I have two time series of data variables with each having 111 data points.
The data series should be strongly correlated given the nature of what is being explored.
However, the correlation between the data series is roughly -0.03 which confused me.
To explore further, I checked the correlation between data points 1-62 and 63-111 separately.
The correlation between the data series for points 1-62 was 0.83 and the correlation between the data series for points 63-111 was 0.55.
It is surprising to me that it is -0.03 for the whole series when they're positively correlated when split into two subsections.
Is this possible or am I making a mistake?
Yes, and it would make sense if you think harder about it.
Take extreme and simplest example of 4 points, no pairs on the same horizontal line or vertical line. Split the four points into pairs, each will be perfectly correlated, right?
Now imagine the 4 points are $$ \pm (10,10) \pm (10^{-10},-10^{-10}) $$ so a very thin rectangle rotated at 45 degrees. The four on the whole has $\rho_{xy}\approx +1$, but if you take the pair $$ (10,10)\pm(10^{-10},-10^{-10}) $$ they are perfectly negatively correlated, similarly for $(-10,-10)\pm(10^{-10},-10^{-10})$.
In general, you can only split into two halves and expect the result to work if the two halves are about the same, in your time series case it means with the same model (same mean, same autocovariance).