Chapter 13 Random Utility Models

Slide 13.11Random Random

Utility ModelsUtility ModelsMathematicalMathematicalMarketingMarketing

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



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



A Dichotomous Dependent Variable

Noif0

Yesif1yi

According to the regression model

yi = 0 + xi1 + ei

1i0ii x)y(Ey

We define



How Do Choice Probabilities Fit In?

]YesPr[]1yPr[p i1i

]NoPr[]0yPr[p i2i

1i

2i1ii

p

p)0(p)1()y(E

1i01i xp

From the definition of Expectation of a Discrete Variable



Two Requirements for a Probability

1p01i

1pp2i1i

Logical Consistency

Sum Constraint



A Requirement for Regression

0yfor)x(0

1yfor)x(1e

i1i0

i1i0i

V(ei) = E[ei – E(ei)]2

21i02i

21i01i

2i )x(p)x1(p)e(E

V(e) = 2I Gauss-Markov Assumption

Two possibilities exist

Since E(ei) = 0

by the Definition of E()



Heteroskedasticity Rears It’s Head

)x1)(x(

)p1(p

)p)(p1()p1(p

1i01i0

1i1i

21i1i

21i1i

21i02i

21i01i

2i )x(p)x1(p)e(E

Note that the subscript i appears on the right hand side!



Two Fixes

1xif1

1x0ifx

0xif0

p

1i0

1i01i0

1i0

1i

1i0 x 2

1i01i0p1i

dz2

zexp

2

1

)x()x(Fp

Linear Probability Model

Probit Model



The Logit Model Is A Third Option

1i0

1i0

1i0L1i xe1

xe

)x(Fp

0.0

0.2

0.4

0.6

0.8

1.0

ix

Slide 13.1010

Random Random Utility ModelsUtility Models

MathematicalMathematicalMarketingMarketing

The Expression for Not Buying

i

i

1i ue1

uep

i

i

i

i

i

1i2i

ue1

ue1

ueue1

ue1p1p

1

where ui = 0 + xi1

Slide 13.1111



The Logit Is a Special Case of Bell, Keeney and Little’s (1975) Market Share Theorem

J

mim

ij

ija

ap

ai1 = and ai2 = 1iue

For J = 2 and the logit model,

Slide 13.1212



My Share of the Market Is My Share of the Attraction

21

1

aaa

)1Pr(

where a1 is a function of Marketing Variables brought to bear on behalf of brand 1

Slide 13.1313



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 50:50.

Now a third competitor shows up: The Red Bus Company.

What will the shares be of the three companies now?

Slide 13.1414



The Model Can Be Linearized for Least Squares

i

i

1i ue1

uep

i

i

i

i

2i1i

ue

ueue1

uepp

1

1

1i0i2i1i xu)ppln(

Slide 13.1515



Aggregate Data and Weighted Least Squares

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

Slide 13.1616



Some Definitions

ni = fi1 + fi2

pi1 = fi1 / ni

)ppln(ˆ2i1i12,i l

)ppln( 2i1i12,i l

Slide 13.1717



Assumptions About Error

)ˆ(x 12,i12,i1i012,i lll

12,i12,iˆ)(E ll

2i1ii

12,i12,i12,i

ppn

1

)ˆ(V)(V

lll

2i1ii12,i12,i ppn1,0N~ll

Slide 13.1818



Weighted Least Squares: Scalar Presentation

2

i1i0iiError )xy(wSS

2i1iii ppnw

Slide 13.1919



Weighted Least Squares: Matrix Presentation

N

i

2

1

2N1N

2i1i

2221

1211

xx1

xx1

xx1

xx1

x

x

x

x

X

2

1

0

β

12,N12,212,1ˆˆˆ lll l

12,N12,212,1 lll l

Slide 13.2020



The Model Expressed in Matrix Terms

βx

βx

i

i

22i11i0

22i11i0

1i

e

e

e

e

1

xx1

xxp

Xβl

Slide 13.2121



Putting the Weights in Weighted Least Squares

2N1NN

22212

12111

ppn100

0ppn10

00ppn1

V

N

2

1

1

w00

0w0

00w

V

Slide 13.2222



Minimizing f leads to the WLS Estimator

)()(

)ˆ()ˆ(f

1

1

XβVXβ

V

ll

llll

l111 ][ˆ VXXVXβ

Slide 13.2323



The Variance of the WLS Estimator

][

][][)ˆ(V

1

111111

XVX

VXXVXVVXXVXβ

So this allows us to test hypotheses of the form H0: a - c = 0

aXVXa

a11 )(

cˆˆ

β

t

Slide 13.2424



Multiple DF Tests Under WLS

H0: A - c = 0

)ˆ(])([)ˆ(SS 111

HcβAAXVXAcβA

)ˆ()ˆ(SS 1

ErrorβXVβX ll

kn,q

Error

H F~kn/SS

q/SS

Slide 13.2525



ML Estimation of the Logit Model

βx

βx

i

i

11i0

11i0

1i

e

e

e

e

1

x1

xp

1

11 1210 ˆˆ

N

i

N

Niii ppl

N

i

ii

ii

yp

yp

1210

1)ˆ()ˆ(l

Two equivalent ways of writing the likelihood

We will use the left one, but isn't the right one clever?

Slide 13.2626



Likelihood Derivations

N

1Ni 11i0

N

1i 11i0

11i0

01

1

x1

1x

1

x

ee

el

.x

1lnx

)ln(L

1N

1i

N

1i

1i0

1i0

00

e

l

0LL

1

0

0

0

These first order conditions must be met:

Slide 13.2727



Second Order ML Conditions

01

0

2

10

0

2

1

0

0

LLL

00

0

00

0

2

0

0

0

11

LLLh

1)](E[)ˆ(V Hβ

When arranged in a matrix, the second order derivatives are called the Hessian.

Minus the expectation of the Hessian is called the Information Matrix.

Slide 13.2828



Three Choice Options

pi1 + pi2 + pi3 = 1

3i

2i

3i

1i

2i

1i

p

pln

p

pln

p

pln

Slide 13.2929



Multinomial Logit Model

J

m

mim

jij

ij

e

ep

βx

βx

J

mim

ij

ij

a

ap

)exp(ajijij

βx

the above model is a special case of the Fundamental Theorem of Marketing Share

Slide 13.3030



The Likelihood for the MNL Model

N

1NiiJ

N

1Ni2i

N

1i1i0

1J

2

1

1

ppp l

Slide 13.3131



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

Classic Example

j

iC

Slide 13.3232



MNL Example Model

4i3

3

i

1

i1i1

3i

1i CAV)CC(Ip

pln

5i3

3

i

2

i2i2

3i

2i BTR)CC(Ip

pln

J

m

jim

jij

ij

e

ep

βx

βx

Slide 13.3333



GLS Estimation of the Transportation Example

5

4

3

2

1

2

1

32

232

222

131

211

31

232

122

131

111

23,

23,2

23,1

13,

13,2

13,1

0010

0010

0010

0001

0001

0001

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

NNNN

NNNN

N

N

BTRCCI

BTRCCI

BTRCCI

CAVCCI

CAVCCI

CAVCCI

l

l

l

l

l

l

Xβl

Slide 13.3434



Other Choice Models

J

mim

ij

ij

a

ap

)xexp(ak

kijkij

k

kijkij

xa

)xexp(ak

jkijkij

k

jk

ijkij

xa

)xexp(aJ

m kmjkimkij

k

mjk

imkm

ijxa

Simple Effects Differential Effects Fully Extended

MNL

MCI

Slide 13.3535



Share Elasticity

ij

ij

ij

ij

ij p

x

x

pe

)xx(expaiJijjij

.)xexp(aijjij

1

mimij

ijij

ij aaxx

p

or

Slide 13.3636



The Derivative Looks Like

dea/da = ea )x(f)x(f

edx

)x(fde

)p1(p

pp

aaaaadx

pd

ijij

ij

2

ij

1

mimijij

2

mimij

ij

ij

Here we have used the following two rules:

Slide 13.3737



The Elasticity for the Simple Effects MNL Model

)p1(xeijijij

Putting the derivative back into the expression for the elasticity yields:

Chapter 13 Random Utility Models

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Transcript of Chapter 13 Random Utility Models