Calculation of Correlation in Time Series

When we observe numerical data in relation to time the set of observations so obtained is known as time series. Time series depicts two types of fluctuations: (i) long-term, and (ii) short-term. While studying correlation between two time series, it is necessary to study separately the correlation of long – term changes and short- term changes. The reason is that the relationship between the long-term changes of two series may be quite different from that between the short-term change of these series. It is quite likely in two time series there may be negative correlation between long-term changes and positive correlation between short-term changes or vice versa. Hence it becomes necessary to study separately correlation between long-term changes and short-term changes as otherwise misleading results may be obtained.

A. Correlation of Long-term Changes

For finding out the correlation of long – term changes, the only thing required is to determine trend values for both the series by the moving average method or the method of least squares. After determining these trend values correlation can be obtained by the methods discussed above and no special method is necessary. The only thing to remember is that the coefficient of correlation shall be computed of the trend values of the two series and not of the original data.

B. Calculation of Correlation in Short – term Changes or Oscillations

Steps. (i) Determine the trend values by the moving average method.

         (ii)  Deduct from the actual values the corresponding trend values obtained in step(i). This would give the short-term fluctuations. Denote these short-term fluctuations by the symbol
 x for X series and y for Y series.

            (iii) Square the short – term fluctuations for X series and obtain the total ∑x2 .

            (iv) Square the short – term fluctuations for Y series and obtain the total ∑y2 .

            (v) Multiply x with y for each value and obtain the total ∑xy.

            (vi) Now apply the formula

                                            _{∑xy____}

                           ^{_{r =}      √(∑x2 x ∑ y2)}

Here, x denotes deviation of X series from moving average and not from arithmetic mean. Similarly, y denotes deviation of Y series from moving average and not from arithmetic mean.

Thus we find that the only difference between correlation explained earlier and correlation in short- term changes is that whereas in the former we take deviations from arithmetic mean, in the latter we take deviations from the trend values.

Note.  It should be carefully noted that x and y in the above formula are different from the x and y of
the Pearsonian formula.