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Statistics and Data Analysis

Professor William Greene

Stern School of Business

IOMS Department

Department of Economics

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Statistics and Data Analysis

Part 22 – Multiple Regression: 2

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Multiple Regression Models

Using Minitab To Compute A Multiple Regression Basic Multiple Regression Using Binary Variables Logs and Elasticities Trends in Time Series Data Using Quadratic Terms to Improve the Model Mini-seminar: Cost benefit test with a dynamic

model

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Application: WHO

Data Used in Assignment 1: WHO data on 191 countries in 1995-1999. Analysis of Disability Adjusted Life Expectancy = DALE EDUC = average years of education PCHexp = Per capita health expenditure

DALE = α + β1EDUC + β2HealthExp + ε

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The (Famous) WHO Data

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Specify the Variables in the Model

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Graphs

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Regression Results

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Practical Model Building

Understanding the regression: The left out variable problem

Using different kinds of variables Dummy variables Logs Time trend Quadratic

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A Fundamental Result What happens when you leave a crucial

variable out of your model? Nothing good.

Regression Analysis: g versus GasPrice (no income)The regression equation isg = 3.50 + 0.0280 GasPricePredictor Coef SE Coef T PConstant 3.4963 0.1678 20.84 0.000GasPrice 0.028034 0.002809 9.98 0.000Regression Analysis: G versus GasPrice, Income The regression equation isG = 0.134 - 0.00163 GasPrice + 0.000026 IncomePredictor Coef SE Coef T PConstant 0.13449 0.02081 6.46 0.000GasPrice -0.0016281 0.0004152 -3.92 0.000Income 0.00002634 0.00000231 11.43 0.000

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Using Dummy Variables Dummy variable = binary variable

= a variable that takes values 0 and 1. E.g. OECD Life Expectancies compared

to the rest of the world: DALE = α + β1 EDUC + β2 PCHexp

+ β3 OECD + ε

Australia, Austria, Belgium, Canada, Czech Republic, Denmark, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Japan, Korea, Luxembourg, Mexico, The Netherlands, New Zealand, Norway, Poland, Portugal, Slovak Republic, Spain, Sweden, Switzerland, Turkey, United Kingdom, United States.

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OECD Life Expectancy

According to these results, after accounting for education and health expenditure differences, people in the OECD countries have a life expectancy that is 1.191 years shorter than people in other countries.

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A Binary Variable in Regression

We set PCHExp to 1000, approximately the sample mean.

The regression shifts down by 1.191 years for the OECD countries

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Academic Reputation

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Dummy Variable in a Log Regression

E.g., Monet’s signature equation

Log$Price = α + β1 logArea + β2 Signed

Unsigned: PriceU = exp(α) Areaβ1

Signed: PriceS = exp(α) Areaβ1 exp(β2)

Signed/Unsigned = exp(β2)

%Difference = 100%(Signed-Unsigned)/Unsigned

= 100%[exp(β2) – 1]

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The Signature Effect: 253%

100%[exp(1.2618) – 1] = 100%[3.532 – 1] = 253.2 %

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Monet Paintings in Millions

Square Inches

Price

70006000500040003000200010000

30

25

20

15

10

5

0

01

Signed

Scatterplot of Price vs Square Inches

Predicted Price is exp(4.122+1.3458*logArea+1.2618*Signed) / 1000000

Difference is about 253%

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Logs in Regression

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Elasticity

The coefficient on log(Area) is 1.346 For each 1% increase in area, price goes up by

1.346% - even accounting for the signature effect. The elasticity is +1.346 Remarkable. Not only does price increase with

area, it increases much faster than area.

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Monet: By the Square Inch

Area

price

70006000500040003000200010000

20000000

15000000

10000000

5000000

0

Scatterplot of Price vs Area

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Logs and Elasticities

Theory: When the variables are in logs:

change in logx = %change in x

log y = α + β1 log x1 + β2 log x2 + … βK log xK + ε

Elasticity = βk

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Elasticities

Price elasticity = -0.02070 Income elasticity = +1.10318

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A Set of Dummy Variables

Complete set of dummy variables divides the sample into groups.

Fit the regression with “group” effects. Need to drop one (any one) of the

variables to compute the regression. (Avoid the “dummy variable trap.”)

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Rankings of 132 U.S.Liberal Arts CollegesReputation = α + β1Religious + β2GenderEcon + β3EconFac + β4North + β5South + β6Midwest + β7West + ε

Nancy Burnett: Journal of Economic Education, 1998

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Minitab does not like this model.

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Too many dummy variables If we use all four region dummies, a is redundant

Reputation = a + bn + … if north Reputation = a + bm + … if midwest Reputation = a + bs + … if south Reputation = a + bw + … if west

Only three are needed – so Minitab dropped west Reputation = a + bn + … if north Reputation = a + bm + … if midwest Reputation = a + bs + … if south Reputation = a + … if west

Why did it drop West and not one of the others? It doesn’t matter which one is dropped. Minitab picked the last.

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Unordered Categorical Variables

House price data (fictitious)

Style 1 = Split levelStyle 2 = RanchStyle 3 = ColonialStyle 4 = Tudor

Use 3 dummy variables for this kind of data. (Not all 4)

Using variable STYLE in the model makes no sense. You could change the numbering scale any way you like. 1,2,3,4 are just labels.

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Transform Style to Types

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House Price Regression

Each of these is relative to a Split Level, since that is the omitted category. E.g., the price of a Ranch house is $74,369 less than a Split Level of the same size with the same number of bedrooms.

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Better Specified House Price Model Using Logs

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Time Trends in Regression

y = α + β1x + β2t + ε β2 is the year to year increase not explained by anything else.

log y = α + β1log x + β2t + ε (not log t, just t) 100β2 is the year to year % increase not explained by anything else.

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Time Trend in Multiple Regression

After accounting for Income, the price and the price of new cars, per capita gasoline consumption falls by 1.25% per year. I.e., if income and the prices were unchanged, consumption would fall by 1.25%. Probably the effect of improved fuel efficiency

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A Quadratic Income vs. Age Regression+----------------------------------------------------+| LHS=HHNINC Mean = .3520836 || Standard deviation = .1769083 || Model size Parameters = 3 || Degrees of freedom = 27323 || Residuals Sum of squares = 794.9667 || Standard error of e = .1705730 || Fit R-squared = .7040754E-01 |+----------------------------------------------------++--------+--------------+--+--------+|Variable| Coefficient | Mean of X|+--------+--------------+-----------+ Constant| -.39266196 AGE | .02458140 43.5256898 AGESQ | -.00027237 2022.85549 EDUC | .01994416 11.3206310+--------+--------------+-----------+

Note the coefficient on Age squared is negative. Age ranges from 25 to 65.

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Implied By The Model

Careful: This shows the incomes of people of different ages, not the path of income of a particular person at different ages.

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Candidate Models for CostThe quadratic equation is the appropriate model.

Logc = a + b1 logq + b2 log2q + e

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A Better Model?

Log Cost = α + β1 logOutput + β2 [logOutput]2 + ε

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Case Study Using A Regression Model: A Huge Sports Contract

Alex Rodriguez hired by the Texas Rangers for something like $25 million per year in 2000.

Costs – the salary plus and minus some fine tuning of the numbers

Benefits – more fans in the stands. How to determine if the benefits exceed the

costs? Use a regression model.

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The Texas Deal for Alex Rodriguez

2001 Signing Bonus = 10M 2001 21 2002 21 2003 21 2004 21 2005 25 2006 25 2007 27 2008 27 2009 27 2010 27 Total: $252M ???

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The Real Deal Year Salary Bonus Deferred Salary 2001 21 2 5 to 2011 2002 21 2 4 to 2012 2003 21 2 3 to 2013 2004 21 2 4 to 2014 2005 25 2 4 to 2015 2006 25 4 to 2016 2007 27 3 to 2017 2008 27 3 to 2018 2009 27 3 to 2019 2010 27 5 to 2020 Deferrals accrue interest of 3% per year.

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Costs

Insurance: About 10% of the contract per year (Taxes: About 40% of the contract) Some additional costs in revenue sharing revenues from

the league (anticipated, about 17.5% of marginal benefits – uncertain)

Interest on deferred salary - $150,000 in first year, well over $1,000,000 in 2010.

(Reduction) $3M it would cost to have a different shortstop. (Nomar Garciaparra)

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PDV of the Costs

Using 8% discount factor Accounting for all costs including insurance Roughly $21M to $28M in each year from

2001 to 2010, then the deferred payments from 2010 to 2020

Total costs: About $165 Million in 2001 (Present discounted value)

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Benefits

More fans in the seats Gate Parking Merchandise

Increased chance at playoffs and world series Sponsorships (Loss to revenue sharing) Franchise value

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How Many New Fans?

Projected 8 more wins per year. What is the relationship between wins

and attendance? Not known precisely Many empirical studies (The Journal of

Sports Economics) Use a regression model to find out.

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Baseball Data 31 teams, 17 years (fewer years for 6 teams) Winning percentage: Wins = 162 * percentage Rank Average attendance. Attendance = 81*Average Average team salary Number of all stars Manager years of experience Percent of team that is rookies Lineup changes Mean player experience Dummy variable for change in manager

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Baseball Data (Panel Data – 31 Teams, 17 Years)

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A Regression Model

team

1

Attendance(team,this year) = α

+ γ Attendance(team, last year)

+ β Wins (team,this year)

2

3

+ β Wins(team, last year)

+ All_Stars(team, this year)

+ (team, this year)

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A Dynamic Equationy(this year) = f[y(last year)…]

Fans(t)=a+bWins(t)+cFans(t-1)+ (Loyalty effect)

Suppose Fans(0) = Fans0 (Start observing in a base year)

Suppose we fix Wins(t) at some Wins* and at 0 (no information).

What values do

es Fans(t) take in a sequence of years?

Fans(1) = a + bWins* + cFans0

Fans(2) = a + bWins* + c(a + bWins* + cFans0)

Fans(3) = a + bWins* + c(a + bWins* + c(a + bWins* + cFans0))

Fans(4) = a + bWins* + c(a

2 t-1 2 t-1 t

+ bWins* + c(a + bWins* + c(a + bWins* + cFans0)))

etc.

Collect terms: Fans(t) = a(1+c+c ... c ) bWins*(1+c+c ... c )+c Fans0

Suppose 0 < c < 1.

a bFans finally settles down at Fans* = + Wi

1-c 1-c

b dFans*ns*. =

1-c dWins *

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Example : Fans(t) = 227969+.6Fans(t-1) +11093Wins

Wins = 85/year

Fans(1990)=2,500,000

Fans(1991) = 227969 + .6(2.5M) + 11093(85) = 2.671M

Fans(1992) = ..... = 2.773M

...

Fans(2006) and years after = 2.93M

(This is the 'equilibrium')

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Example : Fans(t) = 227969+.6Fans(t-1) +11093Wins

Wins = 85/year to 2000, 93/year 2001...

Fans(1990) = 2,500,000

Fans(1991) = 2.671M

Fans(1992 to 2000) about 2.9M

...

Fans(2001) and years after about 3.15M

(This is the 'equilibrium)

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About 220,000 fans

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Marginal Value of One More Win

1 2 3

1 2 3

Our Model is Fans(t) = α + β Wins(t) + β Wins(t-1) + β AllStars + γFans(t-1)

Using the formula for the value of Fans*

α+β Wins*+β Wins*+β AllStarsFans*=

1-γ

The effect of one more Win every year would be 1 2

3

dFans*/dWins* = 1

The new player will definitely be an All Star, so we add this effect as well.

The effect of adding an All Star player to the team would be 1

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= .54914

1 = 11093.7

2 = 2201.2

3 = 14593.5

Effect of 1 more win

11093.7 2201.2= 32757

1 .59414Effect of adding an All Star

14593.5= 35957

1 .59414

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Marginal Value of an A Rod (8 games * 32,757 fans) + 1 All Star = 35957

= 298,016 new fans 298,016 new fans *

$18 per ticket $2.50 parking etc. (Average. Most people don’t park) $1.80 stuff (hats, bobble head dolls,…)

$6.67 Million per year !!!!! It’s not close. (Marginal cost is at least $16.5M / year)

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Summary Using Minitab To Compute a Regression Building a Model

Logs Dummy variables Qualitative variables Trends Effects across time Quadratic