Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: estimators of variance, covariance and correlation Original citation:

Christopher Dougherty

EC220 - Introduction to econometrics (review chapter)Slideshow: estimators of variance, covariance and correlation

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

This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms. http://creativecommons.org/licenses/by-sa/3.0/

http://learningresources.lse.ac.uk/

1

ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION

We have seen that the variance of a random variable X is given by the expression above.

Variance

22)var( XX XEX

2


Given a sample of n observations, the usual estimator of the variance is the sum of the squared deviations around the sample mean divided by n – 1, typically denoted s2

X.

Variance

Estimator .11

1

22

n

iiX XX

ns

22)var( XX XEX

3


Since the variance is the expected value of the squared deviation of X about its mean, it makes intuitive sense to use the average of the sample squared deviations as an estimator. But why divide by n – 1 rather than by n?

Variance

Estimator .11

1

22

n

iiX XX

ns

22)var( XX XEX

4


The reason is that the sample mean is by definition in the middle of the sample, while the unknown population mean is not, except by coincidence.

Variance

Estimator .11

1

22

n

iiX XX

ns

22)var( XX XEX

5


As a consequence, the sum of the squared deviations from the sample mean tends to be slightly smaller than the sum of the squared deviations from the population mean.

Variance

Estimator .11

1

22

n

iiX XX

ns

22)var( XX XEX

6


Hence a simple average of the squared sample deviations is a downwards biased estimator of the variance. However, the bias can be shown to be a factor of (n – 1)/n. Thus one can allow for the bias by dividing the sum of the squared deviations by n – 1 instead of n.

Variance

Estimator .11

1

22

n

iiX XX

ns

22)var( XX XEX

Variance

Estimator

Covariance

Estimator

7


A similar adjustment has to be made when estimating a covariance. For two random variables X and Y an unbiased estimator of the covariance XY is given by the sum of the products of the deviations around the sample means divided by n – 1.

.11

1

22

n

iiX XX

ns

.11

1

n

iiiXY YYXX

ns

YXXY YXEYX ),(cov

22)var( XX XEX

Correlation

8


The population correlation coefficient XY for two variables X and Y is defined to be their covariance divided by the square root of the product of their variances.

22YX

XYXY

9


The sample correlation coefficient, rXY, is obtained from this by replacing the covariance and variances by their estimators.

22YX

XYXY

22

2222

11

11

11

YYXX

YYXX

YYn

XXn

YYXXn

ss

sr

YX

XYXY

Correlation

Estimator

Correlation

Estimator

10


The 1/(n – 1) terms in the numerator and the denominator cancel and one is left with a straightforward expression.

22YX

XYXY

22

2222

11

11

11

YYXX

YYXX

YYn

XXn

YYXXn

ss

sr

YX

XYXY

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.7 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

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Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: estimators of variance, covariance and correlation Original citation: