Illustrate the relationship between
filter by 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.
For details on the variables, enter
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 of
filter, it is important to take these actions.
Set the same random number seed using
Specify the same presample response data using the
'Y0' name-value pair 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',Data);
YSim is a 100-by-4 matrix of simulated responses. Columns correspond to the columns of the variables in
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',Data);
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,
Z by the lower triangular Cholesky factor of the model covariance in
Compare the resulting responses between
(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.