Skip to content

VAR process

3 messages · Mark Leeds, John C Frain, Eric Zivot

Original
Mark Leeds
# 
I didn't respond earlier because I'm not clear on what the problem is 
with rewriting it as VAR(1) ? Lutkepohl text shows how this is done on 
pages 15 an 16 of his text. Except for the first row, the rest of the A 
matrix is composed of identity matrices. They y_t* below the first 
element play no role essentially because they are already known because 
they are in {t-1,t-2,t-3.... }. The only noise
term is the first element, u_t associated with the first element y_t.

I agree that ithe Cov is not of full rank when you write it that way but 
I don't know of any negative repurcussions of that. I think it's more of 
a tool that he uses  to show what the stability condition reduces to for 
a VAR(p) and nothing more than that. This same type of technique is used 
when
writing AR models in state space form.

Hopefully Eric or Bernhard or someone else can say more but I think it's 
just used for
deriving the stability condition in a easier way.
On Mon, Jan 26, 2009 at 9:42 PM, RON70 wrote:

            

      
      
      
    

        

      
      
    

  
  


  


    

    

      
      

        


  

        

      John C Frain
    

    

      
      #
    

  


  

      
I agree.  Reducing a VAR(p) to a VAR(1) in this way is simply a device
to generalise certain properties of a VAR(1) to a VAR(p) or possibly
to complete certain computations.  In a VAR(p) the covariance matrix
or the contemporaneous errors is in general non-diagonal.  The
Choleski decomposition was the original way of transforming the
contemporaneous variables so that the covariance of the disturbances
is diagonal and has nothing to do with the VAR(1) representation.   It
would be possible to work in terms of the VAR(1) representation but
this would be an unnecessary complication.  There are of course
various problems with this kind of analysis (e.g. uniqueness) and
structural VARs, relying on restrictions from economic theory are more
used today.

Best Regards

John

2009/1/27  <markleeds at verizon.net>:

            

      
      
      
    

        

      
      
    

            

      
      
      
    

        

      
      
    

        

  
    

      
      
    

  


  


      

    

    

      
      

        


  

        

      Eric Zivot
    

    

      
      #
    

  


  

      
The transformation of the VAR(p) to a VAR(1) is a mechanical device used to
help understand the dynamics of the model. The VAR(1) form is called the
companion form of the VAR(p) and is an equivalent representation. In the
VAR(1) form the model is first order markov and the eigenvalues of the
transition matrix determines the stability properties of the system. This is
a common trick used for systems of stochastic differential equations.
Lutkepohl's book on multivariate time series gives a very clear explanation
of this.


-----Original Message-----
From: r-sig-finance-bounces at stat.math.ethz.ch
[mailto:r-sig-finance-bounces at stat.math.ethz.ch] On Behalf Of
markleeds at verizon.net
Sent: Monday, January 26, 2009 9:38 PM
To: RON70
Cc: r-sig-finance at stat.math.ethz.ch
Subject: Re: [R-SIG-Finance] [R-sig-finance] VAR process

I didn't respond earlier because I'm not clear on what the problem is with
rewriting it as VAR(1) ? Lutkepohl text shows how this is done on pages 15
an 16 of his text. Except for the first row, the rest of the A matrix is
composed of identity matrices. They y_t* below the first element play no
role essentially because they are already known because they are in
{t-1,t-2,t-3.... }. The only noise term is the first element, u_t associated
with the first element y_t.

I agree that ithe Cov is not of full rank when you write it that way but I
don't know of any negative repurcussions of that. I think it's more of a
tool that he uses  to show what the stability condition reduces to for a
VAR(p) and nothing more than that. This same type of technique is used when
writing AR models in state space form.

Hopefully Eric or Bernhard or someone else can say more but I think it's
just used for deriving the stability condition in a easier way.

        
On Mon, Jan 26, 2009 at  9:42 PM, RON70 wrote:

        

            

      
      
      
    

        

      
      
    

        
_______________________________________________
R-SIG-Finance at stat.math.ethz.ch mailing list
https://stat.ethz.ch/mailman/listinfo/r-sig-finance
-- Subscriber-posting only.
-- If you want to post, subscribe first.