Todays lecture

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Todays lecture

• To introduce simultaneity
• To discuss why the standard OLS model doesnt
  work in the presence of simulaneity
• To introduce identification issues
• Methods of estimation

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Simultaneity

• Most economic models are simultaneous i.e. At
  least two relationships between the variables in
  the regression.
• Good to think of cause and effect.
• Macro example
• YCI
• OLS will mix up the two relationships

c ?1 ?2 y
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Macro Example
1. Consumption, c, is function of income,
y. c is endogenous ???is MPC 2. y
consumption investment. y is
endogenous 3. Investment assumed independent of
income. i is exogenous
c ?1 ?2 y
y c i
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Single vs. Simultaneous Equations
Simultaneous Equations
Single Equation
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ct ?1 ?2 yt et
yt ct it
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• 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

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• 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

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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

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The Failure of Least Squares
The least squares estimators of parameters in a
structural simul- taneous equation is biased
and inconsistent because of the cor- relation
between the random error and the endogenous
variables on the right-hand side of the equation.
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Estimation of simultaneous equations

• Systems method (full information maximum
  likelihood method) All equations estimated
  simultaneously. Highly complex and not much used
• Single equation method (limited information
  method) each equation estimated individually.
• Indirect least squares
• 2stage least squares

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Indirect Least Squares

• One way to estimate is to do OLS on the reduced
  form
• re-write the system of equations in their 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
• This works because no endogenous variable on the
  right hand side i.e. unbiased and consistent

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ct ?1 ?2 yt et
yt ct it
ct ?1 ?2(ct it) et
(1 ? ?2)ct ?1 ?2 it et
ct ?11 ?21 it ?t
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• We can do the same for the equation in Y
• We get the reduced form of the system
• Note the conceptual difference between structural
  and reduced equations

ct ?11 ?21 it ?t
yt ?12 ?22 it wt
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• We can then use the formulae that link the
  parameters of the reduced and structural forms to
  calculate the estimates of ?

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• 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 ?
• Indirect Least Squares linked to the notion of
  Exact Identification

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Identification problem

• Whether numerical estimates of parameters of the
  structural equation can be obtained from the
  estimated reduced form coefficients.
• If it can be , equation is identified
• If not, under- or un-identified
• Equation is over-identified if more than one
  value can be obtained for the parameters of the
  structural coefficients based on the reduced form
  estimates.

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But Dd function cannot be identified
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Conditions of Identification

• Order condition Necessary but not sufficient
• Rank Condition Sufficient
• Mno of endogenous variables in model
• mno of endogenous variable in equation
• Kno of predetermined variables in model incl
  intercept
• kno of predetermined variables in equation incl
  intercept
• Order condition for identification
• K-kgtm-1

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Rank condition

• In a model with M endogenous variables, an
  equation is identified if and only if at least
  one non-zero determinant of order (M-1)(M-1) can
  be constructed from the coefficient of variables
  (both endogenous and predetermined) excluded from
  that particular equation but included in the
  other equations of the model

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Principles of Identifiability

• If K-kgtm-1 rank of matrix is M-1, eqn is over
  identified
• If K-km-1 rank of matrix is M-1, eqn is just
  identified
• If K-kgtm-1 rank of matrix is less than M-1,
  eqn is under identified
• If K-kltm-1, eqn is unidentified
• If equation exactly identified, ILS may be used
• If over identified, 2SLS (instrument variable
  method) may be used

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Estimation- 2SLS

• Two stage least squares useful when equation is
  over identified.
• 1. Regress Y1 on all the predetermined variables
  in the system.
• 2. Y1 in Over-identified Money supply eqn can be
  substituted with estimated y

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• Although Y1 in original MS equation is correlated
  with u2, estimated Y1 is unlikely to be
  correlated with u2. OLS will give consistent
  estimate

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Properties of 2SLS

• Very practical single equation method
• Even in over-identified eqn, 2SLS gives unique
  estimate
• Estimates are consistent but biased
• Estimates are asymptotically normal
• Standard errors are not same formula as OLS --
  usually built into software
• Also known as Instrumental Variables (IV)