Wednesday October 20
• Summary of Monday’s class: R2
• Today’s class material
Summary of Monday’s class:
R2, the coefficient of determination
A measure between 0 and 1 of how well the
data fit the model
R2: 0: poor fit; 1: perfect fit
R2 is the fraction of the variation in Yi that is
explained by the model
Definition:
R2 = 1 −
P 2
ei
ESS
=1−P
TSS
(Yi − Ȳ )2
Extreme values for R2:
P
1. R2 = 1 : ⇒
e2
i = 0 ⇒ all residuals are
zero ⇒ all points are on the regression line
P 2
P
=0: ⇒
ei = (Yi − Ȳ )2 ⇒ β̂0 = Ȳ
2.
and β̂1 = 0 ⇒ regression line is horizontal
R2
Examples
Faculty salaries as a function of “productivity”
(1973 article)
Ŝi = 11, 155 + 230Bi + 18Ai + 102Ei
+489Di + 189Yi + . . .
where
Si = salary of the ith professor in
dollars per year
Bi = number of books published
Ai = number of articles published
Ei = number of excellent articles published
Di = number of dissertations supervised
Yi = number of years of teaching experience
1. coefficient signs?
2. book, two excellent articles, or three dissertations?
3. size of coefficients?
GRE scores: are GRE and SAT biased against
women and ethnic groups?
d i = 172.4+
GRE
39.7 Gi +
(10.9)
78.9 GP Ai
(10.4)
+0.203 SAT M i + 0.110 SAT V i
(0.071)
(0.058)
where
GRE i
Gi
GP Ai
SAT M i
SAT V i
=
=
=
=
=
score of ith student on test
1 if student is male, 0 otherwise
GPA in economics classas
score on SAT-mathematical
score on SAT-verbal
Things to note on previous transparency
1. Dummy variable: a variable that takes on
the values 0 and 1 only (male/female dummy)
This means that if Gi = 1 - i.e. for a man
- we predict a higher GRE score than if
Gi = 0 (i.e. a woman)
2. Between parentheses: standard errors (we
get those from Eviews output too!)
Standard errors measure the statistical uncertainty in the coefficient
cf. margin of error in the Bush-Kerry election
95% confidence interval (likely values for the
coefficient):
coefficient plus or minus 1.96 times the standard error
Example: estimated GRE equation
coefficient for Gi (gender: 1 male, 0 female):
39.7
suggests 39.7 extra GRE points for males
standard error: 10.9
Note: 39.7 is more than 1.96 standard errors
away from 0
Conclusion: the positive coefficient for Gi is
statistically “remote” from 0
Conclusion: we have statistical evidence that
men score higher on the GRE
Revisit GRE example:
d i = 172.4+
GRE
39.7 Gi +
(10.9)
78.9 GP Ai
(10.4)
+0.203 SAT M i + 0.110 SAT V i
(0.071)
(0.058)
where
GRE i
Gi
GP Ai
SAT M i
SAT V i
=
=
=
=
=
score of ith student on test
1 if student is male, 0 otherwise
GPA in economics classes
score on SAT-mathematical
score on SAT-verbal
Mileage per gallon of various cars
Ĝi = 22.008 −0.002Wi − 2.76Ai + 3.28Di + 0.415Ei
(0.001) (0.71)
(1.41) (0.097)
where
Gi = mileage per gallon as tested
Wi = gross weight of car, in pounds
Ai = 1 if car has automatic transmission,
0 otherwise
Di = 1 if car has diesel engine,
0 otherwise
Ei = EPA estimate of mileage per gallon
Material for Midterm, Wednesday October 27:
1. All your classnotes, exercises
2. Chapters 1, 2 and 3 of Studenmund; not:
standard errors
3. Know where to find β̂0 and β̂1 in Eviews
output; know interpretation of these coefficients
4. You will get a question asking you to calculate β̂0 and β̂1