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### 45-734 PROBABILITY AND STATISTICS II (4th Mini AY1997-98)Note Set 5 Handouts

1. MILITARY.WF1 Examples

LS GNP C MILMOB
```============================================================
LS // Dependent Variable is GNP
Date: 04/04/98   Time: 21:54
Sample: 1915 1988
Included observations: 74
============================================================
Variable      CoefficienStd. Errort-Statistic  Prob.
============================================================
C           3.025348   0.547696   5.523772   0.0000
MILMOB        3.697863   0.547363   6.755782   0.0000
============================================================
R-squared            0.387966    Mean dependent var 3.061841
Adjusted R-squared   0.379466    S.D. dependent var 5.980692
S.E. of regression   4.711230    Akaike info criter 3.126553
Sum squared resid    1598.089    Schwarz criterion  3.188825
Log likelihood      -218.6839    F-statistic        45.64060
Durbin-Watson stat   1.266900    Prob(F-statistic)  0.000000
============================================================
```
To test whether or not the period 1915-1945 has the same linear specification as the period 1946-1988, we perform the Chow Breakpoint Test.

CHOW 1946
```============================================================
Chow Breakpoint Test: 1946
============================================================
F-statistic          0.079328    Probability        0.923820
Log likelihood ratio 0.167533    Probability        0.919646
============================================================
```
So we do not reject the null hypothesis.

2. WINDMILL.WF1 Examples.

```============================================================
LS // Dependent Variable is DCOUT
Date: 04/04/98   Time: 22:01
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
============================================================
```

HIST RESID

The p-value of the Jarque-Bera statistic is 0.341451. Hence, we would not reject the null hypothesis that the residuals are normally distributed. However, it is clear from residual plot that they are not normally distributed.

Indeed, doing a Chow breakpoint test here:

CHOW 12
```============================================================
Chow Breakpoint Test: 12
============================================================
F-statistic          24.48672    Probability        0.000003
Log likelihood ratio 30.08983    Probability        0.000000
============================================================
```
So we would reject the null hypothesis that the first 11 observations have the same linear structure as the last 14 observations.

The correct specification is:

LS DCOUT C 1/WIND
```============================================================
LS // Dependent Variable is DCOUT
Date: 04/04/98   Time: 22:13
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
============================================================
```

This looks a lot better. Now the Chow breakpoint test does not reject the null hypothesis that the first 11 observations have a different linear model from the last 14 observations.
```============================================================
Chow Breakpoint Test: 12
============================================================
F-statistic          1.053475    Probability        0.366442
Log likelihood ratio 2.390274    Probability        0.302662
============================================================
```
3. CIGGY.WF1 Examples.

Under View select Representation
```Estimation Command:
=====================
LS CARMON C TAR NICO WEIGHT

Estimation Equation:
=====================
CARMON = C(1) + C(2)*TAR + C(3)*NICO + C(4)*WEIGHT

Substituted Coefficients:
=====================
CARMON = 3.2021885 + 0.96257379*TAR - 2.6316601*NICO - 0.13048044*WEIGHT
```
Under View select Coefficient Test then select Wald Test. Note that this is testing a null hypothesis that the coefficients on nicotine and weight are both equal to zero. We do not reject this null hypothesis.
```====================================================
Wald Test:
Equation: Untitled
====================================================
Null HypothesisC(3)=0
C(4)=0
====================================================
F-statistic     0.233600    Probability     0.793708
Chi-square      0.467199    Probability     0.791679
====================================================
```