Page 143 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Unit 9: Correlation: Definition, Types and its Application for Economists

Notes
σ
and X = M 1 1 ( + r – M 2 )Y
σ 2

σ σ
The coefficients r 2 and r 1 are called the coefficients of regression of Y upon X and of X upon Y
σ 1 σ 2
⎛ σ ⎞ ⎛ σ ⎞
respectively. The first coefficient r 2 ⎟ and the reciprocal of the second ⎜ 2 ⎟ are the slopes of the
⎜
⎝ σ 1 ⎠ ⎝ σ r 1 ⎠
lines of regression. If X and Y be measured in terms of their respective standard deviations as units

1
the slopes of the lines of regression will be r and . In other words, the slope of the line of regression of Y
r
upon X, each series being measured in terms of its standard deviation, is equal to the coefficient of correlation
for the two series. For perfect positive correlation the line would make an angle of 45° with the X axis for perfect
negative correlation the line would make an angle of 135° with the x axis, and for no correlation the line would
be parallel to the x axis.
The correlation coefficients show that there is a very great difference in the degree of correlation of
different pairs of series of statistics. The full significance of the “probable error,” which is used as a
measure of unreliability of any determination, cannot be developed at this point. It is sufficient to
note that, “When r is not greater than its probable error we have no evidence that there is any
correlation, for the observed phenomena might easily arise from totally unconnected causes; but,
when r is greater than, say, six times its probable error, we may be practically certain that the
phenomena are not independent of each other, for the chance that the observed results would be
obtained from unconnected causes is practically zero.”
The high degree of correlation (+0.98) between money in circulation inclusive of bank reserves and
bank reserves is due to the tendency of the two items to vary together during the long time period
and not due to correspondence of minor fluctuations. The reasons for the great increase of money in
circulation in the United States during the period 1879-1904 are the great increase of population and
the industrial expansion. Likewise the number of banks increased in order to serve the increased
population and this meant an increase of total reserves. It is self-evident that the long time tendency
of the two series of statistics must be upward in a growing country. It seemed to me that the bank
reserves during the 26 years, 1879-1904, would be as closely correlated with the population as with
total circulation. The computation of the correlation coefficient between bank reserves and population
gave +0.98. It is the variation upwards of both series during the entire period that causes the high
coefficient.
The correlation coefficient between the index numbers of business distrust and the rates of bank
reserves to check circulation for the same years is 0.53. When the index numbers of business distrust
for one year are correlated with the ratio of bank reserves to check circulation the following year the
coefficient is 0.72. As Dr. Kemmerer has suggested (but not verified), there is a closer correlation
“when proper allowance is made for the time required for alterations in business confidence to exert
their influence on bank reserves.”* The lowest correlation (+ 0.23), that between relative circulation
and general prices, is not high enough to warrant a conclusion that the items vary together. The
smallness of the correlation indicated may have resulted either because the quantity theory is in
error or because the statistics are not adequate to test the theory. Whatever may be the fact, the
statistics and the method of measuring correlation presented by Dr. Kemmerer do not demonstrate
that general prices move in sympathy with relative circulation.
The amount of correlation indicated in each case is small—considering the number of years taken, so
small that no conclusion as to the connection between the two series can be drawn. The correlation
coefficient in the last instance, i. e., between per cent. of successful strikes and business distrust,
suggests an opposite conclusion to that indicated by the other coefficients and that of Mr. Cross. The

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