instrumental variables
play

Instrumental Variables Brady Neal causalcourse.com <latexit - PowerPoint PPT Presentation

Instrumental Variables Brady Neal causalcourse.com <latexit


  1. <latexit sha1_base64="kM2Kw9peVD0/n13t7Mo4wEYSnXc=">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</latexit> Unobserved Confounding Week 5: • Frontdoor adjustment • Unconfounded children criterion • Some other fancy application of U do-calculus T Y Brady Neal 2 / 33

  2. <latexit sha1_base64="kM2Kw9peVD0/n13t7Mo4wEYSnXc=">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</latexit> Unobserved Confounding Week 5: • Frontdoor adjustment • Unconfounded children criterion • Some other fancy application of U do-calculus Week 7: T Y Brady Neal 2 / 33

  3. <latexit sha1_base64="kM2Kw9peVD0/n13t7Mo4wEYSnXc=">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</latexit> Unobserved Confounding Week 5: • Frontdoor adjustment • Unconfounded children criterion • Some other fancy application of U do-calculus Week 7: T Y • Set identification (bounds) Brady Neal 2 / 33

  4. <latexit sha1_base64="kM2Kw9peVD0/n13t7Mo4wEYSnXc=">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</latexit> Unobserved Confounding Week 5: • Frontdoor adjustment • Unconfounded children criterion • Some other fancy application of U do-calculus Week 7: T Y • Set identification (bounds) • Sensitivity analysis Brady Neal 2 / 33

  5. <latexit sha1_base64="kM2Kw9peVD0/n13t7Mo4wEYSnXc=">AWq3icrVhZb9tGEN7YTZu4ba4CfukLG9tAsiq7LZIgMBACdBUTRFgjpHYxkBj5VEiFfJpRVH0E/oD+xT/0pnvl2KpC7KQSIXA5nvpmda3flJIGfqU7n3ysbm19c/fKra9e3v7m2xs3b92+8zqL89SVr9w4iNO3jp3JwI/kK+WrQL5NUmHTiDfOMNjfv/mXKaZH0cn6iKRZ6Hdj/ye79qKSO9vb3zoOrLvR2PlDz8mvqvyVE62LPZs/6IPZlNn7sRPZ5mylay5dnZQHpn1r1uHsVOJtNz6blx1IvzyCOF9y1bWfc6rc59a7y7mGV38mgRMkMqmoMKZaTuW6eODOKRFcieso7inrUY64y1TKVWIMe5cuNQTnFTvz9oBDZCVdi9GVw/ylSaG5NtJz6XFZOnhgGtZC0AKw5/6vWrDk9sNVjuYknMNW89uqRk4Y15uUoElvLuFczVyS+xqSsjr5Zk72/tdNodfKz5wYEZ7AjzeRHfv6P6ApPxMIVuQiFJFQNA6ELTL6noD0REJ0c7EmGgpjXy8l2Iitkg2Jy5JHDZRh3Tt09OpoUb0zJgZpF3SEtAvJUlL7Bkej8Y9UPWd9VsV3mU6xsBmGy/o7hjMkKhKDIjaJFdwri/n0DdsmLWiOTzEbH2aSQIK+8GtzblH94CeFc2QrxfEKWnkVRKI5doAVE1hXWkdNeZ98MEAkbfJGbPUqu1dZze9j+g4Jy6ZxBkvZVks8M3GJoFnCVuYJENPliB9ohtq+Vi96Rx8xF3PgnlPiDIUH2lUR16ls5pdy7mUQZ4s0MX8ChwRcWdEiZH5PvnPJ456rqEyTpsRLKPuSTIoLZB/g3vi2xOKJ7sCdD1Czkmw89iams0u7CzoDuDrBTkh/TuwH0sC9aZI8EdorsLbzYAndKTyM8ubDRnaG3UZkczxbNibOrRbgxvfGnWNoT2p+FRdq6MdEs890Xv0O/pLi34BXO6xbJxrBQZ3EGXZGJ/RH1FfZLSFemsmcLSgs6QtgxQLYMiVbq0is70eaRYb6CQzqmO4BvJYApQXt7J8RxiHmxBHNoXtEYw96WNqiXtcWD4wdk5pW9o2P7jSv1UL+ce3qzjirleUiVKplsvImjriZzNjgDbXfWa7qh7skZjeZscqB/WX4Byt28dSZXzqsfKQjQNI1XHZi4vnuWguh6iTFmLJvu8TxGkeoaeoTYTo2mrUn/HpkpixMPFPTQ5oeudtY1m3oyJHhpPBcSn65zXJH7zN2HayKkuvFd9vwrxqZEvVgPu+GNQV8u9RmXVZVMajadvVsv7GEn4R484hovga8TzOWH6c+qvFkX+/PgRXOInLHKIK2L7iE3Fs/8ROxQL9ihrKr7kyPNlBS7k4TGu1POXcJl3iG0REThCh9P308aY/CEKBPjsR7eFXVY2uCj83ng7JrM0xy6otQMB1f1aq2cl4vyRtNXy3rTfK/LavpqWacIzd90w8yYL1tkFOIA0uElSw9aZCKsSZourb5rzXsi7DW5iYvUmPhmwbJED1Gzyo2Vfm80T6FLpgan7DMu0/w4Dn63wQ7oMv6sZRVl/JmKXfxST4t5UeX9GwpGV7Sv6XkxwZJibXBn9JY5mVjTHXi0YunU9Na4Kk1aTYv+hqa+HqkBd5zfZMZ7LMesNZoVfHI/SA1lr9eX7d4Uo+aKzjHN3XQR9KodlbUTuvPhtaGcF8jflyPfyfLlOfS2Sm82bcg9R7P2zyuriA7s8F4yxhxuZdajYze0jfjoCv1Ine047JB4fi59od3ZA12e1NWtdVD4l5abVXEPCfkBrTZPsCpdHjcxZ5n92omnruMpaSh+E3M2LkDs4NfhnQZvYuRdc54cyeJWU16rzgLn1+umhALNwMd7q+C5G1v9zxNi/l6eoz4FcPSGuh1fmzPLzfNO/DMvPvgnde3QN0I8zw/sEWRegL0vqEPp8rvc+x3P/NHD+HlJVcO6O796Z0VfHnz279nH2zImj5NFnEAVpTasiLjsFJ5BRZq+X4cRYnovbtXOuxpu1LavkVWjwjkwvkqbXb72/tXMw+9/Z/OD1Yfvgl3bn5eHO4fmf7Vr4ntxV9wjPz0Qj6kyX5Bf3Y3/Nq9u3ti8ub2/ef2u+2uZt24YmS+E7XPtvwfPIi3+Q=</latexit> Unobserved Confounding Week 5: This week: • Frontdoor adjustment Instrumental variables • Unconfounded children criterion • Some other fancy application of U do-calculus Week 7: T Y • Set identification (bounds) • Sensitivity analysis Brady Neal 2 / 33

  6. <latexit sha1_base64="uSnWmzenLZoB/62kgJqyZD0I1U=">AWqXicrVjbtGEN3YvSRu0iYp4Je+sLEDJKisym6LBAgMBHASFEVTJICTuLEMg5eVRIi3ku7jqBP6Af2of/SmbNLkdSNUhAJIpfDmTOzc9tdOUngZ6rT+faxuZn3/x5fUbW1/dvPX1N7fv3H2bxXnqyjduHMTpiWNnMvAj+Ub5KpAnSrt0AnkO2d4xO/fXcg08+PoWF0l8iy0+5Hf81bEen8zsZF15F9Pxopf/gh8V2Vp3K8ZnPfeuP2JPZ5Lkb0eNpmwlW56dDaR3Zj3o5lHsZDK9kJ4bR704jzxS+NCylfWg0+o8tEa781l2x0/mITOkojmoUEbqoXqyC+tALZU9Zh3LPmY52xlonUEuQ4V24cyglu6vcHjcBGaAmsH2UqzY3FthNfyIrFE7sAVrIWeBV/P/f6VX8nthos9rAk5pqznqwpWThjVq4SgEbe6twXmNSVkVdLsfPbO512Bx9rdrBvBjvCfF7Fd278I7rCE7FwRS5CIUkFI0DYuMvqdiX3REQrQzMSJaSiMf76UYiy2SzYlLEodN1CFd+/R0aqgRPTNmBmXtAT0S0nSEvcNj0fjHqj6zvqtCu8iHSNgs41XdHcMZkhUJQZEbZIrOFeXc+gbNsxa0RweY7Y+zSQBhf3g1ubco3tAz4pmyNcr4pQ08kgqpZFLtIComsI6Urprz7NvBoiEDT5JI7Z6md3LrOb3MX2HhGXTOIOlbKslXpi4RNAsYSvzBIjpYsS/aYbavmVYvckcfMRdz4J5j4kyFB9oVEdeprOaXYu5lEez9HF/AocEXFnRImR+T75zyeOeq6hMk6bESyj7kyKC2Qf4N74tsTie7AnQ9Qs5JsPYmprNLuws6A7g6wU5If0bsB9LAvWmSPBHaK7C282AJ3Sk+XeHJhoztFb6MyOZ4tmhNnV4twY3rjT7C0J7Q/C4u0dSOiWea7J36Hfklxb8ErnNctko1hoc7iDLoiE/tD6ivsl5CuTGXPFpQWdISwY4BsGRKt1KeRWN+PNIsM9RMY1BHdA3gtAUoL2tk/lxiHmBNHNIfuSxp70MPSFvW6tnhk7BjXtLJvfHSnWa0W8o9rV3fGa0sF6FSLZOVh9DUET+bGXME2O6q1zTDXVPzjC78ZRVDuwvuwbnaN0+lirjU4+Vh2wcQKqOy16cP895czlAnbQS/Z9nzgOIdUz9Ay1mRhNW5X6OzJVEiMeLu6hyQld76ztcurNiOih8VRAfLrOeU3iN38Rpo2c6sJ71fLEJ8b+WI14I4/AnW53FtUVl02pdFo8ma5vI+RhH/0iCOu8RL4OsFcvp/8rMqbVbE/DV40g8gZqwzSqugecmP+zI/FDvWCHcquj850kxJsTtJaLw74dwlXOYdQktEFK7w0eT9uDEGz4gyNh7r4V1Rh6UNPjqfB86uyTzNoStKTXFwVS/Xynk5L280fbmsN8n3uqymL5dlygVy0zf9IAPWSYOcQhxYIqxk6XGDVIw1QdO1zX+uYF+EtTY3eZEaC981SIboMXpWsanKl432KXTB1PiEZd5/hAcv0P/G2AGt68dSVq3lzVLu6qN8WspfrunZUjJc07+l5IcGSYm1wZ/QWOZ1Y0x16tGLp1PTWuCpNWk2L/oamvh6pAXec32TGeyzHrDWaFXx0P0gNZK/Xl23eFK3m+s4xzd10EfSqHZW1I7bz4ZWhnBfIX5Zcj38ny5Sn3Nk5vOm3IPUez9s8rq4gO7PBeMsIe7NOtQsZvbQ/x0BH6lTvaSdkg8PhI/0e5sn64vamvWqh8SspNt6viHhDyI1ptnmFVWh83MWeZvdqJp67jOWkofmNzNiy5A7ODX4S0jt75yDpnvJmTxLQmvVcEJc+P101IJZOB9veXznI+v/OWLs38tT1KdArp4QV8Mrc2bxeb7pX4bFZ9+E7j26BujHmeF9hqwL0JcldQh9Ptd7n6OZfxo4fw+oKjh3R/fuTumr40+fXfs4e+bEUfLoM4iCtKZVERedghPIKLPXy3BiLM/F7do5V+N25ZV8io0eIemF0nT67fOb+/sT/93Njt4e9De/6XdeX2w8/Sx+V/tuvhO3BMPyE+PxFOqzFfkV3fjv83NzZubt7Z/2H69fbL9XrNuXDMy34raZ9v9H7Int5s=</latexit> Unobserved Confounding Week 5: This week: • Frontdoor adjustment Instrumental variables • Unconfounded children criterion • Some other fancy application of Z U do-calculus Week 7: T Y • Set identification (bounds) • Sensitivity analysis Brady Neal 2 / 33

  7. What is an Instrument? No Nonparametric Identification of the ATE Warm-Up: Linear Setting Nonparametric Identification of Local ATE More General Settings for the ATE Brady Neal 3 / 33

  8. What is an Instrument? No Nonparametric Identification of the ATE Warm-Up: Linear Setting Nonparametric Identification of Local ATE More General Settings for the ATE Brady Neal What is an Instrument? 4 / 33

  9. <latexit sha1_base64="/hLeD4FcMEV8euIoCVN+BKXDP0=">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</latexit> Assumption 1: Relevance Z has a causal effect on T U Z T Y Brady Neal What is an Instrument? 5 / 33

  10. <latexit sha1_base64="/hLeD4FcMEV8euIoCVN+BKXDP0=">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</latexit> Assumption 1: Relevance Z has a causal effect on T U Z T Y Brady Neal What is an Instrument? 5 / 33

  11. <latexit sha1_base64="/hLeD4FcMEV8euIoCVN+BKXDP0=">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</latexit> Assumption 2: Exclusion Restriction The causal effect of Z on Y is fully mediated by T U Z T Y Brady Neal What is an Instrument? 6 / 33

  12. <latexit sha1_base64="/hLeD4FcMEV8euIoCVN+BKXDP0=">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</latexit> Assumption 2: Exclusion Restriction The causal effect of Z on Y is fully mediated by T U Z T Y Brady Neal What is an Instrument? 6 / 33

  13. <latexit sha1_base64="SXUJchWKlejMrSU7143taV/9zHo=">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</latexit> Assumption 2: Exclusion Restriction The causal effect of Z on Y is fully mediated by T U Z T Y Brady Neal What is an Instrument? 6 / 33

  14. <latexit sha1_base64="SXUJchWKlejMrSU7143taV/9zHo=">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</latexit> Assumption 2: Exclusion Restriction The causal effect of Z on Y is fully mediated by T U Z T Y Recall: Removing edges corresponds to adding assumptions Brady Neal What is an Instrument? 6 / 33

  15. <latexit sha1_base64="SXUJchWKlejMrSU7143taV/9zHo=">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</latexit> Assumption 3: Instrumental Unconfoundedness Z is unconfounded (no unblockable backdoor paths to Y) U Z T Y Brady Neal What is an Instrument? 7 / 33

  16. <latexit sha1_base64="SXUJchWKlejMrSU7143taV/9zHo=">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</latexit> Assumption 3: Instrumental Unconfoundedness Z is unconfounded (no unblockable backdoor paths to Y) U Z T Y Brady Neal What is an Instrument? 7 / 33

  17. <latexit sha1_base64="K53ughT9jty+/kw1QKV+eOgpqA=">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</latexit> Assumption 3: Instrumental Unconfoundedness Z is unconfounded (no unblockable backdoor paths to Y) U Z T Y Brady Neal What is an Instrument? 7 / 33

  18. <latexit sha1_base64="0iqNsrzTQ+P1Sl71NXqyPFhwHs=">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</latexit> Conditional Instruments W U Z T Y Brady Neal What is an Instrument? 8 / 33

  19. <latexit sha1_base64="TgeuzHqWqDHtSNbnWDpV8XNDN/o=">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</latexit> Conditional Instruments W U Z T Y Brady Neal What is an Instrument? 8 / 33

  20. <latexit sha1_base64="TgeuzHqWqDHtSNbnWDpV8XNDN/o=">AXxHicrVhZbxs3EGZit03cK0kf/FCg2MYO6gBrVXIbOEAgICToCiaIgGco7UMYw9KIrRXd7lWnYX6D/rQn9d/0sfODLmXjl0piARpqeHMN8O5SMqOPJHIbvfa9e3tj/6+JMbN3c+/ezL768dfvO6yRMY4e/ckIvjN/aVsI9EfBXUkiPv41ibvm2x9/YkxOcf3PJ40SEwam8ivi5b40CMRSOJYF0cXvrn4HNRyLIpJi8i4Qj05jPdgz9umf8Gro8KX4PAvh5lkhLcnMoPK/vhIErEkEo0Oy5tw4GAB1GKaBC3rvG5Y0Drpm976R7Vcn9mePlsK6VjLmLqKkQWgnPL7kbg3vzOZeODX64dCo4p0j/lKRFZpQhQRXSZ8HsoD1+FAq7KVY5lUyFkPZP+w5PmksEBq0hKl0Qp8XOmIxGm+kRAM0qBIuNUr8Syw0teWUlhI4GVrDleJdxP3VE13JElx6sjwYG5sRHG0rmjlmUW8ZdWeMaAuDrwCUn9I3eg/NGbZU0aLWs6ukGB9xr4K+D8DOWvVd3Nrdr0MhYHPT3Y/r1Irx98282YC4LmcNS5jPOAiZh7DGLJfA+Yz3WZRHQzlkGtBhGguY5m7EdkE2BiwOHBdQJfI/g15mBvAbMROSdkCLB58YJA12T/O4MB4SVT1Rv1HhXaUjI2y08Qqetsb0gSrZGKhtcjn+nI2vP2WVUtYw0NarYCVRERBPzi1NQ/h6cFvCSvE7yvg5DByQSqGkQM0D6iKgjpieCrPo2/GFAmL+DiM0Oomu5usxvkQ3hPAsmCckKVoq8Ge6bgEpJmTrcjUxXI/4JK1T2NWENizUIirtaBfKeAmXC3sGojtyks5pdq7mkRp4t0YX8kjgC4E6AElLmC/CfAI56rjqAiTosiuSI1hJRBnU08s80n2dzBPE8JHsSipB+SZIT6Qrq7Q7t9ODp03YMchnMDcmPegLE+zhB1T9uZeNIk7hl9T+uWQjc4cvUOVifE0YU2YXSbghjAjCizlCeXP3CJlXQY0Q78P2S+kn0PcTfIK5rUJsiFZqLI4IV2Bjn0f+gr6xYdvpKJnc4pJOnyY0zZMgFaqU8hob7vYRUJ1Y+nUTN4euS1iFBM0o7+mdLYpzVhRFPSPYWxS3pQ2oBe12H2o5ZTSv6RlB3WtRqUP5h7arOK8V5QKqVENnZ80dmPesUYAbS76jVHd0PVkxNa3WzOKpvsL7sG5mjdPpQq41OPlUvZOCapOi56cfk6l63liOrEpFi70fA0SepoaYnVJuR1rRTqb8TXSUhxcOhp69zQtU7apvOzWRA97WnPOBTdY57Es78AZgW5dSAvFedb0J8quXz3QA7fkbUZrnXVFl12RhGWTHTLC9oxMk/aoQRV3gR+TqitXxbfIzKzLrYHwYvWEDEjJUaV10l3Jj+cpP2R70gj3Iqro/MdJIiel0EsF4v+DcB1zknZCWAChY4VkxP2uNwROgzLTHhjSX12Fpg6DO5xLnQGe4lAVJec4sKqbtWJeLsbRW+WdYt8r8sqerMsUi4pN4XuBwlhvW2RkxQHlPArWXraIhXSnqDoyubf1rAvoL021XkRawvftEj61GPUqkJdlc9b7ZPUBWPtE5T5/T08eEn9b0YnoE39WMrKjbxZyl29l09L+emGni0l/Q39W0q+a5HktDeIgoYyL1trCrletHKpfGrbEzjsJvn5RVWbSd82eBH3bFd3JkPvN5gVanfsUw8w1+rPi/sOVnKvtY5T6r429aGYNLsNtfPqg6GVEUzXWF9C+V7eL9epr2Vy83lTniHys39S2V0EYZf3gozOcFO9D+WnuUOKn4rAT9DJnsMJCcn7Ac4nfXg+1ltz1oXFW9Jqe52VdwjQD6G3eYJ7Uqb40b6LnNYu/HUdTwFDflnpu+GJbenT/CrkDbRuxZ5Yy7cJOY16TOimPgUvenqxbEMguX4zXHdzmy+p8jpPN7eYv6EMjVG+J6eGXOrL7Pt/3LsPruG8FzCN8e9eNE8z6hrPOoL3PoEOp+rs4+Jwv/NGD+HkFVYO5md+/M6avjz9dR3T3TIGj5F3EnSilZFXHULjkhG6rNeQjfG8l7cqd1zFd68bUklr3yN19e9iOtev3Nxa683/9/Z4uD1Uaf3oN9ebT3+KH+X+0G+5rdZQfgp2P2GCrzBfjV2fpv+5vt7YPdp/tervJbqpYr1/TMl+x2mv3r/8Bj2cWig=</latexit> Conditional Instruments Slightly weaker version of Assumption 3: W Unconfoundedness after conditioning on observed variables U Z T Y Brady Neal What is an Instrument? 8 / 33

  21. Question: What are the 3 assumptions we need to say that a given variable is an instrument, and what do they correspond to graphically?

  22. What is an Instrument? No Nonparametric Identification of the ATE Warm-Up: Linear Setting Nonparametric Identification of Local ATE More General Settings for the ATE Brady Neal No Nonparametric Identification of the ATE 10 / 33

  23. No Nonparametric Identification of the ATE Brady Neal No Nonparametric Identification of the ATE 11 / 33

  24. No Nonparametric Identification of the ATE Why didn’t we see instruments in Week 5 Identification? Brady Neal No Nonparametric Identification of the ATE 11 / 33

  25. No Nonparametric Identification of the ATE Why didn’t we see instruments in Week 5 Identification? Week 5 was about nonparametric identification Brady Neal No Nonparametric Identification of the ATE 11 / 33

  26. <latexit sha1_base64="uSnWmzenLZoB/62kgJqyZD0I1U=">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</latexit> No Nonparametric Identification of the ATE Why didn’t we see instruments in Week 5 Identification? Week 5 was about nonparametric identification Recall necessary condition for nonparametric identification: For each backdoor path from T to any child that is an ancestor of Y, it is possible to block that path Z U T Y Brady Neal No Nonparametric Identification of the ATE 11 / 33

  27. What is an Instrument? No Nonparametric Identification of the ATE Warm-Up: Linear Setting Nonparametric Identification of Local ATE More General Settings for the ATE Brady Neal Warm-Up: Linear Setting 12 / 33

  28. Another Assumption: Linear Outcome <latexit sha1_base64="UeVu8hQ8nYOpqwWbqcmxjVPKxk=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 13 / 33

  29. <latexit sha1_base64="uSnWmzenLZoB/62kgJqyZD0I1U=">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</latexit> Another Assumption: Linear Outcome <latexit sha1_base64="UeVu8hQ8nYOpqwWbqcmxjVPKxk=">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</latexit> Y := δ T + α u U Z doesn’t appear in this structural equation because of the exclusion Z U restriction assumption T Y Brady Neal Warm-Up: Linear Setting 13 / 33

  30. <latexit sha1_base64="uSnWmzenLZoB/62kgJqyZD0I1U=">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</latexit> Another Assumption: Linear Outcome <latexit sha1_base64="UeVu8hQ8nYOpqwWbqcmxjVPKxk=">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</latexit> Y := δ T + α u U Z doesn’t appear in this structural equation because of the exclusion Z U restriction assumption T Y Linear Brady Neal Warm-Up: Linear Setting 13 / 33

  31. <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> Warm-Up: Binary Linear Setting Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 14 / 33

  32. <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> Warm-Up: Binary Linear Setting Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 14 / 33

  33. <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> Warm-Up: Binary Linear Setting Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 14 / 33

  34. <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> Warm-Up: Binary Linear Setting <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> E [ Y | Z = 1] − E [ Y | Z = 0] Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 14 / 33

  35. <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> Warm-Up: Binary Linear Setting <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> E [ Y | Z = 1] − E [ Y | Z = 0] | | <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> E − E (exclusion restriction and linear outcome assumptions) = E [ δ T + α u U | Z = 1] − E [ δ T + α u U | Z = 0] Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 14 / 33

  36. <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> Warm-Up: Binary Linear Setting <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> E [ Y | Z = 1] − E [ Y | Z = 0] | | <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> E − E (exclusion restriction and linear outcome assumptions) = E [ δ T + α u U | Z = 1] − E [ δ T + α u U | Z = 0] | | − = δ ( E [ T | Z = 1] − E [ T | Z = 0]) + α u ( E [ U | Z = 1] − E [ U | Z = 0]) Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 14 / 33

  37. <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">AXknic1VhbxtFJ42XNpwawpSHnhZmlIV4Rg7BAUhWSpKWiHUSqmUtoE4ita743iVvbE7G5Na/gn8Lf4D/4FXeOacb8Zery+7dhWEsGXv+Mw537nMOWdm3Il9L1WNxh83bq69fY796vf7e+x98+NGdjbsv0yhLHPnCifwoOe7YqfS9UL5QnvLlcZxIO+j48lXnYp/nX13KJPWi8EhdxfI0sM9Dr+s5tiLS2cbamUWv9uMT64HVjLlRIG02oHnWm0vTFWSBTJUVstqnlrbzFbG0zi12u1xnvQAq8rfWVbUGKfB8abVtP+7Z+0sjDqpTC6l60RhN8pCl4y8tH1r/kypSf+umqJXRpcvu+ohlOday0ws5WINiXfeU18sZXmue/VQreS1sel/5X2po/+BT+tnd7Ya9QZe1uygaQZbwrwOo43bv4m2cEUkHJGJQEgRCkVjX9gipfeJaIqGiIl2KgZES2jkYV6KoVgn2Yy4JHYRL2g73P6dWKoIf1mzBTSDmnx6ZOQpCU+Nzwujbug6ifrtyZ4F+kYAJtvKJnx2AGRFWiR9QquRHn8nIdegcVXivy4Vt465EnMSgcB6fgc5ePv1W5CF/XxGnpJFLUgmNHKL5RNU1pHQU0eY9PDStjgkzRiq8vsLrOa5yN6XxCWTeMUlrKtlnhi1iWEZglbmcfHmi5G/JU81PaVYXHPnhYd+0F8x4R5UK8plERuUznZHYt5lIGeThHF/MrcITEnRIlQuZ7FD+POIq56hAm67CxkufwJUYG1Q3yj5gfZXNM67kNe1KsmoV86AnNpWV2z2y06dnB9gJyQ9orgc9HIsa2SOBnSB7R1GsgTuhX38cmCjM0WvozJ5PWvkE2dXjXAjmvHGWDoSOp4ji7R1A6JZ5r0tnkK/pHWvISqc1zWSjWChzuIUukKz9i3qKxyXgL6ZypEdUWrQEcCOHrLlgmi5Po3E+r4iL1LUj29QB/T0EbUYKDVo5/j0MQ7gE69oBt19GrvQw9IW9bq62DN2DAtaOTYeutOsVgv5x7WrO+O0VpYLUamWycoWNDXErvGYV4DtnoyaY7qh7skpvBtOWdWB/XnX4Bwt2sdS+foU18pFNvYgVcTlKM73c54vO6iTGtaSY39OHC1IdQ09RW3GRtP6RP3tmyqJsB4OnoHJCV3vrK0/NTMgemAi5ROfrnPek3jmF8K0kVNtRG9yvgzxsZEf7Qbc8Qeglsu9RGUVZRMaDcYz5fIeRhLx0SNecY0XI9YxfPls/LEmZpbFvh68cAaRM1YZpGXRXeTGfM+PxBb1gi3KqmI8eaWZkuB0EtP4/pjzPuEy7wW0hEThCh+M54eVa3BAlKGJWBdzozrMbfDQ+Vxwtk3maQ5dUWqKg6u6XCvn5by80fRyWXec70VZTS+XZcolctMz/SAF1nGFnMI6sEQwkaVHFVIR9gRN1zb/tIR9IfbazORFYix8VSEZoMdoryJTlc8q7VPogomJCcv8/AYRvET/G+IEtGoc1m1UjRzuas3imku318xsrlksGJ8c8nXFZISe4M3prHM8qaYq7DSi6dT1V7gqTdZHR+0dVWw3eHosh7tms6k2X2G84KvTu20ANqS/Xn2X2HK7lZWcZum8HfSiBZrekdl5cG1q+gtkS/qXI9/x+uUx9zZObzpv8DE6+6cTu4sH7PxeMAZrm/2odFpbhvrp1fgB+pkz+iExON98TWdzpr0/aSwZy2LyrekzHS7SdwdQt6j3eYAu9LquLG5y2wXbjxFHY9Jw+gzNHfDnNs3J/hFSKvonY+sc8aduUlMa9JnxR5x6fvTVQVinoXz8crXdz6y/p8jwvk9v0VdB/LkDXE5vDxnFt/nq/5lWHz3jenZpW8f/Tg1vAfIOh9WVKH0PdzfbZn/mngfN3h6qCc3dw7+6UviL+9N31HfPjDhyHn0HUZDWtEnERbfgGDLKnPVS3Bjze3G9cM/VeNO2pRN5FRi8lulF0vR6/jetOf3f2ezg5U69+U298Xx369Gu+V/tlvhU3BMPKU574hFV5iHF1Vn7fe3Ptb/W/t78ZPO7ze839zXrzRtG5mNReG0+/QdzpBoq</latexit> <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> Warm-Up: Binary Linear Setting <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> E [ Y | Z = 1] − E [ Y | Z = 0] | | <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> E − E (exclusion restriction and linear outcome assumptions) = E [ δ T + α u U | Z = 1] − E [ δ T + α u U | Z = 0] | | − = δ ( E [ T | Z = 1] − E [ T | Z = 0]) + α u ( E [ U | Z = 1] − E [ U | Z = 0]) | | | | − − (instrumental unconfoundedness assumption) = δ ( E [ T | Z = 1] − E [ T | Z = 0]) + α u ( E [ U ] − E [ U ]) Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 14 / 33

  38. <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">AWqXicrVjbtGEN3YvSRu0iYp4Je+sLEDJKisym6KFAgMBHASFEVTJICTuLEMg5eVRIi3ku7jqBP6Af2of/SmbNLkdSNUhAJIpfDmTOzc9tdOUngZ6rT+faxuZn3/x5fUbW1/dvPX1N7fv3H2bxXnqyjduHMTpiWNnMvAj+Ub5KpAnSrt0AnkO2d4xO/fXcg08+PoWF0l8iy0+5Hf81bEen8zsZF15F9Pxopf/gh8V2Vp3K8ZnPfeuP2JPZ5Lkb0eNpmwlW56dDaR3Zj3o5lHsZDK9kJ4bR704jzxS+NCylfWg0+o8tEa781l2x0/mITOkojmoUEbqoXqyC+tALZU9Zh3LPmY52xlonUEuQ4V24cyglu6vcHjcBGaAmsH2UqzY3FthNfyIrFE7sAVrIWeBV/P/f6VX8nthos9rAk5pqznqwpWThjVq4SgEbe6twXmNSVkVdLsfPbO512Bx9rdrBvBjvCfF7Fd278I7rCE7FwRS5CIUkFI0DYuMvqdiX3REQrQzMSJaSiMf76UYiy2SzYlLEodN1CFd+/R0aqgRPTNmBmXtAT0S0nSEvcNj0fjHqj6zvqtCu8iHSNgs41XdHcMZkhUJQZEbZIrOFeXc+gbNsxa0Rx+wWx9mkCvBrc25R/eAnhXNkK9XxClp5JFUSiOXaAFRNYV1pHTXnmfDBAJG3ySRmz1MruXWc3vY/oOCcumcQZL2VZLvDBxiaBZwlbmCRDTxYh/0wy1fcuwepM5+Ii7ngXzHhNlKD7QqI68TGc1uxZzKYM8nqOL+RU4IuLOiBIj83yn08c9Vx1CZN12IhkH3NJkEFtg/wb3hfZnFA892BPhqhZyDcfehJTWaXdhZ0B3R1gpyQ/oncD6GFftMgeCewU2Vt4sQXulJ4u8eTCRneK3kZlcjxbNCfOrhbhxvTGn2BpT2h/FhZp60ZEs8x3T/wO/ZLi3oJXOK9bJBvDQp3FGXRFJvaH1FfYLyFdmcqeLSgt6AhxwDZMiRaqU8jsb4faRYZ6icwqCO6B/BaApQWtLN/LjEOMSeOaA7dlzT2oIelLep1bfHY2DGuaWXf+OhOs1ot5B/Xru6M01pZLkKlWiYrD6GpIx6ZGXME2O6q1zTDXVPzjC78ZRVDuwvuwbnaN0+lirjU4+Vh2wcQKqOy16cP895czlAnbQS/Z9nzgOIdUz9Ay1mRhNW5X6OzJVEiMeLu6hyQld76ztcurNiOih8VRAfLrOeU3iN38Rpo2c6sJ71fLEJ8b+WI14I4/AnW53FtUVl02pdFo8ma5vI+RhH/0iCOu8RL4OsFcvp/8rMqbVbE/DV40g8gZqwzSqugecmP+zI/FDvWCHcquj850kxJsTtJaLw74dwlXOYdQktEFK7w0eT9uDEGz4gyNh7r4V1Rh6UNPjqfB86uyTzNoStKTXFwVS/Xynk5L280fbmsN8n3uqymL5dlygVy0zf9IAPWSYOcQhxYIqxk6XGDVIw1QdO1zX+uYF+EtTY3eZEaC981SIboMXpWsanKl432KXTB1PiEZd5/hAcv0P/G2AGt68dSVq3lzVLu6qN8WspfrunZUjJc07+l5IcGSYm1wZ/QWOZ1Y0x16tGLp1PTWuCpNWk2L/oamvh6pAXec32TGeyzHrDWaFXx0P0gNZK/Xl23eFK3m+s4xzd10EfSqHZW1I7bz4ZWhnBfIX5Zcj38ny5Sn3Nk5vOm3IPUez9s8rq4gO7PBeMsIe7NOtQsZvbQ/x0BH6lTvaSdkg8PhI/0e5sn64vamvWqh8SspNt6viHhDyY1ptnmFVWh83MWeZvdqJp67jOWkofmNzNiy5A7ODX4S0jt75yDpnvJmTxLQmvVcEJc+P101IJZOB9veXznI+v/OWLs38tT1KdArp4QV8Mrc2bxeb7pX4bFZ9+E7j26BujHmeF9hqwL0JcldQh9Ptd7n6OZfxo4fw+oKjh3R/fuTumr40+fXfs4e+bEUfLoM4iCtKZVERedghPIKLPXy3BiLM/F7do5V+N25ZV8io0eIemF0nT67fOb+/sT/93Njt4e9De/7ndeX2w8/SR+V/tuvhO3BMPyE+PxVOqzFfkV3fjv83NzZubt7Z/2H69fbL9XrNuXDMy34raZ9v9H7Dzt5c=</latexit> <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> Warm-Up: Binary Linear Setting <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> E [ Y | Z = 1] − E [ Y | Z = 0] | | <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> E − E (exclusion restriction and linear outcome assumptions) = E [ δ T + α u U | Z = 1] − E [ δ T + α u U | Z = 0] | | − = δ ( E [ T | Z = 1] − E [ T | Z = 0]) + α u ( E [ U | Z = 1] − E [ U | Z = 0]) | | | | − − (instrumental unconfoundedness assumption) = δ ( E [ T | Z = 1] − E [ T | Z = 0]) + α u ( E [ U ] − E [ U ]) | | − − = δ ( E [ T | Z = 1] − E [ T | Z = 0]) Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 14 / 33

  39. <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> <latexit sha1_base64="3cuWur6hLCfODW+HoFEKOFHnbPs=">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</latexit> Warm-Up: Binary Linear Setting <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> E [ Y | Z = 1] − E [ Y | Z = 0] | | <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> E − E (exclusion restriction and linear outcome assumptions) = E [ δ T + α u U | Z = 1] − E [ δ T + α u U | Z = 0] | | − = δ ( E [ T | Z = 1] − E [ T | Z = 0]) + α u ( E [ U | Z = 1] − E [ U | Z = 0]) | | | | − − (instrumental unconfoundedness assumption) = δ ( E [ T | Z = 1] − E [ T | Z = 0]) + α u ( E [ U ] − E [ U ]) | | − − = δ ( E [ T | Z = 1] − E [ T | Z = 0]) δ = E [ Y | Z = 1] − E [ Y | Z = 0] Z U E [ T | Z = 1] − E [ T | Z = 0] T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 14 / 33

  40. <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">AXknic1VhbxtFJ42XNpwawpSHnhZmlIV4Rg7BAUhWSpKWiHUSqmUtoE4ita743iVvbE7G5Na/gn8Lf4D/4FXeOacb8Zery+7dhWEsGXv+Mw537nMOWdm3Il9L1WNxh83bq69fY796vf7e+x98+NGdjbsv0yhLHPnCifwoOe7YqfS9UL5QnvLlcZxIO+j48lXnYp/nX13KJPWi8EhdxfI0sM9Dr+s5tiLS2cbamUWv9uMT64HVjLlRIG02oHnWm0vTFWSBTJUVstqnlrbzFbG0zi12u1xnvQAq8rfWVbUGKfB8abVtP+7Z+0sjDqpTC6l60RhN8pCl4y8tH1r/kypSf+umqJXRpcvu+ohlOday0ws5WINiXfeU18sZXmue/VQreS1sel/5X2po/+BT+tnd7Ya9QZe1uygaQZbwrwOo43bv4m2cEUkHJGJQEgRCkVjX9gipfeJaIqGiIl2KgZES2jkYV6KoVgn2Yy4JHYRL2g73P6dWKoIf1mzBTSDmnx6ZOQpCU+Nzwujbug6ifrtyZ4F+kYAJtvKJnx2AGRFWiR9QquRHn8nIdegcVXivy4Vt465EnMSgcB6fgc5ePv1W5CF/XxGnpJFLUgmNHKL5RNU1pHQU0eY9PDStjgkzRiq8vsLrOa5yN6XxCWTeMUlrKtlnhi1iWEZglbmcfHmi5G/JU81PaVYXHPnhYd+0F8x4R5UK8plERuUznZHYt5lIGeThHF/MrcITEnRIlQuZ7FD+POIq56hAm67CxkufwJUYG1Q3yj5gfZXNM67kNe1KsmoV86AnNpWV2z2y06dnB9gJyQ9orgc9HIsa2SOBnSB7R1GsgTuhX38cmCjM0WvozJ5PWvkE2dXjXAjmvHGWDoSOp4ji7R1A6JZ5r0tnkK/pHWvISqc1zWSjWChzuIUukKz9i3qKxyXgL6ZypEdUWrQEcCOHrLlgmi5Po3E+r4iL1LUj29QB/T0EbUYKDVo5/j0MQ7gE69oBt19GrvQw9IW9bq62DN2DAtaOTYeutOsVgv5x7WrO+O0VpYLUamWycoWNDXErvGYV4DtnoyaY7qh7skpvBtOWdWB/XnX4Bwt2sdS+foU18pFNvYgVcTlKM73c54vO6iTGtaSY39OHC1IdQ09RW3GRtP6RP3tmyqJsB4OnoHJCV3vrK0/NTMgemAi5ROfrnPek3jmF8K0kVNtRG9yvgzxsZEf7Qbc8Qeglsu9RGUVZRMaDcYz5fIeRhLx0SNecY0XI9YxfPls/LEmZpbFvh68cAaRM1YZpGXRXeTGfM+PxBb1gi3KqmI8eaWZkuB0EtP4/pjzPuEy7wW0hEThCh+M54eVa3BAlKGJWBdzozrMbfDQ+Vxwtk3maQ5dUWqKg6u6XCvn5by80fRyWXec70VZTS+XZcolctMz/SAF1nGFnMI6sEQwkaVHFVIR9gRN1zb/tIR9IfbazORFYix8VSEZoMdoryJTlc8q7VPogomJCcv8/AYRvET/G+IEtGoc1m1UjRzuas3imku318xsrlksGJ8c8nXFZISe4M3prHM8qaYq7DSi6dT1V7gqTdZHR+0dVWw3eHosh7tms6k2X2G84KvTu20ANqS/Xn2X2HK7lZWcZum8HfSiBZrekdl5cG1q+gtkS/qXI9/x+uUx9zZObzpv8DE6+6cTu4sH7PxeMAZrm/2odFpbhvrp1fgB+pkz+iExON98TWdzpr0/aSwZy2LyrekzHS7SdwdQt6j3eYAu9LquLG5y2wXbjxFHY9Jw+gzNHfDnNs3J/hFSKvonY+sc8aduUlMa9JnxR5x6fvTVQVinoXz8crXdz6y/p8jwvk9v0VdB/LkDXE5vDxnFt/nq/5lWHz3jenZpW8f/Tg1vAfIOh9WVKH0PdzfbZn/mngfN3h6qCc3dw7+6UviL+9N31HfPjDhyHn0HUZDWtEnERbfgGDLKnPVS3Bjze3G9cM/VeNO2pRN5FRi8lulF0vR6/jetOf3f2ezg5U69+U298Xx369Gu+V/tlvhU3BMPKU574hFV5iHF1Vn7fe3Ptb/W/t78ZPO7ze839zXrzRtG5mNReG0+/QdzpBoq</latexit> <latexit sha1_base64="3cuWur6hLCfODW+HoFEKOFHnbPs=">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</latexit> Warm-Up: Binary Linear Setting <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> E [ Y | Z = 1] − E [ Y | Z = 0] | | <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> E − E (exclusion restriction and linear outcome assumptions) = E [ δ T + α u U | Z = 1] − E [ δ T + α u U | Z = 0] | | − = δ ( E [ T | Z = 1] − E [ T | Z = 0]) + α u ( E [ U | Z = 1] − E [ U | Z = 0]) | | | | − − (instrumental unconfoundedness assumption) = δ ( E [ T | Z = 1] − E [ T | Z = 0]) + α u ( E [ U ] − E [ U ]) | | − − = δ ( E [ T | Z = 1] − E [ T | Z = 0]) δ = E [ Y | Z = 1] − E [ Y | Z = 0] Z U E [ T | Z = 1] − E [ T | Z = 0] T Y (non-zero due to relevance assumption) <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 14 / 33

  41. <latexit sha1_base64="NIoWxOqWvkAXO6A/wEjPiAFLHZo=">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</latexit> <latexit sha1_base64="fyZ1p9ikpkndGU282pak6zlpWw=">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</latexit> Multiplying Path Coefficients in Linear Setting Z U α z δ T Y δ = E [ Y | Z = 1] − E [ Y | Z = 0] E [ T | Z = 1] − E [ T | Z = 0] Brady Neal Warm-Up: Linear Setting 15 / 33

  42. <latexit sha1_base64="NIoWxOqWvkAXO6A/wEjPiAFLHZo=">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</latexit> <latexit sha1_base64="fyZ1p9ikpkndGU282pak6zlpWw=">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</latexit> Multiplying Path Coefficients in Linear Setting Z U α z δ T Y δ = E [ Y | Z = 1] − E [ Y | Z = 0] E [ T | Z = 1] − E [ T | Z = 0] Brady Neal Warm-Up: Linear Setting 15 / 33

  43. <latexit sha1_base64="NIoWxOqWvkAXO6A/wEjPiAFLHZo=">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</latexit> <latexit sha1_base64="OdpuZrnfnZUoyBfY5qJFgjG/9yc=">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</latexit> Multiplying Path Coefficients in Linear Setting Z U α z δ T Y α z δ δ = E [ T | Z = 1] − E [ T | Z = 0] Brady Neal Warm-Up: Linear Setting 15 / 33

  44. <latexit sha1_base64="NIoWxOqWvkAXO6A/wEjPiAFLHZo=">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</latexit> <latexit sha1_base64="OdpuZrnfnZUoyBfY5qJFgjG/9yc=">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</latexit> Multiplying Path Coefficients in Linear Setting Z U α z δ T Y α z δ δ = E [ T | Z = 1] − E [ T | Z = 0] Brady Neal Warm-Up: Linear Setting 15 / 33

  45. <latexit sha1_base64="NIoWxOqWvkAXO6A/wEjPiAFLHZo=">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</latexit> <latexit sha1_base64="OdpuZrnfnZUoyBfY5qJFgjG/9yc=">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</latexit> Multiplying Path Coefficients in Linear Setting Z U α z δ T Y α z δ δ = E [ T | Z = 1] − E [ T | Z = 0] Brady Neal Warm-Up: Linear Setting 15 / 33

  46. <latexit sha1_base64="NIoWxOqWvkAXO6A/wEjPiAFLHZo=">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</latexit> <latexit sha1_base64="8fxaAO836FB6EaO7JISLrsyoqlM=">AUW3icrVjbtGEN0kvSRu2jqJ+9QXNkqBFqBVyU3hPlRADtBUTSA9iO2ygwKJKyCPFWkrJqE/qEol/T1/Y7+tB/6czZpXiREqBJUhczc6cmZ3b7moQuk6cdDr/3rp9573P/jw7r2tj+5/Mmn2w8ensbBJDLtEzNwg+hsYMS26/j2SeIkrn0WRrbhDVz79WB8wPOvL+0odgL/OLkK7beceE7Q8c0EiKdb3/dt2w3MbSe1h9Ghpn2DTcGefXmqTP5oTZ+Xar0+7gpS0OumrQEup1FDy494foC0sEwhQT4Qlb+CKhsSsMEdP7jeiKjgiJ9lakRIto5GDeFjOxRbIT4rKJwyDqmL4v6NcbRfXpN2PGkDZJi0ufiCQ18aXisWg8BFU+Wb9W4F2lIwU23hFz4HC9IiaiBFRm+QyzvXlBvT2Glad0Bq+x2odWkICvBLK15SE+Xfie0Qv6+Ik6bRhZJRTQyieYSVJYR0RP6Xn2zQiRMBn04itrO7zmqeD+g9JiyDxjEsZVs18ULFxYdmG7Yyj4uYrkb8nVYo7avDGs7X4CDuchXMe0yUsbimURm5Tmcxu1ZzJQp5tkQX8yfg8Ik7JkqAzHfIfw5xlHPVJEzWYSCSF1hLiAxqK+SfMJ9lc0jx3IU9MaKmId8c6AlVZeV2Z3a69BwAOyL5lOZG0MO+0MkeG9gRsjfzog7uiH5N8cuEjWaF3kZlcjx1WhNnl064Ac04cyzpCenPzCJpXUo0Tb13xc/Qb1PcdXiF81on2QAWyiyOoctXse9RX2G/ePTNVPZsRtGhw4MdI2TLmGi5PonE+r6hVcSoH1ehpvR04bUQKDq0s3+mGHtYE0d0At1TGlvQw9Ia9bq2Fd2zEpa2TcOutOiVg35x7UrO2NVK8v5qFRNZWUPmjriqVoxR4DtLnrNVN1Q9uQYq5tVrBrA/rxrcI6W7WOpPD7lWFnIxhGkyrjsxeXrXLaWPdSJjliy7y+IowepoaLHqM1Qadoq1N+BqpIA8TDx9FROyHpnbdPKTEp0T3nKJT5Z57wn8cxvhGkgp/rwXnG+DvG5ks92A+74Kaj1cqeorLJsRKN0PlMv72Bkwz9yxBGXeCF8HWItX8w/WmFmXeybwfMXEDljE4W0LrqF3Fi+8mPRol7Qoqwq+5MjzZQIp5OQxk/mnE8Il3nH0OIThSs8nc/PGmNwSJSZ8tgQc1kd5jY46HwWOPsq8ySHrKikwsFVXa+V83JZ3kh6vaw1z/eyrKTXyzLlErnpqH4QA+usQS5BHFjCK2TpcYNUgD1B0qXNv6xhn4+9dqLyIlIWvm6Q9NBj5KoCVZUvG+1L0AUj5ROW+fUdPHiJ/jfDCWhTP+ayUbezOWu3smnufx0Q8/mkt6G/s0lrxskbewNzpzGMq8a4q5jhq5ZD417Qk27SbZ+UVWm47vAXmR92xLdSZN7TecFXJ37KEH6Gv158V9hyu521jHE3TfAfpQBM1WTe2c3BhaHsHJGuLke/5/XKd+lomV82b/AyRnf3jwu7iADu/F6Q4w03VPpSd5nYRPxmBH6mTvaQTEo8PxLd0OuvS94vSnrUuKt+SJqrbFXH3CHmfdptD7Eqb4bqLrNbuvGUdTwnDdlnpu6GOberTvCrkDbRuxZ5oy1cJOoapJnxRFxyfvTVQNinoXL8erjuxZ/s8R4Pye36JuArl4Q1wPL8+Z1f5pn8ZVt9Q3oO6dtFP4V7yGyzkVftqlDyPu5PscLPzTwPm7R1XBuZs+fljRV8av3l0vcPecEfOI+8gCaQlrYi46hYcQiZRZ70YN8b8Xtwu3XMlXtW2uJBXnsLrqV5kq16/db7d6lb/O1scnO61u9+1O6+etp79oP5Xuys+F4/FV+SnfGMKvOI/GqKP8Vf4m/xz6P/du7sbO3cl6y3bymZR6L02vnsf/DX9rk=</latexit> Multiplying Path Coefficients in Linear Setting Z U α z δ T Y δ = α z δ α z Brady Neal Warm-Up: Linear Setting 15 / 33

  47. <latexit sha1_base64="3cuWur6hLCfODW+HoFEKOFHnbPs=">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</latexit> Wald Estimator Wald estimand: δ = E [ Y | Z = 1] − E [ Y | Z = 0] E [ T | Z = 1] − E [ T | Z = 0] Brady Neal Warm-Up: Linear Setting 16 / 33

  48. <latexit sha1_base64="3cuWur6hLCfODW+HoFEKOFHnbPs=">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</latexit> <latexit sha1_base64="BCiNb02P+Gj7F9mYtsGPolA0Ja8=">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</latexit> Wald Estimator Wald estimand: δ = E [ Y | Z = 1] − E [ Y | Z = 0] E [ T | Z = 1] − E [ T | Z = 0] Wald estimator: 1 1 P P i : z i =1 Y i − i : z i =0 Y i n 1 n 0 ˆ δ = 1 1 P P i : z i =1 T i − i : z i =0 T i n 1 n 0 Brady Neal Warm-Up: Linear Setting 16 / 33

  49. <latexit sha1_base64="3cuWur6hLCfODW+HoFEKOFHnbPs=">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</latexit> <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> Continuous Linear Setting δ = E [ Y | Z = 1] − E [ Y | Z = 0] E [ T | Z = 1] − E [ T | Z = 0] Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 17 / 33

  50. <latexit sha1_base64="3cuWur6hLCfODW+HoFEKOFHnbPs=">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</latexit> <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> Continuous Linear Setting δ = E [ Y | Z = 1] − E [ Y | Z = 0] E [ T | Z = 1] − E [ T | Z = 0] What if T and Z are continuous? Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 17 / 33

  51. <latexit sha1_base64="3cuWur6hLCfODW+HoFEKOFHnbPs=">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</latexit> <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> <latexit sha1_base64="+eCRfgBW3aclHm3N1hl5+O3MQK8=">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</latexit> Continuous Linear Setting δ = E [ Y | Z = 1] − E [ Y | Z = 0] δ = Cov( Y, Z ) E [ T | Z = 1] − E [ T | Z = 0] Cov( T, Z ) What if T and Z are continuous? Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 17 / 33

  52. <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> Continuous Linear Setting Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 17 / 33

  53. <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> Continuous Linear Setting Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 17 / 33

  54. <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> Continuous Linear Setting <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> Cov( Y, Z ) = Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 17 / 33

  55. <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">AWqXicrVjbtGEN3YvSRu0iYp4Je+sLEDJKisym6KFAgMBHASFEVTJICTuLEMg5eVRIi3ku7jqBP6Af2of/SmbNLkdSNUhAJIpfDmTOzc9tdOUngZ6rT+faxuZn3/x5fUbW1/dvPX1N7fv3H2bxXnqyjduHMTpiWNnMvAj+Ub5KpAnSrt0AnkO2d4xO/fXcg08+PoWF0l8iy0+5Hf81bEen8zsZF15F9Pxopf/gh8V2Vp3K8ZnPfeuP2JPZ5Lkb0eNpmwlW56dDaR3Zj3o5lHsZDK9kJ4bR704jzxS+NCylfWg0+o8tEa781l2x0/mITOkojmoUEbqoXqyC+tALZU9Zh3LPmY52xlonUEuQ4V24cyglu6vcHjcBGaAmsH2UqzY3FthNfyIrFE7sAVrIWeBV/P/f6VX8nthos9rAk5pqznqwpWThjVq4SgEbe6twXmNSVkVdLsfPbO512Bx9rdrBvBjvCfF7Fd278I7rCE7FwRS5CIUkFI0DYuMvqdiX3REQrQzMSJaSiMf76UYiy2SzYlLEodN1CFd+/R0aqgRPTNmBmXtAT0S0nSEvcNj0fjHqj6zvqtCu8iHSNgs41XdHcMZkhUJQZEbZIrOFeXc+gbNsxa0Rx+wWx9mkCvBrc25R/eAnhXNkK9XxClp5JFUSiOXaAFRNYV1pHTXnmfDBAJG3ySRmz1MruXWc3vY/oOCcumcQZL2VZLvDBxiaBZwlbmCRDTxYh/0wy1fcuwepM5+Ii7ngXzHhNlKD7QqI68TGc1uxZzKYM8nqOL+RU4IuLOiBIj83yn08c9Vx1CZN12IhkH3NJkEFtg/wb3hfZnFA892BPhqhZyDcfehJTWaXdhZ0B3R1gpyQ/oncD6GFftMgeCewU2Vt4sQXulJ4u8eTCRneK3kZlcjxbNCfOrhbhxvTGn2BpT2h/FhZp60ZEs8x3T/wO/ZLi3oJXOK9bJBvDQp3FGXRFJvaH1FfYLyFdmcqeLSgt6AhxwDZMiRaqU8jsb4faRYZ6icwqCO6B/BaApQWtLN/LjEOMSeOaA7dlzT2oIelLep1bfHY2DGuaWXf+OhOs1ot5B/Xru6M01pZLkKlWiYrD6GpIx6ZGXME2O6q1zTDXVPzjC78ZRVDuwvuwbnaN0+lirjU4+Vh2wcQKqOy16cP895czlAnbQS/Z9nzgOIdUz9Ay1mRhNW5X6OzJVEiMeLu6hyQld76ztcurNiOih8VRAfLrOeU3iN38Rpo2c6sJ71fLEJ8b+WI14I4/AnW53FtUVl02pdFo8ma5vI+RhH/0iCOu8RL4OsFcvp/8rMqbVbE/DV40g8gZqwzSqugecmP+zI/FDvWCHcquj850kxJsTtJaLw74dwlXOYdQktEFK7w0eT9uDEGz4gyNh7r4V1Rh6UNPjqfB86uyTzNoStKTXFwVS/Xynk5L280fbmsN8n3uqymL5dlygVy0zf9IAPWSYOcQhxYIqxk6XGDVIw1QdO1zX+uYF+EtTY3eZEaC981SIboMXpWsanKl432KXTB1PiEZd5/hAcv0P/G2AGt68dSVq3lzVLu6qN8WspfrunZUjJc07+l5IcGSYm1wZ/QWOZ1Y0x16tGLp1PTWuCpNWk2L/oamvh6pAXec32TGeyzHrDWaFXx0P0gNZK/Xl23eFK3m+s4xzd10EfSqHZW1I7bz4ZWhnBfIX5Zcj38ny5Sn3Nk5vOm3IPUez9s8rq4gO7PBeMsIe7NOtQsZvbQ/x0BH6lTvaSdkg8PhI/0e5sn64vamvWqh8SspNt6viHhDyY1ptnmFVWh83MWeZvdqJp67jOWkofmNzNiy5A7ODX4S0jt75yDpnvJmTxLQmvVcEJc+P101IJZOB9veXznI+v/OWLs38tT1KdArp4QV8Mrc2bxeb7pX4bFZ9+E7j26BujHmeF9hqwL0JcldQh9Ptd7n6OZfxo4fw+oKjh3R/fuTumr40+fXfs4e+bEUfLoM4iCtKZVERedghPIKLPXy3BiLM/F7do5V+N25ZV8io0eIemF0nT67fOb+/sT/93Njt4e9De/7ndeX2w8/SR+V/tuvhO3BMPyE+PxVOqzFfkV3fjv83NzZubt7Z/2H69fbL9XrNuXDMy34raZ9v9H7Dzt5c=</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> Continuous Linear Setting <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> Cov( Y, Z ) = ) = E [ Y Z ] − E [ Y ] E [ Z ] Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 17 / 33

  56. <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> Continuous Linear Setting <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> Cov( Y, Z ) = ) = E [ Y Z ] − E [ Y ] E [ Z ] − (exclusion restriction and linear outcome assumptions) = E [( δ T + α u U ) Z ] − E [ δ T + α u U ] E [ Z ] Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 17 / 33

  57. <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> Continuous Linear Setting <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">AYEnicrVjNbtGEN4kbpu4f0l68KEXNk4Lp5VU2XGRXgwEcBIURQMkgPTmIZBkSuLMP9KUnYdQW/Ra1+hD9Bb0WtfoG/TmW+XIilRJVagsjV7Mw3Pzszy+Ug8twk7f/vXL12tp739w/cb6hx9/MmnN2/dfpmE49iWL+zQC+PXAyuRnhvIF6mbevJ1FEvLH3jy1eB0n+dfnck4cPgIL2I5JFvnQTu0LWtlEjHt679bu6HZ1tmOE7t0Jcdw3SDJI3HvgzSe8ZXe4b5+DCbLM4dGd3i1BH+FKdNc92gj0LYMh3pZhpmRbygzGN4ZpedHIOjbHQThIZHwmHTsMhuE4cMjeM8szqmfuVZlxefC1nmg1xJBrKvG2UvSlZMLflVoW7Su+380tnHWk8N0q87nbpOJZuyejCib2gQn19YiRqR6Ne0IYtOogZmLsxVQRu7AVA5PQe2mu7jm5v9Xh8fY3GwrQebQn+ehbdu/CFM4YhQ2GIsfCFIFIae8ISCX0Pxboi4hoR2JCtJhGLualmIp1kh0TlyQOi6indD2hf4eaGtB/xkwgbZMWj34xSRriS83j0HgIqrqzfqPAu0zHBNhs4wXdBxrTJ2oqRkRtks428sN6Os3eJ2SD9/DW5c8iUDhONgln4d09+h/Sh7y9YI4JY0ckopZBPNI6qisI6Y7iryHJsRVsICn6QRW1nd53VPB/S95SwLBonsJRtNcQTvS4BNEvYyjwe1nQ54q/kobKvDms48HFuisvmPeAKfiLY3KyHU6i9m1nCvVyNMKXcyfgiMg7oQoITLfpfi5xFHOVZswWYeFlTyBLxEyqKeRf8R8ls0RrWcX9iRYNQP5kJPpCsrtzuz06P7ANgxyU9obgQ9HIsO2SOBHSN7syh2wB3Tv3P8s2GjPUfvoTJ5PTvkE2dXh3BDmnFnWCoSKp6ZRcq6CdEM/e2Kn6Bf0rp3EBXO6w7JhrBQZXECXYFe+z3qKxwXn65M5chmlA50+LBjhGw5JVquTyGxvm/JiwT142nUCd09RC0CSgfaOT7nGPvwiVd0DN3nNHagh6UN6nU98UDbMS1p5di46E6LWg3kH9eu6ozWlkuQKUaOiv3oKkvdrXHvAJsdzFqtu6Gqicn8G46Z9UA9udg3O0bB9L5etTXisH2TiCVBmXo1jtZ5UvO6iTDtaSY39CHuQGmp6gtqMtKb1Qv3t6yoJsR427r7OCVXvrO18bmZCdF9HyiM+Ve8J/HML4RpIadMRK84X4f4WMtnuwF3/Amo9XIvUVl2ZhGk9lMvbyLkUR81IhXOFiHUEX76Y/YzCTFvsy8ELFhA5Y1ON1BbdQW5Ue34gNqkXbFJWlePJK82UGE8nEY3vzjvEi7znkJLQBSu8Mlsftq4Bo+IMtURG2Iuq8PcBhedzwGnqTNPcaiKSuc4uKrtXJeVuWNotfLOrN8L8sqer0sU86Qm67uBwmwXjfIpVgHlvALWXrQIBViT1B0ZfPLewLsNeOdV7E2sJXDZI+eozyKtRV+bTRvhRdMNYxYZk37xDBM/S/KZ6AVo1jLpuFM1c7uKdYprLn68Y2VzSXzG+ueTbBkmJvcGd0VjmeWNMdezRi6VT017gqTdJHt+UdXWwXVAUeQ929GdydD7DWeF2h30AM6rfrz4r7DlbzdWMdjdN8B+lAMzU5N7by4NLR8Bct/EuQ7/n5sk19Vcm1z5t9rENVd8xm1BNI/hSnR6Swv7kAj8/WUzwFHiud7LsebCLDFBr+AP1wqf0jDWFpv0fLdN1yelXa8tKp+zxrpfFnF3CPkB7VePsK+tjhvp01C3dGYq63hMGrLfVJ8uc25PnwGWIa2itxpZ2zcBaZ16SeNkfEpU5gFw2IeR5X49WvbzWyelMS4gSQn8MuA7l4xmyHl+fM8jcCTe8plp+eI7oP6eqhoyea9xGyzkNnl9Rj1AlfPT3tL7yr4Pzdoarg3J3cuT2nr4w/f/o9wel1TBw5jzrFpJBWtCLisnN0BJlUPy0mOHPmJ+te6aSs8OZtSwp5Wu8Pd2PpN4t1o9vbm7Pv31bHLzc6W1/1+s/3918uKvfzF0Xn4s7Yovi9EA8pMp8RnG19bWvl67v7a78dvGnxt/bfytWK9e0TKfidJn45/ABQSsM=</latexit> Cov( Y, Z ) = ) = E [ Y Z ] − E [ Y ] E [ Z ] − (exclusion restriction and linear outcome assumptions) = E [( δ T + α u U ) Z ] − E [ δ T + α u U ] E [ Z ] − = δ E [ TZ ] + α u E [ UZ ] − δ E [ T ] E [ Z ] − α u E [ U ] E [ Z ] Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 17 / 33

  58. <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> Continuous Linear Setting <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> Cov( Y, Z ) = ) = E [ Y Z ] − E [ Y ] E [ Z ] − (exclusion restriction and linear outcome assumptions) = E [( δ T + α u U ) Z ] − E [ δ T + α u U ] E [ Z ] − = δ E [ TZ ] + α u E [ UZ ] − δ E [ T ] E [ Z ] − α u E [ U ] E [ Z ] − − = δ ( E [ TZ ] − E [ T ] E [ Z ]) + α u ( E [ UZ ] − E [ U ] E [ Z ]) Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 17 / 33

  59. <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> Continuous Linear Setting <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> Cov( Y, Z ) = ) = E [ Y Z ] − E [ Y ] E [ Z ] − (exclusion restriction and linear outcome assumptions) = E [( δ T + α u U ) Z ] − E [ δ T + α u U ] E [ Z ] − = δ E [ TZ ] + α u E [ UZ ] − δ E [ T ] E [ Z ] − α u E [ U ] E [ Z ] − − = δ ( E [ TZ ] − E [ T ] E [ Z ]) + α u ( E [ UZ ] − E [ U ] E [ Z ]) − − = δ Cov( T, Z ) + α u Cov( U, Z ) Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 17 / 33

  60. <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> Continuous Linear Setting <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> Cov( Y, Z ) = ) = E [ Y Z ] − E [ Y ] E [ Z ] − (exclusion restriction and linear outcome assumptions) = E [( δ T + α u U ) Z ] − E [ δ T + α u U ] E [ Z ] − = δ E [ TZ ] + α u E [ UZ ] − δ E [ T ] E [ Z ] − α u E [ U ] E [ Z ] − − = δ ( E [ TZ ] − E [ T ] E [ Z ]) + α u ( E [ UZ ] − E [ U ] E [ Z ]) − − = δ Cov( T, Z ) + α u Cov( U, Z ) (instrumental unconfoundedness assumption) = δ Cov( T, Z ) Z U T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 17 / 33

  61. <latexit sha1_base64="+eCRfgBW3aclHm3N1hl5+O3MQK8=">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</latexit> <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> Continuous Linear Setting <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> Cov( Y, Z ) = ) = E [ Y Z ] − E [ Y ] E [ Z ] − (exclusion restriction and linear outcome assumptions) = E [( δ T + α u U ) Z ] − E [ δ T + α u U ] E [ Z ] − = δ E [ TZ ] + α u E [ UZ ] − δ E [ T ] E [ Z ] − α u E [ U ] E [ Z ] − − = δ ( E [ TZ ] − E [ T ] E [ Z ]) + α u ( E [ UZ ] − E [ U ] E [ Z ]) − − = δ Cov( T, Z ) + α u Cov( U, Z ) (instrumental unconfoundedness assumption) = δ Cov( T, Z ) Z U δ = Cov( Y, Z ) Cov( T, Z ) T Y <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 17 / 33

  62. <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="hm9P+ROKGa+XIlenBfnjn0ePYA=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="+eCRfgBW3aclHm3N1hl5+O3MQK8=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> Continuous Linear Setting <latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit> Cov( Y, Z ) = ) = E [ Y Z ] − E [ Y ] E [ Z ] − (exclusion restriction and linear outcome assumptions) = E [( δ T + α u U ) Z ] − E [ δ T + α u U ] E [ Z ] − = δ E [ TZ ] + α u E [ UZ ] − δ E [ T ] E [ Z ] − α u E [ U ] E [ Z ] − − = δ ( E [ TZ ] − E [ T ] E [ Z ]) + α u ( E [ UZ ] − E [ U ] E [ Z ]) − − = δ Cov( T, Z ) + α u Cov( U, Z ) (instrumental unconfoundedness assumption) = δ Cov( T, Z ) Z U δ = Cov( Y, Z ) Cov( T, Z ) T Y (non-zero due to relevance assumption) <latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jX83w=">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</latexit> Y := δ T + α u U Brady Neal Warm-Up: Linear Setting 17 / 33

  63. <latexit sha1_base64="oEV3IjO/P2oPp2MoXoeqsf1CQ7s=">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</latexit> Continuous Linear Setting: Estimator 1 δ = Cov( Y, Z ) Cov( T, Z ) Brady Neal Warm-Up: Linear Setting 18 / 33

  64. <latexit sha1_base64="1UfPzxDyErL8XNDr0YdqGu72nvA=">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</latexit> Continuous Linear Setting: Estimator 1 d Cov( Y, Z ) ˆ δ = d Cov( T, Z ) Brady Neal Warm-Up: Linear Setting 18 / 33

  65. <latexit sha1_base64="uSnWmzenLZoB/62kgJqyZD0I1U=">AWqXicrVjbtGEN3YvSRu0iYp4Je+sLEDJKisym6LBAgMBHASFEVTJICTuLEMg5eVRIi3ku7jqBP6Af2of/SmbNLkdSNUhAJIpfDmTOzc9tdOUngZ6rT+faxuZn3/x5fUbW1/dvPX1N7fv3H2bxXnqyjduHMTpiWNnMvAj+Ub5KpAnSrt0AnkO2d4xO/fXcg08+PoWF0l8iy0+5Hf81bEen8zsZF15F9Pxopf/gh8V2Vp3K8ZnPfeuP2JPZ5Lkb0eNpmwlW56dDaR3Zj3o5lHsZDK9kJ4bR704jzxS+NCylfWg0+o8tEa781l2x0/mITOkojmoUEbqoXqyC+tALZU9Zh3LPmY52xlonUEuQ4V24cyglu6vcHjcBGaAmsH2UqzY3FthNfyIrFE7sAVrIWeBV/P/f6VX8nthos9rAk5pqznqwpWThjVq4SgEbe6twXmNSVkVdLsfPbO512Bx9rdrBvBjvCfF7Fd278I7rCE7FwRS5CIUkFI0DYuMvqdiX3REQrQzMSJaSiMf76UYiy2SzYlLEodN1CFd+/R0aqgRPTNmBmXtAT0S0nSEvcNj0fjHqj6zvqtCu8iHSNgs41XdHcMZkhUJQZEbZIrOFeXc+gbNsxa0RweY7Y+zSQBhf3g1ubco3tAz4pmyNcr4pQ08kgqpZFLtIComsI6Urprz7NvBoiEDT5JI7Z6md3LrOb3MX2HhGXTOIOlbKslXpi4RNAsYSvzBIjpYsS/aYbavmVYvckcfMRdz4J5j4kyFB9oVEdeprOaXYu5lEez9HF/AocEXFnRImR+T75zyeOeq6hMk6bESyj7kyKC2Qf4N74tsTie7AnQ9Qs5JsPYmprNLuws6A7g6wU5If0bsB9LAvWmSPBHaK7C282AJ3Sk+XeHJhoztFb6MyOZ4tmhNnV4twY3rjT7C0J7Q/C4u0dSOiWea7J36Hfklxb8ErnNctko1hoc7iDLoiE/tD6ivsl5CuTGXPFpQWdISwY4BsGRKt1KeRWN+PNIsM9RMY1BHdA3gtAUoL2tk/lxiHmBNHNIfuSxp70MPSFvW6tnhk7BjXtLJvfHSnWa0W8o9rV3fGa0sF6FSLZOVh9DUET+bGXME2O6q1zTDXVPzjC78ZRVDuwvuwbnaN0+lirjU4+Vh2wcQKqOy16cP895czlAnbQS/Z9nzgOIdUz9Ay1mRhNW5X6OzJVEiMeLu6hyQld76ztcurNiOih8VRAfLrOeU3iN38Rpo2c6sJ71fLEJ8b+WI14I4/AnW53FtUVl02pdFo8ma5vI+RhH/0iCOu8RL4OsFcvp/8rMqbVbE/DV40g8gZqwzSqugecmP+zI/FDvWCHcquj850kxJsTtJaLw74dwlXOYdQktEFK7w0eT9uDEGz4gyNh7r4V1Rh6UNPjqfB86uyTzNoStKTXFwVS/Xynk5L280fbmsN8n3uqymL5dlygVy0zf9IAPWSYOcQhxYIqxk6XGDVIw1QdO1zX+uYF+EtTY3eZEaC981SIboMXpWsanKl432KXTB1PiEZd5/hAcv0P/G2AGt68dSVq3lzVLu6qN8WspfrunZUjJc07+l5IcGSYm1wZ/QWOZ1Y0x16tGLp1PTWuCpNWk2L/oamvh6pAXec32TGeyzHrDWaFXx0P0gNZK/Xl23eFK3m+s4xzd10EfSqHZW1I7bz4ZWhnBfIX5Zcj38ny5Sn3Nk5vOm3IPUez9s8rq4gO7PBeMsIe7NOtQsZvbQ/x0BH6lTvaSdkg8PhI/0e5sn64vamvWqh8SspNt6viHhDyI1ptnmFVWh83MWeZvdqJp67jOWkofmNzNiy5A7ODX4S0jt75yDpnvJmTxLQmvVcEJc+P101IJZOB9veXznI+v/OWLs38tT1KdArp4QV8Mrc2bxeb7pX4bFZ9+E7j26BujHmeF9hqwL0JcldQh9Ptd7n6OZfxo4fw+oKjh3R/fuTumr40+fXfs4e+bEUfLoM4iCtKZVERedghPIKLPXy3BiLM/F7do5V+N25ZV8io0eIemF0nT67fOb+/sT/93Njt4e9De/6XdeX2w8/Sx+V/tuvhO3BMPyE+PxFOqzFfkV3fjv83NzZubt7Z/2H69fbL9XrNuXDMy34raZ9v9H7Int5s=</latexit> Two-Stage Least Squares Estimator Z U T Y Brady Neal Warm-Up: Linear Setting 19 / 33

  66. <latexit sha1_base64="uSnWmzenLZoB/62kgJqyZD0I1U=">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</latexit> Two-Stage Least Squares Estimator 1. Linearly regress T on Z to estimate . This gives us the <latexit sha1_base64="qUoqhn4TXm2OMAEZLKEMQePRaE=">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</latexit> E [ T | Z ] <latexit sha1_base64="awFEcaFpPfD49FIWpreA6aLRo=">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</latexit> ˆ projection of T onto Z: T Z U T Y Brady Neal Warm-Up: Linear Setting 19 / 33

  67. <latexit sha1_base64="uSnWmzenLZoB/62kgJqyZD0I1U=">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</latexit> <latexit sha1_base64="sKGswFhJDYMf/mqsKR1vFLuK4/4=">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</latexit> Two-Stage Least Squares Estimator 1. Linearly regress T on Z to estimate . This gives us the <latexit sha1_base64="qUoqhn4TXm2OMAEZLKEMQePRaE=">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</latexit> E [ T | Z ] <latexit sha1_base64="awFEcaFpPfD49FIWpreA6aLRo=">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</latexit> ˆ projection of T onto Z: T <latexit sha1_base64="awFEcaFpPfD49FIWpreA6aLRo=">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</latexit> ˆ E [ Y | ˆ 2. Linearly regress Y on to estimate . Obtain our estimate as <latexit sha1_base64="VsSCPqNMkULYp0hGZrw3VuIwzA=">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</latexit> ˆ T ] T δ <latexit sha1_base64="awFEcaFpPfD49FIWpreA6aLRo=">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</latexit> ˆ the fitted coefficient in front of . T Z U T Y Brady Neal Warm-Up: Linear Setting 19 / 33

  68. <latexit sha1_base64="sKGswFhJDYMf/mqsKR1vFLuK4/4=">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</latexit> <latexit sha1_base64="uSnWmzenLZoB/62kgJqyZD0I1U=">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</latexit> Two-Stage Least Squares Estimator 1. Linearly regress T on Z to estimate . This gives us the <latexit sha1_base64="qUoqhn4TXm2OMAEZLKEMQePRaE=">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</latexit> E [ T | Z ] <latexit sha1_base64="awFEcaFpPfD49FIWpreA6aLRo=">AUhHicrVjbtGEN2kbZq4Nyf1W1/UKAX6ICuSk8B5cRHATlAUDeAtuM2DgyJoixCvJWkrDqE/qSv7WP/p3/TM2eXoqgLKQeWIHI1O3Nmdm7LZTd0nThptf67dfuTz+78/ndextfPnV19s3n9wEgejyLKPrcANotNuJ7Zdx7ePEydx7dMwsjte17Xfdof7Mv/20o5iJ/CPkqvQfu91Lnyn71idBKTzc2zQSdJzxLIJ7tJ5PzXqr2eKntjhom0Fdmc9hcP/ev+pM9VSgLDVSnrKVrxKMXdVRMb7vVFu1VAjae5WCFmHkcN5WE7UB2RG4bHB0QB3ieoF/7wzVx3/BjCltQYuLXwTJmvrB8PQw7pOq76K/NsO7SkdKbLHxCveuwfRATdQA1Cq5jHN9uS6+XsWqE6zhOVfrYCUhKeIHq7DmPu4u/idYoVyvwGlj1INUhJEFmguqpoiOCHftefHNgJHokM/GSKwus7vMapkP8B0Cq4NxTEvF1p6ZeLiU7NW4XHZUxXI/6JFWr7yrD60zU4jLtehfAegTJUHzAqIpfpnM2u1VyJQZ4s0SX8CTl8cMegBMx8B/5zwFHMVQuYoqPDSF5wLSEzqGmQf+F8ls0h4rlNe2JGrcZ8c6gnNJWV253Z6eLeJXYE+RzA+oRXzRgj03siNmbebFB7gj/xvxn0UZrjt5kZUo8G1iTZFcDuAFmnCmW9oT2Z2aRti4FrWa+2+pX6rcR9wa9IndgGxAC3UWx9Tlm9jvoa+IXzxchSqezSgN6vBox4DZMgQt16eRN9jrCJm/bgGNcXdpdCojSoXfwz5tjmiSiI+oeY9yjHpGuodc1a6xY1LQKr5x2J0WtdaYf1K7ujPOaxU5n5VaM1m5R0t9dSsWCIgds96zTLdUPfkmKubzFnVpf1515AcLdonUnl8irHqMRsHlCriheXr3PZWnZYJw3GUnx/AY49SvUNPWZthkbTxkz97ZsqCRgPi3fP5ISud9E2nptJQfeMp1zw6TqXPUlm/gBmhzl1Ru/NzpchvjTy2W4gHT8ltVzuhJVlI0wSqcz5fIORzb9o0cScY0X0tch1/L9FebmVkX+2bw/AVEydjEIK2L3mNuLF/5kaqjF9SRVUV/SqSFEvHpJMT40ZTzEXCFd0gtPihS4el0flIZgwNQJsZjfc5ldZjb4LDz9ch5ZjJPc+iKSuY4pKrLtUpeLsbTS+X7U3zvSir6eWyQrlkbjqmH8TEOq2QSxgHkfBmsvSoQirgnqDp2ubf1rDP5147MnkRGQvfVkh67DF6VYGpyteV9iXsgpHxicj8/hEevGT/m/AJ6Lp+zGWTa3kzl7v6KJ/m8uNrejaX9K7p31zyQ4Wkzb3BmdJE5k1lTQnXYSWXzqeqPcHGbpI9v+hqa/DahRdlz+6ZzlQz+41khd4d9gDGmv158V9Ryq5XVnHI3bfLvtQRM29kto5vjG0PIKjNdYXM9/z8+U69bVMbv282WclnXHbEY/geRPIdnpIZ7Znxzi5yeLlE+BY7OTZc+D28wAHcOf0Qtf4xlrQk1P8HzXxvVYdbF1XOWSPTL2dxd4C8i/3qgPva9XFDcxraLpyZijpeQkP2m5jTZc7tmjPAKqTr6F2OrLOut3AWmdeknzYH4NInsKsKxDyPl+OVx3c5sn5TEvAEkJ/DbgJ59oy5Hl6eM6vfCFS9p1h9eg5x7+PqsqPHhveAWeys9voMfqEr5+e9hfeVUj+7qAqJHfThw/m9BXx50+/Fzy9jsCR8+hTEJpTZtFXHWODimTmKfFmGfO/GTdLJyUNd68bfFMXnkGb8/0I9vsFhvnm/X2/Nu3xcHJTrP9rNl687T+4rl5M3dXfaceqh/hp131ApV5CL9a6Ft/qb/VP1t3thpbT7aeadbt4zMt6rw2frpfx86+jM=</latexit> ˆ projection of T onto Z: T <latexit sha1_base64="awFEcaFpPfD49FIWpreA6aLRo=">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</latexit> ˆ E [ Y | ˆ 2. Linearly regress Y on to estimate . Obtain our estimate as <latexit sha1_base64="VsSCPqNMkULYp0hGZrw3VuIwzA=">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</latexit> ˆ T ] T δ <latexit sha1_base64="awFEcaFpPfD49FIWpreA6aLRo=">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</latexit> ˆ the fitted coefficient in front of . T Z U <latexit sha1_base64="7hJNLSbvPbIVSt2QBaKnRXtOqY=">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</latexit> ˆ T = ˆ T Y E [ T | Z ] Brady Neal Warm-Up: Linear Setting 19 / 33

  69. <latexit sha1_base64="eC2b4dDBm4pmiwPXP6gE0iIizc=">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</latexit> <latexit sha1_base64="sKGswFhJDYMf/mqsKR1vFLuK4/4=">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</latexit> Two-Stage Least Squares Estimator 1. Linearly regress T on Z to estimate . This gives us the <latexit sha1_base64="qUoqhn4TXm2OMAEZLKEMQePRaE=">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</latexit> E [ T | Z ] <latexit sha1_base64="awFEcaFpPfD49FIWpreA6aLRo=">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</latexit> ˆ projection of T onto Z: T <latexit sha1_base64="awFEcaFpPfD49FIWpreA6aLRo=">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</latexit> ˆ E [ Y | ˆ 2. Linearly regress Y on to estimate . Obtain our estimate as <latexit sha1_base64="VsSCPqNMkULYp0hGZrw3VuIwzA=">AUfnicrVjbtGEN2kt8S9xEn91hc1Sos+yIrkprBfDASwExRFAziAb20UBJFWYR4K0lZdQT9Rl/bL+j/9G965uxSFHUhpcASRK5mZ87Mzm257ISuEyeNxn937n708Sefnbv/tbnX3z51YPth4/O42AYWfaZFbhBdNlpx7br+PZ4iSufRlGdtvruPZFZ3Ak8xfXdhQ7gX+a3IT2W695Ts9x2onILVa/XYybnVtN2lP3m1XG/UGP5XFQdMqsp8ToKH9/9VLdVgbLUHnKVr5KMHZVW8X4vlFN1VAhaG/VGLQI4fztpqoLcgOwWDow3qANcr/HtjqD7+C2ZMaQtaXPwiSFbUd4ani3GPVH0X/ZUZ3lU6xsQWG29w7xhMD9RE9UEtk0s515fr4OuVrDrBGg64WgcrCUkRP1i5Nfdwd/E/wQrlegNOG6MupCKMLNBcUDVFdES4a8+Lb/qMRJt8NkZidZHdRVbLfIDvAFhtjGNaKrZW1EsTF5+abdoqPC5juhrxT6xQ21eE1ZuwWHc9SqE9xSUgXqPUR65SOdsdq3mSgzyZIku4U/I4YM7BiVg5jvwnwOfK5awBQdbUbyimsJmUF1g/wL59NsDhHPXdoTM2oV5ptDPaGprMzu1E4X9w6xI8iPMdenHvFDfbYxI6YvakXa+SO8G/EfxZtObodVamxLOGNUl21YAbYMaZYmlPaH+mFmnrxqBVzHdX/Ur9NuJeo1ckr2uQDWihzuKYunwT+0P0FfGLh6tQxbMpUYdHu3oM1sGoGX6NJLoe4pVxKwf16COcXfptZAoNWoX/4w49rgmieiQukcYd6lHpCvodXW1b+yY5LSKbx2p0WtFeaf1K7ujPNaRc5npVZMVh5SU0M9MyuWCIjds16zTDfUPTnm6iZzVnVof9Y1JEfz9olUFp98rLrMxj6l8rjixeXrXLaWPdZJjbEU31+B45BSPUOPWZuh0bQ1U39HpkoCxsPi3TM5oetdtI3mZsage8ZTLvh0ncueJDN/ALPNnGrRe7PzRYgvjHy6G0jH5NaLHfOysrLRhiNpzPF8g5HNv2jRxJxjRfS1yHX8u30V5mZWRf7dvD8BUTJ2MQgrYveZW4sX/mpqIXVJFVeX9KpIUS8ekxPjJlPMJcIV3QC0+KFLh4+n8pDQGx6BMjMd6nEvrMLPBYefrkrNlMk9z6IpK5jikqou1Sl4uyxtNL5btTvM9L6vpxbJCuWZuOqYfxMS6LJFLGAeR8Gay9LREKuCeoOna5t/WsM/nXjs0eREZCy9KJD32GL2qwFTlq1L7EnbByPhEZH7/A9es/9N+AS0qR8z2WQjb2ZyNx/k0x+tKFnM0lvQ/9mku9LJG3uDc6UJjKvS2tKuE5KuXQ+le0JNnaT9PlFV1uN1w68KHt213SmitlvJCv07njIHlBbqz8v7jtSyc3SOh6y+3bYhyJq7hbUztmtoWURHK6xvpj5np0v16mvZXLr580R47CsO6Yz+gkewpJTw/xzP7kED87WYz5FDgyO1n6PLjLDNAx/Bm98BWesSbU9COe75q4vszteuiyjlraPrlLO4ekPexXx1zX9scNzSnod3cmSmv4wU0pL+JOV1m3K45A6xC2kTvcmSd2Fs8i8Jv202QeXPoHdlCBmebwcrzi+y5H1m5KAJ4DsHYbyLNnzPXwspxZ/Uag7D3F6tNziHsPV5cdPTa8x8w6l53dRo/RJ3z9HS08K5C8ncPVSG5O378aE5fHn/+9HvF0+sQHBmPsUklNa0WcRV5+iQMol5Wox5sxO1vXcSVnjzdsWz+SVZ/AOT+yzW6x9W672px/+7Y4ON+rN3+qN14/qz4/MG/m7qlv1GP1A/y0r56jMk/gVwta/lJ/q3921M73O7s7TzXr3TtG5muV+wc/A+OdPgQ</latexit> ˆ T ] T δ <latexit sha1_base64="awFEcaFpPfD49FIWpreA6aLRo=">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</latexit> ˆ the fitted coefficient in front of . T Z U <latexit sha1_base64="7hJNLSbvPbIVSt2QBaKnRXtOqY=">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</latexit> ˆ T = ˆ Y E [ T | Z ] ˆ T Brady Neal Warm-Up: Linear Setting 19 / 33

  70. <latexit sha1_base64="eC2b4dDBm4pmiwPXP6gE0iIizc=">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</latexit> <latexit sha1_base64="sKGswFhJDYMf/mqsKR1vFLuK4/4=">AUlnicrVhZb9tGEN6kV+IecVI/FOgLa6VAH2RVclM4Ly4C2E6LogEcwFdrGQZFrixCvEpSVh1C7/01eW3/Sv9NZ75diqQOUgoiQeRqduab2bmWy17oOnHSbv937/4H3708ScPHm58+tnXzafPzkLA5GkSVPrcANoueGUvX8eVp4iSuvAgjaXo9V573hgc8f34ro9gJ/JPkLpRXnjO3HMhMiXW9ud48u8EosQJPGl3PsY3uwEzSbkIgiSf9ZHJ1vdlot9r4GPODjh40hP4cB48fvhVdYtAWGIkPCGFLxIau8IUMX0vRUe0RUi0K5ESLaKRg3kpJmKDZEfEJYnDJOqQrjf071JTfrPmDGkLdLi0i8iSUN8q3lsGvdBVXfWbxR4l+lIgc023tG9pzE9oiZiQNQ6uYxzdbkefb2aVSe0hudYrUMrCUFhP1ilNfp7tL/hFbI1zvilDSySqikU0l6iKwjoiuivPs28GiIQJPkjtrK7iqreT6g75CwTBrHsJRtNcRLHRcfmiVsZR4XMV2O+BetUNlXhdWfrsFB3NUqmPeEKEPxhkZl5CqdxexazpVo5MkCXcyfgMn7pgoATLfIf85xFHOVYswWYeJSN5gLSEyqKWRf8V8ls0hxXMH9sSImoF8c6An1JWV253Z6dK9B+yI5FOaG0AP+6J9khgR8jezItNcEf0b4x/Fmy0ZugtVCbHs0lr4uxqEm5AM84US3lC+TOzSFmXEs3Q3x3xG/RLinsTXuG8bpJsAtVFsfQ5evY71NfYb94dGUqezajNKHDgx0DZMuQaLk+hcT6vqdVxKgfV6OmdHfhtRAoTWhn/4wx9rAmjugIusc0tqGHpQ3qdS2xp+2YlLSybx0p3mtBvKPa1d1xlmtLOejUg2dlfvQ1BbP9Io5Amx30WuW7oaqJ8dY3WTGqh7sz7sG52jZPpbK41OlY1sHECqjMteXLzORWvZRZ0EUv2/Q1x7EOqr+kxajPUmjYK9XegqyRAPCzcPZ0Tqt5Z23hmJiW6pz3lEp+qc96TeOZPwjSRU14rzhfhXik5bPdgDt+Cmq13Bkqywb0SidzlTLOxhJ+EeNOIKL4SvQ6zlm+nPKMysiv1+8Pw5RM7YRCOtim4jNxav/EQ0qBc0KvK/uRIMyXC0lI46dTzqeEy7xDaPGJwhWeTucntTE4JMpEe6yPuawOcxscdD4bnF2deYpDVQyw8FVXa2V83JR3ih6taw9zfeyrKJXyzLlFrnp6H4QA+uiRi5BHFjCK2TpSY1UgD1B0ZXNv69gn4+9dqTzItIWntdIeugxalWBrspXtfYl6IKR9gnL/PEOHrxF/5vgCWhdP+ayVrezOXu3smnufx4Tc/mkt6a/s0l39RISuwNzpTGMq9ra4q5jmu5VD7V7QmSdpPs+UVWxPXHnmR92xbdyZD7zecFWp3EcPaK7Un+f3Ha7kTm0dj9B9e+hDETbFbVz+t7Q8giOVlhfjHzPz5er1NciudXz5gBxWNQdsxn1BJI/hWSnh7iwPznAz08WKZ4Cx3ony54Hd5ABKoa/UC98Rc9YE2j6gZ7vOnR9Wdr1VkXlc9ZI98si7i4h79F+dYh9bX3cUJ+GdkpnprKOI9KQ/Sb6dJlzu/oMsAxpHb2LkVXW2XNnkVlN6mlzQFzqBHZXg5jn8WK86vguRlZvSgKcAPJz2PtALp4xV8PLc2b5G4G69xTLT8h3ft0dHRY817iKxz0dkl9Rh1wldPTwdz7yo4f3epKjh30+0nM/rK+LOn3xucXkfEkfOoU0wCaUrIi47R4eQSfTYowzZ36ybpVOygpv1ra4kFextvX/Ujq3WLjerPRmX37Nj8421fmy1Xz9rvHiu38w9EF+LbfEd+WlPvKDKPCa/WuJv8Vb8I/7d+mrp62jrZ8V6/17WuZLUfpsHf8Pou4BfQ=</latexit> Two-Stage Least Squares Estimator 1. Linearly regress T on Z to estimate . This gives us the <latexit sha1_base64="qUoqhn4TXm2OMAEZLKEMQePRaE=">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</latexit> E [ T | Z ] <latexit sha1_base64="awFEcaFpPfD49FIWpreA6aLRo=">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</latexit> ˆ projection of T onto Z: T <latexit sha1_base64="awFEcaFpPfD49FIWpreA6aLRo=">AUhHicrVjbtGEN2kbZq4Nyf1W1/UKAX6ICuSk8B5cRHATlAUDeAtuM2DgyJoixCvJWkrDqE/qSv7WP/p3/TM2eXoqgLKQeWIHI1O3Nmdm7LZTd0nThptf67dfuTz+78/ndextfPnV19s3n9wEgejyLKPrcANotNuJ7Zdx7ePEydx7dMwsjte17Xfdof7Mv/20o5iJ/CPkqvQfu91Lnyn71idBKTzc2zQSdJzxLIJ7tJ5PzXqr2eKntjhom0Fdmc9hcP/ev+pM9VSgLDVSnrKVrxKMXdVRMb7vVFu1VAjae5WCFmHkcN5WE7UB2RG4bHB0QB3ieoF/7wzVx3/BjCltQYuLXwTJmvrB8PQw7pOq76K/NsO7SkdKbLHxCveuwfRATdQA1Cq5jHN9uS6+XsWqE6zhOVfrYCUhKeIHq7DmPu4u/idYoVyvwGlj1INUhJEFmguqpoiOCHftefHNgJHokM/GSKwus7vMapkP8B0Cq4NxTEvF1p6ZeLiU7NW4XHZUxXI/6JFWr7yrD60zU4jLtehfAegTJUHzAqIpfpnM2u1VyJQZ4s0SX8CTl8cMegBMx8B/5zwFHMVQuYoqPDSF5wLSEzqGmQf+F8ls0h4rlNe2JGrcZ8c6gnNJWV253Z6eLeJXYE+RzA+oRXzRgj03siNmbebFB7gj/xvxn0UZrjt5kZUo8G1iTZFcDuAFmnCmW9oT2Z2aRti4FrWa+2+pX6rcR9wa9IndgGxAC3UWx9Tlm9jvoa+IXzxchSqezSgN6vBox4DZMgQt16eRN9jrCJm/bgGNcXdpdCojSoXfwz5tjmiSiI+oeY9yjHpGuodc1a6xY1LQKr5x2J0WtdaYf1K7ujPOaxU5n5VaM1m5R0t9dSsWCIgds96zTLdUPfkmKubzFnVpf1515AcLdonUnl8irHqMRsHlCriheXr3PZWnZYJw3GUnx/AY49SvUNPWZthkbTxkz97ZsqCRgPi3fP5ISud9E2nptJQfeMp1zw6TqXPUlm/gBmhzl1Ru/NzpchvjTy2W4gHT8ltVzuhJVlI0wSqcz5fIORzb9o0cScY0X0tch1/L9FebmVkX+2bw/AVEydjEIK2L3mNuLF/5kaqjF9SRVUV/SqSFEvHpJMT40ZTzEXCFd0gtPihS4el0flIZgwNQJsZjfc5ldZjb4LDz9ch5ZjJPc+iKSuY4pKrLtUpeLsbTS+X7U3zvSir6eWyQrlkbjqmH8TEOq2QSxgHkfBmsvSoQirgnqDp2ubf1rDP5147MnkRGQvfVkh67DF6VYGpyteV9iXsgpHxicj8/hEevGT/m/AJ6Lp+zGWTa3kzl7v6KJ/m8uNrejaX9K7p31zyQ4Wkzb3BmdJE5k1lTQnXYSWXzqeqPcHGbpI9v+hqa/DahRdlz+6ZzlQz+41khd4d9gDGmv158V9Ryq5XVnHI3bfLvtQRM29kto5vjG0PIKjNdYXM9/z8+U69bVMbv282WclnXHbEY/geRPIdnpIZ7Znxzi5yeLlE+BY7OTZc+D28wAHcOf0Qtf4xlrQk1P8HzXxvVYdbF1XOWSPTL2dxd4C8i/3qgPva9XFDcxraLpyZijpeQkP2m5jTZc7tmjPAKqTr6F2OrLOut3AWmdeknzYH4NInsKsKxDyPl+OVx3c5sn5TEvAEkJ/DbgJ59oy5Hl6eM6vfCFS9p1h9eg5x7+PqsqPHhveAWeys9voMfqEr5+e9hfeVUj+7qAqJHfThw/m9BXx50+/Fzy9jsCR8+hTEJpTZtFXHWODimTmKfFmGfO/GTdLJyUNd68bfFMXnkGb8/0I9vsFhvnm/X2/Nu3xcHJTrP9rNl687T+4rl5M3dXfaceqh/hp131ApV5CL9a6Ft/qb/VP1t3thpbT7aeadbt4zMt6rw2frpfx86+jM=</latexit> ˆ E [ Y | ˆ 2. Linearly regress Y on to estimate . Obtain our estimate as <latexit sha1_base64="VsSCPqNMkULYp0hGZrw3VuIwzA=">AUfnicrVjbtGEN2kt8S9xEn91hc1Sos+yIrkprBfDASwExRFAziAb20UBJFWYR4K0lZdQT9Rl/bL+j/9G965uxSFHUhpcASRK5mZ87Mzm257ISuEyeNxn937n708Sefnbv/tbnX3z51YPth4/O42AYWfaZFbhBdNlpx7br+PZ4iSufRlGdtvruPZFZ3Ak8xfXdhQ7gX+a3IT2W695Ts9x2onILVa/XYybnVtN2lP3m1XG/UGP5XFQdMqsp8ToKH9/9VLdVgbLUHnKVr5KMHZVW8X4vlFN1VAhaG/VGLQI4fztpqoLcgOwWDow3qANcr/HtjqD7+C2ZMaQtaXPwiSFbUd4ani3GPVH0X/ZUZ3lU6xsQWG29w7xhMD9RE9UEtk0s515fr4OuVrDrBGg64WgcrCUkRP1i5Nfdwd/E/wQrlegNOG6MupCKMLNBcUDVFdES4a8+Lb/qMRJt8NkZidZHdRVbLfIDvAFhtjGNaKrZW1EsTF5+abdoqPC5juhrxT6xQ21eE1ZuwWHc9SqE9xSUgXqPUR65SOdsdq3mSgzyZIku4U/I4YM7BiVg5jvwnwOfK5awBQdbUbyimsJmUF1g/wL59NsDhHPXdoTM2oV5ptDPaGprMzu1E4X9w6xI8iPMdenHvFDfbYxI6YvakXa+SO8G/EfxZtObodVamxLOGNUl21YAbYMaZYmlPaH+mFmnrxqBVzHdX/Ur9NuJeo1ckr2uQDWihzuKYunwT+0P0FfGLh6tQxbMpUYdHu3oM1sGoGX6NJLoe4pVxKwf16COcXfptZAoNWoX/4w49rgmieiQukcYd6lHpCvodXW1b+yY5LSKbx2p0WtFeaf1K7ujPNaRc5npVZMVh5SU0M9MyuWCIjds16zTDfUPTnm6iZzVnVof9Y1JEfz9olUFp98rLrMxj6l8rjixeXrXLaWPdZJjbEU31+B45BSPUOPWZuh0bQ1U39HpkoCxsPi3TM5oetdtI3mZsage8ZTLvh0ncueJDN/ALPNnGrRe7PzRYgvjHy6G0jH5NaLHfOysrLRhiNpzPF8g5HNv2jRxJxjRfS1yHX8u30V5mZWRf7dvD8BUTJ2MQgrYveZW4sX/mpqIXVJFVeX9KpIUS8ekxPjJlPMJcIV3QC0+KFLh4+n8pDQGx6BMjMd6nEvrMLPBYefrkrNlMk9z6IpK5jikqou1Sl4uyxtNL5btTvM9L6vpxbJCuWZuOqYfxMS6LJFLGAeR8Gay9LREKuCeoOna5t/WsM/nXjs0eREZCy9KJD32GL2qwFTlq1L7EnbByPhEZH7/A9es/9N+AS0qR8z2WQjb2ZyNx/k0x+tKFnM0lvQ/9mku9LJG3uDc6UJjKvS2tKuE5KuXQ+le0JNnaT9PlFV1uN1w68KHt213SmitlvJCv07njIHlBbqz8v7jtSyc3SOh6y+3bYhyJq7hbUztmtoWURHK6xvpj5np0v16mvZXLr580R47CsO6Yz+gkewpJTw/xzP7kED87WYz5FDgyO1n6PLjLDNAx/Bm98BWesSbU9COe75q4vszteuiyjlraPrlLO4ekPexXx1zX9scNzSnod3cmSmv4wU0pL+JOV1m3K45A6xC2kTvcmSd2Fs8i8Jv202QeXPoHdlCBmebwcrzi+y5H1m5KAJ4DsHYbyLNnzPXwspxZ/Uag7D3F6tNziHsPV5cdPTa8x8w6l53dRo/RJ3z9HS08K5C8ncPVSG5O378aE5fHn/+9HvF0+sQHBmPsUklNa0WcRV5+iQMol5Wox5sxO1vXcSVnjzdsWz+SVZ/AOT+yzW6x9W672px/+7Y4ON+rN3+qN14/qz4/MG/m7qlv1GP1A/y0r56jMk/gVwta/lJ/q3921M73O7s7TzXr3TtG5muV+wc/A+OdPgQ</latexit> ˆ T ] T δ <latexit sha1_base64="awFEcaFpPfD49FIWpreA6aLRo=">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</latexit> ˆ the fitted coefficient in front of . T Z U <latexit sha1_base64="7hJNLSbvPbIVSt2QBaKnRXtOqY=">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</latexit> ˆ T = ˆ Y E [ T | Z ] ˆ T Brady Neal Warm-Up: Linear Setting 19 / 33

  71. <latexit sha1_base64="MITRZaN0AvA2gq0L+sFNIgCr/E=">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</latexit> <latexit sha1_base64="sKGswFhJDYMf/mqsKR1vFLuK4/4=">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</latexit> Two-Stage Least Squares Estimator 1. Linearly regress T on Z to estimate . This gives us the <latexit sha1_base64="qUoqhn4TXm2OMAEZLKEMQePRaE=">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</latexit> E [ T | Z ] <latexit sha1_base64="awFEcaFpPfD49FIWpreA6aLRo=">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</latexit> ˆ projection of T onto Z: T <latexit sha1_base64="awFEcaFpPfD49FIWpreA6aLRo=">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</latexit> ˆ E [ Y | ˆ 2. Linearly regress Y on to estimate . Obtain our estimate as <latexit sha1_base64="VsSCPqNMkULYp0hGZrw3VuIwzA=">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</latexit> ˆ T ] T δ <latexit sha1_base64="awFEcaFpPfD49FIWpreA6aLRo=">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</latexit> ˆ the fitted coefficient in front of . T Z U <latexit sha1_base64="7hJNLSbvPbIVSt2QBaKnRXtOqY=">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</latexit> ˆ T = ˆ Y E [ T | Z ] ˆ T Brady Neal Warm-Up: Linear Setting 19 / 33

  72. <latexit sha1_base64="MITRZaN0AvA2gq0L+sFNIgCr/E=">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</latexit> <latexit sha1_base64="sKGswFhJDYMf/mqsKR1vFLuK4/4=">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</latexit> Two-Stage Least Squares Estimator 1. Linearly regress T on Z to estimate . This gives us the <latexit sha1_base64="qUoqhn4TXm2OMAEZLKEMQePRaE=">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</latexit> E [ T | Z ] <latexit sha1_base64="awFEcaFpPfD49FIWpreA6aLRo=">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</latexit> ˆ projection of T onto Z: T <latexit sha1_base64="awFEcaFpPfD49FIWpreA6aLRo=">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</latexit> ˆ E [ Y | ˆ 2. Linearly regress Y on to estimate . Obtain our estimate as <latexit sha1_base64="VsSCPqNMkULYp0hGZrw3VuIwzA=">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</latexit> ˆ T ] T δ <latexit sha1_base64="awFEcaFpPfD49FIWpreA6aLRo=">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</latexit> ˆ the fitted coefficient in front of . T Also works as an estimator in the Z U binary setting <latexit sha1_base64="7hJNLSbvPbIVSt2QBaKnRXtOqY=">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</latexit> ˆ T = ˆ Y E [ T | Z ] ˆ T Brady Neal Warm-Up: Linear Setting 19 / 33

  73. <latexit sha1_base64="8qrl2ZQIvUvYlZfeqMm9sVinJ8=">AU0nicrVhLb9tGEF6lbZq4Lyf1rRc2SoEeaFVynboXAwFsB0XRA7gV2sZBkVSFiG+SlJWbUKHotf2P/Rv9d905tulSOpBSoElSFzNznwzO6/dVS90nThpt/9rPrgw48ef/zk6cYn372+Rebz56fxcEoMu1TM3CD6KJnxLbr+PZp4iSufRFGtuH1XPu8Nzg+fNbO4qdwD9J7kL7yjNufKfvmEZCpOvNf7uW7SaGtq91+5Fhpt2jy24wSszAs7Wu51ha1/HjJBp5tp8QU+dK29ZqeNpXE8AkZEcCWhVQJRdBXW82602Xtr8oKMGTaFex8Gzp/+IrBEIEwxEp6whS8SGrvCEDG9L0VHtEVItCuREi2ikYN5W0zEBsmOiMsmDoOoQ/q+oV+XiurTb8aMIW2SFpc+EUlq4hvFY9G4D6p8sn6twLtMRwpstvGOnj2F6RE1EQOi1slnKvL9ejt1aw6oTX8iNU6tJIQFPaDWVpzn54u/U5ohfx9R5w2jSySimhkEs0lqSwjoie0vPsmwEiYDPphFbXWV3ldU8H9B7SFgGjWNYyrZq4o2Kiw/NmxlHhcxXY74B61Q2leF1Z+uwUHc5SqY94QoQ3FPozJylc5idi3nShTyZIEu5k/A4RN3TJQAme+Q/xziKOeqSZisw0Akb7CWEBnUsg/Yz7L5pDiuQ17YkRNQ7450BOqysrtzux06dkDdkTyKc0NoId9oZM9NrAjZG/mR3cEf0a45cJG80ZeguVyfHUaU2cXTrhBjTjTLGkJ6Q/M4ukdSnRNPXeFr9Av01x1+EVzmudZANYKLM4hi5fxX6f+gr7xaNvprJnM4oOHR7sGCBbhkTL9Uk1vcdrSJG/bgKNaWnC6+FQNGhnf0zxtjDmjiI+ge09iCHpbWqNe1xJ6yY1LSyr5x0J3mtWrIP65d2RlntbKcj0rVFbuQ1Nb7KoVcwTY7qLXTNUNZU+OsbrJjFU92J93Dc7Rsn0slcenHCsL2TiAVBmXvbh4nYvWsoM60RFL9v0NcexDq/oMWozVJo2CvV3oKokQDxMPD2VE7LeWdt4ZiYluqc85RKfrHPek3jmd8I0kFNdeK84X4V4pOSz3YA7fgpqtdwZKqsG9Eonc5UyzsY2fCPHEJV4IX4dYy9fTj1aYWRX7YfD8OUTO2EQhrYpuITcWr/xENKkXNCmryv7kSDMlwukpPHLKedLwmXeIbT4ROEKT6fzk9oYHBJlojzWx1xWh7kNDjqfBc6uyjzJISsqmeHgq7Wynm5KG8kvVrWmuZ7WVbSq2WZcovcdFQ/iIF1USOXIA4s4RWy9KRGKsCeIOnS5l9XsM/HXjtSeREpC89rJD30GLmqQFXl21r7EnTBSPmEZX57Dw/eov9NcAJa14+5bLKWN3O5u/fyaS4/XtOzuaS3pn9zyfsaSRt7gzOlscy72piruNaLplPdXuCTbtJdn6R1abju0de5D3bUp1JU/sNZ4XcHfRA/SV+vP8vsOV3Kmt4xG6bw9KIJmq6J2Th8MLY/gaIX1xcj3/H65Sn0tkpvNm/wMkZ3948Lu4gA7vxekOMON1T6Unea2ET8ZgZ+ok72lExKPD8T3dDr0Peb0p61Kirfkaq2xVxdwh5j3abQ+xK6+OG6i6zXbrxlHUckYbsM1F3w5zbVSf4ZUjr6F2MLHPGmrtJzGqSZ8UBcn7010NYp6Fi/Gq47sYWf7PEeD8nt+iHgK5eENcDS/PmeX3+bp/GZbfUN69unbRT+OFe8hs5FX7apQ8j7uTz7HMz908D5u0NVwbmbvng+o6+MP3t3vcHdc0QcOY+8gySQlrQi4rJbcAiZRJ31YtwY83txq3TPlXiztsWFvPIU3r7qRbq9RvXm83O7H9n84OznVbnVav9brf5elf9r/ZEfCVeiG/JT3viNVXmMfnVbDxu6I1XjR+2Trbut/7c+kuyPmomS9F6bX19/FEx+f</latexit> Question: In the binary linear, setting where is each assumption used in the proof below? <latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit> E [ Y | Z = 1] − E [ Y | Z = 0] = E [ δ T + α u U | Z = 1] − E [ δ T + α u U | Z = 0] = δ ( E [ T | Z = 1] − E [ T | Z = 0]) + α u ( E [ U | Z = 1] − E [ U | Z = 0]) = δ ( E [ T | Z = 1] − E [ T | Z = 0]) + α u ( E [ U ] − E [ U ]) = δ ( E [ T | Z = 1] − E [ T | Z = 0]) δ = E [ Y | Z = 1] − E [ Y | Z = 0] E [ T | Z = 1] − E [ T | Z = 0]

  74. <latexit sha1_base64="WEcbnDFOAVNR4xJgWQS71CyS6k0=">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</latexit> Question: In the continuous linear setting, prove the following: δ = Cov( Y, Z ) Cov( T, Z )

  75. What is an Instrument? No Nonparametric Identification of the ATE Warm-Up: Linear Setting Nonparametric Identification of Local ATE More General Settings for the ATE Brady Neal Nonparametric Identification of Local ATE 22 / 33

  76. Linear Outcome Assumption as Homogeneity Linear outcome assumption: <latexit sha1_base64="UeVu8hQ8nYOpqwWbqcmxjVPKxk=">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</latexit> Y := δ T + α u U Brady Neal Nonparametric Identification of Local ATE 23 / 33

  77. Linear Outcome Assumption as Homogeneity Linear outcome assumption: <latexit sha1_base64="UeVu8hQ8nYOpqwWbqcmxjVPKxk=">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</latexit> Y := δ T + α u U There are other variants of the linear outcome assumption that all require the treatment effect to be homogeneous (the same for all units) in some way (see, e.g., Section 16.3 of Hernán & Robins (2020)) Brady Neal Nonparametric Identification of Local ATE 23 / 33

  78. Linear Outcome Assumption as Homogeneity Linear outcome assumption: <latexit sha1_base64="UeVu8hQ8nYOpqwWbqcmxjVPKxk=">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</latexit> Y := δ T + α u U There are other variants of the linear outcome assumption that all require the treatment effect to be homogeneous (the same for all units) in some way (see, e.g., Section 16.3 of Hernán & Robins (2020)) Very restricting! Brady Neal Nonparametric Identification of Local ATE 23 / 33

  79. Can we get identification without parametric assumptions?

  80. <latexit sha1_base64="uSnWmzenLZoB/62kgJqyZD0I1U=">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</latexit> Potential Outcomes Notation with Instruments Z U T Y Brady Neal Nonparametric Identification of Local ATE 25 / 33

  81. <latexit sha1_base64="uSnWmzenLZoB/62kgJqyZD0I1U=">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</latexit> Potential Outcomes Notation with Instruments <latexit sha1_base64="M63Vb4/ai+tR0Jrq/f+NR10cms=">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</latexit> Y (1) and Y (0) are short for Y ( T = 1) and Y ( T = 0) Z U T Y Brady Neal Nonparametric Identification of Local ATE 25 / 33

  82. <latexit sha1_base64="uSnWmzenLZoB/62kgJqyZD0I1U=">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</latexit> Potential Outcomes Notation with Instruments <latexit sha1_base64="M63Vb4/ai+tR0Jrq/f+NR10cms=">AUrHicrVjbtGEF2nbZq4lzip3/rCRimQALIiuSnchwoIYCcoigZwAN/ayDAkcmUR4q3kyo4t6BP6gX3rp3Tm7FIkdSGlwBIkrmZnzszObXfVizw3Uc3mvxv3Pv8i/tfPni4+dX3z7aOvxk5MkHMW2PLZDL4zPet1Eem4gj5WrPHkWxbLr9zx52hvu8/zplYwTNwyO1E0kz/3uZeD2XburiHSxdsJR8oOfm89cLqKPlRja1u4FgTazrRzCZiaSWDMFZWP4zLB1FOpUvA2W1raVABa7mi4utWrPRxMuaH7TMoCbM6zB8/PAf0RGOCIUtRsIXUgRC0dgTXZHQ+4NoiaIiHYuxkSLaeRiXoqJ2CTZEXFJ4ugSdUjfl/Trg6EG9JsxE0jbpMWjT0ySlvjR8Dg07oOqn6zfyvEu0zEGNt4Q8+ewfSJqsSAqFVyKefqcj16+xWrVrSGX7Bal1YSgcJ+sAtr7tPTo9+KVsjfN8QpaeSQVEwjm2geUTWFdcT01J5n3wQiS74JI3Y6jK7y6zm+ZDeQ8Lq0jiBpWyrJd6auATQLGEr83iI6XLEj7RCbV8ZVn+6Bhdx16tg3iOiDMUtjYrIZTrz2bWcSxnkyQJdzK/AERB3QpQme+S/1ziKOaqTZiso4tIXmItETKoYZB/x3yazRHFcwf2JIiahXxzoScylZXZndrp0bMH7JjkxzQ3gB72RZ3skcCOkb2pF+vgjunXNX7ZsNGeoTdQmRzPOq2Js6tOuCHNuFMs7Qntz9Qibd2YaJZ574g/oF9S3OvwCud1nWRDWKizOIGuwMS+TX2F/eLTN1PZsymlDh0+7BgW4ZEy/RpJNb3klaRoH48gzqmpwevRUCpQzv75xpjH2viI6g+5rGDvSwtEW9riH2jB2Tglb2jYvuNK/VQv5x7erOKuV5QJUqmWysg1NTfHKrJgjwHbnvWabqh7coLVTWas6sH+rGtwjhbtY6ksPsVYOcjGAaSKuOzFxetctJZd1EkdsWTfXxJHG1J9Q09Qm5HRtJmrv31TJSHiYePpm5zQ9c7armdmxkT3jac84tN1znsSz/xNmF3kVAfey8+XIb4x8uluwB1/DGq53Akqygb02g8nSmXdzGS8I8ecQ1XgRfR1jLD9OPlZtZFftu8I5RM5YZBWRXeQG4tXfiRq1AtqlFVFf3KkmRLjdBLR+NmU8xnhMu8QWgKicIWPp/OTyhgcEGViPNbHXFqHmQ0uOp8Dzo7JPM2hK0rNcHBVl2vlvFyUN5peLutM870oq+nlsky5Qm6ph8kwDqrkFOIA0v4uSw9qpAKsSdourb5zxXsC7DXjkxexMbC0wpJHz1Gryo0Vfmu0j6FLhgbn7DMX5/gwSv0vwlOQOv6MZNVa3kzk7v5J9m8tdrejaT9Nf0byZ5WyEpsTe4UxrLvK+sKeY6rOTS+VS1J0jaTdLzi62Or575EXesx3TmSyz3BW6N2xjR5QX6k/z+87XMmtyjoeofv20IdiaHZKauf4ztCyCI5WF+CfM/ul6vU1yK52bzJzhDp2T/J7S4usLN7wRhnuGuzD6WnuR3ET0fgN+pk7+iExON98ROdzlr0/bawZ62Kyrekel2edxdQt6j3eYAu9L6uJG5y+wUbjxFHW9IQ/qZmLthxu2ZE/wypHX0LkbWOePM3SRmNemz4oC49P3pgIxy8LFeOXxXYys/+cIcX7PblF3gZy/Ia6Gl+XM8vt81b8My+ET379O2hHyeG9wBZ56EvS+oQ+n6uz7c/80cP7uUlVw7o6fPpnRV8Sfvbte4u45Io6MR9BFKQ1LY+47BYcQUaZs16CG2N2L24U7rkab9a2JdXvsFrm14kTa/fvNiqtWb/O5sfnOw2Wj83mu9f1V7/av5XeyC+F0/Fc/LTnhNlXlIfrXFfxv3Nx5tbG03to+2P2yfa9Z7G0bmO1F4bf/B7YxEQ8=</latexit> Y (1) and Y (0) are short for Y ( T = 1) and Y ( T = 0) <latexit sha1_base64="d6IaN9EZSd+xwhLp8Bud/kYpVUg=">AU0nicrVhbxtFB4XKG24pSVvCx1kYrkGDu0hAcsVUpaIUSlVMoN6iqyd8fxyntjdxw3WfkB8Qr/gb/Fv+Gcb8Zery+7dhVb9o7PnPOdM+c2M+5GnpuoRuO/yp0Pvzo7sf37m98uln3+x/eDhaRIOY1ue2KEXxufdTiI9N5AnylWePI9i2fG7njzrDg54/uxKxokbBsfqOpJv/c5l4PZcu6OIdLH9b1vJdyo9k1YQjqx+50paY6utCEL5MlBP2m6QqHjIY6tlNb+1NL/VCZwixsaUMYzfAUQmVCPpJ+GCtrfLFdbdQbeFmLg6YZVIV5HYUP7v8j2sIRobDFUPhCikAoGnuiIxJ6vxFN0RAR0d6KlGgxjVzMSzEWyQ7JC5JHB2iDuj7kn69MdSAfjNmAmbtHj0iUnSEt8YHofGPVD1k/VbM7yrdKTAZhuv6dk1mD5RlegTtUxuwrm+XJfefsmqFa3hR6zWpZVEoLAf7Nyae/T06LeiFfL3NXFKGjkFdPIJpHVE1hHTE9tefZN31EogM+SO2usjuIqt5PqT3gLA6NE5gKdtqiZcmLgE0S9jKPB5iuhrxHa1Q21eE1ZuwUXc9SqY95goA3FDozxykc7Z7FrNpQzyeIku5lfgCIg7IUqIzHfJfy5x5HPVJkzW0UEkL7GWCBlUN8i/YH6SzRHFcxf2JIiahXxzoScylZXZPbHTo2cX2DHJpzTXhx72RY3skcCOkb0TL9bAHdOvEX7ZsNGeo9dRmRzPGq2Js6tGuCHNuFMs7Qntz4lF2rqUaJZ574pfoV9S3GvwCud1jWRDWKizOIGuwMS+RX2F/eLTN1PZsxNKDTp82NFHtgyIlunTSKzvO1pFgvrxDGpKTw9ei4BSg3b2zwhjH2viA6he0RjB3pY2qJeVxf7xo5xTiv7xkV3WtRqIf+4dnVnNfKcgEq1TJZ2YKmhnhqVswRYLtnvWabqh7coLVjes6sL+rGtwjubtY6ksPvlYOcjGPqTyuOzF5etctpY91EkNsWTfXxJHC1I9Q09Qm5HRtDVTfwemSkLEw8bTNzmh6521jeZmUqL7xlMe8ek65z2JZ/4gzA5yqg3vzc4XIb4w8pPdgDt+Cmqx3CkqKy8b0yidzhTLuxhJ+EePOIaL4KvI6zl6+nHmplZF/t28IFRM5YZDWRXeQG8tXfiyq1AuqlFV5f3KkmRLjdBLR+PGU8zHhMu8AWgKicIWn0/lxaQwOiTI2HuthblKHmQ0uOp8DzrbJPM2hK0rNcXBVF2vlvFyWN5peLOtM8z0vq+nFsky5Qm6ph8kwDovkVOIA0v4M1l6XCIVYk/QdG3zb2vYF2CvHZq8iI2FZyWSPnqMXlVoqvJVqX0KXTA2PmGZ39/Dg1fof2OcgDb1YyarNvJmJnf9Xj7N5EcbejaT9Df0byZ5UyIpsTe4UxrLvC6tKeY6KuXS+VS2J0jaTSbnF1tNXx3yYu8ZzumM1lmv+Gs0LtjCz2gtlZ/Xtx3uJKbpXU8RPftog/F0OwU1M7JraFlERyusb4E+Z7dL9epr2Vy83mTnSEmZ/9kZndxgZ3dC1Kc4UZmH5qc5nYRPx2Bn6mTvaITEo8PxPd0OmvS98vcnrUuKt+ShqbzeLuEfI+7TaH2JU2x43MXWY3d+PJ63hBGiafsbkbZtyeOcGvQtpE73JknTPOwk1iXpM+K/aJS9+frksQsyxcjlc3+XI+n+OEOf37BZ1G8izN8T18LKcWX2fL/uXYfXdN6Jnj749OPE8B4i6z0ZUkdQt/P9dnYOGfBs7fPaoKzt30cM5fXn8+bvrJe6eQ+LIePQdREFa02YRV92CI8goc9ZLcGPM7sX13D1X483blszklW/wWqYXSdPrty62q835/84WB6d79eazeuP10+rzn8z/avfEV+KReEJ+2hfPqTKPyK925W6lVnlW+WHneOdm58+dvzTrnYqR+VLkXjt/w9RVSBF</latexit> We now have T ( Z = 1) and T ( Z = 0) or T (1) and T (0) for short Z U T Y Brady Neal Nonparametric Identification of Local ATE 25 / 33

  83. <latexit sha1_base64="uSnWmzenLZoB/62kgJqyZD0I1U=">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</latexit> Potential Outcomes Notation with Instruments <latexit sha1_base64="M63Vb4/ai+tR0Jrq/f+NR10cms=">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</latexit> Y (1) and Y (0) are short for Y ( T = 1) and Y ( T = 0) <latexit sha1_base64="def7f/hUKSNd3e3nkF4vCLcuX5A=">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</latexit> We also have Y ( Z = 1) and Y ( Z = 0) <latexit sha1_base64="d6IaN9EZSd+xwhLp8Bud/kYpVUg=">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</latexit> We now have T ( Z = 1) and T ( Z = 0) or T (1) and T (0) for short Z U T Y Brady Neal Nonparametric Identification of Local ATE 25 / 33

  84. <latexit sha1_base64="7Wpmtk4gGKaYaQv+qBHDO4V28=">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</latexit> Principal Strata Break data into 4 strata (groups) based on how the instrument affects the treatment they take Z U T Y Brady Neal Nonparametric Identification of Local ATE 26 / 33

  85. <latexit sha1_base64="7Wpmtk4gGKaYaQv+qBHDO4V28=">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</latexit> Principal Strata Break data into 4 strata (groups) based on how the instrument affects the treatment they take Z U T Y Brady Neal Nonparametric Identification of Local ATE 26 / 33

  86. <latexit sha1_base64="7Wpmtk4gGKaYaQv+qBHDO4V28=">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</latexit> Principal Strata Break data into 4 strata (groups) based on how the instrument affects the treatment they take Z U • Compliers: <latexit sha1_base64="oeupem7J3nlOID/wbAq7V7swhcs=">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</latexit> T ( Z = 1) = 1 , T ( Z = 0) = 0 T Y Brady Neal Nonparametric Identification of Local ATE 26 / 33

  87. <latexit sha1_base64="7Wpmtk4gGKaYaQv+qBHDO4V28=">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</latexit> Principal Strata Break data into 4 strata (groups) based on how the instrument affects the treatment they take Z U • Compliers: <latexit sha1_base64="oeupem7J3nlOID/wbAq7V7swhcs=">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</latexit> T ( Z = 1) = 1 , T ( Z = 0) = 0 T Y • Defiers: <latexit sha1_base64="xD8zMCKqXrs71+Hc7nLNY0XPcuc=">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</latexit> T ( Z = 1) = 0 , T ( Z = 0) = 1 Brady Neal Nonparametric Identification of Local ATE 26 / 33

  88. <latexit sha1_base64="7Wpmtk4gGKaYaQv+qBHDO4V28=">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</latexit> Principal Strata Break data into 4 strata (groups) based on how the instrument affects the treatment they take Z U • Compliers: <latexit sha1_base64="oeupem7J3nlOID/wbAq7V7swhcs=">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</latexit> T ( Z = 1) = 1 , T ( Z = 0) = 0 T Y • Defiers: <latexit sha1_base64="xD8zMCKqXrs71+Hc7nLNY0XPcuc=">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</latexit> T ( Z = 1) = 0 , T ( Z = 0) = 1 • Always-takers: <latexit sha1_base64="lpzpnPZaGkQ0k+6ItoP5qSHQ4oQ=">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</latexit> T ( Z = 1) = 1 , T ( Z = 0) = 1 Brady Neal Nonparametric Identification of Local ATE 26 / 33

  89. <latexit sha1_base64="7Wpmtk4gGKaYaQv+qBHDO4V28=">AWr3icrVhZb9tGEN7YPRL3ygH4pS9s7AJIKuSmyIFAgMBnARF0RQJ4MRpLdeVyJVEiFfJpVH1U/oD+xz/0hnvl2KpA5SCiJB5HI483sXLurXuS5iWq1/r2tf3Rx598ev3Gzmef/HlVzdv3X6ThGlsy9d26IXx2143kZ4byNfKVZ58G8Wy6/c8edobHfP70sZJ24YnKirSJ73UHg9l27q4h0cWvr705PDtxgotzRu8i1VRrL6Y5lPvesX0JHJrPnTkCPZ4nqKtlwuslQOufW/U4ahL1ExpfSscOgH6aBQwofWF1l3W81Wg+syf5ylv3p42XIDKloDsqXgXpgnfWkF4tT/aVdRT2reVY56xlJlWBHKbKDn05w43dwbAW2AhVwLpBouLUWLyOrblAhlrw+jNnUPR61FXD1X6WxFxy2eMNJTOXLMoVwlDLW/TACpM6MnBKiXZxc6/VbOFjLQ7aZrAnzOdleOvGP6IjHBEKW6TCF1IEQtHYE12R0PdMtEVLREQ7FxOixTRy8V6Kqdgh2ZS4JHF0iTqi64Cezgw1oGfGTCBtkxaPfjFJWuKe4XFo3AdV31m/VeBdpWMCbLbxiu49g+kTVYkhUevkMs715Xr09WtmrWgOP2C2Ls0kAoX9YJfm3Ke7R8+KZsjXK+KUNHJIKqaRTSPqJrCOmK6a8+zb4aIRBd8kZsdZXdVbz+5C+I8Lq0jiBpWyrJZ6buATQLGEr83iI6WrEv2iG2r4qrP5sDi7irmfBvCdEGYl3NCojV+ksZtdqLmWQp0t0Mb8CR0DcCVFCZL5L/nOJo5yrNmGyji4iOcBcImRQ0yD/hPdZNkcUzwPYkyBqFvLNhZ7IVFZud2anR/cesGOSn9C7IfSwLxpkjwR2jOzNvNgAd0xPYzZsNGeozdRmRzPBs2Js6tBuCG9cWdY2hPan5lF2roJ0SzPRA/Q7+kuDfgFc7rBsmGsFBncQJdgYn9EfUV9otPV6ayZzNKAzp82DFEtoyIluvTSKzvW5pFgvrxDOqE7h68FgGlAe3snzHGPubEU2he0xjB3pY2qJe1xSPjB3Tklb2jYvutKjVQv5x7erOK+V5QJUqmWy8giaWuKhmTFHgO0ues023VD35ASzm85Z1YP9edfgHC3bx1J5fMqxcpCNQ0iVcdmLy+e5bC6HqJMGYsm+HxDHEaT6hp6gNiOjadQf8emSkLEw8bdNzmh6521jefeTIjuG095xKfrnNckfvMnYXaRUx14r/i+CvGZkc9WA+74E1Cr5d6gsqyMY0mszfV8i5GEv7RI464xovg6whz+Wb2swpv1sX+MHjBAiJnrDJI6I7yI3lMz8Re9QL9iryv7kSDMlxu4kovH+jHOfcJl3BC0BUbjCJ7P309oYPCXK1Hisj3dZHeY2uOh8Djg7JvM0h64oNcfBV2tlfNyWd5oerWsM8v3sqymV8sy5RK56Zp+kADrbY2cQhxYwi9k6UmNVIg1QdO1zb+uYV+AtTY1eREbC09rJH30GD2r0FTli1r7FLpgbHzCMr+9hwcv0f+m2AFt6sdcVm3kzVzu6r18msuPN/RsLulv6N9c8l2NpMTa4M5oLPOqtqaY62Utl86nujVB0mqS7V90tTVw7ZEXec12TGeyzHrDWaFXxyP0gMZa/Xlx3eFKbtfWcYru20MfiqHZqaid1x8MLY9gusb8EuR7fr5cp76Wyc3nTb6HyPb+SWF1cYGdnwsm2MONzTqU7eYOED8dgR+pk72gHRKPj8V3tDtr0/V5ac1aF5VPSanpdkXcQ0J+RKvNU6xKm+NG5ixzUDrxlHU8Iw3Zb2rOhjm3Z3bwq5A20bscWeMs3CSmNek94pD4tLnp6saxDwLl+NVx3c5sv6fI8T+PT9FfQjk4glxPbw8Z1af5+v+ZVh9o3o3qerh36cGN6nyDoPfVlSh9Dnc73OV74p4Hz95CqgnN3cvf2nL4y/vzZdYCzZ0ocOY8+gyhIa1oRcdUpOIKMnu9BCfG/FzcLJ1zNd68bUkhr3yDd2R6kTS9fufi5l57/r+zxcGbw2b7+2br1eHek4fmf7Xr4mtxV9wnPz0ST6gyX5Jf7a3/tm9s396+s9vePd39fcPzbp1zcjcEaXPrvs/H5e6tQ=</latexit> Principal Strata Break data into 4 strata (groups) based on how the instrument affects the treatment they take Z U • Compliers: <latexit sha1_base64="oeupem7J3nlOID/wbAq7V7swhcs=">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</latexit> T ( Z = 1) = 1 , T ( Z = 0) = 0 T Y • Defiers: <latexit sha1_base64="xD8zMCKqXrs71+Hc7nLNY0XPcuc=">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</latexit> T ( Z = 1) = 0 , T ( Z = 0) = 1 • Always-takers: <latexit sha1_base64="lpzpnPZaGkQ0k+6ItoP5qSHQ4oQ=">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</latexit> T ( Z = 1) = 1 , T ( Z = 0) = 1 • Never-takers: <latexit sha1_base64="NqOPrwRK/pv6KXEPBk7WPSbRkQ=">AUfHicrVjbtGEN2kt8S9OYnf+sJGLpCitCq5ad2HCghgJyiCBnA39oMCSKsgjzFpKy6wj6hP5Lv6ZAn9ovKXrm7EoUdaMc2ILJ1ezMmdm57a7ase+lWa32963b73/wYcf3bm79vEn372+fq9+0dp1E8c9CJ/Cg5abdS1/dC9zDzMt89iRO3FbR97h9vivzxduknpReJBdxe7roHUWel3PaWUgna4/b2ZgzwI3zB41vTDNkr6MrYZV/xqPmtW0Ldtqvum3OtZC1hpZT9crtWqNf9bsoG4GFWX+9qN7d/9QTdVRkXJUXwXKVaHKMPZVS6X4vFJ1VMxaK/VALQEI4/zrhqNcj2weWCowXqOZ5n+PbKUEN8F8yU0g60+PhPIGmprwxPB+Muqfot+q0J3kU6BsQWG6/wbhvMANRM9UAtkxtxri7XxicoWXWGNfzI1XpYSUyK+MEprLmLt4/vGVYozytwuh1IJVg5IDmg6opoiPBW3tefNjJFrkczESq5fZvcxqmY/wOQdWC+OUloqtlnpm4hJSs0tbhcdnTBcj/o4VavuWYXHa/AYd70K4T0A5Vy9xaiIvEznZHYt5soM8nCOLuHPyBGCOwUlYuZ78J8HjmKuOsAUHS1G8oxriZlBVYP8nPOjbI4Rzy3akzJqFvPNo57YVFZu98hOH+82sRPIDzDXox7xhQ17XGInzN6RF21yJ/h2yW8ObXSm6FVWpsTxpoku2zgRpjxljaE9qfI4u0dQPQLPZUr9Qv4u42/SK5LUN2YgW6ixOqSs0sW+gr4hfAjyFKp4dUWzqCGhHj9lyDlquTyOJvm+xipT14xvUAd4+vRYTxaZ28c8lxwHXJBHtU/clxh3qEWkLva6qdowdw4JW8Y3H7jSr1WL+Se3qzjitVeRCVqplsrJBTX12KxYIiB2T3rNMd1Q9+SUqxtOWdWm/XnXkBwt2idSeXyKseowG3uUKuKF+evc95atlknNmMpvj8DR4NSXUNPWZux0bQ2UX+7pkoixsPhOzA5oetdtF1OzQxAD4ynfPDpOpc9SWbeALPFnGrSe5PzyxCfGvnRbiAdf0DqcrkjVlZRNsFoMJ5ZLu9x5NI/eiQR13gxfR1zLV+O/62JmVWxbwYvnEGUjM0M0qroHebG/JUfqAp6QVZVfSnRFoCU8nMcabY85N4ArvObWEoEiFD8bzw9IY7IEyNB7rcm5Uh7kNHjtfh5xNk3maQ1dUNsUhVb1cq+TlvLzR9OWynXG+F2U1fbmsUC6Ym57pBymxTkrkMsZBJIKJLD0okYq4J2i6tvnXFewLudf2TV4kxsLjEsmAPUavKjJV+aLUvoxdMDE+EZnf3sGDF+x/Q56AruvHXDa7ljdzuat38mkuf3lNz+aSwTX9m0u+LZF0uTd4Y5rIvCytKeHaL+XS+VS2J7jYTUbnF1tNp9teFH27I7pTJbZbyQr9O7YA+wV+rPs/uOVHK9tI7L5t9qGEmjtLaufwxtDyCPZXWF/KfM/vl6vU1zy56bzJzxCjs386sbt4xM7vBQOe4S7NPjQ6zW0xfjoCP6OTvcAJSca76juczup4PivsWauiyi2pb7rdJO42kHew2+xV7o+bmzuMluFG09Rx1NoGP0Pzd0w5/bNCX4R0nX0zkfWOdOZuUlMa9JnxR649P3pqgQxz8L5eMvjOx9Z/84R8fye36JuAnyhrgaXp4zi+/zZb8yL7xnh38fTZj1PDu8es89mXQIfT/XZ5/dmV8aJH+3URWSu4OH96f0FfGn765nvHv2wZHz6DtIRmlNm0RcdAuOKZOZs17KG2N+L64W7rkab9q2dCKvAoPXML3INb1+7XS9Up/+7Wx2cLRdrX9frb18XHnyk/ld7Y76Qj1Uj+CnHfUElbkPvzrqT/WX+kf9+C/jc2Nbza2NOvtW0bmgSr8bfzwP47MAKc=</latexit> T ( Z = 1) = 0 , T ( Z = 0) = 0 Brady Neal Nonparametric Identification of Local ATE 26 / 33

  90. <latexit sha1_base64="uJ8AEZwBJ4xhnUEPOEHwHt79XA=">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</latexit> <latexit sha1_base64="7Wpmtk4gGKaYaQv+qBHDO4V28=">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</latexit> Principal Strata Break data into 4 strata (groups) based on how the instrument affects the treatment they take Z U • Compliers: <latexit sha1_base64="oeupem7J3nlOID/wbAq7V7swhcs=">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</latexit> T ( Z = 1) = 1 , T ( Z = 0) = 0 T Y • Defiers: <latexit sha1_base64="xD8zMCKqXrs71+Hc7nLNY0XPcuc=">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</latexit> T ( Z = 1) = 0 , T ( Z = 0) = 1 • Always-takers: <latexit sha1_base64="lpzpnPZaGkQ0k+6ItoP5qSHQ4oQ=">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</latexit> T ( Z = 1) = 1 , T ( Z = 0) = 1 Z U • Never-takers: <latexit sha1_base64="NqOPrwRK/pv6KXEPBk7WPSbRkQ=">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</latexit> T ( Z = 1) = 0 , T ( Z = 0) = 0 T Y Brady Neal Nonparametric Identification of Local ATE 26 / 33

  91. <latexit sha1_base64="uJ8AEZwBJ4xhnUEPOEHwHt79XA=">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</latexit> <latexit sha1_base64="7Wpmtk4gGKaYaQv+qBHDO4V28=">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</latexit> Principal Strata Break data into 4 strata (groups) based on how the instrument affects the treatment they take Z U • Compliers: <latexit sha1_base64="oeupem7J3nlOID/wbAq7V7swhcs=">AUfHicrVjbtGEN2kt8S9OYnf+sJGLpCitCq5ad2HCghgJyiCBnA39oMCSKsgjzFpKy6wj6hP5Lv6ZAn9ovKXrm7EoUdaMc2ILJ1ezMmdm57a7ase+lWa32963b73/wYcf3bm79vEn372+fq9+0dp1E8c9CJ/Cg5abdS1/dC9zDzMt89iRO3FbR97h9vivzxduknpReJBdxe7roHUWel3PaWUgna4/b2ZgzwI3zB41vTDNkr6MrYZV/1oeVtO2bKv5pt/qWAtZa8JaO12v1Ko1/lmzg7oZVJT524/u3f1DNVHRcpRfRUoV4Uqw9hXLZXi80rVU3FoL1WA9ASjDzOu2qo1iDbB5cLjhao53ie4dsrQw3xXTBTSjvQ4uM/gaSlvjI8HYy7pOq36LcmeBfpGBbLzCu20wA1Az1QO1TG7EubpcG5+gZNUZ1vAjV+thJTEp4gensOYu3j6+Z1ihPK/A6WLUgVSCkQOaD6qmiI4Eb+158U2PkWiRz8VIrF5m9zKrZT7C5xYLYxTWiq2WuqZiUtIzS5tFR6fMV2M+DtWqO1bhtUdr8Fj3PUqhPcAlHP1FqMi8jKdk9m1mCszyM5uoQ/I0cI7hSUiJnvwX8eOIq56gBTdLQYyTOuJWYGVQ3yc86PsjlGPLdoT8qoWcw3j3piU1m53SM7fbzbxE4gP8Bcj3rEFzbscYmdMHtHXrTJneDbJb85tNGZoldZmRJPG2uS7LKBG2HG2NpT2h/jizS1g1As8xnS/1C/S7ibtMrktc2ZCNaqLM4pa7QxL6BviJ+CfAUqnh2RLGpI6AdPWbLOWi5Po0k+r7FKlLWj29QB3j79FpMFJvaxT+XHAdck0S0T92XGHeoR6Qt9Lq2jF2DAtaxTceu9OsVov5J7WrO+O0VpELWamWycoGNdXUY7NiYDYPek1x3RD3ZNTrm4ZVWb9udQ3K0aJ9I5fEpxqrDbOxRqogrXpy/znlr2Wad2Iyl+P4MHA1KdQ09ZW3GRtPaRP3tmiqJGA+H78DkhK530XY5NTMAPTCe8sGn61z2Jl5A8wWc6pJ703OL0N8auRHu4F0/AGpy+WOWFlF2QSjwXhmubzHkUv/6JFEXOPF9HXMtXw5/rcmZlbFvhm8cAZRMjYzSKuid5gb81d+oCroBRVkVdGfEmhJDydxBhvjk3gSu859QSgiIVPhjPD0tjsAfK0Hisy7lRHeY2eOx8HXI2TeZpDl1R2RSHVPVyrZKX8/JG05fLdsb5XpTV9OWyQrlgbnqmH6TEOimRyxgHkQgmsvSgRCrinqDp2uZfV7Av5F7bN3mRGAuPSyQD9hi9qshU5YtS+zJ2wcT4RGR+ewcPXrD/DXkCuq4fc9nsWt7M5a7eyae5/OU1PZtLBtf0by75tkTS5d7gjWki87K0poRrv5RL51PZnuBiNxmdX3S12Xy24UXZszumM1lmv5Gs0Ltjgz3AXqk/z+47Usn10jrus/u2YcSau4sqZ3DG0PLI9hfYX0p8z2/X65SX/PkpvMmP0OMzv7pxO7iETu/Fwx4hrs0+9DoNLfF+OkI/IxO9gInJBnvqu9wOqvj+aywZ62Krekvul2k7jbQN7BbrPHXen6uLG5y2wVbjxFHU+hYfQ/NHfDnNs3J/hFSNfROx9Z50xn5iYxrUmfFXvg0venqxLEPAvn4y2P73xk/TtHxPN7fou6CeTJG+JqeHnOL7Pl/3KsPjuG+PdxdNnP04N7x6zmdfdtEh9P1cn312Z35pkPzdRlVI7g4e3p/SV8Sfvrue8e7ZB0fOo+8gGaU1bRJx0S04pkxmznopb4z5vbhauOdqvGnb0om8Cgxew/Qi1/T6tdP1Sn36t7PZwdF2tf59tfbyceXJT+Z3tTvqC/VQPYKfdtQTVOY+/OqoP9Vf6h/174P/NjY3vtnY0qy3bxmZB6rwt/HD/6LEAKg=</latexit> T ( Z = 1) = 1 , T ( Z = 0) = 0 T Y • Defiers: <latexit sha1_base64="xD8zMCKqXrs71+Hc7nLNY0XPcuc=">AUfHicrVjbtGEN2kt8S9OYnf+sJGLpCitCq5ad2HCghgJyiCBnA39oMCSKsgjzFpKy6wj6hP5Lv6ZAn9ovKXrm7EoUdaMc2ILJ1ezMmdm57a7ase+lWa32963b73/wYcf3bm79vEn372+fq9+0dp1E8c9CJ/Cg5abdS1/dC9zDzMt89iRO3FbR97h9vivzxduknpReJBdxe7roHUWel3PaWUgna4/b2ZgzwI3zB41vTDNkr6MrYZV/xqPmtW0Ldtqvum3OtZC1pqw1k/XK7VqjX/W7KBuBhVl/vaje3f/UE3VUZFyVF8FylWhyjD2VUul+LxSdVTMWiv1QC0BCOP864aqjXI9sHlgqMF6jmeZ/j2ylBDfBfMlNIOtPj4TyBpqa8MTwfjLqn6LfqtCd5FOgbEFhuv8G4bzADUTPVALZMbca4u18YnKFl1hjX8yNV6WElMivjBKay5i7eP7xlWKM8rcLoYdSCVYOSA5oOqKaIjwVt7XnzTYyRa5HMxEquX2b3MapmP8DkHVgvjlJaKrZ6ZuISUrNLW4XHZ0wXI/6OFWr7lmF1x2vwGHe9CuE9AOVcvcWoiLxM52R2LebKDPJwji7hz8gRgjsFJWLme/CfB45irjrAFB0tRvKMa4mZQVWD/Jzo2yOEc8t2pMyahbzaOe2FRWbvfITh/vNrETyA8w16Me8YUNe1xiJ8zekRdtcif4dslvDm10puhVqbE08aJLts4EaY8cZY2hPanyOLtHUD0Cz2VK/UL+LuNv0iuS1DdmIFuosTqkrNLFvoK+IXwI8hSqeHVFs6ghoR4/Zcg5ark8jib5vsYqU9eMb1AHePr0WE8WmdvHPJcB1yQR7VP3JcYd6hFpC72uqnaMHcOCVvGNx+40q9Vi/knt6s4rVXkQlaqZbKyQU019disWCIgdk96zTHdUPfklKsbTlnVpv1515AcLdonUnl8irHqMBt7lCrihfnr3PeWrZJzZjKb4/A0eDUl1DT1mbsdG0NlF/u6ZKIsbD4TswOaHrXbRdTs0MQA+Mp3zw6TqXPUlm3gCzxZxq0nuT8sQnxr50W4gHX9A6nK5I1ZWUTbBaDCeWS7vceTSP3okEd4MX0dcy1fjv+tiZlVsW8GL5xBlIzNDNKq6B3mxvyVH6gKekEFWVX0p0RaKAlPJzHGm2POTeAK7zm1hKBIhQ/G8PSGOyBMjQe63JuVIe5DR47X4ecTZN5mkNXVDbFIVW9XKvk5by80fTlsp1xvhdlNX25rFAumJue6QcpsU5K5DLGQSCiSw9KJGKuCdourb51xXsC7nX9k1eJMbC4xLJgD1GryoyVfmi1L6MXTAxPhGZ397Bgxfsf0OegK7rx1w2u5Y3c7mrd/JpLn95Tc/mksE1/ZtLvi2RdLk3eGOayLwsrSnh2i/l0vlUtie42E1G5xdbTafbXhR9uyO6UyW2W8kK/Tu2GAPsFfqz7P7jlRyvbSO+y+bfahJo7S2rn8MbQ8gj2V1hfynzP75er1Nc8uem8yc8Qo7N/OrG7eMTO7wUDnuEuzT40Os1tMX46Aj+jk73ACUnGu+o7nM7qeD4r7FmrosotqW+63STuNpB3sNvscVe6Pm5s7jJbhRtPUcdTaBj9D83dMOf2zQl+EdJ19M5H1jnTmblJTGvSZ8UeuPT96aoEMc/C+XjL4zsfWf/OEfH8nt+ibgJ58oa4Gl6eM4v82W/Miy+8Z4d/H02Y9Tw7vHrPZl10CH0/12ef3ZlfGiR/t1EVkruDh/en9BXxp+uZ7x79sGR8+g7SEZpTZtEXHQLjimTmbNeyhtjfi+uFu65Gm/atnQirwKD1zC9yDW9fu10vVKf/u1sdnC0Xa1/X629fFx58pP5Xe2O+kI9VI/gpx31BJW5D786k/1l/pH/fvgv43NjW82tjTr7VtG5oEq/G38D+imQCo</latexit> T ( Z = 1) = 0 , T ( Z = 0) = 1 • Always-takers: <latexit sha1_base64="lpzpnPZaGkQ0k+6ItoP5qSHQ4oQ=">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</latexit> T ( Z = 1) = 1 , T ( Z = 0) = 1 Z U • Never-takers: <latexit sha1_base64="NqOPrwRK/pv6KXEPBk7WPSbRkQ=">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</latexit> T ( Z = 1) = 0 , T ( Z = 0) = 0 T Y Brady Neal Nonparametric Identification of Local ATE 26 / 33

  92. Monotonicity Assumption (No Defiers) T i ( Z = 1) ≥ T i ( Z = 0) <latexit sha1_base64="uzQcI2iod6GX+k3WLvmwihepPrs=">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</latexit> ∀ i, Brady Neal Nonparametric Identification of Local ATE 27 / 33

  93. <latexit sha1_base64="7Wpmtk4gGKaYaQv+qBHDO4V28=">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</latexit> Monotonicity Assumption (No Defiers) T i ( Z = 1) ≥ T i ( Z = 0) <latexit sha1_base64="uzQcI2iod6GX+k3WLvmwihepPrs=">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</latexit> ∀ i, Z U T Y Brady Neal Nonparametric Identification of Local ATE 27 / 33

  94. <latexit sha1_base64="7Wpmtk4gGKaYaQv+qBHDO4V28=">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</latexit> Monotonicity Assumption (No Defiers) T i ( Z = 1) ≥ T i ( Z = 0) <latexit sha1_base64="uzQcI2iod6GX+k3WLvmwihepPrs=">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</latexit> ∀ i, Z U • Compliers: <latexit sha1_base64="oeupem7J3nlOID/wbAq7V7swhcs=">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</latexit> T ( Z = 1) = 1 , T ( Z = 0) = 0 T Y • Defiers: <latexit sha1_base64="xD8zMCKqXrs71+Hc7nLNY0XPcuc=">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</latexit> T ( Z = 1) = 0 , T ( Z = 0) = 1 • Always-takers: <latexit sha1_base64="lpzpnPZaGkQ0k+6ItoP5qSHQ4oQ=">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</latexit> T ( Z = 1) = 1 , T ( Z = 0) = 1 • Never-takers: <latexit sha1_base64="NqOPrwRK/pv6KXEPBk7WPSbRkQ=">AUfHicrVjbtGEN2kt8S9OYnf+sJGLpCitCq5ad2HCghgJyiCBnA39oMCSKsgjzFpKy6wj6hP5Lv6ZAn9ovKXrm7EoUdaMc2ILJ1ezMmdm57a7ase+lWa32963b73/wYcf3bm79vEn372+fq9+0dp1E8c9CJ/Cg5abdS1/dC9zDzMt89iRO3FbR97h9vivzxduknpReJBdxe7roHUWel3PaWUgna4/b2ZgzwI3zB41vTDNkr6MrYZV/xqPmtW0Ldtqvum3OtZC1hpZT9crtWqNf9bsoG4GFWX+9qN7d/9QTdVRkXJUXwXKVaHKMPZVS6X4vFJ1VMxaK/VALQEI4/zrhqNcj2weWCowXqOZ5n+PbKUEN8F8yU0g60+PhPIGmprwxPB+Muqfot+q0J3kU6BsQWG6/wbhvMANRM9UAtkxtxri7XxicoWXWGNfzI1XpYSUyK+MEprLmLt4/vGVYozytwuh1IJVg5IDmg6opoiPBW3tefNjJFrkczESq5fZvcxqmY/wOQdWC+OUloqtlnpm4hJSs0tbhcdnTBcj/o4VavuWYXHa/AYd70K4T0A5Vy9xaiIvEznZHYt5soM8nCOLuHPyBGCOwUlYuZ78J8HjmKuOsAUHS1G8oxriZlBVYP8nPOjbI4Rzy3akzJqFvPNo57YVFZu98hOH+82sRPIDzDXox7xhQ17XGInzN6RF21yJ/h2yW8ObXSm6FVWpsTxpoku2zgRpjxljaE9qfI4u0dQPQLPZUr9Qv4u42/SK5LUN2YgW6ixOqSs0sW+gr4hfAjyFKp4dUWzqCGhHj9lyDlquTyOJvm+xipT14xvUAd4+vRYTxaZ28c8lxwHXJBHtU/clxh3qEWkLva6qdowdw4JW8Y3H7jSr1WL+Se3qzjitVeRCVqplsrJBTX12KxYIiB2T3rNMd1Q9+SUqxtOWdWm/XnXkBwt2idSeXyKseowG3uUKuKF+evc95atlknNmMpvj8DR4NSXUNPWZux0bQ2UX+7pkoixsPhOzA5oetdtF1OzQxAD4ynfPDpOpc9SWbeALPFnGrSe5PzyxCfGvnRbiAdf0DqcrkjVlZRNsFoMJ5ZLu9x5NI/eiQR13gxfR1zLV+O/62JmVWxbwYvnEGUjM0M0qroHebG/JUfqAp6QVZVfSnRFoCU8nMcabY85N4ArvObWEoEiFD8bzw9IY7IEyNB7rcm5Uh7kNHjtfh5xNk3maQ1dUNsUhVb1cq+TlvLzR9OWynXG+F2U1fbmsUC6Ym57pBymxTkrkMsZBJIKJLD0okYq4J2i6tvnXFewLudf2TV4kxsLjEsmAPUavKjJV+aLUvoxdMDE+EZnf3sGDF+x/Q56AruvHXDa7ljdzuat38mkuf3lNz+aSwTX9m0u+LZF0uTd4Y5rIvCytKeHaL+XS+VS2J7jYTUbnF1tNp9teFH27I7pTJbZbyQr9O7YA+wV+rPs/uOVHK9tI7L5t9qGEmjtLaufwxtDyCPZXWF/KfM/vl6vU1zy56bzJzxCjs386sbt4xM7vBQOe4S7NPjQ6zW0xfjoCP6OTvcAJSca76juczup4PivsWauiyi2pb7rdJO42kHew2+xV7o+bmzuMluFG09Rx1NoGP0Pzd0w5/bNCX4R0nX0zkfWOdOZuUlMa9JnxR649P3pqgQxz8L5eMvjOx9Z/84R8fye36JuAnyhrgaXp4zi+/zZb8yL7xnh38fTZj1PDu8es89mXQIfT/XZ5/dmV8aJH+3URWSu4OH96f0FfGn765nvHv2wZHz6DtIRmlNm0RcdAuOKZOZs17KG2N+L64W7rkab9q2dCKvAoPXML3INb1+7XS9Up/+7Wx2cLRdrX9frb18XHnyk/ld7Y76Qj1Uj+CnHfUElbkPvzrqT/WX+kf9+C/jc2Nbza2NOvtW0bmgSr8bfzwP47MAKc=</latexit> T ( Z = 1) = 0 , T ( Z = 0) = 0 Brady Neal Nonparametric Identification of Local ATE 27 / 33

  95. <latexit sha1_base64="7Wpmtk4gGKaYaQv+qBHDO4V28=">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</latexit> Monotonicity Assumption (No Defiers) T i ( Z = 1) ≥ T i ( Z = 0) <latexit sha1_base64="uzQcI2iod6GX+k3WLvmwihepPrs=">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</latexit> ∀ i, Z U • Compliers: <latexit sha1_base64="oeupem7J3nlOID/wbAq7V7swhcs=">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</latexit> T ( Z = 1) = 1 , T ( Z = 0) = 0 T Y • Defiers: <latexit sha1_base64="xD8zMCKqXrs71+Hc7nLNY0XPcuc=">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</latexit> T ( Z = 1) = 0 , T ( Z = 0) = 1 • Always-takers: <latexit sha1_base64="lpzpnPZaGkQ0k+6ItoP5qSHQ4oQ=">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</latexit> T ( Z = 1) = 1 , T ( Z = 0) = 1 • Never-takers: <latexit sha1_base64="NqOPrwRK/pv6KXEPBk7WPSbRkQ=">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</latexit> T ( Z = 1) = 0 , T ( Z = 0) = 0 Brady Neal Nonparametric Identification of Local ATE 27 / 33

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend


More recommend