Teste para autocorrelação usando a estatística Durbin-Watson

Use a estatística Durbin-Watson para testar a presença de autocorrelação nos erros de um modelo de regressão. A autocorrelação significa que os erros de observações adjacentes são correlacionados. Se os erros estiverem correlacionados, a regressão de mínimos quadrados pode subestimar o erro padrão dos coeficientes. Os erros padrão subestimados podem fazer com que seus preditores pareçam significativos quando eles não são.

Por exemplo, os erros de um modelo de regressão dos dados de preços diários de ações podem depender da observação anterior porque o preço das ações em um dia afeta o preço do dia seguinte.

A estatística Durbin-Watson (D) está condicionada à ordem das observações (linhas). O Minitab supõe que as observações estão em uma ordem significativa, como ordem de tempo. A estatística de Durbin-Watson determina se ou não a correlação entre os termos de erro adjacente é zero.

Para obter uma conclusão do teste, é possível comparar os valores exibidos para a estatística de Durbin-Watson com os limites superior e inferior corretos na tabela a seguir, elaborada por Savin e White1. Se D > DU, não existe correlação; se D < DL, existe uma correlação positiva; se D estiver entre os dois limites, o teste é inconclusivo. A tabela fornece valores a serem testados quanto à autocorrelação positiva de primeira ordem. O nível de significância para o teste é 0,05. A tabela se destina a modelos com um intercepto.

Também é possível usar esta tabela para realizar testes quanto à autocorrelação negativa de primeira ordem. O teste estatístico é (4 – D). Se (4 – D) > DU, não existe correlação; se (4 – D) < DL, existe uma correlação negativa; se (4 – D) estiver entre os dois limites, o teste é inconclusivo. 2

Para calcular a estatística de Durbin-Watson, selecione Estat > Regressão > Regressão > Ajuste de Modelo de Regressão, clique em Resultados e marque Estatística de Durbin-Watson.

Tamanho amostral Número de termos (incluindo o intercepto) DL DU
6 2 0,61018 1,40015
7 2 0,69955 1,35635
7 3 0,46723 1,89636
8 2 0,7629 1,33238
8 3 0,55907 1,77711
8 4 0,36744 2,28664
9 2 0,82428 1,31988
9 3 0,6291 1,69926
9 4 0,45476 2,12816
9 5 0,29571 2,5881
10 2 0,87913 1,31971
10 3 0,69715 1,64134
10 4 0,52534 2,01632
10 5 0,37602 2,41365
10 6 0,24269 2,82165
11 2 0,92733 1,32409
11 3 0,75798 1,60439
11 4 0,59477 1,92802
11 5 0,44406 2,28327
11 6 0,31549 2,64456
11 7 0,20253 3,00447
12 2 0,97076 1,33137
12 3 0,81221 1,57935
12 4 0,65765 1,86397
12 5 0,51198 2,17662
12 6 0,37956 2,50609
12 7 0,26813 2,83196
12 8 0,17144 3,1494
13 2 1,00973 1,3404
13 3 0,86124 1,56212
13 4 0,71465 1,81593
13 5 0,57446 2,09428
13 6 0,44448 2,38967
13 7 0,32775 2,69204
13 8 0,23049 2,98506
13 9 0,14693 3,26577
14 2 1,04495 1,35027
14 3 0,90544 1,55066
14 4 0,76666 1,77882
14 5 0,63206 2,02955
14 6 0,50516 2,29593
14 7 0,38897 2,57158
14 8 0,28559 2,84769
14 9 0,20013 3,11121
14 10 0,12726 3,36038
15 2 1,07697 1,36054
15 3 0,94554 1,54318
15 4 0,81396 1,75014
15 5 0,68519 1,97735
15 6 0,56197 2,21981
15 7 0,44707 2,47148
15 8 0,3429 2,72698
15 9 0,2509 2,97866
15 10 0,17531 3,21604
15 11 0,11127 3,43819
16 2 1,10617 1,37092
16 3 0,98204 1,5386
16 4 0,85718 1,72773
16 5 0,734 1,93506
16 6 0,61495 2,15672
16 7 0,50223 2,38813
16 8 0,39805 2,62409
16 9 0,30433 2,86009
16 10 0,22206 3,08954
16 11 0,15479 3,30391
16 12 0,09809 3,50287
17 2 1,13295 1,38122
17 3 1,01543 1,53614
17 4 0,89675 1,71009
17 5 0,77898 1,90047
17 6 0,66414 2,10414
17 7 0,55423 2,31755
17 8 0,45107 2,5366
17 9 0,35639 2,75688
17 10 0,27177 2,97455
17 11 0,19784 3,184
17 12 0,13763 3,37817
17 13 0,08711 3,55716
18 2 1,15759 1,39133
18 3 1,04607 1,53525
18 4 0,9331 1,69614
18 5 0,82044 1,87189
18 6 0,70984 2,06
18 7 0,60301 2,2575
18 8 0,50158 2,46122
18 9 0,40702 2,66753
18 10 0,32076 2,87268
18 11 0,24405 3,07345
18 12 0,17732 3,26497
18 13 0,12315 3,44141
18 14 0,07786 3,60315
19 2 1,18037 1,40118
19 3 1,0743 1,53553
19 4 0,96659 1,68509
19 5 0,85876 1,84815
19 6 0,75231 2,02262
19 7 0,6487 2,20614
19 8 0,54938 2,39602
19 9 0,45571 2,58939
19 10 0,36889 2,78312
19 11 0,29008 2,97399
19 12 0,22029 3,1593
19 13 0,15979 3,33481
19 14 0,11082 3,49566
19 15 0,07001 3,64241
20 2 1,20149 1,41073
20 3 1,1004 1,53668
20 4 0,99755 1,67634
20 5 0,89425 1,82828
20 6 0,79179 1,99079
20 7 0,69146 2,16189
20 8 0,59454 2,33937
20 9 0,5022 2,52082
20 10 0,41559 2,70374
20 11 0,33571 2,88535
20 12 0,26349 3,06292
20 13 0,19978 3,23417
20 14 0,14472 3,3954
20 15 0,10024 3,5425
20 16 0,06327 3,67619
21 2 1,22115 1,41997
21 3 1,12461 1,53849
21 4 1,02624 1,66942
21 5 0,92719 1,81157
21 6 0,82856 1,9635
21 7 0,73149 2,12355
21 8 0,6371 2,28988
21 9 0,54645 2,46051
21 10 0,46055 2,63324
21 11 0,38035 2,80588
21 12 0,30669 2,976
21 13 0,24033 3,14129
21 14 0,18198 3,29979
21 15 0,13166 3,44827
21 16 0,09111 3,58322
21 17 0,05747 3,70544
22 2 1,23949 1,42888
22 3 1,14713 1,54079
22 4 1,05292 1,66398
22 5 0,95783 1,79744
22 6 0,86285 1,93996
22 7 0,76898 2,09015
22 8 0,67719 2,24646
22 9 0,58843 2,40718
22 10 0,50363 2,57051
22 11 0,42363 2,73452
22 12 0,34926 2,89726
22 13 0,28119 3,05662
22 14 0,22003 3,21061
22 15 0,16642 3,35756
22 16 0,12028 3,49463
22 17 0,08315 3,6188
22 18 0,05242 3,73092
23 2 1,25665 1,43747
23 3 1,16815 1,54346
23 4 1,07778 1,65974
23 5 0,98639 1,78546
23 6 0,89488 1,91958
23 7 0,8041 2,06093
23 8 0,71493 2,20816
23 9 0,62821 2,35988
23 10 0,54478 2,51449
23 11 0,46541 2,67038
23 12 0,39083 2,82585
23 13 0,32172 2,97919
23 14 0,25866 3,12852
23 15 0,20216 3,27216
23 16 0,15274 3,40865
23 17 0,11029 3,53549
23 18 0,07619 3,65007
23 19 0,04801 3,75327
24 2 1,27276 1,44575
24 3 1,18781 1,54639
24 4 1,101 1,65649
24 5 1,01309 1,77526
24 6 0,92486 1,90184
24 7 0,83706 2,03522
24 8 0,75048 2,17427
24 9 0,66589 2,31774
24 10 0,584 2,46431
24 11 0,50554 2,6126
24 12 0,43119 2,76111
24 13 0,36156 2,90835
24 14 0,29723 3,05282
24 15 0,23869 3,19285
24 16 0,18635 3,327
24 17 0,14066 3,45402
24 18 0,1015 3,57167
24 19 0,07006 3,67769
24 20 0,04413 3,77297
25 2 1,28791 1,45371
25 3 1,20625 1,54954
25 4 1,12276 1,65403
25 5 1,03811 1,76655
25 6 0,95297 1,88634
25 7 0,86803 2,01252
25 8 0,784 2,14412
25 9 0,70154 2,28007
25 10 0,62133 2,41924
25 11 0,54401 2,56041
25 12 0,47019 2,70229
25 13 0,40046 2,8436
25 14 0,33536 2,983
25 15 0,27536 3,11913
25 16 0,2209 3,25058
25 17 0,17231 3,37604
25 18 0,12995 3,49447
25 19 0,09371 3,60384
25 20 0,06465 3,7022
25 21 0,0407 3,79041
26 2 1,30219 1,46139
26 3 1,22358 1,55281
26 4 1,14319 1,65225
26 5 1,06158 1,75911
26 6 0,97937 1,87274
26 7 0,89717 1,9924
26 8 0,81561 2,11722
26 9 0,73529 2,24629
26 10 0,65683 2,37862
26 11 0,58079 2,51315
26 12 0,50775 2,64877
26 13 0,43825 2,78436
26 14 0,37279 2,91872
26 15 0,31182 3,05067
26 16 0,25578 3,17904
26 17 0,20499 3,30253
26 18 0,15977 3,42006
26 19 0,12041 3,53067
26 20 0,08677 3,63257
26 21 0,05983 3,72404
27 2 1,31568 1,46878
27 3 1,23991 1,5562
27 4 1,16239 1,65101
27 5 1,08364 1,75274
27 6 1,00421 1,86079
27 7 0,92463 1,97449
27 8 0,84546 2,09313
27 9 0,76726 2,21588
27 10 0,69057 2,3419
27 11 0,61593 2,47026
27 12 0,54385 2,59997
27 13 0,47482 2,73007
27 14 0,40933 2,8595
27 15 0,3478 2,98721
27 16 0,29062 3,11215
27 17 0,23816 3,23327
27 18 0,19072 3,34944
27 19 0,14853 3,45967
27 20 0,11188 3,56318
27 21 0,08057 3,65833
28 2 1,32844 1,47589
28 3 1,25534 1,55964
28 4 1,18051 1,65025
28 5 1,10444 1,74728
28 6 1,02762 1,85022
28 7 0,95052 1,95851
28 8 0,87366 2,07148
28 9 0,79754 2,18844
28 10 0,72265 2,30862
28 11 0,64947 2,43122
28 12 0,57848 2,5554
28 13 0,51013 2,68025
28 14 0,44486 2,80489
28 15 0,38308 2,92838
28 16 0,32517 3,04976
28 17 0,27146 3,16812
28 18 0,22228 3,28249
28 19 0,17787 3,39189
28 20 0,13843 3,49546
28 21 0,10421 3,59248
29 2 1,34054 1,48275
29 3 1,26992 1,56312
29 4 1,19762 1,64987
29 5 1,12407 1,7426
29 6 1,04971 1,84088
29 7 0,97499 1,9442
29 8 0,90036 2,05196
29 9 0,82626 2,16358
29 10 0,75316 2,27837
29 11 0,68148 2,39562
29 12 0,61166 2,51459
29 13 0,54413 2,63447
29 14 0,47929 2,75449
29 15 0,41753 2,87381
29 16 0,35918 2,9916
29 17 0,30461 3,107
29 18 0,25409 3,21917
29 19 0,2079 3,32728
29 20 0,16625 3,43042
29 21 0,12931 3,52786
30 2 1,35204 1,48936
30 3 1,28373 1,56661
30 4 1,2138 1,64981
30 5 1,14262 1,7386
30 6 1,0706 1,83259
30 7 0,99815 1,93133
30 8 0,92564 2,03432
30 9 0,85351 2,14102
30 10 0,78217 2,2508
30 11 0,71202 2,36307
30 12 0,64345 2,47714
30 13 0,57685 2,59233
30 14 0,51259 2,70793
30 15 0,45105 2,82319
30 16 0,39255 2,93738
30 17 0,3374 3,04971
30 18 0,2859 3,15946
30 19 0,2383 3,26584
30 20 0,19485 3,36811
30 21 0,15572 3,46549
31 2 1,36298 1,49574
31 3 1,29685 1,57011
31 4 1,22915 1,65002
31 5 1,16021 1,73518
31 6 1,0904 1,82522
31 7 1,02008 1,91976
31 8 0,94962 2,01834
31 9 0,8794 2,12046
31 10 0,80979 2,22562
31 11 0,74115 2,33323
31 12 0,67387 2,44273
31 13 0,60828 2,55347
31 14 0,54474 2,66484
31 15 0,48358 2,77618
31 16 0,42513 2,8868
31 17 0,36966 2,99604
31 18 0,31748 3,10322
31 19 0,26882 3,20762
31 20 0,22392 3,30859
31 21 0,18298 3,40545
32 2 1,3734 1,5019
32 3 1,30932 1,57358
32 4 1,24371 1,65046
32 5 1,17688 1,73226
32 6 1,10916 1,81867
32 7 1,04088 1,90931
32 8 0,97239 2,00381
32 9 0,90401 2,10171
32 10 0,83609 2,20255
32 11 0,76897 2,30583
32 12 0,70299 2,41102
32 13 0,63847 2,51758
32 14 0,57573 2,62493
32 15 0,5151 2,73248
32 16 0,45685 2,83963
32 17 0,40129 2,94576
32 18 0,34866 3,05028
32 19 0,29923 3,15253
32 20 0,25319 3,25193
32 21 0,21078 3,34784
33 2 1,38335 1,50784
33 3 1,32119 1,57703
33 4 1,25756 1,6511
33 5 1,19272 1,72978
33 6 1,12698 1,81282
33 7 1,06065 1,89986
33 8 0,99402 1,99057
33 9 0,92743 2,08455
33 10 0,86115 2,18137
33 11 0,79554 2,28061
33 12 0,73086 2,38177
33 13 0,66745 2,48437
33 14 0,60559 2,58789
33 15 0,54558 2,69181
33 16 0,48769 2,79558
33 17 0,43219 2,89865
33 18 0,37933 3,00046
33 19 0,32935 3,10046
33 20 0,28246 3,19808
33 21 0,23887 3,29275
34 2 1,39285 1,51358
34 3 1,33251 1,58045
34 4 1,27074 1,65189
34 5 1,20779 1,7277
34 6 1,14393 1,80758
34 7 1,07944 1,89129
34 8 1,01462 1,97849
34 9 0,94973 2,06882
34 10 0,88506 2,1619
34 11 0,82091 2,25735
34 12 0,75755 2,35473
34 13 0,69527 2,45359
34 14 0,63433 2,55348
34 15 0,57503 2,65392
34 16 0,5176 2,75442
34 17 0,46231 2,85449
34 18 0,40939 2,95361
34 19 0,35907 3,05127
34 20 0,31155 3,14697
34 21 0,26704 3,2402
35 2 1,40194 1,51914
35 3 1,34332 1,58382
35 4 1,2833 1,65282
35 5 1,22214 1,72593
35 6 1,16007 1,80292
35 7 1,09735 1,88351
35 8 1,03424 1,96743
35 9 0,97099 2,05436
35 10 0,90788 2,14395
35 11 0,84516 2,23585
35 12 0,78311 2,32966
35 13 0,72197 2,42501
35 14 0,662 2,52146
35 15 0,60346 2,61858
35 16 0,54659 2,71593
35 17 0,49162 2,81306
35 18 0,43878 2,90951
35 19 0,38829 3,00481
35 20 0,34034 3,09851
35 21 0,29513 3,19013
36 2 1,41065 1,52451
36 3 1,35365 1,58716
36 4 1,2953 1,65387
36 5 1,23583 1,72447
36 6 1,17545 1,79873
36 7 1,11441 1,87643
36 8 1,05294 1,9573
36 9 0,99128 2,04104
36 10 0,92967 2,12737
36 11 0,86836 2,21594
36 12 0,80759 2,30642
36 13 0,74759 2,39844
36 14 0,68861 2,49162
36 15 0,63089 2,58557
36 16 0,57463 2,6799
36 17 0,52008 2,77418
36 18 0,46745 2,868
36 19 0,41692 2,96095
36 20 0,36871 3,05259
36 21 0,32299 3,14249
37 2 1,419 1,52971
37 3 1,36354 1,59044
37 4 1,30678 1,65501
37 5 1,24891 1,72327
37 6 1,19014 1,79499
37 7 1,13071 1,86998
37 8 1,07081 1,94799
37 9 1,01066 2,02876
37 10 0,95051 2,11203
37 11 0,89057 2,19749
37 12 0,83105 2,28481
37 13 0,77219 2,37369
37 14 0,71421 2,46378
37 15 0,65734 2,55471
37 16 0,60177 2,64613
37 17 0,54771 2,73765
37 18 0,49537 2,82891
37 19 0,44494 2,91951
37 20 0,39661 3,00907
37 21 0,35054 3,09719
38 2 1,42702 1,53475
38 3 1,37301 1,59368
38 4 1,31774 1,65625
38 5 1,2614 1,72229
38 6 1,20418 1,79164
38 7 1,14627 1,86409
38 8 1,08787 1,93942
38 9 1,02919 2,01742
38 10 0,97045 2,09782
38 11 0,91183 2,18033
38 12 0,85356 2,2647
38 13 0,79583 2,35061
38 14 0,73886 2,43775
38 15 0,68284 2,52581
38 16 0,62799 2,61444
38 17 0,57448 2,70332
38 18 0,52253 2,79207
38 19 0,47229 2,88036
38 20 0,42396 2,96784
38 21 0,37769 3,05412
39 2 1,43473 1,53963
39 3 1,3821 1,59686
39 4 1,32827 1,65754
39 5 1,27338 1,72152
39 6 1,21761 1,78863
39 7 1,16116 1,8587
39 8 1,10419 1,93153
39 9 1,04692 2,00692
39 10 0,98953 2,0846
39 11 0,9322 2,16437
39 12 0,87514 2,24594
39 13 0,81853 2,32904
39 14 0,76257 2,4134
39 15 0,70743 2,49872
39 16 0,65333 2,58469
39 17 0,60044 2,671
39 18 0,54891 2,75733
39 19 0,49896 2,84336
39 20 0,45072 2,92876
39 21 0,40437 3,0132
40 2 1,44214 1,54436
40 3 1,39083 1,59999
40 4 1,33835 1,65889
40 5 1,28484 1,72092
40 6 1,23047 1,78594
40 7 1,17541 1,85378
40 8 1,11983 1,92426
40 9 1,06391 1,99717
40 10 1,00782 2,07233
40 11 0,95174 2,1495
40 12 0,89585 2,22843
40 13 0,84035 2,30888
40 14 0,78539 2,3906
40 15 0,73115 2,4733
40 16 0,67782 2,55672
40 17 0,62556 2,64056
40 18 0,57454 2,72455
40 19 0,52492 2,80836
40 20 0,47687 2,89172
40 21 0,43054 2,97431
41 2 1,44927 1,54895
41 3 1,39922 1,60307
41 4 1,34803 1,66028
41 5 1,29584 1,72048
41 6 1,2428 1,78353
41 7 1,18907 1,84926
41 8 1,13481 1,91753
41 9 1,08019 1,98813
41 10 1,02536 2,06089
41 11 0,9705 2,13561
41 12 0,91576 2,21204
41 13 0,86132 2,28998
41 14 0,80736 2,36919
41 15 0,75402 2,44941
41 16 0,70146 2,53039
41 17 0,64987 2,61187
41 18 0,5994 2,69358
41 19 0,55018 2,77525
41 20 0,50238 2,8566
41 21 0,45615 2,93734
42 2 1,45615 1,5534
42 3 1,4073 1,60608
42 4 1,35733 1,66172
42 5 1,3064 1,72019
42 6 1,25463 1,78137
42 7 1,20218 1,84512
42 8 1,14918 1,9113
42 9 1,09581 1,97972
42 10 1,04219 2,05023
42 11 0,98851 2,12262
42 12 0,93489 2,1967
42 13 0,88151 2,27227
42 14 0,82852 2,34909
42 15 0,77607 2,42694
42 16 0,72431 2,50558
42 17 0,67341 2,5848
42 18 0,6235 2,66432
42 19 0,57474 2,74389
42 20 0,52726 2,82328
42 21 0,48121 2,9022
43 2 1,46278 1,55773
43 3 1,41507 1,60905
43 4 1,36629 1,66319
43 5 1,31655 1,72002
43 6 1,266 1,77944
43 7 1,21476 1,84132
43 8 1,16298 1,90552
43 9 1,1108 1,97189
43 10 1,05837 2,04027
43 11 1,00581 2,11047
43 12 0,95328 2,18231
43 13 0,90093 2,25562
43 14 0,84891 2,33017
43 15 0,79734 2,40577
43 16 0,74639 2,4822
43 17 0,69619 2,55922
43 18 0,64688 2,63664
43 19 0,5986 2,71419
43 20 0,55149 2,79164
43 21 0,50568 2,86878
44 2 1,4692 1,56193
44 3 1,42257 1,61196
44 4 1,3749 1,66467
44 5 1,32631 1,71996
44 6 1,27692 1,77772
44 7 1,22685 1,83784
44 8 1,17624 1,90017
44 9 1,12522 1,9646
44 10 1,0739 2,03095
44 11 1,02245 2,09907
44 12 0,97099 2,16881
44 13 0,91964 2,23997
44 14 0,86856 2,31237
44 15 0,81787 2,38581
44 16 0,76771 2,46011
44 17 0,71822 2,53505
44 18 0,66953 2,61043
44 19 0,62177 2,68601
44 20 0,57507 2,76161
44 21 0,52954 2,83698
45 2 1,47538 1,56602
45 3 1,4298 1,61482
45 4 1,3832 1,66618
45 5 1,33571 1,71999
45 6 1,28744 1,77618
45 7 1,23849 1,83462
45 8 1,18899 1,8952
45 9 1,13907 1,95778
45 10 1,08886 2,02222
45 11 1,03846 2,08839
45 12 0,98802 2,15611
45 13 0,93765 2,22524
45 14 0,8875 2,29558
45 15 0,83769 2,36698
45 16 0,78833 2,43924
45 17 0,73955 2,51218
45 18 0,69149 2,58559
45 19 0,64427 2,65929
45 20 0,59801 2,73306
45 21 0,55282 2,80672
46 2 1,48136 1,56999
46 3 1,43677 1,61763
46 4 1,39121 1,66769
46 5 1,34477 1,72012
46 6 1,29756 1,77482
46 7 1,24969 1,83167
46 8 1,20127 1,89058
46 9 1,15242 1,95141
46 10 1,10325 2,01404
46 11 1,05388 2,07834
46 12 1,00443 2,14416
46 13 0,95503 2,21134
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