Econ 140
Multiple Regression Applications
Lecture 14
Lecture 14
1
Today’s plan
Econ 140
• Relationship between R2 and the F-test.
• Restricted least squares and testing for the imposition of a
linear restriction in the model
Lecture 14
2
R2
Econ 140
• We know
R2 
Explained S of S
Total S of S
• We can rewrite this as
^
^
b
x
y

b
 1
 x2 y
2
1
2
R 
2
y
• Remember:
– If R2 = 1, the model explains all of the variation in Y
– If R2 = 0, the model explains none of the variation in Y
Lecture 14
3
R2 (2)
Econ 140
• We know from the sum of squares identity that
^
^
^2
 y  b1  x1 y b2  x2 y   e
• Dividing by the total sum of squares we get
^
y2 b
x y  b^
x y


2
y
Lecture 14
1
2
1
y
2
2
^2
2
 ^e

2
y
4
R2 (3)
Econ 140
• Thus
2
e
ˆ

1  R2 
2
y

or
2
e
ˆ

R2  1 
2
y

or
 eˆ 2  1  R 2
2
y

• If we divide the denominator and numerator of the F-test
by the total sum of squares:
bˆ1  x1 y  bˆ2  x2 y
2
2
y

F
2
e

n3
2
y
Lecture 14
5
F-stat in terms of R2
Econ 140
• The F-test for the joint hypothesis: H0: b1 = b2 = 0 can be
written in terms of R2:
R2 2
F
(1  R 2 ) n  3
• Recalling our LINEST (from L12.xls) output, we can
substitute R2 = 0.188
0.188 2
F
 3.82
1  0.188 33
– We would reject the null at a 5% significance level and
accept the null at the 1% significance level
Lecture 14
6
Relationship between R2 & F
Econ 140
• When R2 = 0 there is no relationship between the Y and X
variables
– This can be written as Y = a
– In this instance, we cannot reject the null and F = 0
• When R2 = 1, all variation in Y is explained by the X
variables
– The F statistic approaches infinity as the denominator
would equal zero
– In this instance, we always reject the null
Lecture 14
7
Restricted Least Squares
Econ 140
• Imposing a linear restriction in a regression model and reexamining the relationship between R2 and the F-test.
• Example of Cobb-Douglas production function
• In restricted least squares we want to test a restriction such
as
H0 :    1
Where our model is
ln Y  a   ln L   ln K  e
• We can write  = 1 -  and substitute it into the model
equation so that:
(lnY - lnK) = a + (lnL - lnK) + e
Lecture 14
8
Restricted Least Squares (2)
Econ 140
• We can rewrite our equation as: G = a +Z + e*
Where: G = (lnY - lnK) and Z = (lnL - lnK)
• The model with G as the dependent variable will be our
restricted model
– the restricted model is the equation we will estimate
under the assumption that the null hypothesis is true
Lecture 14
9
Restricted Least Squares (3)
Econ 140
• How do we test one model against another?
• We take the unrestricted and restricted forms and test them
using an F-test
• The F statistic will be
2*
2


e

e
q
ˆ
ˆ


F
 eˆ2  n  k 
–
–
–
–
* refers to the restricted model
q is the number of constraints
in this case the number of constraints = 1 ( + = 1)
n - k is the df of the unrestricted model
Lecture 14
10
Testing linear restrictions
Econ 140
• Estimation when imposing a linear restriction in the CobbDouglas log-linear model:
ln Y  a   ln L   ln K  e
• Test for constant returns to scale, or the restriction:
H0:  +  = 1
• We will use L13_1.xls to test this restriction - worked out
in L14.xls
Lecture 14
11
Testing linear restrictions (2)
Econ 140
• The unrestricted regression equation estimated from the
data (L13_1.xls) is:
ln Yˆ  0.488  0.674 ln L  0.447 ln K  e
(0.185)
0.026
0.030
• Note the t-ratios for the coefficients:
: 0.674/0.026 = 26.01
: 0.447/0.030 = 14.98
– compared to a t-value of around 2 for a 5% significance
level, both  &  are very precisely determined
coefficients
Lecture 14
12
Testing linear restrictions (3)
Econ 140
– adding up the regression coefficients, we have:
0.674 +0.447 = 1.121
– how do we test whether or not this sum is statistically
different from 1?
• Note the restriction:  = 1- 
• Our restricted model is:
(lnY - lnK) = a + (lnL - lnK) + e
or
G = a +Z + e*
Lecture 14
13
Testing linear restrictions (4)
Econ 140
• The procedure for estimation is as follows:
1. Estimate the unrestricted version of the model
2. Estimate the restricted version of the model
2
2*
ê
ê
3. Collect  for the unrestricted model and 
for the restricted model
2*
2
e

e
q
ˆ
ˆ


4. Compute the F-test
F


 eˆ2  n  k 
where q is the number of restrictions (in this case q = 1)
and (n-k) is the degrees of freedom for the unrestricted
model
Lecture 14
14
Testing linear restrictions (5)
Econ 140
• On L14.xls our sample n = 32 and an estimated
unrestricted model provides the following information:
ln Yˆ  0.488  0.674 ln L  0.447 ln K  e
(0.185)
0.026
0.030
R 2  0.996
2
e
ˆ
  .351
Lecture 14
15
Testing linear restrictions (7)
Econ 140
• The restricted model gives us the following information:
Gˆ  1.061  0.679Z
1.061 (0.048)
R 2  0.871
2*
e
ˆ
  1.228
• We can use this information to compute our F statistic:
F* = [(1.228 - 0.351)/1]/(0.359/29) = 72.47
Lecture 14
16
Testing linear restrictions (8)
Econ 140
• The F table value at a 5% significance level is:
F0.05,1,29 = 4.17
– Since F* > F0.05,1,29 we will reject the null hypothesis
that there are constant returns to scale
• Note that the test rejects constant return. As  +  > 1, we
might conclude there are increasing returns to scale.
• NOTE: the dependent variables for the restricted and
unrestricted models are different
– dependent variable in unrestricted version: lnY
– dependent variable in restricted version: (lnY-lnK)
Lecture 14
17
Testing linear restrictions (9)
Econ 140
• We can also use R2 to calculate the F-statistic by first
dividing through by the total sum of squares
• Using our definition of R2 we can write:


1  R   1  R  q R  R  q
F

1  R  n  k
1  R  n  k
2*
2
2
Lecture 14
2
2*
2
18
Testing linear restrictions (10)
Econ 140
• NOTE: we cannot simply use the R2 from the unrestricted
model since it has a different dependent variable
– What we need to do is take the expectation E(G|L,K)
• We need our unrestricted model to have the same
dependent variable G, or:
G  a  Z      1ln K  e
– Where G = (lnY - lnK), Z = (lnL - lnK)
– We can test this because we know that  +  - 1 = 0.121
since  +  = 1
– Estimating this unrestricted model will give us the
unrestricted R2
Lecture 14
19
Testing linear restrictions (11)
Econ 140
• From L14.xls we have :
R2* = 0.871
R2 = 0.963
• Our computed F-statistic will be
0.963  0.871 1 0.092
F

 72.47
1  0.963 29 0.0013
Lecture 14
20
Testing linear restrictions (12)
Econ 140
• On L14.xls we have 32 observations of output, employment,
and capital
– The spreadsheet has regression output for the restricted
and unrestricted models
– The R2 and sum of squares are in bold type
– F-tests on the restriction are on the bottom of the sheet
• We find that Excel gives us an F-statistic of 72.4665
– The F table value at a 5% significance level is 4.1830
– The probability that we could not reject the null given
this F-statistic is very small
Lecture 14
21
Testing linear restrictions (13)
Econ 140
• From this we can conclude that we have a model where
there are increasing returns to scale.
• We don’t know the true value, but we can reject the
restriction that there are constant returns to scale.
Lecture 14
22