The interpretation of VAR models based on
parameter matrices
is clearly restricted.
Therefore concepts and tools were developed to interpret VAR models easily.
The most important are the causality concepts,
forecast error variance decomposition and
the impulse response analysis.
In this section we will deal with the impulse response analysis.
The impulse response analysis quantifies the reaction of every single variable in the model on an exogenous shock to the model. Two special cases of shocks can be identified: The single equation shock and the joint equation shock where the shock mirrors the residual covariance structure. In the first case we investigate forecast error impulse responses, in the latter orthogonalized impulse responses. The reaction is measured for every variable a certain time after shocking the system. The impulse response analysis is therefore a tool for inspecting the inter-relation of the model variables.
We enter the impulse response analysis directly when selecting the menu point Structural Analysis in the main results menu:
When interpreting these charts we have to keep in mind that and
enter the model as
and
.
In the first row of charts we see the response of money growth rate
to a unit impulse in money growth rate, GNP growth rate, and interest,
respectively. The second and third row of charts show the response
pattern of GNP growth rate and interest.
The charts in the last row/last column are the reaction patterns we expected. They show the negative relation of interest and money/GNP. However, all money/GNP charts do not show a definite pattern. We could assume that after the initial impulse the true impulse responses are zero. A measure for checking the accuracy of the estimated impulse responses is desirable. It is provided in Subsection 17.4.2.
In our model it might be particularly interesting to analyze accumulated
impulse responses. Accumulated impulse responses at time horizon
are obtained by summing up all impulse responses from 0 to
.
Selecting this type of impulse responses function gives the following picture:
The impulse response plots in Subsection 17.4.1 are based on the model estimates in Subsection 17.3.2 and therefore also estimates. In order to make inference on statistical grounds we need some measures for the reliability of these estimates. multi provides confidence intervals. There are the asymptotic normal distribution confidence interval (Lütkepohl; 1993, Chapter 3), and two types of bootstrap confidence intervals (e.g. Benkwitz, Lütkepohl, and Wolters; 2000):
Confidence intervals based on the asymptotic normal distribution are known to fail even asymptotically in some cases (Lütkepohl; 1993). Furthermore their small sample properties might be bad (Kilian; 1998). The first problem cannot be solved by the bootstrap confidence intervals implemented in multi (Benkwitz, Lütkepohl, and Neumann; 2000). However, the second might be tackled by the bootstrap.
Here we have decided to compute confidence intervals based on a nominal coverage of 95%.
To sum up we find many insignificant and very little significant impulse responses. This might be due to preliminary data transformation and/or model choice. One might attempt to build a subset or cointegration model.Thesemodels possibly better fit the data which results in betterinterpretation.