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### 45-734 PROBABILITY AND STATISTICS II (4th Mini AY1997-98)Note Set 6 Handouts

1. 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
============================================================
```

2. 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
============================================================
```

3. 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
====================================================
```