Transcript No Slide Title

6.1
Lecture #6
Studenmund(2006) Chapter 7
Objectives:
1. Suppressing the intercept
2. Alternative Functional forms
3. Scaling and units of measurement
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Y
^
Such an effect potentially biases the βs
and inflates their t-values
Estimated relationship
Suppressing the intercept
Yˆi '  ˆ1' X
True relation
Yˆi  ˆ0  ˆ1 X
X
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6.2
6.3
Regression through the origin
The intercept term is absent or zero.
i.e.,
Yi   1 X i   i
Yi
^ ^
SRF : Yi   1 X i
^
1
1
Xi
0
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6.4
Regression through the origin
The estimated model:
~ ~
Y  1 X i
~
~
Y  1 X i   i
or
Applied OLS method:
~
1 
 X iYi
2
 Xi
( )
^2
s
~
and Var 1   X2
i
and
s^ 2 
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 ~2
N -1
Some feature of no-intercept model
~
1.


2.
R2
3.
df
In practice:
i
need not be zero
can be occasions turn out to be negative.
may not be appropriate for the summary
of statistics.
does not include the constant term,
i.e., (n-k)
•1. A very strong priori or theoretical expectation, otherwise stick
to the conventional intercept-present model.
•2. If intercept is included in the regression model but it turns out
to be statistically insignificant, then we have to drop the intercept
to re-run the regression.
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6.5
Regression through origin
Y  
Y   0   1 X  i
 xy

1
 x2
^2
^
s
Var ( 1 ) 
x2
^

s^ 2 

~

X  ’ i
 XY
 X2
^
~
s2
Var ( 1 )
 X2
^ 2
n-2

1
1
N-k-1
[
 (X- X)(Y- Y)]

 (X- X)  (Y- Y)
( xy )
^2 
s
 ’
~2
n -1
N-k
2
R
or
2
2
R 
2
raw R 
2
2
2
x y
2
2
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 ( XY )
2
 X2  Y2
6.6
Example 1: Capital Asset Pricing Model (CAPM) security i’s
expect risk premium=expected market risk premium
( ER
expected
rate of
return on
security i
i
- r
f
)
risk free
of
return
1
( ER
m
- r
f
)
expected
rate of
return on
market
portfolio
 1 as a measure of systematic risk.
 1 >1 ==> implies a volatile or aggressive security.
 1 <1 ==> implies a defensive security.
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6.7
6.8
Example 1:(cont.)
ER i - r f
Security market line
1
1
ERm - f
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Example 2:
6.9
Covered Interest Parity
- e
F
N
 fN
(i - i ) 
e
*
International interest rate differentials equal exchange rate
forward premium.
i.e.,
i - i*
- e
F
(i - i )   1 (
)
e
*
Covered interest
parity line
1
1
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F -e
e
6.10
Example 2:(Cont.)
E (a 0)  0
in regression:
-e
F
(i - i )  a 0   1(
)  ui
e
*
If covered interest parity holds,
a0
is expected to be zero.
Use the t-test to test the intercept to be zero
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6.11
y: Return on A Future Fund, %
X: Return on Fisher Index, %
Formal report:
^
Y 1.0899 X
(5.689)
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N=10
R2=0.714
SEE=19.54
6.12
H0 :  0 = 0
1.279 - 0
7.668
^
Y 1.2797  1.0691X
(0.166)
(4.486)
N=10
R2=0.715
SEE=20.69
The t-value shows that b0 is statistically insignificant different from zero
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6.13
Functional Forms of Regression
The term linear in a simple regression
model means that there are linear in the
parameters; variables in the regression
model may or may not be linear.
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6.14
True model is nonlinear
Y
Income
But run the wrong linear regression model
and makes a wrong prediction
SRF
PRF
X
15
60
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Age
Linear vs. Nonlinear
6.15
Examples of Linear Statistical Models
Yi = 0 + 1Xi + i
ln(Yi) = 0 + 1Xi + i
Yi = 0 + 1 ln(Xi) + i
Yi = 0 + 1X2i + i
Examples of Non-linear Statistical Models
Yi = 0 +
2
1Xi
+ i
2
Yi = 0 + 1Xi + i
Yi = 0 + 1Xi + exp(2Xi) + i
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6.16
Different Functional Forms
1. Linear
Attention to
each form’s
2. Log-Log
slope
and
3. Semilog
elasticity
• Linear-Log or Log-Linear
4. Polynomial
5. Reciprocal (or inverse)
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Functional Forms of Regression models
2. Log-log model:
Y 0 X
- 1
e
This is a nonlinear model
i
Transform into linear log-form:
ln Y ln 0 -  1 ln X  i
==>
ln Y 0 -  1 ln X  i
==>
Y  0   1 X   i
*
*
dY*
dX*
*
*
*
dY
 d ln Y  Y
dX
d ln X
X
*

where 1  -  1
 1*
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elasticity
coefficient
6.17
6.18
Functional Forms of Regression models
Quantity Demand
Quantity Demand
Y
lnY
-


Y
0X
X
price
Y
Y  0X
ln Y ln0 - 1 lnX
1
lnX
lnY
1
ln Y ln0  1 lnX
X
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price
lnX
6.19
Functional Forms of Regression models
3. Semi log model:
Log-lin model or lin-log model:
ln Yi  a0  a 1 X i   i
or
Yi  0  1 ln Xi   i
a1 
dY
relative change in Y
 d ln Y  Y  dY 1
dX
dX dX Y
absolute change in X
1 
absolute change in Y
and
relative change in X

dY
 dY
dX
d ln X
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X
1
Functional Forms of Regression models(Cont.)6.20
4. Polynomial: Quadratic term to capture the nonlinear pattern
Yi= 0 + 1 Xi +2X2i + i
Yi
Yi
1<0, 2>0
1>0, 2<0
Xi
Xi
5. Reciprocal (or inverse) transformations
1
Yi   0   1 (
)  i
Xi
==>
Yi   0   1 ( Xi )   i
*
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1
Where Xi 
Xi
*
6.21
Some features of reciprocal model
Y
Y  0  1
Y
1
X
0
0 > 0 and 1 > 0
0 > 0 and  1 < 0
0
0
Y
+
0
X
Y  0  1
0 < 0 and
0
1
X
1 > 0
X
X
-  1 / 0
Y  < and
0
0
1 < 0
X
0
-
0
-
-0
-  1 / 0
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6.22
Two conditions for nonlinear, non-additive equation
transformation.
1. Exist a transformation of the variable.
2. Sample must provide sufficient information.
Example 1:
Suppose
transforming
rewrite
Y  0   1 X 1  2 X 1   3 X 1 X 2
2
X2* = X12
X3* = X1X2
Y  0   1 X 1  2 X 2   3 X 3
*
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*
6.23
Example 2:
transforming
rewrite
Y  0 
X
*
1


1
X  
2
1
X  2
Y  0  1X
*
1
However, X1* cannot be computed, because  2 is unknown.
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Application of functional form regression
1. Cobb-Douglas Production function:





Y
K e
0L
1
2
Transforming:
==>
d ln Y  
1
d lnL
d ln Y  
2
d lnK
ln Y  ln  0   1 ln L   2 ln K  
ln Y   0   1 ln L   2 ln K  
: elasticity of output w.r.t. labor input
: elasticity of output w.r.t. capital input.
 1   2 > 1 Information about the scale of returns.
<
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6.24
6.25
2. Polynomial regression model:
Marginal cost function or total cost function
costs
2






  (MC)
i.e. Y
0
1X
2X
MC
y
or
costs
TC
2
3








  (TC)
Y 0
1X
2X
3X
y
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Slope
Summary
dY
(
)
dX
Model Equation
linear
Y  0   1 X
dY  
1
dX
Log-log ln Y   0   1 ln X
dY
d ln Y
 Y  1
d ln X dX
X
==>
dY
Y
 1( )
dX
X
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Elasticity
dY
( Y )
dX
X
 1 ( X)
Y
1
6.26
Slope
Summary(Count.)
Log-lin
ln Y   0   1 X
dY
d ln Y
 Y  1
dX
dX
dY  
1Y
dX
dY
dY

 1
Lin-log Y   0   1 ln X
d ln X dX
X
dY   1
==>
1
dX
X
dY
dY
1

 1
Reciprocal Y   0   1
1
1
d
( 2 ) dX
X
X
X
Elasticity
1X
==>
==>
dY   -1
1
dX
X2
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1 1
Y
 1 ( -1 )
XY
6.27
6.28
Linear model
^
GNP  100.304  1.5325M 2
(1.368)
(39.20)
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6.29
Lin-log model
^ = -1.6329.21 + 2584.78 lnM
GNP
2
(-23.44)
(27.48)
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6.30
Log-lin model
^ = 6.8612 + 0.00057 M
lnGNP
2
(100.38)
(15.65)
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6.31
Log-log model
^
ln GNP  0.5529  0.9882 ln M 2
(3.194)
(42.29)
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6.32
Wage(y)
^
wage=10.343-3.808(unemploy)
10.43
(4.862) (-2.66)
SRF
unemp.(x)
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6.33
Reciprocal Model
(1/unemploy)
1
^
Wage = -1.4282+8.7243 ( )
(-.0690) (3.063)
y
x
The 0 is statistically insignificant
Therefore, -1.428 is not reliable
uN: natural rate of unemployment
uN
1
( )
x
-1.428
SRF
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6.34
^
lnwage = 1.9038 - 1.175ln(unemploy)
(10.375)
(-2.618)
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6.35
^
Lnwage = 1.9038 + 1.175 ln
(10.37)
(2.618)
1
( )
X
Antilog(1.9038) = 6.7113, therefore it is a more meaningful
and statistically significant bottom line for min. wage
Antilog(1.175) = 3.238, therefore it means that one unit X increase
will have 3.238 unit decrease in wage
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6.36
(MacKinnon, White, Davidson)
MWD Test for the functional form (Wooldridge, pp.203)
Procedures:
1. Run OLS on the linear model, obtain Y^
^
^ + a
^ X + a
^ X
Y = a
0
1
1
2
2
^
2. Run OLS on the log-log model and obtain lnY
^ + 
^ ln X + ^
lnY = 
 ln X
^
0
1
^
1
2
2
^ - lnY
3. Compute Z1 = ln(Y)
4. Run OLS on the linear model by adding z1
^
Y = ^
a ’ + a^ ’X + a^ ’X + a^ ’ Z
0
1
1
2
2
3
1
and check t-statistic of a3’
If t*a^3 > tc ==> reject H0 : linear model
If t*a^3 < tc ==> not reject H0 : linear model
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6.37
MWD test for the functional form (Cont.)
^
^
5. Compute Z2 = antilog (lnY) - Y
6. Run OLS on the log-log model by adding Z2
^
lnY = ^0’ + ^1’ ln X1 + 2’ ln X2 + ^3’ Z2
^
and check t-statistic of ’
^
3
If t*^3 > tc ==> reject H0 : log-log model
If t*^3 < tc ==> not reject H0 : log-log model
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MWD TEST: TESTING the Functional form of regression
Example:(Table 7.3)
Step 1:
Run the linear model
and obtain
C
^
Y
X1
X2
CV1 =
^
s
_ = 1583.279
24735.33
Y
= 0.064
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6.38
6.39
Step 2:
Run the log-log model
and obtain
^
lnY
C
fitted
or
estimated
LNX1
LNX2
^
0.07481
s
_
CV2 =
=
= 0.0074
Y
10.09653
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6.40
Step 4:
H0 : true model
is linear
C
MWD TEST
X1
X2
tc0.05, 11 = 1.796
Z1
tc0.10, 11 = 1.363
t* < tc at 5%
=> not reject H0
t* > tc at 10%
=> reject H0
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6.41
Step 6:
H0 : true model is
log-log model
MWD Test
tc0.025, 11 = 2.201
C
LNX1
LNX2
Z2
tc0.05, 11 = 1.796
tc0.10, 11 = 1.363
Since t* < tc
Comparing the C.V. =
=> not reject H0
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C.V.1
C.V.2
0.064
=
0.0074
Criterion for comparing two different functional models:
The coefficient of variation:
^
s
C.V. =
Y
It measures the average error of the sample
regression function relative to the mean of Y.
Linear, log-linear, and log-log equations can be
meaningfully compared.
The smaller C.V. of the model,
the more preferred equation (functional model).
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6.42
6.43
Compare two different functional form models:
Model 1
linear model
Coefficient Variation
(C.V.)
Model 2
log-log model
^
s / Y of model 1
2.1225/89.612
=
^
s / Y of model 2
= 4.916
0.0217/4.4891
0.0236
=
0.0048
means that model 2 is better
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Scaling and units of measurement
Y   0   1 X + i
1
: the slope of the regression line.
1 =
Units of change of y
Units of change of x
if
D Y dY
=
or
D X dX
Y* = 1000Y
X* = 1000X
^ ^
then 1000 Y  1000 0  1 1000X  1000^i
==>
^*
^*
^*





Y
X

0
1
*
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6.44
6.45
Changing the scale of X and Y
R2 and the
t-statistics are
no change
in regression
results for 1
but all other
statistics are
change.
Yi = 0 + 1Xi + i
Yi/k = (0/k)+(1)Xi/k + i/k
*
*
Y = +
i
where
0
1Xi*+ *i
Y*i = Yi/k
*i = i/k
*0 = 0/k and X*i = Xi/k
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6.46
Y
25
^
0 *
5
^
0
X
10
50
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6.47
Changing the scale of x
The estimated
coefficient and
standard error
change but the
other statistics
are unchanged.
Yi = 0 + 1Xi + i
Yi = 0 + (k1)(Xi/k) + i
Yi = 0 +
*
 X*+
1
i
i
where
*
 =
1
k1
and
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X*i = Xi/k
6.48
Y
5
^
0
X
10
50
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6.49
Changing the scale of Y
Yi = 0 + 1Xi + i
All statistics
are changed
except for
the t-statistics
and R2.
Yi/k = (1/k) + (1/k)Xi + i/k
*
*
*
Y = + X
i
0
1
*
+

i
i
* =  /k
*

Y
=
Y
/k
where i
i
i
i
*0 = 0/k and 1*= 1/k
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6.50
Y
25
5
^
0
X
10
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Effects of scaling and units change
The values of i, SEE, RSS will be affected.
But
t-statistic
F-statistic
R2
will not be affected.
All properties of OLS estimations are also
unaffected.
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6.51
6.52
^
Both in
GPDIB
billion
measure:
…B: Billion of 1972 dollar
 -37.001  0.1739GNPB
(-0.485)
(3.217)
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6.53
Both in
million
measure:
GPD^IM  -37001.52  0.1739GNPM
(-0.485)
(3.217)
…M: Million of 1972 dollar
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6.54
GPD^IM  -37001.52  173.9491GNPB
(-0.485)
(3.217)
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6.55
GPD^IB  -37.0015  0.00017GNPM
(-0.485)
(3.217)
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The “ex-post” and “ex ante” forecasting:
6.56
For example: Suppose you have data of C and Y from 1947–1999.
And the estimated consumption expenditures for 1947-1995 is
1947 – 1995:
^
Ct = 238.4 + 0.87Yt
Given values of Y96 = 10,419; Y97 = 10,625; … Y99 = 11,286
The calculated predictions or the “ex post” forecasts are:
^
^
1996: C96 = 238.4 + 0.87(10,149) = 9.355
1997: C97 = 238.4 + 0.87(10,625) = 9.535.50
…..
^
1999: C99 = 238.4 + 0.87(11285) = 10,113.70
The calculated predictions or the “ex ante” forecasts base on the
assumed value of Y2000=12000:
^
2000: C2000 = 238.4 + 0.87(12,000) = 10678.4
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