Push-response anomalies in high-frequency S&P 500 price series

We test the hypothesis that consecutive intraday price changes in the most liquid U.S. equity ETF (SPY) are conditionally nonrandom. Using NBBO event-time data for about 1,500 regular trading days, we form for every lag L ordered pairs of a backward price increment ("push") and a forward price increment ("response"), standardize them, and estimate the expected responses on a fine grid of push magnitudes. The resulting lag-by-magnitude maps reveal a persistent structural shift: for short lags (1-5,000 ticks), expected responses cluster near zero across most push magnitudes, suggesting high short-term efficiency; beyond that range, pronounced tails emerge, indicating that larger historical pushes increasingly correlate with nonzero conditional responses. We also find that large negative pushes are followed by stronger positive responses than equally large positive pushes, consistent with asymmetric liquidity replenishment after sell-side shocks. Decomposition into symmetric and antisymmetric components and the associated dominance curves confirm that short-horizon efficiency is restored only partially. The evidence points to an intraday, lag-resolved anomaly that is invisible in unconditional returns and that can be used to define tradable pockets and risk controls.