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Christopher Dougherty
EC220 - Introduction to econometrics
(review chapter)
Slideshow: asymptotic properties of estimators: plims and consistency
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (review chapter). [Teaching Resource]
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ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
The asymptotic properties of estimators are their
properties as the number of observations in a sample
becomes very large and tends to infinity.
We shall be concerned with the concepts of probability
limits and consistency, and the central limit theorem.
These topics are usually given little attention in standard
statistics texts, generally without an explanation of why
they are relevant and useful.
However, asymptotic properties lie at the heart of much
econometric analysis and so for students of
econometrics they are important.
1
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
The asymptotic properties of estimators are their
properties as the number of observations in a sample
becomes very large and tends to infinity.
We shall be concerned with the concepts of probability
limits and consistency, and the central limit theorem.
These topics are usually given little attention in standard
statistics texts, generally without an explanation of why
they are relevant and useful.
However, asymptotic properties lie at the heart of much
econometric analysis and so for students of
econometrics they are important.
2
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Probability limits
lim P X n a 0
n
We will start with an abstract definition of a probability limit and then illustrate it with a simple
example.
3
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Probability limits
lim P X n a 0
n
A sequence of random variables Xn is said to converge in probability to a constant a if,
given any positive , however small, the probability of Xn deviating from a by an amount
greater than tends to zero as n tends to infinity.
4
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Probability limits
lim P X n a 0
n
plim X n a
The constant a is described as the probability limit of the sequence, usually abbreviated as
plim.
5
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
probability density
function of X
n sX
0.08
1
50
0.06
0.04
0.02
n=1
50
100
150
200
We will take as our example the mean of a sample of observations, X, generated from a
random variable X with population mean mX and variance s2X. We will investigate how X
behaves as the sample size n becomes large.
6
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
probability density
function of X
n sX
0.08
1
50
0.06
0.04
0.02
n=1
50
100
150
200
For convenience we shall assume that X has a normal distribution, but this does not affect
the analysis. If X has a normal distribution with mean mX and variance s2X, X will have a
normal distribution with mean mX and variance s2X / n.
7
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
probability density
function of X
n sX
0.08
1
50
0.06
0.04
0.02
n=1
50
100
150
200
For the purposes of this example, we will suppose that X has population mean 100 and
standard deviation 50, as in the diagram.
8
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
probability density
function of X
n sX
0.08
1
50
0.06
0.04
0.02
n=1
50
100
150
200
The sample mean will have the same population mean as X, but its standard deviation will
be 50/ n , where n is the number of observations in the sample.
9
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
probability density
function of X
n sX
0.08
1
50
0.06
0.04
0.02
n=1
50
100
150
200
The larger is the sample, the smaller will be the standard deviation of the sample mean.
10
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
probability density
function of X
n sX
0.08
1
50
0.06
0.04
0.02
n=1
50
100
150
200
If n is equal to 1, the sample consists of a single observation. X is the same as X and its
standard deviation is 50.
11
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
probability density
function of X
n sX
0.08
1
4
50
25
0.06
0.04
n=4
0.02
50
100
150
200
We will see how the shape of the distribution changes as the sample size is increased.
12
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
probability density
function of X
n sX
0.08
1
4
25
50
25
10
0.06
n = 25
0.04
0.02
50
100
150
200
The distribution becomes more concentrated about the population mean.
13
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
probability density
function of X
n sX
n = 100
0.08
1
4
25
100
0.06
50
25
10
5
0.04
0.02
50
100
150
200
To see what happens for n greater than 100, we will have to change the vertical scale.
14
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
probability density
function of X
n sX
0.8
1
4
25
100
0.6
50
25
10
5
0.4
n = 100
0.2
50
100
150
200
We have increased the vertical scale by a factor of 10.
15
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
probability density
function of X
n sX
0.8
1
4
25
100
1000
0.6
n = 1000
50
25
10
5
1.6
0.4
0.2
50
100
150
200
The distribution continues to contract about the population mean.
16
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
probability density
function of X
n sX
n = 5000
0.8
1
4
25
100
1000
5000
0.6
50
25
10
5
1.6
0.7
0.4
0.2
50
100
150
200
In the limit, the variance of the distribution tends to zero. The distribution collapses to a
spike at the true value. The plim of the sample mean is therefore the population mean.
17
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
lim P X m X 0
n
Formally, the probability of X differing from mX by any finite amount, however small, tends to
zero as n becomes large.
18
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
lim P X m X 0
n
plim X m X
Hence we can say plim X = mX.
19
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Consistency
An estimator of a population characteristic is said to be
consistent if it satisfies two conditions:
(1) It possesses a probability limit, and so its
distribution collapses to a spike as the sample size
becomes large, and
(2) The spike is located at the true value of the
population characteristic.
Hence we can say plim X = mX.
20
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
probability density
function of X
n = 5000
0.8
0.6
0.4
0.2
50
100
150
200
The sample mean in our example satisfies both conditions and so it is a consistent
estimator of mX. Most standard estimators in simple applications satisfy the first condition
because their variances tend to zero as the sample size becomes large.
21
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
probability density
function of X
n = 5000
0.8
0.6
0.4
0.2
50
100
150
200
The only issue then is whether the distribution collapses to a spike at the true value of the
population characteristic. A sufficient condition for consistency is that the estimator should
be unbiased and that its variance should tend to zero as n becomes large.
22
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
probability density
function of X
n = 5000
0.8
0.6
0.4
0.2
50
100
150
200
It is easy to see why this is a sufficient condition. If the estimator is unbiased for a finite
sample, it must stay unbiased as the sample size becomes large.
23
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
probability density
function of X
n = 5000
0.8
0.6
0.4
0.2
50
100
150
200
Meanwhile, if the variance of its distribution is decreasing, its distribution must collapse to
a spike. Since the estimator remains unbiased, this spike must be located at the true value.
The sample mean is an example of an estimator that satisfies this sufficient condition.
24
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
probability density
function of Z
n = 20
q
Z
However the condition is only sufficient, not necessary. It is possible that an estimator may
be biased in a finite sample …
25
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
probability density
function of Z
n = 100
n = 20
q
Z
… but the bias becomes smaller as the sample size increases
26
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
n = 100000
probability density
function of Z
n = 1000
n = 100
n = 20
q
Z
… to the point where the bias disappears altogether as the sample size tends to infinity.
Such an estimator is biased for finite samples but nevertheless consistent because its
distribution collapses to a spike at the true value.
27
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Consistency
1 n
Z
Xi .
n 1 i 1
A simple example of an estimator that is biased in finite samples but consistent is shown
above. We are supposing that X is a random variable with unknown population mean mX and
that we wish to estimate mX.
28
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Consistency
1 n
Z
Xi .
n 1 i 1
EZ
n
mX .
n1
The estimator is biased for finite samples because its expected value is nmX/(n + 1). But as
n tends to infinity, n /(n + 1) tends to 1 and the estimator becomes unbiased.
29
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Consistency
1 n
Z
Xi .
n 1 i 1
EZ
var( Z )
n
mX .
n1
n
2
s
n 12 X
The variance of the estimator is given by the expression shown. This tends to zero as n
tends to infinity. Thus Z is consistent because its distribution collapses to a spike at the
true value.
30
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Consistency
In practice we deal with finite samples, not infinite ones. So why
should we be interested in whether an estimator is consistent?
One reason is that sometimes it is impossible to find an estimator
that is unbiased for small samples. If you can find one that is at
least consistent, that may be better than having no estimate at all.
A second reason is that often we are unable to say anything at all
about the expectation of an estimator. The expected value rules are
weak analytical instruments that can be applied in relatively simple
contexts.
In particular, the multiplicative rule E{g(X)h(Y)} = E{g(X)} E{h(Y)}
applies only when X and Y are independent, and in most situations
of interest this will not be the case. By contrast, we have a much
more powerful set of rules for plims.
31
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Consistency
In practice we deal with finite samples, not infinite ones. So why
should we be interested in whether an estimator is consistent?
One reason is that sometimes it is impossible to find an estimator
that is unbiased for small samples. If you can find one that is at
least consistent, that may be better than having no estimate at all.
A second reason is that often we are unable to say anything at all
about the expectation of an estimator. The expected value rules are
weak analytical instruments that can be applied in relatively simple
contexts.
In particular, the multiplicative rule E{g(X)h(Y)} = E{g(X)} E{h(Y)}
applies only when X and Y are independent, and in most situations
of interest this will not be the case. By contrast, we have a much
more powerful set of rules for plims.
32
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Consistency
In practice we deal with finite samples, not infinite ones. So why
should we be interested in whether an estimator is consistent?
One reason is that sometimes it is impossible to find an estimator
that is unbiased for small samples. If you can find one that is at
least consistent, that may be better than having no estimate at all.
A second reason is that often we are unable to say anything at all
about the expectation of an estimator. The expected value rules are
weak analytical instruments that can be applied in relatively simple
contexts.
In particular, the multiplicative rule E{g(X)h(Y)} = E{g(X)} E{h(Y)}
applies only when X and Y are independent, and in most situations
of interest this will not be the case. By contrast, we have a much
more powerful set of rules for plims.
33
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Consistency
In practice we deal with finite samples, not infinite ones. So why
should we be interested in whether an estimator is consistent?
One reason is that sometimes it is impossible to find an estimator
that is unbiased for small samples. If you can find one that is at
least consistent, that may be better than having no estimate at all.
A second reason is that often we are unable to say anything at all
about the expectation of an estimator. The expected value rules are
weak analytical instruments that can be applied in relatively simple
contexts.
In particular, the multiplicative rule E{g(X)h(Y)} = E{g(X)} E{h(Y)}
applies only when X and Y are independent, and in most situations
of interest this will not be the case. By contrast, we have a much
more powerful set of rules for plims.
34
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Plim rules
Plim rule 1
The plim of the sum of several variables is equal to
the sum of their plims. For example, if you have three
random variables X, Y, and Z, each possessing a plim,
plim (X + Y + Z) = plim X + plim Y + plim Z
Plim rule 2
If you multiply a random variable possessing a plim by
a constant, you multiply its plim by the same constant.
If X is a random variable and b is a constant,
plim bX = b plim X
Plim rule 3
The plim of a constant is that constant. For example,
if b is a constant,
plim b = b
35
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Plim rules
Plim rule 1
The plim of the sum of several variables is equal to
the sum of their plims. For example, if you have three
random variables X, Y, and Z, each possessing a plim,
plim (X + Y + Z) = plim X + plim Y + plim Z
Plim rule 2
If you multiply a random variable possessing a plim by
a constant, you multiply its plim by the same constant.
If X is a random variable and b is a constant,
plim bX = b plim X
Plim rule 3
The plim of a constant is that constant. For example,
if b is a constant,
plim b = b
36
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Plim rules
Plim rule 1
The plim of the sum of several variables is equal to
the sum of their plims. For example, if you have three
random variables X, Y, and Z, each possessing a plim,
plim (X + Y + Z) = plim X + plim Y + plim Z
Plim rule 2
If you multiply a random variable possessing a plim by
a constant, you multiply its plim by the same constant.
If X is a random variable and b is a constant,
plim bX = b plim X
Plim rule 3
The plim of a constant is that constant. For example,
if b is a constant,
plim b = b
37
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Plim rules
Plim rule 4
The plim of a product is the product of the plims, if
they exist. For example, if Z = XY, and if X and Y both
possess plims,
plim Z = (plim X)(plim Y)
Plim rule 5
The plim of a quotient is the quotient of the plims, if
they exist. For example, if Z = X/Y, and if X and Y both
possess plims, and plim Y is not equal to zero,
plim X
plim Z =
plim Y
38
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Plim rules
Plim rule 4
The plim of a product is the product of the plims, if
they exist. For example, if Z = XY, and if X and Y both
possess plims,
plim Z = (plim X)(plim Y)
Plim rule 5
The plim of a quotient is the quotient of the plims, if
they exist. For example, if Z = X/Y, and if X and Y both
possess plims, and plim Y is not equal to zero,
plim X
plim Z =
plim Y
39
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Plim rules
Plim rule 6
The plim of a function of a variable is equal to the
function of the plim of the variable, provided that the
variable possesses a plim and provided that the
function is continuous at that point.
plim f(X) = f(plim X)
40
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis
Y Z
To illustrate how the plim rules can lead us to conclusions when the expected value rules
do not, consider this example. Suppose that you know that a variable Y is a constant
multiple of another variable Z
41
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis
Y Z
Z is generated randomly from a fixed distribution with population mean mZ and variance s2Z.
is unknown and we wish to estimate it. We have a sample of n observations.
42
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis
Y Z
X Zw
Y is measured accurately but Z is measured with random error w with population mean zero
and constant variance s2w. Thus in the sample we have observations on X, where X = Z + w,
rather than Z.
43
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis
Y Z
X Zw
Y
X
i
Z
Z
Z w Z w
w
w
Zw
Z w
i
i
i
i
i
i
i
i
i
i
One estimator of l (not necessarily the best) is SYi / SXi
44
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis
Y Z
X Zw
Y
X
i
Z
Z
Z w Z w
w
w
Zw
Z w
i
i
i
i
i
i
i
i
i
i
Substituting from the first two equations, the estimator can be rewritten as shown.
45
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis
Y Z
X Zw
Y
X
i
Z
Z
Z w Z w
w
w
Zw
Z w
i
i
i
i
i
i
i
i
i
i
The expression can be simplified as shown. Hence we have decomposed the estimator into
the true value, , and an error term. To investigate whether the estimator is biased or
unbiased, we need to take the expectation of the error term.
46
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis
Y Z
X Zw
Y
X
i
Z
Z
Z w Z w
w
w
Zw
Z w
i
i
i
i
i
i
i
i
i
i
But we cannot do this. The random quantity appears in both the numerator and the
denominator and the expected value rules are too weak to allow us to investigate the
expectation analytically.
47
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis
Y Z
X Zw
Y
X
i
Z
Z
Z w Z w
w
w
Zw
Z w
i
i
i
i
i
i
i
i
i
i
However, we know that a sample mean tends to a population mean as the sample size tends
to infinity, and so plim w = 0 and plim Z = mZ.
48
ASYMPTOTIC PROPERTIES OF ESTIMATORS: PLIMS AND CONSISTENCY
Example use of asymptotic analysis
Y Z
X Zw
Yi
plim w
0
plim
plim Z plim w
mZ 0
X i
Since the plims of the numerator and the denominator of the error term both exist, we are
able to take the plim of the error term. Thus we are able to show that the estimator is
consistent, despite the fact that we cannot say anything about its finite sample properties.
49
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section R.14 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
