Different measures of covariances

Question

Suppose I have two random time series variables, $X$ and~$Y,$ with a particular realization $x_{t}$ and $y_{t}.$ Suppose I find that $cov\left(X,Y\right)>0.$ What does this imply for i) $cov(\triangle X,\triangle Y)$ where $\triangle$ is the (first) difference operator with respect to time $t$ and ii) $cov(\triangle log(X),\triangle log(Y))$ where the $\triangle log$ corresponds to approximate percentage change in $X$ and $Y$ across adjacent time periods.What is the intuition? An example would be very helpful.

Answer 1

I will work with correlations, as covariances can be altered simply by rescaling all the examples.

With respect to sub-question i: Nothing. Consider the following two examples.

First, $X = Y$. In this case, $\mathrm{cor}(X,Y) = 1$ and $\mathrm{cor}(\Delta X, \Delta Y) = 1$ as well.

Second, a series such as $\Delta X = C + (1,2,\dots, 100)$, with $\Delta Y = C - (1,2, \dots, 100)$. Clearly, $\mathrm{cor}(\Delta X, \Delta Y) = -1$, but as $C \to \infty$, $\mathrm{cor}(X,Y) \to 1$. For a coded version of this example:

dx <- 1000 + 1:100
dy <- 1000 - 1:100
x <- 1 + cumsum(dx)
y <- 1 + cumsum(dy)

> cor(x,y)
[1] 0.9996651
> cor(dx,dy)
[1] -1

Clearly, we can get any correlation for $(\Delta X, \Delta Y)$ given a positive correlation between $X$ and $Y$ by working with examples such as these.

What is happening here? The trend itself dominates the differences between the trends, so the two series are highly correlated because they both trend upwards, but one trend is increasing, and the other may be increasing or decreasing (but not going negative). In the first case, the two trends are both a) positive and b) increasing, so both the original series and the trends are highly positively correlated, but in the second, they are both a) positive and b) one is increasing, one is decreasing, so, although the original series are still positively correlated, the trends are negatively correlated.

With respect to sub-question ii), there is also no relationship. We can engage in a similar pair of examples as above.

First, if $X=Y$, then $\Delta \ln X = \Delta \ln Y$, so both correlations are equal to one.

Second, consider the code fragment that implements an example very similar in spirit to the one above:

dlnx <- 2 + (1:100)/1000
dlny <- 2 - (1:100)/1000

x <- exp(cumsum(dlnx))
y <- exp(cumsum(dlny))

> cor(x,y)
[1] 0.9996189
> cor(dx,dy)
[1] -1

From these examples, it should be clear that simply knowing there is a positive correlation (covariance) between two variables tells us nothing about the correlations between their first differences or their log differences.