Page 142 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes v v
The values 1 , 2 , . . . . v n equal the distances of the various points to the straight
1 + a 1 2 1 + a 1 2 1 + a 1 2

line y = 1 .
ax
The equation of a line such that the sum of the squares of the distances from the given points is a
minimum will now be found. In other words that value of a will be taken which makes v 1 2 + v 2 2 +
1
. . . . v n 2 = a minimum. To find the value of a , for which ( – a x ) 1 2 + y ( 1 1 – a x ) 2 2 + y . . . .
1
1 2
+ ( y – a x ) n 2 will be a minimum, differentiate with respect to a and
1
1 n
y
y
obtain –2x y – ax ) – 2x 2 ( 2 – ax ) . . . . –2x n ( n – a x ) . In order that the original function be a
1
1 2
1 1
( 1
1 n
minimum, this derivative must equal zero. We will then have
( x y 1 1 2 ) 11 + x y – ax 2 ) 2 2 . . . . +( xy – a x 2 ) n n = 0, or
( – ax
1 2
1 n
∑ – 1 ∑xy a x 2 = 0
∑xy
a 1 =
∑x 2
2
Similarly if x = a y , then ∑ – 2 ∑xy a y = 0 will give the value of a for which the sum of the squares
2
2
a
of the distances of the given points to the straight line X = 2 Y+b 2 is a minimum, or
∑xy
a 2 =
∑y 2
∑xy ∑xy
2
2
Let r = = and ∑ x = σn 1 2 ∑y = σ 2 2 .
n
∑ 2. ∑x y 2 n σ σ
12
The equations between the deviations are:
σ
y = r 2 x
σ 1
σ
x = r 1 y
σ 2
It may seem that the two equations just given are inconsistent. But it must be remembered that these
equations do not give the relationship existing between any corresponding pair of deviations unless
all of the points lie exactly on a straight line and there be perfect correlation. For all cases of imperfect
correlation a given deviation x will occur with several different deviations y (if we have a large number
of measurements). If these deviations y are distributed according to the normal law of distribution
then the given value x substituted in the first equation will give the mean of the deviations occuring
with the deviation x and if a given value y be substituted in the second equation the value of x
resulting will be the mean of the deviations of the associated characteristics.
Since y = Y – M and x = X – M 1
2
σ
Y= M 2 2 ( + r – M 1 )X
σ 1

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