17.4 Structural Analysis
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.
17.4.1 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:
Now we have to decide about the time horizon.
Here we have chosen 12 periods which is a time span of three years
in our model.
Next we select forecast error impulse response and
orthogonalized impulse response functions in turn,
which gives the following pictures.
On the left side we see the forecast error impulse responses, on the
right side the orthogonalized impulse responses:
Not surprisingly there is no big difference between the two pictures.
This was expected since looking at the residual correlation structure in
Subsection 17.3.2 we do not see any strong correlation
between the residuals of the equations.
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:
It shows that in the long run the total impulse response from money to
money and from GNP to GNP is slightly below one and the cross total
impulse responses are close to zero.
The negative effect of GNP on money seems not very plausible.
We therefore need some measure for checking the accuracy of the
estimated impulse responses.
17.4.2 Confidence Intervals for Impulse Responses
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):
Let
be an arbitrary impulse response,
its estimate
based on a sample size
, and
a bootstrapped impulse
response.
The bootstrap confidence intervals are based on the statistics
(Efron percentile) and
(Hall Percentile).
A precise description of these confidence
intervals can be found in Efron and Tibshirani (1993) and
Hall (1992). In order to compute bootstrap confidence
intervals we have to set the number of bootstrap drawings.
Here we have set this number to 1,000 drawings.
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%.
The left picture shows the 95% Hall percentile confidence intervals for
the forecast error impulse responses.
The right picture shows the 95% Hall percentile confidence intervals for
the accumulated impulse responses.
Note that the confidence intervals are computed in a pointwise manner.
We are now able to determine which impulse response is significant.
It turns out that the response of interest to an impulse in money growth
rate and GNP growth rate (left picture) is insignificant. Furthermore,
our hypothesis about the zero impulse responses seems to be validated.
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. These models possibly better fit the data which results in better interpretation.