Simultaneous Equations

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Simultaneous Equations

• Main Reading Chapter 18,19 20, Gujarati

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The goal of todays lecture

• To introduce simultaneity
• To discuss why the standard OLS model doesnt
  work in the presence of simulaneity
• To introduce identification issues
• To give empirical examples

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Introduction - 1

• Heteroscedasticity and Autocorrelation
• OLS coefficients are still unbiased though no
  longer most efficient
• A number of cases for which OLS returns biased
  estimates
• Measurement Error in the Regressors
• Omitted Variables
• Lagged Endogenous Variables with Autocorrelation
• Simultaneity

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Introduction - 2

• 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
• Micro example supply and demand
• Lesson simultaneity can appear anywhere
• 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 b2 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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The Structural Formof the Statistical Model
ct ?1 ?2 ytet
yt ct it
Identity
et is a random disturbance term
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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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Single vs. Simultaneous Equations
Simultaneous Equations
Single Equation
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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

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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

ct ?11 ?21 it ?t
yt ?12 ?22 it ?t
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Failure of OLS

• OLS picks best fit --- a mixture of both
  relationships
• Will not yield correct estimate of MPC

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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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ct ?1 ?2 yt et
yt ct it
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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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Formal Proof More detailed version next week

• We can see this explicitly, if we re-write the
  system of equations in their reduced form and use
  the formula for the OLS estimator
• Reminder formula for OLS

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• First step show that the covariance of Y and e
  is not zero i.e. calculate how they move together
  (see p 725 Gujarati).

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• OLS formula is
• Taking expectations we get

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Examples (from Wooldridge)

• Murder Rates and the Size of the Police Force

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Examples (from Wooldridge)

• Romer (1993) Inflation and Openness

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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

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Micro Example

• Book uses a micro economic example i.e. Supply
  and Demand model
• Structural model
• Demand
• Supply
• Price and quantity are endogenous (jointly
  determined) and income is exogenous

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• 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 a1 will pick up the effect of
  the supply curve also
• Cov(p,ed) 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

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Illustrating the Identification Problem

• Suppose we observe the following data
• Is this a supply curve or a demand curve?
• It looks like a supply curve

.
.
.
.
.
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• 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

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• Or it could be movement in both

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• It turns out that we can estimate b 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

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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

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• 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

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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.
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Importance of Identification

• Must check identification before trying 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
• Note necessary vs. sufficient condition

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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.

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Indirect Least Squares

• One way to estimate is to do OLS on the reduced
  form
• This works because no endogenous variable on the
  right hand side i.e. unbiased and consistent

ct ?11 ?21 it ?t
yt ?12 ?22 it ?t
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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 b

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

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Estimation- 2SLS

• Two stage least squares
• 1. Estimate the reduced form using OLS.
  
• 2. Do OLS on the structural form with the
  actual values replaced by the fitted values
  from the first stage

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• 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 e because it is a function
  solely of y which is uncorrelated with e (i.e.
  exogenous)

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• 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

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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

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• Suppose one of the equations you want to estimate
  is
• First check that it is identified i.e. are enough
  x variables excluded from the equation
• Estimate the reduced form for the entire model

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• Replace the endogenous variables in the
  structural equations with their fitted values and
  do OLS
• Note Possible problem with standard errors in
  some computer programs

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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

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Example Market for Truffles

• Structural model
• Demand
• Supply
• ps price of substitute,
• pfrent of pig (i.e. cost of production)
• di per capita disposable income

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Identification

• P and Q are endogenous
• pf, ps and di 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 judgements
• reasonable to say variable is exogenous
• reasonable to exclude it

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Stage 1 Estimate Reduced Form

• Endogenous on left, all exogenous on right
• 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?

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Stage 2 Estimate Structural Form

• Calculate the fitted values for p and q
• Do OLS on
• Note the signs and significance of the coef.

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Macro Example Klein Model

• A simple macro model
• What is endogenous?

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• What is exogenous?
• What are the valid instruments?
• Exogenous variables gov, tax, time
• predetermined variables lagged wages, profits,
  GDP
• reasonable?
• What equations are identified?
• Are exclusions reasonable?
• NB two subjective judgements
• reasonable to say variable is exogenous
• reasonable to exclude it