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

under the trace yields{0 = tr= tr⇒{A i Σ e (V ′ −A i Σ e (V ′ −}N∑∑NA k′ b k′ Γ ′ λ ′ )(I + γ i ) − Σ e (b i A i + γ i A k b k ) ′ λΓA ik=1k=1}N∑∑NA k′ b k′ Γ ′ λ ′ )(I + γ i ) − A i Σ e (b i A i + γ i A k b k ) ′ λΓk=1∑NA i Σ e A k′ b k′ Γ ′ λ ′ (I + γ i ) + A i Σ e (A i′ b i′ +k=1k=1N∑A k′ b k′ γ i′ )λΓ = A i Σ e V ′ (I + γ i ).First-order condition for γ i .⎧⎛⎪⎨⎜0 = tr ⎝ V ′ − ∑ (Nk=1b k A k + γ k ∑ ) ⎞ ⎛ ⎞⎫′Nl=1 bl A l λ′⎟N∑⎪⎩−( ∑ ⎠ ( b k A k ⎜I) ⎝ Σ e 0⎪⎬⎟⎠Nk=1 γk + I) ′ λ ′ k=1 0 Σ u⎪⎭⎧⎛⎞⎛ ⎞⎫⎪⎨∑ N(⎜ l=1 b l A l) ′ λ′⎟∑N−tr ⎝⎠ (b i A i + γ i b k A k γ i ⎜) ⎝ Σ e 0⎪⎬⎟⎠⎪⎩ λ ′k=1 0 Σ u⎪⎭ .Now take the transpose of the second term:⎧⎛⎞⎛ ⎞⎫⎪⎨∑ N(⎜ l=1 b l A l) ′ λ′⎟∑N0 = −tr ⎝⎠ (b i A i + γ i b k A k γ i ⎜) ⎝ Σ e 0⎪⎬⎟⎠⎪⎩ λ ′k=1 0 Σ u⎪⎭⎧⎛⎪⎨⎜= tr ⎝ V ′ − ∑ (Nk=1b k A k + γ k ∑ ) ⎞ ⎛ ⎞⎫′Nl=1 bl A l λ′⎟N∑⎪⎩−( ∑ ⎠ ( b k A k ⎜I) ⎝ Σ e 0⎪⎬⎟⎠Nk=1 γk + I) ′ λ ′ k=1 0 Σ u⎪⎭⎧⎛⎞⎛ ⎞⎫⎪⎨∑ N(⎜ l=1 b l A l) ′⎟∑N−tr ⎝ ⎠λ ′ (b i A i + γ i b k A k γ i ⎜) ⎝ Σ e 0⎪⎬⎟⎠⎪⎩ Ik=1 0 Σ u⎪⎭ ,which, because the transpose is the identity under the trace, we can write as:⎧⎛⎪⎨⎜0 = tr ⎝ V ′ − ∑ (Nk=1b k A k + γ k ∑ ) ⎞ ⎛ ⎞⎫′Nl=1 bl A l λ′⎟N∑⎪⎩−( ∑ ⎠ ( b k A k ⎜I) ⎝ Σ e 0⎪⎬⎟⎠Nk=1 γk + I) ′ λ ′ k=1 0 Σ u⎪⎭⎧⎛⎞ ⎛⎪⎨⎜− tr ⎝ Σ e 0 ⎟ ⎜⎠ ⎝ (bi A i + γ i ∑ ⎞⎫Nk=1 bk A k ) ′⎟N∑ ( ⎪⎬⎠ λ( b l A l) I)⎪⎩ 0 Σ u γ i′l=1⎪⎭ .k=131