Binary Choice Model: Predicting Behavior in Economics, Slides of Econometrics and Mathematical Economics

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Econometric Analysis of Panel Data

15. Nonlinear Effects Models (1) and

Models for Binary Choice

Agenda and References

 Binary choice modeling – the leading example

of formal nonlinear modeling

 Binary choice modeling with panel data

 Models for heterogeneity

 Estimation strategies

 Unconditional and conditional

 Fixed and random effects

 The incidental parameters problem

 JW chapter 15, Baltagi, ch. 11, Hsiao ch. 7,

Greene ch. 21.

Behavioral Assumptions

 Preferences are transitive and complete wrt choice

situations

 Utility is defined over alternatives: U it

 Utility maximization assumption

If Uit1 > Uit2 , consumer chooses alternative 1,

not alternative 2.

 Revealed preference (duality)

If the consumer chooses alternative 1 and not

alternative 2, then Uit1 > Uit.

Random Utility Functions

U it = α + β’x it + γ’z i + u

i

• εit

x it = Attributes of choice presented to person

β = Taste or preference weights

z i = Characteristics of the person

γ = Weights on person specific characteristics

εit = Unobserved random component of utility

Mean: E[εit] = 0, Var[εit] = 1

What Can Be Learned from the Data?

(A Sample of Consumers, i = 1,…,n)

• Are the attributes ‘relevant?’
• Predicting behavior
  • Individual
  • Aggregate
• Analyze changes in behavior when

attributes change

Application

 210 Commuters Between Sydney and

Melbourne

 Available modes = Air, Train, Bus, Car

 Observed:

 Choice

 Attributes: Cost, terminal time, other

 Characteristics: Household income

 First application: Fly or other

A Regression - Like Model

INDEX

.

.

.

.

.

-3.0 -1.8 -.6 .6 1.8 3.

Pr[Fly]

α+β1Cost + β2TTime + γIncome

Econometrics

 How to estimate α, β1, β2, γ?

 It’s not regression

 It looks like regression: If there are many repeated

observations for each x

i

so 0 < P

i

< 1 estimates F( x

i

’β), use

weighted least squares = ‘minimum chi-squared’

 The technique of maximum likelihood

 Prob[y=1] =

Prob[ε > -(α+β1Cost + β2Time + γIncome)]

Prob[y=0] = 1 - Prob[y=1]

 Requires a model for the probability

0 1

Prob[ 0] Prob[ 1]

y y

L y y

= =

= = =

∏ ∏

Log Likelihood for Binary Choice

= =

=

=

n n

i i

y 0 y 1

n

i i i i

i 1

n

i i

i i

i 1

i i

i i

logL log[1 F( )] logF( )

(1 y ) log(1 F ) y logF

y (1 y )

logL

- f , f density

F (1 F )

For optimization results and GMM analogy, note

l

that since E[y ]=F , E

i

x β x β

x

ogL

Special Case: Probit

i

i

n

i i i

i=

n

i

i i i

i=

i

F ( ) standard normal CDF

f ( ) standard normal density

Symmetric density implies 1-F(t) = F(-t)

logL= log (q ), q 2y 1

logL

q , and evaluated at t q

For

= φ =

∂ φ

= φ Φ =

i

i

i

i i

x β

x β

x β

x x β

β

i i i

2 2

2

n

i i i i

i i i

i=

i i i i

2

the Hessian, use (repeatedly), t , so

logL

t. Note, t <0 t

This is the actual Hessian. The expectation is

logL

E

φ = − φ

φ φ φ φ ∂

i i

x x

β β

β β

2

n

i

i=

i i

φ

i i

x x

Special Case: Logit

′

= = Λ

′

′ Λ

= Λ − Λ

∂

′ = − Λ Λ

∂

∑

i i

n

i

i=

i i i

i i i i

exp( )

For the logit model, F

1 exp( )

This is also symmetric.

logL= log (q )

Derivatives have a special form: f (1 ) so

logL

q (1 ) , with evaluated at q.

The Hessian i

i

i

i

i i

x β

x β

x β

x x β

β

∂

′ = −Λ − Λ

′ ∂ ∂

2

i i

i

s even simpler;

logL

(1 )

This is not a function of y , so this is the expectation.

i

x x

β β

Estimated Binary Choice Model

+---------------------------------------------+

| Binomial Probit Model |

| Dependent variable MODE |

| Number of observations 210 |

| Iterations completed 6 |

| Log likelihood function -84.09172 |

| Restricted log likelihood -123.7570 |

| Chi squared 79.33066 |

| Degrees of freedom 3 |

| Prob[ChiSqd > value] = .0000000 |

+---------------------------------------------+

+---------+--------------+----------------+--------+---------+----------+

|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|

+---------+--------------+----------------+--------+---------+----------+

Index function for probability

Constant .43877183 .62467004 .702.

GC .01256304 .00368079 3.413 .0006 102.

TTME -.04778261 .00718440 -6.651 .0000 61.

HINC .01442242 .00573994 2.513 .0120 34.

Estimated Binary Choice Models

LOGIT PROBIT EXTREME VALUE

Variable Estimate t-ratio Estimate t-ratio Estimate t-ratio

Constant 1.78458 1.40591 0.438772 0.702406 1.45189 1.

GC 0.0214688 3.15342 0.012563 3.41314 0.0177719 3.

TTME -0.098467 -5.9612 -0.0477826 -6.65089 -0.0868632 -5.

HINC 0.0223234 2.16781 0.0144224 2.51264 0.0176815 2.

Log-L -80.9658 -84.0917 -76.

Log-L(0) -123.757 -123.757 -123.

Observed empirical regularity: Logit coefficients will approximately

equal 1.6 times probit coefficients.