Post on 14-Apr-2018
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Copyright 2006 The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin
Functional Forms
of Regression Models
chapter nine
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9-2
Time Trends and Growth Rates
Linear Trend Models Time series data
Test for trend over time
Test for breaks in a trend
Absolute changes over
time
Results for U.S.
population 1970-1999from Table 9-4
tt utBBY 21
9987.0
)1243.152)...(2718.743(
3284.29727.201
2
r
t
tYt
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9-3
Table 9-4
Population of United States (millions of people),1970-1999.
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9-4
Modeling Absolute Trends
Example: Appellate80-06.xlsNumber of court of appeals sham litigation decisions by year
1980-2006
Linear trend: Y = B1 + B2t + u
Non-linear trend: Y = B1 + B2t + B3t2 + u
Non-linear trend with break: Y = B1 + B2t + B3t2 +B4D + u
Non-linear trend with break and interaction (add B5Dt)
Test among models using F-test for difference in R
2
[(Ru
2 - Rr2)/m]/[(1 - Ru
2)/(n-k)]~Fm,n-k
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9-5
Compound Growth Rate
The Semilog Model Beginning value Y0Value at t Yt Compound growth rate r
Take natural log (base e)
Let B1 = lnY0 and B2 =ln(1+r)
B2 measures the yearlyproportional change in Y
tt
rYY 10
tt
t
utBBY
rtYY
21
0
ln
1lnlnln
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9-6
Semilog Model Example
Growth rate of USpopulation 1970-1999
US population increasedat a rate of 0.0098 per
yearOr a percentage rate of
100x0.0098 = 0.98%
See Fig. 9-3
Note lnYt is linear in t
9996.0
)98.285)....(39.8739(
0098.03170.5ln
2
r
t
tYt
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9-7
Figure 9-3
Semilog model.
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9-8
Instantaneous vs. Compound Growth Rate
b2 is estimate of ln(1 + r) where r is the compoundgrowth rate
Antilog (b2) = (1 + r) or r = antilog(b2)1
For US population: r = antilog(0.0098)
1Or r = 1.009481 = 0.00948
Compound growth rate of 0.948%
The instantaneous growth rate is usually reported,
unless the compound rate is specifically required.
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9-9
Log-linear Models and Elasticities
Consider this function forLotto expenditure that isnonlinear in X
Convert to a linear form bytaking natural logarithms
(base e) The result is a double-log or
log-linear model
Make a nonlinear model intoa linear one by a suitable
transformation Logarithmic transformation
iii
ii
B
ii
uXBBY
XBAY
AXY
lnln
lnlnln
21
2
2
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9-10
Log-linear Models and Elasticities
The slope coefficient B2 measures theElasticity of Y with respect to X
% change in Y for a % change in X
If Y is quantity demanded and X is price, thenB2 is the price elasticity of demand (Fig. 9-1)
In log form, Y has a constant slope in X, B2So the elasticity is also constant
Sometimes called a constant elasticity model
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9-11
Figure 9-1
A constant elasticity model.
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9-12
Lotto Example
Using data in Table 9-1, runOLS to estimate the log-
linear model
If income increases by one
%, expenditure on lottoincreases by 0.74 % on
average
Lotto exp. is inelastic wrt
income as 0.74 < 1 See Fig. 9-2 8644.0
)0001.0)........(2676.0(
)1440.7).......(1915.1()1015.0)......(5624.0(
)(ln7356.06702.0ln
2
r
p
tse
XY ii
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9-13
Table 9-1
Weekly lotto expenditure (Y
) in relation to weeklypersonal disposable income (X) ($).
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9-14
Figure 9-2
Log-linear model of Lotto expenditure.
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9-15
Example: Electricity Demand
See ElectricExcel2.xls. Calculate natural logarithms
Estimate the log-linear model by OLS
Note:
No change in hypothesis testing for log formOnly POP and PKWH coefficients are significant
R2 cannot be compared directly between linear and log-linear models
How to choose between models? Try not to use R2 alone
E l C bb D l P d ti
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9-16
Example: Cobb-Douglas Production
Function
See data in Table 9-2
Estimate Ln(GDP) as afunction of Ln(Employment)and Ln(Capital)
B2 and B3 are elasticities wrt
output B2 + B3 is the returns to
scale parameter
= 1 constant returns
> 1 increasing returns
< 1 decreasing returns
995.0
)06.9.......().........83.1.().........73.2(ln8460.0ln3397.06524.1
ln
)(ln)(lnln
2
32
33221
3232
R
tXXY
uXBXBBY
XAXY
ttt
tttt
B
t
B
tt
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9-17
Table 9-2
Real GDP, employment, and real fixed capital, Mexico,
1955-1974.
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9-18
Polynomial Regression Models
Estimating cost functions,when total and average cost
must have specific non-
linear shapes
Table 9-8 and Fig. 9-8
Cubic function or third-
degree polynomial
B1, B2, B4 >0
B3 < 0 B3
2
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9-19
Table 9-8
Hypothetical cost-output data.
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9-20
Figure 9-8
Cost-output relationship.
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9-21
Example
Does smoking have anincreasing or decreasing
effect on lung cancer?
Non-linear relationship
between cigarette smokingand lung cancer deaths
Table 9-9, data
Figure 9-9, regression results
Quadratic function or second
degree polynomial
iiii uXBXBBY 2
321
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9-22
Table 9-9
Cigarette smoking and deaths from various types of cancer.
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9-23
Figure 9-9
MINITAB output of regression (9.34).
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9-24
Table 9-11
Summary of functional forms.