# Simulate Responses Using filter

Illustrate the relationship between simulate and filter by estimating a 4-D 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.

For details on the variables, enter Description.

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

Mdl = varm(4,2);
Mdl.SeriesNames = DataTimeTable.Properties.VariableNames;

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

EstMdl = estimate(Mdl,DataTimeTable.Variables);

When reproducing the results of simulate and filter:

• Set the same random number seed using rng.

• Specify the same presample response data using the Y0 name-value argument.

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

rng("default")
YSim = simulate(EstMdl,100,Y0=DataTimeTable.Variables);

YSim is a 100-by-4 matrix of simulated responses. Columns correspond to the columns of the variables in Data.

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

rng("default")
Z = randn(100,4);

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

YFilter = filter(EstMdl,Z,Y0=DataTimeTable.Variables);

YFilter is a 100-by-4 matrix of simulated responses. Columns correspond to the columns of the variables in the data Data. Before filtering the disturbances, filter scales Z by the lower triangular Cholesky factor of the model covariance in EstMdl.Covariance.

Compare the resulting responses between filter and simulate.

(YSim - YFilter)'*(YSim - YFilter)
ans = 4×4

0     0     0     0
0     0     0     0
0     0     0     0
0     0     0     0

The results are identical.