Part 21: Multiple Regression – Part 121-1/37

Statistics and Data Analysis

Professor William GreeneStern School of Business

IOMS DepartmentDepartment of Economics

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Statistics and Data Analysis

Part 21 – Multiple Regression: 1

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Women appear to assess health satisfaction differently from men.

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Or do they? Not when other things are held constant

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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

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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

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Monet in Large and Small

ln (SurfaceArea)

ln (U

S$)

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.

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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.

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Can we separate the two effects?

Average Prices Small LargeUnsigned 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.

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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.

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A Multiple Regression

ln (SurfaceArea)

ln (U

S$)

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

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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.

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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?

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The Real World Data

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U.S. Gasoline Market, 1953-2004

Year

Data

2001199319851977196919611953

5

4

3

2

1

logGlogIncomelogPg

Variable

Time Series Plot of logG, logIncome, logPg

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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

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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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3

Scatterplot of g vs Income

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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.

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A Conspiracy Theory for Art Sales at

Auction

Sotheby’s and Christies, 1995 to about 2000 conspired on commission rates.

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If the Theory is Correct…

ln (SurfaceArea)

ln (U

S$)

9876543

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Scatterplot of ln (US$) vs ln (SurfaceArea)

Sold from 1995 to 2000Sold before 1995 or after 2000

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Evidence

The statistical evidence seems to be consistent with the theory.

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A Production Function Multiple Regression Model

Sales of (Cameras/Videos/Warranties) = f(Floor Space, Staff)

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Production Function for Videos

How should I interpret the negative coefficient on logFloor?

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An Application to Credit Modeling

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Age and Education Effects on Income

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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+--------+--------------+-----------+

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Education and Income Effects

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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 Quadratic relationship between income and education