Chapter 13 Random Utility Models

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Chapter 13 Random Utility Models

• This chapter covers choice models applicable
  where the consumer must pick one brand out of J
  brands. The sequence we will go through
  includes
• Terminology
• Aggregate Data and Weighted Least Squares
• Disaggregate Data and Maximum Likelihood
• Three or More Brands
• A Model for Transportation Mode Choice
• Other Choice Models
• Patterns of Competition

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Key Terminology

• Dichotomous dependent variable
• Polytomous dependent variable
• Income type independent variable and the
  polytomous logit model.
• Price type independent variable and the
  conditional logit model.
• Aggregate data
• Disaggregate dat

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A Dichotomous Dependent Variable
We define
According to the regression model yi ?0
xi?1 ei
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How Do Choice Probabilities Fit In?
From the definition of Expectation of a Discrete
Variable
?
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Two Requirements for a Probability
Logical Consistency
Sum Constraint
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A Requirement for Regression
V(e) ?2I
Gauss-Markov Assumption
Two possibilities exist
Since E(ei) 0
V(ei) Eei E(ei)2
by the Definition of E(?)
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Heteroskedasticity Rears Its Head
Note that the subscript i appears on the right
hand side!
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Two Fixes
Linear Probability Model
Probit Model
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The Logit Model Is A Third Option
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The Expression for Not Buying
where ui ?0 xi?1
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The Logit Is a Special Case of Bell, Keeney and
Littles (1975) Market Share Theorem
For J 2 and the logit model,
ai1 and ai2 1
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My Share of the Market Is My Share of the
Attraction
where a1 is a function of Marketing Variables
brought to bear on behalf of brand 1
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The Story of the Blue Bus and the Red Bus
Imagine a market with two players the Yellow Cab
Company and the Blue Bus Company. These two
companies split the market 5050. Now a third
competitor shows up The Red Bus Company. What
will the shares be of the three companies now?
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The Model Can Be Linearized for Least Squares
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Aggregate Data and Weighted Least Squares
Response Response
Population Yes (yi 1) No (yi 0) x
1 f11 f12 x1
2 f21 f22 x2

i fi1 fi2 xi

N fN1 fN2 xN
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Some Definitions
ni fi1 fi2 pi1 fi1 / ni
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Assumptions About Error
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Weighted Least Squares Scalar Presentation
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Weighted Least Squares Matrix Presentation
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The Model Expressed in Matrix Terms
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Putting the Weights in Weighted Least Squares
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Minimizing f leads to the WLS Estimator
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The Variance of the WLS Estimator
So this allows us to test hypotheses of the form
H0 a?? - c 0
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Multiple DF Tests Under WLS
H0 A? - c 0
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ML Estimation of the Logit Model
Two equivalent ways of writing the likelihood
We will use the left one, but isn't the right one
clever?
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Likelihood Derivations
These first order conditions must be met
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Second Order ML Conditions
When arranged in a matrix, the second order
derivatives are called the Hessian. Minus the
expectation of the Hessian is called the
Information Matrix.
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Three Choice Options
pi1 pi2 pi3 1
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Multinomial Logit Model
the above model is a special case of the
Fundamental Theorem of Marketing Share
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The Likelihood for the MNL Model
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Classic Example
  Ii Income of household i Cost (price) of
alternative j for household i CAVi Cars per
driver for household i BTRi Bus transfers
required for member of household i to get to
work via the bus
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MNL Example Model
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GLS Estimation of the Transportation Example
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Other Choice Models
Simple Effects
Differential Effects
Fully Extended
MNL MCI
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Share Elasticity
or
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The Derivative Looks Like
Here we have used the following two rules
dea/da ea
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The Elasticity for the Simple Effects MNL Model
Putting the derivative back into the expression
for the elasticity yields