I kinda see what you're saying. One could examine changes at each point in time, across N days. Perhaps certain times of the day, there is an anomalous volume relationship between SPY & SH, and then one could dig further into the data to see if there is any predictive power in prices due to the anomaly.
In my likely misguided mind, SPY & SH should be in a leader-follower relationship. I have the view that SPY is just an algorithm that has an overall objective of replicating an investment in the S&P 500 index. So, at every instant of time, it is collecting information about the S&P 500 index, and making adjustments to minimize the tracking error, including issuing new shares or pulling shares off of the market. Additionally, shares of SPY are bouncing around the globe, from individual to individual and from institution to institution, in a chaotic process to achieve collective planetary financial nirvana. And then some clever financial engineers and programmers come along and devise SH, which as an algorithm also uses the S&P 500 as its input, and tries to achieve the inverse return at the end of the trading day. It magically accomplishes this feat by investing in some voodoo financial instruments, rather than by directly shorting the individual S&P 500 constituents stocks. And just like SPY, there are individuals and institutions buying and selling shares of SH throughout the day, hoping to achieve some benefit to themselves or humankind. In this context, when I say "individuals and institutions" I figure we are really talking about automated trading robots. One interesting figure is that the SPY is considerably cheaper than SH, with respective expense ratios of 0.1% versus 0.9%, so the voodoo comes at a price.
Since SPY and SH are both keyed off of the S&P 500 index, then naively one would think that at any instant of time, they would be in perfect harmony by any statistical measure. Maybe if all of the individuals and institutions stopped trading SPY and SH (the zero-volume limit), then the respective ETF algos would bring everything into ideal synchronization?
Weird house of cards. What if one day, Standard & Poor's just decided to stop publishing their index. It's their index, right? They could just stop. The financial world would probably come to a stand-still! It would be hilarious.