X B β W βB K X k 1 X kB βk W βk B 7 However in nonlinear settings this is not

# X b β w βb k x k 1 x kb βk w βk b 7 however in

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X B ( β W - β B ) = K X k =1 ¯ X kB ( β k W - β k B ) (7) However, in nonlinear settings, this is not as simple. First, contributions of a particular covariate will be obtained by computing the marginal impact obtained when the counterfactual is computed by replacing the distribution of this covariate by the one of the other group. Then, there is two possibilities, either the researcher pursues with this strategy for the other covariates without changing back the distribution of the first covariate, or he can change back the first covariate to its observed distribution and changes the distribution of the second covariate instead. The first strategy might be qualified as ”without replacement”, while the second is considered ”with replacement”. But without replacement, the decomposition will be path dependent, while without it will not add up to the aggregate effect. With distribution regressions methods, it is possible to compute a detailed decomposition without replacement. The marginal contribution C k of a particular covariate k will thus be the difference between the aggregate decomposition with and without this covariate for respectively the composition (c) and the structure (s) effects: 15
C k c = Δ c ( K ) - Δ c ( K \ { k } ) (8) and C k s = Δ s ( K ) - Δ s ( K \ { k } ) (9) with Δ s ( K ) the aggregate structure effect for the set of covariates K, while K \ { k } is the same set without the covariate k. 3.5 Shapley Value Shorrocks[49] proposes a solution to the problem of path dependence in detailed decomposition. By averaging all the marginal contributions along all the possible elimination paths, i.e. the possible order in which a particular covariate can be added to other covariates, you obtain a contribution of this covariate which is unique, adds up to the aggregate effect, and is path independent. Contributions are called Shapley Value because they are computed exactly the same way. Initially, the Shapley Value was conceived as the solution to divide fairly the reward of a cooperative game among players according to their efforts. For example, imagine we have three covariates, X = { X 1 , X 2 , X 3 } and we want to know the contribution of X 3 . Then, we have exactly 3! ways to order our covariates. Of course, the order will not matter when the model is evaluated with all the covariates. But it will change the marginal contributions. Let us say that we include first X 1 , then X 2 , and finally X 3 . We will get the particular elimination path {{ X 1 } ; { X 1 , X 2 } ; { X 1 , X 2 , X 3 }} . Then, if we denote the model we will evaluate for a set of covariates by the function V ( . ), we will get three different evaluations of this model, namely { V ( { X 1 } ); V ( { X 1 , X 2 } ); V ( { X 1 , X 2 , X 3 } ) } . But this particular elimination path only interests us for the marginal contribution of X 3 to the evaluation of the model which is in this case V ( { X 1 , X 2 , X 3 } ) - V ( { X 1 , X 2 } ). Finally, we have to repeat this operation for all elimination paths and computing the arithmetic mean of all these marginal contributions.

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