Page 148 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes The table indicates that exports depend upon production and price, and depend equally upon them.
Messrs. Hooker and Yule give the following general solution of the special problem just considered:
To find the maximum correlation coefficient between x and x + bx that results from considering b a
1 2 3
variable, where x , x , and x are the deviations of the series X , X , and X from their respective
1
3
1
2
2
3
arithmetic averages.
Let x + bx = z
2 3
σ
σ + br
then ∑ xz ) = ∑ 12 +xx ∑ bx x σ σnr 1313 )
13 = ( 12 12
( 1
σ+ b
σ
and ∑z 2 = ( σn 2 2 + 2 σ 3 2 2br 23 2 3 )
r σ + br σ
Hence √ xz = 12 2 13 2
1
σ 2 2 + b σ 2 3 2 2br σ 23 2 σ + 3
To find the value of b for which this is a maximum, differentiate with respect to b and equate to zero;
then
.
r
12 23
( 13 − r r ) σ 2
b =
( r − r r ) 12 σ 3
.
13 23
which gives the maximum value
2 + r 2 − r 2r r r
√ xz = 12 13 12 23 31
1
1 − r 23 2
Computing √ xz from the data of Indian production, price, and exports of wheat the value 0.905 is
1
obtained.
Mr. G. U. Yule, in the paper already referred to,* has worked out the general solution of the problem
of the correlation between three variables. In the course of the solution the problem just considered is
solved incidentally. The argument is similar to that used in the case of two variables and so it will not
be repeated here. A concrete notion of the results secured by Mr. Yule can be obtained from the
following explanation taken from Mr. Hooker’s article on the “Correlation of the Weather and the
Crops.”

( ∑ )xy
“I have in the first place formed the ordinary coefficient r = n σ σ between the crop and (a) rainfall,
12
(b) accumulated temperature above 42°. But rainfall and temperature are themselves correlated; hence
an apparent influence of, say, rainfall upon a crop may really be due to rainfall conditions being
dependent upon temperature, or vice versa. Hence it seemed desirable to calculate the partial or net
correlation coefficients, i.e. (following the notation given in Mr. Yule’s paper of 1897).

.
.
ρ 12 = r 12 –r r , ρ = r 13 –r r
12 23
12 23
13
( 13 2 ) 1– r ( 23 2 ) 1 – r . ( 23 2 ) 1– r ( 12 2 ) 1– r
“This partial coefficient() ρ may be regarded as a truer indication of the connection between the
crop and each factor alone, inasmuch as, speaking approximately, we may say that the effect of the
other factor is eliminated. It may be observed, moreover, that the relative influence of rainfall and
ρ
temperature upon the crop is given by 12 ; or, more accurately, this fraction measures the relative
ρ 13

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