NonExperimental Data: Natural Experiments and more on IV

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Non-Experimental DataNatural Experiments and
more on IV
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(No Transcript)
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Non-Experimental Data

• Refers to all data that has not been collected as
  part of experiment
• Quality of analysis depends on how well one can
  deal with problems of
• Omitted variables
• Reverse causality
• Measurement error
• selection
• Or how close one can get to experimental
  conditions

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Natural/ Quasi Experiments

• Used to refer to situation that is not
  experimental but is as if it was
• Not a precise definition saying your data is a
  natural experiment makes it sound better
• Refers to case where variation in X is good
  variation (directly or indirectly via
  instrument)
• A Famous Example London, 1854

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The Case of the Broad Street Pump

• Regular cholera epidemics in 19th century London
• Widely believed to be caused by bad air
• John Snow thought bad water was cause
• Experimental design would be to randomly give
  some people good water and some bad water
• Ethical Problems with this

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Soho Outbreak August/September 1854

• People closest to Broad Street Pump most likely
  to die
• But breathe same air so does not resolve air vs.
  water hypothesis
• Nearby workhouse had own well and few deaths
• Nearby brewery had own well and no deaths
  (workers all drank beer)

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Why is this a Natural experiment?

• Variation in water supply as if it had been
  randomly assigned other factors (air) held
  constant
• Can then estimate treatment effect using
  difference in means
• Or run regression of death on water source
  distance to pump, other factors
• Strongly suggests water the cause
• Woman died in Hampstead, niece in Islington

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Whats that got to do with it?

• Aunt liked taste of water from Broad Street pump
• Had it delivered every day
• Niece had visited her
• Investigation of well found contamination by
  sewer
• This is non-experimental data but analysed in a
  way that makes a very powerful case no theory
  either

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Methods for Analysing Data from Natural
Experiments

• If data is as if it were experimental then can
  use all techniques described for experimental
  data
• OLS (perhaps Snow case)
• IV to get appropriate units of measurement
• Will say more about IV than OLS
• IV perhaps more common
• If can use OLS not more to say
• With IV there is more to say weak instruments

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Conditions for Instrument Validity

• To be valid instrument
• Must be correlated with X - testable
• Must be uncorrelated with error untestable
  have to argue case for this assumption
• These conditions guaranteed with instrument for
  experimental data
• But more problematic for data from
  quasi-experiments

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Bombs, Bones and BreakpointsThe Geography of
Economic Activity Davis and Weinstein, AER, 2002

• Existence of agglomerations (e.g. cities) a
  puzzle
• Land and labour costs higher so why dont firms
  relocate to increase profits
• Must be some compensatory productivity effect
• Different hypotheses about this
• Locational fundamentals
• Increasing returns (Krugman) path-dependence

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Testing these Hypotheses

• Consider a temporary shock to city population
• Locational fundamentals theory would predict no
  permanent effect
• Increasing returns would suggest permanent effect
• Would like to do experiment of randomly assigning
  shocks to city size
• This is not going to happen

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The Davis-Weinstein idea

• Use US bombing of Japanese cities in WW2
• This is a natural experiment not a true
  experiment because
• WW2 not caused by desire to test theories of
  economic geography
• Pattern of US bombing not random
• Sample is 303 Japanese cities, data is
• Population before and after bombing
• Measures of destruction

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Basic Equation

• ?si,47-40 is change in population just before and
  after war
• ?si,60-47 is change in population at later period
• How to test hypotheses
• Locational fundamentals predicts ß1-1
• Increasing returns predicts ß10

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The IV approach

• ?si,47-40 might be influenced by both permanent
  and temporary factors
• Only want part that is transitory shock caused by
  war damage
• Instrument ?si,47-40 by measures of death and
  destruction

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The First-Stage Correlation of ?si,47-40 with Z
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Why Do We Need First-Stage?

• Establishes instrument relevance correlation of
  X and Z
• Gives an idea of how strong this correlation is
  weak instrument problem
• In this case reported first-stage not obviously
  that implicit in what follows
• That would be bad practice

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The IV Estimates
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Why Are these other variables included?

• Potential criticisms of instrument exogeneity
• Government post-war reconstruction expenses
  correlated with destruction and had an effect on
  population growth
• US bombing heavier of cities of strategic
  importance (perhaps they had higher growth rates)
  
• Inclusion of the extra variables designed to head
  off these criticisms
• Assumption is that of exogeneity conditional on
  the inclusion of these variables
• Conclusion favours locational fundamentals view

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An additional piece of supporting evidence.

• Always trying to build a strong evidence base
  many potential ways to do this, not just
  estimating equations

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The Problem of Weak Instruments

• Say that instruments are weak if correlation
  between X and Z low (after inclusion of other
  exogenous variables)
• Rule of thumb - If F-statistic on instruments in
  first-stage less than 10 then may be problem
  (will explain this a bit later)

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Why Do Weak Instruments Matter?

• A whole range of problems tend to arise if
  instruments are weak
• Asymptotic problems
• High asymptotic variance
• Small departures from instrument exogeneity lead
  to big inconsistencies
• Finite-Sample Problems
• Small-sample distirbution may be very different
  from asymptotic one
• May be large bias
• Computed variance may be wrong
• Distribution may be very different from normal

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Asymptotic Problems ILow precision

• asymptotic variance of IV estimator is larger the
  weaker the instruments
• Intuition variance in any estimator tends to be
  lower the bigger the variation in X think of
  s2(XX)-1
• IV only uses variation in X that is associated
  with Z
• As instruments get weaker using less and less
  variation in X

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Asymptotic Problems IISmall Departures from
Instrument Exogeneity Lead to Big Inconsistencies

• Suppose true causal model is
• yXßZ?e
• So possibly direct effect of Z on y.
• Instrument exogeneity is ?0.
• Obviously want this to be zero but might hope
  that no big problem if close to zero a small
  deviation from exogeneity

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But this will not be the case if instruments
weak consider just-identified case

• If instruments weak then SZX small so SZX-1 large
  so ? multiplied by a large number

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An Example The Return to Education

• Economists long-interested in whether investment
  in human capital a good investment
• Some theory shows that coefficient on s in
  regression
• yß0ß1sß2xe
• Is measure of rate of return to education
• OLS estimates around 8 - suggests very good
  investment
• Might be liquidity constraints
• Might be bias

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Potential Sources of Bias

• Most commonly mentioned is ability bias
• Ability correlated with earnings independent of
  education
• Ability correlated with education
• If ability omitted from x variables then usual
  formula for omitted variables bias suggests
  upward bias in OLS estimate

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Potential Solution

• Find an instrument correlated with education but
  uncorrelated with ability (or other excluded
  variables)
• Angrist-Krueger Does Compulsory Schooling
  Attendance Affect Schooling and Earnings , QJE
  1991, suggest using quarter of birth
• Argue correlated with education because of school
  start age policies and school leaving laws
  (instrument relevance)
• Dont have to accept this can test it

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A graphical version of first-stage (correlation
between education and Z)
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In this case

• Their instrument is binary so IV estimator can be
  written in Wald form
• And this leads to following expression for
  potential inconsistency

• Note denominator is difference in schooling for
  those born in first- and other quarters
• Instrument will be weak if this difference is
  small

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Their Results
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Interpretation (and Potential Criticism)

• IV estimates not much below OLS estimates (higher
  in one case)
• Suggests ability bias no big deal
• But instrument is weak
• Being born in 1st quarter reduces education by
  0.1 years
• Means ? will be multiplied by 10

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But why should we have ??0

• Remember this would imply a direct effect of
  quarter of birth on earnings, not just one that
  works through the effect on education
• Bound, Jaeger and Baker argued that evidence that
  quarter of birth correlated with
• Mental and physical health
• Socioeconomic status of parents
• Unlikely that any effects are large but dont
  have to be when instruments are weak

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An example UK data
Effect is small but significantly different from
zero
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A Back-of-the-Envelope Calculation

• Being born in first quarter means 0.01 less
  likely to have a managerial/professional parent
• Being a manager/professional raises log earnings
  by 0.64
• Correlation between earnings of children and
  parents 0.4
• Effect on earnings through this route
  0.010.640.40.00256 i.e. ¼ of 1 per cent
• Small but weak instrument causes effect on
  inconsistency of IV estimate to be multiplied by
  10 0.0256
• Now large relative to OLS estimate of 0.08

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Summary

• Small deviations from instrument exogeneity lead
  to big inconsistencies in IV estimate if
  instruments are weak
• Suspect this is often of great practical
  importance
• Quite common to use odd instrument argue that
  no reason to believe it is correlated with e
  but show correlation with X

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Finite Sample Problems

• This is a very complicated topic
• Exact results for special cases, approximations
  for more general cases
• Hard to say anything that is definitely true but
  can give useful guidance
• Problems in 3 areas
• Bias
• Incorrect measurement of variance
• Non-normal distribution
• But really all different symptoms of same thing

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Review and Reminder

• If ask STATA to estimate equation by IV
• Coefficients compute using formula given
• Standard errors computed using formula for
  asymptotic variance
• T-statistics, confidence intervals and p-values
  computed using assumption that estimator is
  unbiased with variance as computed and normally
  distributed
• All are asymptotic results

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Difference between asymptotic and finite-sample
distributions

• This is normal case
• Only in special cases e.g. linear regression
  model with normally distributed errors are
  small-sample and asymptotic distributions the
  same.
• Difference likely to be bigger
• The smaller the sample size
• The weaker the instruments

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Rule of Thumb for Weak Instruments

• F-test for instruments in first-stage gt10
• Stricter than significant e.g. if one instrument
  F10 equivalent to t3.3

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Conclusion

• Natural experiments useful source of knowledge
• Often requires use of IV
• Instrument exogeneity and relevance need
  justification
• Weak instruments potentially serious
• Good practice to present first-stage regression
• Finding more robust alternative to IV an active
  research area