A Primer on Asymptotics
Eric Zivot
Department of Economics
University of Washington
September 30, 2003
Revised: October 7, 2009
1 Introduction
The two main concepts in asymptotic theory covered in these notes are
•
Consistency
•
Asymptotic Normality
Intuition
•
consistency: as we get more and more data, we eventually know the truth
•
asymptotic normality: as we get more and more data, averages of random
variables behave like normally distributed random variables
1.1 Motivating Example
Let
1
denote an independent and identically distributed (iid ) random sam
ple with
[
]=
and var
(
)=
2
We don’t know the probability density function
(pdf)
(
θ
)
but we know the value of
2
The goal is to estimate the mean value
from the random sample of data. A natural estimate is the sample mean
ˆ
=
1
X
=1
=
¯
Using the iid assumption, straightforward calculation show that
[ˆ
var(ˆ
2
1
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incewedon
’tknow
(
θ
)
we don’t know the pdf of
ˆ
All we know about the pdf
of
is that
[ˆ
]=
and var
(ˆ
)=
2
However, as
→∞
var
2
→
0
and
the pdf of
ˆ
collapses at
Intuitively, as
ˆ
converges in some sense to
In
other words, the estimator
ˆ
is consistent for
Furthermore, consider the standardized random variable
=
ˆ
−
p
var(ˆ
)
=
ˆ
−
q
2
=
√
μ
ˆ
−
¶
For any value of
[
]=0
(
)=1
butwedon’tknowthepdfof
since we
don’t know
(
)
Asymptotic normality says that as
gets large, the pdf of
is
well approximated by the standard normal density. We use the shorthand notation
=
√
μ
ˆ
−
¶
∼
(0
1)
(1)
to represent this approximation. The symbol “
∼
” denotes “asymptotically distrib
uted as”, and represents the asymptotic normality approximation. Dividing both
sides of (1) by
√
and adding
the asymptotic approximation may be rewritten
as
ˆ
=
√
+
∼
μ
2
¶
(2)
The above is interpreted as follows: the pdf of the estimate
ˆ
is asymptotically
distributed as a normal random variable with mean
and variance
2
The quantity
2
iso
ftenre
ferredtoasthe
asymptotic variance
of
ˆ
and is denoted avar
)
The
square root of avar
)
is called the
asymptotic standard error
of
ˆ
and is denoted
ASE(
ˆ
)
With this notation, (2) may be reexpressed as
ˆ
∼
(
avar(ˆ
))
The quantity
2
is sometimes referred to as the asymptotic variance of
√
−
)
The asymptotic normality result (2) is commonly used to constuct a con
f
dence
interval for
For example, an asymptotic 95% con
f
dence interval for
has the form
ˆ
±
1
96
×
p
avar(ˆ
)
This con
f
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 Fall '09
 Zivot
 Economics, Normal Distribution, CLTs, LLNs

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