Identi°cation problem
I
There are
k
+
1
+
m
unknown coe¢ cients
y
i
=
γ
0
+
γ
1
X
1
,
i
+
. . .
+
γ
k
X
k
,
i
+
β
1
Y
1
,
i
+
. . .
+
β
m
Y
m
,
i
+
U
i
.
I
The exogeneity conditions
EU
i
=
0 and
Cov
(
X
1
,
i
,
U
i
) =
. . .
=
Cov
(
X
k
,
i
,
U
i
) =
0 give us only
k
+
1
equations:
E
[
y
i
°
γ
0
°
γ
1
X
1
,
i
°
. . .
°
γ
k
X
k
,
i
°
β
1
Y
1
,
i
°
. . .
°
β
m
Y
m
,
i
] =
0
,
E
[
X
1
,
i
(
y
i
°
γ
0
°
γ
1
X
1
,
i
°
. . .
°
γ
k
X
k
,
i
°
β
1
Y
1
,
i
°
. . .
°
β
m
Y
m
,
i
)] =
0
,
. . .
E
[
X
k
,
i
(
y
i
°
γ
0
°
γ
1
X
1
,
i
°
. . .
°
γ
k
X
k
,
i
°
β
1
Y
1
,
i
°
. . .
°
β
m
Y
m
,
i
)] =
0
.
I
There are more unknowns than equations. Thus, the
knowledge of the true covariances between
X
±s,
Y
±s and
y
is
not su¢ cient to recover the unknown coe¢ cients
γ
0
,
γ
1
,
. . .
,
γ
k
,
β
1
,
. . .
,
β
m
.