Statistics and Data Analysis

1
Statistics and Data Analysis

• Professor William Greene
• Stern School of Business
• IOMS Department
• Department of Economics

2
Statistics and Data Analysis
Part 18 Multiple Regression 2
3
Multiple Regression Models
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Part 17 Multiple Regression2

• Using Minitab To Compute A Multiple Regression
• Basic Multiple Regression
• Using Binary Variables
• Logs and Elasticities
• Hedonic Regression and Interpretation
• Trends in Time Series Data
• Using Quadratic Terms to Improve the Model

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Application WHO
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• 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 a ß1EDUC ß2HealthExp e

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The (Famous) WHO Data
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7
Specify the Variables in the Model
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8

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Graphs? Maybe
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Regression Results
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Practical Model Building
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• 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
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• What happens when you leave a crucial
  variable out of your model? (Bad things)

Regression Analysis g versus GasPrice (no
income) The regression equation is g 3.50
0.0280 GasPrice Predictor Coef SE Coef
T P Constant 3.4963 0.1678 20.84
0.000 GasPrice 0.028034 0.002809 9.98
0.000 Regression Analysis G versus GasPrice,
Income The regression equation is G 0.134 -
0.00163 GasPrice 0.000026 Income Predictor
Coef SE Coef T P Constant
0.13449 0.02081 6.46 0.000 GasPrice
-0.0016281 0.0004152 -3.92 0.000 Income
0.00002634 0.00000231 11.43 0.000
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Using Dummy Variables
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• 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 a ß1 EDUC ß2 PCHexp
  ß3 OECD e

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
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According to these results, after accounting for
education and health expenditure differences,
people in the OECD countries have a life
expectancy that is 1.19 years shorter than people
in other countries.
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Binary Variable in Regression
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The regression shifts down by 1.191 years for the
OECD countries
NonOECD DALE 36.770 2.9962 EDUC
.005079 PCHExp
OECD DALE 36.770 2.9962 EDUC
.005079 PCHExp 1.191
We set PCHExp to 1000, approximately the sample
mean.
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Plotting
For DALE_NonOECD, remove -1.191
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Two Plots
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Dummy Variable in Log Regression
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• E.g., Monets signature equation
• LogPrice a ß1 logArea ß2 Signed
• Unsigned PriceU exp(a) Areaß1
• Signed PriceS exp(a) Areaß1 exp(ß2)
• Signed/Unsigned exp(ß2)
• Difference 100(Signed-Unsigned)/Unsigned
• 100exp(ß2) 1

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The Signature Effect 253
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100exp(1.2618) 1 1003.532 1 253.2
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Monet Paintings in Millions
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Difference is about 253
Predicted Price is exp(4.1221.3458logArea1.2618
Signed) / 1000000
21

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Dummy Variable for One Observation
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ProofsSee p. 40.

• Single out one observation for special attention.
• The equation will predict that observation
  perfectly.
• For the other coefficients, it is the same as
  removing that observation from the sample.

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A London Effect on UK Electronic Store Sales?
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Observation 2 is LondonFit Actual, Residual0.
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Logs in Regression
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Elasticity
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• The coefficient on log(Area) is 1.346
• For each 1 increase in area, price goes up by
  1.34 - even accounting for the signature effect.
• The elasticity is 1.34
• Remarkable. Not only does price increase with
  area, it increases faster than area.

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Monet By the Square Inch
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Elasticities of Demand for Gasoline
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Logs and Elasticities
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• Theory In the equationy a ß1x1 ß2x2
  ßKxK e
• ß (change in y) / (unit change in x)
• Elasticity ß mean of x / mean of y
• When the variables are in logs change in logx
  change in x
• log y a ß1 log x1 ß2 log x2 ßK
  log xK e
• Elasticity ß
• These will often give approximately the same
  answer.
• When in doubt, use logs.

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Elasticities
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Price elasticity -0.02070 Income
elasticity 1.10318
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A Set of Dummy Variables
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• 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 Colleges
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Nancy Burnett Journal of Economic Education,
1998
Reputationaß1Religious ß2GenderEcon
ß3EconFac ß4North
ß5South ß6Midwest ß7West e
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Minitab to the Rescue
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Unordered Categorical Variables
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House price data (fictitious) Type 1 Split
levelType 2 RanchType 3 ColonialType 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
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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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Ordered Categories
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• Health Satisfaction1Poor, 2So_so, 3OK,
  4Good, 5Great
• How to handle such a variable?
• Just use as is? No, So_so Poor 1, but this is
  not equal to Great Good 1 (necessarily)
• Use 4 of the indicator variables.
• Coding. It is not useful to consider
  modifications of the variable, such as
  -2,-1,0,1,2 or 2,4,6,8,10. None make sense as
  this is just a label. Could also use 1,4,8,17,26
  which would also make no sense.
• This needs a special kind of model if it is the
  dependent variable not a regression equation.

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Hedonic Regression
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• A theory of prices
• Price sum of prices for components
• House price
• Land size
• Rooms Fixed amount per room
• Swimming pool
• View
• N car garage
• Etc.
• Computers
• Speed
• Screen size
• Other features

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Fumiro Computer Data
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41
Transform Manufacturer Names to Indicator
Variables
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Calc ? Make Indicator Variables
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Hedonic Regression
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Time Trends in Regression
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• y a ß1x ß2t e ß2 is the year to
  year increase not explained by anything
  else.
• log y a ß1log x ß2t e (not log t,
  just t) 100ß2 is the year to year
  increase not explained by anything else.

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Time Trend Regression
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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. But, of course, these other
things do not remain unchanged.
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Nonlinear Equation
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• Using a quadratic (like using logs)
• y a ß1x ß2x2 e
• Usually ß1 gt 0.
• If ß2 gt 0 If ß2 lt 0

y
y
x
x
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A Quadratic Income vs. Age Regression
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-------------------------------------------------
--- LHSHHNINC 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
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Case Study A Huge Sports Contract
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• Alex Rodriguez hired by the Texas Rangers for
  something like 25 million per year.
• 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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PDV of the Costs
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• Using 8 discount factor
• Accounting for all costs
• 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
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• 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?
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• 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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A Regression Model
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• Based on 10 years of baseball data on wins and
  attendance
• Approximately (depends on your model)
• This years attendance
• team specific constant
• 20,000 Number of Wins
• 6,000 Last Years Number of Wins
• .42 Last Years Attendance
• error

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Marginal Value of a Win
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• Roughly, increase in this years attendance if
  the team wins one more game
• (20,000 6,000) / (1 - .42)
• About 45,000 fans per year per win

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Marginal Value of an A Rod
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• 8 games 45,000 fans 360,000 fans
• 360,000 fans
• 18 per ticket
• 2.50 parking etc.
• 1.80 stuff (hats, bobble head dolls,)
• 8.0 Million per year !!!!! Its not close.
  (Marginal cost is about 16.5M / year)

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Postscripts
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• (1) Texas was not out of last place for a single
  day while A-Rod was on the team. Was it worth it?
  You make the call.
• (2) What about the Yankees they now pay most of
  the same costs. Is it worth it? How would you
  find out?
• (3) What about that David Beckham contract with
  Major League Soccer?

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Summary
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• Using Minitab To Compute a Regression
• Building a Model
• Logs
• Dummy variables
• Qualitative variables
• Trends
• Quadratics
• Effects across time
• All Assuming You Know the Right Variables!