Simultaneous Equations SystemS. K. Bhaumik

Reference: Sankar Kumar Bhaumik, Principles of Econometrics: A

Modern Approach Using EViews, Oxford University Press, 2015,

Ch. 11

1

Single Equation vs. Simultaneous Equation Systems

Under the single equation system ⟶ one variable is expressed as a

function of one or more variables ⟶ it examines one-way causation

between the dependent and independent variable(s).

However, sometimes this one-way or unidirectional causation may not

work ⟶ the simultaneous equations system recognizes the two-way or

simultaneous relationship between Y and some of the Xs.

Features of a Simultaneous Equations System

▪ Presence of more than one equation.

▪ The endogenous variable of one equation may become an

explanatory variable in another equation.

▪ The simultaneous equations system also includes other variables,

called the predetermined or exogenous variables. These variables

are non-stochastic and their values are determined outside the

system. 2

An Example of Simultaneous Equations System (SES)

Simple Keynesian model of income determination

ttt YC ++= (1)

ttt ICY += (2)

where C, Y and I denote consumption, income and investment

respectively. The subscript t refers to time. By assumption, (the

marginal propensity to consume) is greater than zero but less than 1. It

is observed that, while Ct is an endogenous variable in equation (1), it

becomes an explanatory variable in equation (2). Similarly Yt which is

an explanatory variable in (1) becomes endogenous variable in (2).

Thus, Ct and Yt are the two endogenous or jointly-determined

variables of this system while It is an exogenous or pre-determined

variable.

The model comprising equations (1) and (2) represents a simultaneous

equations system.

3

Recursive System

The following is an example of recursive model.

iii uXY 11101 ++= (3)

iiii uXYY 2131102 +++= (4)

Although this system looks like a simultaneous equations system, it is actually a

recursive system.

This system is called ‘recursive’ because each equation can be estimated in

sequential order.

The OLS method can be applied in a straightforward manner to estimate a

recursive system.

OLS Estimation of SES ⟶ Consequence

Generation of simultaneous equations bias⟶ provides an overestimate of .

This bias does not vanish with increasing sample size ⟶ is also asymptotically

biased.

4

Structural and Reduced-form Equations

Structural equation

o A structural equation is any equation of the SES where one

endogenous variable is expressed as a function of other

endogenous variable(s), predetermined variable(s) and

stochastic disturbance term

o The structural equation may also take the form of an identity

o The coefficients of the structural equation are called structural

coefficients

5

Reduced-form equation

o A reduced-form equation expresses an endogenous variable

solely in terms of predetermined variables and stochastic

disturbance

o The reduced-form equations are obtained by solving the

structural equations for the endogenous variables

o The coefficients of the reduced-form equations are called

reduced-form coefficients, which are actually combinations of

the structural coefficients

o There are as many reduced-form equations as there are

endogenous variables in the system

The equations (5) and (6) above are reduced-form equations for

structural equations (1) and (2) respectively 6

Identification Problem

Before estimating any structural equation of a simultaneous equations system, it

is necessary to check whether it would be possible to estimate the parameters of

the equation given the data set

Identification does this by determining whether the parameters of a particular

equation can be estimated meaningfully

If it is not possible to estimate the parameters of a particular structural equation,

we say that the structural equation is unidentified or under-identified

However, if the parameters of the equation can be estimated, we say that the

equation is identified

An identified equation may be either exactly / just identified or over-identified

Exactly identified → when unique numerical values of the parameters of the

structural equation are obtained

Over-identified → when more than one numerical values are obtained for some

of the parameters 7

Rules of Identification

❖ Order or necessary condition: It requires that the number of variables

(endogenous and predetermined) excluded from an equation but included in

other equations must be greater than or equal to the total number of equations

(i.e., the number of endogenous variables in the system) less one.

Symbolically, for gth equation of a simultaneous equations system,

1** −+ GKG where G and **K are respectively the numbers of

endogenous and predetermined variables excluded from the gth equation and

G is total number of endogenous variables in the system.

The status of a structural equation on the basis of this condition is

summarized in Table 1.

Table 1: Status of a Structural Equation on the basis

of Order Condition

Status Number of excluded

variables ( G + **K )

Unidentified or under-

identified < G – 1

Just or exactly identified = G – 1

Over identified > G – 1

8

❖ Rank or sufficient condition: A sufficient condition of identification is the rank

condition. This condition requires that the rank of the matrix containing

structural coefficients of variables excluded from the equation (say gth) but

included in the system must be equal to number of endogenous variables in the

system less one.

Symbolically, 1−= Grank (or 1−=− GGrank ) where is the matrix of

structural coefficients of endogenous and predetermined variables excluded

from the gth equation but included in the system, and G is number of

endogenous variables in the system.

Considering both order and rank conditions, we determine identification

status of a structural equation as follows

Table 2: Status of a Structural Equation on the basis of both

Order and Rank Conditions

Status

Number of

excluded variables

( G + **K )

Rank of matrix

Unidentified or under-

identified G – 1 < G – 1

Just or exactly identified = G – 1 = G – 1

Over-identified > G – 1 = G – 1

9

An Application of Rules of Identification

Consider the following structural model

iiiii

iiii

iiiii

uXYYY

uXYY

uXXYY

33213

2332

12121

2

23

+−−=

++=

++−=

where Ys are endogenous variables and Xs are predetermined variables. Our

objective is to determine the identification status of the equations by applying the

order and rank conditions.

The conditions of identification are:

(i) Order condition: For an equation, the number of excluded variables

(endogenous and exogenous) [ **KG + ] must be greater or equal to total

number of endogenous variables in the system (G) less one

(ii) Rank condition: The rank of the matrix containing structural coefficients of

variables excluded from the equation but included in the system ( ) must be

equal to number of endogenous variables in the system less one. Symbolically,

1−= Grank (or 1−=− GGrank )

10

Examining the order condition

In the above system, we have altogether three endogenous and three

predetermined variables, i.e., G = 3 and K = 3. Now,

• For the first equation, 2KG =+ )( ** )( 1G −= . So the order condition has been

satisfied.

• For the second equation, 3KG =+ )( ** )( 1G− = 2. This shows that the order

condition has been over-fulfilled.

• For the third equation, 2KG =+ )( ** )( 1G −= = 2. So the order condition has

been satisfied.

Examining the rank condition

To examine the rank condition, we rewrite the structural equations as

0200

0000

00203

3321321

2321321

1321321

=+−++−−

=+++++−

=+++−++−

iiiiiii

iiiiiii

iiiiiii

uXXXYYY

uXXXYYY

uXXXYYY

The coefficients of the variables may be tabulated, ignoring the disturbances, as:

Equation

No.

Coefficients of the variables

Y1i Y2i Y3i X1i X2i X3i

1 -1 3 0 -2 1 0

2 0 -1 1 0 0 1

3 1 -1 -1 0 0 -2

11

• For the first equation,

2=G and 1=G

−−=

21

11 and 0

2=rank and 1=− Grank

)()( 11 −==− GGrank . So, the rank condition is satisfied.

As both the order and rank conditions are satisfied, the first equation is just-

identified.

• For the second equation,

2=G and 1=G

−−=

001

121 and 001

211

−−=

2= rank and 1=− Grank

)()( 11 −==− GGrank , which satisfies the rank

condition.

As the order condition is over-fulfilled and rank condition is satisfied, the second

equation is over-identified.

12

• For the third equation,

3=G and 0=G

−=

00

12 and 0=

But rank = 1 as there are non-zero elements in matrix.

Here 101 =−=− Grank and 21=−G

1−− GGrank , which violated the rank condition.

Although the order condition was satisfied, the third equation is an unidentified

equation as the rank condition is violated.

13

Estimation of Simultaneous Equations System

o Two types of methods → ‘single equation methods’ and the

‘system methods’

o Single equation methods → estimate a single structural

equation with limited reference to the rest of the system.

Examples → indirect least squares (ILS), two-stage least

squares (2SLS), instrumental variable method (IV), k-class

estimators etc.

o System methods → all equations of the system are estimated

simultaneously. Examples → three-stage least squares (3SLS),

full information maximum likelihood method (FIML), etc.

We shall discuss two single equation methods → ILS and 2SLS 14

❖ Indirect Least Squares (ILS)

o This method is applied to estimate an exact or just identified

equation of a SES

o Here we apply least squares method to obtain estimates for

unknown parameters of the exactly identified equation

o This is done indirectly using estimates of reduced-form coefficients

Three steps in ILS method:

1. Derive the reduced-form equations for the model

2. Apply OLS method to the reduced-form equations individually

to obtain estimates of reduced-form coefficients

3. Estimates for structural coefficients are computed using

estimated reduced-from coefficients 15

An Illustration

Consider the following structural model:

tttt uXYY 121 ++=

ttt uYY 212 +=

In this model, tY1 and tY2 are the endogenous variables, tX is the

predetermined variable, and tu1 and tu2 are structural disturbances

Applying the rules of identification, it is found that while the first

equation is unidentified, the second equation is exactly identified →

it will not be possible to estimate the parameters of first equation

although we may apply the ILS method to obtain an estimate of the

unknown parameter of the second equation

To estimate the structural parameters in the second equation above

by applying the ILS method, the steps to be followed are given below: 16

Step 1: Derive the reduced–form equations as

tt

tt

t

ttttt

vX

uuX

uXuYY

11

21

1211

11

)(

+=

−

++

−=

+++=

where )1/(1 −= and )1/()( 211 −+= ttt uuv

tt

ttt

ttt

tt

vX

uuX

uuu

XY

22

21

221

2

11

11

+=

−

++

−=

+

−

++

−=

where )1/(2 −= and )1/()( 212 −+= ttt uuv

Step 2: Apply OLS method to obtain estimates for two reduced-

form parameters 1 and 2 → denoted by 1 and 2 respectively

Step 3: Use 1 and 2 to obtain the estimate for structural coefficient

1

2

ˆ

ˆˆ

=

17

What happens if the ILS is applied to estimate an over-identified

equation?

o We know that the ILS method is applied to estimate parameters

of an exactly identified equation

o Let us now examine the consequence of applying this method to

estimate an over-identified equation

Consider the model

ttt

ttttt

uYY

uXXYY

21212

12121112121

+=

+++=

where Ys are endogenous variables, Xs are predetermined variables

and us are structural disturbances

✓ Here the first equation is unidentified and second equation is

over-identified 18

Suppose we apply the ILS method to estimate the structural

parameter of this over-identified equation

The first step is to derive the reduced-form equations, which are

2112

21212

2112

121

2112

111

111

−

++

−+

−= tt

ttt

uuXXY

ttt vXX 1212111 ++=

2112

21212

2112

12211

2112

11212

111

−

++

−+

−= tt

ttt

uuXXY

ttt vXX 2222121 ++=

In the second step, applying the OLS method, estimated reduced-

form equations are obtained:

ttt XXY 2121111ˆˆˆ +=

iit XXY 2221212ˆˆˆ += 19

Finally, the estimate for the structural coefficient of the over-

identified equation ( )21β is computed using estimated reduced-form

coefficients in the manner stated below:

21

2112

11

2112

1121

11

21 ˆˆˆ1

ˆ

ˆˆ1

ˆˆ

ˆ

ˆ

=

−−=

21

2112

12

2112

1221

12

22 ˆˆˆ1

ˆ

ˆˆ1

ˆˆ

ˆ

ˆ

=

−−=

o It is clear that ILS method failed to provide a unique solution

for the structural parameter of the over-identified equation →

problem of choosing between these two estimates → ILS method

is not suitable for estimating the structural parameters of an

over-identified equation

o Another limitation → it does not give the standard error of the

estimates of structural parameters and it is rather complicated

to calculate them 20

Two-Stage Least Squares (2SLS)

• ILS technique → not appropriate to estimate an over-

identified equation

• However, most simultaneous equation models in reality tend

to be over-identified

• To estimate an over-identified equation of a SES → an

appropriate technique is two-stage least squares (2SLS)

The basic idea behind the 2SLS method → replace the

‘endogenous explanatory variables’ or stochastic regressors

(which are correlated with the disturbance term causing

simultaneous equations bias) with one that is non-stochastic and

hence independent of the disturbance term

This is done following two steps: 21

Stage 1: Regress each endogenous variable on all the exogenous

and lagged exogenous variables (if any) of the system by using

OLS and obtain the fitted or estimated values of the endogenous

variables from these regressions (i.e., Y )

Stage 2: Use the fitted values from stage 1 as ‘proxies’ or

‘instruments’ for the ‘endogenous explanatory variables’ in the

original (structural form) equation and estimate the model again

by using OLS

One condition → values of R2 and F-statistics from first-stage

regression should be sufficiently high to ensure that Y and Y are

highly correlated so that Y is a good instrument (or proxy) for Y 22

Stock-Watson (2011) rule of thumb to check the strength of the

instruments → a first-stage F-statistic less than 10 indicates weak

instruments → using such instruments will make 2SLS estimates

biased even in large samples and the corresponding t-statistics

and confidence intervals will become unreliable

Advantages of 2SLS method

o It is very simple and easy to apply

o It’s advantage over the ILS is that it can be used to obtain

consistent structural parameter estimates for over-identified as

well as exactly identified equation

o However, the 2SLS and ILS estimates are identical for an

exactly identified equation 23