1

Page 1

OLS Examples

Page 2

OLS Regression

• Problem– The Kelley Blue Book provides information on

wholesale and retail prices of cars. Following are age and price data for 10 randomly selected Corvettes between 1 and 6 years old. Here, age is in years, and price is in hundreds of dollars.

240340243270230299340210195205price

4154522666age

2

Page 3

OLS Regression

Coding Sheet for Corvette Data

Variable Possible Values Source Mnemonic

Age of Corvettes Years Kelley Blue Book,

various issued age

Price of Corvettes

Hundred of Dollars, Nominal

IBID. price

Page 4

3

Page 5

Note Mean and Std. Dev.

of PRICE

SXX SYY

Note from Stat 101:

( )

41827.539

6.2568111

2

==

−=

−−

= ∑nS

nYY

s YY

Page 6

4

Page 7

OLS SpecificationLS price C age

Form for OLS command:LS dependent variable C independent variable(s)

orEquation eq1.LS dependent variable C independent variable(s)

Page 8

Section 1

Section 2…..Standard table layout

Section 3

5

Page 9

AGE*27.90291 - 371.6019 Price Estimated =

:Yfor Equation

Page 10

160

200

240

280

320

360

0 1 2 3 4 5 6 7AGE

PRIC

E

PRICE of Corvettesvs.

AGE of Corvettes

AGE*27.90291 - 371.6019 Price Estimated =

Residual

6

Page 11

SSE = Sum of Squared Residuals = 1623.709s = Standard Error of Regression = 14.24653

Note:

14.24653202.96363s

202.96363210

1623.7092n

SSEs2

==

=−

=−

=

Page 12

Standard Errors of Estimates

28892.5630.9

14.24653Sss

XXAGE

==

=

Note:

7

Page 13

tc and p-values

Both Highly Significant

10.887292.56288927.90291tAGE −=

−=

Note:

Page 14

8

Page 15

Sum is zero

SSE

Page 16

No Explanatory Variable

9

Page 17

R2

Fc and p-value

0.93677525681.601623.7091

SSSE1

SSTSSE1R

YY

2

=

−=

−=−=

Note:

Note:( ) 60.25681S Y-Y SST YY

2===∑

Page 18

InelasticVery -0.444798

-27.90291257.2

4.1 = ηPrice

=

∗

10

Page 19

-0.444798Elasticity

Inelastic4.1-27.90291PriceClassificationMeanEstimateVariable

Elasticity Summary Table

Interpretation: Price of a corvette is inelastic with respect to the age of the corvette so that a 1% increase in age decreases the price by only 0.4%. Other factors are at play regarding the lower prices, but age is certainly a major factor as evidenced by the R2 of 0.94.

OLS Regression

Page 20

Other ExamplesProblem 1: CAPM

11

Page 21

Problem 1: CAPM

• Problem– Estimate , the systematic risk, for CAPM

• CAPM is

• An empirical version is

β

[ ]ftmifti r )|E(rβ r )|E(r −Ω+=Ω

0= α r - r = r~

r - r = r~ )σN(0,~ε

εr~βαr~

fmm

fii

2imii ++=

Page 22

• beta is key– Each security has a beta– Each portfolio has a beta

• Portfolio beta is weighted average of individual betas

Problem 1: CAPM

12

Page 23

• beta is proportionality factor

– Excess returns for security over riskless return is proportional to excess returns in market overriskless returns

– Excess is extra return to compensate for risk for not holding the market portfolio

• Premium on security is proportional to premium on the market portfolio

ftm

ftii r )|E(r

r )|E(rβ−Ω−Ω

=

Problem 1: CAPM

Page 24

• If

1 r )Ω|E(rr )Ω|E(rβ

ftm

ftii >

−−

= mi Premium Premium >⇒

then asset i is viewed as riskier than the market

Problem 1: CAPM

13

Page 25

beta Interpretation Example1 Moves With Market Conglomerates (AT&T)>1 More Volatile Companies Sensitive

To Macro Events(Autos)

0<beta<1 Less Volatile Mildly Sensitive ToMacro Events (ElectricUtilities)

<0 Move Counter To Goldmining StocksMarket

Problem 1: CAPM

Page 26

Coding Sheet for CAPM Data

Variable Possible Values Source Mnemonic

Excess returns for an overall stock market index of the total market in the UK.

Monthly, 1/80-12/99. Percentages* DataStream

(2000) rendmark

Excess returns on an index of 104 stocks in the cyclical consumer goods sector in

the UK. Monthly, 1/80-12/99. Percentages* IBID. rendcyco

Excess returns on an index of 104 stocks in the noncyclical consumer goods sector

in the UK. Monthly, 1/80-12/99. Percentages* IBID. rendncco

Excess returns on an index of 104 stocks in the information tech sector in the UK.

Monthly, 1/80-12/99. Percentages* IBID. rendit

Excess returns on an index of 104 stocks in the telcom sector in the UK. Monthly,

1/80-12/99. Percentages* IBID. rendtel

*Note: See calculations note.

14

Page 27

Problem 1: CAPM

• Excess returns are calculated as follows– Let pi be the closing price of the index at the

last trading day in month i and let ri be the one-month interest rate at the start of month i. Then the return vi of the index is and the excess return is vi – ri. The reported numbers are 100(vi – ri).

( ) 11 / −−−= iiii pppv

Page 28

15

Page 29

-40

-20

0

20

40

60

RENDTEL RENDNCCO RENDMARK RENDIT RENDCYCO

Exc

ess

Ret

urns

(%)

Excess Returns DistributionsStock Market Sectors (UK)

1980 - 1999

Note: Monthly data

Page 30

-30

-20

-10

0

10

20

80 82 84 86 88 90 92 94 96 98

Exc

ess

Ret

urns

(%)

Excess ReturnsOverall Stock Market Index (UK)

1980 - 1999

Note: Monthly data

16

Page 31

0

10

20

30

40

50

60

70

-30 -20 -10 0 10

Series: RENDMARKSample 1980M01 1999M12Observations 240

Mean 0.808884Median 1.204026Maximum 13.46098Minimum -27.86969Std. Dev. 4.755913Skewness -1.157801Kurtosis 8.374932

Jarque-Bera 342.5191Probability 0.000000

Excess ReturnsOverall Stock Market Index (UK)

1980 - 1999

Page 32

-40

-30

-20

-10

0

10

20

30

80 82 84 86 88 90 92 94 96 98

Exc

ess

Ret

urns

(%)

Excess ReturnsCyclical Consumers Goods Sector (UK)

1980 - 1999

Note: Monthly data

17

Page 33

0

10

20

30

40

-30 -20 -10 0 10 20

Series: RENDCYCOSample 1980M01 1999M12Observations 240

Mean 0.499826Median 0.309320Maximum 22.20496Minimum -35.56618Std. Dev. 7.849594Skewness -0.230900Kurtosis 4.531597

Jarque-Bera 25.59050Probability 0.000003

Excess ReturnsCyclical Consumer Goods Sector (UK)

1980 - 1999

Page 34

-40

-30

-20

-10

0

10

20

80 82 84 86 88 90 92 94 96 98

Exc

ess

Ret

urns

(%)

Excess ReturnsNoncyclical Consumer Goods Sector (UK)

1980 - 1999

Note: Monthly data

18

Page 35

-40

-20

0

20

40

60

80 82 84 86 88 90 92 94 96 98

Exc

ess

Ret

urns

(%)

Excess ReturnsInformation Technology Sector (UK)

1980 - 1999

Note: Monthly data

Page 36

-30

-20

-10

0

10

20

30

40

80 82 84 86 88 90 92 94 96 98

Exc

ess

Ret

urns

(%)

Excess ReturnsTelecommunications Sector (UK)

1980 - 1999

Note: Monthly data

19

Page 37

Page 38

Problem 1: CAPM

• Interpretation– The intercept is highly insignificant suggesting that the line goes

through the origin– The slope is highly significant and positive at the 0.0 level

indicating that the market rate of return (excess returns) have a positive effect on the returns for the cyclical consumer goods sector in the UK

– The R2 is 0.50 suggesting that 50% of the variation in returns in the cyclical consumer goods sector is accounted for by the market excess returns

– The model is significantly different from the naïve model as indicated by the p-value for F

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Page 39

Problem 1: CAPM

• Interpretation– Since this is a CAPM, the slope has the

interpretation of systematic risk• Since it is greater than 1.0, this indicates that the

cyclical consumer goods sector is riskier that the market as a whole

• When the market rises, the excess returns in this sector will rise more than the market

Page 40

Problem 1: CAPM

241.3363(0.0000)

FC

Notes: p-value in parenthese; *=significant

0.5035R2

1.17*(0.0000)

RendMark

-0.477(0.2188)

Intercept

ModelModel Portfolio

21

Page 41

Page 42

22

Page 43

Page 44

Other ExamplesProblem 2: Bank Wages

23

Page 45

Problem 2: Bank Wages

• Problem– Find the relationship between the salary of

employees at a major US bank and their years of education completed

Page 46

Coding Sheet for Bank Salary Data

Variable Possible Values Source Mnemonic

Current yearly salary of bank employees in US Nominal dollars SPSS, version

10 (2000) salary

Number of years of education completed Years IBID. educ

Natural log of salary logs Calculated as natural log of

salary logsalary

24

Page 47

0

20000

40000

60000

80000

100000

120000

140000

6 8 10 12 14 16 18 20 22

Cur

rent

Yea

rly S

alar

y ($

)

Education (Years Completed)

Current Yearly Salaryvs.

Education Completed

Page 48

25

Page 49

0

20000

40000

60000

80000

100000

120000

140000

Cur

rent

Yea

rly S

alar

y ($

)

Distribution of CurrentYearly Salary

Bank Employees (US)

Page 50

0

10

20

30

40

50

60

70

25000 50000 75000 100000 125000

Series: SALARYSample 1 474 IF GENDER=1Observations 258

Mean 41441.78Median 32850.00Maximum 135000.0Minimum 19650.00Std. Dev. 19499.21Skewness 1.629931Kurtosis 5.702920

Jarque-Bera 192.7741Probability 0.000000

Distribution ofCurrent Yearly Salary

Bank Employees (US)

26

Page 51

9.6

10.0

10.4

10.8

11.2

11.6

12.0

Nat

ural

Log

of C

urre

nt Y

ear S

alar

y

Distribution of Log ofCurrent Yearly SalaryBank Employees (US)

Page 52

0

4

8

12

16

20

24

28

32

36

10.0 10.5 11.0 11.5

Series: LOGSALARYSample 1 474 IF GENDER=1Observations 258

Mean 10.54461Median 10.39967Maximum 11.81303Minimum 9.885833Std. Dev. 0.398579Skewness 0.840303Kurtosis 2.805992

Jarque-Bera 30.76731Probability 0.000000

Distribution of Log ofCurrent Yearly Salary

Bank Employees (US)

27

Page 53

• Our model is

or

• Interpret the slope parameter as the percentage increase in salary (S) due to one additional year of education

Problem 2: Bank Wages

( ) ii10i εEDUCββSALARYln ++=

( )dx

SdSdx

Sd=

ln

ii1 εEducβi eAeSalary =

Page 54

Salary rises 9% for each additional year of education

28

Page 55

Problem 2: Bank Wages

• Interpretation– The intercept and slope parameters are highly

significant at the 0.0 level• This indicates that education has a significant,

positive effect on salary– The higher the level of education, the higher the salary

• For each extra 1 year of education, salary rises 9%– Almost 49% of the variation in (log) salary is accounted

for by education– The model is significantly different from the naïve

model as indicated by the p-value for F

Page 56

Problem 2: Bank Wages

225.2383(0.0000)

FC

Notes: p-value in parenthese; *=significant

0.4680R2

0.092*(0.0000)

Educ

9.22*(0.0000)

Intercept

ModelModel Portfolio

29

Page 57

Other ExamplesProblem 3:

Population and Economic Growth

Page 58

Problem 3:Population and Economic Growth

• Population growth is an important factor for long-term economic growth

• Malthus assumed (correctly) that population will grow geometrically (e.g., 1, 3, 9, 27, 81,…) while the food supply will increase arithmetically (e.g., 10, 10, 30, 40 …)

– Population would double every 24 years

• As a result of the faster growth of population, population growth would overtake food supply growth – but he didn’t specify when

30

Page 59

Problem 3:Population and Economic Growth

Food

Out

put a

ndPo

pula

tion

Time

Food

Population

Page 60

• Digression on compound growth– Compounding

– Time to double• What is the implied population growth rate, g, for

Malthus?• How long to double at 1.9%?

Problem 3:Population and Economic Growth

( )t0t g1PVFV +=

31

Page 61

• Curent population projections– The Year 2000 population growth rate, g, was

about 1.4%• World population in 2000 was about 6.1Billion

people• Absolute growth: 85.4 Million in one year

Problem 3:Population and Economic Growth

Page 62

Problem 3:Population and Economic Growth

32

Page 63

Problem 3:Population and Economic Growth

Page 64

• The future– There is some hope ahead

• The UN forecasts a drop in fertility in developing countries

• Developed countries will grow by less than 1% annually

• Predicted world’s population in 2300– Old: 12 Billion– New: 9 Billion

Problem 3:Population and Economic Growth

33

Page 65

• Malthus had checks on population growth– Contraceptives– Abortion– Self-restraint– Wars

• Malthus wrote right at the time of the Industrial and Green Revolutions, so he could not see their implications

Problem 3:Population and Economic Growth

Page 66

• Data from the World Development Indicators, World Bank (2004)– Calculated average annual growth for GDP and

population

Problem 3:Population and Economic Growth

countries 107 ..., 1, i ,11963in Value2003in Valueg

401

i

ii =−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

34

Page 67

Coding Sheet for Growth Data

Variable Possible Values Source Mnemonic

Real GDP in country (1995 $US), 1963 and 2003 Real dollars

World Development

Indicators (World Bank,

2004)

gdp63, gdp03

Average annual growth in GDP Decimal values Calculated g

Population in country, 1963 and 2003 Count

World Development

Indicators (World Bank,

2004)

pop63, pop03

Average annual growth in population Decimal values Calculated p

Page 68

35

Page 69

-.02

.00

.02

.04

.06

.08

.10

.00 .01 .02 .03 .04

Rea

l GD

P A

vera

ge A

nnua

l Gro

wth

Population Average Annual Growthby Country (n = 107)

Real GDP and PopulationAverage Annual Growth

1963 - 2003

Page 70

Highly insignificant

Terrible R2

36

Page 71

-.02

.00

.02

.04

.06

.08

.10

.00 .01 .02 .03 .04

Rea

l GD

P A

vera

ge A

nnua

l Gro

wth

Population Average Annual Growthby Country (n = 107)

Real GDP and PopulationAverage Annual Growth

1963 - 2003