Cross-Asset Speculation in Stock Markets∗ - Econometrics at Illinois ...

⎛( ) ⎛ ⎞⎛⎞⎞⎜ ⎜Σ e 0 ⎟⎜Adding and subtracting ⎝ bA I ⎝ ⎠⎝ A′ b ′⎟⎟⎠⎠0 Σ u I−1Σ u , we write Γ as⎛( ) ⎛ ⎞⎛⎞⎞⎜ ⎜Σ e 0 ⎟⎜Γ = I − ⎝ bA I ⎝ ⎠⎝ A′ b ′ ( )⎟⎟⎠⎠−1 ⎛ ⎞ ⎛ ⎞⎜Σ e 0 ⎟ ⎜bA I ⎝ ⎠ ⎝ A′ b ′⎟⎠0 Σ u I0 Σ u I⎛( ) ⎛ ⎞ ⎛ ⎞⎞−1⎜ ⎜Σ e 0 ⎟ ⎜+ ⎝ bA I ⎝ ⎠ ⎝ A′ b ′⎟⎟⎠⎠Σ u0 Σ u I⎛( ) ⎛ ⎞⎛⎞⎞−1⎛⎜ ⎜Σ e 0 ⎟⎜= I − I + ⎝ bA I ⎝ ⎠⎝ A′ b ′ ( ) ⎛ ⎞ ⎛ ⎞⎞⎟⎟⎜ ⎜Σ e 0 ⎟ ⎜⎠⎠Σ u = ⎝ bA I ⎝ ⎠ ⎝ A′ b ′⎟⎟⎠⎠0 Σ u I0 Σ u IWe use this solution for Γ to simplify the variance-covariance matrix of net order flows,−1Σ u .Γ(bAΣ e A ′ b ′ + Σ u )Γ ′= Γ(bAΣ e A ′ b ′ + Σ u )Σ −1u Σ u Γ ′ = ΓΓ −1 Σ u Γ ′ = Σ u Γ ′ .In the fully-symmetric, two-speculator setting, Γ does not depend on Σ e (see Proposition 6).Proof of Proposition 8 (Unobservable prices). The variance covariance matrix of net orderflows is (bAΣ e A ′ b ′ + Σ u ); and with uncorrelated liquidity trade Σ u is diagonal, so that the offdiagonalterms equal their values in bAΣ e A ′ b ′ . From Corollary 1, with symmetry, speculators onlytrade on their direct signals and they trade with the same intensity on each direct signal. Hence,⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞1 θ1 0 0 0⎛⎞20 021 0 0 0 b 11 0bAΣ e A ′ b ′ ⎜= ⎝ b 11 0 b 11 0⎟0 0 1 01 θ02 200 0 1 00 b 11⎠θ 10 b 11 0 b 11 ⎜0 1 0 0⎟⎜0 ⎝ ⎠ ⎝2 20⎟⎜0 1 0 0⎟⎜b 11 0 ⎟⎠ ⎝ ⎠ ⎝ ⎠θ 10 0 0 120 020 0 0 1 0 b 11⎛ ⎞ ⎛ ⎞1 θ⎛⎞20 02b 11 0⎜= ⎝ b 11 0 b 11 01 θ⎟02 200 b 11⎠θ 10 b 11 0 b 11 ⎜0 ⎝2 20⎟⎜b 11 0 ⎟⎠ ⎝ ⎠θ 120 020 b 11⎛ ⎞= (b 11 ) 2 ⎜⎝ 1 θ ⎟⎠ .θ 141