Time Series Correlation: When a study of correlation is made in two-time series, it becomes necessary to study it separately.
(1) Correlation of a long time changes:
To study the correlation of long-time changes first the trend values of the two series must be obtained. This can be done either by the method of moving average or by the method of fitting straight line trend with least-squares or fitting a parabolic curve to the data. After this, the coefficient of correlation can be calculated between the trend values of the two series. Thus, for studying the correlation between long time changes, the only special thing is, that instead of the original series trend values are used for the calculation of the coefficient of correlation.
(2) Correlation of short-time changes:
To study the correlation of short time change of two series, trend values must be isolated from the time series and the values of short period change. Thus, the special point in the calculation of the coefficient of correlation between short-time changes is that in such cases, deviations are taken from the moving trend values rather than from the arithmetic average. The figures obtained by dividing the sum of the squares of these deviations by their number give the value of the variance. Its square root gives the value of the standard deviation of the series. When the values of Σxy, σ1, and σ2 are thus obtained, the coefficient of correlation can be easily calculated.
(3) Correlation of cyclic change:
We know that short-time fluctuations consist of seasonal change, cyclic fluctuations, and irregular fluctuations. It is necessary to obtain exclusive figures of cyclical fluctuations. After this has been done, these cyclic changes are divided by the standard deviations of the series to which they relate. These figures are then multiplied in pairs and their products are totaled to obtain the value of Σxy. This figure divided by the number of pairs of values and the formula for the correlation coefficient is,
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