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

We substitute (24) for bAΣ e A ′ b ′ γ i′ in (23) and rearrange to obtain2b i A i Σ e A i′ b i′ − bAΣ e A i′ b i′ = Σ u (I + 2γ i ) = Σ u Γ.Subtracting the common b i A i Σ e A i′ b i′on the left-hand side gives usb i A i Σ e A i′ b i′ − b j≠i A j≠i Σ e A i′ b i′ = Σ u Γ = Ψ −1 Σ 2 u,where the final equality follows from substituting for Γ using the first-condition for γ i , equation(11). Multiplying both sides by Ψ −1 then yields(Ψ b i A i Σ e A i′ b i′ − b j≠i A j≠i Σ e A i′ b i′) = Σ 2 u. (25)We now substitute the structure from the setting of Proposition 2,⎛ ⎞A i Σ e A i′ = I and A j≠i ⎜Σ e A i′ = ⎝ 0 θ ⎟⎠ ≡ p,θ 0to simplify the inner term on the left-hand side of equation (25),b i A i Σ e A i′ b i′ − b j≠i A j≠i Σ e A i′ b i′ = b i (I − p)b i′= 2b i (I + p)b i′ 1 2 (I + p)−1 (I − p)= (Ψ − Σ u ) 1 2 (I + p)−1 (I − p). (26)The first equality follows because with two agents, b i = b j≠i . The second follows from commutingI −p with b i′ using symmetry-countersymmetry. The third uses 2b i (I +p)b i′ = bAΣ e A ′ b ′ = Ψ −Σ u .Now substitute for the inner term on the left-hand side of (25) using (26) to obtainΨ(Ψ − Σ u ) 1 2 (I + p)−1 (I − p) = Σ 2 u .Multiplying both sides by (Ψ −1 ) 2 yields(I − Ψ −1 Σ u ) 1 2 (I + p)−1 (I − p) = (Ψ −1 Σ u ) 2or(I − Ψ −1 Σ u )C = (Ψ −1 Σ u ) 2 , (27)37