# Basic IRM Models

## ATE Coverage

The simulations are based on the the make_irm_data-DGP with \(500\) observations. Due to the linearity of the DGP, Lasso and Logit Regression are nearly optimal choices for the nuisance estimation.

```
DoubleML Version 0.8.2
Script irm_ate_coverage.py
Date 2024-08-13 16:44:37
Total Runtime (seconds) 1115.591309
Python Version 3.12.4
```

Learner g | Learner m | Bias | CI Length | Coverage |
---|---|---|---|---|

Lasso | Logistic Regression | 0.123 | 0.557 | 0.935 |

Lasso | Random Forest | 0.147 | 0.720 | 0.956 |

Random Forest | Logistic Regression | 0.150 | 0.616 | 0.877 |

Random Forest | Random Forest | 0.152 | 0.750 | 0.947 |

Learner g | Learner m | Bias | CI Length | Coverage |
---|---|---|---|---|

Lasso | Logistic Regression | 0.123 | 0.468 | 0.875 |

Lasso | Random Forest | 0.147 | 0.604 | 0.906 |

Random Forest | Logistic Regression | 0.150 | 0.517 | 0.796 |

Random Forest | Random Forest | 0.152 | 0.629 | 0.902 |

## ATTE Coverage

As for the ATE, the simulations are based on the the make_irm_data-DGP with \(500\) observations.

```
DoubleML Version 0.8.2
Script irm_atte_coverage.py
Date 2024-08-13 17:15:42
Total Runtime (seconds) 1171.313816
Python Version 3.12.4
```

Learner g | Learner m | Bias | CI Length | Coverage |
---|---|---|---|---|

Lasso | Logistic Regression | 0.135 | 0.635 | 0.940 |

Lasso | Random Forest | 0.181 | 0.868 | 0.956 |

Random Forest | Logistic Regression | 0.150 | 0.656 | 0.924 |

Random Forest | Random Forest | 0.180 | 0.882 | 0.947 |

Learner g | Learner m | Bias | CI Length | Coverage |
---|---|---|---|---|

Lasso | Logistic Regression | 0.135 | 0.533 | 0.888 |

Lasso | Random Forest | 0.181 | 0.729 | 0.895 |

Random Forest | Logistic Regression | 0.150 | 0.551 | 0.874 |

Random Forest | Random Forest | 0.180 | 0.740 | 0.892 |

## Sensitivity

The simulations are based on the the ADD-DGP with \(10,000\) observations. As the DGP is nonlinear, we will only use corresponding learners. Since the DGP includes an unobserved confounder, we would expect a bias in the ATE estimates, leading to low coverage of the true parameter.

The confounding is set such that both sensitivity parameters are approximately \(cf_y=cf_d=0.1\), such that the robustness value \(RV\) should be approximately \(10\%\). Further, the corresponding confidence intervals are one-sided (since the direction of the bias is unkown), such that only one side should approximate the corresponding coverage level (here only the lower coverage is relevant since the bias is positive). Remark that for the coverage level the value of \(\rho\) has to be correctly specified, such that the coverage level will be generally (significantly) larger than the nominal level under the conservative choice of \(|\rho|=1\).