Presentation slides - Social Science Research Commons

Report
Introduction to Instrumental
Variables Methods
Christian Hansen
Booth School of Business
University of Chicago
Introduction
• Many studies in social sciences interested in
inferring structural/causal/treatment effects
–
–
–
–
–
–
Price elasticity of demand
Effect of smoking on birthweight
Effect of 401(k) participation on saving
Effect of job training on wages/employment
Effect of schooling on wages
…
• Only have observational data

Conventional statistical methods may not
recover desired effect
Example 1: Supply and Demand
Supply Curve
Equilibrium
Demand Curve
Example 1: Supply and Demand
Suppose demand and
supply fluctuate from
day to day. We
observe the market
for several days and
want to infer either
the slope of the
supply or demand
curve (from which
economic quantities
such as demand
elasticity are
derived).
Example 1: Supply and Demand
Suppose demand and
supply fluctuate from
day to day. We
observe the market
for several days and
want to infer either
the slope of the
supply or demand
curve (from which
economic quantities
such as demand
elasticity are
derived).
Observed relationship between price and quantity reveals neither supply nor
demand! (Simultaneity)
Example 2: Job Training
• Observe data on earnings for people who have and have not
completed job training.
• Want to infer the causal effect of job training on earnings
• What if people who are more “motivated” are more likely to
get training and on average earn more than less
“motivated”?
‒ Difference between average earnings across the trained
and untrained confounds the effects of motivation and
training
‒ Omitted variables bias: Would like to control for
unobserved (and unobservable?) motivation
Example 3: Classical Measurement Error
•
•
Model:  =  + . Want to know the effect of x on y (β)
Only observe noisy signal for x:  =  + 
•
What do you get from regressing  on ?
 ′   ′ (  +  +   −  )
 +  ′ ( − )
= ′ =
=+
′


 ′
• The last term generally does not have 0 expectation or
converge to 0 even in large samples
• Under “classical measurement error” (  =   =
  = 0; 
converges to
 2 =2
 v
2
2
 ; 
=  ) the second term
(attenuation bias)
2
v x
2
2
Common Structure:
• “Structural” Model:
 =  + 
• y – outcome of interest
• x – observed “treatment” variable
•  – treatment/structural/causal effect (NOT
regression coefficient)
•   ≠ 0. (Endogeneity)
Instrumental variables (IV) offers one approach to
estimating  (when instruments are available…)
What is an instrument?
• Instrumental variable (denoted ) shifts  but is
unrelated to structural error ().
–   ≠ 0 (relevance)
–   = 0 (exclusion)
Key statistical conditions
• Intuition:
Movements in  are unrelated to movements in  but
are related to movements in 

Movements in  “induced” by  are uncontaminated.
Can be used to estimate treatment effect.
How do instruments help?
• Consider supply/demand example:
 =  +  
 =  +  +  
 =  ;  =  
•  is a variable that affects supply ( ≠ 0) but
not demand (say the price of a factor of
production).    =    = 0.
– Valid instrument for demand.
How do instruments help?
• Heuristic: When  changes:
– supply changes
– demand remains the same (on average)
Movements in the
supply curve induced
by changing z trace
out the demand curve
How do instruments help?
• Quasi-mathematically. IV model
–  =  ′  + 
–    ≠ 0
– [  ] = 0
–    ′  
• Recall OLS estimator:  =
( ′ )−1  ′  =  + ( ′ )−1  ′ 
– Neither unbiased nor consistent since   ≠ 0
How do instruments help?
• IV estimator:
 =  ′ 
−1
′ =  + ′
−1
′
– Uses only uncontaminated variation (covariation between
instrument and X and instrument and Y)
– Under conditions above (plus regularity), generally
consistent, asymptotically normal with estimable
covariance matrix.
– Software: (Also really easy to code yourself)
•
•
•
•
•
Stata: ivregress (and variants)
SAS: proc syslin (and others)
R: sem package
SPSS: Analyze -> Regression -> Two-stage least squares
…
Questions?
• Where do instruments come from?
– Intuition, subject matter knowledge, randomization
• Can I assess the validity of the underlying
assumptions?
– Sort of, [  ] = 0 is fundamentally untestable (though
aspects can be tested if extra instruments available)
–    ′   is testable
• What if I have extra instruments (so Z’X is not
invertible)?
– Two-stage least squares (2SLS). Same intuition. Default
implementation in all stats packages. Other options
• Are there things I should look out for?
– Weak instruments. Many instruments. Reasons to doubt
the exclusion restriction. What exactly is estimated when
treatment effects are heterogeneous.
Example: Job Training
• Goal: Assess impact of job training on subsequent
labor market outcomes (e.g. employment, wages)
• Problem: Training receipt is not randomly assigned.
May be endogenous.
– E.g. maybe more motivated people more likely to receive
job training and more likely to find subsequent
employment/have better job performance
• Instrument: Under Job Training Partnership Act (JTPA),
randomized trial conducted in which individuals
offered JTPA services
– Use JTPA offer of services as instrument for receiving
training
Example: Job Training
• Plausibility of instrument:
– Offer of services randomly assigned => Independent
of structural error by construction. (No evidence
about this in data)
– ≈ 60% of those offered training accepted offer =>
Offer strongly related to receipt. (Can look at
correlation between instrument and endogenous
variable to assess this)
• Aside: One could simply regress outcome on offer of
treatment to estimate intention-to-treat (ITT). Our goal is to
estimate effect of treatment, not offer.
Example: Job Training
• Structural Equation:
 =  + ′  + 
• First-Stage Equation:
 = 1,  + ′ 1, + 1,
– Regression of treatment on instrument and controls
– Note: E[zixi] ≠ 0 => π1,z ≠ 0
• Reduced Form Equation:
 = 2,  + ′ 2, + 2,
– Regression of outcome on instrument and controls
– Note: If can’t rule out π2,z = 0, can’t rule out α = 0
• First-stage and reduced form are predictive representations,
not structural . Good practice to report results from all three.
Example: Job Training
• Variables:
– Earn – total earnings over 30 month period following
assignment of offer
– Train – dummy for receipt of job training services
– Offer – dummy for offer to receive training services
– x – 13 additional control variables
• dummies for black and Hispanic persons, a dummy
indicating high-school graduates and GED holders, five agegroup dummies, a marital status dummy, a dummy
indicating whether the applicant worked 12 or more weeks
in the 12 months prior to the assignment, a dummy
signifying that earnings data are from a second follow-up
survey, and dummies for the recommended service strategy
Example: Job Training
• OLS Results (from Stata):
regress earnings train x1-x13 , robust
Linear regression
Number of obs
F( 14, 5087)
Prob > F
R-squared
Root MSE
=
=
=
=
=
5102
38.35
0.0000
0.0909
18659
-----------------------------------------------------------------------------|
Robust
earnings |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------train |
3753.362
536.3832
7.00
0.000
2701.82
4804.904
.
.
.
If intuition about source of endogeneity is correct, this should be an overestimate of the effect of training.
Example: Job Training
• First-Stage Results (from Stata):
regress train offer x1-x13 , robust
Linear regression
Number of obs
F( 14, 5087)
Prob > F
R-squared
Root MSE
=
=
=
=
=
5102
390.75
0.0000
0.3570
.39619
-----------------------------------------------------------------------------|
Robust
train |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------offer |
.6088885
.0087478
69.60
0.000
.591739
.6260379
.
.
.
Strong evidence that E[zixi] ≠ 0
Example: Job Training
• Reduced-Form Results (from Stata):
regress earnings offer x1-x13 , robust
Linear regression
Number of obs
F( 14, 5087)
Prob > F
R-squared
Root MSE
=
=
=
=
=
5102
34.19
0.0000
0.0826
18744
-----------------------------------------------------------------------------|
Robust
earnings |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------offer |
970.043
545.6179
1.78
0.075
-99.60296
2039.689.
.
.
.
Moderate evidence of a non-zero treatment effect
(maintaining exclusion restriction)
Example: Job Training
• IV Results (from Stata):
Note: Some software
reports R2 after IV
regression. This
object is NOT
meaningful and
should not be used.
ivreg earnings (train = offer) x1-x13 , robust
Instrumental variables (2SLS) regression
Number of obs
F( 14, 5087)
Prob > F
R-squared
Root MSE
=
=
=
=
=
5102
34.38
0.0000
0.0879
18689
-----------------------------------------------------------------------------|
Robust
earnings |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------train |
1593.137
894.7528
1.78
0.075
-160.9632
3347.238
.
.
.
Moderate evidence of a positive treatment effect (maintaining
exclusion restriction). Substantially attenuated relative to OLS,
consistent with intuition.
Two-Stage Least Squares
• May have more instruments than endogenous variables
• In principle, many IV estimators can be constructed
• 2SLS is the minimum variance (under homoskedasticity) linear
combination of the potential IV estimators (otherwise may use
GMM)
• 2SLS is the GMM estimator using the full set of orthogonality
conditions implied by [  ] = 0
• 2SLS and IV are numerically equivalent when # of endogenous
variables = # of instruments
• Aside: Some jargon
– r = # of instruments, k = # of endogenous variables
– r = k, “just-identified”
– r > k, “over-identified”
2SLS
• 2SLS estimator:
2 =  ′ 
−1
 ′  =  +  ′ 
−1
 ′ 
– PZ = Z(Z’Z)-1Z’, projection matrix onto Z
– Uses only uncontaminated variation (covariation between
instrument and X and instrument and Y)
– Under conditions above (plus regularity), generally consistent,
asymptotically normal with estimable covariance matrix.
– Can be viewed as two-step procedure where endogenous
variables and outcomes are first projected onto Z and then
projections are used in OLS
– Software: (Also really easy to code yourself)
•
•
•
•
•
Stata: ivregress (and variants)
SAS: proc syslin (and others)
R: sem package
SPSS: Analyze -> Regression -> Two-stage least squares
…
Testing Overidentifying Restrictions
• Have more instruments than we need to estimate
the treatment effect
• If all instruments satisfy exclusion restriction, all
subsets should (asymptotically) return the same
estimate of the treatment effect
• Idea: Obtain multiple estimates of the treatment
effect and test that they are the same.
• Rejection implies some subset of exclusion
restrictions may be invalid
Hansen’s Over-Identification Test
• Also called the “J-Test”, Sargan test, “S-Test”
• Based on GMM criterion function
• For IID, homoskedastic data:
′( ′ )−1  ′ 
=
;  =  − 2
′

 2
 χ  −  , r = dim Z , k = dim(X)
(More generally, J is GMM objective function
evaluated at GMM point-estimate)
Overidentification Tests:
• Can never tell you that the exclusion restriction
(  = 0) is satisfied
– Failure to reject does not imply true
– Even if it did, only learn that probability limits of
various IV estimators are the same. Maybe all the
same and wrong.
– Heterogeneous treatment effects?
• Rejection indicates that some subset of
instruments may be invalid
– Does not indicate which subset
– Does not mean all exclusion restrictions are invalid
– Heterogeneous treatment effects?
Example: Returns to Schooling
• Goal: Estimate the value added of additional
years of schooling in terms of wages
• Problem: Years of completed schooling is not
randomly assigned. May be endogenous.
– E.g. maybe academic ability is related to qualities
that relate to job performance/salary (motivation,
intelligence, task orientation, etc.)
• Instrument: Quarter of birth (Angrist and
Krueger, 1991)
Example: Returns to Schooling
• Plausibility of instrument:
– Compulsory schooling laws in the U.S. are typically based on
age, not number of years of school. People born at different
times of the year can drop out after receiving different amounts
of school. (Can look at correlation between instrument and
endogenous variable to assess this)
– When a person is born is unrelated to inherent traits (e.g.
motivation, intelligence, …) and so should not have a direct
effect on wages but only affect wages through the relationship
to completed schooling induced by compulsory education laws.
• Untestable, but we do have overidentifying restrictions coming from
different birth quarters.
• Validity has been questioned. E.g. winter birth may be correlated to
increased exposure to early health problems; more conscientious
parents may respond by timing birth; …
Example: Returns to Schooling
• Structural Equation:
log( ) = ℎ + ′  + 
• First-Stage Equation:
ℎ = 1,1 1 + 1,2 2 + 1,3 3 + ′ 1, + 1,
– Note: E[zixi] ≠ 0 => π1,1 ≠ 0 or π1,2 ≠ 0 or π1,3 ≠ 0
• Reduced Form Equation:
log( )
= 2,1 1 + 2,2 2 + 2,3 3 + ′ 2, + 2,
Example: Returns to Schooling
• Data from 1980 Census for men aged 40-49 in 1980
• Variables:
–
–
–
–
Wage – hourly wage
School – reported years of completed schooling
Q1-Q3 – dummies for quarter of birth
x – 59 control variables. Dummies for state of birth and year of
birth
Example: Returns to Schooling
• OLS Results (from Stata):
xi: reg lwage educ i.yob i.sob , robust
i.yob
_Iyob_30-39
(naturally coded; _Iyob_30 omitted)
i.sob
_Isob_1-56
(naturally coded; _Isob_1 omitted)
Linear regression
Number of obs
F( 60,329448)
Prob > F
R-squared
Root MSE
=
=
=
=
=
329509
649.29
0.0000
0.1288
.63366
-----------------------------------------------------------------------------|
Robust
lwage |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------educ |
.067339
.0003883
173.40
0.000
.0665778
.0681001
.
.
.
If intuition about source of endogeneity is correct, this should be an over-
estimate of the effect of schooling.
Example: Returns to Schooling
• First-Stage Results (from Stata):
xi: regress educ i.qob i.sob i.yob , robust
Linear regression
Number of obs
F( 62,329446)
Prob > F
R-squared
Root MSE
=
=
=
=
=
329509
292.87
0.0000
0.0572
3.1863
-----------------------------------------------------------------------------|
Robust
educ |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------_Iqob_2 |
.0455652
.015977
2.85
0.004
.0142508
.0768797
_Iqob_3 |
.1060082
.0155308
6.83
0.000
.0755683
.136448
_Iqob_4 |
.1525798
.0157993
9.66
0.000
.1216137
.1835459
.
.
.
testparm _Iqob*
( 1)
( 2)
( 3)
_Iqob_2 = 0
_Iqob_3 = 0
_Iqob_4 = 0
F(
3,329446) =
Prob > F =
First-stage F-statistic.
36.06
0.0000
Example: Returns to Schooling
• Reduced-Form Results (from Stata):
xi: regress lwage i.qob i.sob i.yob , robust
Linear regression
Number of obs
F( 62,329446)
Prob > F
R-squared
Root MSE
=
=
=
=
=
329509
147.83
0.0000
0.0290
.66899
-----------------------------------------------------------------------------|
Robust
lwage |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------_Iqob_2 |
.0028362
.0033445
0.85
0.396
-.0037188
.0093912
_Iqob_3 |
.0141472
.0032519
4.35
0.000
.0077736
.0205207
_Iqob_4 |
.0144615
.0033236
4.35
0.000
.0079472
.0209757
.
.
testparm _Iqob*
( 1)
( 2)
( 3)
_Iqob_2 = 0
_Iqob_3 = 0
_Iqob_4 = 0
F(
3,329446) =
Prob > F =
10.43
0.0000
Example: Returns to Schooling
• 2SLS Results (from Stata):
xi: ivregress 2sls lwage (educ = i.qob) i.yob i.sob , robust
Instrumental variables (2SLS) regression
Number of obs
Wald chi2(60)
Prob > chi2
R-squared
Root MSE
= 329509
= 9996.12
= 0.0000
= 0.0929
= .64652
-----------------------------------------------------------------------------|
Robust
lwage |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------educ |
.1076937
.0195571
5.51
0.000
.0693624
.146025
.
.
.
Bigger than OLS?
Example: Returns to Schooling
• GMM Results (from Stata, efficient under heteroskedasticity):
xi: ivregress gmm lwage (educ = i.qob) i.yob i.sob , robust
Instrumental variables (GMM) regression
Number of obs
Wald chi2(60)
Prob > chi2
R-squared
Root MSE
GMM weight matrix: Robust
= 329509
= 9992.90
= 0.0000
= 0.0927
= .64658
-----------------------------------------------------------------------------|
Robust
lwage |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------educ |
.1077817
.0195588
5.51
0.000
.0694472
.1461163
.
.
.
estat overid
Bigger than OLS?
Test of overidentifying restriction:
Hansen's J chi2(2) = 3.10009 (p = 0.2122)
Fail to reject over-id test
Heterogeneous Treatment Effects: ATE
• Previous discussion posits a constant treatment/structural
effect
• Estimated constant treatment effect = average treatment
effect (ATE) sometimes
 =  ′  +  ,    = 0
 independent of  and  ,  = [ ] =>
 =  ′  +  =  ′  +  +  ′ ( −) =  ′  +  where
   = [  ] + [  ′ ( −)] = 0 + [  ′ ][ −
] = 0
– i.e. IV model holds interpreting parameter as ATE
– Note: Exclusion restriction will not hold in general if
heterogeneous effect is not independent of instrument or
treatment!
–
–
–
–
Heterogeneous Treatment Effects:
LATE
• When heterogeneous effect is not independent, can
still estimate a causal effect among a subpopulation
• Strip model down to binary treatment and binary
instrument
• Four subpopulations:
–
–
–
–
–
Always-takers: zi = 1, xi = 1 and zi = 0, xi = 1
Never-takers: zi = 1, xi = 0 and zi = 0, xi = 0
Complier: zi = 1, xi = 1 and zi = 0, xi = 0
Defier: zi = 1, xi = 0 and zi = 0, xi = 1
Note: Never observe individuals at both instrument states.
Can’t determine observation’s subpopulation
Heterogeneous Treatment Effects:
LATE
• Generally, may estimate ATE among subpopulation of
compliers
– Termed Local Average Treatment Effect (LATE)
• Conditions:
– Independence – instrument independent of all
unobservables affecting outcome and
treatment/endogenous variable state
– Exclusion – instrument only affects outcome though
treatment receipt. (No direct effect of the instrument.)
– First-stage – instrument predicts treatment receipt
– Monotonicity – effect of instrument on probability of
receiving treatment is ≥ 0 for everyone or ≤ 0 for everyone.
(No defiers)
Heterogeneous Treatment Effects:
LATE
• LATE can be generalized (with varying degrees of
difficulty) to multi-valued/continous treatments
or instruments, over-identified models, models
with controls
– 2SLS coefficient estimates (or approximates) a
weighted average of different LATEs
– Each instrument/value of the instrument potentially
gives a different set of compliers => a different LATE
•
Rejection of over-id test does not mean instruments are
invalid as all could be valid but give different complier
populations
Heterogeneous Treatment Effects:
LATE
• Can learn some things about compliers from
data
• A couple simple ones:
– Size of complier population:
   = 1 − [ | = 0]
Probability complier
or always-taker
Probability alwaystaker. No defiers
– Proportion of treated who are compliers:
  = 1
(   = 1 −    = 0 )
  = 1
Example: JTPA
•
•
•
•
•
•
•
•
Recall treatment = dummy for receipt of job training services
Instrument = dummy for offer to receive training services
Size of complier group ≈ 61%
Proportion of treated (those who received training) who are
compliers ≈ (.67/.42)*.61 ≈ .97
IV estimate of training effect: 1593.14 (894.75)
Seems plausible that there are no defiers
Compliers = people who would not receive training if not
offered but choose to have training when offered
Presumably, the remaining 39% of the population are mostly
never-takers
Example: Returns to Schooling
• Returns to schooling more complicated
– Multi-valued treatment
– Multiple instruments
• IV estimand with one binary instrument = weighted
average of effects of increasing schooling by one year
across all schooling levels among compliers
• 2SLS estimand = weighted average of individual IV
estimands
• Monotonicity condition => change in instrument leads
to weakly increasing (decreasing) levels of schooling for
everyone
Example: Returns to Schooling
• Consider dummy for being born quarter j (qj)
• Maintain exclusion restriction as before
• First-stage: Regress schooling on qj
• Can estimate fraction of compliers at each schooling
level as P(si < s|qji = 0) – P(si < s|qji = 1) (Assuming
monotonicity such that changing instrument from 0 to
1 increases schooling)
• Can estimate weight given to each schooling value as
(fraction of compliers)/first-stage
Example: Returns to Schooling
Fractions of
compliers for
different
quarter of
birth
instruments.
Example: Returns to Schooling
Weighting
functions for
different
quarter of
birth
instruments.
Note this is
the fraction of
compliers
scaled by the
first-stage
coefficient.
Example: Returns to Schooling
• OLS Estimate: .067 (.0004)
• IV Estimates:
– Q2: .166 (.071)
– Q3: .209 (.076)
– Q4: .085 (.026)
• 2SLS Estimate: .108 (.020) [Weighted average of
the individual IV estimates]
• Compliers: People who got more school due to
being born later in the year. Margin is mostly in
10-12 years of education.
Many Instruments
• IV Estimates often imprecise
– Only use the variation induced by the instrument
– Often plausible instruments (in the sense of satisfying
the exclusion restriction) have weak predictive power
for outcome
• One approach to increasing precision is to use
more instruments
– Potentially allow extraction of more signal, adding
information helps
• But…
– Are all the instruments really excludable?
– Overfitting is bad
Many Instruments: Overfitting
• Extreme case, # instruments (K) = # observations
(n)
• Recall 2 =  ′ 
Z(Z’Z)-1Z’
−1
 ′  where PZ =
– K=n => (Z’Z)-1 = Z-1(Z’)-1 => PZ = I (the identity matrix)
– So
2 =  ′ 
−1
 ′  =  ′ 
−1
 ′  = 
• Get back original, contaminated object since
perfectly fit both signal AND noise.
– Overfitting fits contaminated noise as well as signal
Example: Returns to Schooling
• Rather than just use 3 quarter of birth effects,
could use quarter of birth interacted with year of
birth x state of birth as instruments (1527
instruments)
• 2SLS estimate: .0712 (.0049)
– Recall OLS gives .067 (.0004) and 2SLS with 3 quarter
of birth dummies gives .108 (.020)
• Theory, simulation evidence, the intuition above
suggests 2SLS is strongly biased towards OLS
when many instruments are used
Many Instruments: Solutions
• Use less instruments
– Use model selection techniques in the first-stage to choose
a good set
• Use an estimator that is less sensitive to first-stage
overfitting
– Limited information maximum likelihood (LIML), Fuller,
Jackknife Instrumental Variables (JIVE)
– Need to adjust standard errors to account for first-stage
overfitting
• Use an estimator that is less sensitive to first-stage
overfitting and regularize the first-stage
– Regularized Jackknife Instrumental Variables (RJIVE)
Example: Returns to Schooling
• 1527 instruments again
– 2SLS estimate: .0712 (.0049)
– 2SLS (3 instrument): .108 (.020)
– OLS gives .067 (.0004)
– Use LASSO to select variables that predict
schooling from among 1527 variables. Use these
variables as instruments: .0862 (.0254)
– JIVE: .0816 (.5168)
– RJIVE: .1067 (.0171)
Example: Eminent Domain
• Estimate economic consequences of the law of takings
or eminent domain
• Potential impacts:
– ``public use'' - removing economic blight and/or promoting
economic development
– redistribution of wealth from groups with little political
power
– distortions in the efficient investment of capital
• underinvestment due to uncertainty induced by potential seizure
• overinvestment when property owners anticipate receiving higher
than market compensation
Example: Eminent Domain
• Want to understand effect of number of decisions that
favor private ownership (go against government
seizure) on economic outcomes
– Real estate prices, GDP
• Legal decisions may be related to these variables.
Potential endogeneity.
– property values provide signal about ``public use''
• low property values: poor prospects, blight
• high property values: viability of a redevelopment or commercial
project
– decisions in other areas of law may affect economic
outcomes and generate precedent/influence decisions
related to takings
Example: Eminent Domain
• US Legal System
– Common law system - judges decide law but also ``make''
law - precedent
– Three layers of courts
• District - Trial Court
• Circuit - Appellate Court (decide issues of new law or if district was
in error)
• Supreme - Very small number of cases
– 12 Circuit Courts. Precedents only binding within circuit
– Judges randomly assigned to cases in Circuit courts
– Circuit court cases are handled by a panel of three judges
• All combinations of demographic characteristics of assigned judges
can be use as instruments for their decisions!
Example: Eminent Domain
• Problem: Too many instruments
– Have between 110-312 observations depending
on outcome
– Have between 138 and 147 instruments
depending on the outcome
– Also have between 30 and 33 controls depending
on outcome
– Use LASSO (variable selection) to select a small set
of instruments
Example: Eminent Domain
• Results:
– log(FHFA House Price Index):
• OLS: .011 (.013)
• 2SLS (after LASSO): .037 (.047) (1 instrument selected)
– log(non-metro House Price Index):
• OLS: .011 (.007)
• 2SLS (after LASSO): .036 (.013) (4 instruments selected)
– log(Case-Shiller House Price Index):
• OLS: .015 (.013)
• 2SLS (after LASSO): .063 (.025) (2 instruments selected)
– log(GDP):
• OLS: .0099 (.0048)
• 2SLS (after LASSO): .013 (.016) (1 instrument selected)
Weak Identification
• Consider the IV estimator:
 =  ′ 
−1
′
• Note the estimator depends on (Z’X)-1
– All IV estimators depend on this (or related) quantities.
– Identification of β depends on E[zixi’] ≠ 0
• ``Weak Identification’’
– E[zixi’] = 0 may hold in population but Z’X will never be 0 in a
finite sample.
• Any estimator that depends on (Z’X)-1 will always suggest you can
learn about β in finite samples
– Z’X may be non-zero but close to zero. Dividing by something
close to 0 causes problems.
Weak Identification
• Extreme case as an illustration:
– dim  = dim  = 1
–
′ 

0,
′

′


•
•
•


 2
,
0


~

′
=+

−1
′



 2
 + −1 
IV estimator inconsistent (so are its variants)
Asymptotic distribution complicated (easy to simulate)

Note if  = 0,  −1  = ( 2), so   = 
–

Inconsistent and centered at probability limit of OLS.
Weak Identification
• What to do?
– Consistent estimation not possible if correlation between Z and
X small
1. Always look at first-stage statistics
–
–
Want strong relationship between instrument (big t/F statistic
on instruments)
Not clear how big. Simulations suggest |t| > 6 not a bad rule
of thumb when dim(z) = dim(x) = 1
2. Forget estimation. Focus on inference that doesn’t
depend on (Z’X)-1
–
–
Various options. See references.
Involves inverting test-statistics (usually by grid search). May
be computationally demanding, especially when dim(x) bigger
than a few
Weak Identification
• One simple approach:
– Suppose you knew the actual treatment effect
• Call it ∗
– Consider the equation
 = ′ ∗ + ′  + 
– From the exclusion restriction, we know  = 0 since we are evaluating
 at the true values
• Algorithm based on this idea:
1.
2.
3.
4.
Hypothesize value for , say 
Form  =  − 
Regress  on  to obtain 
Test  = 0 at your favorite level, α, (say 5%) using your favorite test (say an
F-test)
–
–
5.
If you reject, reject  as a potential value of the treatment effect at the α-level
If you fail to reject, include  in the (1-α)% confidence interval
Repeat steps 1-4 for a set of J candidate values 1 , … ,  . Construct the (1α)% confidence interval as the set of values that are not rejected.
Example: Effects of Institutions on Economic
Growth
• Want to understand the effect of “institutions” on economic
output
• Complicated due to potential joint determination
– Do high quality institutions lead to more growth?
– Does more economic development lead to better
institutions?
• Instrument: European settler mortality (i.e. 1500’s-1600’s)
– Idea: Europeans set up better institutions in places they
wanted to live. Institutions are persistent.
– How often settlers died hundreds of years ago shouldn’t
affect growth except through institutions (maybe need to
control for geography, resources, …)
Example: Effects of Institutions on Economic
Growth
• 64 countries
• Controls 1: latitude
• Controls 2: latitude, Africa dummy, Asia dummy, North America
dummy, South America dummy
• First-Stage 1
Moderate first-stage
regress exprop lnmort latitude , robust
lnmort |
-.5371741
.1544913
-3.48
0.001
-.8460985
-.2282497
• First-Stage 2
Weak first-stage
regress exprop lnmort latitude africa asia samer namer , robust
lnmort |
-.3531495
.1924319
-1.84
0.072
-.7384878
.0321887
Example: Effects of Institutions on Economic
Growth
• IV 1
ivregress 2sls gdp (exprop = lnmort) latitude , robust
exprop |
.9692383
.2077791
4.66
0.000
• IV 2
.5619988
1.376478
Based on strong identification
ivregress 2sls gdp (exprop = lnmort) latitude africa asia namer samer ,
robust
exprop |
1.036001
.450362
2.30
0.021
.1533074
1.918694
Example: Effects of Institutions on Economic
Growth
• Test  = 0:
Reject  = 0
1. Controls 1:
regress gdp lnmort latitude , robust
lnmort |
-.5206497
.0830659
-6.27
0.000
2. Controls 2:
-.6867503
-.3545491
Reject  = 0
regress gdp lnmort latitude africa asia namer samer , robust
lnmort |
-.3658632
.1343732
-2.72
0.009
-.6349408
-.0967855
Example: Effects of Institutions on Economic
Growth
Create new dependent variable
• Test  = 2:
gen newy = gdp - 2*exprop
Reject  = 2
1. Controls 1:
regress newy lnmort latitude , robust
lnmort |
.5536985
.2623656
2.11
0.039
2. Controls 2:
.029066
1.078331
Fail to Reject  = 2
regress newy lnmort latitude africa asia namer samer , robust
lnmort |
.3404359
.3307904
1.03
0.308
-.3219604
1.002832
Example: Effects of Institutions on Economic
Growth
• Repeating the exercise on the previous two
slides for many different hypothesized values
of , we can construct approximate 95%
confidence intervals
– Controls 1:
• Regular Asymptotic: (0.56,1.38)
• Weak identification robust: (0.68,1.83)
– Controls 2:
• Regular Asymptotic: (0.15,1.92)
• Weak identification robust: (-∞,-8.93) U (.41,∞)
Short List of References
• Textbooks:
– Hayashi, Econometrics, Ch. 3
– Wooldridge, Econometric Analysis of Cross Section
and Panel Data, Ch. 5, 6.2
– Angrist and Pischke, Mostly Harmless
Econometrics, Ch. 4
Short List of References
• Papers - Applications:
– Abadie, A., Angrist, J., Imbens, G., 2002. Instrumental variables
estimates of the effect of subsidized training on the quantiles of
trainee earnings. Econometrica, 70, 91–117.
– Acemoglu, D., Johnson, S., and Robinson, J. A., 2001, “The
Colonial Origins of Comparative Development: An Empirical
Investigation,” American Economic Review, 91, 1369-1401.
– Angrist, J. D. and A. Krueger, 1991, “Does Compulsory Schooling
Attendance Affect Schooling and Earnings,” Quarterly Journal of
Economics, 106, 979-1014.
– Belloni, A., Chen, D., Chernozhukov, V., and C. Hansen “Sparse
Models and Methods for Optimal Instruments with an
Application to Eminent Domain” forthcoming Econometrica
Short List of References
• Papers – Heterogeneous Treatment Effects:
– Angrist, J. (2004) “Treatment Effect Heterogeneity in Theory and
Practice,” The Economic Journal 114, C52-C83.
– Angrist, J. (2005) “Instrumental Variables in Experimental
Criminological Research: What, Why, and How,” Journal of
Experimental Criminological Research 2, 1-22.
– Angrist, J., G. Imbens, and D. Rubin, (1996) “Identification of
Causal effects Using Instrumental Variables,” with comments
and rejoinder, Journal of the American Statistical Association.
– Imbens, G. and J. Angrist, (1994) “Identification and Estimation
of Local Average Treatment Effects,” Econometrica.
– Card, D. (1999) "The Causal Effect of Education on Earnings,"
The Handbook of Labor Economics, Volume IIIA, Elsevier Science
Publishers.
Short List of References
•
Papers – Weak and Many Instruments:
–
–
–
–
–
–
–
–
–
–
–
–
Andrews, D. W. K.., M. J. Moreira, and J. H. Stock, 2006, “Optimal Two-Sided Invariant Similar Tests
for Instrumental Variables Regression,” Econometrica, 74, 715-752.
Belloni, A., Chen, D., Chernozhukov, V., and C. Hansen “Sparse Models and Methods for Optimal
Instruments with an Application to Eminent Domain” forthcoming Econometrica
Bekker, P. A., 1994, “Alternative Approximations to the Distributions of Instrumental Variables
Estimators,” Econometrica, 63, 657-681.
Chernozhukov, V. and C. Hansen, 2008, “The Reduced Form: A Simple Approach to Inference with
Weak Instruments,” Economics Letters, 100, 68-71.
Hahn, J., J. A. Hausman, and G. M. Kuersteiner, 2004, “Estimation with Weak Instruments: Accuracy
of Higher-Order Bias and MSE Approximations,” Econometrics Journal, 7, 272-306.
Hansen, C., J. A. Hausman, and W. K. Newey, 2008, “Estimation with Many Instrumental Variables,”
Journal of Business and Economic Statistics, 26, 398-422.
Hansen, C. and D. Kozbur, 2012, “Instrumental Variables Estimation With Many Weak Instruments
Using Regularized JIVE,” working paper (available at
http://faculty.chicagobooth.edu/christian.hansen/research/)
Kleibergen, F., 2002, “Pivotal Statistics for Testing Structural Parameters in Instrumental Variables
Regression,” Econometrica, 70, 1781-1803.
Kleibergen, F., 2007, “Generalizing Weak Instrument Robust IV Statistics towards Multiple
Parameters, Unrestricted Covariance Matrices, and Identification Statistics,” Journal of Econometrics,
139, 181-216.
Moreira, M. J., 2003, “A Conditional Likelihood Ratio Test for Structural Models,” Econometrica, 71,
1027-1048.
Staiger, D. and J. Stock, 1997, “Instrumental Variables Regression with Weak Instruments,”
Econometrica, 65, 557-586.
Stock, J., J. H. Wright, and M. Yogo, 2002, “A Survey of Weak Instruments and Weak Identification in
Generalized Method of Moments,” Journal of Business and Economic Statistics, 20, 518-529.

similar documents