============================================================
White Heteroskedasticity Test:
============================================================
F-statistic 2.872725 Probability 0.000038
Obs*R-squared 54.15035 Probability 0.000094
============================================================
Test Equation:
LS // Dependent Variable is RESID^2
Date: 04/21/98 Time: 13:03
Sample: 1 336
Included observations: 336
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C -5.273037 1.975978 -2.668570 0.0080
TAX 0.004902 0.076273 0.064269 0.9488
TAX^2 -0.014658 0.040564 -0.361361 0.7181
DRKAGE 0.562102 0.206821 2.717822 0.0069
DRKAGE^2 -0.014380 0.005285 -2.720967 0.0069
PCINC -0.074213 0.146670 -0.505982 0.6132
PCINC^2 0.009711 0.019575 0.496121 0.6202
MILES 94.47385 36.46556 2.590769 0.0100
MILES^2 -3898.757 1599.262 -2.437848 0.0153
YNGDRV -1.304460 0.694870 -1.877270 0.0614
YNGDRV^2 0.965221 0.462340 2.087686 0.0376
INSP 0.004712 0.009554 0.493171 0.6222
MORMON -0.677308 0.233042 -2.906385 0.0039
MORMON^2 0.727000 0.321472 2.261470 0.0244
PROT -0.260051 0.236303 -1.100500 0.2720
PROT^2 0.090228 0.418702 0.215494 0.8295
CATH -0.312100 0.129396 -2.411972 0.0164
CATH^2 0.521536 0.192144 2.714302 0.0070
SOBAB -0.533068 0.217292 -2.453235 0.0147
SOBAB^2 1.122543 0.920530 1.219453 0.2236
WET -0.116754 0.847227 -0.137807 0.8905
WET^2 0.086463 0.522267 0.165553 0.8686
============================================================
R-squared 0.161162 Mean dependent var 0.040846
Adjusted R-squared 0.105061 S.D. dependent var 0.072709
S.E. of regression 0.068783 Akaike info criter-5.290355
Sum squared resid 1.485577 Schwarz criterion -5.040425
Log likelihood 434.0164 F-statistic 2.872725
Durbin-Watson stat 1.896893 Prob(F-statistic) 0.000038
============================================================
LS(H) LFT18T20 C TAX DRKAGE PCINC MILES YNGDRV INSP MORMON PROT CATH SOBAB WET
============================================================
LS // Dependent Variable is LFT18T20
Date: 04/21/98 Time: 13:07
Sample: 1 336
Included observations: 336
White Heteroskedasticity-Consistent Standard Errors &
Covariance
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C 3.165057 0.293886 10.76969 0.0000
TAX -0.278794 0.045943 -6.068288 0.0000
DRKAGE -0.035408 0.008922 -3.968661 0.0001
PCINC -0.195457 0.039210 -4.984835 0.0000
MILES 110.1364 8.881711 12.40036 0.0000
YNGDRV 1.524236 0.161228 9.453931 0.0000
INSP -0.075403 0.026855 -2.807803 0.0053
MORMON -0.814448 0.137447 -5.925542 0.0000
PROT -0.712803 0.143324 -4.973384 0.0000
CATH -0.504089 0.160777 -3.135327 0.0019
SOBAB -0.707793 0.244949 -2.889549 0.0041
WET 0.424073 0.106305 3.989223 0.0001
============================================================
R-squared 0.570763 Mean dependent var 4.025630
Adjusted R-squared 0.556190 S.D. dependent var 0.308939
S.E. of regression 0.205812 Akaike info criter-3.126519
Sum squared resid 13.72424 Schwarz criterion -2.990194
Log likelihood 60.49186 F-statistic 39.16620
Durbin-Watson stat 1.156329 Prob(F-statistic) 0.000000
============================================================
LS(WT=WEIGHT) LFT18T20 C TAX DRKAGE PCINC MILES YNGDRV INSP MORMON PROT
CATH SOBAB WET
============================================================
LS // Dependent Variable is LFT18T20
Date: 04/21/98 Time: 13:10
Weighting series: WEIGHT
Sample: 1 336
Included observations: 336
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C 3.222418 0.337739 9.541154 0.0000
TAX -0.274002 0.067616 -4.052302 0.0001
DRKAGE -0.035095 0.009525 -3.684584 0.0003
PCINC -0.196498 0.035992 -5.459494 0.0000
MILES 106.9708 9.091373 11.76618 0.0000
YNGDRV 1.486685 0.151822 9.792312 0.0000
INSP -0.076806 0.026530 -2.895037 0.0040
MORMON -0.804665 0.148464 -5.419917 0.0000
PROT -0.687807 0.146028 -4.710117 0.0000
CATH -0.523134 0.129111 -4.051817 0.0001
SOBAB -0.701809 0.247495 -2.835654 0.0049
WET 0.422053 0.184489 2.287683 0.0228
============================================================
Weighted Statistics
============================================================
R-squared 0.166752 Mean dependent var 4.018315
Adjusted R-squared 0.138463 S.D. dependent var 0.221887
S.E. of regression 0.205953 Akaike info criter-3.125154
Sum squared resid 13.74299 Schwarz criterion -2.988829
Log likelihood 60.26252 F-statistic 5.894548
Durbin-Watson stat 1.167913 Prob(F-statistic) 0.000000
============================================================
Unweighted Statistics
============================================================
R-squared 0.570406 Mean dependent var 4.025630
Adjusted R-squared 0.555821 S.D. dependent var 0.308939
S.E. of regression 0.205898 Sum squared resid 13.73565
Durbin-Watson stat 1.151230
============================================================
Auto Correlation Examples-
The first example is first-order autocorrelation with parameter .5. The
true values for the intercept and slope are both equal to 2. The first
output is for a simple two variable regression. Note the value for the
Durban-Watson statistic and remember that the estimators are unbiased. The
ordinary regression works well here because of low error and a large sample
size.
============================================================
LS // Dependent Variable is YYY
Date: 04/18/98 Time: 14:38
Sample(adjusted): 2 500
Included observations: 499 after adjusting endpoints
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C 2.039942 0.055456 36.78497 0.0000
X 2.019524 0.053867 37.49101 0.0000
============================================================
R-squared 0.738775 Mean dependent var 2.100864
Adjusted R-squared 0.738250 S.D. dependent var 2.420290
S.E. of regression 1.238258 Akaike info criter 0.431411
Sum squared resid 762.0415 Schwarz criterion 0.448295
Log likelihood -813.6873 F-statistic 1405.576
Durbin-Watson stat 0.850402 Prob(F-statistic) 0.000000
============================================================
Given the value of the Durban-Watson statistic, the first step is to try estimating the
first-order autocorrelation model. To do this in EVIEWS, use the command:LS YYY C X AR(1)
The Durban-Watson is now near 2 indicating that our correction has done the trick. Also note that the standard errors are now smaller.
============================================================
LS // Dependent Variable is YYY
Date: 04/18/98 Time: 14:47
Sample(adjusted): 3 500
Included observations: 498 after adjusting endpoints
Convergence achieved after 4 iterations
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C 2.029702 0.107408 18.89720 0.0000
X 1.976182 0.037777 52.31180 0.0000
AR(1) 0.576651 0.036850 15.64846 0.0000
============================================================
R-squared 0.823824 Mean dependent var 2.090885
Adjusted R-squared 0.823113 S.D. dependent var 2.412425
S.E. of regression 1.014617 Akaike info criter 0.035029
Sum squared resid 509.5768 Schwarz criterion 0.060394
Log likelihood -712.3535 F-statistic 1157.349
Durbin-Watson stat 2.029978 Prob(F-statistic) 0.000000
============================================================
Inverted AR Roots .58
============================================================
As a further check, we can run higher order AR models. For example, to add
a second-order autocorrelation issue the EVIEWS command:LS YYY C X AR(1) AR(2)
============================================================
LS // Dependent Variable is YYY
Date: 04/18/98 Time: 14:51
Sample(adjusted): 4 500
Included observations: 497 after adjusting endpoints
Convergence achieved after 4 iterations
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C 2.026031 0.111789 18.12370 0.0000
X 1.981206 0.038435 51.54703 0.0000
AR(1) 0.552113 0.045357 12.17258 0.0000
AR(2) 0.040334 0.045379 0.888813 0.3745
============================================================
R-squared 0.824106 Mean dependent var 2.093776
Adjusted R-squared 0.823036 S.D. dependent var 2.413992
S.E. of regression 1.015496 Akaike info criter 0.038769
Sum squared resid 508.3972 Schwarz criterion 0.072641
Log likelihood -710.8467 F-statistic 769.9438
Durbin-Watson stat 1.974323 Prob(F-statistic) 0.000000
============================================================
Inverted AR Roots .6 -.07
============================================================
To add a second-order and third-order autocorrelation terms issue the EVIEWS
command:LS YYY C X AR(1) AR(2) AR(3)
============================================================
LS // Dependent Variable is YYY
Date: 04/18/98 Time: 14:54
Sample(adjusted): 5 500
Included observations: 496 after adjusting endpoints
Convergence achieved after 4 iterations
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C 2.033835 0.110996 18.32347 0.0000
X 1.981932 0.038471 51.51745 0.0000
AR(1) 0.554496 0.045383 12.21827 0.0000
AR(2) 0.050379 0.052013 0.968582 0.3332
AR(3) -0.015515 0.045462 -0.341276 0.7330
============================================================
R-squared 0.824542 Mean dependent var 2.099213
Adjusted R-squared 0.823113 S.D. dependent var 2.413381
S.E. of regression 1.015019 Akaike info criter 0.039843
Sum squared resid 505.8590 Schwarz criterion 0.082248
Log likelihood -708.6746 F-statistic 576.8487
Durbin-Watson stat 1.979999 Prob(F-statistic) 0.000000
============================================================
Inverted AR Roots .6 .14 -.18
============================================================
This second example is a two variable model with
first and second-order autocorrelation. The intercept and slope are both
equal to 2.0 and the autocorrelation parameters are .5 and .3 respectively.
Again, start with the simple linear regression. Note that the Durban-Watson is quite small so that the next thing we should try is estimating simple first-order autocorrelation.
============================================================
LS // Dependent Variable is YYYY
Date: 04/18/98 Time: 14:57
Sample: 1 500
Included observations: 500
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C 2.127447 0.074982 28.37268 0.0000
X 2.089195 0.072906 28.65600 0.0000
============================================================
R-squared 0.622489 Mean dependent var 2.189925
Adjusted R-squared 0.621731 S.D. dependent var 2.724954
S.E. of regression 1.675945 Akaike info criter 1.036746
Sum squared resid 1398.778 Schwarz criterion 1.053604
Log likelihood -966.6557 F-statistic 821.1663
Durbin-Watson stat 0.463438 Prob(F-statistic) 0.000000
============================================================
First, try estimating first-order autocorrelation.
============================================================
LS // Dependent Variable is YYYY
Date: 04/18/98 Time: 14:59
Sample(adjusted): 2 500
Included observations: 499 after adjusting endpoints
Convergence achieved after 4 iterations
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C 2.091612 0.214367 9.757170 0.0000
X 1.973435 0.036009 54.80381 0.0000
AR(1) 0.777316 0.028582 27.19551 0.0000
============================================================
R-squared 0.847857 Mean dependent var 2.187698
Adjusted R-squared 0.847244 S.D. dependent var 2.727233
S.E. of regression 1.065914 Akaike info criter 0.133659
Sum squared resid 563.5415 Schwarz criterion 0.158985
Log likelihood -738.3982 F-statistic 1382.045
Durbin-Watson stat 2.448294 Prob(F-statistic) 0.000000
============================================================
Inverted AR Roots .78
============================================================
Note that the
Durban-Watson is somewhat high. This is suspicious so the next step is to try
estimating the model with second-order autocorrelation.
============================================================
LS // Dependent Variable is YYYY
Date: 04/18/98 Time: 15:01
Sample(adjusted): 3 500
Included observations: 498 after adjusting endpoints
Convergence achieved after 5 iterations
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C 2.062221 0.296098 6.964657 0.0000
X 2.002662 0.037604 53.25624 0.0000
AR(1) 0.537732 0.043203 12.44650 0.0000
AR(2) 0.308084 0.043190 7.133140 0.0000
============================================================
R-squared 0.860985 Mean dependent var 2.177893
Adjusted R-squared 0.860141 S.D. dependent var 2.721157
S.E. of regression 1.017651 Akaike info criter 0.042993
Sum squared resid 511.5928 Schwarz criterion 0.076813
Log likelihood -713.3367 F-statistic 1019.860
Durbin-Watson stat 1.951773 Prob(F-statistic) 0.000000
============================================================
Inverted AR Roots .8 -.35
============================================================
The model with first and second-order autocorrelation looks good.
The Durban-Watson statistic is near 2. To be cautious, check for a
third-order effect.
============================================================
LS // Dependent Variable is YYYY
Date: 04/18/98 Time: 15:02
Sample(adjusted): 4 500
Included observations: 497 after adjusting endpoints
Convergence achieved after 5 iterations
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C 2.060181 0.279652 7.366943 0.0000
X 2.007865 0.037373 53.72512 0.0000
AR(1) 0.553574 0.045326 12.21324 0.0000
AR(2) 0.337364 0.049643 6.795838 0.0000
AR(3) -0.054397 0.045379 -1.198729 0.2312
============================================================
R-squared 0.861482 Mean dependent var 2.180052
Adjusted R-squared 0.860356 S.D. dependent var 2.723471
S.E. of regression 1.017734 Akaike info criter 0.045167
Sum squared resid 509.6051 Schwarz criterion 0.087507
Log likelihood -711.4364 F-statistic 764.9705
Durbin-Watson stat 1.981769 Prob(F-statistic) 0.000000
============================================================
Inverted AR Roots .8 .14 -.45
============================================================