Two sound waves in phase at different lags with varying signs?

Question

I have two interest rate time series but want to think of this as sound signals for thought purposes.

Imagine you have two time series of audio signals. You run a time lagged cross correlation analysis and find there is a significant correlation between them at lag = 0 and lag = -1. The correlation at lag = 0 is 90%, and -99% at lag = -1.

My inclination is to settle with the strongest lag at -1, and conclude the two series are most in phase at that lag. However, there is a seemingly contradictory correlation at lag = 0, suggesting the two series are in phase but negatively so.

How does one interpret this? I’m struggling with whether or not to stick with the largest correlation, or speaking to both, but I can’t figure out how to explain the significance of having two significant correlations at different lags with different signs (positive correlation and negative correlation.

Answer 1

It looks like you think of the two time series as $$ x(t_i)=\sin(\pi t_i)\,,\quad y(t_i)=\sin(\pi t_i)\,. $$ Clearly, the correlation of $$ [x(t_1),...,x(t_n)]\quad\text{ and }\quad[y(t_1),...,y(t_n)]\quad\text{ (lag $0$) } $$ is $+1\,$, and the correlation of $$ [x(t_1),...,x(t_n)]\quad\text{ and }\quad[y(t_1-1),...,y(t_n-1)]\quad\text{ (lag $-1$) } $$ is $-1$ because $\sin(\pi t)=-\sin(\pi t-\pi)\,.$