˜Binary Choice !LPM !logit !logistic...
Transcript of ˜Binary Choice !LPM !logit !logistic...
(c) Pongsa Pornchaiwiseskul, Faculty of Economics,
Chulalongkorn University
1
(c) Pongsa Pornchaiwiseskul, Faculty of Economics,
Chulalongkorn University
2
� Binary Choice!LPM!logit!logistic regresion!probit
� Multiple Choice!Multinomial Logit
(c) Pongsa Pornchaiwiseskul, Faculty of Economics,
Chulalongkorn University
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(c) Pongsa Pornchaiwiseskul, Faculty of Economics,
Chulalongkorn University
4
(c) Pongsa Pornchaiwiseskul, Faculty of Economics,
Chulalongkorn University
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(c) Pongsa Pornchaiwiseskul, Faculty of Economics,
Chulalongkorn University
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Question: What determines thechoice selection?
Model to determine the probability of an event under a given condition (value of independent variables)
Pr(choice#j)=Fj(X1,X2,1,X
K)
where X3s are determinants for theprobability.
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Note that
1)
2) function Fj() must return a value
between 0 and 1
∑ =j
j 1)#Pr(choice
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Example
JOIN=1 if the observation will join
the government-run health
insurance program
= 0, otherwise
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JA=1 if the observation will join Plan A= 0, otherwise
JB=1 if the observation will join Plan B= 0, otherwise
JC=1 if the observation will join Plan C= 0, otherwise
Note that JA+JB+JC=1 always.
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General Structure
Note that
),...,,()1Pr( 21 KXXXFJOIN ==
),...,,(1)0Pr( 21 KXXXFJOIN −==
1),...,,(0 21 ≤≤ KXXXF
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Define
Assumption of LPM
Linearity of F(.)
Note that there is no error term
KK XXXP βββ +++= L2211
)1Pr( == JOINP
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Formulation of LPM
E(JOIN)=(1)P+(0)(1-P)=P
==> JOIN=P+v
where v is an error term. E(v)=0
=> OLS is valid but not the best. Why?
)1(2211
---- vXXXJOIN KK ++++= βββ L
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Note that V(ν) = V(JOIN) but
V(JOIN) =(1-P)2P+(0-P) 2(1-P)= P(1-P)
==> V(ν) is not constant. It depends on the independent variables (X3s)
==> Violation of a CLRM assumption or ν is heteroscedastic
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Chulalongkorn University
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Define
where
)1(
1
PPw
−=
)2(**
*
22
*
11
*
---- νβ
ββ
+++
+=
KK X
XXJOIN
L
νν w
KkwXX
wJOINJOIN
kk
=
==
=
*
*
*
,...,1
for
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Note that
==> OLS is BLUE for Model (2)
1
)1()1(
1
)()(2*
=
−−
=
=
PPPP
VwV νν
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Estimation of LPM
Step 1 run OLS for unweighted model (1)
==> JOIN = Xββββ
Note that JOIN is the estimate for P
==> OLS is BLUE for Model (2)
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Step 2 compute the weight
Step 3 compute
Step 4 estimate the weighted model (2)using OLS
)1(
1
JOINJOIN
w
−=
**
2
*
1
*,...,,, KXXXJOIN
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Step 5 re-compute JOIN using the
new set of ββββ.
Note that LPM does not assure that
or 1),...,,(0
10
21 ≤≤
≤≤
KXXXF
JOIN
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Chulalongkorn University
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Correction
If Xββββ<0, set JOIN=0
If Xββββ>1, set JOIN=1
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P
Xβ
1
1
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� Less expensive in computer time. No
non-linear equations
� is the effect of X on the
probability. In general, the explanatory
variables should be unitless or are
expressed in percentage
k
kX
Pβ=
∂∂
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Assumption of Logit
F() is a logistic function
Note that always.KK
Z
XXXZ
eP
βββ +++=+
= −
L2211
1
1
1)(0 ≤≤ ZF
No error term
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1.0
0.5
Z
Ze−+1
1
0
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Note that OLS does not applyML Estimation of Logit model
or
Note that Y=JOIN
∑
∏
=
=
−
−−+=
−=
n
i
iiii
n
i
Y
i
Y
i
PYPYL
PPL ii
1
1
)1(
)]1ln()1()ln([lnmax
)1()(max
β
β
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Note that
First-order conditions
For k=1,1,K
0]1
)1([
]1
[ln
1
1
=+
−−
+=
∂∂
∑
∑
=
=−
−
n
i
Z
iki
n
iZ
Z
iki
k
iZ
i
i
i
e
eYX
e
eYX
L
β
Ze
P+
=−1
11
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Solving FOC for ML estimates.
Second-order Conditions
yields Variance-covariance matrix of
∑
∑
=
−
=
−+−−
+−=
∂∂∂
n
i
Z
ikiji
n
iZ
Z
ikiji
kj
iZ
i
i
i
e
eYXX
e
eYXX
L
12
12
2
])1(
)1([
])1(
[ln
ββ
β̂
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Variance-Covariance Matrix for
Note that it is not the estimated VC matrix.Do Z-test or Chi-square test instead of t-
test or F-test on parameters
12
ln)ˆ(
−
∂∂∂
−=kj
LV
βββ
β̂
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Interpretation
sign of βk==>direction of the effect of X
k
on the probability to JOIN.
kk
Z
kiZ
i
e
e
X
Pββ }{
)1(2
+=+
=∂∂
−
−
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Chulalongkorn University
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No R2 for a logit model since there is no
error term.
Define
It is a measure for goodness-of-fit.
JOIN>0.5 ==> predict that JOIN=1
JOIN<0.5 ==> predict that JOIN=0
(n) size samplen predictiocorrect#
=− 2Rpseudo
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Assumption of Logistic Regression
F(.) is a logistic function but the observation(experiment) for eachgiven set of independent variables (X) will be repeated several times.Only the proportion of JOIN=1 can be observed.
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From Logit Model
Note that P is the expected proportion of population JOINing given X3s
KK XXXP
Pβββ +++=
−
L22111
ln
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Define
Ri=observed proportion of observation with the same value of X
ithat JOIN.
Derived Model
iKiKii
i
i XXXR
Rνβββ ++++=
−L
22111
ln
Why? )1(
1)(
iii
iRRN
V−
=ν
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Define
Estimation
where
***
2
*
1
*
21 iKiiiXXXR Ki νβββ ++++=
)1( iii RRNw −=
iii
kiiki
i
ii
w
KkXwX
R
RwR i
νν =
==
−=
*
*
*
,...,1
1ln
for
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=> OLS is BLUE
Interpretation of the parameters same as those for logit model as the
underlying function is also logistic
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Assumption of Probit
F() is a cumulative distribution function of a standard normal.
Note that always.
KK XXXZ
ZP
βββ +++=
Φ=
L2211
)(
1)(0 ≤Φ≤ Z
No error term
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1.0
0.5
Z
Ze−+1
1
0
)(ZΦ
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Chulalongkorn University
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Assumption of Multinomial Logit
Define PAi= Pr(JA
i=1)
PBi= Pr(JB
i=1)
PCi= Pr(JC
i=1)
Choose the choice of plan C as the reference.
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iZA
i
i ePC
PA=
iZB
i
i ePC
PB=
where
where
KiKiii XXXZA ααα +++= L2211
KiKiii XXXZB βββ +++= L2211
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ii
ii
ii
ZBZAi
ZBZA
i
ii
ZBZA
i
ii
eePC
eePC
PBPA
eePC
PBPA
++=
++=+
+
+=+
1
1
11
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ii
i
ii
i
ZBZA
ZB
i
ZBZA
ZA
i
ee
ePB
ee
ePA
++=
++=
1
1
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Chulalongkorn University
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ML Estimation of Multinomial Logit model
Solving FOC yields
)]1ln()1(
)ln()ln([lnmax
)1()()(max
1
1
)1(
iiii
n
i
iiii
n
i
JBJA
ii
JB
i
JA
i
PBPAJBJA
PBJBPAJAL
PBPAPBPAL iiii
−−−−+
+=
−−=
∑
∏
=
=
−−
or β
β
β,α ˆˆ
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Interpretation
kZAZA
ZA
ki
i
ii
i
ee
e
X
PAα
++=
∂∂
1
( )k
ZB
k
ZA
ZAZA
ZA
ii
ii
i
eeee
eβα +
++−
2)1(
kZAZA
ZA
ZAZA
ZA
ii
i
ii
i
ee
e
ee
eα
++−
++=
11
1
kZAZA
ZB
ZAZA
ZA
ii
i
ii
i
ee
e
ee
eβ
++++−
11
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Chulalongkorn University
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Interpretation
own-effect cross-effect
sign of αk==>direction of the own-effect
of Xkon the probability to JOIN A.
sign of βk==>direction of the cross-effect
of Xkon the probability to JOIN A.
kiikii
ki
i PBPAPAPAX
PAβα −−=
∂∂
)1(
+ +
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� Nested Logit /Serial Logit
� Ordered Logit
� Generalized Extreme-Value (GEV)
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Models for Limited Dependent
Varaibles
� Censored Regression
� Tobit Models