How Hasbrouck Information-Share Algorithms Measure Price Discovery Across Markets: Cointegration, VECMs, Common Efficient Prices, Cholesky Bounds and Failure Diagnostics

Reader question: If the same or closely linked asset trades in several markets at once, how can we estimate which market is actually leading the permanent price discovery rather than merely moving a lot?

Hasbrouck information-share algorithms answer that question by modelling several observed prices as noisy views of one underlying efficient price. The method uses cointegration and a Vector Error-Correction Model (VECM) to separate permanent innovations from transitory deviations, then allocates the variance of the common efficient-price innovation across markets.

This article owns the multi-market price-discovery attribution problem: synchronized prices for one economically linked asset across venues + a cointegrated VECM → common-trend innovations, Hasbrouck information shares, Cholesky-ordering bounds, diagnostics and evidence about where permanent information appears to enter first.

It does not own order matching, market-impact estimation, optimal execution or manipulation surveillance. Those are different questions. A venue can execute the most volume without leading price discovery, and a venue can lead price discovery without offering the deepest liquidity.

This is public mathematical and computational education. It is not a recommendation to trade on any venue or instrument.

1. Price discovery is not the same as trading volume

Suppose one security trades on Market A and Market B. Both prices move together because they represent the same economic claim. If A reacts first to new information while B catches up moments later, A may contribute more to price discovery even if B handles more transactions.

The central distinction is:

activity asks where trading occurs; price discovery asks where permanent information enters the common price.

Hasbrouck’s 1995 framework formalises that distinction.

2. Why raw price regressions are dangerous

Financial prices are often non-stationary. Regressing one random-walk-like price on another can produce impressive statistics even when the relationship is spurious.

Closely linked market prices should instead satisfy a long-run equilibrium relation. If p_1,t and p_2,t represent the same asset, the price difference or basis should usually remain bounded around an economically interpretable level after allowing for costs and conventions.

That is the setting for cointegration.

3. Cointegration establishes a common long-run price

Let:

p_t = [p_1,t, p_2,t, …, p_m,t]ᵀ.

If the individual price series are integrated of order one but certain linear combinations are stationary, the vector is cointegrated.

For two near-identical prices, a simple cointegrating relation may resemble:

p_1,t − p_2,t = stationary basis.

The common stochastic trend is interpreted as the permanent efficient-price component; deviations among venues are transitory microstructure effects or temporary dislocations.

4. The Vector Error-Correction Model

A VECM can be written schematically as:

Δp_t = αβᵀp_t−1 + Γ₁Δp_t−1 + … + Γ_k−1Δp_t−k+1 + ε_t.

Here:

• β contains cointegrating relations;
• α contains adjustment speeds;
• Γ matrices capture short-run dynamics;
• ε_t is the vector of contemporaneous innovations.

If one venue’s price moves away from the long-run relation and another venue adjusts strongly afterward, the adjustment coefficients contain evidence about leadership and correction.

5. From the VECM to a common efficient price

The VECM can be represented in a common-trends form. Under the Hasbrouck interpretation, the observed price vector contains:

• a permanent component driven by innovations that accumulate into the efficient price;
• a transitory stationary component that eventually mean-reverts.

The permanent innovation can be expressed as a linear combination of VECM residuals:

w_t = ψ ε_t,

where row vector ψ maps market-specific residual innovations into the innovation of the common efficient price.

The variance of that common innovation is:

Var(w_t) = ψΩψᵀ,

where Ω is the covariance matrix of VECM residuals.

6. Information share as variance attribution

If residual innovations were contemporaneously uncorrelated, contribution from market j could be allocated directly from its squared loading multiplied by its innovation variance.

In practice, venue innovations often occur almost simultaneously and are correlated. Hasbrouck therefore orthogonalizes innovations using a matrix factorization of Ω, commonly a Cholesky decomposition:

Ω = FFᵀ.

The efficient-price innovation can then be written in terms of orthogonal shocks. For a given ordering, market j‘s information share has the form:

IS_j = ([ψF]_j)² / (ψΩψᵀ).

For that decomposition, the information shares sum to one.

7. Why Cholesky ordering creates bounds

Cholesky factorization depends on the ordering of variables. If residual innovations are correlated, whichever market is placed first in the decomposition is credited differently from a market placed later.

This is not merely a coding nuisance. It reflects an identification problem: when two venues move within the same sampling interval, the reduced-form data may not uniquely reveal which contemporaneous innovation arrived first.

Hasbrouck therefore motivates reporting information-share bounds across alternative orderings.

For two markets:

• order A first, B second;
• order B first, A second;
• calculate both shares;
• report the lower and upper bound for each venue.

A narrow bound indicates cleaner identification. A wide bound is evidence that contemporaneous correlation prevents precise attribution.

8. A simple conceptual example

Suppose two cointegrated venues produce a common-price innovation variance of 4 basis-points-squared per interval.

Under one Cholesky ordering, Market A contributes 3 and Market B contributes 1:

IS_A = 3/4 = 75%, IS_B = 25%.

Under the reverse ordering, A receives 2.4 and B 1.6:

IS_A = 60%, IS_B = 40%.

A defensible report is therefore not “A has exactly 67.5% price discovery” unless a midpoint convention is explicitly justified. The stronger statement is:

A’s Hasbrouck information share lies between 60% and 75% under the two residual orderings.

9. Inputs and outputs

Inputs can include:

• venue-specific transaction prices or midquotes;
• timestamps and synchronization rule;
• sampling interval;
• currency, contract and unit conversions;
• cointegration specification;
• VECM lag order;
• trading-hours overlap rule;
• treatment of stale observations;
• residual covariance estimator;
• Cholesky permutations or generalized factorization rule;
• rolling-window length if time variation is studied.

Outputs can include:

• cointegration rank;
• cointegrating vectors;
• error-correction coefficients;
• VECM residual covariance matrix;
• efficient-price loading vector;
• information-share estimates;
• ordering bounds;
• component-share comparisons;
• residual diagnostics;
• rolling price-discovery measures.

10. Synchronization is a modelling decision

High-frequency venue data rarely arrive on perfectly aligned timestamps. One market may update every millisecond while another updates every few hundred milliseconds.

Researchers therefore choose a synchronization rule such as:

• previous-tick interpolation;
• nearest observation;
• fixed-time bars;
• refresh-time sampling;
• event-time matching.

Each choice can alter lead-lag patterns. A slow venue can appear uninformative simply because its prices are sampled as stale, while overly coarse sampling can merge genuinely sequential reactions into one contemporaneous innovation.

Weak link: price-discovery attribution can be sensitive to the clock used to observe the markets.

11. Sampling frequency creates a trade-off

At very high frequency:

• microstructure noise becomes large;
• bid-ask bounce matters;
• asynchronous updates are severe;
• zero returns and stale quotes become common.

At low frequency:

• sequential price responses collapse into the same interval;
• contemporaneous correlation rises;
• Cholesky bounds can widen;
• short-lived leadership disappears.

A robust study therefore repeats the analysis across multiple economically defensible sampling intervals.

12. Midquotes and transaction prices answer slightly different questions

Transaction prices include bid-ask bounce, execution rules and trade direction. Midquotes remove some transaction-level bounce but reflect quoted rather than executed willingness to trade.

If information shares differ sharply between trades and quotes, that difference is not automatically an error. It may reveal where information first enters the quote-setting process versus where it is confirmed in transactions.

The data definition must therefore be part of the page’s model contract.

13. Cointegration failure is a hard boundary

The Hasbrouck framework assumes linked prices share a common long-run stochastic trend.

If two markets are not cointegrated during the sample:

• they may not represent the same economic claim;
• contract specifications may have changed;
• currency or carry adjustments may be wrong;
• the sample may span a structural break;
• one series may contain severe data problems.

Falsifier: if no defensible common-trend relation exists, do not force an information-share decomposition.

14. Lag length matters

Too few VECM lags leave serial dependence in residuals, contaminating the innovation interpretation. Too many lags consume degrees of freedom and can destabilize estimates.

Lag selection can combine information criteria, residual-autocorrelation tests and economic knowledge of venue reaction speeds.

A good implementation stores the chosen lag order and tests whether information-share conclusions survive nearby alternatives.

15. Structural breaks can make one full-sample share meaningless

Market structure changes. A venue can gain electronic liquidity, trading hours can change, a new fee schedule can alter order routing, or a regulatory reform can redistribute price discovery.

A single information share over several years can therefore average together incompatible regimes.

Diagnostic: estimate rolling or subperiod information shares and test for breaks in cointegration, VECM parameters and residual covariance.

16. Information share versus component share

Hasbrouck information share is not the only price-discovery metric.

The Gonzalo–Granger permanent-transitory framework produces component shares, which focus on how observed prices load into the common permanent component. Information shares instead attribute the variance of innovations to that permanent component.

These measures can disagree because they answer related but different mathematical questions.

Research has shown that the interpretation of information share and component share becomes especially delicate when residuals are highly correlated.

17. Generalized and information-leadership measures

Because Cholesky ordering can be inconvenient, later researchers proposed generalized information-share measures and information-leadership measures intended to reduce ordering dependence or distinguish leadership from noise.

A 2025 study by Shen, Zhang and Zivot proposed an improved information-leadership share and documented through simulations that correlated VECM residuals can distort some older metrics.

This does not make Hasbrouck obsolete. It means the original measure should be interpreted as one specific variance-decomposition definition of price discovery, with ordering uncertainty reported honestly.

18. Evidence polarity

Evidence for confidence includes a clear cointegration relation, stable VECM residuals, narrow Cholesky bounds, consistent leadership across reasonable sampling intervals, similar conclusions from midquotes and transactions after accounting for their different meanings, stable rolling estimates, and broad agreement with complementary price-discovery metrics.

Evidence against confidence includes failed cointegration, strong residual autocorrelation, near-singular residual covariance, very wide ordering bounds, high sensitivity to timestamp matching, leadership that reverses under small sampling changes, structural breaks, stale data, or disagreement with every alternative decomposition.

19. Counterexample: volume leader is not information leader

Market B handles 70% of volume but largely follows quotes established first on Market A. B can be highly liquid and operationally important while having a lower information share.

Falsifier: test permanent innovation attribution rather than assuming volume share proxies for price discovery.

20. Counterexample: asynchronous clocks create false leadership

Market A timestamps at microsecond resolution while Market B timestamps only to the second. When synchronized naively, A appears to move first almost every time.

Falsifier: degrade A’s timestamps or use a common coarser/event-time sampling rule. If leadership collapses, the original result was partly a clock artifact.

21. Counterexample: common news causes correlated innovations

A macro announcement reaches both venues nearly simultaneously. Residual innovations become highly correlated within the sampling interval.

Cholesky decomposition cannot uniquely assign the common shock. Information-share bounds widen.

Falsifier: report both orderings. If the bounds overlap heavily, do not claim precise venue leadership.

22. Counterexample: contract basis is not stationary

A futures contract and spot market may be linked by carry, financing, dividends and expiry effects. If those adjustments are ignored, the raw price difference may trend rather than remain stationary.

Falsifier: model the economically correct basis and retest cointegration before interpreting price-discovery shares.

23. Alternatives

Lead-lag regressions ask whether one venue’s returns predict another’s, but do not automatically isolate permanent common-price innovations.

Gonzalo–Granger component shares attribute the common permanent component through adjustment dynamics.

Information leadership measures combine aspects of information and component shares.

Impulse-response analysis studies how venue shocks propagate through a multivariate system.

State-space efficient-price models explicitly model latent efficient prices and venue-specific noise.

24. Diagnostics checklist

1. Validate contract and currency equivalence.
2. Inspect unit roots and stationarity.
3. Test cointegration rank.
4. Choose and justify VECM lag order.
5. Check residual autocorrelation and heteroskedasticity.
6. Inspect residual covariance conditioning.
7. Calculate all relevant Cholesky orderings.
8. Report bounds rather than only a midpoint.
9. Repeat across sampling intervals.
10. Repeat with trades and midquotes where meaningful.
11. Test subperiod and rolling stability.
12. Compare with component-share or leadership measures.

25. Connections to the surrounding Bukit Timah Tutor knowledge estate

Kyle’s lambda algorithms measure how strongly signed order flow moves price; that is a price-impact question, not a multi-market price-discovery decomposition.

Electronic market-matching algorithms explain how orders are allocated once they reach a venue.

Almgren–Chriss algorithms solve an execution scheduling problem under impact and risk. Hasbrouck information share asks a prior structural question: where does the permanent price innovation appear to originate?

The full lane is indexed at Finance & Banking Algorithms | Applied Mathematics in Real Financial Systems.

26. What would falsify confidence?

Confidence should be withdrawn if the price series are not defensibly cointegrated; if VECM residuals retain strong predictable structure; if information-share bounds are too wide for a useful conclusion; if venue leadership changes under small synchronization or lag choices; if contract differences invalidate the common-price assumption; or if structural breaks make one pooled estimate misleading.

27. Verification and update triggers

Preserve raw venue data, synchronization rules, sampling interval, unit conversions, cointegration tests, VECM specification, residual covariance matrix, Cholesky permutations and final bounds. Re-estimate after material changes to trading hours, market design, tick size, fee structure, contract specification, data timestamping or venue participation.

28. Primary and high-quality references

• Joel Hasbrouck, One Security, Many Markets: Determining the Contributions to Price Discovery, Journal of Finance, 1995.
• Gonzalo and Granger, permanent-transitory common-factor framework, foundational comparison for component-share methods.
• A Structural Analysis of Price Discovery Measures, Journal of Financial Markets, on interpretation of information and component shares.
• Shen, Zhang and Zivot, Improving Information Leadership Share for Measuring Price Discovery, Journal of Empirical Finance, 2025.
• Confidence Intervals for Price Discovery, Oxford Bulletin of Economics and Statistics, 2026, for modern inference around efficient-price and price-discovery measures.

Educational boundary: An information share is an econometric attribution under a specific common-price model. It is not proof that a venue is economically superior, safer, cheaper or more profitable to trade.