range of ARMA(2,2) parameters

For econometric discussions not necessarily related to EViews.

Moderators: EViews Gareth, EViews Moderator

Nau2306
Posts: 74
Joined: Thu Nov 17, 2011 11:51 am

range of ARMA(2,2) parameters

Hello

Can someone please tell me within which range must the estimated parameters of an ARMA(2,2) lie for it to be stationary and invertible?

Thanks

startz
Non-normality and collinearity are NOT problems!
Posts: 2933
Joined: Wed Sep 17, 2008 2:25 pm

Re: range of ARMA(2,2) parameters

This isn't a complete answer, but...
EViews computes the inverse roots of the AR and MA polynomials. The requirement is that all the inverse roots be less than one (lie inside the unit circle).

Stationarity depends on combinations of the AR parameters. I think, but may not remember correctly, that the limts are abs(AR(1))<2, abs(ar(2))<1

Nau2306
Posts: 74
Joined: Thu Nov 17, 2011 11:51 am

Re: range of ARMA(2,2) parameters

Is there a way I can compute the range? From what I have read, the roots of the characteristic polynomial must lie outside the unit disc but with 4 parameters, it is a bit complicated to find the roots of the polynomial.

startz
Non-normality and collinearity are NOT problems!
Posts: 2933
Joined: Wed Sep 17, 2008 2:25 pm

Re: range of ARMA(2,2) parameters

Nau2306 wrote:Is there a way I can compute the range? From what I have read, the roots of the characteristic polynomial must lie outside the unit disc but with 4 parameters, it is a bit complicated to find the roots of the polynomial.

That's correct. But an ARMA(2,2) has two characteristic polynomials, one AR and one MA, each of order two. So you can solve for the roots using the quadratic formula.

Nau2306
Posts: 74
Joined: Thu Nov 17, 2011 11:51 am

Re: range of ARMA(2,2) parameters

That means I should estimate the model first, use the estimated parameters to obtain the characteristic polynomial and then solve for it. For stationarity, the roots must be greater than 1. Is that right?

startz
Non-normality and collinearity are NOT problems!
Posts: 2933
Joined: Wed Sep 17, 2008 2:25 pm

range of ARMA(2,2) parameters

No. It means that EViews solves the polynomial for you.

Have you run an ARMA in EViews and looked at the output?

Nau2306
Posts: 74
Joined: Thu Nov 17, 2011 11:51 am

Re: range of ARMA(2,2) parameters

This is the output following an ARMA(2,2) estimation. It seems to me that it gives only the MA roots but not that of the AR. From what I have read, the stationarity of an ARMA model depends solely on the AR part and thus we need only consider the characteristic polynomial of the AR.

Dependent Variable: RETURNS
Method: Least Squares
Date: 12/22/11 Time: 22:41
Convergence achieved after 16 iterations
MA Backcast: 1/02/2000 1/03/2000

Variable Coefficient Std. Error t-Statistic Prob.

C 9.87E-06 3.34E-05 0.295448 0.7677
RETURNS(-1) -0.217256 0.057540 -3.775714 0.0002
RETURNS(-2) 0.748592 0.056903 13.15568 0.0000
MA(1) 0.156197 0.050273 3.107001 0.0019
MA(2) -0.815867 0.049719 -16.40944 0.0000

R-squared 0.010812 Mean dependent var 2.54E-05
Adjusted R-squared 0.009901 S.D. dependent var 0.006483
S.E. of regression 0.006451 Akaike info criterion -7.247997
Sum squared resid 0.180780 Schwarz criterion -7.240665
Log likelihood 15765.77 Hannan-Quinn criter. -7.245409
F-statistic 11.86992 Durbin-Watson stat 2.009067
Prob(F-statistic) 0.000000

Inverted MA Roots .83 -.98

EViews Gareth
Fe ddaethom, fe welon, fe amcangyfrifon
Posts: 10574
Joined: Tue Sep 16, 2008 5:38 pm

Re: range of ARMA(2,2) parameters

That's not the output from an ARMA(2,2) estimation. That's the output from an MA(2) estimation with two lagged dependent variables. If you want the output from an ARMA(2,2) estimation, you should include some AR terms in your specification.

startz
Non-normality and collinearity are NOT problems!
Posts: 2933
Joined: Wed Sep 17, 2008 2:25 pm

range of ARMA(2,2) parameters

Well, it is an ARMA but EViews doesn't know it unless you use the AR command.

Nau2306
Posts: 74
Joined: Thu Nov 17, 2011 11:51 am

Re: range of ARMA(2,2) parameters

Okay that's why i did not obtain the AR roots.