Algorithm 2.

Estimating ψ_0,s using V-fold cross fitting

1: Choose a technique to estimate the conditional means μ0 and μ0,s;

2: Generate a random vector Bn ∈ {1,…, V}n by sampling uniformly from {1,…, V} with replacement, and denote by Dj the subset of observations with index in {i :Bn,i = j} for j = 1,…, V.

3: for υ = 1,…, V do

4: ${\mu }_{\upsilon }$ ← Regress Y on X using the data in ∪j≠υDj using the technique from step (1) to estimate μ0

5: ${\widehat{\mu }}_{s,\upsilon }$ ← Regress ${\widehat{\mu }}_{\upsilon }(X)$ on X−s using the data in ∪j≠υDj to estimate μ0,s;

6: ${\widehat{\psi }}_{n,S}^{\upsilon }\leftarrow \frac{{\sum }_{i\in D_j}{\left\{Y_i-{\widehat{\mu }}_{s,\upsilon }\left(X_i\right)\right\}}^2-{\sum }_{i\in D_j}{\left\{Y_i-{\widehat{\mu }}_{\upsilon }\left(X_i\right)\right\}}^2}{{\sum }_{i\in D_j}{\left(Y_i-{\overline{Y}}_n\right)}^2}$, as in Equation (10);

7: end for

8: ${\widehat{\psi }}_{n,s}^{\text{cv}}\leftarrow \frac{1}{V}{\sum }_{\upsilon =1}^V{\widehat{\psi }}_{n,s}^{\upsilon }$.