Hasbrouck Information Shares in a Two-Market VECM

Worked Solution

How to Think About It: The core question is: when the efficient price moves, which market is driving the move? Both markets observe the same underlying value, but their price updates carry different amounts of "information" vs. "noise." Hasbrouck's insight is to decompose the variance of the permanent (efficient-price) innovation into contributions from each market's price innovation. The VECM gives you the long-run impact of shocks, and Cholesky lets you orthogonalize the correlated innovations so you can attribute variance cleanly. The catch is that Cholesky depends on ordering -- so you run it both ways and report bounds.

Key Insight: The information share for a market is its contribution to the variance of the common efficient price innovation. A market that contributes more variance to the permanent component is more "informationally dominant."

The Method:

*Step 1: The VECM.*

Since $p_{1,t}$ and $p_{2,t}$ are cointegrated with cointegrating vector $(1, -1)$ (same asset, so they share a common stochastic trend), the VECM is:

$\Delta p_t = \alpha (\beta' p_{t-1}) + \sum_{i=1}^{k} \Gamma_i \Delta p_{t-i} + \varepsilon_t$

where $p_t = (p_{1,t}, p_{2,t})'$, $\beta = (1, -1)'$ is the cointegrating vector, $\alpha = (\alpha_1, \alpha_2)'$ is the error-correction loading vector, and $\varepsilon_t \sim (0, \Omega)$ is the innovation vector.

The error-correction term $\beta' p_{t-1} = p_{1,t-1} - p_{2,t-1}$ is just the spread between the two markets. The loadings $\alpha_1 < 0$ and $\alpha_2 > 0$ ensure both prices adjust back toward each other.

*Step 2: The common efficient price.*

By the Granger representation theorem, the VMA($\infty$) form of the price level is:

$p_t = \Psi(1) \sum_{s=1}^{t} \varepsilon_s + \text{stationary terms}$

where $\Psi(1)$ is the long-run impact matrix. Since there is one cointegrating relation, $\Psi(1)$ has rank 1, and all its rows are proportional. Write $\psi = $ the common row vector of $\Psi(1)$, so the permanent component is:

$m_t = m_{t-1} + \psi \varepsilon_t$

where $m_t$ is the common efficient price. The innovation to the efficient price is $\eta_t = \psi \varepsilon_t$, a scalar random walk increment.

Concretely, $\psi = (\psi_1, \psi_2)$ where $\psi_1 + \psi_2 = 1$. These weights come from the error-correction loadings: $\psi$ is proportional to $(\alpha_2, -\alpha_1)$ (or equivalently, the row of $\alpha_{\perp}'$, normalized to sum to 1). If market 1 does not adjust to the spread ($\alpha_1 \approx 0$), then $\psi_1 \approx 1$ -- market 1 is the price leader.

*Step 3: Hasbrouck information shares.*

The variance of the efficient price innovation is:

$\sigma_{\eta}^2 = \text{Var}(\psi \varepsilon_t) = \psi \Omega \psi'$

To attribute this variance to each market, we need to orthogonalize $\varepsilon_t$. Factor $\Omega = F F'$ using the Cholesky decomposition (lower-triangular $F$). Define orthogonalized shocks $u_t = F^{-1} \varepsilon_t$, so $\text{Var}(u_t) = I$. Then:

$\sigma_{\eta}^2 = \psi F F' \psi' = (\psi F)(\psi F)' = \sum_{j=1}^{2} (\psi F)_j^2$

The information share of market $j$ is:

$IS_j = \frac{(\psi F)_j^2}{\psi \Omega \psi'}$

By construction, $IS_1 + IS_2 = 1$.

*Step 4: Ordering dependence and bounds.*

The Cholesky decomposition assigns all the contemporaneous correlation to the variable listed first. If you order $(p_1, p_2)$, market 1 gets "credit" for the correlated component, inflating $IS_1$. If you reverse the ordering to $(p_2, p_1)$, market 2 gets that credit instead.

So each ordering gives a different pair of information shares. Hasbrouck's solution: compute both orderings and report:

• $IS_j^{\text{upper}}$: the share when market $j$ is ordered first
• $IS_j^{\text{lower}}$: the share when market $j$ is ordered second

The midpoint $(IS_j^{\text{upper}} + IS_j^{\text{lower}})/2$ is commonly reported as a point estimate. The bounds are tight when the cross-correlation of $\varepsilon_t$ is small (i.e., $\Omega$ is nearly diagonal) and wide when the markets' innovations are highly correlated.

Practical Considerations:

• If $\Omega$ is nearly diagonal (low cross-correlation), both orderings give nearly the same answer and the bounds are tight. This happens when the two markets operate on different time scales or have independent noise.
• With $n > 2$ markets, there are $n!$ orderings. Practitioners often report the max and min across all permutations, or use the Gonzalo-Granger component share (which does not require orthogonalization) as a complement.
• Lag selection in the VECM matters -- too few lags leave serial correlation in $\varepsilon_t$, which contaminates $\Omega$ and biases the shares.
• In practice, the market with the tightest spreads and highest trade frequency tends to have the largest information share.

Answer: The Hasbrouck information share for market $j$ is $IS_j = (\psi F)_j^2 \, / \, (\psi \Omega \psi')$, where $\psi$ is the long-run impact row from the VECM's Granger representation, $\Omega$ is the innovation covariance, and $F$ is its Cholesky factor. Because Cholesky ordering attributes the correlated component to the first variable, you compute both orderings to get upper and lower bounds on each market's share.