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Discrete Dependent variables: Presentation of Binary
Response Model
Nektarios Aslanidis (Universitat Rovira i Virgili, UNSW)

Aslanidis (URV & UNSW)

Binary Model: Presentation

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Non-linear Models

Consider
P (yi = 1jx) = G (xi β), i = 1, ..., N where G (z ) is a function
0 < G (z ) < 1
At the extremes z z

Aslanidis (URV & UNSW)

!
∞, G (z ) ! 0
! +∞, G (z ) ! 1

Binary Model: Presentation

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Logit Models

Consider
G (xi β) = Λ(xi β) =

exp(xi β)
1 + exp(xi β)

where G (z ) is the cumulative distribution function for a standard logistic random variable, z.
Logit model yi yi

Aslanidis (URV & UNSW)

= G (xi β) + i exp(xi β)
=
+
1 + exp(xi β)

Binary Model: Presentation

i

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Probit Models

Consider
G (xi β) = Φ(xi β) =

Z xi β

φ(v )dv

where Φ(z ) is the cumulative distribution function for a standard normal random variable, z and φ(v ) is the standard normal density. φ(v ) = (2π )

Aslanidis (URV & UNSW)

1/2

exp( v 2 /2)

Binary Model: Presentation

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Probit Models

Probit model

Aslanidis (URV & UNSW)

yi

= Φ(xi β) +

yi

=

Z xi β

i

φ(v )dv +

Binary Model: Presentation

i

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Latent Variable Models
Underlying latent variable model yi = xi β + ei ,

yi = 1[yi > 0]

where 1[.] is an indicator function implying

= 1,
= 0,

yi yi when yi > 0 when yi

0

We also assume

where λ(z ) =
Aslanidis (URV & UNSW)

ei

N (0, σ2 ), if Probit

ei

λ(0, σ2 ), if Logit

exp (z )
(1 +exp (z ))2

is the logistic density function.

Binary Model: Presentation

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Latent Variable Models

Derive the response probability
P (yi

since 1

= 1jxi ) = P (yi > 0jxi ) = P (ei >
= 1 G ( xi β) = G (xi β)

xi βjxi )

G ( z ) = G (z ).

Aslanidis (URV & UNSW)

Binary Model: Presentation

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Calculating partial e¤ects (continuous variable)

Partial e¤ects of a continuous variable
∂P (yi = 1jxi )
∂G (xi β)
∂G (xi β) ∂(xi β)
=
=
= g (xi β) βj
∂xji
∂xji
∂(xi β) ∂xji where g (.) is the density function.
For Logit
∂P (yi = 1jxi )
= λ(xi β) βj
∂xji
For Probit

Aslanidis (URV & UNSW)

∂P (yi = 1jxi )
= φ(xi β) βj
∂xji

Binary Model: Presentation

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Calculating partial e¤ects (continuous variable)

Often we evaluate the partial e¤ects at the mean value of xi
_

_

_

g ( xβ) βj = g ( β1 + β2 x 2 + ... + βK x K ) βj or we calcuate the mean of the partial e¤ects n 1

n

∑ g (xi β) βj

i =1

Ratio of partial e¤ects for xj and xh g (xi β) βj g (xi β) βh

Aslanidis (URV & UNSW)

=

βj βh Binary Model: Presentation

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Calculating partial e¤ects (dummy variable)

Partial e¤ect of a dummy explanatory variable. Assume the response probability is given by G ( β0 + β2 x2i + β3 Di ).The partial e¤ect of Di
P (yi

Aslanidis (URV & UNSW)

= 1jx2i , Di = 1) P (yi = 1jx2i , Di = 0)
= G ( β0 + β2 x2i + β3 ) G ( β0 + β2 x2i )

Binary Model: Presentation

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Calculating partial e¤ects