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Part 17: Multiple Regression – Part 117-1/26
Statistics and Data Analysis
Professor William Greene
Stern School of Business
IOMS Department
Department of Economics
Part 17: Multiple Regression – Part 117-2/26
Statistics and Data Analysis
Part 17 – Multiple Regression: 1
Part 17: Multiple Regression – Part 117-3/26
Part 17: Multiple Regression – Part 117-4/26
Part 17: Multiple Regression – Part 117-5/26
Part 17: Multiple Regression – Part 117-6/26
Part 17: Multiple Regression – Part 117-7/26
Part 17: Multiple Regression – Part 117-8/26
Part 17: Multiple Regression – Part 117-9/26
Part 17: Multiple Regression – Part 117-10/26
Part 17: Multiple Regression – Part 117-11/26
Part 17: Multiple Regression – Part 117-12/26
Part 17: Multiple Regression – Part 117-13/26
Multiple Regression Agenda
The concept of multiple regression Computing the regression equation Multiple regression “model” Using the multiple regression model Building the multiple regression model Regression diagnostics and inference
Part 17: Multiple Regression – Part 117-14/26
Concept of Multiple Regression
Different conditional means Application: Monet’s signature
Holding things constant Application: Price and income effects Application: Age and education Sales promotion: Price and competitors
The general idea of multiple regression
Part 17: Multiple Regression – Part 117-15/26
Monet in Large and Small
ln (SurfaceArea)
ln (
US$)
7.67.47.27.06.86.66.46.26.0
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S 1.00645R-Sq 20.0%R-Sq(adj) 19.8%
Fitted Line Plotln (US$) = 2.825 + 1.725 ln (SurfaceArea)
Log of $price = a + b log surface area + e
Logs of Sale prices of 328 signed Monet paintings
The residuals do not show any obvious patterns that seem inconsistent with the assumptions of the model.
Part 17: Multiple Regression – Part 117-16/26
How much for the signature?
The sample also contains 102 unsigned paintings
Average Sale Price
Signed $3,364,248
Not signed $1,832,712
Average price of a signed Monet is almost twice that of an unsigned one.
Part 17: Multiple Regression – Part 117-17/26
Can we separate the two effects?
Average Prices
Small Large
Unsigned 346,845 5,795,000
Signed 689,422 5,556,490
What do the data suggest?
(1) The size effect is huge
(2) The signature effect is confined to the small paintings.
Part 17: Multiple Regression – Part 117-18/26
Thought experiments: Ceteris paribus
Monets of the same size, some signed and some not, and compare prices. This is the signature effect.
Consider signed Monets and compare large ones to small ones. Likewise for unsigned Monets. This is the size effect.
Part 17: Multiple Regression – Part 117-19/26
A Multiple Regression
ln (SurfaceArea)
ln (
US$)
7.67.47.27.06.86.66.46.26.0
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01
Signed
Scatterplot of ln (US$) vs ln (SurfaceArea)
Ln Price = a + b1 ln Area + b2 (0 if unsigned, 1 if signed) + e
b2
Part 17: Multiple Regression – Part 117-20/26
Part 17: Multiple Regression – Part 117-21/26
Monet Multiple Regression
Regression Analysis: ln (US$) versus ln (SurfaceArea), Signed The regression equation isln (US$) = 4.12 + 1.35 ln (SurfaceArea) + 1.26 SignedPredictor Coef SE Coef T PConstant 4.1222 0.5585 7.38 0.000ln (SurfaceArea) 1.3458 0.08151 16.51 0.000Signed 1.2618 0.1249 10.11 0.000S = 0.992509 R-Sq = 46.2% R-Sq(adj) = 46.0%
Interpretation (to be explored as we develop the topic):(1) Elasticity of price with respect to surface area is 1.3458 – very large
(2) The signature multiplies the price by exp(1.2618) (about 3.5), for any given size.
Part 17: Multiple Regression – Part 117-22/26
Ceteris Paribus in Theory
Demand for gasoline: G = f(price,income)
Demand (price) elasticity:eP = %change in G given %change in P holding income constant.
How do you do that in the real world? The “percentage changes” How to change price and hold income
constant?
Part 17: Multiple Regression – Part 117-23/26
The Real World Data
Part 17: Multiple Regression – Part 117-24/26
U.S. Gasoline Market, 1953-2004
Year
Data
2001199319851977196919611953
5
4
3
2
1
logGlogIncomelogPg
Variable
Time Series Plot of logG, logIncome, logPg
Part 17: Multiple Regression – Part 117-25/26
Shouldn’t Demand Curves Slope Downward?
G
GasP
rice
0.650.600.550.500.450.400.350.30
140
120
100
80
60
40
20
0
Scatterplot of GasPrice vs G
Part 17: Multiple Regression – Part 117-26/26
A Thought Experiment
The main driver of gasoline consumption is income not price
Income is growing over time.
We are not holding income constant when we change price!
How do we do that? Income
g
2750025000225002000017500150001250010000
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Scatterplot of g vs Income
Part 17: Multiple Regression – Part 117-27/26
How to Hold Income Constant?
Multiple Regression Using Price and Income
Regression Analysis: G versus GasPrice, Income
The regression equation isG = 0.134 - 0.00163 GasPrice + 0.000026 Income
Predictor 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
It looks like the theory works.
Part 17: Multiple Regression – Part 117-28/26
A Conspiracy Theory for Art Sales at
Auction
Sotheby’s and Christies, 1995 to about 2000 conspired on commission rates.
Part 17: Multiple Regression – Part 117-29/26
If the Theory is Correct…
ln (SurfaceArea)
ln (
US$)
9876543
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Scatterplot of ln (US$) vs ln (SurfaceArea)
Sold from 1995 to 2000
Sold before 1995 or after 2000
Part 17: Multiple Regression – Part 117-30/26
Evidence
The statistical evidence seems to be consistent with the theory.
Part 17: Multiple Regression – Part 117-31/26
A Production Function Multiple Regression Model
Sales of (Cameras/Videos/Warranties) = f(Floor Space, Staff)
Part 17: Multiple Regression – Part 117-32/26
Production Function for Videos
How should I interpret the negative coefficient on logFloor?
Part 17: Multiple Regression – Part 117-33/26
An Application to Credit Modeling
Part 17: Multiple Regression – Part 117-34/26
Age and Education Effects on Income
Part 17: Multiple Regression – Part 117-35/26
A Multiple 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 = .07040754 |+----------------------------------------------------++--------+--------------+--+--------+|Variable| Coefficient | Mean of X|+--------+--------------+-----------+ Constant| -.39266196 AGE | .02458140 43.5256898 EDUC | .01994416 11.3206310+--------+--------------+-----------+
Part 17: Multiple Regression – Part 117-36/26
Education and Income Effects
Part 17: Multiple Regression – Part 117-37/26
Summary
Holding other things constant when examining a relationship
The multiple regression concept Multiple regression model Applications:
Size and signature Model building for credit applications A cost function for banks Quadratic relationship between income and
education
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