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:
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:
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:
Press OK to enter the menu of VAR estimation results (main results menu) and
select VAR order criteria. Here we are asked to input :
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
.
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:
Selecting Covariance matrix of residuals from the main results menu displays the estimated residual covariance and correlation matrices.
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.
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:
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.
![]() |
(17.2) |
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.