Page 140 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes In the above illustration the numbers were chosen so that for an increase of 1 unit in the X series there
is an increase of 2 units in the Y series. Thus the correlation is perfect and r equals +1. If the Y series
had been 14, 12, 10, 8, 6 (the X series remaining the same) the value of r would have been —1. Thus —
1 stands for perfect negative correlation, an increase in one series corresponding to a decrease in the
other. It should also be noted in this connection that the coefficient of correlation (r) cannot be less
than —1 nor more than + l.
The above illustration suggests the question, “Will a linear relationship between X and Y always give
perfect correlation ?”
Assume the linear relationship
Y= aX + b

Since y = Y – M and x = X – M 2
2
M + y = a( +Mx 1 ) + b or y = ax
2
(since –M – Mba 1 2 = 0)

∑
∑xy ∑ax 2 ax 2
and r = = = = ± 1
2 2
∑ 2. ∑x y 2 ∑ 2. ∑x a x 2 ( a 2 ) ∑x 2
(The sign of r depends upon the sign of a.)
Therefore a linear relationship between two variables will give a correlation coefficient of +1 or —1
depending upon whether large values of one occur with large values of the other or large values of
one occur with small values of the other.
The converse of the above proposition is likewise true, i.e., if the coefficient of correlation (r) equals 1
then the relationship between the X and Y series is linear.
Assume r = l
then ( )∑ 2 ∑xy x 2. ∑– y 2 = 0

x
x
x
Letting 1 = λ y , 2 = λ y . . . n = λ y the above expression becomes
nn
22
11
2
2
yy 2 2 ( 1 1 – λ 2 )λ 2 +y 2 3 2 ( 1 1 λ – 3 )λ y 2 + . . . + yy s 2 ( r r λ s )λ – 2 + . . . = 0
The only way in which this expression can equal zero is by having
λ 1 = λ = λ = . . . = λ n
2
3
and it follows that
x
x
x 1 = λ y , 2 = λ y . . . n = λ y
1n
12
11
or
x = λ y
1
which denotes a linear relationship between X and Y.
That any relation other than a linear one will not lead to r = l is illustrated by the following:
Let Y = X 2
X = 1, 2, 3, 4, 5, M 1 = 3
Y = 1, 4, 9, 16, 25, M 2 = 11

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