Review of Probability and Statistics

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Instrumental Variables and Two Stage Least
Squares

• y b0 b1x1 b2x2 . . . bkxk u
• x1 p0 p1z p2x2 . . . pkxk v

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OUTLINE

• When we need Instrumental Variables?
• What is an Instrumental Variable?
• An Example
• IV Estimation in the Simple RM
• IV Estimator
• Inference
• Poor Instruments
• Some Applications
• IV Estimation in the Multiple RM
• 2SLS Estimator
• Inference
• An Application Institutions and Development
• Addressing Errors-in-Variables with IV Estimation
• Testing for Endogeneity
• Testing for Overidentifying Restrictions
• 2SLS with Heteroscedasticity
• 2SLS with Serial Correlation

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1. When we need Instrumental Variables?

• Instrumental Variables (IV) estimation is used
  when your model has endogenous xs
• That is, whenever Cov(x,u) ? 0
• Thus, IV can be used to address the problem of
  omitted variable bias
• Additionally, IV can be used to solve the classic
  errors-in-variables problem

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2. What is an Instrumental Variable?

• In order for a variable, z, to serve as a valid
  instrument for x, the following three conditions
  must be true
• (1) The instrument must be exogenous
• That is, Cov(z,u) 0
• (2) The instrument must be correlated with the
  endogenous variable x
• That is, Cov(z,x) ? 0
• (3) The instrument should not be a regressor in
  the equation for y, or being perfectly correlated
  with the regressors in that equation.

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What is an Instrumental Variable? (cont.)

• Conditions (2) and (3) are simple to verify. They
  can be tested using the data.
• For condition (2) Just testing H0 p1 0 in x
  p0 p1z v
• For condition (3) We can just look at the
  R-square in the regression of z on all the
  regressors other than x.

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What is an Instrumental Variable? (cont.)

• Condition (1), Cov(z,u) 0, is the key one. And
  it can not be tested
• To justify that condition (1) holds we need to
  have a model with a clear interpretation of which
  are the variables in the error term u.
• We have to use economic theory and common sense
  to decide if it makes sense to assume Cov(z,u) 0

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3. An Example

• Problem Estimate effect of treatment (T) on
  outcome (Y). i.e., estimate ?1 in
• Yi ?0 ?1 Ti ui
• For simplicity, suppose
• Dichotomous treatment variable T1 if treated, 0
  otherwise
• Homogeneous treatment effect (?1)
• No other regressors.

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3. An Example

• For concretness, supose that we are interested in
  the effect of an investment subsidy on firms
  capital investment.
• Ti is the binary variable that indicates if a
  firm has applied for and has been granted the
  subsidy.
• Yi represents the firms investment rate.

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3. An Example

• OLS estimation yields the estimator

However, the key assumption for the consistency
of the OLS estimator (no correlation between T
and u) is unlikely to hold because treatment is
related to omitted factors u influencing
outcome.
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Four Solutions to this Problem

• Randomized Controlled Trial
• Natural Experiments Find similar observations
  with different treatment for arbitrary reasons
  (e.g. regulatory rules, law changes).Difference-
  in-Difference estimates
• Control for Observable Differences
• Attempt to condition on sufficient X's such that
  E(Tu)0
• Then estimate directly by least squares
• (1) Y ?0 ?1 T X? u
• Instrumental Variables (IV)
• Suppose exists instrumental variable (Z) that
  is
• (A1) correlated with treatment E(Z T) ? 0
• (A2) Uncorrelated with residual E(Zu)0

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Example Simple IV

• For instance, in the investment subsidy example,
  suppose that only a random number of firms can
  apply for the subsidy.
• Let Z be the dummy variable (0,1) that indicates
  whether a firm can apply to obtain the subsidy or
  not.
• Because Z is purely random, it is not related to
  u.
• However, Z should be correlated with T because to
  be granted the subsidy (T1) it is necessary to
  be eligible (Z1).

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Example Simple IV

• Then, we have the following moment conditions
• E( ui ) 0 that implies E(Yi - ?0 - ?1
  Ti) 0
• E( Zi ui ) 0 that implies E(Zi Yi -
  ?0 - ?1 Ti) 0
• Using the method of moments, we estimate ?0 and
  ?1 using the the sample moment conditions
  associated with the previous population moment
  conditions.

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Example Simple IV

• In this example, this estimator (IV) is
• (Difference in mean outcomes)/(difference in
  treatment rate)

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4. IV Estimation in the Simple RM

• For y b0 b1x u, and given our assumptions
• Cov(z,y) b1Cov(z,x) Cov(z,u),
• b1 Cov(z,y) / Cov(z,x)
• Therefore, given a random sample of x,y,z, by
  the LLN a consistent estimator of b1 is (the IV
  estimator)

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Inference with IV Estimation

• The homoskedasticity assumption in this case is
  E(u2z) s2 Var(u)
• As in the OLS case, given the asymptotic
  variance, we can estimate the standard error

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Comparison of IV and OLS standard errors

• Standard error in IV case differs from OLS only
  in the R2 from regressing x on z
• Since R2 lt 1, IV standard errors are larger
• However, IV is consistent, while OLS is
  inconsistent, when Cov(x,u) ? 0
• The stronger the correlation between z and x, the
  smaller the IV standard errors

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Poor Instruments

• We have a poor instrument when z and x are weakly
  correlated.
• The problem of weak instruments is not just that
  the variance of the IV estimator is much larger
  than the variance of the OLS.
• A more serious problem is that the IV estimator
  can have a large asymptotic bias even if z and u
  are only moderately correlated.

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Poor Instruments (cont.)

• We can compare the asymptotic bias in OLS and IV

• Prefer IV if Corr(z,u)/Corr(z,x) lt Corr(x,u),
  that is if
• Corr(z,x) gt Corr(z,u)/Corr(x,u)

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Some Applications IV.Estimating treatment
effects in AMI

• McClellan, M., B. McNeil and J. Newhouse, JAMA,
  1994.
• "Does More Intensive Treatment of Acute
  Myocardial Infarction Reduce Mortality?
• ? Medicare claims data, elderly with heart
  attack (AMI), 1987-91
• ? Treatment Cardiac Catheterization (marker for
  aggressive care)
• ? Outcome Survival to 1 day, 30 days, 90 days,
  etc.
• ? Instrument Is nearest hospital a
  catheterization hospital?
• Differential Distance
• (distance to nearest cath) - (distance to
  nearest non-cath)
• based on zipcode of residence, zip code of
  hospital

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Poor Instruments (cont.)

• Suppose that Corr(z,x)0.10 (which in fact is
  larger than in many applications).
• Then, the IV estimator has a smaller bias than
  the OLS estimator only if Corr(z,u) is at least
  10 times smaller than Corr(x,u).
• Suppose that Corr(x,u)0.10 and that
  Corr(x,u)0.01.
• Then, the IV estimator has a smaller bias than
  the OLS estimator only if Corr(z,x)gt0.10.

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Is Differential Distance a Good Instrument?

• Correlated with treatment (Cath)? Yes. ?
  26.2 get Cath if nearest hospital is Cath
  hospital ? 19.5 get Cath if nearest hospital
  is not Cath hospital
• 2. Uncorrelated with unobserved patient severity?
  Never sure! But unrelated to observable patient
  severity in claims

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Major Findings of McClellan et al.

• Least squares dramatically overstates treatment
  effect, because Cath associated with fewer risk
  factors.
• ? 1-year mortality is 30 lower (17 vs. 47)
  if Cath ? OLS estimate is 24, adjusting for
  observable risk factors
• 2. IV estimates suggest Cath associated with 5-10
  percentage point reduction in mortality nearly
  all in 1st day.

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Validating McClellan et al.

• Recent work replicates validates earlier work
  using
• 1. more comprehensive control variables
• 2. alternative instruments
• McClellan and Noguchi, 1998 (Tables 1 2 below)
• Geppert, McClellan and Staiger, 2001 (Table 4
  below)
• -- Data from Cooperative Cardiovascular Project
  (CCP)
• Chart data for appx. 180,000 AMI patients from
  1994-95
• Linked Medicare claims data
•  
• -- Treatments and outcomes of AMI in elderly as
  in earlier work
•  
• -- Instruments
• (1) Differential distance
• (2)    Variation in hospital Cath rate (gt4000
  dummies)
•  

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Key Validation Questions

• Are severity measures unobserved in claims data
  uncorrelated with instrument (differential
  distance)?
• Are OLS results closer to IV with more extensive
  controls?
• Are IV results robust to more extensive controls?
  
• Are IV results robust to alternative instruments?
  

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Conclusions of Validation

• Measured individual covariates can be used to
  assess bias of alternative methods for estimating
  treatment effects with observational data.
• Methods that attempt to adjust for observable
  differences are quite sensitive to the use of
  more detailed chart data, and yield biased
  estimates of treatment effects in commonly
  available datasets.
• IV methods for evaluating AMI treatment are not
  sensitive to the use of more detailed chart data,
  and appear to have minimal bias.

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Growing number of applications of IV using a
variety of instruments

•  n    Geography as an instrument
• (distance, rivers, small area variation)
•  n    Legal/political institutions as an
  instrument
• (laws, election dynamics)
•  n    Administrative rules as an instrument
• (wage/staffing rules, reimbursement rules,
  eligibility rules)
•  n    Naturally occurring randomization
• (draft, birth timing, lottery, roommate
  assignment, weather)

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Example Angrist Krueger

• Data US Census 5 PUMS
• Sample 329,509 men born 1930-39
• ln(earnings) Education? X? e
• Instruments (Quarter of Birth)(Year of
  birth) (Quarter of Birth)(State of birth)
• Instruments 178
• First-stage F 1.869(p-value) (0.000)

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Example Angrist Krueger (cont.)

• Estimate of ? 95 Confidence interval
•  
• OLS 0.063 (0.062,0.063)
•  
• 2SLS 0.081 (0.060,0.102)
•  
• 2SLS 0.060 (0.031,0.089)
• with random
• instruments
•  
• LIML 0.098 (0.068, 0.128)
•  
• Valid Confidence (-0.015,0.240)
• Interval (Anderson-Rubin)

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Example Geppert, McClellan Staiger

• Use between-hospital variation in treatment
  intensity (e.g. cath rate) as instrument to
  estimate treatment effects
• Equivalent to using gt4000 hospital dummies as
  instruments
• But instruments are weak 1st Stage F-statistic
  is 10-25 ? 2SLS estimates have small bias (1/F)
  towards OLS ? 2SLS SEs are too small (many
  instruments, modest F) ? LIML SEs should be
  okay
• Using hierarchical structure, we develop
  alternative GMM estimation procedure to correct
  estimates SEs. (asymptotically equivalent to
  LIML, but simpler)
• Cath effects similar to McClellan et al., but
  more precisely estimated

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4. IV Estimation in the Multiple RM

• IV estimation can be extended to the multiple
  regression case.
• Call the model we are interested in estimating
  the structural model.
• Our problem is that one or more of the variables
  are endogenous.
• We need an instrument for each endogenous variable

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IV Estimation in the Multiple RM (cont.)

• Write the structural model as
• y1 b0 b1y2 b2z1 u1
• where y2 is endogenous and z1 is exogenous.
• Let z2 be the instrument, so Cov(z2,u1) 0 and
• y2 p0 p1z1 p2z2 v2
• where p2 ? 0
• This reduced form equation regresses the
  endogenous variable on all exogenous ones

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Two Stage Least Squares (2SLS)

• Its possible to have multiple instruments
• Consider our original structural model, and let
• y2 p0 p1z1 p2z2 p3z3 v2
• Here were assuming that both z2 and z3 are valid
  instruments they do not appear in the
  structural model and are uncorrelated with the
  structural error term, u1

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2SLS Best Instrument

• Could use either z2 or z3 as an instrument
• The best instrument is a linear combination of
  all of the exogenous variables, y2 p0 p1z1
  p2z2 p3z3
• We can estimate y2 by regressing y2 on z1, z2
  and z3 can call this the first stage
• If then substitute y2 for y2 in the structural
  model, get same coefficient as IV

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More on 2SLS

• While the coefficients are the same, the standard
  errors from doing 2SLS by hand are incorrect, so
  let Stata do it for you.
• Method extends to multiple endogenous variables
  need to be sure that we have at least as many
  excluded exogenous variables (instruments) as
  there are endogenous variables in the structural
  equation

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Institutions as the fundamental cause of
long-run growth

• D. Acemoglu, S. Johnson, J. Robinson (2004) with
  some additions
• Theoretical Framework
• Economic Institutions and Income differences
• Natural Experiments
• The Colonial Origins of Comparative Development
  An Empirical Investigation (2001)
• Why do Institutions differ?
• Sources of inefficiencies
• Political implications
• Summary

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Theoretical Framework
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Economic Institutions and Income Differences

• Economic institutions (vs. geography and culture)
  as fundamental cause of different patterns of
  economic growth
• Good economic institutions
• (to simplify and focus the discussion)
  institutions that provide security of property
  rights and relatively equal access to economic
  resources to a broad cross-section of society

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Economic Institutions and Income Differences
Average Protection Against Risk of Expropriation
1985-95 and log GDP per capita 1995
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Economic Institutions and Income Differences

• Secure property rights cause prosperity?
• Problems with making such an inference!
• It could be reverse causation!
• It could be a problem of omitted variable bias
• What can we do?
• look for a natural experiment
• find a source of variation in economic
  institutions that should have no effect on
  economic outcomes

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Natural Experiment The Korean Experiment

• At the time of separation
• approximately the same GDP per capita
• Few geographic and cultural distinctions
• North followed the model of Soviet socialism and
  the Chinese Revolution in abolishing private
  property. Economic decision not mediated by the
  market
• South system of private property and market and
  private incentives to develop the economy

GDP per capita in North and South Korea 1950-98
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Natural Experiment The Korean Experiment

• The only possible explanation for the radically
  different economic experience
• their very different INSTITUTIONS
• Necessity to look at a larger scale natural
  experiment in institutional divergence!!!

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Natural Experiment The Colonial Experiment

• Europeans imposed different sets of institutions
  in different parts of the globe
• The Reversal of Fortune
• The nation states that coincide today with the
  boundaries of prosperous empires (Incas, Aztecs)
  in 1500 are among the poorer societies today!
• The less developed civilisation in North America,
  Australia are much richer than those in the land
  of Incas and Aztecs

log GDP per capita in 1995 and log Population
Density in 1500
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The Colonial Experiment

• Institutions hypothesis of the Reversal Fortune
• Densely-settled relative developed places
  worse institutions
• Sparsely-settled areas better institutions
• Why?
• Introduce/maintain extraction resources economy
  in densely settled areas (where they could
  exploit the population)
• Protection their own rights in sparsely-settled
  areas where the Europeans were the majority

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The colonial Origins of Comparative Development
An Empirical Investigation

• The disease environment not favourable for the
  attractiveness of European settlement
• Settlement mortality as exogenous variable for
  the subsequent path of institutional development
  to pin the causal effect of economic institutions
  on prosperity
• No impact of this variable on current income
  levels only though economic institutions during
  the colonial period
• Measure Mortality rate faced by Europeans
  (primarily soldiers, sailors and bishops)

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The colonial Origins of Comparative Development
An Empirical Investigation

• Hypothesis
• (potential) settler mortality ? settlements?
• early institutions? current institutions?
• ?current performance
• Empirical Results
• Institutions cause growth!!!

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5. Addressing Errors-in-Variables with IV

• Remember the classical errors-in-variables
  problem where we observe x1 instead of x1
• Where x1 x1 e1, and e1 is uncorrelated with
  x1 and x2
• If there is a z, such that Corr(z,u) 0 and
  Corr(z,x1) ? 0, then IV will remove the
  measurement error bias

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6. Testing for Endogeneity

• Since OLS is preferred to IV if we do not have an
  endogeneity problem, then wed like to be able to
  test for endogeneity
• If we do not have endogeneity, both OLS and IV
  are consistent
• Idea of Hausman test is to see if the estimates
  from OLS and IV are different.

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Testing for Endogeneity (cont)

• While its a good idea to see if IV and OLS have
  different implications, its easier to use a
  regression test for endogeneity
• If y2 is endogenous, then v2 (from the reduced
  form equation) and u1 from the structural model
  will be correlated
• The test is based on this observation

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Testing for Endogeneity (cont)

• Save the residuals from the first stage
• Include the residual in the structural equation
  (which of course has y2 in it)
• If the coefficient on the residual is
  statistically different from zero, reject the
  null of exogeneity
• If multiple endogenous variables, jointly test
  the residuals from each first stage

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7. Testing Overidentifying Restrictions

• If there is just one instrument for our
  endogenous variable, we cant test whether the
  instrument is uncorrelated with the error
• We say the model is just identified
• If we have multiple instruments, it is possible
  to test the overidentifying restrictions to see
  if some of the instruments are correlated with
  the error

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Testing Overidentifying Restrictions

• Estimate the structural model using IV and obtain
  the residuals
• Regress the residuals on all the exogenous
  variables and obtain the R2 to form nR2
• Under the null that all instruments are
  uncorrelated with the error, LM cq2 where q is
  the number of extra instruments

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8. Testing for Heteroskedasticity

• When using 2SLS, we need a slight adjustment to
  the Breusch-Pagan test
• Get the residuals from the IV estimation
• Regress these residuals squared on all of the
  exogenous variables in the model (including the
  instruments)
• Test for the joint significance

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9. Testing for Serial Correlation

• When using 2SLS, we need a slight adjustment to
  the test for serial correlation
• Get the residuals from the IV estimation
• Re-estimate the structural model by 2SLS,
  including the lagged residuals, and using the
  same instruments as originally
• Can do 2SLS on a quasi-differenced model, using
  quasi-differenced instruments