Simultaneous Equations System S. K. Bhaumik Reference: Sankar Kumar Bhaumik, Principles of Econometrics: A Modern Approach Using EViews, Oxford University Press, 2015, Ch. 11 1
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