Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: estimators of variance, covariance and correlation Original citation:
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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]
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/
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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
ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION
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
ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION
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
ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION
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
ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION
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
ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION
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
ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION
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
ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION
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
ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION
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
ESTIMATORS OF VARIANCE, COVARIANCE, AND CORRELATION
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