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
range of ARMA(2,2) parameters
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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
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
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.

 Nonnormality and collinearity are NOT problems!
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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.
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?

 Nonnormality and collinearity are NOT problems!
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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?
Have you run an ARMA in EViews and looked at the output?
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
Sample (adjusted): 1/04/2000 11/30/2011
Included observations: 4349 after adjustments
Convergence achieved after 16 iterations
MA Backcast: 1/02/2000 1/03/2000
Variable Coefficient Std. Error tStatistic Prob.
C 9.87E06 3.34E05 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
Rsquared 0.010812 Mean dependent var 2.54E05
Adjusted Rsquared 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 HannanQuinn criter. 7.245409
Fstatistic 11.86992 DurbinWatson stat 2.009067
Prob(Fstatistic) 0.000000
Inverted MA Roots .83 .98
Dependent Variable: RETURNS
Method: Least Squares
Date: 12/22/11 Time: 22:41
Sample (adjusted): 1/04/2000 11/30/2011
Included observations: 4349 after adjustments
Convergence achieved after 16 iterations
MA Backcast: 1/02/2000 1/03/2000
Variable Coefficient Std. Error tStatistic Prob.
C 9.87E06 3.34E05 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
Rsquared 0.010812 Mean dependent var 2.54E05
Adjusted Rsquared 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 HannanQuinn criter. 7.245409
Fstatistic 11.86992 DurbinWatson stat 2.009067
Prob(Fstatistic) 0.000000
Inverted MA Roots .83 .98

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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.
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 Nonnormality and collinearity are NOT problems!
 Posts: 3009
 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.
Re: range of ARMA(2,2) parameters
Okay that's why i did not obtain the AR roots.
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