Question: I have two time series $P\_i$ and $T\_i$ where $i = 0...t$

Now the correlation between the time series ${\rm corr}(P,T)\sim.95$, but the correlation between the derivative of the time series ${\rm corr}(dP, dT)\sim.5$
where $dP = P\_i - P\_{i-1}$ for all $i$.

I'm trying to interpret this but can't come up with any explanations. How can the time series be correlated but the running differences not be?

Answer: 
There are a variety of situations in which this happens. Indeed, I'd say it's often to be expected.

Consider the very simple example two time series subject to mutually and (internally) independent shocks and a common drift (i.e. of the form $y\_t=\delta\_t+y\_{t-1}+\epsilon\_t$ and $z\_t=\alpha\_t+z\_{t-1}+\eta\_t$), so that their differences are just $\delta+\epsilon\_t$ and $\alpha+\eta\_t$ respectively.

Then the differences are uncorrelated, but over long time frames, the original series will often be *very* highly correlated.
One reason this could happen is the noise. Derivatives, by definition, are much noisier than the series themselves. That's why some people prefer looking at empirical CDF plots instead of histograms.

If you take the differences with higher lags, does the correlation increase?

ADD: see this example 

```
N <- 1000
eX <- rnorm(N) 
r <- 0.9

X <- numeric(N)
for (i in 2:N){
  X[i] <- r*X[i-1] + eX[i]
 } 

 Y <- X + rnorm(N) 

 dX <- X[2:N] - X[1:(N-1)]
 dY <- Y[2:N] - Y[1:(N-1)]

 c(cor(X,Y), cor(dX,dY))

 plot(X)
  lines(Y,col="red")

```

Here, cor(X,Y) = 0.93 and cor(dX,dY) = 0.59. From the graph, it's clear that X, Y are related, but you don't see it until the lag is sufficiently high. So look at the plot of your data, maybe it helps to see if the relationship is "real" or spurious. 
![enter image description here](https://i.stack.imgur.com/7pRm2.jpg)