構造方程式ゼミナール
2012 年 11 月 21 日

k < g - 1
1 k = g - 1 k > g -1
1 rank (A) < g -1 rank (A) = g-1 rank (A) = g -1

(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.

① Consumption
② Investment
③ Private wages
④ Equilibrium demand
⑤ Private profits
⑥ Capital stock
Klein’s Model(1950)
C t
 
0
 
1
P t
 
2
P t

1
 
3
( Wp t

Wg t
)
 e
1 t
I t
 
0
 
1
P t
 
2
P t

1
 
3
K t

1
 e
2 t
Wp t
 
0
 
1
X t
 
2
X t

1
 
3
A t
 e
3 t
X t

C t
  t
G t
P t

X t

WP t

T t
K t

I t

K t

1

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 K t-1
: previous year’s capital stock
P t-1
: corporate profits
X t-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
C t
I t
 
1
P t
 
3
Wp t
 
1
P t
 
0



2

0

P t

1


2
P t

1

3
 
3
Wg t
 e
1 t
K t

1
 e
2 t
Wp t
 
1
X t
  o
X t

C t
  t
G t
 
2
P t

X t

WP t
 
T t
K t

I t

K t

1
X t

1
 
3
A t
 e
3 t
If the structural equation is expressed by a matrix

1
1
0
2 3 4 5 6
0 −1 0 0
0
[ C
1
I t
WP t
X t
P t
K t
] −α
3
0
1
1
0 −1 0 −1
0 1 0
0 0 −γ
1
1 −1 0
−α
1
−β
1
0 0 1 0
0 0 0 0 0 1
= for example, identification of the Equation
C
1 2 3 4 5 6
α
0
β
0
γ
0
0 0 0
α
2
β
2
0 0 0 0
[ 1 P t−1
WG t
K t−1
X t−1
A t
G t
T t
] α
3
0 0 0 0 0 + (e
1
+e
2
+ e
3
) i
0 β
3
0 0 0 1
For example identification of the
0 0 γ
2
0 0 0
0 0 γ
3
0 0 0
0 0 0 1 0 0
0 0 0 0 −1 0 equation C

• Estimation
Klein’s model by 2SLS ;
Klein’s model by 3SLS
• Evaluation;

Estimates of Klein’s Model I
(Estimated Asymptotic Standard Errors in Parentheses)
I
C
Wp
16.6
(1.32)
20.3
(7.54)
1.5
(1.15)
2SLS
0.017
(0.118)
0.150
(0.173)
0.439
(0.036)
0.216
(0.107)
0.616
(0.162)
0.157
(0.039)
0.810
(0.04)
-0.158
(0.036)
0.13
(0.029)
I
C
Wp
16.2
(1.3)
10.1
(5.47)
1.48
(1.27)
OLS
0.193
(0.091)
0.48
(0.097)
0.439
(0.032)
0.09
(0.091)
0.333
(0.101)
0.146
(0.037)
0.796
(0.04)
-0.112
(0.027)
0.13
(0.032)
I
C
Wp
16.4
(1.3)
28.2
(6.79)
1.8
(1.12)
3SLS
0.125
(0.108)
-0.013
(0.162)
0.4
(0.032)
0.163
(0.1)
0.756
(0.153)
0.181
(0.034)
0.79
(0.033)
-0.195
(0.038)
0.15
(0.028)
I
C 16.6
(1.22)
42.9
(10.6)
2.62
I3SLS
0.165
(0.096)
-0.356
(0.26)
0.375
0.177
(0.09)
1.01
(0.249)
0.194
0.766
(0.035)
-0.26
(0.051)
0.168
Wp
(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