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  "Description": "Implements contamination bias diagnostics and alternative\nestimators for regressions with multiple treatments. The\nimplementation is based on Goldsmith-Pinkham, Hull, and Kolesár\n(2024) <doi:10.48550/arXiv.2106.05024>.",
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  "Date/Publication": "2024-12-13 16:12:39 UTC",
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      "title": "Multiple Treatment Effects Regression",
      "author": "Michal Kolesár",
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        "Standard errors",
        "Methods",
        "Oracle standard errors",
        "Overlap sample",
        "Wald and LM tests",
        "The score evaluated at the unrestricted estimator $\\hat",
        "Derivations",
        "Finally, we show \\eqref{eq:overlap_oracle}. The derivative of the moment condition\\eqref{eq:moment_overlap_weights} with respect to $\\pi_{k}=p_{k}$ (assumingcorrect specification of the propensity score) is given by\\begin{equation*}-E[\\lambda\\frac{X_{ik}}{p_{k}^{2}(W_{i})} (\\mu_{k}-\\alpha^{\\text{CW}}{k})\\dot{p}{k}(W_{i})],\\end{equation*}where we write $\\lambda$ for $\\lcw(W_{i})$. Since $p_{k}$ is a projection, byProposition 4 in @newey94ecta, the influence function for$\\hat{\\alpha}^{\\text{CW}}{k}$ is given by\\begin{multline*}\\frac{1}{E[\\lambda]}\\left(  \\lambda \\frac{X{ik}}{p_{k}(W_{i})}(Y_{i}-\\alpha^{\\text{CW}}{k})-\\frac{\\lambda}{p{k}(W_{i})}(\\mu_{k}(W_{i})-\\alpha^{\\text{CW}}{k})(X{ik}-p_{k}(W_{i}))\\right)\\",
        "References"
      ],
      "created": "2023-08-02 13:56:59",
      "modified": "2024-07-10 19:13:54",
      "commits": 7
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