Christopher Dougherty

EC220 - Introduction to econometrics (review chapter)Slideshow: asymptotic properties of estimators: the use of simulation

 

 

 

 

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Dougherty, C. (2012) EC220 - Introduction to econometrics (review chapter). [Teaching Resource]

© 2012 The Author

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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.

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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.

ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

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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.

ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

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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.

ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

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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.

ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

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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 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.

ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

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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 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 Y / X as an estimator of .

ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

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Simulations

We then use as an estimator of .

We have already seen that the estimator is consistent.

The question now is how well it performs in finite samples.

00

plim plim plim

plimZi

i

wZw

X

Y

i

i

X

Y

ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

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We will start by taking samples of size 20. The figure shows the distribution of l, the estimator of , for one million samples.

ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

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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.

n = 20mode = 4.65mean = 5.33

ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

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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.

n = 100mode = 4.94mean = 5.05

ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

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n = 1000mode = 4.99mean = 5.01

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.

ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

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n = 1000mode = 4.99mean = 5.01

We demonstrated analytically that the estimator is consistent, but this is a theoretical property relating to samples of infinite size.

ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

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n = 1000mode = 4.99mean = 5.01

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.

ASYMPTOTIC PROPERTIES OF ESTIMATORS: THE USE OF SIMULATION

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