Chapter 5.pptx

Chapter 5:
Simultaneous Equation Models
By Dereje T. (MSc Biostatistics)

 The models in which there was a single dependent variable
Y and one or more explanatory variables, the X’s.
 In such models the emphasis was on estimating and/or
predicting the average value of Y conditional upon the
fixed values of the X variables.
 The cause-and-effect relationship, if any, in such models
therefore ran from the X’s to the Y.
 But in many situations, such a one-way or unidirectional
cause-and-effect relationship is not meaningful.

Cont’d
 This occurs if Y is determined by the X’s, and some of the X’s
are, in turn, determined by Y.
 In short, there is a two-way, or simultaneous, relationship
between Y and (some of) the X’s.
 It is better to lump together a set of variables that can be
determined simultaneously by the remaining set of variables—
precisely what is done in simultaneous-equation models.
 In such models there is more than one equation—one for
each of the mutually, or jointly, dependent or endogenous
variables.

Cont’d
And unlike the single-equation models, in the
simultaneous-equation models one may not
estimate the parameters of a single equation
without taking into account information
provided by other equations in the system.
So far we have seen regression models of the
form:

Cont’d
Y = f (X1, X2, X3, . . ., Xk) + ɛ
where Y is the dependent variable and X1 , X2 , X3 , . .
., Xk are independent (explanatory) variables.
The assumption so far was that Y depends upon the
Xi’s, but none of the Xi's depends on Y.
But in real life situations we may find variables that
are dependent on each other (simultaneity).

Cont’d
 Example: At the macro level, aggregate consumption
expenditure depends on aggregate disposable income;
aggregate disposable income depends upon the national
income and taxes imposed by the government; national
income depends on aggregate consumption expenditure of
the economy.
 Disregarding these sequences of relationship, if we estimate
a single equation of, say, aggregate consumption on
disposable income, then the estimates will be biased and
inconsistent.

Cont’d
Example: A simple model of the market for a given
commodity may involve a supply and demand function:
Where Q is the equilibrium quantity exchanged
on the market, P is equilibrium price, Y is income
of consumers, and U1t and U2t are the disturbance
terms.
We also have

Cont’d
 Suppose we are interested in the effect of P on Q.
 Can we toss out the second equation and estimate the first
equation alone using OLS?
 Consider the following figure:
1. The equilibrium price and quantity are determined in the market
by the intersection of supply and demand curves.
2. Therefore, we cannot determine equilibrium price by solving the
demand equation independently.

Cont’d
 A shift in the demand function produces a change in both
equilibrium price and quantity if the supply curve has an upward
slope.
 Equations (1) are called the structural form of the model
understudy.
 These equations can be solved for the ‘endogenous’ variables to
give:

Cont’d
 The solution given by equations (2a) and (2b) is
called the reduced form of the model.
 The reduced form equations show explicitly how the
“endogenous” variables are jointly dependent on the
“predetermined” variables and the disturbances of the
system.

Cont’d
 Thus, in the demand equation (1a):
 the variable Pt that appears as an independent or ‘exogenous’
variable is correlated with the disturbance term u1t, and
consequently, estimation of the demand equation using OLS
leads to biased and inconsistent estimators of the parameters.
This is referred to as simultaneity bias.
 The solution is to bring the supply function into the picture and
estimate the supply and demand functions simultaneously.
 Such models are known as simultaneous equations models.

Cont’d
 Where W is rate of change in money wage, U is unemployment rate
(in percentage), P is rate of change in prices, R is rate of change in
cost of capital, and M is money supply.
 Here the price variable P enters into the wage equation (3a) and the
wage variable W enters into the price equation (3b).
 Thus, these two variables are jointly dependent to each other, and
estimation of the two equations individually by OLS yields biased
and inconsistent estimators.

Cont’d
Note:
 Endogenous variables are variables that are jointly determined by the
economic model. (Or are determined by the exogenous variables).
 Loosely speaking, endogenous variables are the equivalent of the
dependent variable in the single-equation regression model.
 Exogenous variables are determined outside of the model and
independently of the endogenous variables.
 Predetermined variables are exogenous variables, lagged exogenous
variables and lagged endogenous variables.
 Predetermined variables are non-stochastic and hence independent of
the disturbance terms.

Structural form and reduced form of simultaneous equations model (SEM)
 Consider the simple Keynesian model of income determination:
 where C is consumption expenditure, Y is income, I is
investment (assumed to be exogenous).
 The above model is said to be the structural form of the SEM,
and the parameters b0 and b1 are said to be structural
parameters.
 Substituting (4a) in place of C in (4b) we get:

Cont’d
 Note that equation (5) is expressed solely as a function of
the exogenous variable and the disturbance term.
 It is referred to as the reduced form of the SEM, and the
parameters and are said to be reduced form parameters.
 Note that the exogenous variable It is not correlated with
the disturbance term, and hence, we can apply OLS to
the reduced form equation to obtain consistent
estimators of and .

Cont’d
 In general, the structural form of a simultaneous
system of equations can be described as:
 where the Y’s are endogenous variables, X’s are
predetermined variables, and the u’s are stochastic
disturbances.

Cont’d
 The β’s and γ’s are the structural coefficients.
 There are G endogenous and K predetermined variables in the
system.
 Not all endogenous and predetermined variables will appear in
every equation (that is, some of β’s and γ’s will be zero).
 In each equation, one of the β’s is taken to be unity, that is, one of
the endogenous variables serves as the ‘dependent’ variable when
the equation is written out as a standard regression equation.
 Some of the equations may be identities, that is, their coefficients
are known and they contain no stochastic disturbance.

Cont’d
The above model in matrix form is:
 The reduced form of the system is obtained by
expressing the Y’s solely as a function of the
predetermined variables X’s and the disturbance
terms:

Cont’d
 Comparing (6) and (7), the relationship between the
structural and reduced form parameters is:
 Example: Suppose we have the following system of
equations:

Cont’d
where q is equilibrium quantity exchanged on the
market, p is equilibrium price, y is income of
consumers, R is the amount of rainfall (Note:
rainfall affects demand, i.e., if there is rain, people
do not go shopping) u1 and u2 and are the error
terms.
This structural form model can be re-written as:

Cont’d
• Note that since the reduced form of the system is
obtained by expressing the endogenous variables solely
as a function of the predetermined variables, OLS
yields consistent estimators of the reduced form
parameters. Thus, the OLS estimators from
equation (b) above are unbiased and consistent.

Identification problem
 Consider the supply and demand equations in the example
above.
 We stated that the parameters of the reduced form model
(that is, the п’s) can be estimated using OLS consistently
 Can we always recover the parameters of the structural
equations (that is , ) uniquely from the п’s?
 In other words, can we always estimate the structural
coefficients via the reduced form coefficients?
 This leads us to the concept of identification.
2
2
1
1
1 b
,
a
,
c
,
b
,
a

Cont’d
• Identification is a problem of model formulation rather than of
model estimation or appraisal.
• We say a model is identified if it is in a unique statistical form,
enabling unique estimators of its parameters to be subsequently
made from sample data.

Note: (Status of identification)
 In econometric theory, three possible situations of
identifiability can arise: equation under consideration is
exactly identified, over identified or under identified.
1. If there is a one to one correspondence between the reduced
form and structural form parameters, then we have exact
identification,
 that is, there is a unique solution for the structural parameters
in terms of the reduced form parameters.

2. If the number of reduced form parameters exceeds the number of
structural parameters, then we have over identification (no unique
solution).
 Here there is more than sufficient information regarding the equation
under consideration.
3. If the number of reduced form parameters is less than the number of
structural parameters, then we have under identification (no solution).
 Here there is no sufficient information regarding the equation under
consideration.

Formal rules of identification
1. The order condition for identification
 Let G be the total number of endogenous variables in
the system and let k be the total number of variables
(both endogenous and predetermined) missing from the
equation under consideration. Then if:
a) k = G-1, the equation is exactly identified.
b) k > G-1, the equation is over identified.
c) k < G-1, the equation is under identified.

Cont’d
 This is known as the order condition for identification.
 It is a necessary but not sufficient condition for the identification
status of an equation.
 Example Wage-price model
 Here U, R and M are predetermined while W and P are exogenous
variables. Thus G = 2.
a) Consider the wage equation. The variables R and M are missing
from this equation.
 Thus, k = 2. The equation is over identified since k = 2 > 1 = G – 1.

Cont’d
 Consider the price equation. The variable U is missing from this equation. Thus,
k=1.
 The equation is exactly identified since k = 1 = G – 1.
 where C = consumption, Y = income, I = investment, r = rate of
interest, M = money supply and G = government expenditure. The
variables are endogenous while the remaining are predetermined
variables. Thus, G = 4.

Cont’d
a) Consider the consumption equation. The variables are
missing from this equation. Thus, k = 5. The equation is over
identified since k = 5 > G – 1 = 3.
b) Consider the investment equation. The variables are
missing from this equation. Thus, k = 5. The equation is over
identified since k = 5 > G – 1 = 3.

Cont’d
2. The rank condition for identification
 The rank condition states that: in a system of G equations any
particular equation is (exactly or over) identified if and only if it is
possible to construct at least one non-zero determinant of order (G-1)
from the coefficients of the variables excluded from that particular
equation but contained in the other equations of the system.

Cont’d
 Note that the coefficient of a variable excluded from an equation is equal to
zero.
 Ignoring the random disturbances and the constants, a table of the
parameters of the model is as follows:
 Now suppose we want to check the identification status of the
consumption function.
a) We eliminate the row corresponding to the consumption function.
b) We eliminate the columns in which the consumption function has
non-zero coefficients

Cont’d
The two steps are shown below:
 Note that by doing steps (a) and (b) above, we are left with the
coefficients of variables not included in the consumption
function, but contained in the other equations of the system.
After eliminating the relevant row and columns, we get the
following table (matrix) of parameters:

Cont’d
 If at least one of these determinants is non-zero, then the
consumption equation is (exactly or over) identified.
 If all determinants of order 3 are zero, then the consumption
equation is under identified.
• For example

Cont’d
For example,
 Thus, we can form at least one non-zero determinant of
order 3, and hence, the consumption equation is exactly or
over identified.
 To see whether the consumption equation is exactly or over
identified, we can use the order condition.

Estimation of simultaneous equations models
1. Indirect least squares (ILS) method
 In this method, we first obtain the estimates of the reduced form parameters by
applying OLS to the reduced form equations and then indirectly get the estimates of
the parameters of the structural model.
 This method is applied to exactly identified equations.
Steps:
1. Obtain the reduced form equations (that is, express the endogenous variables in terms
of predetermined variables).
2. Apply OLS to the reduced form equations individually. OLS will yield consistent
estimates of the reduced form parameters (since each equation involves only non-
stochastic (predetermined) variables that appear as ‘independent’ variables).

Cont’d
3. Obtain (or recover back) the estimates of the original
structural coefficients using the estimates in step (2).
Example
 Consider the following model for demand and supply of pork:
 where Qt is consumption of pork (pounds per capita), Pt is
real price of pork (cents per pound), Yt is disposable personal
income (dollars per capita) and is Zt‘predetermined elements
in pork production’.

Cont’d
 Here P and Q are endogenous variables while Y and Z are
predetermined variables.
 It can easily be shown that both equations are exactly
identified.
 Thus, we can apply ILS to estimate the parameters.
 We first express P and Q in terms of the predetermined
variables and disturbances as:

Cont’d
We can re-write equations (9a) and (9b) as:

Cont’d
2. Instrumental variable (IV) method
 Suppose we have the model (in deviation form):
 Where xi is correlated with εt .
 We cannot estimate β by OLS as it will yield an inconsistent
estimator of β.
 What we do is search for an instrumental variable (IV) that
is Zi uncorrelated with εt but correlated with xi ; that is,
cov(Zi,εt )=0 and cov(Zi,xi)≠0 . The sample counterpart of
cov(Zi,εt )=0 is:
i
i
i ε
βx
y 


Cont’d
 Consider the following simultaneous equations
model:

Cont’d
Thus, OLS method of estimation cannot be
applied.
To find consistent estimators, we look for a
variable that is correlated with xi but not
correlated with u1
Fortunately we have Z3 that satisfies these two
conditions, that is, cov(y2, Z3)≠0 and cov(y2,
u1)=0 . Thus, Z3 can serve as an IV for y2.

Cont’d
The procedure for estimation of the first
equation is as follows:
a) Regress y2 on z1 ,z2 ,z3; that is, using OLS estimate
the model:
b) obtain
Regress on
Note that since z1 ,z2 ,z3 are predetermined
variables, and hence, not correlated with we
have:

Cont’d
 Thus, the OLS estimation using the above procedure
yields consistent estimators.
 Consider the second equation.
Since z3 is predetermined, it is not correlated
with u2, that is, Cov (z3, u2) = 0.
y1 is not independent of u2, that is, cov(y1, u2)
≠0

Cont’d
 Again OLS cannot be applied to estimate the parameters.
 To find consistent estimators, we look for a variable that
is correlated with y1 but not correlated with u2
 Here we have two choices, namely, z1 and z1 that can
serve as instruments.
 Note: We have more than enough instrumental variables
since the second equation is over identified.

Cont’d
In order to estimate the second equation:
a) Regress y1 on z1, z3 and (if z1 is considered as an IV
for y1 ) or y1 on z2, z3 and (if z2 is considered as an IV
for y1 ) using OLS and obtain .
b) Regress
 Note that the solution is not unique, that is, depending
on whether is z1 considered as an IV for y1 or z2 is
considered as an IV for y1 , we may get different
results.

Cont’d
3. Two-stage least squares (2-SLS) method
 The main difference between the IV and 2-SLS methods
is that in the former case the are used as instruments,
while in the latter case the are used as regressors.
 Both methods yield the same result if the equation under
consideration is exactly identified.
 The 2-SLS procedure is generally applicable for
estimation of over-identified equations as it provides
unique estimators.

Cont’d
Steps:
a) Estimate the reduced form equations by OLS and
obtain the predicted .
b) Replace the right hand side endogenous variables in the
structural equations by the corresponding and
estimate them by OLS.
 Consider the above simultaneous equations model:

Since cov(y2,u1)≠0 and cov(y2,u1)≠0 , we
cannot apply OLS.
Since equation (a) is exactly identified, the 2-
SLS procedure is the same as the IV method.
The 2-SLS procedure of estimation of equation
(b) (which is over-identified) is:
We first estimate the reduced form equations
by OLS; that is, we regress y1on z1, z2, z3 using
OLS and obtain .

Cont’d
 We then y1 replace by and estimate equation (b) by
OLS, that is, we apply OLS to:
Note
a) Unlike ILS, 2-SLS provides only one estimate per
parameter for over-identified models.
b) In case of exactly identified equations, both ILS
and 2-SLS produce the same parameter
estimates.
2
3
3
1
2
2 u
z
c
ŷ
b
y 



Cont’d
c) If the coefficient of determination (R2) from a reduced form equation
assumes a value close to one, then OLS and 2-SLS estimates will be
very close.
d) If R2 is close to zero, then 2-SLS estimates will be meaningless (since
a small value of R2 means that the are poor estimates of the
which they are going to replace).
i
ŷ

Chapter 5.pptx

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Chapter 5.pptx