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Part 17: Multiple Regression – Part 1 17-1/26 Statistics and Data Analysis Professor William Greene Stern School of Business IOMS Department Department of Economics

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Part 17: Multiple Regression – Part 1 17-2/26 Statistics and Data Analysis Part 17 – Multiple Regression: 1

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Part 17: Multiple Regression – Part 1 17-3/26

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Part 17: Multiple Regression – Part 1 17-4/26

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Part 17: Multiple Regression – Part 1 17-5/26

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Part 17: Multiple Regression – Part 1 17-6/26

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Part 17: Multiple Regression – Part 1 17-7/26

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Part 17: Multiple Regression – Part 1 17-8/26

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Part 17: Multiple Regression – Part 1 17-9/26

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Part 17: Multiple Regression – Part 1 17-10/26

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Part 17: Multiple Regression – Part 1 17-11/26

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Part 17: Multiple Regression – Part 1 17-12/26

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Part 17: Multiple Regression – Part 1 17-13/26 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

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Part 17: Multiple Regression – Part 1 17-14/26 Concept of Multiple Regression Different conditional means Application: Monets 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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Part 17: Multiple Regression – Part 1 17-15/26 Monet in Large and Small 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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Part 17: Multiple Regression – Part 1 17-16/26 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.

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Part 17: Multiple Regression – Part 1 17-17/26 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.

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Part 17: Multiple Regression – Part 1 17-18/26 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.

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Part 17: Multiple Regression – Part 1 17-19/26 A Multiple Regression Ln Price = a + b1 ln Area + b2 (0 if unsigned, 1 if signed) + e b2

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Part 17: Multiple Regression – Part 1 17-20/26

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Part 17: Multiple Regression – Part 1 17-21/26 Monet Multiple Regression Regression Analysis: ln (US$) versus ln (SurfaceArea), Signed The regression equation is ln (US$) = 4.12 + 1.35 ln (SurfaceArea) + 1.26 Signed Predictor Coef SE Coef T P Constant 4.1222 0.5585 7.38 0.000 ln (SurfaceArea) 1.3458 0.08151 16.51 0.000 Signed 1.2618 0.1249 10.11 0.000 S = 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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Part 17: Multiple Regression – Part 1 17-22/26 Ceteris Paribus in Theory Demand for gasoline: G = f(price,income) Demand (price) elasticity: e P = %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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Part 17: Multiple Regression – Part 1 17-23/26 The Real World Data

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Part 17: Multiple Regression – Part 1 17-24/26 U.S. Gasoline Market, 1953-2004

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Part 17: Multiple Regression – Part 1 17-25/26 Shouldnt Demand Curves Slope Downward?

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Part 17: Multiple Regression – Part 1 17-26/26 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?

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Part 17: Multiple Regression – Part 1 17-27/26 How to Hold Income Constant? Multiple Regression Using Price and Income 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 It looks like the theory works.

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Part 17: Multiple Regression – Part 1 17-28/26 A Conspiracy Theory for Art Sales at Auction Sothebys and Christies, 1995 to about 2000 conspired on commission rates.

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Part 17: Multiple Regression – Part 1 17-29/26 If the Theory is Correct… Sold from 1995 to 2000 Sold before 1995 or after 2000

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Part 17: Multiple Regression – Part 1 17-30/26 Evidence The statistical evidence seems to be consistent with the theory.

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Part 17: Multiple Regression – Part 1 17-31/26 A Production Function Multiple Regression Model Sales of (Cameras/Videos/Warranties) = f(Floor Space, Staff)

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Part 17: Multiple Regression – Part 1 17-32/26 Production Function for Videos How should I interpret the negative coefficient on logFloor?

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Part 17: Multiple Regression – Part 1 17-33/26 An Application to Credit Modeling

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Part 17: Multiple Regression – Part 1 17-34/26 Age and Education Effects on Income

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Part 17: Multiple Regression – Part 1 17-35/26 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 +--------+--------------+-----------+

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Part 17: Multiple Regression – Part 1 17-36/26 Education and Income Effects

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Part 17: Multiple Regression – Part 1 17-37/26 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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