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DEWEY
Technology Department of Economics Working Paper Series
Massachusetts
Institute of
INSTRUMENTAL VARIABLES AND THE SEARCH FOR IDENTIFICATION FROM SUPPLY AND DEMAND TO NATURAL EXPERIMENTS*
Joshua Angrist,
MIT
Alan B. Krueger, Princeton University
Working Paper 01-33 August 2001
Room
E52- 251
50 Memorial Drive Cambridge^ MA 02142
This paper can be downloaded without charge from the Social Science Research Network Paper Collection at http://papers.ssrn.com/abstract id=xxxxx
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
AUG 2
9 2001
LIBRARIES
Technology Department of Economics Working Paper Series
Massachusetts
Institute of
INSTRUMENTAL VARIABLES AND THE SEARCH FOR IDENTIFICATION FROM SUPPLY AND DEMAND TO NATURAL EXPERIMENTS*
Joshua Angrist,
MIT
Alan B. Krueger, Princeton University
Working Paper 01-33 August 2001
Room E52-251 50 Memorial Drive Cambridge, MA 02142
This paper can be downloaded without charge from the Social Science Research Network Paper Collection at http://papers.ssrn.com/abstract id=xxxxx
INSTRimiENTAL VARIABLES AND THE SEARCH FOR IDENTIFICATION: FROM SUPPLY AND DEMAND TO NATURAL EXPERIMENTS*
Joshua D. Angrist,
MIT
Alan B. Krueger, Princeton University
August 2001
*This paper was prepared for the Journal of Economic Perspectives symposium on econometric and presented at the January 200 1 meetings of the American Economic Association. We are
tools
grateful
to
comments.
Brad DeLong, David Freedman, Tim Taylor, and Michael Waldman
for helpfiil
Abstract
The method of instrumental variables was first used in the 1920s to estimate supply and demand elasticities, and later used to correct for measurement error in single -equation models.
Recently,
instrumental
variables
have
been widely used to
reduce bias
from
omitted variables in estimates of causal relationships such as the effect of schooling on earnings, in
portion
variables
methods use only a portion of the variabihty
unrelated to the omitted variables.
is
variables,
to
instrumental
hituitively,
key variables to estimate the relationships of
and the
qualities that
make
for a
interest;
We
if the
good instrument, devoting
instruments that are derived from "natural experiments."
experiments approach
is
the transparency
and
also discuss the use of instrumental variables in
MIT
Department
Of Economics
50 Memorial Drive
Cambndge, and
reflitability
MA 02142
A
particular attention
key feature of the natural
of identifying assumptions.
randomized experiments.
Alan
Joshua Angrist
instniments are valid, that
discuss the mechanics of instaimental
P.
Krueger
Princeton University Firestone Library
Princeton,
N J 08544
NBER
angrist(a)mit.edu
akrueger(g),princeton
.
edu
We
The method of tool
instrumental variables
The canonical example, and
kit.
attempts
involved
a signature technique in the econometrics
earliest
apphcations.
demand and supply
estimate
to
is
of instrumental
variables
Exonomists such as P.G.
curves.
Wright, Henry Schultz, Ehner Working, and Ragnar Frisch were interested in estimating
the elasticities of
demand and supply
data
on
and
quantities
prices
Consequently, an ordinary
- that
is,
trace out
-
demand and supply
If both the
with time- series data.
least
for products ranging
a
reflect
set
from herrmg
to butter, usually
curves shift over time, the observed
of equilibrium
points
on both
squares regression of quantities on prices
either the supply or
demand
fails
curves.
to identify
relationship.
P.G. Wright (1928) confronted this issue in the seminal appUcation of instrumental
variables:
linseed
oil.
estimating
the
of supply and demand
elasticities
Wright noted the
for
flaxseed,
of obtaining estimates of the
difficulty
elasticities
and demand from the relationship between price and quantity alone. however, that certain "curve
shifters"
- what we would now
can be used to address the problem
which (A)
1.
affect
demand
(p.
312):
"Such
the
He
source of
of supply suggested,
call instrumental variables
additional factors
conditions without affecting cost conditions or
may
-
be factors
which (B)
affect
See Goldberger (1972) and Morgan (1990) for a discussion of the origins of instmmental and related methods. Bowden and Turkington (1984) provide a more technical
variables
discussion of instmmental variables. Reiers0l (1945);
Ragnar 2. In
Morgan
sites
The
first
an interview
in
use of the term "instrumental variables" was in which Riersol attributed the term to his teacher,
Frisch..
the early 1920s, Wright's son, Sewall Wright, developed "causal path analysis," a method-
of-moments-type equations.
technique
P.G. Wright
for
showed
his simultaneous equations application.
credit for his father's use
stmctural models and simultaneous and instrumental variables were equivalent in quite likely that Sewall Wright deserves much of the
estimating
recursive
that path analysis It is
of instmmental variables.
cost conditions without affecting
demand
A
conditions."
variable he used for the
demand
curve shifter was the price of substitute goods, such as cottonseed, while a vanable he
used for the supply curve
shifter
was
yield per acre,
which can be thought of as primarily
determined by the weather.
an instrumental vanables estimate of the demand
Specifically,
can be
elasticity
constructed by dividing the sample covariance between the log quantity of flaxseed and
the yield per acre
by the sample covariance between the log price of flaxseed and the
This estimate
yield per acre.
demand
the error in the
is
consistent as long as yield per acre
is
uncorrected with
Replacing yield per acre with
equation and correlated with price.
the price of substitutes in this calculation generates an instrumental variables estimate of
the
supply
demand
weather- related shifts in yield are
Intuitively,
elasticity.
while changes in the price of substitutes are used to
curve,
curve so as to trace out the supply curve.
Wright (1928,
p.
demand
resulting
curve,
shift
the
demand
.
314) observed: "Success with this method depends on success in
discovering factors of the type
the
used to trace out the
A
He
and B."
and then averaged the
average elasticity of
demand
for
literature.
instrumental
six
was
flaxseed
variables estimate of the elasticity of supply
unnoticed by the subsequent
used six different supply
was
Not
variables
-.80.
estimates.
The
His average instrumental
Wright's econometric advance was
2.4.
until
shifters to estimate
the
1940s were instrumental variables
and related methods rediscovered and extended. Wright's
(1928)
method
of
averaging
the
different
instrumental
variables
estimates does not necessarily produce the most efficient estimate; other estimators
combine
the
information
in
different
instruments
to
produce
an
estimate
with
may less
sampling
The most
variability.
two-stage
way
to
combine multiple instruments
developed by Theil (1953).^
originally
squares,
least
efficient
"endogenous" right-hand side variable (price in
is
instruments and the coefficients estimated from the
plugged directly
into
the
equation of interest
m
first- stage
usually
the
stage,
on the data
the
all
for the
regression, are then either
place of the endogenous regressor
used as an instrument for the endogenous regressor. In
least squares takes the
first
regressed on
In the second stage, the predicted values of price, based
instruments.
equivalently,
appUcation)
this
In the
is
or,
way, two- stage
this
infomiation in a set of instruments and neatly boils
it
down
to a
single instrument.'*
Instrumental Variables and Measurement Error
Instrumental
error
problems
variables
methods were also pioneered
in explanatory variables.^
including the limited ability of
deviation
between
practice.
If
3.
The
staristical
variables
the
of two-stage
is
overcome measurement
error can arise for
many
reasons,
agencies to collect accurate information, and the
specified
an explanatory variable
relative efficiency
Measurement
to
in
economic theory and those collected
measured with additive random
least squares turns
errors,
in
then the
on a number of auxiliary assumptions,
such as homoscedastic errors. See Wooldridge (2001) for a discussion of alternative generalized method of moments estimators. 4.
Typically, a
demand
number of "exogenous" conditioning
nevertheless should be included in both the
squares can also be used there are at least as
variables also appear in both the supply
These exogenous covariates do not play the
equations.
if there is
many
first-
role
and
of instniments but
and second-stage regressions. Two-stage least regressor in an equation, provided
more than one endogenous
instruments as endogenous regressors (see, for example,
Bowden and
Turkington, 1984). 5.
Wald's (1940) method of fitting straight lines was specifically developed to overcome errorsDurbin (1954) showed that Wald's method is a special case of instmmental
in-variables problems. variables.
See also Geary (1949). Hausman (2001) provides a recent overview of measurement
error problems.
on
coefficient
toward zero
that variable in a bivariate ordinary least squares regression will
in
a
sample.
large
Given instrument
errors, the greater the bias.
error
the proportion of vanability that
The higher
and the equation error
(that
is,
measured data) but correlated with the correctly measured
is
due
to
with the measurement
that is uncorrelated
the equation error from the
be biased
model with
the correctly
then
instrumental
variable,
variables provides a consistent estimate even in the presence of measurement error.
Friedman's
celebrated
(1957)
analysis
of
the
consumption
interpreted as an appUcation of instrumental variables in this context.
function
can
Annual income
be
is
a
noisy measure of permanent income, so a regression of consumption on annual income
yields
too
small
an estimate of the marginal propensity to consume
income.
To overcome
which
equivalent
is
measurement problem, Friedman grouped
this
using a two- stage
to
least
squares
impHcitly a regression of annual income on a set of
The
fitted
same
While
two -stage
they
are
income
values, as
done
is
and are
unbiasedness
squares
unbiased.
ratio
or have a simple form.
typically exist
data
by
The
first
stage
city,
is
by
in
city,
so that regressing
two -stage
least squares, is
where the weights
are the
city.
least
not
because they involve a
between
fitted
his
indicating each of the cities.
as a weighted regression using city average data,
number of observations per
consistent,
dummies
values from this regression would be average income
micro consumption data on the
procedure.
from permanent
and
fristrumental
of random
In
contrast,
quantities,
instrumental
variables
for
expectations
easily calculated.
and
other
consistency,
variables
estimates
estimators
are
not
unbiased
which expectations need not
of ordinary
least
are
exist
squares estimates
Textbooks sometimes gloss over the distinction but
the
difference
can
matter
in
practice.
Unbiasedness means the estimator has a sampling distribution centered on the parameter
of
in
interest
a
sample of any
size,
while
only means that the
consistency
converges to the population parameter as the sample size grows.
variables
estimates
variables should aspire to
The estimator
technical
(i.e.,
work with
form
precise
but
consistent,
are
not
unbiased,
researchers
Since
using
of the
Most
instrumental
instrumental
large samples.
asymptotic
distribution
of an
instrumental
the sampling distnbution in very large samples) depends
conditions.
estimator
modem
packages
software
on a number of
options
include
variables
"robust
for
standard errors" that are asymptotically vaUd under reasonably general assumptions.
important
to
remember,
however,
that
practice
in
these
standard
are
errors
It
is
only
approximate.
Instrumental Variables and Omitted Variables
Although instrumental variables methods are of simultaneous equations and recent
work uses
estimates
of causal
to counteract bias
instrumental
variables
widely used to estimate systems
from measurement
overcome
to
Studies
relationships.
still
of
this
type
omitted
are
error,
a flowering of
variables
primarily
problems
in
concerned with
estimating a narrowly defined causal relationship such as the effect of schooling, training,
or military service on earnings; the impact of smoking or medical treatments on health;
the effect of social insurance programs
The observed these
association
on labor supply; or
the effect
of policing on crime.
between the outcome and explanatory variable of
and many other examples
is
likely
to
be misleading
in
the
interest
sense that
it
in
partly
omitted
reflects
measured
and
held
are
that
factors
constant
related
in
a
to
both
variables.
omitted
the
regression,
If these
could be
factors
variables
would be
bias
In practice, however, economic theory typically does not specify
eliminated.
variables that should be held constant while estimating a relationship,
accurately measure
One
all
of the relevant variables even
solution to the omitted variable
and
it
all
of the
difficult to
is
if they are specified.
problem
is
to
randomly assign the variable of
For example, social experiments are sometimes used to assign people to a job
interest.
Random
program or a control group.
training
program (among those
in the assignment pool)
Randomized experiments
social factors.
assignment assures that participation in the
is
not correlated with omitted personal or
are not always possible, however.
would not
It
be feasible to force a randomly chosen group of people to quit smoking or attend school for an extra year, or to
the other hand,
variation
randomly assign the value of the minimum wage across
may be
it
variables
in
states.
possible to find, or even to create, a degree of exogenous
schooling,
like
smoking,
and
minimum
wages.
Instrumental
variables offer a potential solution in these situations.
To
see
how instrumental variables can
suppose that
we would like to use the
measure the
"rate
solve the omitted variables problem,
following cross- sectional regression equation to
of retum to schooling," denoted
Yi
=
a
+
?S,
+ BA, +
?:
Ei
log wage.
or her highest grade of schooling
In this equation, Yi
is
person
completed, and Ai
is
a measure of ability or motivation. (For simplicity,
i's
Si is his
we take Ai
to
could be a vector of variables.) Although the problem of
a single variable, although
it
estimating this equation
straightforward in principle, data
is
On
on
A are typically
be
unavailable, and researchers are unsure what the right controls for abUity or motivation
would be
in
any case.^ Without additional infomiation, the parameter of interest,
identified; that
is,
we
cannot deduce
it
from the joint
distribution
?, is
not
of earnings and
schooling alone.
Suppose, however,
we have a third variable, the instrument,
correlated with schooling, but otherwise unrelated to earnings. That
with the omitted variables and the regression
estimate of the payoff to schooling
is
error, Ej.
denoted ^, which
is,
Zj is
Then an instrumental
is
uncorrected
variables
the sample analog of Cov(Yi, Zi)/Cov(Si, Zi).
The
instmmental variables methods allow us to consistently estimate the coefficient of
interest free
from bias from omitted
variables, without actually
omitted variables or even knowing what they
are.
(If there is
instrument the coefficient of interest can be estimated
Intuitively, instrumental variables solve the
of the
variability in schooling
variables
-
-
having data on the
more than one
by two-stage
least squares.)
omitted variables problem by using only part
specifically, a part that is uncorrelated
to estimate the relationship
valid
with the omitted
between schooUng and earnings.
Instruments that are used to overcome omitted variables bias are sometimes said
to
denve from
"natural experiments".^
instrumental variables in this way; that
Recent years have seen a resurgence
is,
to exploit situations
in the
use of
where the forces of nature
or government policy have conspired to produce an environment somewhat akin to a
randomized
experiment.
This
type
of application
has
generated
some of
the
The expected coefficient on schooling from a regression that omits the A variable where ? is the regression coefficient from a hypothetical regression of A, on schooling. 6.
be apparent A,
is
?+B?,
should
that if the omitted variable is uncorrelated with schooling, or uncorrelated with
earnings conditional on schooling, the coefficient on schooling if
is It
most
omitted from the equation.
is
an unbiased estimate of ? even
provocative
empirical
findings
economics,
in
along
some
with
controversy
over
substance and methods.
A
good instrument
is
correlated
with the endogenous regressor for reasons the
researcher can verify and explain, but uncorrelated with the
beyond
effect
its
"Where do you an answer.
on the endogenous
regressor.
economic mechanism and
Maddala (1977,
Like most econometrics
get such a variable?"
In our view,
outcome variable
good instruments often come from
institutions
for reasons
p.
154) nghtflilly asks,
texts,
he does not provide
detailed
knowledge of the
determining the regressor of interest.
hi the case of schooling, theory suggests that schooling choices are detemiined
comparing the costs and benefits of
would be
instruments
vary
independently
Thus, one possible source of
differences in costs due, say, to loan pohcies or other subsidies that
of abOity or earnings
educational attainment
choices.
alternative
by
is
A
potential.
second source of variation
in
Angrist and Krueger (1991) exploit this
institutional constraints.
kind of variation in a paper that typifies the use of "natural experiments" to try to eliminate
omitted variables bias.
The
rationale for the Angrist
states required students to enter
start
age
grade,
is
fri
and Krueger (1991) approach
that,
because most
school in the calendar year in which they turned 6, school
a fimction of date of birth.
states
is
Those
bom
late
in the
with a December 31st birthday cutoff, children
enter school at age 5 3/4, while those
bom
year are young for their
bom
in the fourth quarter
in the first quarter enter school at
age 6 3/4.
Furthermore, because compulsory schooling laws typically require students to remain in
school unto their 16th birthday, these groups of students will be in different grades
7.
A
natural experiment can be studied without application
this case,
reduced form estimates would be presented.
when
of instrumental variables methods;
in
they reach the legal dropout age. In essence, the combination of school
and
compulsory
schooling
laws
creates
a
experiment
natural
compelled to attend school for different lengths of time depending on
Using
from
data
the
1980
educational attainment and quarter of birth for
pattem
is
younger
that
birth
cohorts
education-quarter-of- birth pattem for
that
men bom
eamings
men bom
men bom
early in the calendar year tend to
But the pattem
profile.
in
1940s and
the
more
men
in
to
1959.
Figure
The
figure clearly
the
shows
We
levels.
age tend to have a relatively
men bom
overall
displays
1
have lower average schooling this
are
between
relationship
The
1930s.
the
cf less education for
1950s as well.
schooling.
children
their birthdays.
the
at
which
in
men bom from 1930
finished
10- year birth cohort because
selected this
we looked
census,
age pohcies
start
age-
flat
early in the year holds for
Because an individual's date of
birth
is
probably unrelated to his or her innate abihty, motivation, or family connections (ruling
out astrological effects), date of birth should provide a valid instrument for schooling.
Figure 2 displays average earnings by quarter of birth for the same sample. In
essence, this
figure
shows the "reduced form"
the dependent variable.
with work experience.
quarters
of the
Importantly,
schoohng.
this
An
year
relationship
between the instruments and
Older cohorts tend to have higher earnings, because earnings
But the figure also shows ahnost
reduced
always
form
examination
eamed
relationship
of the
less
that,
on average, men
than
those
parallels
reduced- form
and
the
bom
later
bom
quarter-of-birth
first- stage
in early
the
in
rise
year.
pattem
estimates,
either
in
in
graphical or tabular form, often provides insights concerning the causal story motivating
a
particular
set
of instrumental variable
estimates.
In
this
case,
it
is
clear
that
the
differences in education and earnings associated with quarter of birth are discrete blips,
rather than
smooth changes
The
intuition
of aging.
related to the gradual effects
behind
variables
instrumental
case
this
in
is
difiFerences
that
in
earnings by quarter of birth are assumed to be accounted for solely by differences in
by
schooling
quarter
of
so that the estimated return to
birth,
appropriately rescaled difference in average earnings
part
of the variabihty in schooling
identify the return to
bom
education.
in the first quarter
later quarters,
bom
in
and
about
schooling,
The
quarters.
later
.10,
is
by quarter of
-- the part associated
In our fonnal statistical estimates,
in the first quarter
ratio
is
simply the
Only a small
birth.
with quarter of birth --is used to
have about one-tenth of a year
men bom
that
schooling
eam
of the difference
we
found that
less schooling than
men bom
about 0.1 percent less than
in
earnings
to
the
men in
men
difference
in
an instrumental variables estimate of the proportional earnings
gain from an additional year of schooling.
As differs
turns out, this estimate of the change in earnings
from a simple ordinary
Uttle
our data.
the
it
least
least
squares
estimate
to additional education
squares regression of education on earnings in
This finding suggests that there
ordinary
due
is
of the
little
effect
bias
from omitted abihty variables
of education on earnings,
in
probably
because omitted variables in the earnings equation are uncorrelated with education.
In
other appUcations, such as Angrist and Lavy's (1999) analysis of the effect of class-size
on student achievement based on
maximum
class
size rule,
differences
in
class- size
stemming from Maimonides
the instrumental variables estimates and ordinary least squares
estimates are quite different.
10
A common is
that
does
It
of the
criticism
not
folly
natural experiments approach to instrumental variables
out
spell
Rosenzweig and Wolpin, 2000).
the
underlying
the
In
Angrist
theoretical
relationships
and Krueger (1991)
example, the theoretical relationship between education and earnings
from an elaborate
motivate
to
instrumental
usually
a
Importantly,
in
variables
depends on the
Moreover,
economics.
is
fundamentally
well- developed
these
it
e.g.,
application,
developed
not
is
for
institutional details
of the
Nevertheless, interest in the causal effect of education on earnings
education system.
easy
model; instead,
theoretical
(see,
stories
story
have
or
grounded
model
the
theory,
in
motivating
approach
experiments
natural
sense
the
in
choice
the
imphcations that can be used to
that
of
support
is
to
there
is
instruments.
or
refote
a
behavioral interpretation of the resulting instrumental variables estimates.
For example, the interpretation of the patterns
the
interaction
laws, so if quarter of birth
was
This group
that
the
is
is
resulting
strategy
identification
supported by our
instruments
was not
sample the
would have been refoted
by
this
rational
refoted.
test
suggests
The that
other than compulsory schooling are not responsible for the correlation between
education and the instrument in the
The nch
implications
full
and
sample, and adds credibility to the exercise.
potential
refotability
of instrumental variables analyses
based on natural experiments are an important part of what makes the approach
We
from
unconstrained by compulsory schooling
related to education or earnings in this
motivating the use of quarter of birth as
factors
and 2 as
unrelated to earnings and educational attainment for those
is
with a college degree or higher.
that
1
of school- start- age policy and compulsory schooling
finding that quarter of birth
finding
in Figures
would argue
that this
approach contrasts favorably with studies
11
attractive.
that provide detailed
but
abstract
unexamined choices about which about
what
mechanical
and
atheoretical
and
hard- to- assess
assumptions
from lagged variables
instruments
endogenous
variables
time
in
instruments
as
is
to
dynamic
about
or
series
choice
the
problematic
panel
if
one
Indeed,
follow.
common, approaches
yet
naive,
of the most
of instruments
relationships
equation
the
uses
construct
to
The use of lagged
data.
error
hi this regard, Wright's (1928) use of the
variables are serially correlated,
and
implausible
exclude from the model and assumptions
variables
certain
on
based
identification
variables to
distribution
statistical
by
followed
models,
theoretical
omitted
or
more
plausible
exogenous instrument "yield per acre" seems well ahead of its time.
Interpreting Estimates v^th Heterogeneous Responses
One
observation's behavior
variables
in
difiiculty
interpreting
affected
is
instrumental
variables
by the instrument.
estimates
As we have
methods can be thought of as operating by using only
explanatory variable
--
that
is,
part
is
that
not
every
stressed, instrumental
of the variation in an
by changing the behavior of only some people.
For
example, in the Angrist and Krueger (1991) study just discussed, the quarter- of- birth instrument
is
most relevant
soon as possible, with
Uttle
for those
lottery
numbers as
earnings later in Ufe.
early
are
or no effect on those
Another example that makes draft
who
The
an
this
instrument
draft lottery
who
point
to
high probabihty of quitting school as
at
are likely to proceed
is
Angrist's
estimate
the
on
to college.
(1990) use of Vietnam-era
effect
of military
numbers randomly assigned
to
service
young men
1970s were highly correlated with the probabihty of being drafted into the
12
on
in the
military,
but not correlated with other factors that might change earnings
presumably
affected
otherwise.
But most of those
have served regardless of
numbers
as instruments
is
who
their
who would
of those
behavior
the
draft
number.
lottery
therefore based
An
military draft
have joined the
not
served in Vietnam were
The
later.
mihtary
tme volunteers who would estimate
on the experience of
using
draft
lottery
This
draftees only.
may
not capture the effects of mihtary service on volunteers' civihan earnings.
—
In other words, instrumental variables provide an estimate for a specific group
The
namely, people whose behavior can be mampulated by the instrument. birth
whose
instruments used
by Angrist and Kmeger (1991) generate an estimate
of schooling was changed by
level
quarter-of-
that
instrument.
Similarly,
the
for those
draft
lottery
instrument provides estimates of a well- defined causal effect for a subset of the treated
group:
men whose behavior was changed by the This issue arises in
many
draft lottery "experiment".
studies using instrumental variables,
dummy
those
endogenous
variable,
for short,
trial.
hi
The
some
effect
entire
response
particular
distinction
a
between
known
treated group.
And
intervention
LATE
and
instrument
the
as the Local
cases, the experiences
of those of the
to
is
is
discussed
that
with
instrumental variables methods estimate causal effects for
whose behavior would be changed by
randomized
it
They show
formally in papers by Imbens and Angrist (1994), and related work.
a
and
or
other
of
if
this
it
were assigned
Average Treatment
Effect, or
in
a
LATE
group of "compliers" are representative
everyone in the population has the same
treatment,
parameters
13
if
as
does
is
commonly not
matter.
assumed,
But
the
with
may
"heterogeneous treatment effects," the parameter identified by instrumental variables
differ
from the average be
should
It
All
research.
effect
of interest.
noted
that
this
of estimates
specificity
from
methods,
statistical
^
the
simplest
heterogeneous
using
studied
fruitfully
many
Nevertheless,
responses.
estiinated
effects
much progress
Our view
made
in that field.
that
instrumental
is
problem of eliminating omitted variables bias sample
and
size
exfrapolation
range
other
to
heavily on theory and
probably
have
Moreover,
the
analyzing
more
a
and explored.
of
common
beneficial
existence
natural
variability
populations
is
sense.
effect
in
methods
variables
for
naturally
(A
often
empirical
somewhat
fertilizer that
Cahfomia as
of heterogeneous
can
be
Indeed, this lack of
clinical trials, yet
solve
the
a well-defined population.
many
in
phenomena with
relationships
probably the norm in medical research based on
is
has been
and
most complex
the
to
empirical
subsamples, provided the possible
specific
for
endemic to
to analyze
interventions
limitations to generalizing the results are understood
immediate generahty
regressions
when used
stmctural models, have elements of this limitation
is
treatment
com
to
hmited,
and often rehes
grow
though one
effects
Since the
quite
is
speculative
helps
well,
studies
first- order
in
can't
Iowa wiU be
sure.)
would be a reason
for
experiments, not fewer, to understand the source and extent of
heterogeneity in the effect of interest.
methods identify LATE requires a technical assumption known as "monotonicity". This means that the instrument only moves the endogenous regressor in one direction. With draft lottery instmments, for example, monotonicity implies that being draft-eligible makes a person at least as likely to serve in the military as he would be if he were draft-exempt. This seems reasonable, and is automatically satisfied by traditional latent index models for endogenous treatments. 8.
The
theoretical result that instrumental variables
14
experiment
worth emphasizing
also
is
It
often of intrinsic interest.
is
instrumental variables estimate, which
by
affected
that
compulsory
the
is
schooling
one leams about
the population
identified
level,
by
differences in schooling for people
relevant
is
for
if the
In the case of the draft
effect
of military experience on earnings for both volunteers and
effect
of being drafted on
later
civUian earnings
the instrumental variable analysis often opens
Angrist
(1990)
interpreted
the
civilian
is
draftees,
up other avenues of penalty
service as due to a loss of labor market experience.
us about the
tell
knowing
the
Moreover, the story behind
important.
earnings
economic
the
changes from policies
do not necessarily
variables estimates
instrumental
assessing
institutional
designed to keep children from dropping out of high school.
even
natural
For example, the Angnst and Krueger (1991)
rewards to increases in schooling induced by legal and
lottery,
a
in
For example,
inquiry.
Vietnam- era
with
associated
If true, the resulting estimates have
predictive vaUdity for the consequences of compulsory military service in other times
and
places.
Potential Pitfalls
What can go wrong with problem variables
is
a bad instrument, that
(or
simultaneous
the
error
equations).
term
in
instrumental variables?
is,
the
Especially
an instrument that
stmctural
worrisome
equation
is
the
The most important is
correlated with the omitted
of
that
is
much
greater
than
the
15
bias
in
interest
possibility
between the instrumental variable and omitted variables can lead estimates
potential
ordinary
in
that
the
an
case
of
association
to a bias in the resulting
least
squares
estimates.
Moreover, seemingly appropriate instruments can turn out to be correlated with omitted
on
variables
For example, the weather
closer examination.
in Brazil
probably
shifts the
supply curve for coffee, providing a plausible instrument to estimate the effect of price on
demanded.
quantity
But weather
sophisticated commercial
where coffee
may
the
not materialize in
Another concern
is
the
endogenous
correlated with the
at
New
York
shift
the
adjust holdings
variables estimates with very
in
coffee
if
anticipation of
of bias when instruments are
possibihty
This possibihty was
regressor(s).
weak
for
fact.
and emphasized by Bound, Jaeger and Baker (1995).
(1959),
demand
Sugar and Coca Exchange,
Coffee,
are traded, use weather data to
fixtures
price increases that
buyers
might also
Brazil
in
first
In
only
weakly
noted by Nagar
fact,
instrumental
instruments tend to be centered on the corresponding
ordinary least squares estimate (Sawa, 1969).
weak instruments problem have been proposed.
Several solutions to the
bias of
two -stage
other words, if
the bias
if the
is
least
K
squares
proportional to the degree of over- identification,
instruments are used to estimate
proportional to KrG.
number of instruments
approximately
is
A
zero.
is
he
effect
of
G
endogenous
Using fewer instruments therefore reduces equal to the
number of endogenous
of technical fixes
variety
First,
and diagnostic
bias.
the
hi
variables,
In fact,
variables, the bias is
tests
have also been
proposed for the weak instrument problem.^
9.
One
solution
Although
is
LIML
algebraically distributions
to
use the Limited Information
and two-stage
Maximum
Likelihood (LIML) estimator.
least squares have the same asymptotic distributions and are
equivalent in just-identified models, in over-identified models their finite-sample
can be very
sense that the median of
different.
Most
importantly,
LIML
is
approximately unbiased in the
sampling distribution is generally close to the population parameter being estimated (Anderson et a!., 1982). Altematives to LIML include the approximately its
unbiased split-sample and jackknife instrumental variables estimators (Angrist and Krueger, 1995;
Angrist,
Imbens, and Krueger,
1999;
Blomquist and Dahlberg,
16
1999),
bias-corrected
Concerns about weak instruments can be mitigated most simply by looking reduced form equation, that
of
variable
unbiased,
if
squares
least
on the instruments and exogenous
interest
even
ordinary
the
is,
the
instruments
of the outcome
regression
These estimates are
variables.
Because the reduced form
weak.
are
at the
effects
are
proportional to the coefficient of interest, one can detemiine the sign of the coefficient of
and
interest
assumptions
guestimate
about
the
by rescahng
magnitude
its
of the
size
first- stage
the
form
reduced
Most
coefficient(s).
using
plausible
importantly,
if
the
reduced form estimates are not sigmficantly different from zero, the presumption should
be
of
that the effect
The
plausibility
We both the
interest is either absent or the instruments are too
of the magnitude of the reduced form
conclude our review of
first
and second stages
pitfalls
in
and
may even do some
estimates does not turn
with a
least
getting the
Moreover, using a nonlinear
endogenous
For example,
squares estimation.
regressor.
first
But
stage in
this is not
two -stage necessary
In two-stage least squares, consistency of the second stage
harm.
on
it.
with a discussion of flmctional fomi issues for
two -stage
dummy
to detect
be considered.
effects should also
researchers are sometimes tempted to use probit or logit for the
least squares application
weak
first
first-stage
functional
form
right
(Kelejian,
1971).
stage does not generate consistent estimates unless the
nonlinear model happens to be exactly
right,
so the dangers of mis -specification are high.
Nonlinear second stage estimates with continuous or multi- valued regressors are
similarly
tricky,
requiring
easily
interpreted.
linear
instrumental
correctly- specified
And even variables
estimation (Sawa, 1973; ( 1
a
997) and Hahn and
if
the
estimates
flmctional
underlying
form
second- stage
for
the
estimates
relationship
is
to
be
nonlinear,
such as two -stage least squares typically capture
Bekker, 1994), inference procedures discussed by Staiger and Stock
Hausman
(2000), and Bayesian smoothing of the
17
first
stage (Chamberlain
an average
effect
of economic
interest
analogous to the
LATE
dummy
parameter for
endogenous regressors (Angrist, Graddy, and Imbens, 2000; Card, 1995; Heckman and Vytlacil, 2000).
a
natural
Thus, two-stage least squares
starting
form
functional
point
can
issues
instrumental
for
be
assessed
a robust estimation method that provides
is
variables
a
in
The importance
applications.
more
detailed
secondary
analysis
of
by
experimenting with alternative instruments and examining suitable graphs.
Nature's Stream of Experiments
Trygve
Haavelmo
experiments, "those
is steadily
we
designs
appUcations
p. 14)
drew
an
two
between
analogy
sorts
of
should hke to make", and "the stream of experiments that nature
own enormous
turning out fi^om her
He
passive observers."
their
(1944,
laboratory,
also lamented, "unfortunately
of experiments
explicitly."
The
—
and which
we
merely watch as
most economists do not describe
defining
characteristic
of many
of instrumental variables to the omitted variables problem
is
the
devoted to describing and assessing the underlying quasi- experimental design.
recent
attention
This can
be seen as an expUcit attempt to use observational data to mimic randomized experiments as closely as possible.
-
Some economists number of
useflil
natural
are
pessimistic
experiments.
about
the
prospects
of finding
a
substantial
Michael Hurd (1990), for example, called the
search for natural experiments to test the effect of Social Security on labor supply "overly
cautious" and warned, "if apphed to other areas of empirical
effectively stop estimation."
and Imbens, 1996).
We
make no claim
work
[this
method] would
that natural experiments are the only
way
to
obtain
useful
only
results,
have
they
that
potential
the
understanding of important economic relationships.
Table
greatly
to
provides
1
increase
a sampling of
our
some
recent studies that have used instrumental variables to analyze a natural experiment or a
researcher- generated randomized experiment.
hard to conclude that empincal work
is
It
has effectively stopped.
The
first
panel
Table
in
illustrates
1
the
by Meyer (1995) and Rosenzweig and Wolpin But
underlying
attempts
many
The second panel instrumental
experiments
variables
when,
in
training
while
programs,
some
in
for
control
but not
impose,
the examples are
There
is
more
models where the
more
to substantiate the
"theory"
justification
behind these
including
for
neither explicitly described nor evaluated.
Table
in
illustrates
1
another important development: the use of
randomized experiments. because
of
Instrumental
or
practical
ethical
example,
some
treatment
members may
Similarly,
incentives
avail
group
are
useful
in
considerations,
there
is
members may
decline
training
themselves of training through channels
in medical trials,
that
variables
In randomized evaluations of
the treatment or control groups.
group
outside the experiment.
offer,
is
either
incomplete compliance
stmctural
Some of
by a serious attempt
causaUty.
infer
ostensibly
or excluding certain variables
(2000).
are distinguished
used to
assumptions
than in
all
of the natural
Other examples can be found in the surveys
experiments idea in recent empirical work.
convincing that others.
breadth of application
doctors
change behaviors
may be
like
willing to randomly
smoking or taking a new
medication.
Even
in
experiments
used to estimate the
effect
with
compliance
problems,
instrumental
variables
can
be
of interventions such as job training or medical treatments.
19
The
instrumental
variable
in
such
cases
dummy
a
is
variable
indicating
randomized
assignment to the treatment or control group and the endogenous right-hand-side variable
is
an indicator of actual treatment
variable
member who training.
for
the
dummy
would be a
For example, the actual treatment
status.
status
variable that equals one for each treatment and control group
participated in training,
and zero for
all
those
who
did not participate in
This approach yields a consistent estimate of the causal effect of the treatment
population that
comphes with
their
random assignment,
As
"compliers" (see Imbens and Angrist, 1994).
-
instruments
to estimate the effect of interest.
growing
is
and
reflects
the
the
population
of
in natural experiments, the instrument is
used to exploit an exogenous source of variation - created by in these cases
i.e.,
explicit
The use of such
accelerating
random assignment researcher- generated
convergence
of
classical
experimentation and observational research methods.
Our view
is
that
depends mostly on the
progress
gritty
in
work of
the
apphcation of instrumental vanables methods
finding or creating plausible experiments that can
be used to measure important economic relationships (1991) has called "shoe-leather" research.
in
the
detailed
sense
of requiring
institutional
forces at work.
Of
new
knowledge,
--
statistician
David Freedman
Here the challenges are not primarily technical
theorems or estimators.
and the
what
careflil
Rather,
investigation
course, such endeavors are not really new.
the heart of good empirical research.
20
and
progress
comes
quantification
fi^om
of the
They have always been
at
Figure
5.94
1
:
Mean Years
of Completed Education by, Quarter of Birth
1
30
31
32
33
34
35
Quarter of Birth Source; Authors' calculations from the 1980 Census
21
36
Figure
2:
Mean Log Weekly
Earnings, by Quarter of
Birth 5.94
5.86
30
31
32
33
35
34
Quarter of Birth Source: Authors' calculations from the 1980 Census
22
36
37
38
39
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http://www.archive.org/details/instrumentalvariOOangr
DEWEY
Technology Department of Economics Working Paper Series
Massachusetts
Institute of
INSTRUMENTAL VARIABLES AND THE SEARCH FOR IDENTIFICATION FROM SUPPLY AND DEMAND TO NATURAL EXPERIMENTS*
Joshua Angrist,
MIT
Alan B. Krueger, Princeton University
Working Paper 01-33 August 2001
Room
E52- 251
50 Memorial Drive Cambridge^ MA 02142
This paper can be downloaded without charge from the Social Science Research Network Paper Collection at http://papers.ssrn.com/abstract id=xxxxx
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
AUG 2
9 2001
LIBRARIES
Technology Department of Economics Working Paper Series
Massachusetts
Institute of
INSTRUMENTAL VARIABLES AND THE SEARCH FOR IDENTIFICATION FROM SUPPLY AND DEMAND TO NATURAL EXPERIMENTS*
Joshua Angrist,
MIT
Alan B. Krueger, Princeton University
Working Paper 01-33 August 2001
Room E52-251 50 Memorial Drive Cambridge, MA 02142
This paper can be downloaded without charge from the Social Science Research Network Paper Collection at http://papers.ssrn.com/abstract id=xxxxx
INSTRimiENTAL VARIABLES AND THE SEARCH FOR IDENTIFICATION: FROM SUPPLY AND DEMAND TO NATURAL EXPERIMENTS*
Joshua D. Angrist,
MIT
Alan B. Krueger, Princeton University
August 2001
*This paper was prepared for the Journal of Economic Perspectives symposium on econometric and presented at the January 200 1 meetings of the American Economic Association. We are
tools
grateful
to
comments.
Brad DeLong, David Freedman, Tim Taylor, and Michael Waldman
for helpfiil
Abstract
The method of instrumental variables was first used in the 1920s to estimate supply and demand elasticities, and later used to correct for measurement error in single -equation models.
Recently,
instrumental
variables
have
been widely used to
reduce bias
from
omitted variables in estimates of causal relationships such as the effect of schooling on earnings, in
portion
variables
methods use only a portion of the variabihty
unrelated to the omitted variables.
is
variables,
to
instrumental
hituitively,
key variables to estimate the relationships of
and the
qualities that
make
for a
interest;
We
if the
good instrument, devoting
instruments that are derived from "natural experiments."
experiments approach
is
the transparency
and
also discuss the use of instrumental variables in
MIT
Department
Of Economics
50 Memorial Drive
Cambndge, and
reflitability
MA 02142
A
particular attention
key feature of the natural
of identifying assumptions.
randomized experiments.
Alan
Joshua Angrist
instniments are valid, that
discuss the mechanics of instaimental
P.
Krueger
Princeton University Firestone Library
Princeton,
N J 08544
NBER
angrist(a)mit.edu
akrueger(g),princeton
.
edu
We
The method of tool
instrumental variables
The canonical example, and
kit.
attempts
involved
a signature technique in the econometrics
earliest
apphcations.
demand and supply
estimate
to
is
of instrumental
variables
Exonomists such as P.G.
curves.
Wright, Henry Schultz, Ehner Working, and Ragnar Frisch were interested in estimating
the elasticities of
demand and supply
data
on
and
quantities
prices
Consequently, an ordinary
- that
is,
trace out
-
demand and supply
If both the
with time- series data.
least
for products ranging
a
reflect
set
from herrmg
to butter, usually
curves shift over time, the observed
of equilibrium
points
on both
squares regression of quantities on prices
either the supply or
demand
fails
curves.
to identify
relationship.
P.G. Wright (1928) confronted this issue in the seminal appUcation of instrumental
variables:
linseed
oil.
estimating
the
of supply and demand
elasticities
Wright noted the
for
flaxseed,
of obtaining estimates of the
difficulty
elasticities
and demand from the relationship between price and quantity alone. however, that certain "curve
shifters"
- what we would now
can be used to address the problem
which (A)
1.
affect
demand
(p.
312):
"Such
the
He
source of
of supply suggested,
call instrumental variables
additional factors
conditions without affecting cost conditions or
may
-
be factors
which (B)
affect
See Goldberger (1972) and Morgan (1990) for a discussion of the origins of instmmental and related methods. Bowden and Turkington (1984) provide a more technical
variables
discussion of instmmental variables. Reiers0l (1945);
Ragnar 2. In
Morgan
sites
The
first
an interview
in
use of the term "instrumental variables" was in which Riersol attributed the term to his teacher,
Frisch..
the early 1920s, Wright's son, Sewall Wright, developed "causal path analysis," a method-
of-moments-type equations.
technique
P.G. Wright
for
showed
his simultaneous equations application.
credit for his father's use
stmctural models and simultaneous and instrumental variables were equivalent in quite likely that Sewall Wright deserves much of the
estimating
recursive
that path analysis It is
of instmmental variables.
cost conditions without affecting
demand
A
conditions."
variable he used for the
demand
curve shifter was the price of substitute goods, such as cottonseed, while a vanable he
used for the supply curve
shifter
was
yield per acre,
which can be thought of as primarily
determined by the weather.
an instrumental vanables estimate of the demand
Specifically,
can be
elasticity
constructed by dividing the sample covariance between the log quantity of flaxseed and
the yield per acre
by the sample covariance between the log price of flaxseed and the
This estimate
yield per acre.
demand
the error in the
is
consistent as long as yield per acre
is
uncorrected with
Replacing yield per acre with
equation and correlated with price.
the price of substitutes in this calculation generates an instrumental variables estimate of
the
supply
demand
weather- related shifts in yield are
Intuitively,
elasticity.
while changes in the price of substitutes are used to
curve,
curve so as to trace out the supply curve.
Wright (1928,
p.
demand
resulting
curve,
shift
the
demand
.
314) observed: "Success with this method depends on success in
discovering factors of the type
the
used to trace out the
A
He
and B."
and then averaged the
average elasticity of
demand
for
literature.
instrumental
six
was
flaxseed
variables estimate of the elasticity of supply
unnoticed by the subsequent
used six different supply
was
Not
variables
-.80.
estimates.
The
His average instrumental
Wright's econometric advance was
2.4.
until
shifters to estimate
the
1940s were instrumental variables
and related methods rediscovered and extended. Wright's
(1928)
method
of
averaging
the
different
instrumental
variables
estimates does not necessarily produce the most efficient estimate; other estimators
combine
the
information
in
different
instruments
to
produce
an
estimate
with
may less
sampling
The most
variability.
two-stage
way
to
combine multiple instruments
developed by Theil (1953).^
originally
squares,
least
efficient
"endogenous" right-hand side variable (price in
is
instruments and the coefficients estimated from the
plugged directly
into
the
equation of interest
m
first- stage
usually
the
stage,
on the data
the
all
for the
regression, are then either
place of the endogenous regressor
used as an instrument for the endogenous regressor. In
least squares takes the
first
regressed on
In the second stage, the predicted values of price, based
instruments.
equivalently,
appUcation)
this
In the
is
or,
way, two- stage
this
infomiation in a set of instruments and neatly boils
it
down
to a
single instrument.'*
Instrumental Variables and Measurement Error
Instrumental
error
problems
variables
methods were also pioneered
in explanatory variables.^
including the limited ability of
deviation
between
practice.
If
3.
The
staristical
variables
the
of two-stage
is
overcome measurement
error can arise for
many
reasons,
agencies to collect accurate information, and the
specified
an explanatory variable
relative efficiency
Measurement
to
in
economic theory and those collected
measured with additive random
least squares turns
errors,
in
then the
on a number of auxiliary assumptions,
such as homoscedastic errors. See Wooldridge (2001) for a discussion of alternative generalized method of moments estimators. 4.
Typically, a
demand
number of "exogenous" conditioning
nevertheless should be included in both the
squares can also be used there are at least as
variables also appear in both the supply
These exogenous covariates do not play the
equations.
if there is
many
first-
role
and
of instniments but
and second-stage regressions. Two-stage least regressor in an equation, provided
more than one endogenous
instruments as endogenous regressors (see, for example,
Bowden and
Turkington, 1984). 5.
Wald's (1940) method of fitting straight lines was specifically developed to overcome errorsDurbin (1954) showed that Wald's method is a special case of instmmental
in-variables problems. variables.
See also Geary (1949). Hausman (2001) provides a recent overview of measurement
error problems.
on
coefficient
toward zero
that variable in a bivariate ordinary least squares regression will
in
a
sample.
large
Given instrument
errors, the greater the bias.
error
the proportion of vanability that
The higher
and the equation error
(that
is,
measured data) but correlated with the correctly measured
is
due
to
with the measurement
that is uncorrelated
the equation error from the
be biased
model with
the correctly
then
instrumental
variable,
variables provides a consistent estimate even in the presence of measurement error.
Friedman's
celebrated
(1957)
analysis
of
the
consumption
interpreted as an appUcation of instrumental variables in this context.
function
can
Annual income
be
is
a
noisy measure of permanent income, so a regression of consumption on annual income
yields
too
small
an estimate of the marginal propensity to consume
income.
To overcome
which
equivalent
is
measurement problem, Friedman grouped
this
using a two- stage
to
least
squares
impHcitly a regression of annual income on a set of
The
fitted
same
While
two -stage
they
are
income
values, as
done
is
and are
unbiasedness
squares
unbiased.
ratio
or have a simple form.
typically exist
data
by
The
first
stage
city,
is
by
in
city,
so that regressing
two -stage
least squares, is
where the weights
are the
city.
least
not
because they involve a
between
fitted
his
indicating each of the cities.
as a weighted regression using city average data,
number of observations per
consistent,
dummies
values from this regression would be average income
micro consumption data on the
procedure.
from permanent
and
fristrumental
of random
In
contrast,
quantities,
instrumental
variables
for
expectations
easily calculated.
and
other
consistency,
variables
estimates
estimators
are
not
unbiased
which expectations need not
of ordinary
least
are
exist
squares estimates
Textbooks sometimes gloss over the distinction but
the
difference
can
matter
in
practice.
Unbiasedness means the estimator has a sampling distribution centered on the parameter
of
in
interest
a
sample of any
size,
while
only means that the
consistency
converges to the population parameter as the sample size grows.
variables
estimates
variables should aspire to
The estimator
technical
(i.e.,
work with
form
precise
but
consistent,
are
not
unbiased,
researchers
Since
using
of the
Most
instrumental
instrumental
large samples.
asymptotic
distribution
of an
instrumental
the sampling distnbution in very large samples) depends
conditions.
estimator
modem
packages
software
on a number of
options
include
variables
"robust
for
standard errors" that are asymptotically vaUd under reasonably general assumptions.
important
to
remember,
however,
that
practice
in
these
standard
are
errors
It
is
only
approximate.
Instrumental Variables and Omitted Variables
Although instrumental variables methods are of simultaneous equations and recent
work uses
estimates
of causal
to counteract bias
instrumental
variables
widely used to estimate systems
from measurement
overcome
to
Studies
relationships.
still
of
this
type
omitted
are
error,
a flowering of
variables
primarily
problems
in
concerned with
estimating a narrowly defined causal relationship such as the effect of schooling, training,
or military service on earnings; the impact of smoking or medical treatments on health;
the effect of social insurance programs
The observed these
association
on labor supply; or
the effect
of policing on crime.
between the outcome and explanatory variable of
and many other examples
is
likely
to
be misleading
in
the
interest
sense that
it
in
partly
omitted
reflects
measured
and
held
are
that
factors
constant
related
in
a
to
both
variables.
omitted
the
regression,
If these
could be
factors
variables
would be
bias
In practice, however, economic theory typically does not specify
eliminated.
variables that should be held constant while estimating a relationship,
accurately measure
One
all
of the relevant variables even
solution to the omitted variable
and
it
all
of the
difficult to
is
if they are specified.
problem
is
to
randomly assign the variable of
For example, social experiments are sometimes used to assign people to a job
interest.
Random
program or a control group.
training
program (among those
in the assignment pool)
Randomized experiments
social factors.
assignment assures that participation in the
is
not correlated with omitted personal or
are not always possible, however.
would not
It
be feasible to force a randomly chosen group of people to quit smoking or attend school for an extra year, or to
the other hand,
variation
randomly assign the value of the minimum wage across
may be
it
variables
in
states.
possible to find, or even to create, a degree of exogenous
schooling,
like
smoking,
and
minimum
wages.
Instrumental
variables offer a potential solution in these situations.
To
see
how instrumental variables can
suppose that
we would like to use the
measure the
"rate
solve the omitted variables problem,
following cross- sectional regression equation to
of retum to schooling," denoted
Yi
=
a
+
?S,
+ BA, +
?:
Ei
log wage.
or her highest grade of schooling
In this equation, Yi
is
person
completed, and Ai
is
a measure of ability or motivation. (For simplicity,
i's
Si is his
we take Ai
to
could be a vector of variables.) Although the problem of
a single variable, although
it
estimating this equation
straightforward in principle, data
is
On
on
A are typically
be
unavailable, and researchers are unsure what the right controls for abUity or motivation
would be
in
any case.^ Without additional infomiation, the parameter of interest,
identified; that
is,
we
cannot deduce
it
from the joint
distribution
?, is
not
of earnings and
schooling alone.
Suppose, however,
we have a third variable, the instrument,
correlated with schooling, but otherwise unrelated to earnings. That
with the omitted variables and the regression
estimate of the payoff to schooling
is
error, Ej.
denoted ^, which
is,
Zj is
Then an instrumental
is
uncorrected
variables
the sample analog of Cov(Yi, Zi)/Cov(Si, Zi).
The
instmmental variables methods allow us to consistently estimate the coefficient of
interest free
from bias from omitted
variables, without actually
omitted variables or even knowing what they
are.
(If there is
instrument the coefficient of interest can be estimated
Intuitively, instrumental variables solve the
of the
variability in schooling
variables
-
-
having data on the
more than one
by two-stage
least squares.)
omitted variables problem by using only part
specifically, a part that is uncorrelated
to estimate the relationship
valid
with the omitted
between schooUng and earnings.
Instruments that are used to overcome omitted variables bias are sometimes said
to
denve from
"natural experiments".^
instrumental variables in this way; that
Recent years have seen a resurgence
is,
to exploit situations
in the
use of
where the forces of nature
or government policy have conspired to produce an environment somewhat akin to a
randomized
experiment.
This
type
of application
has
generated
some of
the
The expected coefficient on schooling from a regression that omits the A variable where ? is the regression coefficient from a hypothetical regression of A, on schooling. 6.
be apparent A,
is
?+B?,
should
that if the omitted variable is uncorrelated with schooling, or uncorrelated with
earnings conditional on schooling, the coefficient on schooling if
is It
most
omitted from the equation.
is
an unbiased estimate of ? even
provocative
empirical
findings
economics,
in
along
some
with
controversy
over
substance and methods.
A
good instrument
is
correlated
with the endogenous regressor for reasons the
researcher can verify and explain, but uncorrelated with the
beyond
effect
its
"Where do you an answer.
on the endogenous
regressor.
economic mechanism and
Maddala (1977,
Like most econometrics
get such a variable?"
In our view,
outcome variable
good instruments often come from
institutions
for reasons
p.
154) nghtflilly asks,
texts,
he does not provide
detailed
knowledge of the
determining the regressor of interest.
hi the case of schooling, theory suggests that schooling choices are detemiined
comparing the costs and benefits of
would be
instruments
vary
independently
Thus, one possible source of
differences in costs due, say, to loan pohcies or other subsidies that
of abOity or earnings
educational attainment
choices.
alternative
by
is
A
potential.
second source of variation
in
Angrist and Krueger (1991) exploit this
institutional constraints.
kind of variation in a paper that typifies the use of "natural experiments" to try to eliminate
omitted variables bias.
The
rationale for the Angrist
states required students to enter
start
age
grade,
is
fri
and Krueger (1991) approach
that,
because most
school in the calendar year in which they turned 6, school
a fimction of date of birth.
states
is
Those
bom
late
in the
with a December 31st birthday cutoff, children
enter school at age 5 3/4, while those
bom
year are young for their
bom
in the fourth quarter
in the first quarter enter school at
age 6 3/4.
Furthermore, because compulsory schooling laws typically require students to remain in
school unto their 16th birthday, these groups of students will be in different grades
7.
A
natural experiment can be studied without application
this case,
reduced form estimates would be presented.
when
of instrumental variables methods;
in
they reach the legal dropout age. In essence, the combination of school
and
compulsory
schooling
laws
creates
a
experiment
natural
compelled to attend school for different lengths of time depending on
Using
from
data
the
1980
educational attainment and quarter of birth for
pattem
is
younger
that
birth
cohorts
education-quarter-of- birth pattem for
that
men bom
eamings
men bom
men bom
early in the calendar year tend to
But the pattem
profile.
in
1940s and
the
more
men
in
to
1959.
Figure
The
figure clearly
the
shows
We
levels.
age tend to have a relatively
men bom
overall
displays
1
have lower average schooling this
are
between
relationship
The
1930s.
the
cf less education for
1950s as well.
schooling.
children
their birthdays.
the
at
which
in
men bom from 1930
finished
10- year birth cohort because
selected this
we looked
census,
age pohcies
start
age-
flat
early in the year holds for
Because an individual's date of
birth
is
probably unrelated to his or her innate abihty, motivation, or family connections (ruling
out astrological effects), date of birth should provide a valid instrument for schooling.
Figure 2 displays average earnings by quarter of birth for the same sample. In
essence, this
figure
shows the "reduced form"
the dependent variable.
with work experience.
quarters
of the
Importantly,
schoohng.
this
An
year
relationship
between the instruments and
Older cohorts tend to have higher earnings, because earnings
But the figure also shows ahnost
reduced
always
form
examination
eamed
relationship
of the
less
that,
on average, men
than
those
parallels
reduced- form
and
the
bom
later
bom
quarter-of-birth
first- stage
in early
the
in
rise
year.
pattem
estimates,
either
in
in
graphical or tabular form, often provides insights concerning the causal story motivating
a
particular
set
of instrumental variable
estimates.
In
this
case,
it
is
clear
that
the
differences in education and earnings associated with quarter of birth are discrete blips,
rather than
smooth changes
The
intuition
of aging.
related to the gradual effects
behind
variables
instrumental
case
this
in
is
difiFerences
that
in
earnings by quarter of birth are assumed to be accounted for solely by differences in
by
schooling
quarter
of
so that the estimated return to
birth,
appropriately rescaled difference in average earnings
part
of the variabihty in schooling
identify the return to
bom
education.
in the first quarter
later quarters,
bom
in
and
about
schooling,
The
quarters.
later
.10,
is
by quarter of
-- the part associated
In our fonnal statistical estimates,
in the first quarter
ratio
is
simply the
Only a small
birth.
with quarter of birth --is used to
have about one-tenth of a year
men bom
that
schooling
eam
of the difference
we
found that
less schooling than
men bom
about 0.1 percent less than
in
earnings
to
the
men in
men
difference
in
an instrumental variables estimate of the proportional earnings
gain from an additional year of schooling.
As differs
turns out, this estimate of the change in earnings
from a simple ordinary
Uttle
our data.
the
it
least
least
squares
estimate
to additional education
squares regression of education on earnings in
This finding suggests that there
ordinary
due
is
of the
little
effect
bias
from omitted abihty variables
of education on earnings,
in
probably
because omitted variables in the earnings equation are uncorrelated with education.
In
other appUcations, such as Angrist and Lavy's (1999) analysis of the effect of class-size
on student achievement based on
maximum
class
size rule,
differences
in
class- size
stemming from Maimonides
the instrumental variables estimates and ordinary least squares
estimates are quite different.
10
A common is
that
does
It
of the
criticism
not
folly
natural experiments approach to instrumental variables
out
spell
Rosenzweig and Wolpin, 2000).
the
underlying
the
In
Angrist
theoretical
relationships
and Krueger (1991)
example, the theoretical relationship between education and earnings
from an elaborate
motivate
to
instrumental
usually
a
Importantly,
in
variables
depends on the
Moreover,
economics.
is
fundamentally
well- developed
these
it
e.g.,
application,
developed
not
is
for
institutional details
of the
Nevertheless, interest in the causal effect of education on earnings
education system.
easy
model; instead,
theoretical
(see,
stories
story
have
or
grounded
model
the
theory,
in
motivating
approach
experiments
natural
sense
the
in
choice
the
imphcations that can be used to
that
of
support
is
to
there
is
instruments.
or
refote
a
behavioral interpretation of the resulting instrumental variables estimates.
For example, the interpretation of the patterns
the
interaction
laws, so if quarter of birth
was
This group
that
the
is
is
resulting
strategy
identification
supported by our
instruments
was not
sample the
would have been refoted
by
this
rational
refoted.
test
suggests
The that
other than compulsory schooling are not responsible for the correlation between
education and the instrument in the
The nch
implications
full
and
sample, and adds credibility to the exercise.
potential
refotability
of instrumental variables analyses
based on natural experiments are an important part of what makes the approach
We
from
unconstrained by compulsory schooling
related to education or earnings in this
motivating the use of quarter of birth as
factors
and 2 as
unrelated to earnings and educational attainment for those
is
with a college degree or higher.
that
1
of school- start- age policy and compulsory schooling
finding that quarter of birth
finding
in Figures
would argue
that this
approach contrasts favorably with studies
11
attractive.
that provide detailed
but
abstract
unexamined choices about which about
what
mechanical
and
atheoretical
and
hard- to- assess
assumptions
from lagged variables
instruments
endogenous
variables
time
in
instruments
as
is
to
dynamic
about
or
series
choice
the
problematic
panel
if
one
Indeed,
follow.
common, approaches
yet
naive,
of the most
of instruments
relationships
equation
the
uses
construct
to
The use of lagged
data.
error
hi this regard, Wright's (1928) use of the
variables are serially correlated,
and
implausible
exclude from the model and assumptions
variables
certain
on
based
identification
variables to
distribution
statistical
by
followed
models,
theoretical
omitted
or
more
plausible
exogenous instrument "yield per acre" seems well ahead of its time.
Interpreting Estimates v^th Heterogeneous Responses
One
observation's behavior
variables
in
difiiculty
interpreting
affected
is
instrumental
variables
by the instrument.
estimates
As we have
methods can be thought of as operating by using only
explanatory variable
--
that
is,
part
is
that
not
every
stressed, instrumental
of the variation in an
by changing the behavior of only some people.
For
example, in the Angrist and Krueger (1991) study just discussed, the quarter- of- birth instrument
is
most relevant
soon as possible, with
Uttle
for those
lottery
numbers as
earnings later in Ufe.
early
are
or no effect on those
Another example that makes draft
who
The
an
this
instrument
draft lottery
who
point
to
high probabihty of quitting school as
at
are likely to proceed
is
Angrist's
estimate
the
on
to college.
(1990) use of Vietnam-era
effect
of military
numbers randomly assigned
to
service
young men
1970s were highly correlated with the probabihty of being drafted into the
12
on
in the
military,
but not correlated with other factors that might change earnings
presumably
affected
otherwise.
But most of those
have served regardless of
numbers
as instruments
is
who
their
who would
of those
behavior
the
draft
number.
lottery
therefore based
An
military draft
have joined the
not
served in Vietnam were
The
later.
mihtary
tme volunteers who would estimate
on the experience of
using
draft
lottery
This
draftees only.
may
not capture the effects of mihtary service on volunteers' civihan earnings.
—
In other words, instrumental variables provide an estimate for a specific group
The
namely, people whose behavior can be mampulated by the instrument. birth
whose
instruments used
by Angrist and Kmeger (1991) generate an estimate
of schooling was changed by
level
quarter-of-
that
instrument.
Similarly,
the
for those
draft
lottery
instrument provides estimates of a well- defined causal effect for a subset of the treated
group:
men whose behavior was changed by the This issue arises in
many
draft lottery "experiment".
studies using instrumental variables,
dummy
those
endogenous
variable,
for short,
trial.
hi
The
some
effect
entire
response
particular
distinction
a
between
known
treated group.
And
intervention
LATE
and
instrument
the
as the Local
cases, the experiences
of those of the
to
is
is
discussed
that
with
instrumental variables methods estimate causal effects for
whose behavior would be changed by
randomized
it
They show
formally in papers by Imbens and Angrist (1994), and related work.
a
and
or
other
of
if
this
it
were assigned
Average Treatment
Effect, or
in
a
LATE
group of "compliers" are representative
everyone in the population has the same
treatment,
parameters
13
if
as
does
is
commonly not
matter.
assumed,
But
the
with
may
"heterogeneous treatment effects," the parameter identified by instrumental variables
differ
from the average be
should
It
All
research.
effect
of interest.
noted
that
this
of estimates
specificity
from
methods,
statistical
^
the
simplest
heterogeneous
using
studied
fruitfully
many
Nevertheless,
responses.
estiinated
effects
much progress
Our view
made
in that field.
that
instrumental
is
problem of eliminating omitted variables bias sample
and
size
exfrapolation
range
other
to
heavily on theory and
probably
have
Moreover,
the
analyzing
more
a
and explored.
of
common
beneficial
existence
natural
variability
populations
is
sense.
effect
in
methods
variables
for
naturally
(A
often
empirical
somewhat
fertilizer that
Cahfomia as
of heterogeneous
can
be
Indeed, this lack of
clinical trials, yet
solve
the
a well-defined population.
many
in
phenomena with
relationships
probably the norm in medical research based on
is
has been
and
most complex
the
to
empirical
subsamples, provided the possible
specific
for
endemic to
to analyze
interventions
limitations to generalizing the results are understood
immediate generahty
regressions
when used
stmctural models, have elements of this limitation
is
treatment
com
to
hmited,
and often rehes
grow
though one
effects
Since the
quite
is
speculative
helps
well,
studies
first- order
in
can't
Iowa wiU be
sure.)
would be a reason
for
experiments, not fewer, to understand the source and extent of
heterogeneity in the effect of interest.
methods identify LATE requires a technical assumption known as "monotonicity". This means that the instrument only moves the endogenous regressor in one direction. With draft lottery instmments, for example, monotonicity implies that being draft-eligible makes a person at least as likely to serve in the military as he would be if he were draft-exempt. This seems reasonable, and is automatically satisfied by traditional latent index models for endogenous treatments. 8.
The
theoretical result that instrumental variables
14
experiment
worth emphasizing
also
is
It
often of intrinsic interest.
is
instrumental variables estimate, which
by
affected
that
compulsory
the
is
schooling
one leams about
the population
identified
level,
by
differences in schooling for people
relevant
is
for
if the
In the case of the draft
effect
of military experience on earnings for both volunteers and
effect
of being drafted on
later
civUian earnings
the instrumental variable analysis often opens
Angrist
(1990)
interpreted
the
civilian
is
draftees,
up other avenues of penalty
service as due to a loss of labor market experience.
us about the
tell
knowing
the
Moreover, the story behind
important.
earnings
economic
the
changes from policies
do not necessarily
variables estimates
instrumental
assessing
institutional
designed to keep children from dropping out of high school.
even
natural
For example, the Angnst and Krueger (1991)
rewards to increases in schooling induced by legal and
lottery,
a
in
For example,
inquiry.
Vietnam- era
with
associated
If true, the resulting estimates have
predictive vaUdity for the consequences of compulsory military service in other times
and
places.
Potential Pitfalls
What can go wrong with problem variables
is
a bad instrument, that
(or
simultaneous
the
error
equations).
term
in
instrumental variables?
is,
the
Especially
an instrument that
stmctural
worrisome
equation
is
the
The most important is
correlated with the omitted
of
that
is
much
greater
than
the
15
bias
in
interest
possibility
between the instrumental variable and omitted variables can lead estimates
potential
ordinary
in
that
the
an
case
of
association
to a bias in the resulting
least
squares
estimates.
Moreover, seemingly appropriate instruments can turn out to be correlated with omitted
on
variables
For example, the weather
closer examination.
in Brazil
probably
shifts the
supply curve for coffee, providing a plausible instrument to estimate the effect of price on
demanded.
quantity
But weather
sophisticated commercial
where coffee
may
the
not materialize in
Another concern
is
the
endogenous
correlated with the
at
New
York
shift
the
adjust holdings
variables estimates with very
in
coffee
if
anticipation of
of bias when instruments are
possibihty
This possibihty was
regressor(s).
weak
for
fact.
and emphasized by Bound, Jaeger and Baker (1995).
(1959),
demand
Sugar and Coca Exchange,
Coffee,
are traded, use weather data to
fixtures
price increases that
buyers
might also
Brazil
in
first
In
only
weakly
noted by Nagar
fact,
instrumental
instruments tend to be centered on the corresponding
ordinary least squares estimate (Sawa, 1969).
weak instruments problem have been proposed.
Several solutions to the
bias of
two -stage
other words, if
the bias
if the
is
least
K
squares
proportional to the degree of over- identification,
instruments are used to estimate
proportional to KrG.
number of instruments
approximately
is
A
zero.
is
he
effect
of
G
endogenous
Using fewer instruments therefore reduces equal to the
number of endogenous
of technical fixes
variety
First,
and diagnostic
bias.
the
hi
variables,
In fact,
variables, the bias is
tests
have also been
proposed for the weak instrument problem.^
9.
One
solution
Although
is
LIML
algebraically distributions
to
use the Limited Information
and two-stage
Maximum
Likelihood (LIML) estimator.
least squares have the same asymptotic distributions and are
equivalent in just-identified models, in over-identified models their finite-sample
can be very
sense that the median of
different.
Most
importantly,
LIML
is
approximately unbiased in the
sampling distribution is generally close to the population parameter being estimated (Anderson et a!., 1982). Altematives to LIML include the approximately its
unbiased split-sample and jackknife instrumental variables estimators (Angrist and Krueger, 1995;
Angrist,
Imbens, and Krueger,
1999;
Blomquist and Dahlberg,
16
1999),
bias-corrected
Concerns about weak instruments can be mitigated most simply by looking reduced form equation, that
of
variable
unbiased,
if
squares
least
on the instruments and exogenous
interest
even
ordinary
the
is,
the
instruments
of the outcome
regression
These estimates are
variables.
Because the reduced form
weak.
are
at the
effects
are
proportional to the coefficient of interest, one can detemiine the sign of the coefficient of
and
interest
assumptions
guestimate
about
the
by rescahng
magnitude
its
of the
size
first- stage
the
form
reduced
Most
coefficient(s).
using
plausible
importantly,
if
the
reduced form estimates are not sigmficantly different from zero, the presumption should
be
of
that the effect
The
plausibility
We both the
interest is either absent or the instruments are too
of the magnitude of the reduced form
conclude our review of
first
and second stages
pitfalls
in
and
may even do some
estimates does not turn
with a
least
getting the
Moreover, using a nonlinear
endogenous
For example,
squares estimation.
regressor.
first
But
stage in
this is not
two -stage necessary
In two-stage least squares, consistency of the second stage
harm.
on
it.
with a discussion of flmctional fomi issues for
two -stage
dummy
to detect
be considered.
effects should also
researchers are sometimes tempted to use probit or logit for the
least squares application
weak
first
first-stage
functional
form
right
(Kelejian,
1971).
stage does not generate consistent estimates unless the
nonlinear model happens to be exactly
right,
so the dangers of mis -specification are high.
Nonlinear second stage estimates with continuous or multi- valued regressors are
similarly
tricky,
requiring
easily
interpreted.
linear
instrumental
correctly- specified
And even variables
estimation (Sawa, 1973; ( 1
a
997) and Hahn and
if
the
estimates
flmctional
underlying
form
second- stage
for
the
estimates
relationship
is
to
be
nonlinear,
such as two -stage least squares typically capture
Bekker, 1994), inference procedures discussed by Staiger and Stock
Hausman
(2000), and Bayesian smoothing of the
17
first
stage (Chamberlain
an average
effect
of economic
interest
analogous to the
LATE
dummy
parameter for
endogenous regressors (Angrist, Graddy, and Imbens, 2000; Card, 1995; Heckman and Vytlacil, 2000).
a
natural
Thus, two-stage least squares
starting
form
functional
point
can
issues
instrumental
for
be
assessed
a robust estimation method that provides
is
variables
a
in
The importance
applications.
more
detailed
secondary
analysis
of
by
experimenting with alternative instruments and examining suitable graphs.
Nature's Stream of Experiments
Trygve
Haavelmo
experiments, "those
is steadily
we
designs
appUcations
p. 14)
drew
an
two
between
analogy
sorts
of
should hke to make", and "the stream of experiments that nature
own enormous
turning out fi^om her
He
passive observers."
their
(1944,
laboratory,
also lamented, "unfortunately
of experiments
explicitly."
The
—
and which
we
merely watch as
most economists do not describe
defining
characteristic
of many
of instrumental variables to the omitted variables problem
is
the
devoted to describing and assessing the underlying quasi- experimental design.
recent
attention
This can
be seen as an expUcit attempt to use observational data to mimic randomized experiments as closely as possible.
-
Some economists number of
useflil
natural
are
pessimistic
experiments.
about
the
prospects
of finding
a
substantial
Michael Hurd (1990), for example, called the
search for natural experiments to test the effect of Social Security on labor supply "overly
cautious" and warned, "if apphed to other areas of empirical
effectively stop estimation."
and Imbens, 1996).
We
make no claim
work
[this
method] would
that natural experiments are the only
way
to
obtain
useful
only
results,
have
they
that
potential
the
understanding of important economic relationships.
Table
greatly
to
provides
1
increase
a sampling of
our
some
recent studies that have used instrumental variables to analyze a natural experiment or a
researcher- generated randomized experiment.
hard to conclude that empincal work
is
It
has effectively stopped.
The
first
panel
Table
in
illustrates
1
the
by Meyer (1995) and Rosenzweig and Wolpin But
underlying
attempts
many
The second panel instrumental
experiments
variables
when,
in
training
while
programs,
some
in
for
control
but not
impose,
the examples are
There
is
more
models where the
more
to substantiate the
"theory"
justification
behind these
including
for
neither explicitly described nor evaluated.
Table
in
illustrates
1
another important development: the use of
randomized experiments. because
of
Instrumental
or
practical
ethical
example,
some
treatment
members may
Similarly,
incentives
avail
group
are
useful
in
considerations,
there
is
members may
decline
training
themselves of training through channels
in medical trials,
that
variables
In randomized evaluations of
the treatment or control groups.
group
outside the experiment.
offer,
is
either
incomplete compliance
stmctural
Some of
by a serious attempt
causaUty.
infer
ostensibly
or excluding certain variables
(2000).
are distinguished
used to
assumptions
than in
all
of the natural
Other examples can be found in the surveys
experiments idea in recent empirical work.
convincing that others.
breadth of application
doctors
change behaviors
may be
like
willing to randomly
smoking or taking a new
medication.
Even
in
experiments
used to estimate the
effect
with
compliance
problems,
instrumental
variables
can
be
of interventions such as job training or medical treatments.
19
The
instrumental
variable
in
such
cases
dummy
a
is
variable
indicating
randomized
assignment to the treatment or control group and the endogenous right-hand-side variable
is
an indicator of actual treatment
variable
member who training.
for
the
dummy
would be a
For example, the actual treatment
status.
status
variable that equals one for each treatment and control group
participated in training,
and zero for
all
those
who
did not participate in
This approach yields a consistent estimate of the causal effect of the treatment
population that
comphes with
their
random assignment,
As
"compliers" (see Imbens and Angrist, 1994).
-
instruments
to estimate the effect of interest.
growing
is
and
reflects
the
the
population
of
in natural experiments, the instrument is
used to exploit an exogenous source of variation - created by in these cases
i.e.,
explicit
The use of such
accelerating
random assignment researcher- generated
convergence
of
classical
experimentation and observational research methods.
Our view
is
that
depends mostly on the
progress
gritty
in
work of
the
apphcation of instrumental vanables methods
finding or creating plausible experiments that can
be used to measure important economic relationships (1991) has called "shoe-leather" research.
in
the
detailed
sense
of requiring
institutional
forces at work.
Of
new
knowledge,
--
statistician
David Freedman
Here the challenges are not primarily technical
theorems or estimators.
and the
what
careflil
Rather,
investigation
course, such endeavors are not really new.
the heart of good empirical research.
20
and
progress
comes
quantification
fi^om
of the
They have always been
at
Figure
5.94
1
:
Mean Years
of Completed Education by, Quarter of Birth
1
30
31
32
33
34
35
Quarter of Birth Source; Authors' calculations from the 1980 Census
21
36
Figure
2:
Mean Log Weekly
Earnings, by Quarter of
Birth 5.94
5.86
30
31
32
33
35
34
Quarter of Birth Source: Authors' calculations from the 1980 Census
22
36
37
38
39
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