Lecture 5: Instrumental Variable Methods Chapters 15

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Lecture 5 Instrumental Variable Methods
(Chapters 15)

• Eco420Z
• Dr. S. Chen

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Omitted-variables bias

• Example estimate the return to schooling
  Suppose the true model is
• y?0 ?1x1 ?2x2e
• But we dont observed ability, so we run the
  following regression
• y?0 ?1x1u
• Omitted variables bias arises because
• Cov(x1,u)?0
• Leading OLS estimators to be biased

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Use instrumental variable to deal with
omitted-variables problems

• Suppose we have a third variable, the instrument
  variable z,
• Correlated with education x1
• Cov(z,x1)?0
• Uncorrelated with u (i.e. unrelated to the
  unobserved ability and the error term e)
• Cov(z,u)0
• Then the parameter of interest is identified by

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• Intuitively, instrumental variables (IV) solve
  the omitted variable problem by using only part
  of the variability in schooling x1
  ---specifically, a part that is uncorrelated with
  the omitted variables --- to estimate the
  relationship between schooling and earnings.

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Unobserved variables (ability x2)
Years of Schooling (x1)
Log wage (y)
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Instrumental Variable (Z)
Log wage (y)
Years of Schooling (x1)
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Statistical inference with the IV Estimator (in
large sample)

• IV estimators have the similar asymptotic
  properties as OLS estimators
• Asymptotic variance of OLS estimator
• Asymptotic variance of the IV estimator

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Examples

• Ex15.1 Estimate the return to schooling for
  married women (MROZ)
• x1 married womens education can be correlated
  with wage residuals
• IVfathers education (fatheduc)
• Required conditions for this IV to work
• fatheduc is uncorrelated with wage residual but
  correlated with the womens education

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• Ex15.2 effects of being a veteran in the Vietnam
  War on earnings
• Veteran status (x1) can be correlated with
  earnings residual
• IVVietnam draft lottery
• Required conditions for this IV to work
• The lottery number is uncorrelated with earnings
  residual but correlated with the veteran status

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More Examples about IV Estimation

• Ex 15.4 (Estimate the return to education) Using
  College Proximity as an IV for Education
• Education can be correlated with earning residual
• IVCollege Proximity
• Required conditions for this IV to work
• College proximity is uncorrelated with earnings
  residual but correlated with educational choices.
  
• CARD

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Two Stage Least Squares

• Ex15.4
• Regress education on IV and covariates x
• Because IV is exogenous and uncorrelated with
  ability, the predicted education based on IV
  captures the variation in education not caused by
  unobserved ability
• Regress lwage on the predicted education and
  covariates x
• Note that the set of covariates should be the
  same in both stages. And the IV only shows up in
  the 2nd stage, not the 1st stage (exclusion
  restriction).

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Application IV Solutions to Errors-in-Variables
Problems

• Example (wage regressions)
• y?0 ?1x1 ?2x2 u
• where we dont observe x1 but suppose we have an
  observed proxy x1 x1e1. We call e1 the
  measurement error. This linear relationship
  between the proxy and measurement error is
  referred to as the classical error-in-variable
  assumption. The above equation can be written as
• y?0 ?1x1 ?2x2 (u- ?1e1)
• OLS will be biased because the combined residual
  is correlated with x1 the proxy.

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Solution

• Find another proxy z1 for x1 and treat it as an
  instrument for x1
• Required assumptions for z1 to be a valid
  instrument
• Both proxies are correlated (z1 is correlated x1)
• but their measurement errors are uncorrelated and
  z1 is uncorrelated with u.

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Example (Ex 15.6)

• Wage regression
• lwage?0 ?1educ abil u
• Using test scores as proxy for IQ test1 ?
  abile1
• Thus,
• lwage?0 ?1educ ?1test1 (u-?1e1)
• OLS is biased because test1 is correlated with
  wage residual. Consider test2 as the 2nd proxy
  for IQ. We can use test2 to instrument for test1
  assuming
• test2 is correlated with test1
• test2 is uncorrelated with u and e1 (the
  measurement error of test1)
• WAGE2

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Properties of IV with a Poor Instrumental Variable

• y?0 ?1xu
• IV estimator
• If z is correlated residual u, the bias of IV
  estimates can be very serious when z and x are
  only weakly correlated.

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Example

• Ex15.3 (Estimating the effect of smoking on birth
  weight BWGHT)
• Log(bweight) ?0 ?1packsu
• where packs can be correlated with u (consisting
  of other health factors or the availability of
  good prenatal care).
• Is cigprice (the price of cigs in the sate of
  residence) a good instrument? Is cigprice
  important to explain packs? If not, we shouldnt
  use cigprice as an instrument.