Simultaneous Equations
Introduction
• Most economic models are simultaneous i.e.
at least two relationships between the
variables in the regression.
• Macro example: c = 1 + 2 y
• Micro Example: Supply and Demand
• Lesson: Simultaneity can appear anywhere
• OLS will mix up the two relationships
Macro Example
1. Consumption, c, is function of income, y.
c = 1 + 2 y
c is “endogenous” 2 is MPC
2. y = consumption + investment.
y=c+i
y is endogenous
3. Investment assumed independent of income.
i is “exogenous”
The Structural Form
of the Statistical Model
ct = 1 + 2 yt+et
Identity
yt = ct + it
et is a random disturbance term
• The model is simultaneous because we
cannot determine C or Y without knowing
the other
• Jargon: C and Y are :
• endogenous
• jointly determined
• jointly endogenous
• But I (investment) is exogenous
• We rely on economic intuition to tell us
whether a variable is endogenous or
exogenous -- not really a statistical issue
Single vs. Simultaneous
Equations
Single Equation:
Simultaneous Equations:
yt
ct
et
yt
et
ct
it
Reduced Form
• For use later, useful to re-write the system
of equations in their reduced form
– “Solve” the model
– Reduced form: each equation has only one
endogenous variable on the left
– method: substitute one equation into the other
• Easy for this simple Macro example, more
difficult in real world cases
• Note the conceptual difference between
structural and reduced forms
ct = 1 + 2 yt + et
yt = ct + it
ct = 1 + 2(ct + it) + et
(1  2)ct = 1 + 2 it + et
2
1
1
ct =
+
it +
et
(12)
(12) (12)
ct = 11 + 21 it + t
• We can do the same for the equation in Y
• We get the reduced form of the system
• Note the conceptual difference
ct = 11 + 21 it + t
yt = 12 + 22 it + t
Failure of OLS
C
Y
• OLS picks best fit --- a mixture of both
relationships
• Not get correct estimate of MPC
• OLS is biased and inconsistent because the
right hand side variable (y) is correlated
with the disturbance term.
1. Any change in e, leads to a change in C via
consumption equation
2. Change in consumption leads to a change in
income via the identity
3. This change in income will feed back into a
change in consumption via the consumption
equation
• Thus any time there is a change in e there is
a simultaneous change in Y
1.
ct = 1 + 2 yt + et
3.
yt = ct + it
2.
Fundamental Problem of OLS
• OLS will give credit to Y for changes in e
i.e. the estimated effect of Y on C will
include also the effect of e on C
• OLS will act as if a change in consumption
brought about by some random effect (e),
was due to a change in income
• OLS will overstate the effect of income on
consumption i.e. the MPC
• OLS will be biased and inconsistent
The Failure of Least
Squares
The least squares estimators of
parameters in a structural simultaneous equation is biased and
inconsistent because of the correlation between the random error
and the endogenous variables on
the right-hand side of the equation.
Indirect Least Squares
• One way to estimate is to do OLS on the
reduced form
ct = 11 + 21 it + t
yt = 12 + 22 it + t
• This works because no endogenous variable
on the right hand side i.e. unbiased and
consistent
• We can then use the formulae that link the
parameters of the reduced and structural
forms to calculate the estimates of 
11 = 12 =
1
(12)
22 = (121) =
ˆ1, ILS
ˆ11

ˆ 22
1
(12)
• In practice, this method is not used because
usually the link between the reduced form
and structural form is very complicated in
more realistic models
• Several different structural forms may have
the same reduced form.
• Difficult to get standard errors on 
Identification
• Biggest issue in simultaneous equations,
biggest issue in econometrics
• OLS cannot distinguish between effect of Y
and effect of e
• Problem is to separate these two effects or
literally “identify” the effect of Y on C
Micro Example
• We use a micro economic example i.e.
Supply and Demand model
• Structural model:
Demand: q  1P   2 y   d
Supply:
q  1P   s
• Price and quantity are endogenous (jointly
determined) and income is exogenous
• The model is simultaneous because:
– q is a function of p (demand curve)
– p is a function of q (supply curve)
• OLS estimation of the demand equation will
be biased and inconsistent
• The OLS estimate of 1 will pick up the
effect of the supply curve also
• Cov(p,d) is not equal to zero
• Problem of identification is to separate the
effect of the supply curve from that of the
demand curve
• Have to do this to have hope of estimating
25
20
10
15
price of truffle
30
35
Illustrating the Identification Problem
5
10
15
20
quantity of truffles demanded
25
• Is this a supply curve or a demand curve?
• It looks like a supply curve
• It could be a supply curve, i.e data is
generated by movements of the demand
curve along a supply curve -- so trace out
the supply curve
p
S
D
q
• Or it could be movement in both
p
S
S
D
q
• It turns out that we can estimate 
consistently, but cannot estimate the
demand curve
• The reason for this is that y income is in the
demand curve but excluded from the supply
curve
• As income changes we know the demand
curve will shift but the supply curve will be
fixed
• Therefore if we can concentrate on those
changes in p and q that are caused by
changes in income, we can trace out the
supply curve
Exclusion Restrictions
• We can identify (trace out) the supply curve
only because y is in the demand curve
equation but not in the supply curve
• It is because y is excluded from the supply
curve that we can be sure that changes in y
move the demand curve only
• If y was in the supply curve we could not do
this
• We cannot identify (trace out) the demand
curve, because there is no variable in the
supply curve that is not in the demand curve
• “exclusion restrictions”
General Condition for Identification of an
equation
An equation containing M endogenous
variables must exclude at least M1
exogenous variables from a given
equation in order for the parameters of
that equation to be identified and to be
consistently estimated.
Importance of Identification
• Must check identification before try to
estimate
• If equation is unidentified, will not be able
to get consistent estimates of the structural
parameters
• Always try to design models so that the
equations are identified
Beware of Artificial Restrictions
• Must justify exclusion restrictions using
economic intuition
• For example: Is it reasonable that income
affects demand but not supply
• Most cases are not so obvious
• if a restriction is wrong -- no hope of getting
correct answers
• most arguments in applied economic papers
are over the validity of these restrictions
Estimation- 2SLS
• Two stage least squares
1. Estimate the reduced form using
OLS.
1
pt   1 yt  vt
qt   2 y t  v
2. Do OLS on the structural form with
the actual values replaced by the fitted
values from the first stage
2
t
• Why this works for the supply equation
– The fitted values from the first stage are by
definition the part of the variation in p and q
that is due to changes in income
– Therefore we are sure that the fitted values lie
along the supply curve --- so we just do OLS on
these values
– More formally: the fitted value of p is
uncorrelated with  because it is a function
solely of y which is uncorrelated with  (i.e.
exogenous)
Pˆt  ˆ1 yt
qˆt  1Pˆt   s
• Why does it not work on the demand
equation?
– Computer will generate an error at second stage
estimation of demand equation because
effectively the income variable will appear
twice
– Perfect multicolinearity
d
ˆ
qˆt  1 Pt   2 yt   t
Pˆ  ˆ y
t
1 t
General 2SLS Procedure
• The 2SLS procedure can be used for a
system of any degree of complication
• M equations
• M endogenous variables (y1 .... yM)
• K exogenous variables (x1 .... xk)
• Remember: can only estimate those
equations that pass the identification
condition
• Suppose one of the equations you want to
estimate is:
y1  1 x1   2 x2   2 y2  1
• First check that it is identified i.e. are
enough x variables excluded from the
equation
• Estimate the reduced form for the entire
model
y1   11 x1  ....  1k xk  v1
y2   21 x1  ....  1k xk  v2
yM   M 1 x1  ....  Mk xk  vM
• Replace the endogenous variables in the
structural equations with their fitted values
and do OLS
yˆ1  1 x1   2 x2   2 yˆ 2  1
• Note: Possible problem with standard errors
in some computer programs
Properties of 2SLS
•
•
•
•
Estimates are consistent
Estimates are biased
Estimates are asymptotically normal
Standard errors are not same formula as
OLS -- usually built into software
• Also known as Instrumental Variables (IV)
• Beware of false restrictions
Example: Market for Truffles
• Structural model:
Demand: q     P   ps   y  
qt  1  2 Pt  3ct   ts
Supply:
• ps= price of substitute,
c=rent of pig (i.e. cost of production)
y= per capita disposable income
• Estimate by OLS
t
1
2 t
3
t
4 t
– Note the sign of price coefficient
d
t
Identification
•
•
•
•
•
P and Q are endogenous
c, ps and y are exogenous ? Plausible?
Is supply identified? Why?
Is demand identified? Why?
Are the restrictions plausible? ---- very
important
• Can we use 2SLS?
• N.B: two subjective judgments
• reasonable to say variable is exogenous
• reasonable to exclude it
Stage 1: Estimate Reduced Form
• Endogenous on left, all exogenous on right
qt   11   21 pst   31 yt   41ct  vtq
pt   12   22 pst   32 yt   21ct  vtp
• See the results: note exogenous variables
are significant , R2 is high
– this is close to being the “sufficient condition”
– a.k..a “rank” condition
– what happens if insignificant?
Stage 2: Estimate Structural Form
• Calculate the fitted values for p and q
• Do OLS on
ˆq     Pˆ   ps   y   d
t
1
2 t
3
t
4 t
t
qˆt  1  2 Pˆt  3ct   ts
• Note the signs and significance of the coef.