Problem 1 (15 points) Consider the following directed acyclic graph which models dependence over time between treatments D, outcomes Y and a unobserved variables U which affect Y but not D. The subscript t denotes time (so D,_1 is the random variable denoting treatment at the time period before t, D, is the treatment at time period t and D,\" is the treatment one period after t) Dt2 > Dtl ' Dz > Dt+1 : I :3 -> > :5 (3.2 UH Ut Ut+1 Part A (3 points) Suppose we are interested in identifying the effect of an intervention on D, on Y,, that is, the \"instantaneous\" effect of a treatment on the outcome. Does the empty set satisfy the backdoor criterion? In other words, can we identify the effect without conditioning on any other variables? Explain why or why not. Part B (5 points) Is {Y,_1, Y,_2} a valid adjustment set under the backdoor criterion? In other words, is conditioning on both of the lagged outcomes Y,_2 and Y,_1 sufficient to identify the effect of D, on Y,? Explain why or why not. Part C (5 points) Find the minimal sufficient adjustment set for the effect of D, on Y,. In other words, what is the smallest set of variables that would be sufficient to satisfy the backdoor criterion and to identify the effect of D, on Y,? Explain your reasoning. Part D (2 points) Suppose you were to add Y,_1 to the set you found in Part C. Would that set still be an admissible adjustment set under the backdoor criterion? That is, would conditioning on that larger set still identify the effect of D, on Y,? Explain why or why not