Quantitative Microstructure VECM & Price Discovery Hasbrouck (1995)

Hasbrouck Information Share & Price Discovery Lab

Decompose multi-venue price discovery into permanent common random-walk innovations and transitory noise using Joel Hasbrouck's (1995) Vector Error Correction Model (VECM) and Gonzalo-Granger (1995) Component Share bounds.

Market Microstructure Archetypes:
Market 1 Information Share
82.4%
Bound: [71.2% - 93.6%] | Width: 22.4%
Market 2 Information Share
17.6%
Bound: [6.4% - 28.8%]
Gonzalo-Granger CS (M1)
91.7%
Market 2 CS: 8.3% (Ratio: 11.00x)
Price Discovery Regime
Market 1 Leads
Basis Std: $0.184 | Max Gap: $0.42
Model Parameters
VECM & Cholesky
Initial Security Price ($P_0$) $100.00
Market 1 Adjustment Speed ($\alpha_1$) -0.05
Error correction rate ($\alpha_1 \le 0$). Near zero indicates strong price leadership.
Market 2 Adjustment Speed ($\alpha_2$) 0.55
Error correction rate ($\alpha_2 \ge 0$). Higher values mean rapid satellite adjustment.
Market 1 Innovation Volatility ($\sigma_1$) $0.35
Venue 1 innovation noise per discrete time interval ($e_{1,t}$).
Market 2 Innovation Volatility ($\sigma_2$) $0.45
Venue 2 innovation noise per discrete time interval ($e_{2,t}$).
Cross-Market Innovation Correlation ($\rho$) 0.65
$\rho = \text{Corr}(e_1, e_2)$. Determines the spread between Cholesky upper/lower bounds.
Real-Time Canvas

Econometric Output & Cointegration Matrix

Analytical Decompositions
Parameter / Econometric Measure Theoretical Formula Calculated Equilibrium Value
Information Share Sensitivity Matrix

Market 1 Midpoint Information Share ($\overline{IS}_1$ %) across Innovation Correlation ($\rho$) and Error Correction Ratio ($\alpha_2 / |\alpha_1|$).

Cholesky Bound Width Uncertainty Matrix

Hasbrouck Bound Spread ($IS_1^{\max} - IS_1^{\min}$ %) across Volatility Ratio ($\sigma_1 / \sigma_2$) and Cross-Market Correlation ($\rho$).

Price Discovery Econometrics Masterclass

1. The Law of One Price and Cointegration in Multi-Venue Trading

In contemporary fragmented financial markets, the identical underlying financial asset trades simultaneously on multiple competing execution venues—for example, the S&P 500 index via the Chicago Mercantile Exchange (CME) E-mini futures contract versus the New York Stock Exchange (NYSE Arca) SPDR S&P 500 ETF (SPY), or cross-listed American Depositary Receipts (ADRs) versus local home market shares, or perpetual swaps versus spot order books.

Because arbitrageurs exploit pricing differentials between venues, prices cannot wander arbitrarily far apart. Let $p_{1,t}$ and $p_{2,t}$ represent log price series on Venue 1 and Venue 2. Each series is non-stationary and integrated of order one, $I(1)$. However, their difference (the basis or price differential) is stationary, $I(0)$:

$$z_t = p_{1,t} - p_{2,t} \sim I(0), \quad \text{Cointegrating Vector } \beta = [1, -1]'$$

By the Granger Representation Theorem, cointegrated series can be written as a Vector Error Correction Model (VECM). For a bivariate price system with one lag, the model is formulated as:

$$\begin{pmatrix} \Delta p_{1,t} \\ \Delta p_{2,t} \end{pmatrix} = \begin{pmatrix} \alpha_1 \\ \alpha_2 \end{pmatrix} (p_{1,t-1} - p_{2,t-1}) + \begin{pmatrix} e_{1,t} \\ e_{2,t} \end{pmatrix}, \quad \text{Var}\begin{pmatrix} e_{1,t} \\ e_{2,t} \end{pmatrix} = \Omega = \begin{pmatrix} \sigma_1^2 & \rho \sigma_1 \sigma_2 \\ \rho \sigma_1 \sigma_2 & \sigma_2^2 \end{pmatrix}$$
2. Hasbrouck (1995) Information Share vs. Gonzalo-Granger (1995) Component Share

While both methodologies isolate the common long-run price factor, their economic definitions of price leadership diverge fundamentally:

Gonzalo-Granger (1995) Component Share ($CS$)

Gonzalo and Granger define the common efficient price as a linear combination of contemporaneous prices $f_t = \gamma_1 p_{1,t} + \gamma_2 p_{2,t}$ such that the transitory noise contains no permanent random-walk innovations. The vector $\gamma = (\gamma_1, \gamma_2)'$ is orthogonal to the error correction vector $\alpha$:

$$CS_1 = \gamma_1 = \frac{\alpha_2}{\alpha_2 - \alpha_1}, \quad CS_2 = \gamma_2 = \frac{-\alpha_1}{\alpha_2 - \alpha_1}$$

Component Share relies exclusively on the speed-of-adjustment parameters $\alpha$. If Market 1 does not adjust to basis discrepancies ($\alpha_1 = 0$), $CS_1 = 100\%$, denoting that Market 1 sets the price.

Hasbrouck (1995) Information Share ($IS$)

Hasbrouck focuses on the variance of the common permanent innovation. The common random walk component is $w_t = \psi e_t = \gamma_1 e_{1,t} + \gamma_2 e_{2,t}$, with total variance:

$$\sigma_w^2 = \psi \Omega \psi' = \gamma_1^2 \sigma_1^2 + 2 \gamma_1 \gamma_2 \rho \sigma_1 \sigma_2 + \gamma_2^2 \sigma_2^2$$

Information Share measures the proportion of $\sigma_w^2$ originating from shocks on each exchange. It synthesizes both error correction speeds $\alpha$ and the innovation covariance structure $(\sigma_1, \sigma_2, \rho)$.

3. The Cholesky Factorization and Ordering Uncertainty

When contemporaneous innovations are correlated ($\rho \ne 0$), the covariance matrix $\Omega$ is non-diagonal. To attribute variance to orthogonal factors, Hasbrouck applies a lower triangular Cholesky factorization $\Omega = M M'$ where:

$$M^{[1,2]} = \begin{pmatrix} \sigma_1 & 0 \\ \rho \sigma_2 & \sigma_2 \sqrt{1 - \rho^2} \end{pmatrix}$$

Because Cholesky orthogonalization attributes all common covariance $\rho \sigma_1 \sigma_2$ to whichever variable is ordered first, ordering $[1, 2]$ yields the maximum possible share for Market 1, while ordering $[2, 1]$ yields its minimum possible share:

$$IS_1^{[1,2]} = \frac{(\gamma_1 \sigma_1 + \gamma_2 \rho \sigma_2)^2}{\sigma_w^2}, \quad IS_1^{[2,1]} = \frac{(\gamma_1 \sigma_1 \sqrt{1 - \rho^2})^2}{\sigma_w^2}$$

Baillie, Booth, Tse, and Zabotina (2002) proposed taking the simple arithmetic mean of the upper and lower bounds as the consensus estimator:

$$\overline{IS}_1 = \frac{IS_1^{\min} + IS_1^{\max}}{2}$$

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Frequently Asked Questions

Hasbrouck's (1995) Information Share (IS) is the foundational econometric metric used in market microstructure to determine which trading venue contributes new information to the true fundamental value of an asset traded across multiple markets. Under the law of one price, prices on competing exchanges are cointegrated and share a single permanent common random-walk factor (the efficient price). Hasbrouck decomposes the total variance of this permanent efficient price innovation into shares attributable to each exchange, isolating informational innovation from transitory liquidity noise.

Gonzalo-Granger (1995) Component Share (CS) measures price leadership strictly through the error correction adjustment vectors alpha: CS1 = alpha_2 / (alpha_2 - alpha_1). It identifies which venue adjusts least to temporary pricing discrepancies (the venue with alpha near zero leads). Unlike Hasbrouck IS, Gonzalo-Granger CS ignores the innovation covariance matrix Omega and volatility differences between venues. Hasbrouck IS incorporates both the error correction speeds and the relative volatility and contemporaneous correlations of the innovation errors, providing a variance-based attribution.

When innovation errors between two trading venues are correlated (contemporaneous correlation rho != 0), the covariance matrix Omega is non-diagonal. To isolate independent shocks, Hasbrouck applies a lower triangular Cholesky factorization Omega = M M'. However, the Cholesky decomposition depends on the variable ordering: the venue ordered first is arbitrarily credited with all common contemporaneous covariance, maximizing its Information Share. By permuting orderings [1, 2] and [2, 1], Hasbrouck establishes rigorous upper and lower bounds [IS_min, IS_max], with the Baillie et al. (2002) midpoint serving as a pragmatic central estimate.

In a bivariate cointegrated system where the basis is defined as (p1 - p2), alpha_1 <= 0 and alpha_2 >= 0. If alpha_1 is close to zero and alpha_2 is large and positive, Market 1 does not adjust to temporary price divergences, while Market 2 rapidly corrects toward Market 1. This reveals that Market 1 is the dominant price leader driving price discovery, while Market 2 acts as a price satellite or follower. Conversely, if alpha_1 is negative and large while alpha_2 is zero, Market 2 is the primary price leader.

The width of the Hasbrouck bounds (IS_max - IS_min) is directly proportional to the absolute magnitude of cross-market innovation correlation |rho|. When rho = 0, the innovation covariance matrix is diagonal, Cholesky ordering becomes irrelevant, and the upper and lower bounds collapse to a single point. As rho approaches 1 or -1, common unassigned variance expands, causing the bounds to widen significantly and creating greater ordering ambiguity in econometric attribution.