Lecture 17:
Asymptotics
(Chapter 12.2–12.5)
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Agenda
• Review of Consistency and Probability
Limits (Chapter 12.2)
• Consistency of OLS (Chapter 12.3)
• Replacing Fixed X ’s with Stochastic X ’s
(Chapter 12.4)
• Asymptotic Efficiency (Chapter 12.5)
• Review of Chapter 12.1–12.5
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17-2
Consistency (Chapter 12.2)
• We would love to be guaranteed that
our estimator will be exactly right, or at
least very, very close to exactly right.
• Certainty is too much to ask for in a
stochastic world.
• However, if our estimator is consistent,
we can be “almost always almost right”
in very, very large samples.
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17-3
Consistency (cont.)
• Consistency: as the sample size
approaches infinity, the estimator falls
arbitrarily close to the true value with
probability approaching 1.
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17-4
Consistency (cont.)
• An estimator is consistent if, provided we get
the sample size high enough, the probability
can be as close to 1 as we like that our
estimate is as close to being right as we like.
• With finite sample sizes, there is always some
probability that a consistent estimator is very
wrong. But as the sample size grows, that
probability becomes very, very small.
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17-5
Figure 12.3 The Collapse of a Consistent
Estimator’s Distribution as n Grows
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17-6
Consistency
• What does it take to get consistency?
• One often-encountered way is to have an
asymptotically unbiased estimator whose
variance shrinks to zero as the sample
size grows.
• In large samples, the estimator on average
is right.
• In large samples, no single estimate is likely
to be very far from the estimator’s average.
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17-7
Probability Limits (Chapter 12.2)
• How can we determine if an estimator is
consistent or not?
• We need a new mathematical tool, the
probability limit (or plim).
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17-8
Probability Limits (cont.)
• A random variable b converges in
probability to a constant value c if, as
the sample size grows very large, the
probability approaches 1 that b takes on
a value very close to c.
• We call c the probability limit of b.
• plim(b ) = c
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17-9
Probability Limits (cont.)
An estimator of b is consistent if
p lim( bˆ )  b
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17-10
Algebra of Probability Limits
Let k be a constant.  , b are random variables.
1. p lim(k )  k
2. p lim(  b )  p lim( )  p lim( b )
3. p lim(b )  p lim( ) p lim( b )
 b  p lim( b )
4. p lim   
if p lim( )  0
   p lim( )
5. p lim( g ( b ))  g ( p lim( b ))
for continuous function g ( ).
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17-11
Checking Understanding


2 
1 2 
Let  n ~ N  3,  . Let b n ~ N  2  , 2 
n 
n n 


i. What is p lim  n  b n  ?
 n 
ii. What is p lim   ?
 bn 
iii. What is p lim  b n 2  ?
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17-12
Checking Understanding (cont.)


2 
1 2 
Let  n ~ N  3,  . Let b n ~ N  2  , 2 
n 
n n 


i. What is p lim  n  b n  ?
E ( n )  3. Because  n's variance shrinks to 0
as n  , p lim( n )  3.
1
E ( b n )  2  . As n  , E ( b n ) approaches 2.
n
Because b n's variance shrinks to 0 as n  ,
p lim( b n )  2.
p lim( n  b n )  p lim( n )  p lim( b n )  3  2  1.
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17-13
Checking Understanding (cont.)


2 
1 2 
Let  n ~ N  3,  . Let b n ~ N  2  , 2 
n 
n n 


 n 
ii. What is p lim   ?
 bn 
  n  p lim( n )
3
p lim   

2
 b n  p lim( b n )
iii. What is p lim  b n 2  ?
X 2 is a continuous function, so
p lim( b n 2 )  ( p lim( b n )) 2  2 2  4
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17-14
Consistency of OLS
• Is OLS consistent under Gauss–Markov?
Yi  b 0  b1 X i   i
E ( i )  0
Cov( i ,  j )  0 if i  j
Var ( i )   2
X 's fixed across samples
We know OLS is BLUE.
xi yi
ˆ
b1 
xi 2
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17-15
Consistency of OLS (cont.)
 xiYi 
 xi ( b 0  b1 X i   i ) 
ˆ
p lim( b1 )  p lim 
 p lim 

2 
2

x

x
i
 i 



b 0 p lim xi  b1 p lim xi2  p lim xi i
p lim xi2
1


x

n i i
 b1  p lim 
1 2 
 xi 
 n

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17-16
Consistency of OLS (cont.)
1

xi i 

p lim( bˆ1 )  b1  p lim  n
1 2 
 xi 
 n

1

p lim  xi i 
n


 b1 
1 2
p lim  xi 
n

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17-17
Consistency of OLS (cont.)
• We now need to add an assumption to the
Gauss–Markov DGP, to guarantee that
1 2
p lim  xi   Q for some Q  0, Q  
n

• We need to assume that, as n grows large,
1 2
xi  Q, a finite, non-zero constant.
n
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17-18
Consistency of OLS (cont.)
1

1

p lim  xi i 
p lim  xi i 
n
n




ˆ
Then p lim( b1 )  b1 
 b1 
Q
1

p lim  xi 2 
n

To show that OLS is consistent, we need to show that
1

p lim  xi i   0
n

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17-19
Consistency of OLS (cont.)
We need to show that
1

p lim  xi i   0
n

We will use the strategy of showing that its expectation
is 0 and that its variance declines to 0 as n increases.
1
 1
E  xi i   E(xi i )
n
 n
1
 xi E( i )  0
n
where we have used the assumption that X's are fixed
across samples (so we can treat xi as a constant).
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17-20
Consistency of OLS (cont.)
We need to show that
1

Var  xi i   0 as n grows large
n

1
 1
1
Var  xi i   2 Var(xi i )  2 Var(xi i )  Cov 's
n
n
 n
2
1

1 2
2
 2 xi Var( i ) 
xi
n n
n
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17-21
Consistency of OLS (cont.)
We need to show that
1

Var  xi i   0 as n grows large
n

1
 2 1 2
Var  xi i  
xi
n
 n n
We have already assumed that
1 2
xi  Q as n grows large.
n
We also require that  2  .
1
  2Q
As n grows large, Var  xi i  
0
n
n

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17-22
Consistency of OLS (cont.)
1

1

Because E  xi i   0 and Var  xi i   0
n

n

1

as n grows large, p lim  xi i   0.
n

0
ˆ
Thus p lim( b )  b   b
Q
bˆ is consistent.
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17-23
Consistency of OLS (cont.)
• When determining the assumptions
needed for a consistent estimator,
notice that we need two new types
of assumptions.
• First, we typically need to assume
that  2 is finite, in order to show that
the variance of a term goes to 0 as
n increases.
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17-24
Consistency of OLS (cont.)
• Second, we typically need to assume
something about what happens to the
explanators as the sample size
approaches infinity.
1 2
p lim  xi   Q, a non-zero constant.
n

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17-25
Stochastic X ’s (Chapter 12.4)
• It is crucial to pin down the assumptions
needed to achieve a consistent
estimator when the X ’s are not fixed,
because the expectations needed to
show unbiasedness will generally
be intractable.
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17-26
Stochastic X ’s (cont.)
 xiYi 
 xi i 
ˆ
E ( b1 )  E 
 b1  E 
2 
2 
 xi 
 xi 
When the X 's are fixed, we can easily pull the weights
to the other side of the expectations operator:
 xi i 
xi
b1  E 
 b1  2 E ( i )
2 
xi
 xi 
When the X 's are stochastic, this step becomes
 xi i 
impossible, and we cannot easily eliminate E 
.
2 
 xi 
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17-27
Stochastic X ’s (cont.)
When working with consistency, we can take
advantage of the properties of probability limits:
 xi i 
 xiYi 
ˆ
 b1  p lim 
p lim( b1 )  p lim 
2 
2 
 xi 
 xi 
Because the p lim of the ratio is the ratio of the p lim ,
 xi i 
p lim xi i
 b1 
b1  p lim 
2
2 
x

lim
p
x

i
 i 
Then, if we multiply both the numerator and denominator
1
by , we can work with each component separately.
n
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17-28
Stochastic X ’s (cont.)
• What key properties do we rely on for
OLS to be consistent?
1

p lim  xi i   0
n

1 2
p lim  xi   Q, a non-zero constant
n

• We have simply added the second
property to our list of assumptions.
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17-29
Stochastic X ’s (cont.)
1

p lim  xi i   0
n

• For the Gauss–Markov DGP, we established this
property using:
– The assumption that the X ’s are fixed, to establish
the expectation was zero, and
– Our usual assumptions about the variance and
covariance of  and our new assumptions that  2
is finite and xi2 approaches some finite non-zero
limit, to show that the variance approaches 0 as
n increases
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17-30
Stochastic X ’s (cont.)
1

p lim  xi i   0
n

• When we relax the assumption that
X ’s are fixed across samples, the key
to achieving consistency will be finding
some alternative assumption that makes
this plim 0.
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17-31
Stochastic X ’s (cont.)
• Econometricians generally prefer
to base a DGP on statements
about expectations.
E(xi i )  0
plus some assumption such that
1

Var  xi i 
n

approaches 0 as n increases.
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17-32
Stochastic X ’s (cont.)
• Under the Gauss–Markov conditions,
E(xii ) = 0 follows trivially from fixing the
X ’s across samples.
• When we relax that assumption, we
will have a much harder time achieving
this property.
• The next lecture examines several
routine econometric difficulties that
violate this key property.
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17-33
Stochastic X ’s (cont.)
• So far, we have shown that OLS is a
consistent estimator when we have a
single explanator.
• What if we have multiple explanators?
• The same conditions apply, but for
each explanator.
• Also, the X ’s must not be multicollinear
in the probability limit.
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17-34
Stochastic X ’s (cont.)
• Conditions for OLS to be consistent with
multiple explanators:
1

p lim  x Ri i   0 for all R
n

1 2 
p lim  x Ri   Q, a non-zero constant, for all R
n 
X's are not multicollinear as n grows large
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17-35
Asymptotic Efficiency
• Earlier in the course, when we focused
on cross-sample properties, we looked
for the BLUE Estimator: the unbiased
linear estimator that was “best” (i.e. with
the smallest cross-sample variance).
• Now we’re looking for consistent
estimators; is there a comparable
concept of “best”?
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17-36
Asymptotic Efficiency (cont.)
• Efficiency: for a fixed sample size,
the estimator has the smallest crosssample variance.
• Asymptotic Efficiency: the consistent
estimator has the smallest cross-sample
variance for all very large sample
sizes (for all n > N, where N is some
finite number).
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17-37
Asymptotic Efficiency (cont.)
• Problem: nearly all consistent
estimators have variances that collapse
to 0 in the limit.
• We cannot define asymptotic efficiency
as having the smallest variance as n
approaches infinity, because by the
time n is approaching infinity all the
variances have gotten vanishingly close
to 0.
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17-38
Asymptotic Efficiency (cont.)
• Instead of comparing variances as
consistent estimators approach infinity,
we compare the speed at which the
variances collapse towards 0.
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17-39
Asymptotic Efficiency (cont.)
Suppose we have two estimators, ˆ and bˆ.
E (ˆ )  E ( bˆ )  
Var (ˆ ) 
2
and Var ( bˆ ) 
2
2
.
n
n
As n  , both estimators collapse around  .
However, the distribution of bˆ collapses
much more quickly.
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17-40
Asymptotic Efficiency (cont.)
Var (ˆ ) 
2
and Var ( bˆ ) 
2
2
.
n
n
The variance of bˆ shrinks to 0 more quickly than ˆ .
At n  10, Var (ˆ ) 
2
10
At n  100, Var (ˆ ) 
while Var ( bˆ ) 
2
100
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2
100
while Var ( bˆ ) 
.
2
10, 000
.
17-41
Asymptotic Efficiency (cont.)
Var (ˆ ) 
2
and Var ( bˆ ) 
2
2
.
n
n
As n  , both collapse around their expectation,  .
As n gets very large, we cannot easily distinguish
between ˆ and bˆ. The differences between them
eventually get too small for us to conceptualize.
We need a "magnifying glass" to tell them apart.
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17-42
Asymptotic Efficiency (cont.)
Var (ˆ ) 
2
and Var ( bˆ ) 
2
2
.
n
n
As n  , both collapse around their  .
We need a "magnifying glass" to tell them apart.
What if we compared instead n ˆ and n bˆ ?
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17-43
Asymptotic Efficiency (cont.)


ˆ
Var (ˆ ) 
and Var ( b )  2 .
n
n
What if we compared instead n ˆ and n bˆ ?
Problem: as n  , E (n ˆ )  n E (ˆ )  n .
The expectation of n ˆ and n bˆ is growing
2
2
without bound as n approaches infinity!
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17-44
Asymptotic Efficiency (cont.)
What if we compared instead n ˆ and n bˆ ?
Problem: as n  , E (n ˆ )  n E (ˆ )  n .
Solution: compare the ERRORS of the estimators,
which remain centered around 0.
E[n (ˆ -  )]  E[n ( bˆ -  )]  n - n  0
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17-45
Asymptotic Efficiency (cont.)
Compare the ERRORS of the estimators,
which remain centered around 0.
Var (n (ˆ   ))  n 2 Var (ˆ   )
 n Var (ˆ )  n
2
2

2
 n 2
n
As n  , the distribution of n (ˆ   )
explodes. The variance increases
without bound.
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17-46
Asymptotic Efficiency (cont.)
Var (n ( bˆ   ))  n 2 Var ( bˆ   )
2 
2
ˆ
 n Var ( b )  n 2  
n
As n  , the variance of n ( bˆ   )
2
2
converges to  .
2
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17-47
Asymptotic Efficiency (cont.)
As n  , the variance of n ( bˆ   )
converges to  2 .
While the variance of bˆ is shrinking as
n increases, we have magnified bˆ by
just enough to "keep pace."
n ( bˆ   ) has a stable asymptotic distribution
with mean 0 and variance  2 .
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17-48
Asymptotic Efficiency (cont.)
Var (n (ˆ -  ))  n 2 and Var (n ( bˆ -  ))   2 .
As n  , the distribution of n (ˆ -  ) explodes
while the distribution of n ( bˆ -  ) converges.
A factor of n is "fast enough" to "keep pace"
with bˆ -  , but it's too fast to keep pace
with ˆ -  . We need a smaller magnifying
glass to look at ˆ -  .
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17-49
Asymptotic Efficiency (cont.)
Magnifying ˆ -  by n is too much
to reach a stable asymptotic distribution.
What about
n?
E[ n (ˆ -  )]  E[ n ( bˆ -  )] 
n -
n  0
Var ( n (ˆ   ))  n Var (ˆ   )
 n Var (ˆ )  n
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
2
n
2
17-50
Asymptotic Efficiency (cont.)
As n  , the variance of
n (ˆ   )
converges to  . The asymptotic
2
distribution of
n (ˆ   ) has
expectation 0 and variance  .
2
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17-51
Asymptotic Efficiency (cont.)
Var ( n ( bˆ   ))  n Var ( bˆ   )


ˆ
 n Var ( b )  n 2 
n
n
As n  , the distribution of n ( bˆ   )
collapses around 0.
2
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2
17-52
Asymptotic Efficiency (cont.)
• In general, if we “blow up” a variable
whose distribution is collapsing around 0
by some power of n, we can stabilize the
process at some asymptotic distribution.
• We have to pick just the right power of n.
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17-53
Asymptotic Efficiency (cont.)
We call the power of n that stabilizes
the distribution the "convergence rate."
The convergence rate of (ˆ   ) is root n.
The convergence rate of ( bˆ   ) is n.
bˆ is clearly more efficient than ˆ.
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17-54
Asymptotic Efficiency (cont.)
Example: let's consider the variance of OLS
under the Gauss–Markov assumptions when
Yi  b 0  b1 X i   i
Var ( bˆ1 ) 
2
xi 2
 2 
2
p lim(Var )  p lim  2  
2

x
p
lim(

x
i )
 i 
1 2
2
We can re-write xi as n xi .
n
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17-55
Asymptotic Efficiency (cont.)
1 2
We can re-write xi as n xi .
n
(Multiplying and dividing by n is a standard
2
trick when working with probability limits.)
1 2
xi is the variance of X. If, as we keep
n
increasing the sample size, the variance of X
converges to some constant (a reasonable thought,
1

in many applications), then p lim  xi   Q.
n

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17-56
Asymptotic Efficiency (cont.)

ˆ
Var ( b1 ) 
2
xi
2
 2 
2
p lim(Var )  p lim  2  
2
 xi  p lim(xi )

2
 1 2
p lim  n xi 
 n

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
2
p lim(n) Q
0
17-57
Asymptotic Efficiency (cont.)
Now, instead of looking at Var ( bˆ1 ),
let's look at n Var ( bˆ ).
1
2
2
n

 
ˆ
n p lim(Var ( b 1 ))  n
 p lim  
p lim(n) Q
n Q
If we blow up the variance by n, then
the variance stabilizes. We can move the n
inside the variance: p lim(Var ( n b1 ) 
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2
Q
.
17-58
Asymptotic Efficiency (cont.)

ˆ
p lim(Var ( n b1 )) 
Q
However, E ( n bˆ ) grows
2
1
without bound.
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17-59
Asymptotic Efficiency (cont.)
Look at the distribution of the ERRORS.
p lim( E ( n ( bˆ1  b1 ))  0.
p lim(Var ( n ( bˆ1  b1 ))) 
2
Q
.
The convergence rate of bˆ1-b1 is root n.
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17-60
Asymptotic Efficiency (cont.)
• The convergence rate of the OLS errors
is root n.
• We say that OLS is root n consistent.
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17-61
Asymptotic Efficiency (cont.)
• The variance of an estimator’s magnified
errors is called its asymptotic variance.
• The asymptotic variance of OLS is

2
Q
Q  lim n xi 2
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17-62
Asymptotic Efficiency (cont.)
• Among estimators with the same
convergence rate, the consistent
estimator with the smallest asymptotic
variance is called asymptotically
efficient among estimators with that
convergence rate.
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17-63
Checking Understanding
ˆ1 ~ N ( ,
2
n x
3
2
i
)
2
ˆ2 ~ N ( ,
 1
2 2
n 1   (xi )
 n
where xi2  Q, with 1  Q  
3
What is the convergence rate of (ˆ1   )?
What is the converge rate of (ˆ   )?
2
Which estimator is more efficient?
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17-64
Checking Understanding (cont.)

ˆ
Var (1   )  3 2
n xi
2

ˆ
n Var (1   )  2
xi
2
3
3
2
Var (n (ˆ1   )) 
2
x
2
i

2
Q
as n  
3
2
The convergence rate of (ˆ1   ) is n .
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17-65
Checking Understanding (cont.)
Var (ˆ2   ) 
2
 1
n 1   (xi2 ) 2
 n
3
n3Var (ˆ2   ) 
3
2
2
 1
2 2
1

(

x

 i)
 n
Var (n (ˆ2   )) 
2
Q
2
as n grows large.
3
2
The convergence rate of (ˆ2   ) is n .
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17-66
Checking Understanding (cont.)
3
2
Var (n (ˆ1   )) 
3
2
Var (n (ˆ2   )) 
2
Q
2
Q2
If one estimator converged more quickly than
the other, then it would be more efficient.
Both estimators converge at the same rate.
Because Q  1, ˆ   has a smaller asymptotic
2
variance than ˆ1   .
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17-67
Asymptotic Efficiency
• Efficiency is a subtler concept when
working at the limit as n approaches
infinity.
• The distribution of a consistent estimator
tends to collapse; the variance shrinks
to 0.
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17-68
Asymptotic Efficiency (cont.)
• We cannot usefully compare the
distributions of different estimators or
their errors as n approaches infinity.
• Every consistent estimator’s distribution
has the same asymptotic “destination.”
• We CAN compare how quickly they
approach that destination.
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17-69
Asymptotic Efficiency (cont.)
• Because the distributions of consistent
estimators tend to get very close
together as n grows, we need to
magnify the differences so we can see
them better.
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17-70
Asymptotic Efficiency (cont.)
• We examine the distributions of the
errors magnified by their convergence
rate, as n approaches infinity.
• We can work with this asymptotic
distribution.
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17-71
Asymptotic Efficiency (cont.)
• As n approaches infinity, the errors (as blown
up by the appropriate magnifier) converge to
some distribution.
• If the Central Limit Theorem holds, the
asymptotic distribution of the errors
is Normal.
• The most often violated requirement of the
CLT is that variances are bounded.
• See Appendix 12.4.
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17-72
Figure 12.4 The Distributions of Errors for bg1,
bg2, bg3, and bg4 for Various n and Normal and
Skewed, Discrete Disturbances (Panel A)
17-73
Figure 12.4 The Distributions of Errors for bg1,
bg2, bg3, and bg4 for Various n and Normal and
Skewed, Discrete Disturbances (Panel B)
17-74
Figure 12.4
The Distributions
of Errors for bg1,
bg2, bg3, and bg4
for Various n and
Normal and
Skewed, Discrete
Disturbances
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Asymptotic Efficiency
• In moderate sized samples (say,
n > 100), we can use the asymptotic
distribution for inference when the
finite-sample distribution is unknown.
• When OLS is consistent and
asymptotically normally distributed,
we can use t and F tests.
• When OLS is inconsistent, tests
are biased.
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17-76
Review
• We have a new strategy for estimation.
• When our sample size is reasonably large,
we can look not for unbiased estimators but
for consistent estimators.
• When we relax the assumption that X ’s are
fixed across samples, it will be much easier to
work with probability limits (for consistency)
than with expectations (for unbiasedness).
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17-77
Review (cont.)
• As the sample size grows very large,
consistent estimators have a very high
probability of giving estimates very close
to the correct value.
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17-78
Review (cont.)
• One method for demonstrating that an
estimator is consistent:
– Demonstrate that the estimator is
asymptotically unbiased
– Demonstrate that the estimator’s variance
converges to 0 as n approaches infinity
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17-79
Review (cont.)
• To show that an estimator is consistent,
we require assumptions that bound
key variances.
• We also require that error terms and
explanators be contemporaneously
uncorrelated, i.e. E(xii ) = 0.
• This assumption is weaker than requiring
the X ’s to be fixed across samples.
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17-80
Review (cont.)
• To evaluate the efficiency of an estimator and
draw inferences (i.e. run hypothesis tests),
we need to be able to describe an estimator’s
distribution as n approaches infinity.
• However, the distribution collapses as
n grows.
• The answer: magnify the errors by just the
right amount (the convergence rate) such that
the distribution converges.
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17-81
Review (cont.)
• When the Central Limit Theorem applies
(as occurs frequently), the asymptotic
distribution is Normal.
• With asymptotic normality, we can
use the same hypothesis tests we
developed for unbiased estimators
to draw inferences from consistent
estimators.
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17-82
Review (cont.)
• We will be working with probability limits
as we develop consistent estimators.
• You need to be familiar with the algebra
of plims.
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17-83
Algebra of Probability Limits
Let k be a constant.  , b are random variables.
1. p lim(k )  k
2. p lim(  b )  p lim( )  p lim( b )
3. p lim(b )  p lim( ) p lim( b )
 b  p lim( b )
4. p lim   
if p lim( )  0
   p lim( )
5. p lim( g ( b ))  g ( p lim( b ))
for continuous function g ( ).
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17-84