1. Identify each counterfactual succinctly. Figure 2: Causal graph D. This exercise presents a causal structure for trading in dark pools. Consider the causal structure D from Figure 2. The nodes have the following interpretation. (a) Q,,P are like the prediction bias causal graph. (b) An unobserved external flow M exhibits crowding with the trading process Q. (c) D captures trades between M and Q that cross, making these trades observable. This graph differs from the synchronization trading graph in two ways: first, the external flow M interacting with Q over the dark pool is unobserved. Second, there is no leakage: Q does not trigger M. The only source of confounding is crowding, represented by node {. 1. Show that Q and M are independent conditional on . 2. Show that Q and M are dependent conditional on ,D. Explain the reason for this dependence. 3. Identify E[Pdo(Q),D], the impact of Q conditional on observing a cross. How does it differ from the unconditional impact E[Pdo(Q)] ? Comment. 1. Identify each counterfactual succinctly. Figure 2: Causal graph D. This exercise presents a causal structure for trading in dark pools. Consider the causal structure D from Figure 2. The nodes have the following interpretation. (a) Q,,P are like the prediction bias causal graph. (b) An unobserved external flow M exhibits crowding with the trading process Q. (c) D captures trades between M and Q that cross, making these trades observable. This graph differs from the synchronization trading graph in two ways: first, the external flow M interacting with Q over the dark pool is unobserved. Second, there is no leakage: Q does not trigger M. The only source of confounding is crowding, represented by node {. 1. Show that Q and M are independent conditional on . 2. Show that Q and M are dependent conditional on ,D. Explain the reason for this dependence. 3. Identify E[Pdo(Q),D], the impact of Q conditional on observing a cross. How does it differ from the unconditional impact E[Pdo(Q)] ? Comment