Response Model
Nektarios Aslanidis (Universitat Rovira i Virgili, UNSW)
Aslanidis (URV & UNSW)
Binary Model: Presentation
1 / 14
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
2 / 14
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
3 / 14
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
4 / 14
Probit Models
Probit model
Aslanidis (URV & UNSW)
yi
= Φ(xi β) +
yi
=
Z xi β
∞
i
φ(v )dv +
Binary Model: Presentation
i
5 / 14
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
6 / 14
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
7 / 14
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
8 / 14
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
9 / 14
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
10 / 14
Calculating partial e¤ects