Fit a VAR(4) model to the consumer price index (CPI) and unemployment rate data. Then, infer the model innovations using the estimated 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');

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.

rcpi = price2ret(DataTable.CPIAUCSL);
unrate = DataTable.UNRATE(2:end);

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

Mdl = varm(2,4);

Estimate the model using the entire data set.

EstMdl = estimate(Mdl,[rcpi unrate]);

EstMdl is a fully specified, estimated varm model object.

Infer innovations from the estimated model.

E = infer(EstMdl,[rcpi unrate]);

E is a 241-by-2 matrix of inferred innovations. The first and second columns contain the residuals corresponding to the CPI growth rate and unemployment rate, respectively.

Alternatively, you can return residuals when you call estimate by supplying an output variable in the fourth position.

Plot the residuals on separate plots. Synchronize the residuals with the dates by removing any missing observations from the data and removing the first Mdl.P dates.

idx = all(~isnan([rcpi unrate]),2);
datesr = DataTable.Time(idx);

figure;
plot(datesr((Mdl.P + 1):end),E(:,1));
ylabel('Consumer price index');
xlabel('Date');
title('Residual plot');
hold on
plot([min(datesr) max(datesr)],[0 0],'r--');
hold off

figure;
plot(datesr((Mdl.P + 1):end),E(:,2));
ylabel('Unemployment rate');
xlabel('Date');
title('Residual plot');
hold on
plot([min(datesr) max(datesr)],[0 0],'r--');
hold off

The residuals corresponding to the CPI growth rate exhibit heteroscedasticity because the series appears to cycle through periods of higher and lower variance.

Estimate a VAR(4) model of the consumer price index (CPI), the unemployment rate, and the gross domestic product (GDP). Include a linear regression component containing the current quarter and the last four quarters of government consumption expenditures and investment (GCE). Infer model innovations.

Load the Data_USEconModel data set. Compute the real GDP.

load Data_USEconModel
DataTable.RGDP = DataTable.GDP./DataTable.GDPDEF*100;

Plot all variables on separate plots.

figure;
subplot(2,2,1)
plot(DataTable.Time,DataTable.CPIAUCSL);
ylabel('Index');
title('Consumer Price Index');
subplot(2,2,2)
plot(DataTable.Time,DataTable.UNRATE);
ylabel('Percent');
title('Unemployment Rate');
subplot(2,2,3)
plot(DataTable.Time,DataTable.RGDP);
ylabel('Output');
title('Real Gross Domestic Product');
subplot(2,2,4)
plot(DataTable.Time,DataTable.GCE);
ylabel('Billions of $');
title('Government Expenditures');

Stabilize the CPI, GDP, and GCE by converting each to a series of growth rates. Synchronize the unemployment rate series with the others by removing its first observation.

inputVariables = {'CPIAUCSL' 'RGDP' 'GCE'};
Data = varfun(@price2ret,DataTable,'InputVariables',inputVariables);
Data.Properties.VariableNames = inputVariables;
Data.UNRATE = DataTable.UNRATE(2:end);

Expand the GCE rate series to a matrix that includes its current value and up through four lagged values. Remove the GCE variable from Data.

rgcelag4 = lagmatrix(Data.GCE,0:4);
Data.GCE = [];

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

Mdl = varm(3,4);
Mdl.SeriesNames = ["rcpi" "unrate" "rgdpg"];

Estimate the model using the entire sample. Specify the GCE matrix as data for the regression component.

EstMdl = estimate(Mdl,Data.Variables,'X',rgcelag4);

Infer innovations from the estimated model. Supply the predictor data. Return the loglikelihood objective function value.

[E,logL] = infer(EstMdl,Data.Variables,'X',rgcelag4);
logL

logL = 1.7056e+03

E is a 240-by-3 matrix of inferred innovations. The columns contain the residuals corresponding to the CPI growth rate, unemployment rate, and GDP growth rate, respectively.

Plot the residuals on separate plots. Synchronize the residuals with the dates by removing any missing observations from the data and removing the first Mdl.P dates.

idx = all(~isnan([Data.Variables rgcelag4]),2);
datesr = DataTable.Time(idx);

figure;
for j = 1:Mdl.NumSeries
    subplot(2,2,j)
    plot(datesr((Mdl.P + 1):end),E(:,j));
    ylabel(Mdl.SeriesNames{j});
    xlabel('Date');
    title('Residual plot');
    hold on
    plot([min(datesr) max(datesr)],[0 0],'r--');
    hold off
end

The residuals corresponding to the CPI and GDP growth rates exhibit heteroscedasticity because the CPI series appears to cycle through periods of higher and lower variance. Also, the first half of the GDP series seems to have higher variance than the latter half.