1998 Flex-Time and Flex-Mode Prob-Stat II Note Set 6 Handouts Page
LEGACY CONTENT.
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45-734 PROBABILITY AND STATISTICS II (4th Mini AY1997-98)
Note Set 6 Handouts
KEPLER.WF1 Examples
LS DAYS C DIST
HIST RESID
Note that the Probability here is 0.668929 so we do not reject the null
hypothesis that residuals are normally distributed.
SCAT(R) LOG(DAYS) LOG(DIST)
LS LOG(DAYS) C LOG(DIST)
============================================================
LS // Dependent Variable is LOG(DAYS)
Date: 04/04/98 Time: 17:18
Sample: 1 9
Included observations: 9
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C 5.901252 0.001346 4385.273 0.0000
LOG(DIST) 1.498890 0.000617 2430.848 0.0000
============================================================
R-squared 0.999999 Mean dependent var 8.081784
Adjusted R-squared 0.999999 S.D. dependent var 2.586347
S.E. of regression 0.003009 Akaike info criter-11.41893
Sum squared resid 6.34E-05 Schwarz criterion -11.37510
Log likelihood 40.61473 F-statistic 5909021.
Durbin-Watson stat 1.598728 Prob(F-statistic) 0.000000
============================================================
SCAT RESID LOG(DIST)
Note the scale on the residuals!
To directly test Kepler's 3rd Law:
LS DAYS^2 C DIST^3
============================================================
LS // Dependent Variable is DAYS^2
Date: 04/04/98 Time: 17:27
Sample: 1 9
Included observations: 9
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C 7012275. 9104212. 0.770223 0.4664
DIST^3 131103.7 396.1369 330.9556 0.0000
============================================================
R-squared 0.999936 Mean dependent var 1.44E+09
Adjusted R-squared 0.999927 S.D. dependent var 2.81E+09
S.E. of regression 24038144 Akaike info criter 34.18343
Sum squared resid 4.04E+15 Schwarz criterion 34.22726
Log likelihood -164.5959 F-statistic 109531.6
Durbin-Watson stat 2.174360 Prob(F-statistic) 0.000000
============================================================
WINDMILL.WF1 Examples
LS DCOUT C WIND
============================================================
LS // Dependent Variable is DCOUT
Date: 04/04/98 Time: 17:30
Sample: 1 25
Included observations: 25
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C 0.130875 0.125989 1.038779 0.3097
WIND 0.241149 0.019049 12.65927 0.0000
============================================================
R-squared 0.874493 Mean dependent var 1.609600
Adjusted R-squared 0.869036 S.D. dependent var 0.652278
S.E. of regression 0.236052 Akaike info criter-2.810787
Sum squared resid 1.281573 Schwarz criterion -2.713277
Log likelihood 1.661381 F-statistic 160.2571
Durbin-Watson stat 0.536610 Prob(F-statistic) 0.000000
============================================================
SCAT RESID WIND
Ramsey Reset Test With One Fitted Term. Note that we reject
the null hypothesis that we have a linear specification (that is, the
y-hat-squared variable is statistically significant).
============================================================
Ramsey RESET Test:
============================================================
F-statistic 63.16035 Probability 0.000000
Log likelihood ratio 33.83734 Probability 0.000000
============================================================
Test Equation:
LS // Dependent Variable is DCOUT
Date: 04/04/98 Time: 17:33
Sample: 1 25
Included observations: 25
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C -1.144670 0.173341 -6.603572 0.0000
WIND 0.764313 0.066569 11.48152 0.0000
Fitted^2 -0.655530 0.082484 -7.947349 0.0000
============================================================
R-squared 0.967577 Mean dependent var 1.609600
Adjusted R-squared 0.964630 S.D. dependent var 0.652278
S.E. of regression 0.122674 Akaike info criter-4.084281
Sum squared resid 0.331077 Schwarz criterion -3.938016
Log likelihood 18.58005 F-statistic 328.2660
Durbin-Watson stat 1.633283 Prob(F-statistic) 0.000000
============================================================
Ramsey Reset Test With Three Fitted Terms. Note that we reject
the null hypothesis that we have a linear specification (that is, the
(y-hat)2, (y-hat)3, and (y-hat)4
variables are jointly significant!).
============================================================
Ramsey RESET Test:
============================================================
F-statistic 39.22341 Probability 0.000000
Log likelihood ratio 48.22822 Probability 0.000000
============================================================
Test Equation:
LS // Dependent Variable is DCOUT
Date: 04/04/98 Time: 17:37
Sample: 1 25
Included observations: 25
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C -4.499507 1.028285 -4.375738 0.0003
WIND 4.052803 1.096749 3.695287 0.0014
Fitted^2 -13.20964 4.522482 -2.920883 0.0085
Fitted^3 4.820264 1.900918 2.535756 0.0197
Fitted^4 -0.659612 0.286779 -2.300075 0.0323
============================================================
R-squared 0.981767 Mean dependent var 1.609600
Adjusted R-squared 0.978120 S.D. dependent var 0.652278
S.E. of regression 0.096483 Akaike info criter-4.499916
Sum squared resid 0.186180 Schwarz criterion -4.256141
Log likelihood 25.77549 F-statistic 269.2287
Durbin-Watson stat 2.511711 Prob(F-statistic) 0.000000
============================================================
LS DCOUT C WIND WIND^2 WIND^3
============================================================
LS // Dependent Variable is DCOUT
Date: 04/04/98 Time: 17:43
Sample: 1 25
Included observations: 25
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C -2.358550 0.438464 -5.379120 0.0000
WIND 1.426866 0.246758 5.782440 0.0000
WIND^2 -0.160366 0.042056 -3.813172 0.0010
WIND^3 0.006458 0.002211 2.920922 0.0082
============================================================
R-squared 0.976944 Mean dependent var 1.609600
Adjusted R-squared 0.973650 S.D. dependent var 0.652278
S.E. of regression 0.105881 Akaike info criter-4.345226
Sum squared resid 0.235428 Schwarz criterion -4.150206
Log likelihood 22.84186 F-statistic 296.6100
Durbin-Watson stat 2.122609 Prob(F-statistic) 0.000000
============================================================
Here is the correct specification:
LS DCOUT C 1/WIND
============================================================
LS // Dependent Variable is DCOUT
Date: 04/04/98 Time: 17:45
Sample: 1 25
Included observations: 25
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C 2.978860 0.044902 66.34096 0.0000
1/WIND -6.934547 0.206434 -33.59215 0.0000
============================================================
R-squared 0.980025 Mean dependent var 1.609600
Adjusted R-squared 0.979156 S.D. dependent var 0.652278
S.E. of regression 0.094171 Akaike info criter-4.648660
Sum squared resid 0.203970 Schwarz criterion -4.551150
Log likelihood 24.63479 F-statistic 1128.433
Durbin-Watson stat 2.269181 Prob(F-statistic) 0.000000
============================================================
SCAT RESID 1/WIND
Ramsey Reset Test With One Fitted Term. Note that we do
not reject
the null hypothesis that we have a linear specification (that is, the
y-hat-squared variable is not statistically significant).
============================================================
Ramsey RESET Test:
============================================================
F-statistic 0.773032 Probability 0.388784
Log likelihood ratio 0.863364 Probability 0.352799
============================================================
Test Equation:
LS // Dependent Variable is DCOUT
Date: 04/04/98 Time: 17:49
Sample: 1 25
Included observations: 25
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C 2.671045 0.352996 7.566788 0.0000
1/WIND -6.072507 1.002167 -6.059378 0.0000
Fitted^2 0.046004 0.052323 0.879222 0.3888
============================================================
R-squared 0.980703 Mean dependent var 1.609600
Adjusted R-squared 0.978949 S.D. dependent var 0.652278
S.E. of regression 0.094639 Akaike info criter-4.603194
Sum squared resid 0.197046 Schwarz criterion -4.456929
Log likelihood 25.06647 F-statistic 559.0351
Durbin-Watson stat 2.400475 Prob(F-statistic) 0.000000
============================================================
MPG.WF1 Examples
LS MPG C SPEED SPEED^2
============================================================
LS // Dependent Variable is MPG
Date: 04/04/98 Time: 17:52
Sample: 1 12
Included observations: 12
============================================================
Variable CoefficienStd. Errort-Statistic Prob.
============================================================
C -182.5821 17.67703 -10.32878 0.0000
SPEED 8.983214 0.761563 11.79575 0.0000
SPEED^2 -0.091071 0.007993 -11.39408 0.0000
============================================================
R-squared 0.947388 Mean dependent var 32.00000
Adjusted R-squared 0.935696 S.D. dependent var 6.809085
S.E. of regression 1.726658 Akaike info criter 1.304694
Sum squared resid 26.83214 Schwarz criterion 1.425921
Log likelihood -21.85543 F-statistic 81.03174
Durbin-Watson stat 2.102570 Prob(F-statistic) 0.000002
============================================================
SCAT RESID SPEED^2
Under View select Representation
Estimation Command:
=====================
LS MPG C SPEED SPEED^2
Estimation Equation:
=====================
MPG = C(1) + C(2)*SPEED + C(3)*(SPEED^2)
Substituted Coefficients:
=====================
MPG = -182.58214 + 8.9832143*SPEED - 0.091071429*(SPEED^2)
Under View select Coefficient Test then select
Wald Test. Note that this is testing a null hypothesis
that the most efficient speed is 50 mph. We do not reject this
null hypothesis.
====================================================
Wald Test:
Equation: Untitled
====================================================
Null HypothesisC(3)=-C(2)/100
====================================================
F-statistic 3.068953 Probability 0.113713
Chi-square 3.068953 Probability 0.079801
====================================================