Lecture 18:
Major Challenges
for Consistency
Measurement Error,
Omitted Variables
Bias, Simultaneous
Causality, and Lagged
Dependent Variables
(Chapter 8.2, 13.1, 13.3)
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Agenda
• Stochastic X ’s
• Omitted Variables Bias
(Chapter 8.2, 13.3)
• Measurement Error (Chapter 13.1)
• Simultaneous Causality (Chapter 13.3)
• Using Lagged Values of the Dependent
Variable as Explanators (Chapter 13.3)
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18-2
Review: Fixed X ’s
• The Gauss–Markov DGP
Yi   0  1 X i   i
E ( i )  0
Var ( i )   2
Cov( i ,  j )  0 if i  j
X ’s fixed across samples
OLS is BLUE.
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18-3
Review: Fixed X ’s (cont.)
 xiYi 
 xiYi 
ˆ
E ( 1 )  E   2    E  2 
 xi 
 xi 
 xi

  E  2 [  0  1 X i   i ] 
 xi

 xi
 xi i  
xi
   2  0  2 1 X i  E  2  
xi
 xi  
 xi
xi
xi X i xi E ( i )
 0
 1

 0  1  0
2
2
2
xi
xi
xi
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18-4
Stochastic X ’s
• The assumption that X ’s are fixed across
samples is very strong.
• We are assuming that the process that
generates the i ’s is completely separate from
the process that generates the Xi ’s.
• What happens to the expectation of OLS if we
do not make this assumption?
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18-5
Stochastic X ’s (cont.)
 xiYi 
 xiYi 
ˆ
E ( 1 )  E   2    E  2 
 xi 
 xi 
 xi

 E  2 [  0  1 X i   i ] 
 xi

 xi
 xi i  
xi
   2  0  2 1 X i  E  2  
xi
 xi  
 xi
 xi i 
 0  1  E  2 
 xi 
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18-6
Stochastic X ’s (cont.)
• Consistency is an easier goal.
• We need to assume
2 
1 2
p lim  xi   Q, where Q is a non-zero constant.
n

E(xi i )  0
which, by the Law of Large Numbers,
1

implies p lim  xi i   0.
n

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18-7
Stochastic X ’s (cont.)
 xiYi 
 xi [  0  1 X 1   i ] 
ˆ
p lim( 1 )  p lim   2   p lim 

2

x

x
i
 i 


1

 n xi i 
 xi i 
 1  p lim 
 1  p lim 

2 
1
 xi 
 xi2 
 n

1

p lim  xi i 
0
n


 1 
 1   1
Q
1 2
p lim  xi 
n

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18-8
Stochastic X ’s (cont.)
• Asking that key variances be non-zero
and bounded is often reasonable
(except in certain time series cases).
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18-9
Stochastic X ’s (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.”
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18-10
Stochastic X ’s (cont.)
• This lecture outlines 4 major causes of
correlation between Xi and i
• One or more of these problems is at
least a plausible concern in most
applications.
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18-11
Stochastic X ’s (cont.)
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
•
Any of these conditions will make OLS
inconsistent (and biased).
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18-12
Omitted Variables Bias
• We have already learned that omitting
an explanator of Y from the regression
can create an Omitted Variables Bias
in OLS.
• The omitted explanator must be
correlated with an included explanator.
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18-13
Omitted Variables Bias (cont.)
• The DGP includes X2i.
Yi   0  1 X 1i   2 X 2i   i
• Y depends on both X1 and X2 , but we
regress Y on only X1.
Yi   0   1 X 1i   i
Is E (ˆ 1)   1 ?
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18-14
Omitted Variables Bias (cont.)
E (ˆ1 )  E ( wiYi )   E ( wiYi )
  wi E (Yi )   wi E (  0  1 X 1i   2 X 2i  1 )
  0  wi  1  wi X 1i   2  wi X 1i  E ( i )
 1   2  wi X 2i
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18-15
Omitted Variables Bias (cont.)
• We can reconceptualize OVB as case of
concurrent correlation between X and
the error term:
xii  0
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18-16
Omitted Variables Bias (cont.)
When Y  0  1 X1  2 X 2  
but we instead regress
Y  0  1 X1  
then  = 2 X 2  
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18-17
Omitted Variables Bias (cont.)
Y  0  1 X1  
  2 X 2  
Cov( X1 , )  Cov( X1 , 2 X 2   )
 Cov( X1 , 2 X 2 )  Cov( X1 , )
 2Cov( X1 , X 2 )
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18-18
Omitted Variables Bias (cont.)
• X is correlated with 
• Contemporaneous correlation
implies inconsistency.
• Using probability limits, we can
re-derive the formula for omitted
variables bias and OLS.
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18-19
Omitted Variables Bias (cont.)
1

p lim  xii 
n

p lim(ˆ1 )  1 
1 2
p lim  xi 
n

E ( xii )
Cov ( X 1 , )
 1 
 1 
2
E ( xi )
Var ( X 1 )
Cov( X 1 , X 2 )
 1   2
Var ( X 1 )
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18-20
Omitted Variables Bias (cont.)
Cov( X 1 , X 2 )
p lim(ˆ1 )  1   2
Var ( X 1 )
Cov( X 1 , X 2 )
Note:
is the OLS formula
Var ( X 1 )
for  1 in the regression X 2   0   1 X 1  v
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18-21
Omitted Variables Bias (cont.)
• Example: heart attacks and treatments
• Do cardiac catheterization,
revascularization, and other intensive
medical procedures reduce mortality
among the elderly?
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18-22
Omitted Variables Bias (cont.)
• Patients who receive more intensive
cardiac treatment are on average
younger, urban, white males.
• These traits may account for the
lower mortality rates among the
intensively treated.
• We can easily include these
observable variables.
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18-23
Omitted Variables Bias (cont.)
Mortalityi  0  1 D
IntensivelyTreated
i
 2 D
Female
i
... kUrbani   i
• If these observables 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.
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18-24
Omitted Variables Bias (cont.)
• However, we reasonably believe that
a doctor’s choice to perform intensive
cardiac procedures are correlated with
many other variables.
• In general, whenever an explanator
is actively chosen by an economic
agent, we worry a great deal about
selection biases.
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18-25
Omitted Variables Bias (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.
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18-26
Omitted Variables Bias (cont.)
• OLS will give an inconsistent estimate of
the benefits of intensive treatment.
• The extent of the bias depends on
the extent to which health status
affects mortality and the extent to
which intensive treatment predicts
health status.
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18-27
Measurement Error (Chapter 13.1)
• Another source of correlation between
X and  is the mis-measurement of X
(also called “errors in variables”).
• To the extent that the X we observe
differs from the X we are modeling, our
coefficients will be wrong.
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18-28
Measurement Error (cont.)
• Another source of correlation between X
and  is the mis-measurement of X.
• Measurement error might arise because:
– The data are flawed (for example, survey
respondents mis-remember how long
they’ve worked at their current jobs)
– The economist is using an imperfect proxy
(for example, using total taxable income as
a measure of total income)
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18-29
Measurement Error (cont.)
• Another source of correlation between
X and  is the mis-measurement of X.
• Measurement error might arise because:
– The economist does not properly
understand the nature of X (for example,
focusing on changes in annual income
instead of changes in total lifetime income,
the relevant variable in Friedman’s
“permanent income hypothesis”)
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18-30
Measurement Error (cont.)
We model Yi  0  1 X i   i
but instead of observing X i we
observe M i  X i  vi
where vi is some error
E(vi )  0, Cov( X i ,vi )  0
We regress Yi   0  1 M i  i
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18-31
Measurement Error (cont.)
We model Yi  0  1 X i   i
We regress Yi  0  1 M i  i
Substituting, our regression equation
is equivalent to 0  1 ( X i  vi )  i
 0  1 X i  1vi  i
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18-32
Measurement Error (cont.)
We model Yi   0  1 X i   i
We regress Yi   0  1M i  i
which is, in essence,  0  1 X i  1vi  i
ˆ1 has to perform two jobs.
It estimates the coefficient on X i (1 )
but it ALSO estimates the coefficient
on vi . Since vi does not appear in our model,
its true coefficient is 0.
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18-33
Measurement Error (cont.)
In essence, we regress Yi   0  1 X i  1vi  i
ˆ1 has to perform two jobs.
It estimates the coefficient on X i ( 1 )
but it ALSO estimates the coefficient
on vi . Since vi does not appear in our model,
its true coefficient is 0.
ˆ1 will be a weighted average of 1 and 0.
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18-34
Measurement Error (cont.)
• How can measurement error be
conceptualized as a correlation between
X and  ?
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18-35
Measurement Error (cont.)
Our model is Yi  0  1 X i   i
Instead of X i , we observe M i  X i   i
Yi  0  1 ( M i  vi )   i
 0  1 M i  1vi   i
 0  1 M i  i
where i  1vi   i
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18-36
Measurement Error (cont.)
Our model is Yi   0  1 X i   i
Instead of X i , we observe M i  X i  vi
We regress Yi   0  1M i  i
where i   1vi   i
Cov( M i ,i )  Cov( X i  vi ,  1vi   i )
 Cov(vi ,  1vi )
  1 Cov(vi , vi )   1 Var (vi )
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18-37
Measurement Error (cont.)
• Measurement error implies a correlation
between our observed explanator
M i  Xi  vi and the error term i  -1vi   i
• What bias does this correlation create
in OLS?
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18-38
DGP with Measurement Error
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
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18-39
Measurement Error
Regress Yi   0  1M i  i
Is ˆ1 a consistent estimator of 1 ?
Yi mi
p lim(ˆ1 )  p lim 
where mi  M i  M
2
m j
(  0  1M i  1vi   i )mi
 p lim
m j 2
1
p lim (  0  1M i  1vi   i )mi
n

1
p lim ( x j  v j ) 2
n
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18-40
Measurement Error (cont.)
1
p lim (  0  1 ( X i  vi )  1vi   i )mi
n
p lim(ˆ1 ) 
1
p lim ( x j  v j ) 2
n
1
1
1
p lim  0 mi  1 X i mi  mi i
n
n
n


1 2 1 2 1
p lim  x j  v j  xi vi 
n
n

n
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18-41
Measurement Error (cont.)
1
1 2 1
1
p lim  0 mi  1 xi  xi vi  mi i
n
n
n
n
p lim(ˆ1 ) 
1 2 1 2 1

p lim  x j  v j  xi vi 
n
n
n

Applying the Law of Large Numbers....

1Var ( X i )  1Cov( X i , vi )  Cov( xi  vi ,  i )
Var ( X i )  Var (vi )  Cov( X i , vi )
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18-42
Measurement Error (cont.)
p lim(ˆ1 ) 
1Var ( X i )  1Cov( X i , vi )  Cov( xi  vi ,  i )
Var ( X i )  Var (vi )  Cov( X i , vi )
Var ( X i )
 1
Var ( X i )  Var (vi )
 1
 X2
 X2   v2
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18-43
Measurement Error (cont.)
p lim(ˆ1  1 )  1
 X2
 
2
X
2
v
 1
 v2
 2
1
2
 X v
 v2
Notice that 2
1
2
 X v
The bias will always pull ˆ1 PARTWAY from 1 to 0.
We call this bias ATTENUATION BIAS.
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18-44
Measurement Error (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.
• A small amount of random
measurement noise will not bias the
estimate very much.
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18-45
Measurement Error (cont.)
• Mismeasuring X leads to ATTENUATION
BIAS. The estimated coefficient is biased
towards 0.
• Note: mismeasuring Y does NOT lead
to measurement error bias (though it
does increase the variance of the error
term, thus increasing standard errors).
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18-46
Checking Understanding (cont.)
• Suppose you are advising policy makers on
the effect of a one-time tax rebate on
consumption. Using a cross section of 50,000
households, you regress reported current
consumption against reported current income.
• Your results suggest that the tax rebate in
question will NOT have a large enough
impact on consumption to justify the policy.
(continued on next slide)
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18-47
Checking Understanding (cont.)
• Suppose your results found that the marginal
propensity to consume was too small to justify
the tax rebate.
• A proponent of the measure argues that you
should be regressing consumption against
Friedman’s “permanent income,” not current
income. The resulting measurement error
renders your results irrelevant.
• Assess this argument.
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18-48
Checking Understanding (cont.)
• Answer: If your regression suffers from
measurement error, then the true effect
of the tax rebate on consumption is
larger than what you found.
• The coefficient with attenuation bias is
too small to justify the policy. The larger,
true coefficient might or might not be
large enough to support the tax rebate.
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18-49
Simultaneous Causality (Chapter 13.3)
• Another common source of correlation
between Xi and i is SIMULTANEOUS
CAUSALITY
• This complication is also called
JOINTLY DETERMINED VARIABLES or
ENDOGENEITY
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18-50
Simultaneous Causality (cont.)
• Both X and Y are jointly determined
• The process that generates Y also
generates X at the same time
• Because X and Y are determined
simultaneously, X can adjust in
response to shocks to Y ()
• Thus X will be correlated with 
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18-51
Simultaneous Causality (cont.)
• The classic example of simultaneous
causality in economics is supply
and demand.
• Both prices and quantities adjust until
supply and demand are in equilibrium.
• A shock to demand or supply causes
BOTH prices and quantities to move.
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18-52
Simultaneous Causality (cont.)
• Thus, any attempt to estimate the relationship
between prices and quantities (say, to
estimate a demand elasticity) suffers from
SIMULTANEITY BIAS.
• Econometricians have a frequent interest in
estimating elasticities resulting from such an
equilibrium process. Simultaneity bias is a
MAJOR problem.
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18-53
Simultaneous Causality (cont.)
• For example, consider the market
for wheat.
• The quantity demanded for wheat is
a function of the price consumers pay
and the income of the population:
QiD  0  1Pi D  2 I i  iD
• i indexes separate markets
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18-54
Simultaneous Causality (cont.)
• The quantity of wheat supplied is
a function of the price suppliers
receive and the weather (which affects
crop yields).
Q  0  1Pi  2Wi  
S
i
S
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S
i
18-55
Simultaneous Causality (cont.)
S
D
Q

Q
• In equilibrium,
and Pi  Pi
S
i
D
i
• Let’s focus on the demand equation.
Is PiD correlated with iD ?
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18-56
Simultaneous Causality (cont.)
Q  0  1Pi  2Wi  
S
i
S
S
i
QiD  0  1Pi D  2 I i  iD
• Suppose iD > 0 (there is a positive shock to
demand). This shock makes QiD greater than
usual. In equilibrium, QiD = QiS
• To balance the supply equation, PiS must
increase. Suppliers must be paid a higher
price to supply the greater demanded quantity.
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18-57
Simultaneous Causality (cont.)
QiS  0  1Pi S  2Wi  iS
QiD  0  1Pi D  2 I i  iD
• In equilibrium, PiS = PiD. The consumers
must pay a higher price to enjoy the
higher quantity of wheat they demand
• Thus, a positive shock to iD induces a
higher PiD
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18-58
Simultaneous Causality (cont.)
E(Pi •  )  0
D
D
i
• A positive demand shock increases
the quantity demanded. In order to
increase supply, the price must go up.
The demand shock and the price
are correlated.
• OLS will be inconsistent.
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18-59
Simultaneous Causality (cont.)
• When we have a system of equations
(as with supply and demand), all the
variables that are jointly determined are
called endogenous variables. Price
and quantity are endogenous variables.
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18-60
Simultaneous Causality (cont.)
• Variables that are determined outside
the system of equations are called
exogenous variables. The weather is
an exogenous variable. In partial
equilibrium (such as the supply and
demand for wheat), the population’s
income is also exogenous.
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18-61
Simultaneous Causality (cont.)
• It is arbitrary which endogenous
variables we write on the left-hand side.
• We could write both equations
with either Price or Quantity on the lefthand side.
• For convenience, let us use Price
on the LHS of the Supply equation
and Quantity on the LHS of the
Demand equation.
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18-62
Simultaneous Causality (cont.)
(1) Q  0  1 Pi  2 I i  
D
i
D
D
i
(2) Pi S  0  1QiS   2Wi   iS
  D determines QiD .
i
 QiD  Qi S , so Cov( iD ,QiS )  0
 Q determines Pi
S
i
S
 Pi  Pi , so Cov(Pi ,Q )  0
S
D
D
S
i
 Therefore, Cov(Pi D , iD )  0
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18-63
Simultaneous Causality (cont.)
(1) Q  0  1 Pi  2 I i  
D
i
D
D
i
(2) Pi   0  1Q   2Wi  
S
S
i
  Q  Q  Pi  Pi
D
i
D
i
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S
i
S
S
i
D
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Lagged Dependent Variables
(Chapter 13.3)
• Using lagged dependent variables as
explanators is another potential source of
correlation between an explanator and the
error term.
• For example, you try to predict next period’s
inflation as a function of this period’s inflation.
Inft   Inft 1   t
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18-65
Lagged Dependent Variables (cont.)
• Lagged dependent variables present a
problem in the presence of serial correlation.
• Example: suppose there is first order
serial correlation:
Inft   Inft 1   t
 t   t 1  vt
0 |  | 1
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18-66
Lagged Dependent Variables (cont.)
Inft   Inft 1  t
Substituting in  t   t 1  vt :
Inft   Inft 1  t 1  vt
However,
Inft 1   Inft 2   t 1
 t 1 is a determinant of BOTH t and Yt 1
Cov(Inft 1 , t )  0
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18-67
Lagged Dependent Variables (cont.)
• Including lagged dependent variables as
an explanator does NOT lead to
inconsistency in the absence of firstorder serial correlation.
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18-68
Review
• Instead of assuming fixed X ’s, we can
instead assume 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.”
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18-69
Review (cont.)
• This lecture outlined 4 major causes of
correlation between Xi and i
• One or more of these problems is
at least a plausible concern in
most applications.
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18-70
Review (cont.)
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)
•
Any of these conditions will make OLS
inconsistent (and biased).
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18-71
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).
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18-72
Review (cont.)
Yi   0  1 X 1i   2 X 2i   i
Regress: Yi   0  1 X 1i  i
Cov( X 1 , X 2 )
p lim(ˆ1 )  1   2
Var ( X 1 )
Cov( X 1 , X 2 )
Note:
is the OLS formula
Var ( X 1 )
for  1 in the regression X 2   0   1 X 1  v
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18-73
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
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18-74
Review (cont.)
• Measurement error implies a correlation
between our observed explanator
M i  Xi  vi
and the error term,
i  -1vi   i
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18-75
Review (cont.)
Instead of observing X i , we observe M i  X i  vi
Var ( X i )
 X2
p lim(ˆ1)  1
 1 2
Var ( X i )  Var (vi )
 X   v2
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18-76
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.
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18-77
Review (cont.)
• Under simultaneous causality, Q and P
are jointly determined
• Because Q and P are determined
simultaneously, P can adjust in
response to shocks to Q ( D)
• Thus P will be correlated with  D
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18-78
Simultaneous Causality (cont.)
(1) QiD  0  1 Pi D  2 I i   iD
(2) Pi S   0  1QiS   2Wi   iS
  Q  Q  Pi  Pi
D
i
D
i
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S
i
S
D
18-79
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
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18-80