Testing the Square Root Law with Limit Order Data

Summary

The document raises a problem in testing the square root market impact law when a dataset does not identify metaorders directly. It describes using activity during an early market window and unusually large order size as proxies, then observes that the regression produces a negative coefficient. The proposed explanation is that the selected population may contain many limit orders, for which signed price changes can move against the order direction.

The question asks whether impact should instead be measured using the absolute price change. It does not include an answer, supporting analysis, or empirical results, so it does not establish that absolute impact is the right dependent variable or resolve the identification issue. Its useful contribution is highlighting how proxy construction and order type can affect the sign and interpretation of estimated impact. A proper test would need to define the target impact measure and distinguish order direction, execution type, and subsequent price movement.

Key ideas

• When metaorders are unavailable, time windows and large order sizes can serve as imperfect proxies.
• A negative signed-impact estimate may reflect the mix of order types in the selected sample.
• Limit orders can coincide with price moves opposite to the order direction.
• Using absolute price changes changes the measured quantity and does not by itself resolve proxy or identification concerns.

Tags

# Square root law and limit orders


# Square root law and limit orders












I am working on this assignment, whose goal is to test the square root law for a given dataset. I did not have explicitly metaorders, but the given hints is detecting them by proxying T (first two opening hours of the market) and order size (90th quantile f.e.). The issue I am having is that when I perform regression I end up with a negative coefficient (i.e. a negative Y). This is at first incoherent meaning that actually the impact of metaorders is negative on the price. But, I was wondering that could be due to the fact that maybe my metaorders sub-populations is dominated by limit orders (?). In other words Δpi​=+1⋅[logp(t1​)−logp(t0​)] tends to be negative for buys for p(t1​)<p(t0​) when there's a large limit order. What do you recommend for this, to consider ∣Δpi​∣?

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