Algorithm 3.

Causal MARS.

Require: Data (Xi, Yi, Yi), parameter D

Define ℱ ≡ {{(xj − c)+, (c −xj)+} : c ∈ {Xij}, j ∈ {1,…,p}}

Initialize ℬ = {1}

$\widehat{\beta }\leftarrow \arg {min}_{\beta }{{\displaystyle {\sum }_{i=1}^n\left(Y_i-{\beta }_h^1{\mathbb{I}}_{T_i=1}-{\beta }_h^0{\mathbb{I}}_{T_i=0}\right)}}^2$

$R_i\leftarrow Y_i-{\widehat{\beta }}_h^1{\mathbb{I}}_{T_i=1}-{\widehat{\beta }}_h^0{\mathbb{I}}_{T_i=0}$

for d in 1,…,D do

for {f, g} in {{b(x)f*(x), b(x)g*(x)} : b ∈ ℬ, {f*,g*} ∈ ℱ} do

$RS{S}_{\mu }\leftarrow {min}_{\beta }{\displaystyle {\sum }_{i=1}^n{\left(R_i-{\displaystyle {\sum }_{h\in \left\{f,g\right\}}{\beta }_{h}h\left(X_i\right)}\right)}^2}$

$RS{S}_{\tau }\leftarrow {min}_{\beta }{\displaystyle {\sum }_{i=1}^n{\left(R_i-{\displaystyle {\sum }_{h\in \left\{f,g\right\}}{\beta }_h^{1}h\left(X_i\right){\mathbb{I}}_{T_i=1}}-{\displaystyle {\sum }_{h\in \left\{f,g\right\}}{\beta }_h^{0}h\left(X_i\right){\mathbb{I}}_{T_i=0}}\right)}^2}$

dRSS = RSSτ − RSSμ

end for

Choose {f,g} which maximize dRSS

$\widehat{\beta }\leftarrow \arg {min}_{\beta }{\displaystyle {\sum }_{i=1}^n{\left(Y_i-{\displaystyle {\sum }_{h\in \left\{f,g\right\}}{\beta }_h^{1}h\left(X_i\right){\mathbb{I}}_{T_i=1}}-{\displaystyle {\sum }_{h\in \left\{f,g\right\}}{\beta }_h^{0}h\left(X_i\right){\mathbb{I}}_{T_i=0}}\right)}^2}$

$R_i\leftarrow Y_i-{\displaystyle {\sum }_{h\in \left\{f,g\right\}}{\widehat{\beta }}_h^{1}h\left(X_i\right){\mathbb{I}}_{T_i=1}}-{\displaystyle {\sum }_{h\in \left\{f,g\right\}}{\widehat{\beta }}_h^{0}h\left(X_i\right){\mathbb{I}}_{T_i=0}}$

$\mathcal{B}\leftarrow \mathcal{B}\cup \left\{f,g\right\}$

end for

Backward deletion: delete terms one at a time, using the same criterion dRSS = RSSτ − RSSμ

Use out-of-hag dRSS to estimate the optimal model size.