In many cases, there is an unobservable heterogeneity in the probit model. For instance, when modelling the consumption choice of a certain brand, consumers’ personal preference is unobserved but needs to be considered in the model. Owing to omitted variable or measurement error, endogeneity issue also could arise. A probit model including both of these two issues can be represented as:
y
i
t
=
1
[
y
i
t
∗
>
0
]
y
i
t
∗
=
x
i
t
(
1
)
β
+
z
i
t
δ
+
c
i
+
u
i
t
z
i
t
=
x
i
t
(
1
)
γ
1
+
x
i
t
(
2
)
γ
2
+
v
i
t
where
c
i
is the unobservable heterogeneity effect and
u
i
t
∣
x
i
∼
N
(
0
,
1
)
,
v
i
t
|
x
i
∼
N
(
0
,
σ
2
)
. If
v
i
t
and
u
i
t
are independent, this model will degenerate to a probit model with unobservable heterogeneity. In this case, we can just integrate
P
(
y
i
T
,
…
,
y
i
0
∣
x
i
,
c
i
)
against the density of
c
i
conditional on
x
i
, then
P
(
y
i
T
,
…
,
y
i
0
|
x
i
)
can be obtained and the objective for the conditional Maximum Likelihood Estimation is
∑
i
=
1
N
log
[
P
(
y
i
T
,
…
,
y
i
0
|
x
i
)
]
If
v
i
t
and
u
i
t
are correlated, under the normality assumption, it can be assumed that
v
i
t
=
ρ
u
i
t
+
ϵ
i
t
, where
ϵ
i
t
∼
i
i
d
N
(
0
,
σ
2
−
ρ
2
)
and
ϵ
i
is independent with
v
i
and
u
i
. Then the model can be rewritten as:
y
i
t
=
1
[
x
i
t
(
1
)
(
β
+
δ
γ
1
)
+
x
i
t
(
2
)
δ
γ
2
+
c
i
+
ω
i
t
>
0
]
where
ω
i
t
=
(
1
+
ρ
δ
)
u
i
t
+
δ
ϵ
i
t
,
ω
i
t
∼
N
(
0
,
(
1
+
ρ
δ
)
2
+
δ
2
(
σ
2
−
ρ
2
)
)
and
corr
(
ω
i
t
,
ω
i
,
t
−
s
)
=
(
1
+
ρ
δ
)
2
corr
(
u
i
t
,
u
i
,
t
−
s
)
(
1
+
ρ
δ
)
2
+
δ
2
(
σ
2
−
ρ
2
)
.
Based on this, following the same Maximum Likelihood Estimation procedure and the scaled parameter
(
β
+
δ
γ
1
,
δ
γ
2
)
/
(
1
+
ρ
δ
)
2
+
δ
2
(
σ
2
−
ρ
2
)
can be consistently estimated, then the APE can be consistently estimated correspondingly.
