Regression, Correlation, and the Time Interval: Additive-Multiplicative Framework
研究当一个变量是加性、另一个是乘性时，时间区间划分对相关系数和回归系数的影响，发现相关系数随区间数增加而递减但实际变化很小，而回归系数变化显著。
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 2 n ) 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 2 n 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.