Chapter 3
Extensions of Simple
Regression Model
Lectured by Mr. CHHAY Khunlong
E-mail:chhayk@gmail.com
1
CK@Econometrics
1 Units of measurement and scaling
.
y Interpretation of regression results must
include the units of measurement
y Changing the units a measurement
affects the magnitude of estimated
coefficients, but not r2
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Example
The example compares measuring GDPI and
G P
f $ 992
f $ 992
D in millions o 1
to billions o 1
Gross Private Domestic Investment and GDP, USA, 1988-1997
YEAR GPDIBL
GPDIM
GDPB
GDPM
1988
828 2
828200
5865 2
5865200
.
.
1989
863.5
863500
6062
6062000
1990
815
815000
6136.3
6136300
1991
738 1
738100
6079 4
6079400
.
.
1992
790.4
790400
6244.4
6244400
1993
863.6
863600
6389.6
6389600
1994
975.7
975700
6610.7
6610700
1995
996.1
996100
6761.6
6761600
1996
1084.1
1084100
6994.8
6994800
1997
1206.4
1206400
7269.8
7269800
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Example
If Y* = w1Y and X* = w2X then from OLS
formulae
b2* = (w1/w2) b2 and se(b2*) = (w1/w2) se(b2)
b1* = (w1) b1
and se(b1*) = (w1) se(b1)
s* = (w1) s
Example: GDPIBL=1000*GDPIM, GDPB=1000*GDPM
Both In billions of dollars:
GDPI-hat =-1026.50+
0.3016GDPt
Se =
257.587
0.0399
GDPI in billions and GDP In Millions of dollars:
GDPI-hat =-1026.50+
0.000301GDPt
S
257 587
0 0000399
e =
.
.
4
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Example
y If coefficient magnitudes are too large or small to report,
f
then you can rescale them by changing the units o the
variables for example changing Y (GDPI) from billions to
millions (w = 1000 and w = 1) multiplies b and b by
1
2
1
2
1000 (column 4 ↓)
Both in
Both in
GDPI in bill GDPI in mill
billions
millions
GDP in mill
GDP in bill
Intercept
-1026.5
(257 6)
-1,026,500
(257 600)
-1026.5
(257 6)
-1,026,500
(257 600)
(b1)
.
-3.985
,
-3.985
.
-3.985
,
-3.985
0.3016
0.3016
0.00030
301.6
Slope
(b2)
(0.0399)
7.559
(0.0399)
7.559
(0.00004)
7.559
(39.90)
7.559
r2
0 877
0 87