Transcript Simultaneous Equations Model

Econometrics 1
Lecture 15
Simultaneous Equation Models –
Reduced Form and Simultaneity Bias
1
Main Feature of Simultaneous Equation System
 So far we only discussed regression model in which a
dependent variable Y is explained by one or a set of
explanatory variables with assumption that there one
way causation from independent variables to the
dependent variables.
 However, many
interdependent.
variables
in
economics
are
 Consider a market model with demand and supply. How
much price can a firm charge for a particular product
from its customers depends on the quantity sold in the
market and how much quantity is demanded by
customers depend on the market price. Price determines
quantity and quantity determines price. Same is true in
national income determination model.
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Main Feature of Simultaneous Equation System-2
 Consumption is a component of income that determines
income, but income is the major determinant of
consumption. Both quantities and prices and income and
consumption are determined simultaneously.
 We need to estimates a system of equations, not a single
equation, in order to be able to capture this
interdependency among variables.
 The main features of a simultaneous equation model are:
1. two or more dependent (endogenous) variables
2. A set of equations
3. Computationally cumbersome, highly non-linearity
in parameters and errors in one equation transmitted
through the whole system
3
Simultaneity-an Example
 Consider a relation between quantity and price
 Qt    Pt  ut
0
1
 A priory it is impossible to say whether this a demand or
supply model, both of them have same variables.
 If we estimate a regression model like this how can we
be sure whether the parameters belong to a demand or
supply model?
We need extra information. Economic theory suggests that
demand is related with income of individual and supply
may be respond
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Example of an Identified Model
Demand
Qtd    Pt  It  u
0 1
2
1,t
(1)
Supply
Qts     Pt   P  u
0 1
2 t 1 2,t
d
(2)
s
where Qt is quantity demanded andQt is quantity
supplied, Pt is the price of commodity,Pt 1 is price
lagged by one period,It is income of an individual,
u and u are independently and identically distributed
1,t
2,t
(iid) error terms with a zero mean and a constant
variance.
Qt
P
I
and t are endogenous variables andPt 1 andt are
  
 are six
exogenous variables, 0 , 1 , 2 ,0 , 1 and
2
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parameters defining the system.
Example of an Identified Model
 In equilibrium Qtd =Qts , however note that quantities
bought and sold (Q) depends upon market prices (P)
and the equilibrium prices is determined by quantity
supplied and demanded.
 The random terms of quantity and price equations
above, u1,t and u2,t , are not independent of each
other. There is simultaneity problem.
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A Simple Keynesian Model of Income
Determination
A simple version of Keynesian income
determination model of the following form:
Ct     Yt  ut
0 1
Yt  Ct  It
(3)
(4)
where Yt is income, C is consumptionI
t
ut
investment, and
is the random error term. The
subscript t refers to time period.
0
and are
1
structural parameters, and 0  1  1 .
Yt
It
endogenous variables and
variable.
is
t
Ct
and are
is an exogenous
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Reduced form of the Market Model
a. Reduced for is obtained by expressing endogenous
variable in terms of exogenous variables. In the
demand and supply equation in example 1 the
reduced form takes the following form:
Qt  
 P

I v
10
11 t 1
12 t
1,t
(5)
Pt  

P

I v
20
21 t 1
22 t
2,t
(6)
Where the reduced form coefficients are defined as:

  0  0 
  2

 2
10
   ; 11    ; 12    ;
1
1
1
1
1
1
u
u
1,t ;
v  2,t
1,t
 
1
1


    
 11
1 0
0 1
; 21   
 

  2 1
12
  ;
1
1
;
1
1
1
 u
 u
1 1,t
v
 1 2,t
2,t
;
 
1
1
These coefficients are obtained by using equation (2)
in (1) for quantity (Q) and using that information in
(2) for price level.
2,0
1
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Reduced form of the National Income Model
In the income determination model (example 2) the
reduced form is obtained by expressing C and Y
endogenous variables in terms of I which is the
only exogenous variable in the model.
Ct     It  v
11 12
1,t
Yt     It  v
21 22
2,t

(7)
(8)

 0   1   0
Where 11  ; 12 1  ; 21 1  and
1
1
1
  1 .
22 1 
1

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Simultaneity bias
c y

y
t
b1
t
2
t
t
 C  C y
=
y
t
t
t
2
t
t
C y  C  y
=
y
t
t
t
2
t
t
C y

y
t
b1
t
2
t
t
t
C y
=
y
t
=> b1 
 
t
t
t
0
t
t
t
2
t
t
  1Yt  et  y t
y
2
t
t
e
 e 
cov Yt , et   E Yt  E Yt et  E et   E  t  et 
1  1
1   1 
2


  et y t n 
 e2 1   1 
t


p limb1    1 
 1 
 y2


2
  yt n 
 t

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