National open university of nigeria introduction to econometrics II eco 356

INTRODUCTION TO ECONOMETRICS II

ECO 306

NOUN
30 15 15 5.00 16 12 21.63 17 16 12.10 18 12 5.55 19 12 7.50 20 14 8.00








Table 1.2.0(b)Covariance table
Observation S Y
̅
̅
( ̅)( ̅)
1 15 17.24 1.75 3.016 5.277 2
16 15.00 2.75 0.775 2.133 3
8 14.91
-5.25 0.685
-3.599 4
6 4.50
-7.25
-9.725 70.503 5
15 18.00 1.75 3.776 6.607 6
12 6.29
-1.25
-7.935 9.918 7
12 19.23
-1.25 5.006
-6.257 8
18 18.69 4.75 4.466 21.211 9
12 7.21
-1.25
-7.015 8.768 10 20 42.06 6.75 27.836 187.890 11 17 15.38 3.75 1.156 4.333 12 12 12.70
-1.25
-1.525 1.906 13 12 26.00
-1.25 11.776
-14.719

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).

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