# [Gretl-users] AR(7)-GARCH(1,1)-t Roots

Riccardo (Jack) Lucchetti r.lucchetti at univpm.it
Sat Mar 31 08:30:32 EDT 2012

```On Sat, 31 Mar 2012, Daniel Bencik wrote:

> Dear all,
>
> I tried to google on the internet what the inverted roots problem is, as you
> Following up our discussion, the "best" model for my problem (under given
> circumstances) is AR(7)-GARCH(1,1)-t but the common inverse roots show up
> again. Can you please tell me what this means?
>
> The Eviews estimation result is http://eubie.sweb.cz/gretl_forum/ar7.PNG

There's no such a thing as an "inverted roots" problem. The problem I
mentioned in my earlier message was the common roots problem, which may
arise when you have both an AR part and an MA part. A classic reference is
Mcleod (1999), "Necessary and Sufficient Condition for Nonsingular Fisher
Information Matrix in ARMA and Fractional ARIMA Models", The American
Statistician, although the problem was known long before that.

In short: suppose you have a difference equation

A(L) y_t = C(L) x_t

(of which an arma model is a special case, with x_t a white noise
process), where the order of A() is p and the order of C() is q. Of course
the above equality still holds if you apply the polynomial (1-bL) on both
sides of the equation, whatever the value of b.

This means that, for each value of b, you have an equivalent
representation

P(L) y_t = Q(L) x_t

where P(L) = A(L) (1-bL) and Q(L) (1-bL). In turn, this means that for
every ARMA(p,q) process there exists an uncountable infinity of equivalent
ARMA(p+1, q+1) representations. These representations are called
"redundant".

If you read the argument backwards, when you apply an ARMA(p,q) model to a
data series, you may have common factors between the A() and C()
polynomials: if so, then your representation is just one of the possible
equivalent redundant representations and you'll be much better of with an
ARMA(p-1, q-1) model.

Of course, all this does not apply, for obvious reasons, to pure AR or MA

Hope this helps,

Riccardo (Jack) Lucchetti
Dipartimento di Economia

Università Politecnica delle Marche
(formerly known as Università di Ancona)

r.lucchetti at univpm.it
http://www2.econ.univpm.it/servizi/hpp/lucchetti
```