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Regression ModelsProfessor William Greene

Stern School of Business

IOMS Department

Department of Economics

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Regression and Forecasting Models

Part 6 – Multiple Regression

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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 (

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.

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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 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.

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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 (

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 = b0 + 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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6

5

4

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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Application: WHO

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?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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An Elaborate Multiple Loglinear Regression Model

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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 (

US$)

9876543

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

Sold from 1995 to 2000

Sold 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 Age Effects on Income

Effect on log Income of 8 more years of education