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
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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
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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)
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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
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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:
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Page 13
tc and p-values
Both Highly Significant
10.887292.56288927.90291tAGE −=
−=
Note:
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8
Page 15
Sum is zero
SSE
Page 16
No Explanatory Variable
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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
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Other ExamplesProblem 1: CAPM
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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.
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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
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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
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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
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-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
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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
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-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
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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
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-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
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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
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Page 43
Page 44
Other ExamplesProblem 2: Bank Wages
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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
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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
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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)
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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)
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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)
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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)
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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
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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
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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
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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 +=
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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
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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 =−⎟⎟
⎠
⎞⎜⎜⎝
⎛=
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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
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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
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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