I am working with two time series and I am interested in understanding the relationship between them. As a first step, I checked the cross correlation function (using `ccf()` in R).

The output of my code is shown below, where I'm running `ccf(x,y)`. As I understand it, the conclusion here is that `x` at time `t+k` is negatively correlated with `y` at time `t`, where the lag, `k = 2,3,4,5,6`.

[![The output of CCF function](https://i.stack.imgur.com/DY382.png)](https://i.stack.imgur.com/DY382.png)

I'm really surprised by this because my hypothesis was that `x` would lead to negative changes in `y`, and therefore I expected to see the negative correlation at negative values of the lag, `k`. In fact, I can't think of how changes in `y` might induce a change in `x` at some point in the future, despite what I see here.

Is my interpretation of the output correct? I assume so after running some tests of the form: 

```
x <- rnorm(10)
y <- -lead(x)
ccf(x, y, na.action=na.omit)

```

But given how surprised by the result I feel, I want to double check. 

Assuming that my interpretation is correct, could there any other explanation for this almost sinusoidal shape I see in the results? (I'm thinking that perhaps there is some feature to my data that could artificially cause these shapes - I've no intuition around this!). 

I've got multiple independent experiments featuring these time series and I see a sinusoidal pattern in a majority of them (although the 'sine wave' has a different amplitude and frequency from plot to plot).

I'd also love to hear of alternative approaches to quantifying the relationship between two time-series.