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

Substituting for λ 11 − c = λ 11−λ 122and λ 12 − c = λ 12−λ 112, and rearranging yieldsd 1 = cov(λ 11−λ 122(q 1 − q 2 ),q 1 − q 2 )var(q 1 − q 2 )= λ 11 − λ 12.2The respective shares of the total variation attributed to P X+u and P w areS(P X+u ) = 2c 2 var(P X+u )var(p 1 ) + var(p 2 ) = 2c2 var(q 1 + q 2 )var(p 1 ) + var(p 2 ) = 2c2 var(q i ) + cov(q 1 ,q 2 )(λ 2 ii + λ2 ij )var(q i) + 2λ ii λ ij cov(q i ,q j )S(P w ) = 2d 2 ivar(P w )var(p 1 ) + var(p 2 ) = 2d2 iwhich we can calculate numerically.var(q 1 − q 2 )var(p 1 ) + var(p 2 ) = 2d2 ivar(q i ) − cov(q 1 ,q 2 )(λ 2 ii + λ2 ij )var(q i) + 2λ ii λ ij cov(q i ,q j ) ,Calibration. Our model is sufficiently stylized that it is not directly comparable to the Hasbrouckand Seppi framework. To appreciate the stylization, note that in our model only two factors drivepricing (reflecting the two net order flows) 6 , whereas Hasbrouck and Seppi find three significantfactors. Further, in our two-asset model, the ratio of the first two eigenvalues is entirely drivenby the correlation structure of the matrices, whereas in the data, the correlation structure is notthe sole determinant. We also consider a symmetric setting, whereas Hasbrouck and Seppi averageover heterogeneous stocks.Our findings in Propositions 4 and 5 that (a) speculative trade is proportional to (AΣ e A ′ ) −1/2 Σ 1/2uand (b) the correlation of price changes is unaffected by the cross-asset correlation of liquidity tradeη, imply that as long as the correlation in liquidity trade is high enough that cross-asset order flowsare positively correlated,S(P X+u )S(P X+u )+S(P w )depends only on the cross-asset signal correlation θ, andnot on either η or the scales of asset value innovations and liquidity trade. Raising θ increases thefraction explained by commonality in order flow. We find that a relatively small signal correlationof θ = 0.2 generates the two-thirds share ratio that Hasbrouck and Seppi find. 7 For θ = 0.2,the cross-asset correlation in liquidity trade must be at least η = 0.135 for net order flows to bepositively correlated (and, for example, η = 0.25 generates a net order flow correlation of 0.12).Taking our model further, we can compare the ratios of the first two eigenvalues in the covariancematrices of order flow and price changes with their empirical counterparts. Fixing θ = 0.2, aliquidity trade correlation of η = 0.5 allows us to match the 2.183 eigenvalue ratio for order flowsthat Hasbrouck and Seppi find. Because the correlation of price changes is unaffected by the correlationof liquidity trade, η only influences the eigenvalue ratio for order flow; and θ = 0.2 implies19