Instrumental variables
Lukˊaˇs Laffˊers
Matej Bel University, Dept. of Mathematics
MUNI Brno
12.11.2021
Causal graph
D Y
U
Z
Y is the outcome
D is a variable of interest
(treatment)
Z is an instrument
U is an unobserved variable
In this situation, it is not possible to (non-parametrically) identify the causal
effect of D on Y.
Things are not completely hopeless though.
Homogenous treatment effects
Let us simplify it a little bit.
We will assume:
homogeneity of the effect
linearity of the function forms
Thus, we will assume a lot...
But it makes it possible to proceed in a rather straightforward manner.
D Y
U
ε
The true relationship is
Yi = α +δDi +γUi +εi
=ηi
But Ui is unobserved
Yi = α +δDi +ηi
ˆδ =
Cov(Y,D)
Var(D)
=
E[YD]−E[Y]E[D]
Var(D)
=
E[αDi +δD2
i +γUiDi +εiDi]−E[α +δDi +γUi +εi]E[D]
Var(D)
= δ +γ
Cov(U,D)
Var(D)
D Y
U
Z
ε
The true relationship is
Yi = α +δDi +γUi +εi
=ηi
But Ui is unobserved
Yi = α +δDi +ηi
New variable Z
Notice: no Z → U or Z → Y
Cov(Y,Z) = Cov(α +δD +γU +ε,Z) = E[(α +δD +γU +ε)Z]−E[D]E[Z]
= δCov(D,Z)+γ Cov(U,Z)
=0
+Cov(ε,Z)
=0
=⇒
δ =
Cov(Y,Z)
Cov(D,Z)
Exclusion restriction
There are no arrows
Z → U
Z → Y
This is called an exclusion restriction
Z provides us with the much needed exogenous source of variation
The regression coefficient δ = Cov(Y,Z)
Cov(D,Z) can be estimated by
ˆδ =
Cov(Y,Z)
Cov(D,Z)
=
1
n ∑n
i=1(Zi − ¯Z)(Yi − ¯Y)
1
n ∑n
i=1(Zi − ¯Z)(Di − ¯D)
If we assume
Yi = α +δDi +ηi
Di = β0 +βzZi +υi
Then
ˆδ =
1
n ∑n
i=1(Zi − ¯Z)
=α+δDi +ηi
Yi
1
n ∑n
i=1(Zi − ¯Z)Di
= δ +
→P0
1
n ∑n
i=1(Zi − ¯Z)ηi
1
n ∑n
i=1(Zi − ¯Z)Di
Yi = α +δ
β0+βz Zi +υi
Di +ηi = α0 +
δ·βZ
αZ Zi +ωi
Reduced form eq.
Di = β0 +βzZi +υi
First stage eq.
Take a closer look at ˆδ
ˆδ =
Cov(Y,Z)
Cov(D,Z)
=
Cov(Y,Z)
Var(Z)
Cov(D,Z)
Var(Z)
=
ˆαZ
ˆβZ
Two-stage least squares
ˆδ =
Cov(Y,Z)
Cov(D,Z)
=
ˆβZ Cov(Y,Z)
ˆβZ Cov(D,Z)
=
Cov(Y, ˆβZ Z)
ˆβ2
Z Var(Z)
=
Cov(Y, ˆβZ Z)
Var(ˆβZ Z)
= ··· =
Cov(Y, ˆD)
Var(ˆD)
where ˆD = ˆβ0 + ˆβZ Z
This suggest the following two-stage strategy:
Step 1 Estimate (ˆβ0, ˆβZ ) from Di = β0 +βzZi +υi and obtain ˆD = ˆβ0 + ˆβZ Z
Step 2 Plug ˆD and estimate (ˆα, ˆδ) from Yi = α +δ ˆDi +ηi
Such regression coefficient ˆδ will be identical to ˆαZ
ˆβZ
Additional covariates?
D Y
U
Z
X
It is important to close all these paths (D ← X → Y) too.
Wald estimator
In case of abinary instrument and no covariates, the IV estimator is
ˆδIV =
ˆE[Y|Z = 1]− ˆE[Y|Z = 0]
ˆE[D|Z = 1]− ˆE[D|Z = 0]
Additional covariates?
D Y
U
Z
ε
X
υ
Yi = α +δDi +δX Xi +
=ηi
δUUi +εi
= α0 +αZ Zi +αX Xi +ωi
Reduced form eq.
Di = β0 +βzZi +βX Xi +υi
First stage eq.
Step 1 Estimate (ˆβ0, ˆβZ , ˆβX ) from Di = β0 +βzZi +βx Xi +υi and obtain
ˆD = ˆβ0 + ˆβZ Z + ˆβX X
Step 2 Plug ˆD and estimate (ˆα, ˆδ, ˆδX ) from Yi = α +δ ˆDi +δX Xi +ηi
Instrument
There are two qualities that the instrument needs to have:
Validity - instrument Z has no direct effect on Y. It only operates via D.
Z needs to be uncorrelated with ηi and therefore with both Ui and εi
Relevance - Z is correlated with D
Where are we now:
So far, we were unable to non-parametrically identify ATE. We could
not close the paths going via confounder U.
By simplifying a lot, we can at least identify and estimate the regression
coefficient δ within a linear model.
This is a ratio of coefficients from two regression OR we can look at it
as two stage estimator
That is all great as long as the linear model is correct and effects are
homogenous.
Let us see it in action.
Example: children and labor supply
We wish to understand the causal link between the family size and the labor
supply.
Do parents of bigger families work more?
A lot of literature found negative correlation between family size and female
labor supply.
How to estimate these? Clearly, the family size is not ”randomly assigned”.
Angrist, Joshua, and William Evans. ”Children and Their Parents’ Labor Supply: Evidence from Exogenous Variation in Family Size.” American Economic
Review 88.3 (1998): 450-77.
Example: children and labor supply
Where do we find a proper instrument, that would provide an exogenous
variation in the family size?
Parents have preference for
mixed genders
The gender ”assignment” itself
is as good as random
Parents with these kids
{(♀,♀),(♂,♂)} are more likely
to have another one in
comparison to parents with
{(♀,♂),(♂,♀)} kids
Exogenous variation in the
probability of having a third
child!
Gender of the first kid does not predict the probability of having the second
child.
Table 3 from Angrist and Evans (1998)
Gender composition predicts the probability of having a third child.
Table 3 from Angrist and Evans (1998)
Ordinary least squares
estimator (for comparison
purposes)
Yi = α +δDi +δX Xi +ηi
Instrumental variable
estimation
Yi = α +δDi +δX Xi +ηi
Di = β0 +βzZi +βX Xi +υi
Y is one of these
worked
weeks worked
hours/week
log family income
non-wife income
D is an indicator of having more than 2
children
X consists of: age, age at first birth, black
indicator, hispanic indicator, boy 1st
indicator, boy 2nd indicator
z is one of these
same sex
two boys, two girls (as separate
instruments)
No covariates - Wald estimates
Table 5 from Angrist and Evans (1998)
Instrument is relevant
Table 6 from Angrist and Evans (1998)
With covariates
Magnitude of the effect is smaller than under OLS
Table 7 from Angrist and Evans (1998)
Mechanics
Step 1 Estimate (ˆβ0, ˆβZ , ˆβX ) from Di = β0 +βzZi +βx Xi +υi and obtain
ˆD = ˆβ0 + ˆβZ Z + ˆβX X
Step 2 Plug ˆD and estimate (ˆα, ˆδ, ˆδX ) from Yi = α +δ ˆDi +δX Xi +ηi
Can be translated as
Step 1 Regress D on all sources of exogenous variation (Z and X)
Step 2 Regress Y on the predicted values ˆD of D and exogenous variables X
(not instruments!)
Mechanics (it is a simple projection)
X = [1,X,D] Z = [1,X,Z] yi = Xiβ +ei
ˆβOLS = (XT
X)−1
XT
Y →P β
because E(XT
e) = 0
Regress all the columns of X onto Z to obtain ˆX
ˆX = Z(ZT
Z)−1
ZT
X = PZ X
(note that projecting X on Z will give us the same X because it is in Z !)
Regress y on ˆX
ˆβIV = (ˆX
T
ˆX)−1 ˆX
T
y = ((PZ X)T
PZ X)−1
(PZ X)T
y = (XT
PZ
=PT
Z PZ
X)−1
XT
PZ y
Careful with the standard errors
The second-stage regression does not give you the correct standard errors.
(It ignores the first stage uncertainty).
Notice that IV estimator is weighted least squares estimator:
ˆβIV = (ˆX
T
ˆX)−1 ˆX
T
y = (XT
PZ X)−1
XT
PZ y
and thus ˆσ2
(XT
PZ X)−1
is a consistent estimator of covariance matrix of ˆβIV
under homoscedasticity.
Weak instruments
D Y
U
Z
X
We relied on the fact that there exists this connection: Z → D
But what if the link is only weak?
Weak instruments
So what if the correlation is very small(?)
ˆδ =
Cov(Y,Z)
Cov(D,Z)
very small
=
Cov(Y,Z)
Var(Z)
Cov(D,Z)
Var(Z)
=
ˆαZ
ˆβZ
Then the ˆβZ is very imprecisely estimated. And this leads to an imprecise
estimator for ˆδ itself.
Weak instruments
ˆδ = δ +
→P0???
1
n ∑n
i=1(Zi − ¯Z)ηi
1
n
n
∑
i=1
(Zi − ¯Z)Di
very small
Even a tiny small deviation from the exogeneity Cov(Z,η) = 0 may severely
bias our estimator(!)
This is a huge deal.
Bound, John, David A. Jaeger, and Regina M. Baker. ”Problems with instrumental variables estimation when the correlation between the instruments and
the endogenous explanatory variable is weak.” Journal of the American statistical association 90.430 (1995): 443-450.
Weak instruments
Luckily, we can check if we have this problem simply by looking at the first
stage.
Common rule of thumb is to have the value of F-statistic from the first stage
regression at least 10.
There is a huge strain of literature on weak instruments, many weak instruments etc.
Older Survey: Stock, James H., Jonathan H. Wright, and Motohiro Yogo. ”A survey of weak instruments and weak identification in generalized method of
moments.” Journal of Business & Economic Statistics 20.4 (2002): 518-529.
Newer survey Andrews, Isaiah, James H. Stock, and Liyang Sun. ”Weak instruments in instrumental variables regression: Theory and practice.” Annual
Review of Economics 11 (2019): 727-753.
Statistical Inference: Staiger, Douglas O., and James H. Stock. ”Instrumental variables regression with weak instruments.” (1994).
Heterogenous effects
https://www.nobelprize.org/uploads/2021/10/fig4 ek en 21 LATE.pdf
Heterogenous effects
A natural question to ask is the following:
Do all people have the same effect from the treatment?
If not, who are these people who benefit from the treatment?
Interpretation
We now drop the linearity assumption and consider binary treatment and
binary instrument.
Every individual i may have her own effect δi = Yi(1)−Yi(0) depending on
the treatment
Every individual i may also react different in terms of treatment Di(1)−Di(1)
on the instrument
Z - randomly offered training
D - actual training
Y - outcome
always-taker Di(1) = 1 and Di(0) = 1
complier Di(1) = 1 and Di(0) = 0
defier Di(1) = 0 and Di(0) = 1
never-taker Di(1) = 0 and Di(0) = 0
Denote Yi(d,z) as a potential outcome under Di = d and Zi = z.
If
Instrument is independent of potential outcomes:
(Yi (Di (1),1).Yi (Di (0),0),Di (1),Di (0)) ⊥⊥ Zi
Exclusion restriction: Yi (d) ≡ Yi (d,1) = Yi (d,0)
Relevance restriction: E[Di (1)−Di (0)] = 0
Monotonicity: Di (1) ≥ Di (0)
Stable Unit Treatment Value Assumption: There are no interaction between
individuals and there is no hidden variation in the treatment
then
δIV =
E[Y|Z = 1]−E[Y|Z = 0]
E[D|Z = 1]−E[D|Z = 0]
= E[Y(1)−Y(0)|D(1) > D(0)]
Local average treatment effect
Imbens, G. W. and Angrist, J. D. (1994). Identication and Estimation of Local Average Treatment Effects. Econometrica
Proof
E[Y|Z = 1] =
exclusion
E[Y(0)+(Y(1)−Y(0))D|Z = 1] =
Ind.
E[Y(0)+(Y(1)−Y(0))D(1)]
and also
E[Y|Z = 0] = E[Y(0)+(Y(1)−Y(0))D(0)]
so
E[Y|Z = 1]−E[Y|Z = 0] =
mono
E[(Y(1)−Y(0))(D(1)−D(0))]
= E[(Y(1)−Y(0))|D(1) > D(0)]P(D(1) > D(0))
Similarly
E[D|Z = 1]−E[D|Z = 0] = E[D(1)−D(0)] = P(D(1) > D(0))
Effects on the compliers
Effects on the compliers
LATE interpretation is specific for the instrument
no restrictions were placed on the homogeneity of the effects
no linearity was assumed
Extensions:
Further discussions: Angrist, J. D., Imbens, G. W. and Rubin, D. B. (1996). Identification of Causal Effects Using Instrumental Variables. Journal
of the American Statistical Association.
Multiple valued treatment: Angrist, Joshua D., and Guido W. Imbens. ”Two-stage least squares estimation of average causal effects in models
with variable treatment intensity.” Journal of the American statistical Association 90.430 (1995): 431-442.
Non-parametric LATE with covariates: Fr¨olich, Markus. ”Nonparametric IV estimation of local average treatment effects with covariates.”
Journal of Econometrics 139.1 (2007): 35-75.
Further applications
Returns to schooling - Quarter of birth instrument (Andgrist and
Krueger, 1991)
Returns to schooling - Nearby college instrument (Card, 1995)
Returns to schooling - Different instruments (Ichino and Winter-Ebmer,
1999)
Classroom size - Legislative rule as instrument (Angrist and Lavy 1999)
Effect of military service on labor market outcomes - Draft lottery
instrument (Angrist, 1990)
Impact of institutions on economic growth - Mortality instrument
(Acemoglu, Johnson and Robinson, 2001), Comment (Albouy, 2012),
Reply (AJR, 2012)
Impact of economic conditions on prob. of a conflict - rainfall
instrument (Miguel, Satyanath and Segenti, 2004)
Further applications
Demand for fish - Weather as an IV (Angrist, Graddy and Imbens)
Childbearing on labor supply - twin births as a natural experiment
(Jacobsen, Pearce and Rosenbloom. 1999) and (Black, Devereux and
Salvanes, 2015)
Using economic theory to estimate supply and demand curves using
variation in a single tax rate(!) (Zoutman, Gavrilova and Hopland. 2018)
Parental Meth Abuse and Foster Care - use supply shock on meth
market as instrument (Cunningham and Finlay, 2013)
Measurement error
Suppose that X is measured with error:
Yi = β0 +βX (X∗
i +ui)
Xi
+εi
ˆβX =
Cov(X,Y)
Var(X)
=
Cov(X∗ +u,β0 +βX (X∗ +u)+ε)
Var(X∗ +u)
→P βX
σ2
X
σ2
X +σ2
u
which is attenuated even if ui is uncorrelated with both X∗
i and εi
AJR 2001
Institutions - with more secure property rights people will invest more
in physical and human capital. Also includes indpendent judiciary, equal
access to education and ensuring civil liberties
Do institutions matter? well, they do: North/South Korea, West/East
Germany.
Different colonization policies: extractive (Kongo) vs strong property
rights (Australia, Canada, USA)
Higher mortality made it more difficult to set up settlements with
strong property rights
Settler mortality → Settlements → Early institutions → Current
institutions → Current performance
Reduced form
AJR 2001
Exclusion restriction: mortality more that 100yrs ago have no direct
impact on GDP per capita today (apart the channel via institutions).
Why? Mortality mainly due to malaria and yellow fever.
Insensitive to outliers (USA, Canada, NZ, Australia)
Africa dummy and distance to equator insignificant
Results robust to different covariates added: identify of main colonizer,
climate, religion, geography, natural resources, current disease. (in DAG
language: closing all the backdoor paths)
Table 1 in AJR 2001
Model
AJR 2001
Fig 3 in AJR 2001
IV estimates
Table 4 in AJR 2001
First stage
Table 4 in AJR 2001
OLS
Table 4 in AJR 2001
This is compatible with attenuation bias explanation.
Example: Meth, Parents and Foster Care
(Cunningham and Finley, 2013)
effect of drug abuse on parenting
In 1994 - regulation on ephedrine → more difficult to produce meth
Fig 3 from Cunningham and Finley (2013)
Fig 4 from Cunningham and Finley (2013)
Fig 5 from Cunningham and Finley (2013)
s - state
t - specific month
γs,δs - state fixed effects
φs,λt - month fixed effects
tst ,ωst - state specific linear time trends
Xst - log of state population of whites aged 0-19, 15-49, cigarette tax,
state unemployment rate, log of alcohol treatment cases for whites
Part of Table 3 from Cunningham and Finley (2013)
Overidentifying restrictions test
Z may be multidimensional.
Two stage least squares procedure still can be used.
Say we have 2 instruments: Under instrument exogeneity, both of them are
fine and hence ˆβIV1 should be similar to ˆβIV2
Under exogeneity, both Z1 and Z2 should have zero coefficients in a
regression with residuals (using original X and ˆβIV )
F-statistic that jointly tests this multiplied with m is called J-statistic ∼ χ2
q .
Where m is the number of instruments, q is the number of endogenous
variables and q = m −k is the number of over-identifying restrictions.
See row Sargan-row in summary table of ivreg.
Wrap up
IV approach allows to make use of quasi-experimental variation in the
treatment that is induced by the instrument.
IV provides this exogenous variation
IV needs to be strong enough otherwise estimates are sensitive
Under monotonicity condition, results informs us only about a specific
subpopulation (compliers).
(*) More on IVs
Testable implications on IVs
Balke and Pearl (1997) for binary Y - based on linear programming
Huber and Mellace, (2015) - under LATE assumptions
Kitagawa, (2021) extends Balke and Peal (1997) results to continuous Y
Zhang. Tian and Bareinboim (2021) - general algorithm for
identification of distributions of counterfactual outcomes
Fig 1 in Balke and Pearl (1997)
If Y,D,Z are discrete, we have that
max
d
∑
y
max
z
P(y,d|z) ≤ 1
Furthermore ATE = E[Y(1)−Y(0)] is bounded.
Thank you for your attention!
References
Imbens, Guido W., and Joshua D. Angrist. ”Identification and Estimation of Local Average Treatment Effects.” Econometrica 62.2 (1994): 467-475.
Angrist, Joshua D., and Alan B. Keueger. ”Does compulsory school attendance affect schooling and earnings?.” The Quarterly Journal of Economics 106.4 (1991):
979-1014.
Angrist, Joshua D. ”Lifetime earnings and the Vietnam era draft lottery: evidence from social security administrative records.” The American Economic Review (1990):
313-336.
Card, David. ”Using geographic variation in college proximity to estimate the return to schooling.” (1993).
Acemoglu, Daron, Simon Johnson, and James A. Robinson. ”The colonial origins of comparative development: An empirical investigation.” American economic review
91.5 (2001): 1369-1401.
Miguel, Edward, Shanker Satyanath, and Ernest Sergenti. ”Economic shocks and civil conflict: An instrumental variables approach.” Journal of political Economy 112.4
(2004): 725-753.
Ichino, Andrea, and Rudolf Winter-Ebmer. ”Lower and upper bounds of returns to schooling: An exercise in IV estimation with different instruments.” European
Economic Review 43.4-6 (1999): 889-901.
Bound, John, David A. Jaeger, and Regina M. Baker. ”Problems with instrumental variables estimation when the correlation between the instruments and the
endogenous explanatory variable is weak.” Journal of the American statistical association 90.430 (1995): 443-450.
Stock, James H., Jonathan H. Wright, and Motohiro Yogo. ”A survey of weak instruments and weak identification in generalized method of moments.” Journal of
Business & Economic Statistics 20.4 (2002): 518-529.
Andrews, Isaiah, James H. Stock, and Liyang Sun. ”Weak instruments in instrumental variables regression: Theory and practice.” Annual Review of Economics 11
(2019): 727-753.
Staiger, Douglas O., and James H. Stock. ”Instrumental variables regression with weak instruments.” (1994).
Fr¨olich, Markus. ”Nonparametric IV estimation of local average treatment effects with covariates.” Journal of Econometrics 139.1 (2007): 35-75.
”Two-stage least squares estimation of average causal effects in models with variable treatment intensity.” Journal of the American statistical Association 90.430
(1995): 431-442.
Angrist, J. D., Imbens, G. W. and Rubin, D. B. (1996). Identification of Causal Effects Using Instrumental Variables. Journal of the American Statistical Association.
References
Jacobsen, Joyce P., James Wishart Pearce III, and Joshua L. Rosenbloom. ”The effects of childbearing on married women’s labor supply and earnings: using twin births
as a natural experiment.” Journal of Human Resources (1999): 449-474.
Angrist, Joshua D., Kathryn Graddy, and Guido W. Imbens. ”The interpretation of instrumental variables estimators in simultaneous equations models with an
application to the demand for fish.” The Review of Economic Studies 67.3 (2000): 499-527.
Zoutman, Floris T., Evelina Gavrilova, and Arnt O. Hopland. ”Estimating both supply and demand elasticities using variation in a single tax rate.” Econometrica 86.2
(2018): 763-771.
Cunningham, Scott, and Keith Finlay. ”Parental substance use and foster care: Evidence from two methamphetamine supply shocks.” Economic Inquiry 51.1 (2013):
764-782.
Overidentification test, the very first paper: Sargan, John D. ”The estimation of economic relationships using instrumental variables.” Econometrica: Journal of the
Econometric Society (1958): 393-415.
Balke, Alexander, and Judea Pearl. ”Bounds on treatment effects from studies with imperfect compliance.” Journal of the American Statistical Association 92.439
(1997): 1171-1176.
Huber, Martin, and Giovanni Mellace. ”Testing instrument validity for LATE identification based on inequality moment constraints.” Review of Economics and
Statistics 97.2 (2015): 398-411.
Kitagawa, Toru. ”The identification region of the potential outcome distributions under instrument independence.” Journal of Econometrics (2021).
Zhang, Junzhe, Jin Tian, and Elias Bareinboim. ”Partial Counterfactual Identification from Observational and Experimental Data.” arXiv preprint arXiv:2110.05690
(2021).