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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.