Summary Econometrics 2 UvA week 4

Week 4: paragraaf 6.1

Linear probability model non -
linear model for probabilities
h
'

P E E- [ Ei ] tahe monotcnically deoeasing
yi xi -1
Ei =P +
Pjxji + Ei 0 We non
= =
can a -


, ,
g- = 2



which takes Values and function F such the CDF to force
can
only zero are . ,
as




xi
'

P = E- [ yi ] = 0 P[
yi =o ] -11 P [ yi =
] =P [
,
-
=
, ] Ptyi =
, ] =
Flxi '
Ps )
y,
. -




the interval to
In this model
,
✗ ÍP measures the prcbability
to fail in zero are .




'
In this model Xi P can be explained as the
that an individual with character istics ✗i


the Strength of the stimulus for the outccme
wilt make the Choice =\ so that
yi ,



=L with
the i
marginat effect of jth explanatory
,




'
P[ ] s 1 if Xi P ' as
yi
=
,
to
variable is
eqval
]
'
P [ yi = 1 so
if Xi P ) -




J P [ yi ] h
Bj
=
,
j
= = 2 . . .




, , ,


J i The
margin at effect of the jth explanatory

Disaduantages linear model variable is

'
it impos.es the restricties o ←
Xi PEI as
JP [ yi =
]
'

f- ( xi P ) Pj 2 k
g-
, = = _ . _




, , ,



J i
we have to deal with
probabilities ,


The ME smalle for individual s
E- [ ] =P [y ]
'
Also
a re
usually
yi ,
-

=
, =
Xi P .




,
[i are



for which P [ ] to or P [ ] = I
yi y
=
= , ,
-




, .




not normally distributed , but as follows :




'
'
with prob P [ ] P
Ei 1- ✗i
/3 yi Xi
=
=
.
=
,

Restricties needed for parameter Identification
[i = 0 -



xi
'

13 with
prcb .
P [yi =
0 ] =
1- Xi
'

P
The a
of the
density should be fixed .


If
so
Ei has a discrete Bernoulli distribution
have (t) f- Cont ) then G ( t) Flat) ,
.




we
g -
-

= -



,



( Ei ) 13 (
'
P) the terms
'
since Xi
)
Xi
'
er ror

(
i
( xi B)
va r '
P [ yi
=


]
-




, and =
, = F- =
G Xi P .
So
5

are heteroshedastic as
they depesd on P .




F ( Xi 13 )
'
is
equivalent to the model with
'
as OLS ignores OE Xi 13<-1 ,
we can
endup
' ' function G and parameter vector Plo .




with probabilities smaller than ze ro or

Uariance of distribution f- should be fixed ,



trigger than a re .




other Wise not identified .




S. Veeling

ijij

, Interpretation in terms of latent variable

"
margin at effects of explanatcry variable s
IID [ Ei ]
'

Xi P + Ei Ei E-
~


yi
=

, ,


Since the ME depend om ✗i
, they a re
This
'
is the index function ,
with Xi P the

different among individual s . The effects
Systematic prefereren and Ei the individual -




of the jth variable can be svmmarized by
specific effect .




the effects the sample
mean
marginaal over

The obseved Choice is related to the
y
of n individuals
index "

b
y
*
÷
IÈ J P [ yi =
, ] =


pj ÷ È .
1- Lxip ) , j = 1
,
. . .




,
h

yi
= 1 if yi
Zo J i
*

if probabilih.es the Odds ratio
yi
=
o
yi
< 0
Campari Son
of and


The
It is assumed that f- ( Ei ) =

f- ( -
[i ) so that Odds ratio is de fired as


a t
'
P [ ] ( xi p)
§ / [ Ei F
P [ Ei t ] f- (s ) ds f- ( s ) =P < t]
=
,
ds
-




= =
> - = ,


P [ yi )
'
.

-
as
=
o ] 1 -




Flxi P

and the relative prefeence of option I
and then gives

as
compared to option 0 which depends.cn
P[ [ Ei P ] =P [ Ei E Xi P ] Flxi P )
' '
] =P
' ,

yi
=
, 2 -




✗i =




the values of Xi .




with F CDF of Ei .




For the logit model we have F- =
A = et
,

1 + et
model for mutaties
and 1 -
Ilt ) = 1 so that

[ ] Flxi P )
'
et
depend 1 +
P does not only en
= =
yi
,




1 ( t) = et and
the explanatory variable s but also on the
1 ( f)
,
y -




distribution of the unobseved individual
' '

log
1 ( xi 13 ) =
xi P
effects Ei This detemines the Shape ( x: p)
'
.




1 -




1



of the
margin al respons via f- ( Ei ) .
In So in the
logit model the
log
-
odds is


the prcbit Linear of
practica ,
a re choses often a
function Xi .




model with Standard normal density
,

'
maximum likelihood for probit and
logit
It
( t) ¢ ( t) e-
f-
#
= =

If the probability of succes is the same




for an observations so P [ ] =p then
yi
=
,


logistic density
, ,
the model
or
logit with


the probability distribution for the ith
f- ( t )
=
✗ ( t) = et
2
( + et ) " '
Yi
-




observation is
p ( i -




p ) . If observations

that
an
advangtage of the
logit model is
are
mutvally independent ,
then we have



¥
there exist an explicit former latior for " '
Yi
(p )
-




L (i ) and
log likelihood
p
=

p
-
-




,



the CDF
t
( Llp ) ) 2 (p ) 2 log (i p)
| et log
=
log + -




1 ( f) =
✗ ( s ) ds = =
1
{ i : g. }
t
= ,
{ i yio }:


→ 1 + et y + e-

=

È .
yi log (p ) +
È( i -




yi ) log ( 1-
p)


maximizing this gives  = ÷ Ê ,
yi .




S. Veeling

ij ijij