Interpreting Cross-Correlations at Multiple Lags

Summary

This document considers how to interpret significant cross-correlations with opposite signs at adjacent lags in two time series. Its example frames interest rate data as sound signals and reports a strong positive correlation at zero lag alongside a stronger negative correlation at a one-step offset. The answer explains that lagged correlations describe different alignments, so their signs can differ without contradiction.

A sinusoidal example illustrates the point: shifting one repeated wave by half a cycle reverses its sign relative to the other, while the unshifted series align positively. This helps explain periodic signals, but it does not establish that the same interpretation applies to arbitrary interest rate series. The document gives no treatment of statistical significance testing, trend or autocorrelation effects, or how to select lags for inference. Those issues matter before treating a lagged correlation as evidence of a meaningful relationship.

Key ideas

• Correlations at different lags describe distinct alignments between the series.
• A positive correlation at one lag and a negative correlation at another can both occur for periodic signals.
• A sinusoidal series shifted by half a cycle can reverse the sign of its correlation.
• The example clarifies lag interpretation but does not provide a general inference procedure for financial time series.

Tags

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


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












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 by Kurt G. (score 1)

https://quant.stackexchange.com/a/69170

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 $\sin(\pi t)=-\sin(\pi t-\pi)\,.$

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