Regression, correlation, and the time interval: Additive-multiplicative framework

When two random variables are both additive or multiplicative, the effect of the way one "slices" the available period to subperiods (time intervals) is well documented in the literature. In this paper, we investigate the time interval effect when one of the variables is additive and one is multiplicative. We prove that the squared multiperiod correlation coefficient (ρ_n²) decreases monotonically as n increases, and approaches zero when n goes to infinity. However, for relevant data corresponding to the U.S. stock market index, when shifting from weekly parameters to quarterly parameters the decrease in ρ_n² is negligible. The effect on the regression coefficient is much more dramatic and even a shift from weekly data to quarterly data affects the regression coefficient substantially. The regression slope generally approaches zero, minus infinity or plus infinity, as the number of periods increases. Montonicity, however, exists only in certain cases.