[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 
> recently told me about the ARMA-GARCH problem I had. 
> 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 

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 

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 
models (like in your case).

Hope this helps,

Riccardo (Jack) Lucchetti
Dipartimento di Economia

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

r.lucchetti at univpm.it

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