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Technical Details

The computational task of determining the convex hull membership is made feasible even for large numbers of explanatory variables and observations by the solution proposed in King & Zeng (2006), which eliminates the most time-consuming part of the problem: the characterization of the convex hull itself. In addition, they show that the remaining (implicit) point location problem can be expressed as a linear programming exercise, making it possible to take advantage of existing well-developed algorithms designed for other purposes to speed up the computation. Specifically, a counterfactual $x$ is in the convex hull of the explanatory variables $X$ if there exists a feasible solution to the following standard form linear programming problem:

$$C'\eta$$ min 

$$A'\eta=B'$$ s.t.  (1)

$$\eta \geq 0$$

where $C$ is a vector of zeros (so that there is no objective function to minimize); $\eta$ is a vector of coefficients; $A'$ is $X'$ with an additional, final row of $1$'s; and $B'$ is $x'$ with an additional, final element equal to $1$.

The default Gower distance (which is suitable for both quantitative and qualitative data) between a pair of $K$ dimensional points $x_i$ and $x_j$ is defined simply as the average absolute distance between the elements of the two points divided by the range of the data:

$$G_{ij} = \frac{1}{K}\sum_{k=1}^K \frac{\left\vert x_{ik} - x_{jk}\right\vert}{r_k}$$ (2)

where the range is $r_k =$   max$(X_{.k}) -$   min$(X_{.k})$ and the min and max functions return the smallest and largest elements respectively in the set including the $k$th element of the explanatory variables $X$. The optional squared Euclidian distance (which is suitable only for quantitative data) between points $x_{i}$ and $x_{j}$ is given by the familiar definition, i.e. the sum of the squared differences between the elements of the two points:

$$E_{ij} = \sum_{k=1}^K (x_{ik} - x_{jk})^2$$ (3)