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Forecast VAR Model Conditional Responses

This example shows how to forecast responses conditional on the current values of other responses in the forecast horizon. To illustrate conditional forecasting, the example models quarterly measures of the consumer price index (CPI) and the unemployment rate as a VAR(4) model.

Load the Data_USEconModel data set.

load Data_USEconModel

Plot the two series on separate plots.

figure;
plot(DataTable.Time,DataTable.CPIAUCSL);
title('Consumer Price Index');
ylabel('Index');
xlabel('Date');

figure;
plot(DataTable.Time,DataTable.UNRATE);
title('Unemployment Rate');
ylabel('Percent');
xlabel('Date');

The CPI appears to grow exponentially.

Stabilize the CPI by converting it to a series of growth rates. Synchronize the two series by removing the first observation from the unemployment rate series. Create a new data set containing the transformed variables.

rcpi = price2ret(DataTable.CPIAUCSL);
unrate = DataTable.UNRATE(2:end);
Data = array2timetable([rcpi unrate],'RowTimes',DataTable.Time(2:end),...
    'VariableNames',{'rcpi','unrate'});

Create a default VAR(4) model using the shorthand syntax.

Mdl = varm(2,4)
Mdl = 
  varm with properties:

     Description: "2-Dimensional VAR(4) Model"
     SeriesNames: "Y1"  "Y2" 
       NumSeries: 2
               P: 4
        Constant: [2×1 vector of NaNs]
              AR: {2×2 matrices of NaNs} at lags [1 2 3 ... and 1 more]
           Trend: [2×1 vector of zeros]
            Beta: [2×0 matrix]
      Covariance: [2×2 matrix of NaNs]

Mdl is a varm model object. It serves as a template for model estimation.

Fit the model to the data.

EstMdl = estimate(Mdl,Data.Variables)
EstMdl = 
  varm with properties:

     Description: "AR-Stationary 2-Dimensional VAR(4) Model"
     SeriesNames: "Y1"  "Y2" 
       NumSeries: 2
               P: 4
        Constant: [0.00171639 0.316255]'
              AR: {2×2 matrices} at lags [1 2 3 ... and 1 more]
           Trend: [2×1 vector of zeros]
            Beta: [2×0 matrix]
      Covariance: [2×2 matrix]

EstMdl is a varm model object. EstMdl is structurally the same as Mdl, but all parameters are known.

Suppose that an economist expects the unemployment rate to remain the same as the last observed rate for the next two years. Create an 8-by-2 matrix in which the first column contains NaN values and the second column contains the last observed unemployment rate.

YF = [nan(8,1) repmat(Data.unrate(end),8,1)];

Using the estimated model, forecast a path of the quarterly CPI growth rate for the next two years. Assume that the unemployment rate remains the same for the next two years. Specify the entire data set as presample observations and the known values in the forecast horizon.

Y = forecast(EstMdl,8,Data.Variables,'YF',YF)
Y = 8×2

   -0.0076    8.5000
   -0.0136    8.5000
   -0.0009    8.5000
   -0.0051    8.5000
   -0.0050    8.5000
   -0.0007    8.5000
   -0.0009    8.5000
    0.0003    8.5000

Y is an 8-by-2 array of response paths. Although the first column contains conditional forecasts, the second column consists entirely of the last observed unemployment rate.

Given that the unemployment rate is 8.5 for the next two years, the CPI growth rate is expected to shrink.

See Also

Objects

Functions

Related Topics