Introduction to Econometrics

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Introduction to Econometrics
Lecture 10 Simultaneous equations models
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Single equations or systems of equations?

• Even if we are only interested in one particular
  equation (e.g. a demand function or a consumption
  function) we may have to consider it as part of a
  system of equations.

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Endogenous regressors and bias

• Bias. Single equation (OLS) estimators will be
  biased if one or more regressors is endogenous
  (jointly dependent).
• Consistency. But other methods may be available
  to obtain consistent estimators, such as Indirect
  Least Squares, Instrumental Variables or Two
  Stage Least Squares estimation.

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The identification problem

• Even before we worry about bias we must be sure
  that we can identify the equation of interest in
  the system.

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The Failure of Least Squares - BIAS
The least squares estimators of the parameters
in a structural simultaneous equation are biased
and inconsistent because of the correlation
between the random error and the endogenous
variables on the right-hand side of the equation.
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Simple example the consumption function as part
of a Keynesian Macro Model
Assumptions of Simple Keynesian Model

• 1. Consumption, c, is function of income, y.
• 2. Total expenditures consumption
  investment.
• 3. Investment assumed independent of income.

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The Structural Equations
consumption is a function of income
c b1 b2 y
income is either consumed or invested
y c i
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The Statistical Model
The consumption equation
ct b1 b2 yt et
The income identity
yt ct it
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The Simultaneous Nature of Simultaneous Equations
ct b1 b2 yt et
Since yt contains et they are correlated
yt ct it
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Single vs. Simultaneous Equations
Single Equation
Simultaneous Equations
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Deriving the Reduced Form
ct b1 b2 yt et
yt ct it
ct b1 b2(ct it) et
(1 - b2)ct b1 b2 it et
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Deriving the Reduced Form
(1 - b2)ct b1 b2 it et
ct p11 p21 it nt
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Reduced Form Equation
ct p11 p21 it nt
1
nt et
and
(1-b2)
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yt ct it
where ct p11 p21 it nt
yt p11 (1p21) it nt
It is sometimes useful to give this equation its
own reduced form parameters as follows
yt p12 p22 it nt
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ct p11 p21 it nt
yt p12 p22 it nt
Since ct and yt are related through the
identity yt ct it , the error term, nt, of
these two equations is the same, and it is easy
to show that
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Identification
The structural parameters are b1 and b2.
The reduced form parameters are p11 and p21.
Once the reduced form parameters are
estimated, the identification problem is to
determine if the original structural parameters
can be expressed uniquely in terms of the reduced
form parameters.
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Identification
An equation is under-identified if its structural
(behavioural) parameters cannot be expressed in
terms of the reduced form parameters.
An equation is exactly identified if its
structural (behavioural) parameters can be
uniquely expressed in terms of the reduced form
parameters.
An equation is over-identified if there is
more than one solution for expressing its
structural (behavioural) parameters in terms of
the reduced form parameters.
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The Identification Problem
A system of M equations containing M endogenous
variables must exclude at least M-1 variables
from a given equation in order for the parameters
of that equation to be identified and to be able
to be consistently estimated.
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Two Stage Least Squares
yt1 b1 b2 yt2 b3 xt1 et1
yt2 a1 a2 yt1 a3 xt2 et2
Problem right-hand endogenous variables yt2 and
yt1 are correlated with the error terms.
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Problem right-hand endogenous variables yt2 and
yt1 are correlated with the error terms.
Solution First, derive the reduced form
equations.
yt1 b1 b2 yt2 b3 xt1 et1
yt2 a1 a2 yt1 a3 xt2 et2
Solve two equations for two unknowns, yt1, yt2
yt1 p11 p21 xt1 p31 xt2 nt1
yt2 p12 p22 xt1 p32 xt2 nt2
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2SLS Stage I
yt1 p11 p21 xt1 p31 xt2 nt1
yt2 p12 p22 xt1 p32 xt2 nt2
Use least squares to get fitted values
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2SLS Stage II
and
Substitute in for yt1 , yt2
yt1 b1 b2 yt2 b3 xt1 et1
yt2 a1 a2 yt1 a3 xt2 et2
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2SLS Stage II (continued)
Run least squares on each of the above
equations to get 2SLS estimates






b1 , b2 , b3 , a1 , a2 and a3
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2SLS and Instrumental Variables (IV) estimation
an example from microeconomics

• Take a simple model of demand and supply
• The demand function here includes two exogenous
  variables Y (income) and W (Wealth). We expect
  g1 gt 0, g2 gt 0.

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OLS estimation of the supply function is biased
because P is endogenous (jointly determined)

• So

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Biased and inconsistent

• So with
• OLS produces biased and inconsistent estimates.
• But we can use an IV estimator.

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IV estimation

• We could eliminate the problem by finding a proxy
  to P
• The proxy should be
• highly correlated with P
• uncorrelated with the error term in the supply
  equation.
• If such a proxy was to available we could
  directly estimate the identified supply equation
  using IV estimation.

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IV estimation

• How do we find an instrumental variable?
• There are two methods
• Arbitrary search and test.
• Two stage least squares.
• 2SLS offers an excellent direct estimation method
  in the case of over-identified equations.

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Two stage least squares (2SLS)

• While it is still a single equation estimation
  technique, 2SLS uses the information available
  from the specification of the entire equation
  system.
• In doing so, it is able to provide unique
  estimates of each structural parameter in the
  over-identified equation.

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Two stage least squares

• The first stage involves the creation of an
  instrument.
• The second stage involves a variant of
  instrumental variables estimation.
• So it is in fact a special way and perhaps less
  arbitrary way of doing instrumental variables
  estimation.

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2SLS in the supply and demand model

• Is the demand equation identified?
• Two equations - (G - 1) 2 - 1 1. The demand
  equation imposes R 0 restrictions.
• Therefore it is not identified.

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2SLS in the supply and demand model

• Is the supply equation identified?
• Two equations - (G - 1) 1. The supply equation
  imposes R 2 restrictions, whereas it only needs
  to impose R 1.
• Therefore it is over-identified.

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• The demand curve is not identified, but shifts in
  the demand curve allow the supply curve to be
  identified

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2SLS in the supply and demand model

• As it is over-identified, indirect estimation of
  the reduced form will not generate unique
  estimates of the structural parameters.

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2SLS in the supply and demand model

• So what if we were to use IV directly?
• Natural instruments would include Yt and Wt.
• Both would produce consistent estimates.
• But how would we choose between them?

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2SLS in the supply and demand model

• A reasonable (and efficient) way out of this
  dilemma is to choose as an instrument a weighted
  average of the two exogenous variables.
• The weights being chosen to maximise the
  correlation between the new instrument and Pt.

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2SLS in the supply and demand model

• How do we get this instrument?
• We simply regress Pt on Yt and Wt and calculate
  the fitted values.

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First Stage of 2SLS

• You would regress the endogenous variable on all
  the predetermined variables in the system.
• This just means you estimate the RF equation for
  Pt .
• You can use OLS for this purpose.

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First Stage of 2SLS

• The first-stage gives you the fitted values
• The fitted values are independent of either error
  term in the system.
• So we construct an instrument which via OLS is
  linearly related to the endogenous variable but
  which is independent of the error term in the
  supply equation.

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Second stage of 2SLS

• We directly estimate the supply equation of the
  structural model by replacing the variable Pt
  with the fitted value .
• So we regress using OLS
• This yields consistent estimates for all the
  coefficients in the equation.

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2SLS - summary

• We directly estimate the over-identified
  equation.
• 2SLS eliminates the problem of an oversupply of
  instruments by using combinations of
  predetermined variables to create a new
  instrument.
• It is applied to a single equation without
  considering other equations. This is very useful
  in dealing with large models.

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2SLS - summary

• It provides one estimate per parameter and thus
  overcomes identification.
• We need only know the predetermined variables in
  the system rather than the exact structure of the
  other equations.
• If the R2 in the RF regression (the first stage)
  is high (gt 0.80) then the OLS and 2SLS estimates
  will be close because the fitted values will be
  close to the actual values.

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Comparing 2SLS, ILS and IV

• When an equation is exactly identified 2SLS, ILS
  and IV are identical.
• ILS will provide a unique estimate and the RF has
  only the exact number of exogenous variables
  required for identification.
• IV will use only one instrument.
• 2SLS will also use only one instrument.

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Unidentified equations

• 2SLS cannot be used to estimate an unidentified
  equation.
• This is because the fitted values for Pt would be
  included in a second-stage regression (the demand
  equation) along with Yt and Wt which would be
  perfectly collinear.