Nested logit and GEV models

Example: Demand for Pharmaceuticals, anti-

inflammatory drugs

Drug 11 Drug 12 Drug 13

Drug21 Drug22

Group 1 Group 2

Anti-inflammatory drugs

• Level1A:Eddiksyrederivater:• Level Ak:Confortid, Indocid,,,,,• Level 1B: Oksikamer• Level Bk:Brexidol,,,,• Level 1C: Propionsyrederivater• Level Ck: Iboprofen,Naproxen,,,• Level 1D:Koksiber• Level Dk: Celebra,,,

Other examples

• To evade taxes or not

• Given evasion, how many hours of work in regular and irregular jobs

• Given no tax evasion, how many hours of work in regular jobs

Other examples

• Travels; public or private

• Given public; train, bus or airplane

• Given private; own car or rental car

Other examples

• Wine; from Spain or Italy

• Given Spain; what brand

• Given Italy; what brand

Why nested logit

• A natural tree decision structure

• Within one branch, correlation across alternatives (with drugs, sideffect may be correlated)

• No correlation across branches

Software programs

• Stata, not so good,

• SAS seems ok

• Gauss, of course

• TSP also good

• LIMDEP, perhaps

The generalized extreme value model: GEV

• G is homogenous of degree 1

• The kth partial derivative of the G-function exist, is continuous, non-negative if k is odd, and non-positive if k is even, and

iy 1 i mlim G(y , , y , , , y ) , i 1, 2, , ,m

Then if

1 i nx x x1 i nF(x , , , x , , , x ) exp G(e , , , e , , , e )

• is a multivariate distribution function, the choice probabilities that result from the maximization of the random utilities for which the multivariate distribution function is given by F(.) are equal to

j

j j1

j j1

j j1

v vv

j j k m k kj

v vvj

v vv

lnG(e ,,,e ,,,e )P P(v max (v )v

G(e ,,,e ,,,e ) / v; j 1,2,,,m

G(e ,,,e ,,,e )

Example 1

• Multinomial Logit

j1 m kmv vv v

k 1G(e ,,,e ,,,e ) e

j

k

v

mjv

k 1

e

eP

j j j

j j

j

v = v (q )

withv (q )

= -b < 0q

¶

·

k k

k k

j j

j j j

jm

v (q )j j

k 1

j

sj

j jjj j

j j

sjs

mv (q )

k 1

v (q )

j

s

j

s

0

ln v (q ) ln

P1 1b P ) 0P q e

P1 bPP q

q PP bq (1 ) 0

P q

P bq P 0

P e

e ( b) b(1

P

Example 2

• A nested structure

• Two branches,

• In branch 1, one alternative

• In branch 2, two alternatives, with correlations in the tasteshifters

2 1V31 k3

v v /v v

k 2G(e ,e ,e ) e e

2corr( ) 1

corr( ) 0,k 2,3

Choice probailities

• The GEV model

k

3v /

k 2

11

1

j

jk

1 k

vv

11 v

/j

j

j

k

13 v /v /

k 23

v /v

k 2

3 vv /

k 2

e

j 2,3

, j 2,3

1 G(.) 1 eeG(.) v G

e

1 G(.) 1P { } ,G(.) v G

P

P

e e

e e

e e

Derivaties and elasticities

• The nested- or rather the corrlation structure- has a strong impact on the price elasticities

1

1 k

jk

j

k1 k

v11

31 1 v /v

k 2

13 v /v /

v /k 2j

33v /j j v /v

k 2k 2

jj| j 2,3 j

j j

P1 1b e ( b) b(1 )P q

e e

e eP1 e 1 1 1( 1) ( b) ( b) ( b)

P qe e e

P1 1( 1)P 1 P ( b); j 2,3P q

fo

P

jj

j j

rP1 b(1 ), j 2,3

P qP

jk

1 k

1

1 k

3 v /v /

1 k 2j3

v /v1 j

k 2

j v13

v /v1j

k 2

jss|s 2,3

sj

;

{ e } eP1 b( ) b ; j 2,3

P qe e

P1 1 e ( b) b j 2,3P q

e e

P1 b( 1)P P ( ); j,s 2,3, j sP q

P

P

Nested logit.

• Ujk=vjk+jk• j: indicates upper level (Level 1: Groups of

pharmaceutical, Lj)

• k: indicates drugs at lower level

• kLj

• We will use the GEV structure:

j

jk /1n 2n11 1 21 2

j

j2 vv vv v

j 1 k L

jk jr j

jk ir 0

G(e ,,,e ,e ,,,e ) e

corr( ) 1 ;r k

corr( ) ; j i,allk&r

j

j j

jk j jr j jk j

j j

ik i ik i

i i

jrjr jr i 1,2 k L ik

1

v / v / v /

vk L k L

jr 2 2v / v /

i 1 k L i 1 k L

1n 2n11 1 21 2

1n 2n11 1 21 2

v vv v

v vv vP P U max max U

e e ee

P

e e

G(e ,,,e ,e ,,,e ) / v

G(e ,,,e ,e ,,,e )

i

jr j

jk j

j

j

jk j

j

ik i

i

jr j

jk j

j

/

j r| jv /

k L

v /

k L

j2

v /

i 1 k L

v /

r| jv /

k L

PP

e

e

P

e

eP

e

Two stage version of nested logit

jk j

jj

j j

i i

jr jjr j

jk j jk j

j j

v /j k L jk j

k L

S /

j 2S /

i 1v /

v /jr|j j jv / v /

jr jk L k L

rj r|j j

S E max U ln e ; j 1,2

ePe

S 1 1 eP e ;r Lv e e

P P P

i

jv /jk j jk j

jk Lj

j

iv /ik ii ik ik Li

i

j

jk j jr j jk j

j j

ik i

i

v /ln e

k L

j2ln e v /2

i 1 k Li 1

1

v / v / v /

k L k L

jr2

v /

i 1 k L

eeP

ee

e e e

P

e

i

j

jr j

jk jik i

ji

v /

j r|j2 v /v /

k Li 1 k L

e P P

ee

The Likelihood

s s1 2

s,1,r s,2,rs N s N

L P ( ) P (