


SimultaneousEquations.pdf
SimultaneousEquations.pdf
Showing 13 out of 12
ECON  5370  Advanced Econometric Theory III  le...
ECON  5370  Advanced Econometric Theory III  lecture_notes Simultaneous Equation Models 1 Introduction Many economic problems involve the interaction of multiple endogenous variables within a system of equa tions. Estimating the parameters of such as system is typically not as simple as doing OLS equationby equation. Issues such as identication (whether the parameters are even estimable) and endogenei
ECON  5370  Advanced Econometric ...
ECON  5370  Advanced Econometric Theory III  lecture_notes Simultaneous Equation Models 1 Introduction Many economic problems involve the interaction of multiple endogenous variables within a system of equa tions. Estimating the parameters of such as system is typically not as simple as doing OLS equationby equation. Issues such as identication (whether the parameters are even estimable) and endogenei
Page 1
Simultaneous Equation Models
1
Introduction
Many economic problems involve the interaction of multiple endogenous variables within a system of equa
tions.
Estimating the parameters of such as system is typically not as simple as doing OLS equationby
equation.
Issues such as identication (whether the parameters are even estimable) and endogeneity bias
are the primary topics in this chapter.
2
The Model
The simultaneous system can be written as
Y
+
XB
=
E
(1)
where the variable matrices are
Y
T
M
=
2
6
6
6
6
6
6
6
4
Y
11
Y
12
Y
1
M
Y
21
Y
22
Y
2
M
.
.
.
.
.
.
.
.
.
Y
T
1
Y
T
2
Y
TM
3
7
7
7
7
7
7
7
5
;
X
T
K
=
2
6
6
6
6
6
6
6
4
X
11
X
12
X
1
K
X
21
X
22
X
2
K
.
.
.
.
.
.
.
.
.
X
T
1
X
T
2
X
TK
3
7
7
7
7
7
7
7
5
;
E
T
M
=
2
6
6
6
6
6
6
6
4
11
12
1
M
21
22
2
M
.
.
.
.
.
.
.
.
.
T
1
T
2
TM
3
7
7
7
7
7
7
7
5
and the coe¢ cient matrices are
M
M
=
2
6
6
6
6
6
6
6
4
11
21
M
1
12
22
M
2
.
.
.
.
.
.
.
.
.
1
M
2
M
MM
3
7
7
7
7
7
7
7
5
;
B
K
M
=
2
6
6
6
6
6
6
6
4
11
21
M
1
12
22
M
2
.
.
.
.
.
.
.
.
.
1
K
2
K
MK
3
7
7
7
7
7
7
7
5
:
Some denitions.
Y
t;j
is the
jth
endogenous variable.
X
t;j
is the
jth
exogenous or predetermined variable
Equations (1) are referred to as
structural
equations.
and
B
are the
structural
parameters.
To examine the assumptions about the error terms, rewrite the
E
matrix as
~
E
=
vec
(
E
)=(
11
;
21
; :::;
T
1
;
12
;
22
; :::;
T
2
; :::;
1
M
;
2
M
; :::;
TM
)
0
:
1
Page 2
We assume
E
(
~
E
)
=
0
E
(
~
E
~
E
0
)
=
I
T
where the variancecovariance matrix for
t
=(
t
1
;
t
2
; :::;
tM
)
0
is
=
2
6
6
6
6
6
6
6
4
11
21
M
1
12
22
M
2
.
.
.
.
.
.
.
.
.
1
M
2
M
MM
3
7
7
7
7
7
7
7
5
:
2.1
Reduced Form
The
reducedform solution to (1) is
Y
=
XB
1
+
E
1
=
X
+
V
where
=
B
1
,
V
=
E
1
and the error vector
~
V
=
vec
(
V
)
satises
E
(
~
V
)
=
0
E
(
~
V
~
V
0
)
=
(
1
0
1
I
T
) = (
I
T
)
where
=
0
.
2.2
Demand and Supply Example
Consider the following demand and supply equations
Q
s
t
=
0
+
1
P
t
+
2
W
t
+
3
Z
t
+
s
t
Q
d
t
=
0
+
1
P
t
+
3
Z
t
+
d
t
Q
s
t
=
Q
d
t
2
Page 3
where
Q
s
t
,
Q
d
t
and
P
t
are endogenous variables and
W
t
and
Z
t
are exogenous variables.
Let
Q
=
Q
s
t
=
Q
d
t
.
In matrix form, the system can be written as
Y
=
2
6
6
6
6
6
6
6
4
Q
1
P
1
Q
2
P
2
.
.
.
.
.
.
Q
T
P
T
3
7
7
7
7
7
7
7
5
;
X
=
2
6
6
6
6
6
6
6
4
1
W
1
Z
1
1
W
2
Z
2
.
.
.
.
.
.
.
.
.
1
W
T
Z
T
3
7
7
7
7
7
7
7
5
;
E
=
2
6
6
6
6
6
6
6
4
s
1
d
1
s
2
d
2
.
.
.
.
.
.
s
T
d
T
3
7
7
7
7
7
7
7
5
and
=
2
6
4
1
1
1
1
3
7
5
;
B
=
2
6
6
6
6
4
0
0
2
0
3
3
3
7
7
7
7
5
3
Identication
Identication
Question. Given data on
X
and
Y
, can we identify
,
B
and
?
3.1
Estimation of
and
Begin by making the standard assumptions about the reduced form
Y
=
X
+
V
:
plim
(
1
T
X
0
X
)=
Q
plim
(
1
T
X
0
V
)=0
plim
(
1
T
V
0
V
)=
:
These assumptions imply that the equationbyequation OLS estimates of
and
will be consistent.
3.2
Relationship Between (
;
) and (
;B;
)
With these estimates (
^
and
^
) in hand, the question is whether we can map back to
,
B
and
?
We
know the following
1.
=
B
1
and
2.
=
1
0
1
.
To see if identication is possible, we can count the number of known elements on the lefthand side and
compare with the number of unknown elements on the righthand side.
Number
of
Known
Elements
KM
elements in
3
Ace your assessments! Get Better Grades
Browse thousands of Study Materials & Solutions from your Favorite Schools
University of Wyoming
University_of_Wyoming
School:
Advanced_Econometric_Theory_III
Course:
Introducing Study Plan
Using AI Tools to Help you understand and remember your course concepts better and faster than any other resource.
Find the best videos to learn every concept in that course from Youtube and Tiktok without searching.
Save All Relavent Videos & Materials and access anytime and anywhere
Prepare Smart and Guarantee better grades
Students also viewed documents
lab 18.docx
lab_18.docx
Course
Course
3
Module5QuizSTA2023.d...
Module5QuizSTA2023.docx.docx
Course
Course
10
Week 7 Test Math302....
Week_7_Test_Math302.docx.docx
Course
Course
30
Chapter 1 Assigment ...
Chapter_1_Assigment_Questions.docx.docx
Course
Course
5
Week 4 tests.docx.do...
Week_4_tests.docx.docx
Course
Course
23
Week 6 tests.docx.do...
Week_6_tests.docx.docx
Course
Course
106