ECONOMETRICS DARMANTO STATISTICS UNIVERSITY OF BRAWIJAYA PREFACE In contrast to singleequation models in simultaneousequation models more than one dependent or endogenous variable is involved ID: 402714
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Slide1
SIMULTANEOUS EQUATION MODELS
ECONOMETRICS
DARMANTO
STATISTICS
UNIVERSITY OF BRAWIJAYASlide2
PREFACE…
In contrast to single-equation models, in simultaneous-equation models more than one dependent, or
endogenous
, variable is involved,
necessitating
as many equations as the number of endogenous variables.
A unique feature of simultaneous-equation models is that the endogenous variable (i.e.,
regressand
) in one equation may appear as an explanatory variable (i.e.,
regressor
) in another equation of the system.
As a consequence, such an endogenous explanatory variable becomes stochastic and is usually correlated with the disturbance term of the equation in which it appears as an explanatory variable.
In this situation the classical OLS method may not be applied because the estimators thus obtained are not consistent, that is, they do not converge to their true population values no matter how large the sample size.Slide3
EXAMPLES : 1.1
DEMAND-AND-SUPPLY MODEL
As is well known, the price P of a commodity and the quantity Q sold are determined by the intersection of the demand-and-supply curves for that commodity.
Thus, assuming for simplicity that the demand-and-supply curves are linear and adding the stochastic disturbance terms u
1
and u2,
… (1)
… (2)Slide4
EXAMPLES : 1.2Slide5
EXAMPLES : 2.1
KEYNESIAN MODEL OF INCOME DETERMINATION
Consider the simple Keynesian model of income determination:
Consumption function: C
t
= β0 + β1Yt
+ ut
; 0 <β1 < 1 …(3)Income identity:
Yt = Ct + I
t
( = S
t
) …(4)
where C = consumption expenditure
Y = income
I = investment (assumed exogenous)
S = savings
t = time
u = stochastic disturbance term
β
0
and
β
1
=
parametersSlide6
EXAMPLES : 2.2Slide7
EXAMPLES: 3
…(5)
…(6)Slide8
EXAMPLES: 4
THE IS MODEL OF MACROECONOMICS
The celebrated IS, or goods market equilibrium, model of macroeconomics in its
nonstochastic
form can be expressed asSlide9
EXAMPLES: 5
ECONOMETRIC MODEL:
Klein’s model I
(Professor Lawrence Klein of the Wharton School of the University of Pennsylvania). His initial
model`is
as follows:Slide10
SIMULTANEOUS EQUATION MODELS
THE IDENTIFICATION PROBLEMSlide11
PREFACE…
The problem of identification precedes the problem of estimation.
The identification problem asks whether one can obtain unique numerical estimates of the structural coefficients from the estimated reduced form coefficients.
If this can be done, an equation in a system of simultaneous equations is identified. If this cannot be done, that equation is un- or under-identified.
An identified equation can be just identified or over-identified. In the former case, unique values of structural coefficients can be obtained; in the latter, there may be more than one value for one or more structural parameters.Slide12
NOTATION AND DEFINITION: 1
The general M equations model in M endogenous, or jointly dependent,
variables may be written as Eq. (7):Slide13
NOTATION AND DEFINITION: 2
Where
Y
1
, Y
2, ... , YM = M endogenous, or jointly dependent, variablesX
1, X2
, ... , XK = K predetermined variables (one of these X variables may take a value of unity to allow for the intercept term in each equation)
u1, u2, ... ,
u
M
=
M stochastic disturbances
t = 1, 2, ... , T =
total number of observations
β’s
=
coefficients of the endogenous variables
γ
’s
=
coefficients of the predetermined variablesSlide14
NOTATION AND DEFINITION: 3
The variables entering a simultaneous-equation model are of two types:
Endogenous,
that is, those (whose values are) determined within the model; and
Predetermined
, that is, those (whose values are) determined outside the model. The predetermined variables are divided into two categories:Exogenous
, current as well as lagged, and Thus, X1t is a current (present-time) exogenous variable, whereas X1(t−1) is a lagged exogenous variable, with a lag of one time period.
Lagged endogenous. Y(t−1) is a lagged endogenous variable with a lag of one time period, but since the value of Y1(t−1) is known at the current time t, it is regarded as non-stochastic, hence, a predetermined variable.Slide15
NOTATION AND DEFINITION: 4
The equations appearing in (7) are known as
the structural
, or
behavioral
, equations because they may portray the structure (of an economic model) of an economy or the behavior of an economic agent (e.g., consumer or producer). The
β’s and γ’s
are known as the structural parameters or coefficients.
From the structural equations one can solve for the M endogenous variables and derive the reduced-form equations
and
the associated reduced form coefficients
.
A reduced-form equation
is one that expresses an endogenous variable solely in terms of the predetermined variables and the stochastic disturbances. Slide16
NOTATION AND DEFINITION: 5
Consider the simple Keynesian model of income determination:
Consumption function: C
t
=
β0 + β1Yt + u
t ; 0 <β
1 < 1 …(3)Income identity: Y
t = Ct + It ( = S
t
) …(4)
If (3) is substituted into (4), we obtain, after simple algebraic manipulation,
…(8)
where
…(9)
r
educed-form equation
reduced-form coefficientSlide17
NOTATION AND DEFINITION: 6
Substituting the value of Y from (8) into C of (3), we obtain another reduced-form equation:
The reduced-form coefficients, such as
∏
1
and
∏3, are also known as
impact, or short-run, multipliers, because they measure the immediate impact on the endogenous variable of a unit change in the value of the exogenous variable.
where
…(10)
…(11)Slide18
NOTATION AND DEFINITION: 7
If in the preceding Keynesian model the investment expenditure is increased by, say, $1 and if the MPC is assumed to be 0.8, then from (9) we obtain ∏
1
= 5. This result means that increasing the investment by $1 will immediately (i.e., in the current time period) lead to an increase in income of $5, that is, a fivefold increase.
Similarly, under the assumed conditions, (11) shows that ∏
3 = 4, meaning that $1 increase in investment expenditure will lead immediately to $4 increase in consumption expenditure.Slide19
THE IDENTIFICATION PROBLEM
By the identification problem we mean whether numerical estimates of the parameters of a structural equation can be obtained from the estimated reduced-form coefficients.
If this can be done, we say that the particular equation is
identified
.
If this cannot be done, then we say that the equation under consideration is unidentified, or under-identified.
An identified equation may be either:Exactly (or fully or just) identified
. if unique numerical values of the structural parameters can be obtainedOver-identified. If more than one numerical value can be obtained for some of the parameters of the structural equations. Slide20
UNDER-IDENTIFICATION: 1
Consider once again the demand-and-supply model (1) and (2), together with the market-clearing, or equilibrium, condition that demand is equal to supply. By the equilibrium condition, we obtain
…(12)
where
…(13)
…(14)
…(12.a)Slide21
UNDER-IDENTIFICATION: 2
Substituting P
t
from (12.1) into (1) or (2), we obtain the following equilibrium quantity:
…(15)
…(16)
…(17)
whereSlide22
UNDER-IDENTIFICATION: 3
Equations (12.a) and (15) are reduced-form equations. Now our demand-and-supply model contains four structural coefficients α
0
, α
1
, β0, and β1, but there is no unique way of estimating them.
WHY…?
The answer lies in the two reduced-form coefficients given in (13) and (16). These reduced-form coefficients contain all four structural parameters, but there is no way in which the four structural unknowns can be estimated from only two reduced-form coefficients. Slide23
JUST OR EXACT IDENTIFICATION: 1
Consider the following demand-and-supply model
By the market-clearing mechanism we have
…(18)
…(19)
…(20)Slide24
JUST OR EXACT IDENTIFICATION: 2
Solving this equation, we obtain the following equilibrium price:
…(21)
…(22)
whereSlide25
JUST OR EXACT IDENTIFICATION: 3
Substituting the equilibrium price into the demand or supply equation, we obtain the corresponding equilibrium quantity:
…(23)
…(24)
whereSlide26
Let modify the demand function (18) as follows, keeping the supply function as before (R represents wealth):
OVER-IDENTIFICATION
…(25)
…(26)
…(27)
…(28)
(29)…Slide27
RULES OF IDENTIFICATION
Fulfill
the
order and rank conditions,
N
otations:M = number of endogenous variables in the modelm = number of endogenous variables in a given equation
K
= number of predetermined variables in the model including the
interceptk
= number of predetermined variables in a given
equationSlide28
THE ORDER CONDITION OF IDENTIFIABILITY
: 1
DEFINITION
1
:
In a model of M simultaneous equations in order for an equation to be identified, it must exclude at least M − 1 variables (endogenous as well as
pre-determined
) appearing in the model. If it excludes exactly M − 1 variables, the equation is just identified. If it excludes more
than M − 1 variables, it is over-identified.
DEFINITION 2
:
In a model of M simultaneous equations, in order for an equation to be identified, the
number
of pre
-
determined
variables excluded from the equation must not be less than the number
of
endogenous
variables included in that equation less 1, that is,
K
− k ≥ m− 1
(
...
3
0
)
If K − k = m − 1, the equation is just identified, but if K − k > m − 1, it is
over
-
identified
.Slide29
THE ORDER CONDITION OF IDENTIFIABILITY
:
2Slide30
THE ORDER CONDITION OF IDENTIFIABILITY
:
3Slide31
THE RANK CONDITION OF IDENTIFIABILITY
: 1
RANK CONDITION OF IDENTIFICATION
In a model containing M equations in M endogenous variables, an equation is identified if
and
only if at least one nonzero determinant of order (M − 1)(M − 1) can be constructed from the coefficients
of the variables (both endogenous and predetermined) excluded from that
particular equation but included in the other equations of the model.Slide32
THE RANK CONDITION OF IDENTIFIABILITY
:
2
C
onsider
the following hypothetical system of simultaneous equations in which the Y variables
are endogenous and the X variables are predetermined
Consider the first equation, which
excludes variables Y4
, X2, and X
3Slide33
THE RANK CONDITION OF IDENTIFIABILITY
:
3Slide34
THE RANK
CONDITION OF IDENTIFIABILITY
: 3
The first equation:
Since the determinant is zero, the rank of the
matrix, denoted by
ρ(A), is less than 3. Therefore, Eq. (19.3.2) does not satisfy the rank condition
and hence is not identified.As noted, the rank condition is both a necessary and sufficient condition
for identification. Therefore, although the order condition shows that Eq
. (19.3.2) is identified, the rank condition shows that it is not. Apparently,Slide35
THE RANK CONDITION OF IDENTIFIABILITY
:
4
To apply the rank condition one may proceed as follows:
Write
down the system in a tabular formStrike out the coefficients of the row in which the equation
under consideration
appears.Also strike out the columns corresponding to those coefficients in 2
which are nonzero.The entries left in the table will then give only the coefficients of
the
variables
included in the system but not in the equation under
consideration
. Slide36
THE RANK CONDITION OF IDENTIFIABILITY
:
5