構造方程式ゼミナール
2012年11月21日
構造方程式の作成と応用
Identification
Order condition
Underidentified
Exactly identified Overidentified
k  g 1
k=g-1
k > g -1
rank (A) = g-1
rank (A) = g -1
k<g-1
Rank condition
k  g 1
rank (A) < g -1
事例3 Klein’s Model(1950)
1950 年のクラインのモデル1 は僅
か8 本の方程式で構成され、1921
年から40 年にかけて大不況の経緯
を見事にトレースし、これを持って、
経済学は一人前の科学として船出を
遂げたと言われる。
In a book published in 1950, Lawrence Klein reported a
model of the U.S. economy for the period 1921-41, which
is widely known as Klein’s Model I.
Klein’s Model(1950)
①Consumption
Ct  0  1 Pt  2 Pt 1  3 (Wpt  Wgt )  e1t
②Investment
It  0  1 Pt  2 Pt 1  3 Kt 1  e2t
③Private wages
Wpt   0   1 X t   2 X t 1   3 At  e3t
④Equilibrium demand
X t  Ct  It  Gt
⑤Private profits
Pt  X t  WPt  Tt
⑥Capital stock
Kt  It  Kt 1
Data for Klein’s Model, 22 yearly
Observations, 1920-1941
•
•
•
•
•
•
•
•
•
•
•
Year=date
C=consumption,
P=corporate profits,
Wp=private wage bill,
Wg=government wage bill,
I=investment,
Kt-1=previous year’s capital stock,
X=GNP,
G=government nonwage spending,
T=indirect business taxes plus net exports,
A=time trend measured as years from 1931
Exogenous variables
G : government nonwage spending
T : indirect business taxes plus net exports
Wg : government wage bill
A : time trend measured as years from 1931
Predetermined variables Kt-1 : previous year’s capital stock
Pt-1 : corporate profits
Xt-1 : gross demand
The endogenous variables are each on the left-hand side of an equation
Put all the endogenous variable to the left-hand side of an equation
Ct  1 Pt  3Wpt  0  2 Pt 1  3Wgt  e1t
It  1 Pt  0  2 Pt 1  3 Kt 1  e2t
Wpt   1 X t   o   2 X t 1   3 At  e3t
X t  Ct  It  Gt
Pt  X t  WPt  Tt
If the structural equation is expressed
Kt  It  Kt 1
by a matrix
[ C1 It
WPt Xt Pt Kt ]
1
1
0
−α3
0
−α1
0
2
0
1
0
0
−β1
0
3
0
0
1
−γ1
0
0
4
−1
−1
0
1
0
0
5
0
0
1
−1
1
0
for example, identification of the Equation
C
[ 1 Pt−1 WGt Kt−1 Xt−1 At Gt
For example
identification of the
equation C
1
α0
α2
Tt ] α3
0
0
0
0
0
2
β0
β2
0
β3
0
0
0
0
3
γ0
0
0
0
γ2
γ3
0
0
4
0
0
0
0
0
0
1
0
5
0
0
0
0
0
0
0
−1
6
0
−1
0
0
0
1
6
0
0
0
1
0
0
0
0
=
+ (e1 +e2 + e3 )i
Estimates of Klein’s Model I
• Estimation
Klein’s model by 2SLS;
Klein’s model by 3SLS
• Evaluation;
Estimates of Klein’s Model I
(Estimated Asymptotic Standard Errors in Parentheses)
2SLS
C
I
Wp
3SLS
16.6
0.017
0.216
0.810
(1.32)
(0.118)
(0.107)
(0.04)
20.3
0.150
0.616
-0.158
(7.54)
(0.173)
(0.162)
(0.036)
1.5
0.439
0.157
0.13
(1.15)
(0.036)
(0.039)
(0.029)
C
I
Wp
16.4
0.125
0.163
0.79
(1.3)
(0.108)
(0.1)
(0.033)
28.2
-0.013
0.756
-0.195
(6.79)
(0.162)
(0.153)
(0.038)
1.8
0.4
0.181
0.15
(1.12)
(0.032)
(0.034)
(0.028)
OLS
C
I
Wp
I3SLS
16.2
0.193
0.09
0.796
(1.3)
(0.091)
(0.091)
(0.04)
10.1
0.48
0.333
-0.112
(5.47)
(0.097)
(0.101)
(0.027)
1.48
0.439
0.146
0.13
(1.27)
(0.032)
(0.037)
(0.032)
C
I
Wp
16.6
0.165
0.177
0.766
(1.22)
(0.096)
(0.09)
(0.035)
42.9
-0.356
1.01
-0.26
(10.6)
(0.26)
(0.249)
(0.051)
2.62
0.375
0.194
0.168
(1.2)
(0.031)
(0.032)
(0.029)
I3SLS（iterative three-stage least squares）
References:
• W. Greene (2000), Econometric Analysis, 4th edition, Prentice-Hall.
• L. Klein (1950), Economic Fluctuations in the United States 1921-1941,
(preface), Cowles Foundation Monograph
• Robert Dixon, Simulation with Klein's Model I Using TSP, Department of
Economics at the University of Melbourne
• L. Klein, The dynamics of Price Flexibility: Comment, AER, Vol.40,
No.4, 605-609.
• L. Klein (1947), The Use of Econometric Models as a Guide to conomic
Policy, Econometrica, Vol.15, No.2.
• David A. Freedman (2005). Statistical Models: Theory and Practice,
Cambridge University press.
• David R. Brillinger(2001). Time Series-Data Analysis and Theory,
Society for Industrial and Applied Mathematics Philadelphia
•N. Zhang (2012) A Statistical Model for Global-Flow-of-Funds Analysis
Social Systems Solutions Applied by Economic Sciences and
Mathematical Solutions , Vol.3, No.1, 77-97

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