Mark

This should be reasonably straightforward. In the simplest case you wih to
draw a random complex number in the unit circle. This is best done in polar
coordinates.

If r is a random mumber on (0,1) and theta a random number on (0, 2 Pi)
then if x=r cos(theta) and y= r sin(theta), x + i y is inside the unit
circle.  As such roots come in conjugate pairs a second is x-iy. If you
then need an odd number of roots the final can simply be a random number on
(0,1). You do not need to use a uniform distribution but can use any
distribution on the required intervals or restrain more or the eigenvalues
to be real.

John

On Sunday, 15 January 2012, Mark Leeds <markleeds2@gmail.com> wrote:
> hi  john. I think I follow you. but , in your algorithm, it is
straightforward to
> generate a set of eigenvalues with modulus less than 1 ?  thanks.
>
>
> On Sat, Jan 14, 2012 at 5:31 PM, John C Frain <frainj@gmail.com> wrote:
>
> Mark, statquant2
>
> As I understand the question it is not to test if a VAR is stable but how
to construct a VAR that is stable and automatically satisfies the condition
Mark has taken from Lutkohl. The algorithm that I have set out will
automatically satisfy that condition.The matrix that should be "estimated
by the algorithm is A on the last line of page 15 of Lutkepohl.
 Incidentally the corresponding matrix for the example on page 15 is
singular. The algorithm that I have set out will only lead to systems with
a non-singular matrix.
>
> I still don't see how a matrix generated in this way corresponds to a
real economic system.  Of course you may have some other constraints in
mind that would make the generated system correspond to something more real.
>
> John
>
> On Saturday, 14 January 2012, Mark Leeds <markleeds2@gmail.com> wrote:
>> Hi statquant2 and john: In the first chapter of Lutkepohl, it is shown
that stability f
>> a VAR(p) is the same as
>>
>> det(I_k - A1z - .... Ap Z^p )  does not equal zero for z < 1.
>>
>> where I_k - A1z - ... Ap z^p is referred to as the reverse
characteristic polynomial.
>>
>> So, statquant2,  given your A's,  one way to do it but be would be to
check the roots of the
>> polynomial implied by taking the determinant of the your polynomial.
>>
>> There's an example on pg 17 of lutkepohl if you have it. If you don't, I
can fax it to you
>> over the weekend if you want it.
>>
>>
>>
>> On Fri, Jan 13, 2012 at 8:34 PM, John C Frain <frainj@gmail.com> wrote:
>>>
>>> I think that you must approach this in a different way.
>>>
>>> 1 Draw a set of random eigenvalues with modulus < 1
>>> 2 Draw a set of random eigenvalues vectors.
>>> 3 From these you can, with some matrix manipulations, derive the
>>> corresponding Var coefficients.
>>>
>>> If your original coefficients were drawn at random I suspect that the
VAR
>>> would not be stable. I am curious about what you are trying to do.
>>>
>>> John
>>>
>>> On Friday, 13 January 2012, statquant2 <statquant@gmail.com> wrote:
>>> > Hello Paul
>>> > Thanks for the answer but my point is not how to simulate a VAR(p)
process
>>> > and check that it is stable.
>>> > My question is more how can I generate a VAR(p) such that I already
know
>>> > that it is stable.
>>> >
>>> > We know a condition that assure that it is stable (see first message)
but
>>> > this is not a condition on coefficients etc...
>>> > What I want is
>>> > generate say a 1000 random VAR(3) processes over say 500 time periods
that
>>> > will be STABLE (meaning If I run stability() all will pass the test)
>>> >
>>> > When I try to do that it seems that none of the VAR I am generating
pass
>>> > this test, so I assume that the class of stable VAR(p) is very small
>>> > compared to the whole VAR(p) process.
>>> >
>>> >
>>> >
>>> > --
>>> > View this message in context:
>>>
http://r.789695.n4.nabble.com/simulating-stable-VAR-process-tp4261177p4291835.html
>>> > Sent from the R help mailing list archive at Nabble.com.
>>> >
>>> > ______________________________________________
>>> > R-help@r-project.org mailing list
>>> > https://stat.ethz.ch/mailman/listinfo/r-help
>>> > PLEASE do read the posting guide
>>> http://www.R-project.org/posting-guide.html
>>> > and provide commented, minimal, self-contained, reproducible code.
>>> >
>>>
>>> --
>>> John C Frain
>>> Economics Department
>>> Trinity College Dublin
>>> Dublin 2
>>> Ireland
>>>

-- 
John C Frain
Economics Department
Trinity College Dublin
Dublin 2
Ireland
www.tcd.ie/Economics/staff/frainj/home.html
mailto:frainj@tcd.ie
mailto:frainj@gmail.com

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