Lecture 19:
Instrumental
Variables
(Chapter 13.2–13.3)
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
Agenda
• Review
• Randomization
• Instrumental Variables (Chapter 13.2)
• Using Instrumental Variables
(Chapter 13.2)
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-2
Review
• If we relax the assumption that explanators
are fixed across samples, we confront a
number of serious complications.
• Stochastic explanators render the
mathematics of finding unbiased estimators
intractable.
• Instead, econometricians search for
consistent estimators.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-3
Review (cont.)
• The most problematic condition for OLS
to be consistent is
E(xi i )  0
which, by the Law of Large Numbers,
1

implies p lim  xi i   0.
n

Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-4
Review (cont.)
 xiYi 
 xi [  0  1 X 1   i ] 
ˆ
p lim( 1 )  p lim   2   p lim 

2

x

x
i
 i 


1


x

n i i
 xi i 
 1  p lim 
 1  p lim 

2 
1

x
 i 
 xi2 
 n

1

p lim  xi i 
0
n


 1 
 1   1
Q
1 2
p lim  xi 
n

Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-5
Review (cont.)
• Asking that key variances be non-zero
and bounded is often reasonable
(except in certain time series cases).
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-6
Review (cont.)
• How reasonable an assumption is E(Xii )  0
(i.e. Xi and i are uncorrelated)?
• In practice, this assumption is often terrible.
• When an explanator is correlated with the
error term, we call the explanator a
“troublesome variable.”
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-7
Review (cont.)
• We have examined four complications
that induce correlation between X and 
1. Omitted Variables Bias
2. Measurement Error
3. Simultaneous Causality
4. Using Lagged Values of the Dependent
Variable as Explanators, in the presence
of serial correlation
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-8
Review (cont.)
• Omitting a variable creates a bias only if:
1. X2 is an explanator of Y (so, when
omitted, it becomes a component of the
error term).
2. X2 is correlated with X1 (so that X2
creates a correlation between X1 and the
error term).
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-9
Review (cont.)
• Measurement error also induces a
correlation between our included
explanator and the error term
• Instead of observing Xi , we observe
M i  Xi  vi
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-10
Review (cont.)
Model: Yi   0  1 X 1   i
Instead of observing X i , we observe M i  X i  vi
We regress Yi   0  1M i  i
Var ( X i )
 X2
 1 2
p lim(ˆ1 )  1
 X   v2
Var ( X i )  Var (vi )
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-11
Review (cont.)
• Mismeasuring X leads to
ATTENUATION BIAS. The estimated
coefficient is biased towards 0.
• The magnitude of the bias depends on
the relative variances of X and v.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-12
Review (cont.)
• Under simultaneous causality, X and
Y are jointly determined.
• Because X and Y are determined
simultaneously, X can adjust in
response to shocks to Y ().
• Thus X will be correlated with 
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-13
Review (cont.)
• Using lagged dependent variables as
explanators is another potential source
of correlation between an explanator
and the error term.
• If there is first-order serial correlation in
the error terms, then t-1 is correlated
with t
• Therefore Yt-1 is correlated with t
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-14
Randomization
• In practice, correlation between X and
 is endemic.
• Much of econometric work involves
studying the process determining the
explanators, to see how they might be
correlated with .
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-15
Randomization (cont.)
• The ideal X variable has been
randomly assigned.
• If X has been randomly assigned, then
it contains no information about .
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-16
Randomization (cont.)
• If X has been randomly assigned, then
E( | X)  0
• It follows that E( X)  0
• Moreover, if E( | X)  0 , then OLS
is unbiased.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-17
Randomization (cont.)
• Laboratory scientists typically work with
randomly assigned explanators.
• In an experiment, the scientist
randomly assigns different subjects to
different treatments.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-18
Randomization (cont.)
• Economists have been making increasing use
of experiments.
• Some economists construct carefully
controlled markets in computer laboratories
(usually with undergraduate subjects).
• Other economists insert randomization into
actual programs in the field, for example
randomly assigning participants to different
health care plans.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-19
Randomization (cont.)
• See Chapter 15 for a discussion of
controlled experiments.
• Most econometric research, however,
studies field data.
• The econometrician observes variables
as they are naturally determined, and
attempts to draw inferences.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-20
Randomization (cont.)
• Occasionally, variables are naturally
determined by random chance.
• When nature assigns X randomly,
econometricians can study a
“natural experiment.”
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-21
Randomization (cont.)
• For example, Alvin Roth studies centralized
computer matching programs to assign
medical school graduates to hospital training
programs. Graduates and hospitals each
submit preference lists, and a computer sorts
through the lists and creates matches.
• Game theory suggests key properties for
such a matching program.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-22
Randomization (cont.)
• Roth noticed that in the United
Kingdom, each region of the country ran
its own centralized matching program.
• In some of the regions, organizers had
happened to adopt a matching program
that possessed good game theoretic
properties. Other regions did not adopt
good designs.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-23
Randomization (cont.)
• Roth speculated that the choice of program
design was reasonably random, at least in the
sense that program design was not
systematically linked to any other properties
of the region.
• Programs with good game theoretic
properties were much more likely to be
successful. Centralized programs with poor
properties fell into disuse.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-24
Randomization (cont.)
• In Roth’s study, advanced econometric
methods were unnecessary.
• The data naturally provided the
“experiment” he needed to observe.
• Of course, it is Roth’s duty to argue
compellingly that the program rules
really are uncorrelated with other
aspects of the regions.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-25
Randomization (cont.)
• Nature often provides a certain degree
of randomization.
• For example, suppose we are studying the
effect of military service on earnings.
• During the Vietnam War, the draft lottery was
a largely random process.
• Thus, a study of Vietnam War veterans
would be more random than a study of
veterans from the more recent all-volunteer
American army.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-26
Randomization (cont.)
• However, service in Vietnam was determined
by more than simple random chance.
• Some men voluntarily enlisted; others
received draft deferments (or simply evaded
the draft).
• The decision to volunteer, or to avoid being
drafted, would undoubtedly be correlated with
many other variables.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-27
Randomization (cont.)
• Military service in Vietnam, then,
consists of two elements:
1. A “clean” random element determined by
the draft lottery, that would be
uncorrelated with  in an earnings
regression; and
2. An “unclean” element, determined by
active decision-making on the agent’s
part, that could easily be correlated with .
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-28
Randomization (cont.)
• Vietnam War military service is
partially determined by an obviously
random process.
• Nonetheless, a simple OLS estimate of
the effect of military service on earnings
would be inconsistent. Military service is
also partially determined non-randomly.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-29
Randomization (cont.)
• Fortunately, econometricians have
discovered a method for separating out
the random elements of explanators
from the elements that may be
correlated with .
• Unfortunately, this method requires the
data to include an instrumental
variable with certain key properties.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-30
Instrumental Variables (Chapter 13.2)
• An Instrumental Variable is a variable that is
correlated with X but uncorrelated with .
• If Zi is an instrumental variable:
1. E( Zi Xi ) ≠ 0
2.
E( Zi i ) = 0
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-31
Instrumental Variables (cont.)
• For example, a Vietnam War lottery
number is:
1. Correlated with X (“winning” the lottery is
highly correlated with military service), but
2. Is uncorrelated with  (the lottery number
is not connected to cultural, psychological,
or economic factors that might affect the
decision to join the military, and that might
also affect earnings).
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-32
Instrumental Variables (cont.)
• The econometrician can use an
instrumental variable Z to estimate the
effect on Y of only that part of X that is
correlated with Z.
• Because Z is uncorrelated with , any
part of X that is correlated with Z must
also be uncorrelated with .
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-33
Instrumental Variables (cont.)
• An instrumental variable lets the
econometrician find a part of X that
behaves as though it had been
randomly assigned.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-34
Instrumental Variables (cont.)
• For example, let’s revisit the question of
how much mortality can be reduced by
intensive cardiac treatment.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-35
Instrumental Variables (cont.)
Mortalityi  0  1 DiIntensivelyTreated  2 DiFemale
... kUrbani   i
• If our observable control variables, such as
DFemale, were the only differences between
patients who received intensive treatment and
those who did not, then DIntensivelyTreated would
not tell us anything about . OLS would
be consistent.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-36
Instrumental Variables (cont.)
• However, we reasonably believe that a
doctor’s choice to perform intensive
cardiac procedures are correlated with
many other variables.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-37
Instrumental Variables (cont.)
• Doctors might select patients to receive
treatment based on their underlying
health status.
• If health status is an unobservable
determinant of mortality, then it is a
component of .
• OLS will give an inconsistent estimate of
the benefits of intensive treatment.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-38
Instrumental Variables (cont.)
• To eliminate Omitted Variables Bias,
we need to find some determinant of a
patient’s receiving intensive cardiac
care that is unrelated to mortality.
• Is there any element of the cardiac care
process that is reasonably random?
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-39
Instrumental Variables (cont.)
• Not every hospital is equipped to
provide intensive cardiac care.
• Some patients live near cardiac care
centers. When they have a heart attack,
they are more likely to be transported to
a center, and thus more likely to receive
intensive treatment.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-40
Instrumental Variables (cont.)
• McClellan, McNeil, and Newhouse
argue that the distance from a patient’s
home to the nearest hospital equipped
for intensive cardiac care is a valid
instrumental variable.
• They argue that geographic location is
likely to be uncorrelated with unobserved
traits that influence mortality.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-41
Instrumental Variables (cont.)
• A major task for McClellan, McNeil, and
Newhouse is to determine whether or not
distance to a cardiac care center really is
uncorrelated with mortality (except through
the increased likelihood of receiving
cardiac treatment).
• As a reader of such a study, you should ask
yourself whether the authors have convinced
you that their instrument is uncorrelated with 
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-42
Instrumental Variables (cont.)
• When McClellan, McNeil, and
Newhouse use their instrumental
variable, they find only a small benefit
from intensive cardiac care for the
marginal patient (some patients may
still benefit quite a lot!).
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-43
Instrumental Variables (cont.)
• When the economist is worried about
measurement error, a good choice of
instrument is simply a different measure
of the same variable.
• The new measure may have its own
errors, but these errors are unlikely to
be correlated with the mistakes in the
first measure, or with any other
component of .
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-44
Instrumental Variables (cont.)
• For example, Ashenfelter and Rouse
were studying the effect of education
on earnings.
• Their data came from a survey of twins.
• They were concerned that individuals
might mis-report their own years of
schooling, leading to measurement
error biases.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-45
Instrumental Variables (cont.)
• However, Ashenfelter and Rouse had two
separate measures for each individual’s years
of schooling.
• The survey asked each individual to list both
his/her own years of schooling, and also the
years of schooling for his/her twin.
• The twin’s report of an individual’s schooling
served as an instrumental variable for the
individual’s self-report.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-46
Instrumental Variables (cont.)
• Another example: policy makers are
greatly interested in the effects of tax
rates on labor force participation (and
other taxpayer behaviors).
• They would like to run regressions with
an individual’s tax rate as an explanator.
• However, an individual has some choice
over his/her tax rate.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-47
Instrumental Variables (cont.)
• Taxpayers who are close to the income
threshold for a new tax bracket can choose to
limit their taxable income.
• For example, they might take more of their
pay in the form of untaxed benefits or
deferred 401(k) compensation rather than pay
higher taxes on the extra compensation.
• The ability and desire to adjust taxable
income may well be correlated with .
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-48
Instrumental Variables (cont.)
• When the government changes the
tax rates, the individual’s new tax rate is
determined by two elements:
1. The change in tax rates (which is
uncorrelated with anything else about
the individual), and
2. The individual’s decisions about how to
respond to the tax change (which could
well be correlated with ).
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-49
Instrumental Variables (cont.)
• Public finance economists construct an
instrumental variable that captures only
the change in tax rates, not the change
in behavior.
• They use the new tax tables to look up
the tax rate individuals would face IF
they did NOT change their behavior
from before the tax change.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-50
Instrumental Variables (cont.)
• The constructed tax rate is correlated with
the tax rate the individuals face after the
tax change.
• The constructed tax rate is uncorrelated with
the behavioral adjustments individuals make
in response to the tax rate.
• Such a constructed instrument is called a
simulated instrumental variable.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-51
Checking Understanding
• Suppose you are studying the effect of price
on the demand for cigarettes, using a
cross-section of different states’ cigarette
consumption and average price.
• You would like to regress
CigarettesSoldi  0  1 Pricei  i
where i indexes each state
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-52
Checking Understanding (cont.)
CigarettesSoldi  0  1 Pricei  i
where i indexes each state
• Because Pricei is endogenous, you need to
instrument. Which of these variables would
be suitable?
1.
Each state’s cigarette excise tax
2.
A measure of each state’s anti-smoking laws
3.
Each state’s sales tax
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-53
Checking Understanding (cont.)
1. Each state’s cigarette excise tax
– Cigarette excise taxes are surely
correlated with cigarette prices. However,
they also reflect the level of anti-smoking
sentiment in the state (MA has a tax of
$1.51 per pack, NC has a tax of $0.05
per pack). Anti-smoking sentiment is an
omitted determinant of consumption, so
excise taxes are correlated with . Excise
taxes are not a valid instrument.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-54
Checking Understanding (cont.)
2. A measure of state anti-smoking laws
– State anti-smoking laws might be
correlated with price, but only through
their effect on cigarette demand in
the state. Such measures are an
explanator of cigarette consumption;
moreover, they are also a proxy for state
anti-smoking sentiment. Anti-smoking
laws are a component of , and would
make a terrible instrument.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-55
Checking Understanding (cont.)
3. Each state’s sales tax
– State sales taxes are correlated with
cigarette prices. Higher sales taxes raise
the prices of all goods.
– There is no reason to expect sales taxes
to have any other effect on cigarette
consumption, or to be correlated with any
other determinant of consumption.
– State sales taxes are a reasonable
instrument.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-56
Using Instrumental Variables
(Chapter 13.2)
• Instrumental variables are NOT the
explanator of interest.
• We want to know the effect of military service
on earnings, not just the effect of “winning”
the draft lottery.
• We want to know the effect of intensive
cardiac care on mortality, not just the effect of
living near a cardiac care center.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-57
Using Instrumental Variables (cont.)
• We do NOT simply use instrumental
variables as proxies for the explanator
of interest.
• Instead, we use IV’s as a tool to
tease out the “random” (or at least
uncorrelated) component of X.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-58
Using Instrumental Variables (cont.)
• Let’s construct a consistent IV estimator
for the case of measurement error.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-59
DGP with E(Xii ) ≠ 0
Yi  0  1 X i   i
E( i )  0
Var( i )   2  
Cov( i , j )  0 for i  j
E( X i i )  0,
M i  X i  vi
1
(xi 2 )   2X <
n
E(vi )  0
Var(vi )   v2
Cov(vi ,v j )  0 for i  j
Cov(vi , X i )  0
Cov(Z i , X i )  0
Cov(Z i , i )  0
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-60
Using Instrumental Variables
• If Xi were uncorrelated with i , we
would want to weight more heavily
observations with a high xi value.
• We know that Zi is correlated with the
“clean” part of Xi , so now we want to
weight more heavily observations with
a high zi value.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-61
Using Instrumental Variables (cont.)
ˆ

IV
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
ziYi

zi mi
19-62
Using Instrumental Variables (cont.)
Generalizing away from measurement error:
Yi  0  1 X i   i
E( X i i )  0
E(Z i i )  0
E(zi xi )  0
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-63
Using Instrumental Variables (cont.)
ˆ

IV
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
ziYi

zi xi
19-64
Checking Understanding
ˆ
IV
ziYi

zi xi
IV
OLS
ˆ
ˆ
How does  differ from the 
that would result from regressing
Yi   0  1Z i   i
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-65
Checking Understanding (cont.)
ziYi
IV
ˆ
 
zi xi
ziYi
OLS
ˆ
 
2
zi
The estimators differ in the denominators.
Only z appears in the denominator of ˆ OLS ,
i
in the form of zi2 .
Both zi and xi appear in the denominator of
ˆ IV , in the form of z x .
i i
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-66
Using Instrumental Variables
• What is the expectation of IV?
 ziYi 
 zi (  0  1 X i   i ) 
IV
ˆ
E ( 1 )  E 
  E

zi xi
 zi xi 


 zi X i 
 zi i 
 1 E 
 E

 zi xi 
 zi xi 
 zi i 
 1   E 

 zi xi 
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-67
Using Instrumental Variables (cont.)
• What is the expectation of IV?
 zi i 
IV
ˆ
E (  )  1   E 

 zi xi 
Because Cov( X i ,  i )  0, the
bias term cannot be eliminated.
IV is biased in the same direction
as the bias in OLS.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-68
Using Instrumental Variables (cont.)
• What is the probability limit of IV?
1
p lim ziYi


ziYi
IV
n
ˆ
p lim(  )  p lim 



z
x
 i i  p lim 1 zi xi
n
1
p lim zi (  0  1 X i   i )
n

1
p lim zi xi
n
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-69
Using Instrumental Variables (cont.)
• What is the probability limit of IV?
1
1
1 p lim zi xi  p lim zi i
IV
n
n
ˆ
p lim( 1 ) 
1
p lim zi xi
n
By the Law of Large Numbers....

1Cov( Z i , X i )  Cov( Z i ,  i )
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
Cov( Z i , X i )
 1
19-70
Using Instrumental Variables (cont.)
• What is the probability limit of IV?
Cov( Z i , X i ) Cov( Z i ,  i )
IV
ˆ
p lim( 1 )  1

Cov( Z i , X i ) Cov( Z i , X i )
If Cov( Z i , X i )  0, the denominator equals 0,
and the p lim does not exist.
If Cov( Z ,  )  0, then ˆ IV is inconsistent.
i
i
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
1
19-71
Using Instrumental Variables (cont.)
• The asymptotic variance of  IV is
1 2
p lim zi
1 2
1 2 Var(Zi )
n

 
2
n
n Cov(Zi , X i )2
1

p lim  zi xi 
n

• The greater the covariance between X
and Z, the lower the asymptotic variance.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-72
Using Instrumental Variables (cont.)
• In the multiple regression case, we may
have more than one explanator that is
correlated with the error term (i.e. more
than one troublesome variable).
• However, an instrumental variable only
needs to instrument for one of the
troublesome variables.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-73
Using Instrumental Variables
• A variable Zi can instrument for a
particular troublesome explanator, XRi, if:
1. Cov( Zi,XRi ) ≠ 0
2. Cov( Zi,i ) = 0
• Zi must be correlated with the
troublesome variable for which it
instruments, but need not be correlated
with all of the troublesome variables.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-74
Using Instrumental Variables (cont.)
• To estimate a multiple regression
consistently, we need at least one
instrumental variable for each
troublesome explanator.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-75
Using Instrumental Variables (cont.)
• When we have just enough instruments
for consistent estimation, we say the
regression equation is exactly identified.
• When we have more than enough
instruments, the regression equation is
over identified.
• When we do not have enough instruments,
the equation is under identified
(and inconsistent).
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-76
Review (cont.)
• In practice, correlation between X and
 is endemic.
• Much of econometric work involves
studying the process determining the
explanators, to see how they might be
correlated with 
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-77
Review (cont.)
• The ideal X variable has been
randomly assigned.
• If X has been randomly assigned,
then it contains no information about 
• However, true randomization is
relatively uncommon.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-78
Review (cont.)
• Often, an explanator is partially
determined in a way that is random, or
at least uncorrelated with 
• However, the explanator is also
influenced by omitted variables, or
determined endogenously, or is in some
other way correlated with 
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-79
Review (cont.)
• Fortunately, econometricians have
discovered a method for separating out
the random elements of explanators
from the elements that may be
correlated with .
• Unfortunately, this method requires the
data to include an instrumental
variable with certain key properties.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-80
Review (cont.)
• An Instrumental Variable is a
variable that is correlated with X but
uncorrelated with .
• If Zi is an instrumental variable:
1. E(ZiXi ) ≠ 0
2. E(Zii ) = 0
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-81
Review (cont.)
• If Xi were uncorrelated with i , we
would want to weight more heavily
observations with a high xi value.
• We know that Zi is correlated with the
“clean” part of Xi , so now we want to
weight more heavily observations with
a high zi value.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-82
Review (cont.)
ˆ

IV
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
ziYi

zi xi
19-83
Review (cont.)
• What is the probability limit of IV?
Cov( Z i , X i ) Cov( Z i ,  i )
IV
ˆ
p lim( 1 )  1

 1
Cov( Z i , X i ) Cov( Z i , X i )
If Cov( Z i , X i )  0, the denominator equals 0,
and the p lim does not exist.
If Cov( Z ,  )  0, then ˆ IV is inconsistent.
i
i
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
1
19-84
Review (cont.)
• The asymptotic variance of  IV is
1 2
p lim zi
1 2
1 2 Var(Zi )
n

 
2
n
n Cov(Zi , X i )2
1

p lim  zi xi 
n

• The greater the covariance between X and Z,
the lower the asymptotic variance.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-85
Review (cont.)
• To estimate a multiple regression
consistently, we need at least one
instrumental variable for each
troublesome explanator.
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-86
Review (cont.)
• When we have just enough instruments for
consistent estimation, we say the regression
equation is exactly identified.
• When we have more than enough
instruments, the regression equation is
over identified.
• When we do not have enough instruments,
the equation is under identified
(and inconsistent).
Copyright © 2006 Pearson Addison-Wesley. All rights reserved.
19-87