Why is the correlation between two data series negative when the top half and bottom half are both strongly correlated?

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Skye Hamilton · Accepted Answer

Step 1Take 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  ±  (  10  ,  10  )  ±  (      10          −      10        ,  −      10          −      10        )so a very thin rectangle rotated at 45 degrees. The four on the whole has       ρ          x      y        ≈  +  1 , but if you take the pair  (  10  ,  10  )  ±  (      10          −      10        ,  −      10          −      10        )they are perfectly negatively correlated, similarly for   (  −  10  ,  −  10  )  ±  (      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).

Why is the correlation between two data series negative if the top and bottom halves are correlated?

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