Page 144 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes analysis shows that the conclusion that there is negative correlation between general prosperity and
per cent. of successful strikes is not warranted.
Finally, what is the degree of correlation between the prices of British Consols and Sauerbeck’s index
numbers of the prices of commodities ? The chart on indicates a greater degree of correlation (negative)
between the minor fluctuations of the two series than shown by any of the pairs of series that we have
considered. The coefficient of correlation based upon statistics for the 57 years from 1851 to 1907,
inclusive, is — 0.58 ± 0.06. A correlation coefficient of — 0.58 based upon 57 pairs of items warrants
the conclusion that the two series have inverse movements.
The relations between the average deviations, x and y, of the two series of statistics being considered
are:
y = – 1.465x and x = – 0.2295x
The equations of regression are:
Y = 225.6 – 1.465 X and X = 19.439 – 0.2295 Y
For certain pairs of time-series (corresponding items occur at same time) of measurements a correlation
coefficient approximating zero may be obtained even though graphs of the statistics show that the
up-and-down fluctuations occur together. This result will come about if the long-time variations show
opposite tendencies, as, for instance, in the statistics of marriages and bank clearings in the United
Kingdom. On the other hand, a high correlation coefficient may be obtained for two series having the
same long-time tendency regardless of the non-correspondence of the short-time fluctuations. For
example, the coefficient for the two series, population and bank reserves, came out to be 0.98. This
high coefficient comes from the fact that the long-time variation of both series is the same.
Consequently, before it is legitimate to draw any conclusions as to the meaning of a lack of correlation,
or amount of correlation between two series of measurements it is necessary to ascertain the periodic
and the secular variations in the two series. This correlation coefficient may be large through the
correspondence of either secular or periodic variation, or both. It may be null because one variation
covers up the other.
Three methods have been used for isolating the short-time variations of time-series of measurements.
They will now be considered.
1. If upon plotting the two series being compared with time as abscissa and the measurements as
ordinates, periodic variations appear at approximately equal intervals of time the curve may be
“smoothed” and the secular variations may be eliminated as follows:
(a) Ascertain the length of the wave by finding the number of time units between
corresponding parts of the waves, i. e., crest to crest, or hollow to hollow. Let 1 represent
the number of time units found.
(b) Average groups of 1 consecutive measurements, placing the points, determined by these
averages at the middle of each group of measurements. Take enough groups so that the
points obtained will indicate the general tendency of the series.
(c) Draw a smooth curve through the points located by the process described in (b). This
curve shows the secular tendency.
(d) Subtract (this can be done graphically on cross-section paper) the ordinates of the
“smoothed” curve from those of the original curve in order to obtain the series of
measurements of the periodic fluctuation. Let d stand for any one of these differences.
(e) The coefficients computed for corresponding ordinates of two smoothed curves, and for
corresponding differences, d and d’, give measures of the secular and periodic correlation,
respectively.
The method described above has been applied by Mr. R. H. Hooker in his paper “On the
Correlation of the Marriage-Rate with Trade,” and by Mr. G. U. Yule in his study of “ Changes
in Marriage and Birth-Rates in England and Wales during the Past Half Century. “ The following
table gives the correlation coefficients computed in the articles named for the periodic variation:

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