While it is easily tractable to compute the Shapley Value for a small number of

While it is easily tractable to compute the shapley

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While it is easily tractable to compute the Shapley Value for a small number of covariates such as 3, it becomes computationally cumbersome to do it for a large number of covariates. Already for 4 covariates, you have 24 elimination paths to compute, with 5 it is 120, and 720 for 6. Simple combinatorics gives you that for n covariates there are n ! elimination paths for each covariate. Note however that, as we are interested only in marginal contributions, there will be redundancy in some marginal contributions. But still the method is computationally heavy. Then researchers have to precisely define which contribution they are interested in. 16
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One way to reduce the computational burden might be to group covariates describing related phenomena together and then adding or removing them by group. For instance, we are interested in the contribution of the housing market to segregation but we only have some characteristics of the dwellings such as the number of rooms, the type of construction (hut, house brick, shacks...), some of their equipments (light, water, toilets...). Then, all these variables together are going to tell us something about the housing market. Thus, it does not make sense to evaluate their contribution individually. We will use this alternative in our analysis. Thus, our final specification is: Segregation ij ( t ) = α + Segregation i ( t - 1) × β 1 + Demographics ij ( t - 1) × β 2 + Income ij ( t - 1) × β 3 + HousingCharacteristics ij ( t - 1) × β 4 + PublicGoods ij ( t - 1) × β 5 + ij (10) with β 1 , β 2 , β 3 , β 4 , β 5 column vectors of coefficients associated to a particular regressor in the corresponding subsets of regressors. Segregation i ( t - 1) is a row vector composed of two regressors, the cumulative distributions of Whites and Blacks of the subplace i at the previous period. Demographics ij ( t - 1) is a row vector composed of six regressors, the mean age, the marriage to divorce ratio, 40 the sex ratio, the average number of years of schooling, the share of individuals speaking ”White” languages, 41 , and the unemployment rate. All these regres- sors are computed by subplace i and population group j at the previous period. Income ij ( t - 1) is the mean income of group j individuals living in subplace i at the previous period. HousingCharacteristics ij ( t - 1) is a row vector composed of five regressors, the mean number of rooms in the dwelling, the share of houses in brick, the share of informal dwellings, the share of owners, and the rural to urban ratio. PublicGoods ij ( t - 1) is a row vector composed of two regressors, the share of households not having access to a refuse disposal, and the share of households not having access to public water. Finally, the dependent variable Segregation ij ( t ) is the cumulative distribution of group j individuals for subplace i a the current period. 3.6 Practical issues We will face some practical issues with the estimation of the segregation curves and their inference. First, observed and counterfactual segregation curves might 40 We aggregate all types of marriage (civil monogamous union, polygamous, and traditional).
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