How to specify ARMA model correctly?

[petrenko]

I am new to statistics. I would like to estimate ARMA(4,4) model. When I estimated this model in EViews-6, I got the following results:

Method: Least Squares
Date: 11/14/12 Time: 13:36
Sample (adjusted): 2011M08 2012M09
Included observations: 14 after adjustments
Convergence achieved after 31 iterations
MA Backcast: 2011M04 2011M07

Variable Coefficient Std. Error t-Statistic Prob.

C 702499.3 26664.66 26.34570 0.0000
AR(1) -0.060124 0.280320 -0.214482 0.8386
AR(2) -0.764413 0.220084 -3.473279 0.0178
AR(3) 0.054642 0.163998 0.333187 0.7525
AR(4) -0.666453 0.181979 -3.662243 0.0146
MA(1) 0.354147 0.081850 4.326794 0.0075
MA(2) 0.268601 0.357232 0.751896 0.4860
MA(3) 1.006474 0.080253 12.54125 0.0001
MA(4) 0.153277 0.387849 0.395198 0.7090

R-squared 0.913361 Mean dependent var 726858.6
Adjusted R-squared 0.774738 S.D. dependent var 201770.6
S.E. of regression 95763.96 Akaike info criterion 26.03326
Sum squared resid 4.59E+10 Schwarz criterion 26.44408
Log likelihood -173.2328 Hannan-Quinn criter. 25.99523
F-statistic 6.588814 Durbin-Watson stat 1.658332
Prob(F-statistic) 0.026367

Inverted AR Roots .45-.75i .45+.75i -.48-.80i -.48+.80i
Inverted MA Roots .39-.92i .39+.92i -.16 -.97

Then I did tests of redundant variables, etc. that showed me that I can safely "delete" some explanatory variables. So, I left with the following equation:

Method: Least Squares
Date: 11/14/12 Time: 14:19
Sample (adjusted): 2011M08 2012M09
Included observations: 14 after adjustments
Convergence achieved after 16 iterations
MA Backcast: 2011M05 2011M07

Variable Coefficient Std. Error t-Statistic Prob.

C 705838.5 16472.49 42.84954 0.0000
AR(2) -0.798345 0.150040 -5.320880 0.0003
AR(4) -0.642053 0.128967 -4.978435 0.0006
MA(3) 0.955344 0.030998 30.81945 0.0000

R-squared 0.863006 Mean dependent var 726858.6
Adjusted R-squared 0.821908 S.D. dependent var 201770.6
S.E. of regression 85149.11 Akaike info criterion 25.77715
Sum squared resid 7.25E+10 Schwarz criterion 25.95974
Log likelihood -176.4401 Hannan-Quinn criter. 25.76025
F-statistic 20.99865 Durbin-Watson stat 1.035324
Prob(F-statistic) 0.000123

Inverted AR Roots .45+.77i .45-.77i -.45+.77i -.45-.77i
Inverted MA Roots .49+.85i .49-.85i -.98

Concerning to this, I have a question: do I have to specify ARMA (4,4) with all ARs and MAs, or I can keep only the variables that have high explanation power (in my case AR(2), AR(4) and MA(4)), is it theoretically correct?

Sincerely yours,
P.