17.3 Specifying a VAR Model
The starting point in the Sims methodology (Sims; 1980)
is the formulation of an unrestricted VAR model.
We will specify a VAR model of order
(VAR(
)) in
the following general form:
 |
(17.1) |
where
-
is the
-dimensional vector of the
time series at time
,
-
is a vector of ones (intercept),
-
is a
-dimensional disturbance
vector with covariance matrix
,
is the parameter matrix of
,
.
To analyze a model in the form (17.1)
we need to set the model type to Full VAR after
selecting Model type in the Main Menu:
In our model
is specified with
.
The variable order in the vector
is given by the data
matrix x.
The next steps include finding a suitable model order
and estimating the model parameter matrices
.
Having done this we should conduct a residual analysis to check the whiteness
assumption.
If the residual analysis is satisfactory we can use the model for interpretation
and forecasting.
17.3.1 Process Order
We can use economic theory or information contained in the data
for specifying the model order
.
Since we have no a priori knowledge from theory we use statistical
tools for choosing an appropriate
.
The quantlib multi provides the FPE, AIC, HQ and SC criteria
(see Severini and Staniswalis; 1994, Chapter 4).
They all compare different VAR(
) models with
with respect
to some objective function.
The order
which optimizes the function is the recommended order.
Before we apply the order selection criteria we must set the highest
possible order
.
This can be difficult:
In order to avoid an optimum at the edge and to restrict
the parameter space not too much
should be reasonable large.
On the other hand
must not be too large since we need at least
presample values which reduces the sample size
and results
in unprecise estimates or worst in a model that cannot be estimated.
Since we deal with quarterly data we should
consider at least the periodicity as a possible process order. Moving
a bit ``away'' from the periodicity we set
.
In the Main Menu we select Model specification and estimation
and prepare the subsequent call to the model selection criteria:
For that we must divide the data set in a presample and sample period.
We can ignore the input fields Order and Mean adjusted (set to 0).
The presample period must contain at least
observations. Since
we have differenced the data once one more observation is 'lost'.
Therefore the Beginning of sample is set to
.
If the sample is not split appropriately an error message appears
in the output window which indicates the problem.
Press OK to enter the menu of VAR estimation results (main results menu) and
select VAR order criteria. Here we are asked to input
:
The results of the order selection criteria
will be presented in a separate window. The optimum for every
criterion is found at the minimum value:
The optimum values are FPE
, AIC
, HQ
, and
SC
.
The recommendation of the SC-criterion is quite different from the others.
Such a result is not uncommon.
For a detailed discussion about the properties of
the criteria see Lütkepohl (1993, Chapter 4).
We start our analysis with
but should keep in mind the other
possible process order.
Thus we start with a VAR(4) which is the most general model
supported by the data.
This also includes a VAR(1) by setting
.
17.3.2 Model Estimation
In order to estimate a VAR(4) we need to go back to the Main Menu
and select Model specification and estimation again.
However, this time we set the Order to 4 for estimating a VAR(4) model:
Press OK to enter the results main menu. The VAR(4) is estimated by
multivariate least squares. Next we view the estimates
and their
-values:
It can be seen that not all elements of the parameter matrices are significant
different from zero. Especially in
and
we observe only one significant
value. This could be the starting point for choosing a subset VAR where
single elements of
are restricted to zero.
Selecting Covariance matrix of residuals from the main results menu
displays the estimated residual covariance and correlation matrices.
The correlation matrix tells us something about the
contemporaneous correlation structure
in the residual vector
.
We note here that there is no correlation in
.
At a later point we will come back to the implications of this feature.
17.3.3 Model Validation
In Subsection 17.3.2 we have estimated a VAR(4)-model.
Since we did not know the ``correct'' order we used statistical tools to
find a reasonable one.
Some estimation results were presented.
Partly they are based on properties of the estimator (limiting normal
distribution) which assume certain conditions.
Whether these conditions hold is checked in this subsection.
One can think of a residual analysis, tests for nonnormality and tests
for structural change.
Here we will consider the residual analysis and
a test for nonnormality in more detail.
Checking the whiteness of the residuals is a prerequisite for drawing
valid conclusions from the
-values presented above.
If we want to compute reliable forecast intervals we need to check
the normality of the residuals in addition.
From the main results menu we select Residual Analysis which enables
us to go through the three steps of residual analysis in multi:
Individual residual analysis
First we have to select one equation.
Then we have the chance to do some transformations to the
estimated residuals
. We selected here Residuals which
leaves
untransformed.
From the following menu we present here the Plot of residuals.
We do this for all equations.
In the residual plots the unit of measurement is one standard deviation.
In other words, the residuals are normalized to have unit variance. Thus,
if many residuals exceed 2 in absolute value this may be evidence for
nonnormality or nonlinear features that are not adequately captured by
the model. Furthermore we might look for distinct patterns in the
residual plots that rule out whiteness.
Multivariate portmanteau statistic
Checking the white noise assumption for the residuals
is a central issue. Many inferential
procedures rely on this assumption.
The menu point Multivariate Portmanteau statistic
provides two tools.
Here we look at the residual vector
at time points
and
.
For these we compute the
-th autocorrelation matrix
.
White noise means zero autocorrelation for all
.
Before checking the autocorrelation functions and
carrying out an overall test we are asked to input a maximum
lag
we want to check autocorrelation for:
For the overall test to work the maximum lag of the autocorrelations
to be included must exceed the order
of the process as otherwise negative degrees of freedom of the
approximating
distribution will result. If a lag
less than or
equal to the VAR order is specified a warning is given and the
statistic is computed for the smallest feasible lag. Generally, the
approximation to the true distribution of the modified
portmanteau statistic may be inappropriate for small lags
.
At the same time we must make sure that
for obvious reasons.
Here we have chosen
. The resulting plots of the autocorrelation
functions appear.
The autocorrelation plots come along with approximate
confidence
bounds. These plots do not exhibit significant autocorrelations. Especially the
with
are much smaller than the approximate confidence
bound which is a good result since the exact confidence bound for smaller
autocorrelation lag can be much smaller than the approximate.
A test for the overall significance of the residual autocorrelations
up to lag
appears in a second display. It is the result of the
test
 |
(17.2) |
The value of the
modified portmanteau statistic
(see Lütkepohl; 1993, Chapter 4) is shown:
As expected we cannot reject
at a
significance level. This
result is in line with the shown residual autocorrelation functions above.
The residuals of our model do not exhibit autocorrelation.
Multivariate normality test
Multivariate Normality test displays the
-statistics
associated with the skewness and kurtosis of the residuals which
may be used for tests of nonnormality.
The result shows that normality of the residuals is not rejected on grounds
of the test of kurtosis and the joint test of skewness and kurtosis.
The implication is that the confidence intervals computed for the forecasts
are reliable.