/
SIMULTANEOUS EQUATION MODELS SIMULTANEOUS EQUATION MODELS

SIMULTANEOUS EQUATION MODELS - PowerPoint Presentation

cheryl-pisano
cheryl-pisano . @cheryl-pisano
Follow
442 views
Uploaded On 2016-07-13

SIMULTANEOUS EQUATION MODELS - PPT Presentation

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

variables equation endogenous model equation variables model endogenous identified condition coefficients identification equations form rank reduced variable structural simultaneous

Share:

Link:

Embed:

Download Presentation from below link

Download Presentation The PPT/PDF document "SIMULTANEOUS EQUATION MODELS" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.


Presentation Transcript

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