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

where once more we commute under the trace. The associated first-order condition is⎧⎛⎪⎨⎜0 = tr ⎝ V ′ − ∑ (Nk=1b k A k + γ k ∑ ) ⎞ ′Nl=1 bl A l λ′⎟⎪⎩−( ∑ ⎠Nk=1 γk + I) ′ λ ′( (N∑N )) ⎛ ⎞⎫∑N∑× − b k A k + γ k b l A l − ( γ k ⎜ Σ e 0⎪⎬⎟+ I) ⎝ ⎠k=1l=1k=1 0 Σ u⎪⎭{( ())N∑ ∑N N∑= tr − b k A k + γ k b l A l − ( γ k + I)k=1l=1⎛ ⎞ ⎛⎜× ⎝ Σ e 0 ⎟ ⎜⎠ ⎝ V ′ − ∑ (Nk=1b k A k + γ k ∑ ) ⎞⎫′Nl=1 bl A l λ′ ⎪⎬⎟0 Σ u −( ∑ ⎠Nk=1 γk + I) ′ λ ′ ⎪⎭ . (19)where we commuted under the trace. We now compute the trace to rewrite (19) compactly as⎛ ⎞ ⎛ ⎞⎜(−ΓbA − Γ) ⎝ Σ e 0 ⎟ ⎜⎠ ⎝ V ′ − A ′ b ′ Γ ′ λ ′⎟⎠ = 00 Σ u −Γ ′ λ ′⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎜⇒ Γ(bA I) ⎝ Σ e 0 ⎟ ⎜⎠ ⎝ A′ b ′⎟⎠ Γ ′ λ ′ ⎜= Γ(bA I) ⎝ Σ e 0 ⎟ ⎜⎠ ⎝ V ′⎟⎠.0 Σ u I0 Σ u 0Completion of the first-order condition for γ: The left-hand side of the first-order conditionfor λ shows up in the first-order condition for γ i , equation (18). Substituting this we write thefirst-order condition for γ i in a simpler form,⎛ ⎞⎛⎞⎜(bA I) ⎝ Σ e 0 ⎟⎜⎠⎝ Ai′ b i′ + A ′ b ′ γ i′⎟⎠λ = 00 Σ u γ i′⎛ ⎞⎛⎞⎛ ⎞⎛⎞⎜⇒ (bA I) ⎝ Σ e 0 ⎟⎜⎠⎝ A′ b ′⎟⎜⎠γ i′ λ = −(bA I) ⎝ Σ e 0 ⎟⎜⎠⎝ Ai′ b i′⎟⎠ λ.0 Σ u I0 Σ u 0Carrying out the matrix multiplications yields the solution in (11):k=1Proof of Proposition 3: Defining p =γ i′ = −(bAΣ e A ′ b ′ + Σ u ) −1 bAΣ e A i′ V ′ .⎛⎜⎝ 0 θθ 0⎞⎟⎠, routine algebra reveals that∑A i Σ e A j′ = p, and A i Σ e V ′ = I + p.j≠i33