ARMA Parameter Estimation - Free Statistics and Forecasting Software (Calculators)

:: ARMA Parameter Estimation ::

The non-commercial (academic) use of this software is free of charge. The only thing that is asked in return is to cite this work when results are used in publications.

:: ARMA Estimation Algorithm 1.2 ::

:: searching starting values ::

Backforecasting (values listed snake-wise)
-0.00932602	-0.007103194	0.001478359	0.00342012	-0.000984232
-1.5822E-5	-0.00370135	-0.005298879	6.5089E-5	0.002497791
-0.000500922	-0.003321091	0.000334671	0
14 backforecasts generated.

Iteration 1
MA(1)	0.3483
SMA(1)	0.5039

Iteration 2
MA(1)	0.3811
SMA(1)	0.5949

Iteration 3
MA(1)	0.382
SMA(1)	0.5704

:: estimation of parameters ::

Backforecasting (values listed snake-wise)
-0.031840825	-0.01799405	-0.000889203	0.004480612	0.007180302
-0.00357537	-0.010944531	-0.010180168	-0.026380782	0.016813962
-0.012122755	0.002835424	0.000318657	6.5466E-5	6.7421E-5
-2.5579E-5	7.925E-5	-1.9635E-5	6.4236E-5	-0.000148487
3.7633E-5	3.271E-5	9.5891E-5	3.5241E-5	0.000124057
-3.1258E-5	-4.1497E-5	7.7586E-5	-0.000180736	5.4165E-5
5.4724E-5	1.7874E-5	5.0625E-5	-3.9141E-5	-8.8662E-5
-5.0001E-5	-4.0153E-5	-5.7447E-5	-1.804E-5	-8.859E-6
-3.0925E-5	5.4233E-5	-5.641E-6	2.392E-5	8.692E-6
-2.3755E-5	7.5143E-5	-4.7868E-5	1.6904E-5	-8.8354E-5
5.3656E-5	-4.3615E-5	-5.9244E-5	6.4398E-5	-0.000105146
1.7868E-5	2.7268E-5	-4.7076E-5	-3.722E-6	2.9122E-5
-7.8614E-5	8.9204E-5	-5.7183E-5	-1.625E-5	3.8514E-5
-4.1123E-5	2.2587E-5	7.6492E-5	-6.2036E-5	9.6232E-5
-0.000200771	0.000100099	9.2324E-5	2.2734E-5	8.7685E-5
-1.6561E-5	3.247E-5	-0.000139023	0.000118079	0.000140402
0.000128771	-7.062E-5	8.548E-5	-0.000188675	9.822E-6
-2.3723E-5	-0.000125248	-6.0841E-5	1.9727E-5	-4.9018E-5
6.6779E-5	-0.000266916	-8.5497E-5	0.000137315	0.000266298
-0.000118733	-4.5368E-5	-1.9124E-5	-2.485E-6	8.5138E-5
-0.00015773	0.000116396	-0.000134028	0.000323239	-9.9081E-5
5.4204E-5	-0.000234107	3.9364E-5	-1.3134E-5	-0.000178051
0.00011479	0.000102694	-1.2558E-5	-6.71E-7	-4.7465E-5
-0.000326014	0.000300186	-0.00010321	0.000134379	-0.000132903
0.000139379	0.000181518	-4.5367E-5	-8.9783E-5	2.5465E-5
-6.7E-7	9.1205E-5	0.000151592	-3.0456E-5	-4.7845E-5
1.59E-7	8.9479E-5	-7.3292E-5	-4.142E-5	-2.046E-6
1.0314E-5	1.6527E-5	-8.23E-6	-2.5193E-5	-2.3434E-5
-6.0722E-5	3.8702E-5	-2.7904E-5	6.526E-6
144 backforecasts generated.

Iteration 4
MA(1)	0.3954
SMA(1)	0.6148

Iteration 5
MA(1)	0.395
SMA(1)	0.602

FINAL PARAMETER ESTIMATES ON CONVERGENCE
NON-SEASONAL DIFFERENCING		d =	1
SEASONAL DIFFERENCING		D =	1
SEASONAL PERIOD		s =	12
TRANSFORMATION		lambda =	0

PARAMETER		S.E.	T-STAT
MA(1)	0.3959	0.0814	4.8625
SMA(1)	0.6055	0.0767	7.8997

First roots of ARMA polynomials
MA(q)	B(1) = 2.5257575455301
SMA(Q)	B(1) = 1.651432051454

Correlation matrix
in %	MA(1)	SMA(1)
MA(1)	100	-7
SMA(1)	-7	100

Statistic	Value
G = (1 - ∑ [φ])(1 - ∑ [Φ])	1
# e[t] > 0	120
# e[t] < 0	142
MAD	6.5445933419257E-5
Mean of e[t]	-0.00070040157896975
S.E. of e[t]	0.0015725914829209
T-STAT (Mean/S.E.)	-0.44538049873501
R-squared Y[t]	1

Scatterplot of Actuals and Interpolation Forecast

Scatterplot of Squared Fit versus Squared Residuals

The above output shows the parameter estimation of the ARIMA model for the Airline data. The non-seasonal MA(1) and the seasonal SMA(1) parameter are both significantly different from zero.

Test for Heteroskedasticty : compute the simple regression analysis of the squared (stationary) residuals versus the squared (stationary) fit.

Send output to:


Data X:
0.014619666980201 0.0027896087499933 0.0086127757214437 0.0027289743777402 1.0313474783798E-6 0.00011847264970549 0.0072485088181445 2.567661849388E-6 0.0044667981855155 0.0034749305590677 0.00011695893371536 0.0086839141677618 0.00314816828798 0.0024592768484081 0.013191306099009 0.0035209996967181 0.0025747163210981 0.001279733563305 0.00012296068526216 0.0044887333707484 0.0038957048898316 0.0028915171223211 0.0010941053725491 0.031158204476367 0.00025516257077347 0.00056381263574383 0.030110339822659 0.0028755497284441 0.0070682508220044 0.0035225511609209 0.0067353369910615 0.0020751461621349 0.00011871849438793 0.010478974536004 0.0072185352020113 0.0099913102182394 0.00011879857931087 0.0056423006411939 0.031754207590112 0.0035632347199558 1.6012843313949E-5 0.00040823606080566 0.0012077026599459 0.0025084632116896 0.0023217501245053 0.00023013690108244 0.0015870236944888 0.00011133217049782 0.0015948564470573 6.1036990523143E-5 1.3071084341748E-5 0.0064022871545119 0.0039156345217186 0.00054160838724664 0.00018276225264929 0.00014948708927651 9.7693385073262E-5 0.0018673681780218 0.0048026166874743 0.0068361784053049 0.0010625913493378 0.0042533157616637 0.0016502505827753 0.00025521476213757 4.7525670785813E-6 0.0024759120943736 4.7334851994526E-5 4.8256132119362E-5 0.0020044570132171 8.1502076125621E-5 2.4109234301124E-5 5.1549737652745E-6 0.00085767887641031 0.00052611279434801 0.00010324456955803 0.0043908103799571 0.001935520941015 8.2184007841211E-8 7.6976314650646E-5 0.003989181337224 3.2971937895231E-5 0.0031171781848736 0.00069049582868695 0.0052658770173853 0.00051624638149563 0.00077138562827967 0.00025643427259094 0.0024409550862204 0.0096419880746682 3.5437284624462E-5 0.00029828726686476 0.0012861983084387 0.002628494974626 0.00046511802624128 0.0023101749840414 0.0025202749268144 0.0030238949144362 0.0030694113579706 0.00023323254139863 1.3274993162447E-6 0.001344203663964 0.0057003505492084 0.00027807868497043 0.0010076526847263 6.785064598126E-5 0.0053682060594509 0.012267081258398 0.0017154118794069 1.063048535583E-5 0.00093421306809879 0.0067006233751953 0.0034996085770658 3.0840226350317E-5 5.2075807060679E-5 0.00030691350386985 0.0021835014932648 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Data Y:
0.01008341747523 0.0043238422304526 0.00071057995686752 0.00015190440556516 1.0313474783798E-6 0.00049455080308888 0.0021551564563729 0.00031739892840079 0.00015168729080283 3.5920646501428E-6 0.0046598347501225 0.001191440461596 0.00013364059841048 0.0065795167964853 0.00070172454043684 0.0015215581855566 0.0025747163210981 0.00093683134752684 0.0011436380851622 0.00028348604717966 0.00021454121364655 0.0035032643877752 0.0025471206061717 0.0056403027763247 6.1879980657384E-5 0.0044222358419344 0.0010760680097858 1.8600897884735E-5 0.0011032011968321 7.8913430533678E-5 0.0020061889280626 0.0021476745679315 0.00035771239052044 0.014831861574544 0.001133579264046 0.00025687325967759 0.0024056465300662 0.0014640278515368 0.0039097048938502 0.00092351701502356 0.00028913088052934 0.0001286705314893 7.4146443067612E-5 1.6303592554887E-5 0.0014568662478843 0.00011260771196528 0.0017506328836668 0.00072226559425963 9.1134670400887E-5 0.0023446892195336 0.0033261925240156 0.00080792599398443 6.0893440440716E-6 7.8767461895734E-5 0.00041709931678536 0.00068428107960268 8.0639489962739E-8 7.9096996522061E-6 0.00065138042757993 1.8080874559364E-5 9.0265288685299E-5 0.0014866369547538 5.0418297801596E-5 3.448188593582E-5 0.0003902488695307 0.0010716183762894 1.2323125130324E-8 0.00030994579092867 3.2789534771588E-5 0.00061676471338975 6.4519853134876E-5 1.2934048905232E-5 8.3262896478648E-5 0.0012302389032383 4.4402863244461E-6 0.0004357137300078 0.00024730688652382 0.00063307632160825 9.7693286312844E-5 0.0015648576806366 0.0010466889052105 0.0023229780655638 0.0021886424850117 0.0015705499333454 0.00016107133357594 0.0010138459299476 0.00071271246642419 0.0022322146380328 0.0055266210786416 5.172744907589E-5 0.00043382247358934 0.0018844566887295 0.00070766386231728 1.9479122784088E-5 0.00070928573948679 0.00014339350706255 0.0014650882992878 0.0010967505081035 0.00054417808967533 0.00052504376974606 0.00016488957957602 7.444743467358E-6 0.00030004881210133 0.00019618086263731 2.5518663843032E-5 0.00036409979167244 0.0090546483578561 0.0069500774569299 0.00031553220550309 0.0001938987669735 0.00031111361261269 0.00095847732488107 0.00019026100099163 0.00075651266621663 0.00080317408727305 0.00029401393528553 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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::Free Statistics and Forecasting Software::

:: ARMA Parameter Estimation ::