Fit a VAR(4) model to the consumer price index (CPI) and unemployment rate data. Then, simulate responses from the estimated model.

Load theData_USEconModeldata set.

loadData_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. Create a new data set containing the transformed variables, and do not include any rows containing at least one missing observation.

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

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,Data.Variables);

EstMdlis a fully specified, estimatedvarmmodel object.

Simulate a response series path from the estimated model with length equal to the path in the data.

rng(1);% For reproducibilitynumobs = size(Data,1); Y = simulate(EstMdl,numobs);

Yis a 245-by-2 matrix of simulated responses. The first and second columns contain the simulated CPI growth rate and unemployment rate, respectively.

Plot the simulated and true responses.

figure; plot(Data.Time,Y(:,1)); holdon; plot(Data.Time,Data.rcpi) title('CPI Growth Rate'); ylabel('Growth rate'); xlabel('Date'); legend('Simulation','True')

figure; plot(Data.Time,Y(:,2)); holdon; plot(Data.Time,Data.unrate) ylabel('Percent'); xlabel('Date'); title('Unemployment Rate'); legend('Simulation','True')

Illustrate the relationship betweensimulateandfilterby estimating a 4-dimensional VAR(2) model of the four response series in Johansen's Danish data set. Simulate a single path of responses using the fitted model and the historical data as initial values, and then filter a random set of Gaussian disturbances through the estimated model using the same presample responses.

Load Johansen's Danish economic data.

loadData_JDanish

For details on the variables, enterDescription.

Create a default 4-D VAR(2) model.

Mdl = varm(4,2);

Estimate the VAR(2) model using the entire data set.

EstMdl = estimate(Mdl,Data);

When reproducing the results ofsimulateandfilter,重要的是take these actions.

Set the default random seed. Simulate 100 observations by passing the estimated model tosimulate. Specify the entire data set as the presample.

rngdefaultYSim =模拟(EstMdl, 100,'Y0',Data);

YSimis a 100-by-4 matrix of simulated responses. Columns correspond to the columns of the variables inData.

Set the default random seed. Simulate 4 series of 100 observations from the standard Gaussian distribution.

rngdefaultZ = randn(100,4);

Filter the Gaussian values through the estimated model. Specify the entire data set as the presample.

YFilter = filter(EstMdl,Z,'Y0',Data);

YFilteris a 100-by-4 matrix of simulated responses. Columns correspond to the columns of the variables in the dataData. Before filtering the disturbances,filterscalesZby the lower triangular Cholesky factor of the model covariance inEstMdl.Covariance.

Compare the resulting responses betweenfilterandsimulate.

(YSim - YFilter)'*(YSim - YFilter)

ans =0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

The results are identical.

估计消费者价格的VAR模型(4)印度x (CPI), the unemployment rate, and the gross domestic product (GDP). Include a linear regression component containing the current and the last 4 quarters of government consumption expenditures and investment. Simulate multiple paths from the estimated model.

Load theData_USEconModeldata set. Compute the real GDP.

loadData_USEconModelDataTable.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. Create a new data set containing the transformed variables.

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

Expand the GCE rate series to a matrix that includes its current value and up through four lagged values. RemoveGCEand observations containing any missing values fromData.

rgcelag4 = lagmatrix(Data.GCE,0:4); idx = all(~ismissing([Data table(rgcelag4)]),2); Data = Data(idx,:); 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);

Simulate 1000 paths from the estimated model. Specify that the length of the paths is the same as the length of the data without any missing values. Supply the predictor data. Return the innovations (scaled disturbances).

numpaths = 1000; numobs = size(Data,1); rng(1);% For reproducibility[Y,E] = simulate(EstMdl,numobs,'X',rgcelag4,'NumPaths',numpaths);

Yis a 244-by-3-by-1000 matrix of simulated responses.Eis a matrix whose dimensions correspond to the dimensions ofY, but represents the simulated, scaled disturbances. The columns correspond to the CPI growth rate, unemployment rate, and GDP growth rate, respectively.simulateapplies the same predictor data to all paths.

For each time point, compute the mean vector of the simulated responses among all paths.

MeanSim = mean(Y,3);

MeanSimis a 244-by-3 matrix containing the average of the simulated responses at each time point.

Plot the simulated responses, their averages, and the data.

figure;forj = 1:Mdl.NumSeries subplot(2,2,j) plot(Data.Time,squeeze(Y(:,j,:)),'Color',[0.8,0.8,0.8]) title(Mdl.SeriesNames{j}); holdonh1 = plot(Data.Time,Data{:,j}); h2 = plot(Data.Time,MeanSim(:,j)); holdoffendhl = legend([h1 h2],'Data','Mean'); hl.Position = [0.6 0.25 hl.Position(3:4)];