Transcript Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: asymptotic properties of estimators: the use of simulation Original citation: Dougherty, C.

Christopher Dougherty
EC220 - Introduction to econometrics
(review chapter)
Slideshow: asymptotic properties of estimators: the use of simulation
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (review chapter). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/141/
Available in LSE Learning Resources Online: May 2012
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ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION
Consistency
Why are we interested in consistency, when in practice we have
finite samples?
As a first approximation, the answer is that if we can show that an
estimator is consistent, then we may be optimistic about its finite
sample properties, whereas is the estimator is inconsistent, we know
that for finite samples it will definitely be biased.
1
ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION
Consistency
Why are we interested in consistency, when in practice we have
finite samples?
As a first approximation, the answer is that if we can show that an
estimator is consistent, then we may be optimistic about its finite
sample properties, whereas is the estimator is inconsistent, we know
that for finite samples it will definitely be biased.
2
ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION
Consistency
However, there are reasons for being cautious about preferring
consistent estimators to inconsistent ones.
First, a consistent estimator may be biased for finite samples.
Second, we are usually also interested in variances. If a consistent
estimator has a larger variance than an inconsistent one, the latter
might be preferable if judged by the mean square error or similar
criterion that allows a trade-off between bias and variance.
How can you resolve these issues? Mathematically they are
intractable, otherwise we would not have resorted to large sample
analysis in the first place.
3
ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION
Consistency
However, there are reasons for being cautious about preferring
consistent estimators to inconsistent ones.
First, a consistent estimator may be biased for finite samples.
Second, we are usually also interested in variances. If a consistent
estimator has a larger variance than an inconsistent one, the latter
might be preferable if judged by the mean square error or similar
criterion that allows a trade-off between bias and variance.
How can you resolve these issues? Mathematically they are
intractable, otherwise we would not have resorted to large sample
analysis in the first place.
4
ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION
Consistency
However, there are reasons for being cautious about preferring
consistent estimators to inconsistent ones.
First, a consistent estimator may be biased for finite samples.
Second, we are usually also interested in variances. If a consistent
estimator has a larger variance than an inconsistent one, the latter
might be preferable if judged by the mean square error or similar
criterion that allows a trade-off between bias and variance.
How can you resolve these issues? Mathematically they are
intractable, otherwise we would not have resorted to large sample
analysis in the first place.
5
ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION
Simulations
The answer is to conduct a simulation experiment, directly
investigating the distributions of estimators under controlled
conditions.
We will do this for the example in the previous subsection. We will
generate Z as a random variable with a normal distribution with
mean 1 and variance 0.25.
We will set l equal to 5, so the value of Y in any observation is 5
times the value of Z: Y = 5Z.
We will generate the measurement error as a normally distributed
random variable with zero mean and unit variance. The value of X in
any observation is equal to the value of Z plus this measurement
error: X = Z + w.
6
ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION
Simulations
The answer is to conduct a simulation experiment, directly
investigating the distributions of estimators under controlled
conditions.
We will do this for the example in the previous subsection. We will
generate Z as a random variable with a normal distribution with
mean 1 and variance 0.25.
We will set l equal to 5, so the value of Y in any observation is 5
times the value of Z: Y = 5Z.
We will generate the measurement error as a normally distributed
random variable with zero mean and unit variance. The value of X in
any observation is equal to the value of Z plus this measurement
error: X = Z + w.
We then use SY / SX as an estimator of l.
7
ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION
Simulations
We then use
Y
X
i
as an estimator of l.
i
We have already seen that the estimator is consistent.
  Yi 
plim w
0
plim 
l
l
  l l
plim Z  plim w
Z  0
 X i 
The question now is how well it performs in finite samples.
8
ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
6
7
8
9
We will start by taking samples of size 20. The figure shows the distribution of l, the
estimator of l, for one million samples.
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ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION
2.5
n = 20
mode = 4.65
mean = 5.33
2
1.5
1
0.5
0
0
1
2
3
4
5
6
7
8
9
Although the mode of the distribution, 4.65, Is lower than the true value, the estimator is
actually upwards biased, the mean estimate in the million samples being 5.33.
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ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION
2.5
n = 100
mode = 4.94
mean = 5.05
2
1.5
1
0.5
0
0
1
2
3
4
5
6
7
8
9
If we increase the sample size to 100, the distribution of the estimates obtained with one
million samples is less skewed. The mode is 4.94 and the mean is 5.05.
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ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION
2.5
n = 1000
mode = 4.99
mean = 5.01
2
1.5
1
0.5
0
0
1
2
3
4
5
6
7
8
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If we increase the sample size to 1000, the distribution of the estimates obtained with one
million samples is less skewed. The mode is 4.99 and the mean is 5.01.
12
ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION
2.5
n = 1000
mode = 4.99
mean = 5.01
2
1.5
1
0.5
0
0
1
2
3
4
5
6
7
8
9
We demonstrated analytically that the estimator is consistent, but this is a theoretical
property relating to samples of infinite size.
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ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION
2.5
n = 1000
mode = 4.99
mean = 5.01
2
1.5
1
0.5
0
0
1
2
3
4
5
6
7
8
9
The simulation shows us that for sample size 1000, the estimator is almost unbiased.
However for smaller sample sizes the estimator is biased, especially when the sample size
is as small as 20.
14
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