INTRODUCTION TO ECONOMETRICS II ECO 306 NOUN 31 14 9
7.50
-1.45
-6.725 28.579 15 15 5.00 1.75
-9.225
-16.143 16 12 21.63
-1.25 7.406
-9.257 17 16 12.10 2.75
-2.125
-5.842 18 12 5.55
-1.25
-8.675 10.843 19 12 7.50
-1.25
-6.725 8.406 20 14 8.00 0.75
-6.225
-4.668
Total 265 284.49
-
-
305.888
Average 13.250 14.225
-
-
15.294 Note from the above example that the association is positive. This is given by the positive covariance.
1.2.3.3 Population Covariance If
X and
Y are random variables, the expected value of the product of their deviations from their means is defined to be the population covariance
:
,(
)(
)-
…[2.20] Where and
are the population means of X and
Y, respectively. As you would expect, if the population covariance is unknown, the sample covariance
will provide an estimate of it, given a sample of observations. However, the estimate will be biased downwards, for
, ( )-
…[2.21] The reason is that the sample deviations are measured from the sample means of
X and
Y and tend to underestimate the deviations from the true means. Therefore, we can construct an unbiased estimator by multiplying
the sample estimate by n/(
n–1).
1.2.3.4 Sample Variance INTRODUCTION TO ECONOMETRICS II ECO 306 NOUN 32
Fora sample of n observations,
, ...,
, the sample variance will be defined as the average squared deviation in the sample
( )
∑
(
̅)
…[2.22]
The sample variance,
thus defined, is a biased estimator of the population variance. The reason for the underestimation is because it is calculated as the average squared deviation from the sample mean rather than the true mean. This is because the sample mean is automatically in the centre of the sample, the deviations from it will tend to be smaller than those from the population mean. Therefore, sample variance as an unbiased estimate of population
variance is given as ∑
(
̅ )
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