SPY’s High-Frequency Price Moves Hide Lagged Push-Response Anomalies

If you watch a market long enough, you might expect yesterday’s price moves to fade into noise. This paper shows that, in SPY’s fastest tick-by-tick data, some of those moves leave a faint fingerprint. Using NBBO event-time data from about 1,500 regular trading days, the authors pair each backward price change, called a push, with the next forward price change, called a response, and measure how the response depends on push size. For short lags of 1 to 5,000 ticks, expected responses cluster near zero across most push magnitudes, which looks like strong short-term efficiency. Past that range, the map changes: pronounced tails appear, and larger historical pushes line up with nonzero conditional responses. The effect is asymmetric too: large negative pushes are followed by stronger positive responses than equally large positive pushes. The paper says this pattern is invisible in unconditional returns and could help define tradable pockets and risk controls.

One tiny move in SPY can vanish in a blink. Or so the usual market story goes. But this study looks at the fund one tick at a time, across about 1,500 regular trading days. It pairs each backward price change, called a push, with the next forward change, called a response. That sounds dry. It is not. The surprise is that the past starts to matter again after a long wait. For short gaps, the next move stays near zero. Later, the link stops looking flat. If you ever assumed that fast markets erase memory right away, this result asks you to look again.

When the market looks blind, then suddenly remembers

The strongest result is not a simple trend. It is a map. For lags from 1 to 5,000 ticks, expected responses sit near zero across most push sizes. That looks like strong short-term efficiency. But beyond that stretch, the map grows tails. Large historical pushes line up with nonzero later responses. The effect is not the same on both sides. Large negative pushes are followed by stronger positive responses than equally large positive pushes. That points to a bounce after sell-side shocks. The pattern also survives a split into symmetric and antisymmetric parts. So the short-horizon calm is only partly restored.

How the lag map was built

The setup uses NBBO event-time data. NBBO means the best bid and offer prices quoted across the market. Event time means the clock moves when trades or quote changes happen, not by the wall clock. The study takes each lag, then forms pairs of a push and a response. It standardizes them, which means it puts them on the same scale. Then it estimates the average response for many push sizes on a fine grid. That grid turns raw ticks into a picture. The picture shows where the market behaves like noise. It also shows where a past shove still leaves a trace.

Why this matters for fast trading

This matters because unconditional returns hide the pattern. A plain return series can look calm even when certain lag and size pairs keep repeating a small edge. The study says those pockets can help define tradable spots and risk controls. That does not mean easy money. It means the market may be efficient on average, yet still leave small, time-shaped gaps. A trader who ignores lag can miss them. A risk system that watches only broad averages can miss them too. The useful message is narrow but sharp. In SPY, timing and push size both seem to matter.

What to test next

The next test is whether the same lag map holds outside SPY. This result comes from the most liquid U.S. equity ETF. That makes it a strong place to start. It also makes the open question clear. Do other liquid assets show the same long-lag tails and the same sell-side bounce? Or is this shape tied to SPY’s own trading flow? A fresh check on other ETF or stock tick data would show how far this delayed echo travels. If it repeats, the map becomes more than a curiosity. It becomes a way to look for where market memory still survives.