treatment
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We consider the following two models with p=3,

-   Model I: $$Y_i=1+(X_{i1}-X_{i2})(X_{i1}+X_{i3})+\{A_i-\pi({{\bm X}}_{i})\}{{\mbox{\boldmath $\beta$}}}_0{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}_i+\sigma({{\bm X}}_{i})\epsilon_i,$$ where ${{\bm X}}_{i}=(X_{i1},X_{i2},X_{i3}){^{\mbox{\tiny {\sf T}}}}$ are multivariate normal with mean 0, variance 1, and $\mathrm{Corr}(X_{ij},X_{ik})=0.5^{|j-k|}$, $\tilde{{{\bm X}}}_i=(1,{{\bm X}}_i{^{\mbox{\tiny {\sf T}}}}){^{\mbox{\tiny {\sf T}}}}$ and ${{\mbox{\boldmath $\beta$}}}_0=(0,1,-1,1){^{\mbox{\tiny {\sf T}}}}$.

-   Model II: $$Y_i={{\mbox{\boldmath $\gamma$}}}_0{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}_i+\{A_i-\pi({{\bm X}}_{i})\}{{\mbox{\boldmath $\beta$}}}_0{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}_i+\sigma({{\bm X}}_{i})\epsilon_i,$$ where ${{\mbox{\boldmath $\gamma$}}}_0{^{\mbox{\tiny {\sf T}}}}=(0.5,4,1,-3)$, and ${{\bm X}}_{i}$, $\tilde{{{\bm X}}}_{i}$ and ${{\mbox{\boldmath $\beta$}}}_0$ are the same as Model I.

We take linear forms for both the baseline and the contrast functions, where $\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})={{\mbox{\boldmath $\gamma$}}}{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}$ and $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})={{\mbox{\boldmath $\beta$}}}{^{\mbox{\tiny {\sf T}}}}\tilde{{{\bm X}}}$. We assume the propensity scores $\pi(\cdot)$ are known, and we study both the constant case $(\pi({{\bm X}}_i)=0.5)$ and the non-constant case $(\pi({{\bm X}}_i)=\mathrm{logit}({{\bm X}}_{i1}-{{\bm X}}_{i2}))$. In addition, We consider two different $\sigma({{\bm X}}_{i})$ functions, i.e., the homogeneous case with $\sigma({{\bm X}}_{i})=1$, and the heterogenous case with $\sigma({{\bm X}}_{i})=0.5+(X_{i1}-X_{i2})^2$. The simulation results under constant and non-constant propensity scores are similar. Thus, for brevity, we only report the constant case and allocate the result of non-constant case to the Appendix B. The results of Model I and II with constant propensity score are given 