|_* SHAZAM code downloaded from UCLA Econ 143 (CAMERON) WebSite:
|_* HTML file called e143sh4.htm, and should have
|_* been downloaded as study.sha
|_* example of omitted variables bias: study time and midterm grade
|_* midterm=percent score on midterm exam
|_* study=hours spent studying for midterm in week before exam
|_* gpa=proxy for quality of advance preparation for coursework
|_sample 1 13
|_read midterm study gpa
3 VARIABLES AND 13 OBSERVATIONS STARTING AT OBS 1
Run a regression of MIDTERM on STUDY and GPA.
|_ols midterm study gpa
REQUIRED MEMORY IS PAR= 2 CURRENT PAR= 500
OLS ESTIMATION
13 OBSERVATIONS DEPENDENT VARIABLE = MIDTERM
...NOTE..SAMPLE RANGE SET TO: 1, 13
R-SQUARE = 0.9319 R-SQUARE ADJUSTED = 0.9182
VARIANCE OF THE ESTIMATE-SIGMA**2 = 14.136
STANDARD ERROR OF THE ESTIMATE-SIGMA = 3.7598
SUM OF SQUARED ERRORS-SSE= 141.36
MEAN OF DEPENDENT VARIABLE = 78.692
LOG OF THE LIKELIHOOD FUNCTION = -33.9577
VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR 10 DF P-VALUE CORR. COEFFICIENT AT MEANS
STUDY 5.1352 0.8352 6.149 0.000 0.889 0.9808 0.3363
GPA 58.514 5.621 10.41 0.000 0.957 1.6605 2.5739
CONSTANT -150.32 23.27 -6.459 0.000-0.898 0.0000 -1.9102
Immediately after the OLS regression, issue a CONFID command, first for the coefficient on the STUDY variable (referred to by the variable it modifies). Note that the point estimate of the slope coefficient is in the middle of the row of numbers, the pair of 10% significance-level cutoffs is on either side, and the pair of 5% significance-level cutoffs is outermost. The output also gives the parameter standard error (which can also be seen in the second column of the regression output, and the 0.025 and 0.05 critical values of the relevant t-distribution.
|_confid study
USING 95% AND 90% CONFIDENCE INTERVALS
CONFIDENCE INTERVALS BASED ON T-DISTRIBUTION WITH 10 D.F.
- T CRITICAL VALUES = 2.228 AND 1.812
NAME LOWER 2.5% LOWER 5% COEFFICIENT UPPER 5% UPPER 2.5% STD. ERROR
STUDY 3.274 3.622 5.1352 6.649 6.996 0.835
Now issue a second CONFID command, this time for the coefficient on the GPA variable.
|_confid gpa
USING 95% AND 90% CONFIDENCE INTERVALS
CONFIDENCE INTERVALS BASED ON T-DISTRIBUTION WITH 10 D.F.
- T CRITICAL VALUES = 2.228 AND 1.812
NAME LOWER 2.5% LOWER 5% COEFFICIENT UPPER 5% UPPER 2.5% STD. ERROR
GPA 45.99 48.33 58.514 68.70 71.04 5.621
Now ask for a joint confidence ellipse, to assess plausible PAIRS of values for b2 and b3, as opposed to implausible values. (If the pair of values (0,0) lies within this confidence ellipse, we cannot reject the joint hypothesis that both slopes are simultaneously zero.) With a plain CONFID command, without a GNU option, you get a crummy dot-matrix plot.
|_confid study gpa
USING 95% AND 90% CONFIDENCE INTERVALS
CONFIDENCE INTERVALS BASED ON T-DISTRIBUTION WITH 10 D.F.
- T CRITICAL VALUES = 2.228 AND 1.812
NAME LOWER 2.5% LOWER 5% COEFFICIENT UPPER 5% UPPER 2.5% STD. ERROR
STUDY 3.274 3.622 5.1352 6.649 6.996 0.835
GPA 45.99 48.33 58.514 68.70 71.04 5.621
CONFIDENCE REGION PLOT FOR study AND gpa
USING F DISTRIBUTION WITH 2 AND 10 D.F. F-VALUE = 4.100
REQUIRED MEMORY IS PAR= 1 CURRENT PAR= 500
FOR MAXIMUM EFFICIENCY USE AT LEAST PAR= 4
205 OBSERVATIONS
Notice that the univariate confidence intervals are indicated by the + signs. Together, these two univariate confidence intervals form a "box." The joint confidence intervals, however, do not exactly cover the box. Some points that are inside the box (i.e. plausible values for each parameter taken individually) are outside the joint confidence ellipse (i.e. they are implausible pairs of values).
M=MULTIPLE POINT
10.000 |
9.5789 |
9.1579 |
8.7368 |
8.3158 |
7.8947 |
7.4737 | * M*
7.0526 | MM*M* M
6.6316 | + MMMM + MM
6.2105 | *MMM MM
5.7895 | MMM MMM
5.3684 | MMM MMM
4.9474 | MM* * MMM
4.5263 | *MM MMM
4.1053 | MM *MM*
3.6842 | MM MMMM
3.2632 | MM + *MMM* +
2.8421 | M M*MM*
2.4211 | **M **
2.0000 |
________________________________________
40.000 50.000 60.000 70.000 80.000
GPA
For a crisp line-plot of the confidence ellipse, Shazam for Windows makes it easy to display a gnuplot in the context of running the program. Here, I have added commfile= and datafile= options so that I can save the gnuplot for later modification.
|_confid study gpa / gnu lineonly commfile=conf.gnu &
| datafile=conf.dat
USING 95% AND 90% CONFIDENCE INTERVALS
CONFIDENCE INTERVALS BASED ON T-DISTRIBUTION WITH 10 D.F.
- T CRITICAL VALUES = 2.228 AND 1.812
NAME LOWER 2.5% LOWER 5% COEFFICIENT UPPER 5% UPPER 2.5% STD. ERROR
STUDY 3.274 3.622 5.1352 6.649 6.996 0.835
GPA 45.99 48.33 58.514 68.70 71.04 5.621
CONFIDENCE REGION PLOT FOR study AND gpa
USING F DISTRIBUTION WITH 2 AND 10 D.F. F-VALUE = 4.100
REQUIRED MEMORY IS PAR= 1 CURRENT PAR= 500
FOR MAXIMUM EFFICIENCY USE AT LEAST PAR= 4
205 OBSERVATIONS
SHAZAM WILL NOW MAKE A PLOT FOR YOU
NO SYMBOLS WILL BE PLOTTED, LINE ONLY
..INPUT FILE COMPLETED..TYPE A NEW COMMAND OR TYPE: STOP
TYPE COMMAND
Here is a screen shot of what you see when you ask for a gnuplot of the joint confidence ellipse. Does it look like the pair of values (0,0) is inside the ellipse? Therefore, is H0: b2=b3=0 a plausible hypotheses? Could it be that neither study time nor gpa has any effect on midterm score?
