Choice Models

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 1

Choice Models

Covered Topics

• Binary Choice

–LPM –logit

–logistic regresion –probit

• Multiple Choice

–Multinomial Logit

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 3

• Yes or No

• Buy or Not Buy

• Join or Not Join

• Own or Not Own

• Switch or Stay

Binary Choice

• Yes, No, Abstain

• Buy, Sell or No Action

• Buy Brand A, B, C or

• Join Plan X, Y or Z None

Multiple Choice

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 5

Note that all the choices

must be mutually exclusive and exhaustive. One and only one choice or event will occur.

Mutual Exclusiveness

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 6

Question: What determines the choice selection?

Model to determine the probability of an event under a given condition (value of independent variables)

Pr(choice#j)=F j (X 1 ,X 2 ,…,X K )

where X’s are determinants for the probability.

Choice Model (1)

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 7

Note that

1)

2) function F j () must return a value between 0 and 1

Choice Model (2)

∑ =

j

j ) 1

# Pr( choice

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 8

Example

JOIN=1 if the observation will join the government-run health insurance program

= 0, otherwise

Quantification of

Binary Choices

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 9

JA=1 if the observation will join Plan A = 0, otherwise

JB=1 if the observation will join Plan B = 0, otherwise

JC=1 if the observation will join Plan C = 0, otherwise

Note that JA+JB+JC=1 always.

Quantification of Multiple Choices

General Structure

Note that

Binary Choice Model

) ,..., ,

( )

1

Pr( JOIN = = F X 1 X 2 X K ) ,..., ,

( 1 ) 0

Pr( JOIN = = − F X 1 X 2 X K 1 ) ,..., ,

(

0 ≤ F X 1 X 2 X K ≤

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 11

Define

Assumption of LPM Linearity of F(.)

Note that there is no error term

Linear Probability Model (1)

K K X X

X

P = β 1 1 + β 2 2 + L + β ) 1

Pr( =

= JOIN P

Formulation of LPM

E(JOIN)=(1)P+(0)(1-P)=P

==> JOIN=P+v

where v is an error term. E(v)=0

=> OLS is valid but not the best. Why?

Linear Probability Model (2)

) 1

2 (

2 1

1 X X X v - - - -

JOIN = β + β + L + β K K +

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 13

Note that

V(ν) = V(JOIN) but

V(JOIN) =(1-P) 2 P+(0-P) 2 (1-P) = P(1-P)

==> V(ν) is not constant. It depends on the independent variables (X’s)

==> Violation of a CLRM assumption or ν is heteroscedastic

Linear Probability Model (3)

Define

where

Linear Probability Model (4)

) 1 (

1 P w P

= −

) 2

* (

*

* 2 2

* 1 1

*

- - - ν -

β β β

+ +

+ +

=

K K X

X X

JOIN

L

ν

ν w

K k

wX X

wJOIN JOIN

k k

=

=

=

=

*

*

*

,...,

for 1

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 15

Note that

==> OLS is BLUE for Model (2)

Linear Probability Model (5)

1

) 1 ) ( 1 (

1 ) ( )

( * 2

=

− −

=

=

P P P

P V w

V ν ν

Estimation of LPM

Step 1 run OLS for unweighted model (1)

==> JOIN = X β

Note that JOIN is the estimate for P

Linear Probability Model (6)

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 17

Step 2 compute the weight

Step 3 compute

Step 4 estimate the weighted model (2) using OLS

Linear Probability Model (7)

) 1

( 1

JOIN JOIN

w = −

*

* 2

* 1

* , X , X ,..., X K JOIN

Step 5 re-compute JOIN using the new set of β .

Note that LPM does not assure that or

Linear Probability Model (8)

1 ) ,..., ,

( 0

1 0

2

1 ≤

≤

≤

≤

X K

X X F

JOIN

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 19

Correction

If X β <0, set JOIN=0 If X β >1, set JOIN=1

Linear Probability Model (9)

Linear Probability Model (10)

P

X β 1

1

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 21

Linear Probability Model (11)

• Less expensive in computer time. No non-linear equations

• is the effect of X on the probability. In general, the explanatory variables should be unitless or are expressed in percentage

k

X k

P = β

∂

∂

Assumption of Logit F() is a logistic function

Note that always.

Logit Model (1)

K K Z

X X

X Z

P e

β β

β + + +

=

= + −

L

2 2 1 1

1 1

1 ) ( 0 ≤ F Z ≤

No error term

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 23

Logit Model (2)

1.0 0.5

Z e − Z

+ 1

1

0

Note that OLS does not apply ML Estimation of Logit model or

Note that Y=JOIN

Logit Model (3)

∑

∏

=

=

−

−

− +

=

−

=

n i

i i

i i n

i

Y i Y

i

P Y

P Y L

P P

L

^{i i}

1 1

) 1 (

)]

1 ln(

) 1 ( ) ln(

[ ln

max

) 1 ( ) ( max

β

β

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 25

Note that

First-order conditions For k=1,…,K

Logit Model (4)

0 1 ]

) 1 ( [

1 ] ln [

1 1

+ =

−

−

= +

∂

∂

∑

∑

=

= −

−

n

i

Z i ki n

i Z

Z i ki k

Zi i i i

e Y e X

e Y e L X

β

e Z

P = +

− 1

1 1

Solving FOC for ML estimates.

Second-order Conditions

yields Variance-covariance matrix of

Logit Model (5)

∑

∑

=

−

=

+

−

−

−

− +

∂ =

∂

∂

n

i

Z i

ki ji n

i Z

Z i ki ji k

j

Zi i i i

e Y e

X X

e Y e X L X

1 2

1 2

2

) ] 1

) ( 1 ( [

) ] 1 [ (

ln β β

β ˆ

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 27

Variance-Covariance Matrix for

Note that it is not the estimated VC matrix.

Do Z-test or Chi-square test instead of t-test or F-test on parameters

Logit Model (6)

2 ln 1

ˆ ) (

−

 





 





∂

∂

− ∂

=

k j

V L

β β β

β ˆ

Interpretation

sign of β k ==>direction of the effect of X k on the probability to JOIN.

Logit Model (7)

k k

Z

k

^Zi

i

e e X

P β { } β

) 1

( 2 = +

= +

∂

∂

−

−

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 29

No R 2 for a logit model since there is no error term.

Define

It is a measure for goodness-of-fit.

JOIN>0.5 ==> predict that JOIN=1 JOIN<0.5 ==> predict that JOIN=0

Logit Model (8)

(n) size sample predictio n correct

= #

− R 2 pseudo

Assumption of Logistic Regression F(.) is a logistic function but the

observation(experiment) for each given set of independent variables (X) will be repeated several times.

Only the proportion of JOIN=1 can be observed.

Logistic Regression (1)

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 31

From Logit Model

Note that P is the expected proportion of population JOINing given X’s

Logistic Regression (2)

K K X X

P X

P  = β + β + + β



 





− 1 1 2 2 L

ln 1

Define

R i =observed proportion of observation with the same value of X i that JOIN.

Derived Model

Logistic Regression (3)

i Ki K i

i i

i X X X

R

R   = β + β + + β + ν

 





− 1 1 2 2 L

ln 1

) Why?

1 ( ) 1

(

i i i

i N R R

V ν = −

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 33

Define Estimation where

Logistic Regression (4)

*

*

* 2

* 1

*

2

1i

X

i

X

Ki i

X

R i = β + β + + β K + ν )

1

( i

i

i R R

N

w = −

i i i

ki i ki

i i i

w

K k

X w X

R w R

R

i

ν ν =

=

=

 

 





= −

*

*

*

,..., 1 ln 1

for

=> OLS is BLUE

Interpretation of the parameters same as those for logit model as the underlying function is also logistic

Logistic Regression (5)

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 35

Assumption of Probit

F() is a cumulative distribution function of a standard normal.

Note that always.

Probit Model (1)

K K X X

X Z

Z P

β β

β + + +

= Φ

=

L

2 2 1 1

) (

1 ) ( 0 ≤ Φ Z ≤

No error term

Probit Model (2)

1.0 0.5

Z e − Z

+ 1

1 0

)

Φ (Z

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 37

Assumption of Multinomial Logit Define PA i = Pr(JA i =1)

PB i = Pr(JB i =1) PC i = Pr(JC i =1)

Choose the choice of plan C as the reference.

Multinomial Logit Model (1)

Multinomial Logit Model (2)

ZA

i

i

i e

PC PA =

ZB

i

i

i e

PC PB =

where where

Ki K i

i

i X X X

ZA = α 1 1 + α 2 2 + L + α

Ki K i

i

i X X X

ZB = β 1 1 + β 2 2 + L + β

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 39

Multinomial Logit Model (3)

i i

i i

i i

ZB i ZA

ZB ZA

i i i

ZB ZA

i i i

e PC e

e PC e

PB PA

e PC e

PB PA

+

= +

+ +

+ = +

+ + =

1

1 1 1

Multinomial Logit Model (4)

i i

i i i

i

ZB ZA

ZB i

ZB ZA

ZA i

e e

PB e

e e

PA e

+

= +

+

= +

1

1

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 41

ML Estimation of Multinomial Logit model

Solving FOC yields

Multinomial Logit Model (5)

)]

1 ln(

) 1

(

) ln(

) ln(

[ ln

max

) 1

( ) ( ) ( max

1 1

) 1

(

i i i

i n

i

i i

i i

n

i

JB JA i i JB

i JA i

PB PA JB

JA

PB JB

PA JA

L

PB PA PB

PA

L

^{i i i i}

−

−

−

− +

+

=

−

−

=

∑

∏

=

=

−

−

or β β

β , α ˆ ˆ

Interpretation

Multinomial Logit Model (6)

ZA k ZA

ZA

ki i

i i

i

e e

e X

PA α

+

= +

∂

∂

1 ( k )

ZB k ZA ZA

ZA ZA

i i

i i

i

e e

e e

e α + β

+

− + 2

) 1

(

ZA k ZA

ZA ZA

ZA ZA

i i

i

i i

i

e e

e e

e

e  α



 





+

− + +

= +

1 1 1

ZA k ZA

ZB ZA

ZA ZA

i i

i

i i

i

e e

e e

e

e β

+ + +

− +

1 1

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 43

Interpretation

own-effect cross-effect sign of α k ==>direction of the own-effect

of X k on the probability to JOIN A.

sign of β k ==>direction of the cross-effect of X k on the probability to JOIN A.

Multinomial Logit Model (6)

k i i k i i

ki

i PA PA PA PB

X

PA = − α − β

∂

∂ ( 1 )

+ +

Other Choice Models

• Nested Logit /Serial Logit

• Ordered Logit

• Generalized Extreme-Value (GEV)

(c) Pongsa Pornchaiwiseskul, Faculty of Economics, Chulalongkorn University 45

LIMDEP

Models for Limited Dependent Varaibles

• Censored Regression

• Tobit Models