdl = varm(EstMdl)
VARMdl = 
  varm with properties:

     Description: "AR-Nonstationary 7-Dimensional VAR(2) Model"
     SeriesNames: "GDP"  "GDPDEF"  "COE"  ... and 4 more
       NumSeries: 7
               P: 2
        Constant: [14.1329 8.77841 -7.20359 ... and 4 more]'
              AR: {7×7 matrices} at lags [1 2]
           Trend: [7×1 vector of zeros]
            Beta: [7×0 matrix]
      Covariance: [7×7 matrix]

VARMdl is a varm model object.

Input Arguments

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VEC model, specified as a vecm model object created by vecm or estimate. Mdl must be fully specified.

Output Arguments

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VAR model equivalent, returned as a varm model object.

Algorithms

Consider the m-D VEC(p – 1) model using lag operator notation.

(1−L)yt=c+dt+Πyt−1+∑j=1p−1Φj(1−L)yt−j+βxt+εt.

  • yt is an m-by-1 vector of values corresponding to m response variables at time t, where t = 1,...,T.

  • Lyt = yt – 1.

  • c is the overall constant.

  • d is the overall time trend coefficient.

  • Π is an m-by-m impact matrix with a rank of r.

  • xt is a k-by-1 vector of values corresponding to k exogenous predictor variables.

  • β is an m-by-k matrix of regression coefficients.

  • εt is an m-by-1 vector of random Gaussian innovations, each with a mean of 0 and collectively an m-by-m covariance matrix Σ. For t ≠ s, εt and εs are independent.

  • Φj is an m-by-m matrix of short-run coefficients.

The equivalent VAR(p) model in difference equation notation is

yt=c+dt+∑j=1pΓjyt−j+βxt+εt.

Γj is an m-by-m matrix of autoregressive coefficients.

Version History

Introduced in R2017b

See Also

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