Summary week 5-6 Econometrics 2 UvA

Week 5+6: paragraaf 6.3
Limited depesdert variable Graphical illustraties truncation


These a re
quantitave ,
continuous variable s het
yi
"
=
Xi +
Ei ,
Ei ~
NID ( 0,1 ) .
In general ,




with out comes that are restricted in some for a
given value of Xi ,
the density is



truncated at the value ✗i
In truncated the observations
as
yi only
samples
-




wa .

,


*
ob Served If y
_




> 0 =) Ei > -


Xi The
be obtained from Limited part ,
.




can only a


truncation effect is
large for small Values
of the undvlying population . Model s
where

and small large value of
the
of Xi for Xi
possible observed outcomes outccmes


truncated standard normaal
Limited to interval called




✗
are an are f

censor ed samples .




a model
for truncated data
I
' '
-2
-1 I
We can sides the situation whee the truncation


is
from below with known truncation point .
Truncated density of the e r ro r terms


It assumed that the truncah.cn point is the truncation
is we
analyse effect of on



' * '

which be achieved b P
*

always p
-

Zero , can =
xi + 5 Ei
y
=
xi + Ei
yi
_




,


0 5

in deviation the known
measuring yi from
" '

yi only ob Served if yi
> 0
,
so Ei > -
✗ i P .




truncation pcint . We Write the model as (o ) or




XÍB complete b
"
+
OEI Ei IID E- [ Ei ] 0 We CDF F
yi
= ~ =
, ,




Ei is an e r ro r term with known symmetrie
{ }
' '
P E t P TE
*
Ei Ei > xi o
if ✗ c. P
- -
=




5
and continuous density f .
The Scale
factor
5




convenant
5 is as b extracting 5 we


/ ] [ ]
' '
* P ziet Ei > xi P P xi /310 < ziet
- -
=




that the
5 PE Ei > Xi
'

1310 ]
density f of the
-




aan now a ss u m e




( n or m ali te r ) Ei is
completely known
F (t) FC '
)
.




=
- -




✗i 310
/
Flxi > Plo )
the model
We assume the data satisfies ,


'
t > P /
if Xi 0

y :*
-




but to are not do Served .




" ' "
This gives trvncated density
yi
=

yi
=
✗i P + •
Ei if i
>0


( t)
'
fi 0
if te ✗ i. Plo
-
=
*
obseved ⇐ 0
yi
not if yi
f- i ( t ) f- ( t ) Plo
'
t
=
íf > xi
-




FLXÍPIO )

so the truncated of
density Ei is


'
'
to with
proportioneel the right part
'
The
t > ✗i 1315 of the
origin at
density f
-




.




b FC 1315 )
'


scoring is Needed to
✗i get

ffiltldt =/ .




S. Veeling

ijijij ij

, Estimation likelihood Tobit model censored data
b maximum for

consistent estimates of B are obtained b Dependent variable is called censored when

ml For the norman distribution we
get the cannot tahe values below
,
response or
.




pl i ) 0 / ( / ) Is )
3
'

xi
g- i
-
=


above a certain threshold the tobit model
⑤ ( xi 310
'
)
.




/

relaties obseved outcomes zo to an
as truncated density .
yi
"

by of
'
as Observations
yi
a re assumed to be index function yi
=
×:
p + 5 Ei means


'

nvutually independent
" "
,
we
get yi
=
yi
=
xi p + • Ei if yi
> 0




log ( L) =

log ( ply , ,
. .
.
, yn ) ) =

Ê ,
109 ( Plyi ) )
yi
= o
if yi
*
Eo



and have
with Scale parameter a
log Co2 )
a a {i
log 4) log (z i t )
-12 f-
= -




know symmetrie density f- with E [ Ei] =
0 .




'
-




÷ È
( yi -
xi
'
/3 )
-




Én log ( ¢ ( xi
>
Plo ) ) In the tcbit model ,
we
usvall Choose


¢ and F- OI
f-
= =
.




The last term comes in addition to the usual

In the truncated model only
"
> o whee
OLS terms and is called the truncation yi
,




obseved whereas in the cessored model
effect .
That term is non -
linear in Band 5



it assumed that response s
integration
is



yi-ocorresponding.to
so we need numerical to sake this .




"
to are also obseved
yi
Marginat effects in trvncated modus
and that values of Xi for such
Parameters P Measure the ME E- [ ]
on
y
observations a re known .




of the explanatory variable s × in the
The tobit model can be seen as a

population .
Therefore they a re
of interest

variaties of the
probit model ,
with a re

for cut -

of -

sample predictions ,
so to estimate

discrete option ( gia ) and whose the
effects for vnabseved
y C- 0 . If we a re



option S u c c e ss is
replaced by the
interest ed in within -


sample effects ,
so in


continuous variable > 0
the trvncated population with
"
then yi .




yi > 0
,




for the nor man distribution the ME are

Graphical
illustrations
[ yil i
E- ] ( Ai Plo B
"
> o = i -
-
✗ ixi
'
)
2x ; If we would simply apply OLS on a



with Ai = E [ Ei
lyi
"
> o ] =
¢ (x : Plo ) >
>0 cersored yi ,
we get inconsistent estimators

☒ Lxi 310
/ ) '

'
as E- [ yi] =/ Xi p .




The correctie term for P lies in (a ,
i )

The ME i n the d-
and is equal for an xi .




E
0
Ò
truncated population clases to than
o
a re ze ro




[
in the untruncated .




§
is


Ratios Bj / 13h continue to have the §
interpretation of the relative effect of I, I to I I to
and ✗
Xj ✗ in on the dependent variable and


untrvncated truncated 17 Xi 0 in
"

Xi + Ei then P[ ]
yi
=

equal for and
= =
a re .
yi ,




P [ Ei to ] =
0,5 and
yi > o have Standard normal



density .




yi
< a is not possible .




S. Veeling
s­ij ijijij ij