Correlation and the time interval over which the variables are measured

When two random variables are multiplicative over time, their correlation coefficient is not invariant under changes of the differencing interval even when each of the random variables is a product of i.i.d. variables over time. It is shown that unless Y = kX, k > 0, the coefficient of determination (ϱ2) decreases monotonically as the differencing interval increases, approaching zero in the limit. In sampling for empirical studies, the differencing interval is often selected arbitrarily. Such a choice may dramatically affect the sample correlation coefficient, as well as its statistical significance.