<latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="ankFpkUc9naoITIsD9NjYr0YunY=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="PeOKuh2RY/b2tuD2rOSag28FZ40=">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</latexit> <latexit sha1_base64="+Rktm3TlQFEu8Sf2KZ4iNZT7NFc=">AEp3icbVNtSxtBED5r2tr0TduP/bKtCApnmohgCw0IpRSqAWNghHZ25skS/aN3TltPI7+kX5t/1P/TecuZzXRPQ6enXnmfTZxSgZst/8uPFhsPHz0eOlJ8+mz5y9eLq+86gWbeQFHwirTxIeQEkDRyhRwYnzwHWi4DgZ75X64wvwQVpziBMHZ5oPjRxIwZFE58vLfYQfmKe2WD9kXdbeOF9ebfa1WF3QacGq1F9Ds5XFn/2UysyDQaF4iGcdtoOz3LuUQoFRbOfBXBcjPkQTgkariGc5VXqBVsjScoG1tNvkFXS2xY51yFMdEJMzXEU5nWl8D7daYaD2e5NC5DMGIaJAphpaVfWCp9CBQTQhw4SXlysSIey6QutWcCZPotlcY4dyfMVq2Wwe01pmREjkG6sAiNIMA7MOpZXdcmCZ4ErNvTcjUKLyF+yUPbATYdDwgMRzJQSI+V79KnkonfpKHEXcQ4hSE9dUoQ8y9t5chFlyJGrc0I8HEmNngyxZlATlWToid3mT0dn8ymkFYp6hjY1Ny84E5EYA63aY0IyblJUgDqAlJSTGcWVHJIT3rYATRdQclJIuQMxSzy9jpqWROtPsUqY4KhertUM+iqmps9Lgf1OWC+lpU65NpTHgqWuS2u3TYEHUqlpaoJ2hNYsdPOidpXI6SAhrf186qeuqUhxGkNRfUrZg3UbYcxowrOTRdBQPCQVlHRs1qfns0EqspstbUCZq7gcv6kvf3i7xfrl6S5PtFMavrcV9rvc7Ly5xemhQcMRx4x/pvy49VlzmeuSYai0S6h05PtjP/QO+C3lars936+H17dfdT/XiXojfRu2g96kQ70W70OTqIjiIRXUS/ot/Rn8ZG41uj1ziZUh8s1Davo5nT4P8Ayh6Fow=</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> Potential outcomes: notation Y i | do( T =1) : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i specific unit/individual do( T = 0) 5 / 41 Brady Neal
<latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="PeOKuh2RY/b2tuD2rOSag28FZ40=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="+Rktm3TlQFEu8Sf2KZ4iNZT7NFc=">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</latexit> <latexit sha1_base64="ankFpkUc9naoITIsD9NjYr0YunY=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="tcq7x+h5H1MVi9e74Wsj/KX91Xo=">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</latexit> Potential outcomes: notation Y i | do( T =1) : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i specific unit/individual Y i | do( T =0) do( T = 0) 5 / 41 Brady Neal
<latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="PeOKuh2RY/b2tuD2rOSag28FZ40=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="ankFpkUc9naoITIsD9NjYr0YunY=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="+Rktm3TlQFEu8Sf2KZ4iNZT7NFc=">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</latexit> <latexit sha1_base64="wHBl5SOG0FiXEwPvpVB6ZgvKM3g=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="tcq7x+h5H1MVi9e74Wsj/KX91Xo=">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</latexit> Potential outcomes: notation ) , Y i (1) Y i | do( T =1) : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i specific unit/individual : potential outcome under treatment Y i (1) Y i | do( T =0) do( T = 0) 5 / 41 Brady Neal
<latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="PeOKuh2RY/b2tuD2rOSag28FZ40=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="ankFpkUc9naoITIsD9NjYr0YunY=">AErnicbVNbaxNBFN7aqDXeWn3wZfRUqiwjUkpVMFAoREfKjQmzQhzM6eJEPmxsxZa7ou/hf9Qf5bzy72dqm7SwL35znfuZxCkZsN3+u3BnsXH3v2lB82Hjx4/ebq8uwo2MwLOBRWX+S8ABKGjhEiQpOnAeuEwXHyWS31B9/Ax+kNQc4dDXfGTkUAqOJBosv/g6kD8GeQ/hO+apLdYPWJd13hSD5dV2q10dhN0arAa1Wd/sL4s5dakWkwKBQP4bTdtjPuUcpFBTNXhbAcTHhIzglaLiG0M+rCgq2RpKUDa2n3yCrpFctcq5DmOqEmJrjOFzXlcLbdKcZDt/1c2lchmDELNAwUwtK9vBUulBoJoS4MJLypWJMfdcIDWtORcm0UWzucYO5OSc1bL5PGa1zImQyJdWARClGQVmHUotz+uSBc8CV2zkuRuHFpE/ZaHsgZtuOB4QGI5loJAeK9+lTyUTz/0D2PuIMQpCOuriYaYe2/PQiy4EjVuaUAeDyXGzgZsigJyrN0RO7yJqOz8ZnTDsQ8Qxsbm5adCciNANbtMKEZNykrQRxAS0pITOLKjkgIb1sBp4qoOSglXYCYpZ6fxUxLI3Wm2ZlMcUyb1W5tk49iZuqsNPjflOVCetqUC1NpDHhqmevS2m1R4KFUapaoB2hNQvdvKhdJXI2SEhrf928qeuKuVhDGnNBXUl5mWUTYcx40qOTFfBkHBQ1pFRs5rfLo3EaoqsNXWC5m7grL7kvb0i75WrlyT5XlHM6464r7Ve5+Xlml6aFBwxHjHeq/Kj1WXazxzQTQWiXQLnZ5s5/oDvQmONludrdb7L1urOx/qx7sUvYxeR+tRJ9qOdqKP0X50GImoiH5Fv6M/jXbjqNFvDGbUOwu1zfNo7jTG/wB+j4je</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="+Rktm3TlQFEu8Sf2KZ4iNZT7NFc=">AEp3icbVNtSxtBED5r2tr0TduP/bKtCApnmohgCw0IpRSqAWNghHZ25skS/aN3TltPI7+kX5t/1P/TecuZzXRPQ6enXnmfTZxSgZst/8uPFhsPHz0eOlJ8+mz5y9eLq+86gWbeQFHwirTxIeQEkDRyhRwYnzwHWi4DgZ75X64wvwQVpziBMHZ5oPjRxIwZFE58vLfYQfmKe2WD9kXdbeOF9ebfa1WF3QacGq1F9Ds5XFn/2UysyDQaF4iGcdtoOz3LuUQoFRbOfBXBcjPkQTgkariGc5VXqBVsjScoG1tNvkFXS2xY51yFMdEJMzXEU5nWl8D7daYaD2e5NC5DMGIaJAphpaVfWCp9CBQTQhw4SXlysSIey6QutWcCZPotlcY4dyfMVq2Wwe01pmREjkG6sAiNIMA7MOpZXdcmCZ4ErNvTcjUKLyF+yUPbATYdDwgMRzJQSI+V79KnkonfpKHEXcQ4hSE9dUoQ8y9t5chFlyJGrc0I8HEmNngyxZlATlWToid3mT0dn8ymkFYp6hjY1Ny84E5EYA63aY0IyblJUgDqAlJSTGcWVHJIT3rYATRdQclJIuQMxSzy9jpqWROtPsUqY4KhertUM+iqmps9Lgf1OWC+lpU65NpTHgqWuS2u3TYEHUqlpaoJ2hNYsdPOidpXI6SAhrf186qeuqUhxGkNRfUrZg3UbYcxowrOTRdBQPCQVlHRs1qfns0EqspstbUCZq7gcv6kvf3i7xfrl6S5PtFMavrcV9rvc7Ly5xemhQcMRx4x/pvy49VlzmeuSYai0S6h05PtjP/QO+C3lars936+H17dfdT/XiXojfRu2g96kQ70W70OTqIjiIRXUS/ot/Rn8ZG41uj1ziZUh8s1Davo5nT4P8Ayh6Fow=</latexit> <latexit sha1_base64="wHBl5SOG0FiXEwPvpVB6ZgvKM3g=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">AElnicbVNbaxNBFN7aqDXeWn0RfBktBR+2MSmB6kOkUIoiPrTQGzSlzM6eJEPmxsxZY7os/gBf9cf5bzy72dom7SwL35znfuZxCkZsN3+u3RvuXH/wcOVR83HT54+e769uI42MwLOBJWX+a8ABKGjhCiQpOnQeuEwUnyXi31J98Bx+kNYc4dXCu+dDIgRQcSXQgL1bX2612dht0KnBelSf/Yu15Z/91IpMg0GheAhnbD85x7lEJB0exnARwXYz6EM4KGawjneZVpwTZIkrKB9fQbZJX0pkXOdQhTnRBTcxyFRV0pvEt3luHgw3kujcsQjJgFGmSKoWVl2SyVHgSqKQEuvKRcmRhxzwVSc5pzYRJdNJsb7FCOL1ktm89jVsucCIl8bRUAUZphYNah1PKyLlnwLHDFhp67UWgR+WsWyh646abjAYHhSAYK6bHyXfpUMvHcT/Mw4g5CnIKwvpciLn3dhJiwZWocUsD8ngMXY2yJFSVCepSNylzcZnc1vHOFHzDO0sbFp2ZmA3AhgvQ4TmnGTshLEAbSkhMQ4ruyIhPC+FXCqiJqDUtIFiFnq+SRmWhqpM80mMsUR69EGbZOPYmbqrDT435TlQnralCtTaQx4apnr0dp1KfBAKjVLTdCO0JqFXl7UrhI5GySktb9eXtVTV5XyMIK05oK6EfM6ypbDmHElh6anYEA4KOvIqFnNb5dGYjVF1po6QXM3MKkveX+vyPvl6iVJvlcU87pj7mut13l5WdBLk4IjhgPvWP9N+bHqsAzV0RjkUh30OnJdhYf6G1wvNXqdFsfD7rO9368a5Er6O30buoE21HO9GXaD86ikQE0a/od/Sn8arxqbHX+Dyj3luqbV5Gc6ex/w80lH/</latexit> <latexit sha1_base64="tcq7x+h5H1MVi9e74Wsj/KX91Xo=">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</latexit> Potential outcomes: notation ) , Y i (1) Y i | do( T =1) : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i specific unit/individual : potential outcome under treatment Y i (1) Y i | do( T =0) do( T = 0) 5 / 41 Brady Neal
<latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="ankFpkUc9naoITIsD9NjYr0YunY=">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</latexit> <latexit sha1_base64="wHBl5SOG0FiXEwPvpVB6ZgvKM3g=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="PeOKuh2RY/b2tuD2rOSag28FZ40=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="+Rktm3TlQFEu8Sf2KZ4iNZT7NFc=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="dZthWb4oBYLiMHMHcUuWJX6FBrI=">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</latexit> <latexit sha1_base64="tcq7x+h5H1MVi9e74Wsj/KX91Xo=">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</latexit> Potential outcomes: notation ) , Y i (1) Y i | do( T =1) : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) ) , Y i (0) Y i | do( T =0) do( T = 0) 5 / 41 Brady Neal
<latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="ankFpkUc9naoITIsD9NjYr0YunY=">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</latexit> <latexit sha1_base64="wHBl5SOG0FiXEwPvpVB6ZgvKM3g=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="PeOKuh2RY/b2tuD2rOSag28FZ40=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="+Rktm3TlQFEu8Sf2KZ4iNZT7NFc=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="dZthWb4oBYLiMHMHcUuWJX6FBrI=">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</latexit> <latexit sha1_base64="tcq7x+h5H1MVi9e74Wsj/KX91Xo=">AErnicbVNbaxNBFN7aqDXeWn3wZfRUqiwjUkpVMFAoREfKjQmzQhzM6eJEPmxsxZa7ou/hf9Qf5bzy72dqm7SwL35znfuZxCkZsN3+u3BnsXH3v2lB82Hjx4/ebq8uwo2MwLOBRWX+S8ABKGjhEiQpOnAeuEwXHyWS31B9/Ax+kNQc4dDXfGTkUAqOJBosv/g6kD8GeQ/hO+apLdYPWJe13xSD5dV2q10dhN0arAa1Wd/sL4s5dakWkwKBQP4bTdtjPuUcpFBTNXhbAcTHhIzglaLiG0M+rCgq2RpKUDa2n3yCrpFctcq5DmOqEmJrjOFzXlcLbdKcZDt/1c2lchmDELNAwUwtK9vBUulBoJoS4MJLypWJMfdcIDWtORcm0UWzucYO5OSc1bL5PGa1zImQyJdWARClGQVmHUotz+uSBc8CV2zkuRuHFpE/ZaHsgZtuOB4QGI5loJAeK9+lTyUTz/0D2PuIMQpCOuriYaYe2/PQiy4EjVuaUAeDyXGzgZsigJyrN0RO7yJqOz8ZnTDsQ8Qxsbm5adCciNANbtMKEZNykrQRxAS0pITOLKjkgIb1sBp4qoOSglXYCYpZ6fxUxLI3Wm2ZlMcVxuVmubfBQzU2elwf+mLBfS06ZcmEpjwFPLXJfWbosCD6VSs9QE7QitWejmRe0qkbNBQlr76+ZVPXVKQ9jSGsuqCsxL6NsOowZV3JkugqGhIOyjoya1fx2aSRWU2StqRM0dwNn9SXv7RV5r1y9JMn3imJed8R9rfU6Ly/X9NKk4IjhwDvWe1V+rLpc45kLorFIpFvo9GQ71x/oTXC02epstd5/2Vrd+VA/3qXoZfQ6Wo860Xa0E32M9qPDSERF9Cv6Hf1ptBtHjX5jMKPeWahtnkdzpzH+B3pbiN0=</latexit> Potential outcomes: notation ) , Y i (1) Y i | do( T =1) : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) ) , Y i (0) Y i | do( T =0) do( T = 0) 5 / 41 Brady Neal
<latexit sha1_base64="qHxz+IbAe3fX/QpP6FnXyGr2itM=">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</latexit> <latexit sha1_base64="tcq7x+h5H1MVi9e74Wsj/KX91Xo=">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</latexit> <latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">AElnicbVNbaxNBFN7aqDXeWn0RfBktBR+2MSmB6kOkUIoiPrTQGzSlzM6eJEPmxsxZY7os/gBf9cf5bzy72dom7SwL35znfuZxCkZsN3+u3RvuXH/wcOVR83HT54+e769uI42MwLOBJWX+a8ABKGjhCiQpOnQeuEwUnyXi31J98Bx+kNYc4dXCu+dDIgRQcSXQgL1bX2612dht0KnBelSf/Yu15Z/91IpMg0GheAhnbD85x7lEJB0exnARwXYz6EM4KGawjneZVpwTZIkrKB9fQbZJX0pkXOdQhTnRBTcxyFRV0pvEt3luHgw3kujcsQjJgFGmSKoWVl2SyVHgSqKQEuvKRcmRhxzwVSc5pzYRJdNJsb7FCOL1ktm89jVsucCIl8bRUAUZphYNah1PKyLlnwLHDFhp67UWgR+WsWyh646abjAYHhSAYK6bHyXfpUMvHcT/Mw4g5CnIKwvpciLn3dhJiwZWocUsD8ngMXY2yJFSVCepSNylzcZnc1vHOFHzDO0sbFp2ZmA3AhgvQ4TmnGTshLEAbSkhMQ4ruyIhPC+FXCqiJqDUtIFiFnq+SRmWhqpM80mMsUR69EGbZOPYmbqrDT435TlQnralCtTaQx4apnr0dp1KfBAKjVLTdCO0JqFXl7UrhI5GySktb9eXtVTV5XyMIK05oK6EfM6ypbDmHElh6anYEA4KOvIqFnNb5dGYjVF1po6QXM3MKkveX+vyPvl6iVJvlcU87pj7mut13l5WdBLk4IjhgPvWP9N+bHqsAzV0RjkUh30OnJdhYf6G1wvNXqdFsfD7rO9368a5Er6O30buoE21HO9GXaD86ikQE0a/od/Sn8arxqbHX+Dyj3luqbV5Gc6ex/w80lH/</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="dZthWb4oBYLiMHMHcUuWJX6FBrI=">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</latexit> <latexit sha1_base64="ankFpkUc9naoITIsD9NjYr0YunY=">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</latexit> <latexit sha1_base64="wHBl5SOG0FiXEwPvpVB6ZgvKM3g=">AEwHicbVNbTxNBF6kKtYb6KMvo4QEkqW2hARNbEIkJMb4gAk3Q0kzO3vaTpgbM2fFsm78Cf4aX/V3+G8u12EArPZ5JtzvnM/kzglA7bf2fuzDbu3rs/96D58NHjJ0/nF57tB5t5AXvCKusPEx5ASQN7KFHBofPAdaLgIDnZKvUHX8EHac0ujh0caz40ciAFRxL151e+9OX3ft5D+IZ5aovlXdZlnZWC9dBLboYKThlRljsr/fnFdqtdHXYTdGqwGNVnp78w+6OXWpFpMCgUD+Go03Z4nHOPUigomr0sgOPihA/hiKDhGsJxXtVUsCWSpGxgPf0GWSW9apFzHcJYJ8TUHEfhuq4U3qY7ynDw5jiXxmUIRkwCDTLF0LKyQSyVHgSqMQEuvKRcmRhxzwVSG5tTYRJdNJtLbFenLNaNp3HpJYpERL50ioAojTDwKxDqeV5XbLgWeCKDT13o9Ai8scslD1w41XHAwLDkQwU0mPlu/SpZOK5H+dhxB2EOAVhfTXjEHPv7VmIBVeixi0NyOBxNjZIEsWJUF5lo7IXd5kdFY/cdqKmGdoY2PTsjMBuRHAuh0mNOMmZSWIA2hJCYmTuLIjEsLrVsCxImoOSkXIGap52cx09JInWl2JlMc0a61Wxvko5iYOisN/jdluZCeNuXCVBoDnlrmurR26xR4IJWapCZoR2jNQjcvaleJnAwS0tpfN6/qatKeRhBWnNBXYl5GWXNYcy4kPTVTAgHJR1ZNSs5rdFI7GaImtNnaC5GzirL3lvu8h75eolSb5dFNO6fe5rd5ebmlyYFRwH3rHey/Jj1eUaz1wQjUi3UKnJ9u5/kBvgv21Vme9fbz+uLmu/rxzkUvolfRctSJNqLN6EO0E+1FIvoZ/Yp+R38a7xujhm2cTqh3Zmqb59HUaZz/A18Jj5g=</latexit> <latexit sha1_base64="+Rktm3TlQFEu8Sf2KZ4iNZT7NFc=">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</latexit> <latexit sha1_base64="PeOKuh2RY/b2tuD2rOSag28FZ40=">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</latexit> Potential outcomes: notation ) , Y i (1) Y i | do( T =1) : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) ) , Y i (0) Y i | do( T =0) do( T = 0) Causal effect Y i (1) − Y i (0) 5 / 41 Brady Neal
<latexit sha1_base64="WfpAwrSe3NbrWyeSl21kioycO4=">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</latexit> <latexit sha1_base64="ankFpkUc9naoITIsD9NjYr0YunY=">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</latexit> <latexit sha1_base64="wHBl5SOG0FiXEwPvpVB6ZgvKM3g=">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</latexit> <latexit sha1_base64="qHxz+IbAe3fX/QpP6FnXyGr2itM=">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</latexit> <latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="PeOKuh2RY/b2tuD2rOSag28FZ40=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="+Rktm3TlQFEu8Sf2KZ4iNZT7NFc=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> Potential outcomes: notation ) , Y i (1) Y i | do( T =1) : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) Y i (0) = 0 do( T = 0) Causal effect Y i (1) − Y i (0) 5 / 41 Brady Neal
<latexit sha1_base64="WfpAwrSe3NbrWyeSl21kioycO4=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="BchGElSW5YZ4EvU3clhKDxkCzM=">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</latexit> <latexit sha1_base64="PeOKuh2RY/b2tuD2rOSag28FZ40=">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</latexit> <latexit sha1_base64="qHxz+IbAe3fX/QpP6FnXyGr2itM=">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</latexit> <latexit sha1_base64="+Rktm3TlQFEu8Sf2KZ4iNZT7NFc=">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</latexit> <latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> Potential outcomes: notation Y i (1) = 1 : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) Y i (0) = 0 do( T = 0) Causal effect Y i (1) − Y i (0) 5 / 41 Brady Neal
<latexit sha1_base64="WfpAwrSe3NbrWyeSl21kioycO4=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="BchGElSW5YZ4EvU3clhKDxkCzM=">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</latexit> <latexit sha1_base64="+uQ3sxdOdEq7LET+FL7NQqyVAg=">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</latexit> <latexit sha1_base64="PeOKuh2RY/b2tuD2rOSag28FZ40=">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</latexit> <latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="+Rktm3TlQFEu8Sf2KZ4iNZT7NFc=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> Potential outcomes: notation Y i (1) = 1 : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) Y i (0) = 0 do( T = 0) Causal effect Y i (1) − Y i (0) = 1 5 / 41 Brady Neal
What are potential outcomes? The fundamental problem of causal inference Getting around the fundamental problem of causal inference A complete example with estimation 6 / 41 Brady Neal
<latexit sha1_base64="BchGElSW5YZ4EvU3clhKDxkCzM=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="+uQ3sxdOdEq7LET+FL7NQqyVAg=">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</latexit> <latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="PeOKuh2RY/b2tuD2rOSag28FZ40=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">AElnicbVNbaxNBFN62UWu8tfoi+DJaCj5sY1IC1YdIoRFfGihN2lCmZ09SYbMjZmzxnQJ/gBf9cf5bzy72dom7SwL35znfuZxCkZsNn8u7S8Urt3/8Hqw/qjx0+ePltbf34SbOYFHAurD9LeAlDRyjRAVnzgPXiYLTZLRX6E+/gw/SmiOcOhpPjCyLwVHEh1+u1jbaDa5WG3QasCG1F1Di7WV352UysyDQaF4iGct5oOezn3KIWCab2bBXBcjPgAzgkariH08jLTKdskScr61tNvkJXSmxY51yFMdEJMzXEYFnWF8C7deYb971cGpchGDEL1M8UQ8uKslkqPQhUEwJceEm5MjHknguk5tTnwiR6Wq9vsiM5umSVbD6PWS1zIiTytVUARGkGgVmHUsvLqmTBs8AVG3juhqFB5C9ZKHrgJluOBwSGQxkopMfSd+FTycRzP8nDkDsIcQrC+nJyIebe23GIBVeiwg0NyO+xNjZIAsWJUF5Fo7IXV5ndLa+coQfMc/QxsamRWcCciOAdVpMaMZNygoQB9CSEhKjuLQjEsK7RsCJImoOSkXIGap5+OYaWmkzjQbyxSHrEMbtEM+pjNTZ6XB/6YsF9LTplyZSmPAU8tch9auTYH7UqlZaoJ2hNYsdPJp5SqRs0FCWvnr5GU9VUpD0NIKy6oGzGvo2w7jBlXcmA6CvqEg7KOjOrl/PZoJFZTZK2pEzR3A+Pqknf3p3m3WL0kyfen03ndCfeV1u8uCzopUnBEcOBd6z7uvhYeVngmSuisUikO+j0ZFuLD/Q2ONlutNqND4ftjd129XhXo1fRm+ht1Ip2ot3oc3QHUciguhX9Dv6U3tZ+1jbr32aUZeXKpsX0dypHfwD8WV/7w=</latexit> <latexit sha1_base64="+Rktm3TlQFEu8Sf2KZ4iNZT7NFc=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="WfpAwrSe3NbrWyeSl21kioycO4=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> Fundamental problem of causal inference Y i (1) = 1 : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) Y i (0) = 0 do( T = 0) Causal effect Y i (1) − Y i (0) = 1 Then, what does imply causation? 7 / 41 Brady Neal
<latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">AElnicbVNbaxNBFN7aqDXeWn0RfBktBR+2MSmB6kOkUIoiPrTQGzSlzM6eJEPmxsxZY7os/gBf9cf5bzy72dom7SwL35znfuZxCkZsN3+u3RvuXH/wcOVR83HT54+e769uI42MwLOBJWX+a8ABKGjhCiQpOnQeuEwUnyXi31J98Bx+kNYc4dXCu+dDIgRQcSXQgL1bX2612dht0KnBelSf/Yu15Z/91IpMg0GheAhnbD85x7lEJB0exnARwXYz6EM4KGawjneZVpwTZIkrKB9fQbZJX0pkXOdQhTnRBTcxyFRV0pvEt3luHgw3kujcsQjJgFGmSKoWVl2SyVHgSqKQEuvKRcmRhxzwVSc5pzYRJdNJsb7FCOL1ktm89jVsucCIl8bRUAUZphYNah1PKyLlnwLHDFhp67UWgR+WsWyh646abjAYHhSAYK6bHyXfpUMvHcT/Mw4g5CnIKwvpciLn3dhJiwZWocUsD8ngMXY2yJFSVCepSNylzcZnc1vHOFHzDO0sbFp2ZmA3AhgvQ4TmnGTshLEAbSkhMQ4ruyIhPC+FXCqiJqDUtIFiFnq+SRmWhqpM80mMsUR69EGbZOPYmbqrDT435TlQnralCtTaQx4apnr0dp1KfBAKjVLTdCO0JqFXl7UrhI5GySktb9eXtVTV5XyMIK05oK6EfM6ypbDmHElh6anYEA4KOvIqFnNb5dGYjVF1po6QXM3MKkveX+vyPvl6iVJvlcU87pj7mut13l5WdBLk4IjhgPvWP9N+bHqsAzV0RjkUh30OnJdhYf6G1wvNXqdFsfD7rO9368a5Er6O30buoE21HO9GXaD86ikQE0a/od/Sn8arxqbHX+Dyj3luqbV5Gc6ex/w80lH/</latexit> <latexit sha1_base64="+uQ3sxdOdEq7LET+FL7NQqyVAg=">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</latexit> <latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="PeOKuh2RY/b2tuD2rOSag28FZ40=">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</latexit> <latexit sha1_base64="+Rktm3TlQFEu8Sf2KZ4iNZT7NFc=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="WfpAwrSe3NbrWyeSl21kioycO4=">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</latexit> <latexit sha1_base64="BchGElSW5YZ4EvU3clhKDxkCzM=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> Fundamental problem of causal inference Y i (1) = 1 : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i Counterfactual specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) Y i (0) = 0 do( T = 0) Causal effect Factual Y i (1) − Y i (0) = 1 Then, what does imply causation? 7 / 41 Brady Neal
<latexit sha1_base64="BchGElSW5YZ4EvU3clhKDxkCzM=">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</latexit> <latexit sha1_base64="WfpAwrSe3NbrWyeSl21kioycO4=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="+uQ3sxdOdEq7LET+FL7NQqyVAg=">AEqnicbVNbSxtBF5r2tr0pvWxL9OKoLCmWRFsoQFBhFL6YItXjMjs7EkyZG7MnG0al9Cf0tf2J/Xf9OxmrUadZeGbc75zP5M6JQO23/nHsw3Hj56vPCk+fTZ8xcvF5deHQWbewGHwirT1IeQEkDhyhRwYnzwHWq4Dgd7pb64+/g7TmAMcOzjXvG9mTgiOJLhaXTy/kWrLONlgJ2usw5KLxZV2q10dhckNViJ6rN/sT/s5tZkWswKBQP4SxpOzwvuEcpFEya3TyA42LI+3BG0HAN4byosp+wVZJkrGc9/QZJb1pUXAdwlinxNQcB+G2rhTepzvLsf+vJDG5QhGTAP1csXQsrIVLJMeBKoxAS68pFyZGHDPBVLDmjNhUj1pNlfZgRxeslo2m8e0lhkREvnaKgCiNP3ArEOp5WVdsuB54Ir1PXeD0CLy5zyUPXDjDcDAsOBDBTSY+W79Klk6rkfF2HAHYQ4A2F9Nc0Qc+/tKMSCK1HjlgbkcU9i7GyQJYuSoDxLR+SuaDI6G184wo+Y52hjY7OyMwG5EcA6CROacZOxEsQBtKSExDCu7IiE8K4VcKyIWoBS0gWIWeb5KGZaGqlzUYywHtVbu1T4mU1NnpcH/pqwQ0tOmXJlKY8BTy1yH1m6LAvekUtPUBO0IrVnoFJPaVSqng4Ss9tcpqnrqjIeBpDVXFA3Yl5H2XQYM65k3QU9AgHZR0ZNav57dJIrKbIWlMnaO4GRvWl6O5Nim65emla7E0ms7oj7mut10V5uaWXJgNHDAfese6b8mPV5RbPXBGNRSLdQ6cnm9x+oHfB0WYr2Wp9+Lq1svOxfrwL0evobQWJdF2tBN9ivajw0hE4+hX9Dv604gb3xqnjbMp9cFcbMczZxG9g+agITj</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="PeOKuh2RY/b2tuD2rOSag28FZ40=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="+Rktm3TlQFEu8Sf2KZ4iNZT7NFc=">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</latexit> Fundamental problem of causal inference Y i (1) = 1 : observed treatment T do( T = 1) : observed outcome Y : used in subscript to denote a i Factual specific unit/individual : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) Y i (0) = 0 do( T = 0) Causal effect Counterfactual Y i (1) − Y i (0) = 1 Then, what does imply causation? 7 / 41 Brady Neal
<latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="FSkNaui+guUCONR9/NkF2jubVLM=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">AEm3icbVNbSxtBF41bW160/axFKYVwcKaJiLYPgQEYr4YEGjxQSZnT1Jps6NmbO1cVn6F/ra/rP+m57drNWosyx8c8537mcSp2TAdv3PxC48HDR4uPm0+ePnv+Ymn5ZS/YzAs4ElZf5LwAEoaOEKJCk6cB64TBcfJ+U6pP/4OPkhrDnHiYKD5yMihFBxJ1Pt6Jtc678+WVtqtdnXYXdCpwUpUn4Oz5YWf/dSKTINBoXgIp52w0HOPUqhoGj2swCOi3M+glOChmsIg7xKt2CrJEnZ0Hr6DbJKetMi5zqEiU6IqTmOw21dKbxPd5rh8OMgl8ZlCEZMAw0zxdCysnaWSg8C1YQAF15SrkyMuecCqUPNmTCJLprNVXYozy9ZLZvNY1rLjAiJfG0VAFGaUWDWodTysi5Z8CxwxUaeu3FoEXkvC2UP3GTd8YDAcCwDhfRY+S59Kpl47id5GHMHIU5BWF+NL8Tce3sRYsGVqHFLA/J4KDF2NsiSRUlQnqUjcpc3GZ31fY7wI+YZ2tjYtOxMQG4EsG6HCc24SVkJ4gBaUkLiPK7siITwoRVwoiag1LSBYhZ6vlFzLQ0UmeaXcgUx6xLG7RFPoqpqbPS4H9TlgvpaVOuTKUx4Klrktrt0mBh1KpaWqCdoTWLHTzonaVyOkgIa39dfOqnrqlIcxpDUX1I2Y1E2HMaMKzkyXQVDwkFZR0bNan47NBKrKbLW1Amau4GL+pL3d4u8X65ekuS7RTGr63Ffa73Oy8stvTQpOGI48I7135Yfqy63eOaKaCwS6R46PdnO7Qd6F/Q2Wp3N1qcvmyvbm/XjXYxeR+itagTbUXb0efoIDqKRPQt+hX9jv403jR2GnuN/Sl1fq62eRXNnMbRPyUlgWs=</latexit> Missing data interpretation : observed treatment T : observed outcome Y : used in subscript to denote a i specific unit/individual Y (1) Y (0) Y (1) − Y (0) i T Y : potential outcome under treatment Y i (1) : potential outcome under no treatment Y i (0) 1 0 0 ? 0 ? 2 1 1 1 ? ? 3 1 0 0 ? ? 4 0 0 ? 0 ? 5 0 1 ? 1 ? 6 1 1 1 ? ? 8 / 41 Brady Neal
Question: What is the fundamental problem of causal inference?
What are potential outcomes? The fundamental problem of causal inference Getting around the fundamental problem of causal inference A complete example with estimation 10 / 41 Brady Neal
<latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="FSkNaui+guUCONR9/NkF2jubVLM=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="bg29vTMGqhBy203y1BQJ/yjmYZM=">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</latexit> Average treatment effect (ATE) Y i (1) − Y i (0) Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 ? 0 ? : observed treatment T 2 1 1 1 ? ? : observed outcome Y 3 1 0 0 ? ? : used in subscript to denote a i specific unit/individual 4 0 0 ? 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 ? 1 ? Y i (0) 6 1 1 1 ? ? 11 / 41 Brady Neal
<latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="FSkNaui+guUCONR9/NkF2jubVLM=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="jOqD2SLongAgEpA+yGOtqvcVHIU=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">AElnicbVNbaxNBFN7aqDXeWn0RfBktBR+2MSmB6kOkUIoiPrTQGzSlzM6eJEPmxsxZY7os/gBf9cf5bzy72dom7SwL35znfuZxCkZsN3+u3RvuXH/wcOVR83HT54+e769uI42MwLOBJWX+a8ABKGjhCiQpOnQeuEwUnyXi31J98Bx+kNYc4dXCu+dDIgRQcSXQgL1bX2612dht0KnBelSf/Yu15Z/91IpMg0GheAhnbD85x7lEJB0exnARwXYz6EM4KGawjneZVpwTZIkrKB9fQbZJX0pkXOdQhTnRBTcxyFRV0pvEt3luHgw3kujcsQjJgFGmSKoWVl2SyVHgSqKQEuvKRcmRhxzwVSc5pzYRJdNJsb7FCOL1ktm89jVsucCIl8bRUAUZphYNah1PKyLlnwLHDFhp67UWgR+WsWyh646abjAYHhSAYK6bHyXfpUMvHcT/Mw4g5CnIKwvpciLn3dhJiwZWocUsD8ngMXY2yJFSVCepSNylzcZnc1vHOFHzDO0sbFp2ZmA3AhgvQ4TmnGTshLEAbSkhMQ4ruyIhPC+FXCqiJqDUtIFiFnq+SRmWhqpM80mMsUR69EGbZOPYmbqrDT435TlQnralCtTaQx4apnr0dp1KfBAKjVLTdCO0JqFXl7UrhI5GySktb9eXtVTV5XyMIK05oK6EfM6ypbDmHElh6anYEA4KOvIqFnNb5dGYjVF1po6QXM3MKkveX+vyPvl6iVJvlcU87pj7mut13l5WdBLk4IjhgPvWP9N+bHqsAzV0RjkUh30OnJdhYf6G1wvNXqdFsfD7rO9368a5Er6O30buoE21HO9GXaD86ikQE0a/od/Sn8arxqbHX+Dyj3luqbV5Gc6ex/w80lH/</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> Average treatment effect (ATE) E [ Y i (1) − Y i (0)] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 ? 0 ? : observed treatment T 2 1 1 1 ? ? : observed outcome Y 3 1 0 0 ? ? : used in subscript to denote a i specific unit/individual 4 0 0 ? 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 ? 1 ? Y i (0) 6 1 1 1 ? ? 11 / 41 Brady Neal
<latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="FSkNaui+guUCONR9/NkF2jubVLM=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="0uWIM7EmbUiNEQbwXBnOxSr8co=">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</latexit> Average treatment effect (ATE) E [ Y (1) − Y (0)] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 ? 0 ? : observed treatment T 2 1 1 1 ? ? : observed outcome Y 3 1 0 0 ? ? : used in subscript to denote a i specific unit/individual 4 0 0 ? 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 ? 1 ? Y i (0) 6 1 1 1 ? ? 11 / 41 Brady Neal
<latexit sha1_base64="2fEdt7FdkgfwU2jYIDligNtIdwM=">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</latexit> <latexit sha1_base64="0uWIM7EmbUiNEQbwXBnOxSr8co=">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</latexit> <latexit sha1_base64="FSkNaui+guUCONR9/NkF2jubVLM=">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</latexit> <latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> Average treatment effect (ATE) E [ Y (1) − Y (0)] = E [ Y (1)] − E [ Y (0)] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 ? 0 ? : observed treatment T 2 1 1 1 ? ? : observed outcome Y 3 1 0 0 ? ? : used in subscript to denote a i specific unit/individual 4 0 0 ? 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 ? 1 ? Y i (0) 6 1 1 1 ? ? 11 / 41 Brady Neal
<latexit sha1_base64="2fEdt7FdkgfwU2jYIDligNtIdwM=">AE1XicbVNLaxRBEJ6YVeP6ShRPXlpDIHJuhsCUXAhECIiHiKYh2SX0NTu9tsv+iuMW6GuYlXf4M30V/kv7FmdqLZJD0MfFX1aOrqhOnZMB2+8/cjfnGzVu3F+407967/+Dh4tKjg2AzL2BfWGX9UcIDKGlgHyUqOHIeuE4UHCbjndJ+Bl8kNZ8xImDvuZDIwdScCTVyeKTLuvtHn9a7az12foUtf6J4vL7Va7Ouwq6NRgOarP3snS/M9eakWmwaBQPITjTthP+cepVBQNHtZAMfFmA/hmKDhGkI/r+ov2ApUjawn6DrNJe9Mi5DmGiE2JqjqNw2VYqr7MdZzh42c+lcRmCEdNEg0wxtKxsBkulB4FqQoAL6lWJkbc4HUsuZMmkQXs7K1Y+RJKBhbYW+odCMFMNIoIM8V9lGOz1hNni14eukZFRK5+OcVAFGaYWDWodTyrO6N4Fngig09d6PQIvK7LJTNcpN1xwNS8pEMlNJjFbuMqWTiuZ/kYcQdhDgFYX01+Bz7+1piAVXosYtDcjgcTY2SBLFhVBdZaBKFzeZHTW3OELzHP0MbGpmULA3JDV+92mNCMm5SVIA6gJRUkxnHlRySEF62AE0XUHJSLkDMUs9PY6alkTrT7FSmOGJdWrUtilFMXZ2VBv+5slxITyt17iqNAU8tc13az01KPJBKTUsTtEy0j6GbF3WoRE4nDmkdr5tX96lvlfIwgrTmgrqQ83+WDYcx40oOTVfBgHBQ1pFTs5rfDo3EasqsNXWC5m7gtBby3m6R98odTZJ8tyhmbQfc1av81K4ZJcmBUcMB96x3rPyY5VwiWfOicYika6h09vuXH7JV8HBRquz2Xr1YXN5+3X9yheip9HzaDXqRFvRdvQ2ov2IxHl0Y/oV/S7cdgoGl8b36bUG3O1z+No5jS+/wUB3pWQ</latexit> <latexit sha1_base64="RMPR9FH5dkW3I90m4IOyDU5t1DM=">AE5XicbVNLbxNJEJ4Q7xK8y5LAkUtBFGkPE2OjSAEJS0hRVghxACkJrGIr6ukp2y3S901BDOaX7DaG+K6V657X4J/4a8cBiQ49Gqsf31aurM69VpH7/08aVzc5P1/dutb95dfrv93Y3rl5Fl0RJ5Kp14lYmIWlk8JUaX/mAwmQaX2bzo9r/8jWGqJw9oYXHsRFTqyZKCmLTxfbeEbH53/CyKgcTmAIgzHsr5n64vt3X6v3xz4Xhi0wm7SnucXO5sfRrmThUFLUosYzwd9T+NSBFJSY9UdFRG9kHMxXMWrTAYx2XTwV7bMlh4gL/lqCxfsohYlxYTJGkGzuO6rjT/ynRc0eTAulfUFoZXLRJNCAzmohwO5CihJL1gQMiuFeRMBCGJR9hdSZOZalV3bk4ixXAHvzBpVslEdikZl7cKLmb6EFrxa8bHrFRAyuvrIiEik7jeA8KaPetrORohCwzQIP4s9Bj8tYj0sv9j3IhIn6nIKQM1seuYWmVBhEUZ8JjTHOULjSLEFMRgruMqRatnLPIl0oij1LqoaxUVwnXUgDld2gc/+M0H4JhUFudS6vB5hJG59eEApAFhc6iFNKJRXJCcpw2PQYT3epEWmqElaq18xBTyIC5TMoqUxi4VDnN6g3sHXKMakn1Tln6SoVSqsAr9YWqrMXAI/ND3s8DTjxRWi9Lk7xMvI9xWFZtqEwtbxzNt6wbPpu8pFnGHeYlF/k/P/LPc9pSC0mtqhxgnLUTvPpG5zf0d8Jc5wZmN4EnzvFi9bpRwdV+Wo3tEsK4+ratV3JkLrDaslTW/sjl6RngMHkZ36g8aZQ1nvwCtIwb9AM5ve7D+kr8Xzu73Bge9hy8Odh8/al/5VnI7uZv8ngySw+Rx8iR5npwmMvkr+Tf5L/nYmXb+7rzrvF9Cr2y0nFvJyun8xldqprc</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="FSkNaui+guUCONR9/NkF2jubVLM=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="0uWIM7EmbUiNEQbwXBnOxSr8co=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> Average treatment effect (ATE) E [ Y (1) − Y (0)] = E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 ? 0 ? : observed treatment T 2 1 1 1 ? ? : observed outcome Y 3 1 0 0 ? ? : used in subscript to denote a i specific unit/individual 4 0 0 ? 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 ? 1 ? Y i (0) 6 1 1 1 ? ? 11 / 41 Brady Neal
<latexit sha1_base64="2fEdt7FdkgfwU2jYIDligNtIdwM=">AE1XicbVNLaxRBEJ6YVeP6ShRPXlpDIHJuhsCUXAhECIiHiKYh2SX0NTu9tsv+iuMW6GuYlXf4M30V/kv7FmdqLZJD0MfFX1aOrqhOnZMB2+8/cjfnGzVu3F+407967/+Dh4tKjg2AzL2BfWGX9UcIDKGlgHyUqOHIeuE4UHCbjndJ+Bl8kNZ8xImDvuZDIwdScCTVyeKTLuvtHn9a7az12foUtf6J4vL7Va7Ouwq6NRgOarP3snS/M9eakWmwaBQPITjTthP+cepVBQNHtZAMfFmA/hmKDhGkI/r+ov2ApUjawn6DrNJe9Mi5DmGiE2JqjqNw2VYqr7MdZzh42c+lcRmCEdNEg0wxtKxsBkulB4FqQoAL6lWJkbc4HUsuZMmkQXs7K1Y+RJKBhbYW+odCMFMNIoIM8V9lGOz1hNni14eukZFRK5+OcVAFGaYWDWodTyrO6N4Fngig09d6PQIvK7LJTNcpN1xwNS8pEMlNJjFbuMqWTiuZ/kYcQdhDgFYX01+Bz7+1piAVXosYtDcjgcTY2SBLFhVBdZaBKFzeZHTW3OELzHP0MbGpmULA3JDV+92mNCMm5SVIA6gJRUkxnHlRySEF62AE0XUHJSLkDMUs9PY6alkTrT7FSmOGJdWrUtilFMXZ2VBv+5slxITyt17iqNAU8tc13az01KPJBKTUsTtEy0j6GbF3WoRE4nDmkdr5tX96lvlfIwgrTmgrqQ83+WDYcx40oOTVfBgHBQ1pFTs5rfDo3EasqsNXWC5m7gtBby3m6R98odTZJ8tyhmbQfc1av81K4ZJcmBUcMB96x3rPyY5VwiWfOicYika6h09vuXH7JV8HBRquz2Xr1YXN5+3X9yheip9HzaDXqRFvRdvQ2ov2IxHl0Y/oV/S7cdgoGl8b36bUG3O1z+No5jS+/wUB3pWQ</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="6UZUlUoNAMm3rBUqYGrWy8+y6ow=">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</latexit> <latexit sha1_base64="0uWIM7EmbUiNEQbwXBnOxSr8co=">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</latexit> <latexit sha1_base64="RMPR9FH5dkW3I90m4IOyDU5t1DM=">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</latexit> Average treatment effect (ATE) E [ Y (1) − Y (0)] = E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? : observed treatment T 2 1 1 1 ? : observed outcome Y 3 1 0 0 ? : used in subscript to denote a i specific unit/individual 4 0 0 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 1 ? Y i (0) 6 1 1 1 ? 11 / 41 Brady Neal
<latexit sha1_base64="2fEdt7FdkgfwU2jYIDligNtIdwM=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="6UZUlUoNAMm3rBUqYGrWy8+y6ow=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="0uWIM7EmbUiNEQbwXBnOxSr8co=">AEz3icbVNtaxNBED7bqDW+Vf3ol9VSaOEaEymoYKAgFRE/WOgrTSh7e5Nkyb6xO2dMjxO/+hv8quBf8t84dzlrk7rHwbw8z8zszGzilAzYbv+trTcuH7j5sqt5u07d+/dX3w8DYzAs4EFZf5zwAEoaOECJCo6dB64TBUfJ+E3pP/oEPkhr9nHqoK/50MiBFBzJdLZ6v7d7erLR2WRb7GSjvdk/W1rt9rVYVeFTi2sRfX5ePZg+VcvtSLTYFAoHsJp+2wn3OPUigomr0sgONizIdwSqLhGkI/ryov2DpZUjawn6DrLJeZuRchzDVCSE1x1FY9JXG/lOMxy87OfSuAzBiFmiQaYWla2gaXSg0A1JYEL6lWJkbc4HUrOZcmkQX87q1Y+RJKBhbZ2+pdCMFMLIoIOY625fjc1aD5wueXrOhAQuLlgBEKUZBmYdSi3P694IngWu2NBzNwotAr/PQtksN91yPCAlH8lAKT1WscuYSiae+2keRtxBiFMQ1lcjDzH3k5CLgStdzSgDweSIydDbJEURFUZxmIwuVNRmfrA0f4HPMbWxsWrYwIDd09W6HCc24SVkpxAG0pILEOK54BEJ41go4VQTNQSnpAsQs9XwSMy2N1JlmE5niHVp1V5QjGJGdVYavKCyXEhPK/WXKo0BTy1zXdrPbUo8kErNShO0TLSPoZsXdahEziYOaR2vm1f3qW+V8jCtMaCupTzX5bnDmPGlRyaroIByUFZR6RmNb83NBKrKbPW1Amau4FJreS93SLvlTuaJPluUcz7DrmvV7npbLglyYFRwgH3rHek/JjlbKAM3+BxiKB/gOnt91ZfMlXhcPnrc5269Xe9trO6/qVr0SPo6fRtSJXkQ70bvoY3QiSiLvkc/op+Nvcak8aXxdQZdulZzHkVzp/HtD4QUk54=</latexit> <latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="RMPR9FH5dkW3I90m4IOyDU5t1DM=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">AElnicbVNbaxNBFN7aqDXeWn0RfBktBR+2MSmB6kOkUIoiPrTQGzSlzM6eJEPmxsxZY7os/gBf9cf5bzy72dom7SwL35znfuZxCkZsN3+u3RvuXH/wcOVR83HT54+e769uI42MwLOBJWX+a8ABKGjhCiQpOnQeuEwUnyXi31J98Bx+kNYc4dXCu+dDIgRQcSXQgL1bX2612dht0KnBelSf/Yu15Z/91IpMg0GheAhnbD85x7lEJB0exnARwXYz6EM4KGawjneZVpwTZIkrKB9fQbZJX0pkXOdQhTnRBTcxyFRV0pvEt3luHgw3kujcsQjJgFGmSKoWVl2SyVHgSqKQEuvKRcmRhxzwVSc5pzYRJdNJsb7FCOL1ktm89jVsucCIl8bRUAUZphYNah1PKyLlnwLHDFhp67UWgR+WsWyh646abjAYHhSAYK6bHyXfpUMvHcT/Mw4g5CnIKwvpciLn3dhJiwZWocUsD8ngMXY2yJFSVCepSNylzcZnc1vHOFHzDO0sbFp2ZmA3AhgvQ4TmnGTshLEAbSkhMQ4ruyIhPC+FXCqiJqDUtIFiFnq+SRmWhqpM80mMsUR69EGbZOPYmbqrDT435TlQnralCtTaQx4apnr0dp1KfBAKjVLTdCO0JqFXl7UrhI5GySktb9eXtVTV5XyMIK05oK6EfM6ypbDmHElh6anYEA4KOvIqFnNb5dGYjVF1po6QXM3MKkveX+vyPvl6iVJvlcU87pj7mut13l5WdBLk4IjhgPvWP9N+bHqsAzV0RjkUh30OnJdhYf6G1wvNXqdFsfD7rO9368a5Er6O30buoE21HO9GXaD86ikQE0a/od/Sn8arxqbHX+Dyj3luqbV5Gc6ex/w80lH/</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> Average treatment effect (ATE) E [ Y (1) − Y (0)] = E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? : observed treatment T 2 1 1 1 ? : observed outcome Y 3 1 0 0 ? : used in subscript to denote a i specific unit/individual 4 0 0 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 1 ? Y i (0) 6 1 1 1 ? 2 / 3 11 / 41 Brady Neal
<latexit sha1_base64="2fEdt7FdkgfwU2jYIDligNtIdwM=">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</latexit> <latexit sha1_base64="0uWIM7EmbUiNEQbwXBnOxSr8co=">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</latexit> <latexit sha1_base64="6UZUlUoNAMm3rBUqYGrWy8+y6ow=">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</latexit> <latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">AFE3icbVRLaxRBEJ4kq8b1lejRS2sIiEzW3RiIguCREQ8KCRyIbQ01O72y/6K5xsxnmP3jxr3jxoIhXDx79N9bMTjQT7WGgHl+9vq6ZxCkZsNv9tbC41Lpw8dLy5faVq9eu31hZvbkfbOYF7AmrH+X8ABKGthDiQreOQ9cJwreJpNnpf/te/BWrOLMweHmo+MHErBkUxHK/cHYei5yDeL/GHBNlit9iq131CPVta6nW512L9CrxbWovq8Plpd+jlIrcg0GBSKh3DQ6zo8zLlHKRQU7UEWwHEx4SM4INFwDeEwr4Yq2DpZUja0nl6DrLKejci5DmGmE0JqjuNw3lca/+c7yHD46DCXxmUIRswLDTPF0LKSIZKDwLVjAQuvKRemRhzogGJx3ajTKLpm7tBHkSCsbW2XNq3UgBjCwKmv0dl7Q2UaWlvBMqsc525eSE1eBm5JydhgkJXPyJCoAozSgw61BqeVKTKHgWuGIjz904dAj8Mgslq2624XhA6nIsA5X0WOUucyqZeO5neRhzByFOQVhfrU2Iufd2GmLBlajljgbk8VBi7GyQJYqaoD7LRJQubzM6G684wnHM7SxsWnJdUBuaPp+jwnNuElZKcQBtKSGxCSu4giE8KATcKYImoNS0gWIWer5NGZaGqkzaYyxTEtbezTmKeaiz0uCfUJYL6Wn3TkOlMeCJMtenRd6iwkOp1Lw1QVtHixv6eVGnSuR8NSCt8/Xzap56qpSHMaQ1FtSZmn+rbDqMGVdyZPoKhiQHZR0Ftav7e0ZXYjV1pqYoHs3MK2VfLBT5INymZMk3ymKpm+f+9rdV4q5/zSpOAI4cA7NrhTPqxSzuHMKdBYJNB/4PQT6J3/5P8V9jc7va3O4zdba0+f1L+D5eh2dDe6F/Wi7ehp9CJ6He1FIvoQfYq+RF9bH1ufW9a3+fQxYU65lbUOK0fvwGprAu</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="RMPR9FH5dkW3I90m4IOyDU5t1DM=">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</latexit> Average treatment effect (ATE) E [ Y (1) − Y (0)] = E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? : observed treatment T 2 1 1 1 ? : observed outcome Y 3 1 0 0 ? : used in subscript to denote a i specific unit/individual 4 0 0 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 1 ? Y i (0) 6 1 1 1 ? 2 / − 1 / 3 3 11 / 41 Brady Neal
<latexit sha1_base64="2fEdt7FdkgfwU2jYIDligNtIdwM=">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</latexit> <latexit sha1_base64="6UZUlUoNAMm3rBUqYGrWy8+y6ow=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">AFE3icbVRLaxRBEJ4kq8b1lejRS2sIiEzW3RiIguCREQ8KCRyIbQ01O72y/6K5xsxnmP3jxr3jxoIhXDx79N9bMTjQT7WGgHl+9vq6ZxCkZsNv9tbC41Lpw8dLy5faVq9eu31hZvbkfbOYF7AmrH+X8ABKGthDiQreOQ9cJwreJpNnpf/te/BWrOLMweHmo+MHErBkUxHK/cHYei5yDeL/GHBNlit9iq131CPVta6nW512L9CrxbWovq8Plpd+jlIrcg0GBSKh3DQ6zo8zLlHKRQU7UEWwHEx4SM4INFwDeEwr4Yq2DpZUja0nl6DrLKejci5DmGmE0JqjuNw3lca/+c7yHD46DCXxmUIRswLDTPF0LKSIZKDwLVjAQuvKRemRhzogGJx3ajTKLpm7tBHkSCsbW2XNq3UgBjCwKmv0dl7Q2UaWlvBMqsc525eSE1eBm5JydhgkJXPyJCoAozSgw61BqeVKTKHgWuGIjz904dAj8Mgslq2624XhA6nIsA5X0WOUucyqZeO5neRhzByFOQVhfrU2Iufd2GmLBlajljgbk8VBi7GyQJYqaoD7LRJQubzM6G684wnHM7SxsWnJdUBuaPp+jwnNuElZKcQBtKSGxCSu4giE8KATcKYImoNS0gWIWer5NGZaGqkzaYyxTEtbezTmKeaiz0uCfUJYL6Wn3TkOlMeCJMtenRd6iwkOp1Lw1QVtHixv6eVGnSuR8NSCt8/Xzap56qpSHMaQ1FtSZmn+rbDqMGVdyZPoKhiQHZR0Ftav7e0ZXYjV1pqYoHs3MK2VfLBT5INymZMk3ymKpm+f+9rdV4q5/zSpOAI4cA7NrhTPqxSzuHMKdBYJNB/4PQT6J3/5P8V9jc7va3O4zdba0+f1L+D5eh2dDe6F/Wi7ehp9CJ6He1FIvoQfYq+RF9bH1ufW9a3+fQxYU65lbUOK0fvwGprAu</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">AFE3icbVRLaxRBEJ4kq8b1lejRS2sIiEzW3RiIguCREQ8KCRyIbQ01O72y/6K5xsxnmP3jxr3jxoIhXDx79N9bMTjQT7WGgHl+9vq6ZxCkZsNv9tbC41Lpw8dLy5faVq9eu31hZvbkfbOYF7AmrH+X8ABKGthDiQreOQ9cJwreJpNnpf/te/BWrOLMweHmo+MHErBkUxHK/cHYei5yDeL/GHBNlit9iq131CPVta6nW512L9CrxbWovq8Plpd+jlIrcg0GBSKh3DQ6zo8zLlHKRQU7UEWwHEx4SM4INFwDeEwr4Yq2DpZUja0nl6DrLKejci5DmGmE0JqjuNw3lca/+c7yHD46DCXxmUIRswLDTPF0LKSIZKDwLVjAQuvKRemRhzogGJx3ajTKLpm7tBHkSCsbW2XNq3UgBjCwKmv0dl7Q2UaWlvBMqsc525eSE1eBm5JydhgkJXPyJCoAozSgw61BqeVKTKHgWuGIjz904dAj8Mgslq2624XhA6nIsA5X0WOUucyqZeO5neRhzByFOQVhfrU2Iufd2GmLBlajljgbk8VBi7GyQJYqaoD7LRJQubzM6G684wnHM7SxsWnJdUBuaPp+jwnNuElZKcQBtKSGxCSu4giE8KATcKYImoNS0gWIWer5NGZaGqkzaYyxTEtbezTmKeaiz0uCfUJYL6Wn3TkOlMeCJMtenRd6iwkOp1Lw1QVtHixv6eVGnSuR8NSCt8/Xzap56qpSHMaQ1FtSZmn+rbDqMGVdyZPoKhiQHZR0Ftav7e0ZXYjV1pqYoHs3MK2VfLBT5INymZMk3ymKpm+f+9rdV4q5/zSpOAI4cA7NrhTPqxSzuHMKdBYJNB/4PQT6J3/5P8V9jc7va3O4zdba0+f1L+D5eh2dDe6F/Wi7ehp9CJ6He1FIvoQfYq+RF9bH1ufW9a3+fQxYU65lbUOK0fvwGprAu</latexit> <latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="RMPR9FH5dkW3I90m4IOyDU5t1DM=">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</latexit> <latexit sha1_base64="0uWIM7EmbUiNEQbwXBnOxSr8co=">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</latexit> Average treatment effect (ATE) E [ Y (1) − Y (0)] = E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? : observed treatment T 2 1 1 1 ? : observed outcome Y 3 1 0 0 ? : used in subscript to denote a i specific unit/individual 4 0 0 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 1 ? Y i (0) 6 1 1 1 ? / 2 / − 1 / / − 1 / 3 − 3 3 3 = 3 11 / 41 Brady Neal
<latexit sha1_base64="2fEdt7FdkgfwU2jYIDligNtIdwM=">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</latexit> <latexit sha1_base64="6UZUlUoNAMm3rBUqYGrWy8+y6ow=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">AEl3icbVNbaxNBFN7aqDXeWn0SX0ZLwYdtTEqg+hAslEoRH1rpDZpQZmdPkiFzY+asMV2Cf8BX/W/+G89utrZJO8vCN+d8534mcUoGbDb/Lt1brt1/8HDlUf3xk6fPnq+uvTgJNvMCjoV1p8lPICSBo5RoIz54HrRMFpMtot9KfwQdpzRFOHPQ0HxjZl4Ijib4d1S9W15uNZnYbdCqwHpUnYOLteWf3dSKTINBoXgI562mw17OPUqhYFrvZgEcFyM+gHOChmsIvbxMdco2SJKyvX0G2Sl9KZFznUIE50QU3MchkVdIbxLd5h/0Mvl8ZlCEbMAvUzxdCyom6WSg8C1YQAF15SrkwMuecCqTv1uTCJntbrG+xIji5ZJZvPY1bLnAiJfG0VAFGaQWDWodTysipZ8CxwxQaeu2FoEPlLFoeuMm4wGB4VAGCumx9F34VDLx3E/yMOQOQpyCsL4cXYi593YcYsGVqHBDA/K4LzF2NsiCRUlQnoUjcpfXGZ3NrxzhR8wztLGxadGZgNwIYJ0WE5pxk7ICxAG0pITEKC7tiITwvhFwoiag1LSBYhZ6vk4ZloaqTPNxjLFIevQBm2Tj+nM1Flp8L8py4X0tClXptIY8NQy16G1a1PgvlRqlpqgHaE1C518WrlK5GyQkFb+OnlZT1VysMQ0oL6kbM6yhbDmPGlRyYjoI+4aCsI6N6Ob9dGonVFlr6gTN3cC4uTdvWneLVYvSfK96XRed8J9pfU6Ly4LemlScMRw4B3rvik+Vl4WeOaKaCwS6Q46PdnW4gO9DU62Gq124+Nhe32nXT3eleh19DZ6F7Wi7Wgn2o8OouNIRP3oV/Q7+lN7VftU+1zbn1HvLVU2L6O5Uzv8By5Uf/4=</latexit> <latexit sha1_base64="RMPR9FH5dkW3I90m4IOyDU5t1DM=">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</latexit> <latexit sha1_base64="0uWIM7EmbUiNEQbwXBnOxSr8co=">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</latexit> Average treatment effect (ATE) E [ Y (1) − Y (0)] = E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? : observed treatment T 2 1 1 1 ? : observed outcome Y 3 1 0 0 ? : used in subscript to denote a i specific unit/individual 4 0 0 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 1 ? Y i (0) 6 1 1 1 ? / 2 / − 1 / / − 1 / 3 − 3 3 3 = 3 11 / 41 Brady Neal
<latexit sha1_base64="2fEdt7FdkgfwU2jYIDligNtIdwM=">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</latexit> <latexit sha1_base64="6UZUlUoNAMm3rBUqYGrWy8+y6ow=">AGL3icjVTdbxtFED+nNRTz1ZRHXhaSoFa6GF9IKUhYRKqKEOKhSElalIuivb2xvfJ+aXeO4J7u/+GVPwYhXhCv/A28MHu+OHqh46159mZ3zszOwWTsmAo9Ffva07d/tvX3vncG73/wYf3tx+cBlt5ASfCKutfFjyAkgZOUKCl84D14WCF8X8adS/+AV8kNYc48LBueZTIydScCTRxXbP5AVMpamRF5XivqmP6ka0REwzyNE6XykY7Mpd9hnbPW6/Py+/D7NHTPafXQlYfuM9qTIc5ZrWbWGWlH3YoUufbvW3ZNeT4IFG2Wi1iI+6LDjO69rQRd/iGcR93mGyFyzbivnyz/FheWESr27PnYMpVeS/u74yGo5bY60zWMTtJR8vtu/8lpdWVBoMCsVDOMtGDs9r7lEKBdShKoDjYs6ncEas4RrCed1ORsP2SFKyifW0DLJWetOi5jqEhS4IqTnOwm1dFG7SnVU4+eq8lsZVCEYsA0qxdCyOGaslB4EqgUxXHhJuTIx454LpGEcrIUpdLO+t3ZO5QoNY3vsO0rdSAGMJArIco8dy/kr1oHXE14ek2EBG5WVgEQpZkGZh1KLV91tRG8ClyxqeduFoYE/qEKsVhuse94QAo+k4FCemx9R59KFp7R1m3EFISxDWt1cqpNx7exlSwZXo+KEG5OlEYupskBFScS7RY7IXT2IU7P/I0f4NeUV2tTYMpYwIDd09HGhGbclCwyaQAtKSExT1s7AiF8Pgy4UAStQSnpAqSs9PwyZVoaqSvNLmWJMzamUXtCPpqlqbPS4MqU1UJ6GqkrU2kMeCqZG9N8HlLgiVRqmZqgYaJ5DGN6IpauCrnsOJSdv3Hdnqc7VcnDMoOC+pGzOsoBw5TxpWcmrGCfFBWUdGg7Z/T6klVlNkrakS1HcDl92mzp81dR5ntCjqZ/HRuqk7jY9aq/W6jptbemlKcIRw4B3LP4k/1m5u4cwV0Fgk0AY43e3s9k1+nTk9GaHw69/Otw5+qa75feSj5NPk4dJljxJjpLvk+fJSJ6f/T+27q71e/3v+z/3f/nyV0q9fZfJSsUf/f/wEQiNvq</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="RMPR9FH5dkW3I90m4IOyDU5t1DM=">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</latexit> <latexit sha1_base64="0uWIM7EmbUiNEQbwXBnOxSr8co=">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</latexit> Average treatment effect (ATE) E [ Y (1) − Y (0)] = E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? : observed treatment T 2 1 1 1 ? : observed outcome Y 3 1 0 0 ? : used in subscript to denote a i specific unit/individual 4 0 0 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 1 ? Y i (0) 6 1 1 1 ? / 2 / − 1 / / − 1 / 3 − 3 3 3 = 3 11 / 41 Brady Neal
<latexit sha1_base64="2fEdt7FdkgfwU2jYIDligNtIdwM=">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</latexit> <latexit sha1_base64="6UZUlUoNAMm3rBUqYGrWy8+y6ow=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="ilQ7lU+sqi5Nob1MokD7r3y8Ug=">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</latexit> <latexit sha1_base64="/d+scwbDPiu9WShO9FsrOFq50c4=">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</latexit> <latexit sha1_base64="35LYrIzCBcqde2ZQomHOlJF5E8s=">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</latexit> <latexit sha1_base64="JQUulebzr/RKdwE8rIO+42c18YA=">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</latexit> <latexit sha1_base64="BVS1cv+/BiNF1N0SRQkUw3wzFI=">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</latexit> <latexit sha1_base64="RMPR9FH5dkW3I90m4IOyDU5t1DM=">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</latexit> <latexit sha1_base64="0uWIM7EmbUiNEQbwXBnOxSr8co=">AEz3icbVNtaxNBED7bqDW+Vf3ol9VSaOEaEymoYKAgFRE/WOgrTSh7e5Nkyb6xO2dMjxO/+hv8quBf8t84dzlrk7rHwbw8z8zszGzilAzYbv+trTcuH7j5sqt5u07d+/dX3w8DYzAs4EFZf5zwAEoaOECJCo6dB64TBUfJ+E3pP/oEPkhr9nHqoK/50MiBFBzJdLZ6v7d7erLR2WRb7GSjvdk/W1rt9rVYVeFTi2sRfX5ePZg+VcvtSLTYFAoHsJp+2wn3OPUigomr0sgONizIdwSqLhGkI/ryov2DpZUjawn6DrLJeZuRchzDVCSE1x1FY9JXG/lOMxy87OfSuAzBiFmiQaYWla2gaXSg0A1JYEL6lWJkbc4HUrOZcmkQX87q1Y+RJKBhbZ2+pdCMFMLIoIOY625fjc1aD5wueXrOhAQuLlgBEKUZBmYdSi3P694IngWu2NBzNwotAr/PQtksN91yPCAlH8lAKT1WscuYSiae+2keRtxBiFMQ1lcjDzH3k5CLgStdzSgDweSIydDbJEURFUZxmIwuVNRmfrA0f4HPMbWxsWrYwIDd09W6HCc24SVkpxAG0pILEOK54BEJ41go4VQTNQSnpAsQs9XwSMy2N1JlmE5niHVp1V5QjGJGdVYavKCyXEhPK/WXKo0BTy1zXdrPbUo8kErNShO0TLSPoZsXdahEziYOaR2vm1f3qW+V8jCtMaCupTzX5bnDmPGlRyaroIByUFZR6RmNb83NBKrKbPW1Amau4FJreS93SLvlTuaJPluUcz7DrmvV7npbLglyYFRwgH3rHek/JjlbKAM3+BxiKB/gOnt91ZfMlXhcPnrc5269Xe9trO6/qVr0SPo6fRtSJXkQ70bvoY3QiSiLvkc/op+Nvcak8aXxdQZdulZzHkVzp/HtD4QUk54=</latexit> Average treatment effect (ATE) E [ Y (1) − Y (0)] = E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] associational difference Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? : observed treatment T 2 1 1 1 ? : observed outcome Y 3 1 0 0 ? : used in subscript to denote a i specific unit/individual 4 0 0 0 ? : potential outcome under treatment Y i (1) : potential outcome under no treatment 5 0 1 1 ? Y i (0) 6 1 1 1 ? / 2 / − 1 / / − 1 / 3 − 3 3 3 = 3 11 / 41 Brady Neal
Association is not causation Sleeping with shoes on is strongly correlated with waking up with a headache 12 / 41 Brady Neal
Association is not causation Sleeping with shoes on is strongly correlated with waking up with a headache Common cause: drinking the night before 12 / 41 Brady Neal
Association is not causation Sleeping with shoes on is strongly correlated with waking up with a headache Common cause: drinking the night before 1. Confounding 12 / 41 Brady Neal
<latexit sha1_base64="uGwb7/40FtxyqwMkvjDesmjJa3g=">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</latexit> Association is not causation Sleeping with shoes on is strongly correlated with waking up with a headache Common cause: drinking the night before 1. Confounding X T Y 12 / 41 Brady Neal
<latexit sha1_base64="uGwb7/40FtxyqwMkvjDesmjJa3g=">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</latexit> Association is not causation Sleeping with shoes on is strongly correlated with waking up with a headache Common cause: drinking the night before 1. Confounding 2. Shoe-sleepers differ from non-shoe- sleepers in a key way X T Y 12 / 41 Brady Neal
<latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> <latexit sha1_base64="g+ReQUPJbhQVm3ZDK3s2eRN7Vk=">AE+XicbVNLaxRBEJ4k62t9RT16aQ2BCJN1NwRiwAUhREQ8RMhLskvo6andbZfdNcY12F+i3gLXv0NXvXsv7FmdvLYmB4Gvqr6tnViVMyYLv9d25+oXHj5q3bd5p3791/8HDx0eP9YDMvYE9YZf1hwgMoaWAPJSo4dB64ThQcJOt0n7wGXyQ1uzixEFf86GRAyk4kup4cbO3fRpfOiz1ZBdsEuxVkPS1TtktS58x6oWr3jxeX2q12dj/oFODpag+O8ePFk57qRWZBoNC8RCOm2H/Zx7lEJB0exlARwXYz6EI4KGawj9vOqxYMukSdnAevoNskp72SPnOoSJToipOY7CVupvM52lOHgVT+XxmUIRkwTDTLF0LJyYCyVHgSqCQEuvKRamRhxzwXSWJszaRJdzMrWjpEnoWBsmb2l0o0UwEijgDyX2a4cf2U1ebgadMzKiRyce4VAFGaYWDWodTyaz0bwbPAFRt67kahReT3WSiH5Sarjgek5CMZKXHKnYZU8nEcz/Jw4g7CHEKwvpqOULMvbcnIRZciRq3NCPBxJjZ4MsWVQE1VkGonB5k9FZ/cARvsQ8Qxsbm5YjDMgNtd7tMKEZNykrQRxASypIjOPKj0gIL1sBJ4qoOSglXYCYpZ6fxExLI3Wm2YlMcVRuYGuDYhRTV2elwXNXlgvpaXOXKUx4Glkrkv7uU6JB1KpaWmClon2MXTzog6VyOmNQ1rH6+ZVP3VXKQ8jSGsuqEs5L7KsOYwZV3JougoGhIOyjpya1f1t0ZVYTZm1pknQvRs4qYW8t13kvXJHkyTfLopZ2z73tdXrvBSu2KVJwRHDgXes96z8WCVc4ZkzorFIpGvo9LY7V1/y/2B/rdVZb21+XF9687p+5bejp9HzaCXqRBvRm+hdtBPtRSL6Fv2Kfkd/Gnje+O08WNKnZ+rfZ5EM6fx8x/+yKCY</latexit> Why? Because the groups are not comparable E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Went to sleep with shoes on Went to sleep without shoes on ( T = 1) ( T = 0) 13 / 41 Brady Neal
<latexit sha1_base64="g+ReQUPJbhQVm3ZDK3s2eRN7Vk=">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</latexit> <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">AE7nicbVRLaxRBEJ4kq8b1lejRS2sIRJisuyEQBRcCISLiIUJekA2hp6d2t9l+0V1jshnmR3jxoIhX/45H/401s5OYTdLDQHXV91V9XV0ziVMyYLv9d2Z2rnHn7r35+80HDx89frKw+HQ/2MwL2BNWX+Y8ABKGthDiQoOnQeuEwUHyWirjB98AR+kNbs4dnCs+cDIvhQcyXWwsu6rP3qZGp3WpXi90OrWxFNVr52Rx7k8vtSLTYFAoHsJRp+3wOcepVBQNHtZAMfFiA/giEzDNYTjvNJbsGXypKxvPb0GWeW9ysi5DmGsE0JqjsNwPVY6b4sdZdh/c5xL4zIEIyaF+pliaFl5eJZKDwLVmAwuvCStTAy5wKpRc2pMokupvfWjpAnoWBsmb0n6UYKYORMK3vrE/5plGlp2w3lVhmu3J0zmrwNHPSnSkXEri4ZAVAlGYQmHUotTyvmyh4FrhiA8/dMLQI/DELZVfdeNXxgKRyKAOV9FjlLnMqmXjux3kYcgchTkFYX01EiLn39jTEgitR2y0NyO+xNjZIEsUiSCdZSJKlzcZrdVPHOEs5hna2Ni07HVAbuj03Q4TmnGTstKIA2hJgsQorngEQnjdCjhWBM1BKekCxCz1/DRmWhqpM81OZYrDclRbG5SjmFCdlQYvqSwX0tPsXVClMeCpZa5Lg7xOhftSqYk0QVNHgxu6eVGnSuRkNCt83Xz6jz1qVIehpDWFBXav6vsuYwZlzJgekq6JMdlHVEalb3t0VXYjV1po6Qfdu4LTe5L3tIu+Vw5wk+XZRTMf2ua+jXufl5lpcmhQcIRx4x3ovyodVm2s4cwE0Fgl0C5x+Ap3rn/xNY3+t1Vlvf28vrT5rv4dzEfPo5fRStSJNqLN6EO0E+1FIhpFX6Pv0Y+Ga3xr/Gz8mkBnZ2rOs2hqNX7/A4LxoMk=</latexit> Why? Because the groups are not comparable E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Went to sleep with shoes on Went to sleep without shoes on ( T = 1) ( T = 0) drunk sober sober sober drunk sober drunk drunk sober sober sober drunk drunk drunk sober sober drunk sober sober sober drunk drunk sober drunk sober drunk sober sober 13 / 41 Brady Neal
<latexit sha1_base64="g+ReQUPJbhQVm3ZDK3s2eRN7Vk=">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</latexit> <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">AE7nicbVRLaxRBEJ4kq8b1lejRS2sIRJisuyEQBRcCISLiIUJekA2hp6d2t9l+0V1jshnmR3jxoIhX/45H/401s5OYTdLDQHXV91V9XV0ziVMyYLv9d2Z2rnHn7r35+80HDx89frKw+HQ/2MwL2BNWX+Y8ABKGthDiQoOnQeuEwUHyWirjB98AR+kNbs4dnCs+cDIvhQcyXWwsu6rP3qZGp3WpXi90OrWxFNVr52Rx7k8vtSLTYFAoHsJRp+3wOcepVBQNHtZAMfFiA/giEzDNYTjvNJbsGXypKxvPb0GWeW9ysi5DmGsE0JqjsNwPVY6b4sdZdh/c5xL4zIEIyaF+pliaFl5eJZKDwLVmAwuvCStTAy5wKpRc2pMokupvfWjpAnoWBsmb0n6UYKYORMK3vrE/5plGlp2w3lVhmu3J0zmrwNHPSnSkXEri4ZAVAlGYQmHUotTyvmyh4FrhiA8/dMLQI/DELZVfdeNXxgKRyKAOV9FjlLnMqmXjux3kYcgchTkFYX01EiLn39jTEgitR2y0NyO+xNjZIEsUiSCdZSJKlzcZrdVPHOEs5hna2Ni07HVAbuj03Q4TmnGTstKIA2hJgsQorngEQnjdCjhWBM1BKekCxCz1/DRmWhqpM81OZYrDclRbG5SjmFCdlQYvqSwX0tPsXVClMeCpZa5Lg7xOhftSqYk0QVNHgxu6eVGnSuRkNCt83Xz6jz1qVIehpDWFBXav6vsuYwZlzJgekq6JMdlHVEalb3t0VXYjV1po6Qfdu4LTe5L3tIu+Vw5wk+XZRTMf2ua+jXufl5lpcmhQcIRx4x3ovyodVm2s4cwE0Fgl0C5x+Ap3rn/xNY3+t1Vlvf28vrT5rv4dzEfPo5fRStSJNqLN6EO0E+1FIhpFX6Pv0Y+Ga3xr/Gz8mkBnZ2rOs2hqNX7/A4LxoMk=</latexit> Why? Because the groups are not comparable E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Went to sleep with shoes on Went to sleep without shoes on ( T = 1) ( T = 0) drunk sober sober sober drunk sober drunk drunk sober sober sober drunk drunk drunk sober sober drunk sober sober sober drunk drunk sober drunk sober drunk sober sober 13 / 41 Brady Neal
<latexit sha1_base64="g+ReQUPJbhQVm3ZDK3s2eRN7Vk=">AE+XicbVNLaxRBEJ4k62t9RT16aQ2BCJN1NwRiwAUhREQ8RMhLskvo6andbZfdNcY12F+i3gLXv0NXvXsv7FmdvLYmB4Gvqr6tnViVMyYLv9d25+oXHj5q3bd5p3791/8HDx0eP9YDMvYE9YZf1hwgMoaWAPJSo4dB64ThQcJOt0n7wGXyQ1uzixEFf86GRAyk4kup4cbO3fRpfOiz1ZBdsEuxVkPS1TtktS58x6oWr3jxeX2q12dj/oFODpag+O8ePFk57qRWZBoNC8RCOm2H/Zx7lEJB0exlARwXYz6EI4KGawj9vOqxYMukSdnAevoNskp72SPnOoSJToipOY7CVupvM52lOHgVT+XxmUIRkwTDTLF0LJyYCyVHgSqCQEuvKRamRhxzwXSWJszaRJdzMrWjpEnoWBsmb2l0o0UwEijgDyX2a4cf2U1ebgadMzKiRyce4VAFGaYWDWodTyaz0bwbPAFRt67kahReT3WSiH5Sarjgek5CMZKXHKnYZU8nEcz/Jw4g7CHEKwvpqOULMvbcnIRZciRq3NCPBxJjZ4MsWVQE1VkGonB5k9FZ/cARvsQ8Qxsbm5YjDMgNtd7tMKEZNykrQRxASypIjOPKj0gIL1sBJ4qoOSglXYCYpZ6fxExLI3Wm2YlMcVRuYGuDYhRTV2elwXNXlgvpaXOXKUx4Glkrkv7uU6JB1KpaWmClon2MXTzog6VyOmNQ1rH6+ZVP3VXKQ8jSGsuqEs5L7KsOYwZV3JougoGhIOyjpya1f1t0ZVYTZm1pknQvRs4qYW8t13kvXJHkyTfLopZ2z73tdXrvBSu2KVJwRHDgXes96z8WCVc4ZkzorFIpGvo9LY7V1/y/2B/rdVZb21+XF9687p+5bejp9HzaCXqRBvRm+hdtBPtRSL6Fv2Kfkd/Gnje+O08WNKnZ+rfZ5EM6fx8x/+yKCY</latexit> <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> Why? Because the groups are not comparable E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Went to sleep with shoes on Went to sleep without shoes on ( T = 1) ( T = 0) drunk sober sober sober drunk sober drunk drunk sober sober sober drunk drunk drunk sober sober drunk sober sober sober drunk drunk sober drunk sober drunk sober sober 13 / 41 Brady Neal
<latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> <latexit sha1_base64="g+ReQUPJbhQVm3ZDK3s2eRN7Vk=">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</latexit> What would comparable groups look like? E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Went to sleep with shoes on Went to sleep without shoes on ( T = 1) ( T = 0) drunk sober sober sober drunk sober drunk drunk sober sober sober drunk drunk drunk sober sober drunk sober sober sober drunk drunk sober drunk sober drunk sober sober 14 / 41 Brady Neal
<latexit sha1_base64="g+ReQUPJbhQVm3ZDK3s2eRN7Vk=">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</latexit> <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> What would comparable groups look like? E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Went to sleep with shoes on Went to sleep without shoes on ( T = 1) ( T = 0) drunk drunk sober sober sober sober sober sober drunk drunk drunk drunk drunk drunk sober sober sober sober drunk drunk 14 / 41 Brady Neal
<latexit sha1_base64="g+ReQUPJbhQVm3ZDK3s2eRN7Vk=">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</latexit> <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> What would comparable groups look like? E [ Y (1)] − E [ Y (0)] = E [ Y | T = 1] − E [ Y | T = 0] Went to sleep with shoes on Went to sleep without shoes on ( T = 1) ( T = 0) drunk drunk sober sober sober sober sober sober drunk drunk drunk drunk drunk drunk sober sober sober sober drunk drunk 14 / 41 Brady Neal
Question: Why is association not causation?
What assumptions would make the ATE equal to the associational difference?
<latexit sha1_base64="8HLHS2O4Ueq6yvnfYygFxEsT8no=">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</latexit> Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T 17 / 41 Brady Neal
<latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="8HLHS2O4Ueq6yvnfYygFxEsT8no=">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</latexit> Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] (ignorability) 17 / 41 Brady Neal
<latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="8HLHS2O4Ueq6yvnfYygFxEsT8no=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] 17 / 41 Brady Neal
<latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">AF83icfVRfbxtFED+XGoqBksIDSLwsuEFuhfFKkgYalSFYQD0VK0lY5K9rbm7NX3n/anSN1T/fC1+AN8coH4hvwMZg9X5tcWrGWrdmZ3/zmt7PjLZySAefzf0a3rk9fve9O+9Pvjwo7sf79375CzY2gs4FVZ/6zgAZQ0cIoSFTxzHrguFDwtNo9j/Olv4IO05gS3Dpar4yspOBIrou93/Pj8+cPsm+W7IB15pzMrxesd7Ncy5KdsAXLriGuvPMly5GvGrky1vNCKonbluX5hNHqad7CMSC42JvOZ/NusTeNrDemSb+eXNy7/Twvrag1GBSKh3CezR0uG+5RCgXtJK8DOC42fAXnZBquISybrlkt2ydPySr6WuQd7rGQ3XIWx1QUjNcR1uxqLzbHzGqvlo0rkYwYleoqhVDy2LnWSk9CFRbMrjwkrQyseaeC6T7mQzKFLod7q3dIC9Cy9g+5GkGymAkUfBUN+LiviGqOiJd0l9tmJ3LxkPXiYuevOwIUEbl9nBUCUZhWYdSi1fNk3UfA6cMVWnrt1mBH45zrErtgeMBSeVaBirpseOnEoWnvtE9bcQUhLEDQ7ncSUe28vQyq4Er0904A8rSmzgYZUSCdEYiomu6STv4hSO8SHmNjW2jL0OyA2dfpExoRk3JYtGkBLEiQ2aZdHIRvZwG3iqANKCVdgJSVnl+mTEsjda3ZpSxHWd19pA42l2qs9Lg61TWCOlp9l6lSmPAU8vcgb5iApXUqmdNEFTR4MbFk3bUxVyNxpQ9nyLpjtPf6qShzWUPRbUtZpXVQ4dpowr+hsuFRkB2UdJU26+3tMV2I1VdaOkH3buCy3zT5cdvkcZiLojlu2HsjPs+6nUTNzfi0pTgCOHAO5Z/GT+s29zAmVdAY5FA/w/fsQL9GCRqb5F604lwXNEQwpXzCnSq4gKS6iuR6bZ9LA/BKgm94q75n503m9zvQFvDnXdxH1LyfQYZTefnjeNs8NZdjT7/tej6aMf+mfpTvJF8lXyIMmSh8mj5KfkSXKaiOTf0d3RZ6Px/X4j/Gf47920FujPufTZLDGf/8HwkH2Ow=</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="8HLHS2O4Ueq6yvnfYygFxEsT8no=">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</latexit> <latexit sha1_base64="FSkNaui+guUCONR9/NkF2jubVLM=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 ? 0 ? 2 1 1 1 ? ? 3 1 0 0 ? ? 4 0 0 ? 0 ? 5 0 1 ? 1 ? 6 1 1 1 ? ? 17 / 41 Brady Neal
<latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="8HLHS2O4Ueq6yvnfYygFxEsT8no=">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</latexit> <latexit sha1_base64="6UZUlUoNAMm3rBUqYGrWy8+y6ow=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? 2 1 1 1 ? 3 1 0 0 ? 4 0 0 0 ? 5 0 1 1 ? 6 1 1 1 ? 17 / 41 Brady Neal
<latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="6UZUlUoNAMm3rBUqYGrWy8+y6ow=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="8HLHS2O4Ueq6yvnfYygFxEsT8no=">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</latexit> Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? 2 1 1 1 ? 3 1 0 0 ? 4 0 0 0 ? 5 0 1 1 ? 6 1 1 1 ? 2 / 3 17 / 41 Brady Neal
<latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">AFE3icbVRLaxRBEJ4kq8b1lejRS2sIiEzW3RiIguCREQ8KCRyIbQ01O72y/6K5xsxnmP3jxr3jxoIhXDx79N9bMTjQT7WGgHl+9vq6ZxCkZsNv9tbC41Lpw8dLy5faVq9eu31hZvbkfbOYF7AmrH+X8ABKGthDiQreOQ9cJwreJpNnpf/te/BWrOLMweHmo+MHErBkUxHK/cHYei5yDeL/GHBNlit9iq131CPVta6nW512L9CrxbWovq8Plpd+jlIrcg0GBSKh3DQ6zo8zLlHKRQU7UEWwHEx4SM4INFwDeEwr4Yq2DpZUja0nl6DrLKejci5DmGmE0JqjuNw3lca/+c7yHD46DCXxmUIRswLDTPF0LKSIZKDwLVjAQuvKRemRhzogGJx3ajTKLpm7tBHkSCsbW2XNq3UgBjCwKmv0dl7Q2UaWlvBMqsc525eSE1eBm5JydhgkJXPyJCoAozSgw61BqeVKTKHgWuGIjz904dAj8Mgslq2624XhA6nIsA5X0WOUucyqZeO5neRhzByFOQVhfrU2Iufd2GmLBlajljgbk8VBi7GyQJYqaoD7LRJQubzM6G684wnHM7SxsWnJdUBuaPp+jwnNuElZKcQBtKSGxCSu4giE8KATcKYImoNS0gWIWer5NGZaGqkzaYyxTEtbezTmKeaiz0uCfUJYL6Wn3TkOlMeCJMtenRd6iwkOp1Lw1QVtHixv6eVGnSuR8NSCt8/Xzap56qpSHMaQ1FtSZmn+rbDqMGVdyZPoKhiQHZR0Ftav7e0ZXYjV1pqYoHs3MK2VfLBT5INymZMk3ymKpm+f+9rdV4q5/zSpOAI4cA7NrhTPqxSzuHMKdBYJNB/4PQT6J3/5P8V9jc7va3O4zdba0+f1L+D5eh2dDe6F/Wi7ehp9CJ6He1FIvoQfYq+RF9bH1ufW9a3+fQxYU65lbUOK0fvwGprAu</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="6UZUlUoNAMm3rBUqYGrWy8+y6ow=">AGL3icjVTdbxtFED+nNRTz1ZRHXhaSoFa6GF9IKUhYRKqKEOKhSElalIuivb2xvfJ+aXeO4J7u/+GVPwYhXhCv/A28MHu+OHqh46159mZ3zszOwWTsmAo9Ffva07d/tvX3vncG73/wYf3tx+cBlt5ASfCKutfFjyAkgZOUKCl84D14WCF8X8adS/+AV8kNYc48LBueZTIydScCTRxXbP5AVMpamRF5XivqmP6ka0REwzyNE6XykY7Mpd9hnbPW6/Py+/D7NHTPafXQlYfuM9qTIc5ZrWbWGWlH3YoUufbvW3ZNeT4IFG2Wi1iI+6LDjO69rQRd/iGcR93mGyFyzbivnyz/FheWESr27PnYMpVeS/u74yGo5bY60zWMTtJR8vtu/8lpdWVBoMCsVDOMtGDs9r7lEKBdShKoDjYs6ncEas4RrCed1ORsP2SFKyifW0DLJWetOi5jqEhS4IqTnOwm1dFG7SnVU4+eq8lsZVCEYsA0qxdCyOGaslB4EqgUxXHhJuTIx454LpGEcrIUpdLO+t3ZO5QoNY3vsO0rdSAGMJArIco8dy/kr1oHXE14ek2EBG5WVgEQpZkGZh1KLV91tRG8ClyxqeduFoYE/qEKsVhuse94QAo+k4FCemx9R59KFp7R1m3EFISxDWt1cqpNx7exlSwZXo+KEG5OlEYupskBFScS7RY7IXT2IU7P/I0f4NeUV2tTYMpYwIDd09HGhGbclCwyaQAtKSExT1s7AiF8Pgy4UAStQSnpAqSs9PwyZVoaqSvNLmWJMzamUXtCPpqlqbPS4MqU1UJ6GqkrU2kMeCqZG9N8HlLgiVRqmZqgYaJ5DGN6IpauCrnsOJSdv3Hdnqc7VcnDMoOC+pGzOsoBw5TxpWcmrGCfFBWUdGg7Z/T6klVlNkrakS1HcDl92mzp81dR5ntCjqZ/HRuqk7jY9aq/W6jptbemlKcIRw4B3LP4k/1m5u4cwV0Fgk0AY43e3s9k1+nTk9GaHw69/Otw5+qa75feSj5NPk4dJljxJjpLvk+fJSJ6f/T+27q71e/3v+z/3f/nyV0q9fZfJSsUf/f/wEQiNvq</latexit> <latexit sha1_base64="8HLHS2O4Ueq6yvnfYygFxEsT8no=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? 2 1 1 1 ? 3 1 0 0 ? 4 0 0 0 ? 5 0 1 1 ? 6 1 1 1 ? 2 / − 1 / 3 3 17 / 41 Brady Neal
<latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">AFE3icbVRLaxRBEJ4kq8b1lejRS2sIiEzW3RiIguCREQ8KCRyIbQ01O72y/6K5xsxnmP3jxr3jxoIhXDx79N9bMTjQT7WGgHl+9vq6ZxCkZsNv9tbC41Lpw8dLy5faVq9eu31hZvbkfbOYF7AmrH+X8ABKGthDiQreOQ9cJwreJpNnpf/te/BWrOLMweHmo+MHErBkUxHK/cHYei5yDeL/GHBNlit9iq131CPVta6nW512L9CrxbWovq8Plpd+jlIrcg0GBSKh3DQ6zo8zLlHKRQU7UEWwHEx4SM4INFwDeEwr4Yq2DpZUja0nl6DrLKejci5DmGmE0JqjuNw3lca/+c7yHD46DCXxmUIRswLDTPF0LKSIZKDwLVjAQuvKRemRhzogGJx3ajTKLpm7tBHkSCsbW2XNq3UgBjCwKmv0dl7Q2UaWlvBMqsc525eSE1eBm5JydhgkJXPyJCoAozSgw61BqeVKTKHgWuGIjz904dAj8Mgslq2624XhA6nIsA5X0WOUucyqZeO5neRhzByFOQVhfrU2Iufd2GmLBlajljgbk8VBi7GyQJYqaoD7LRJQubzM6G684wnHM7SxsWnJdUBuaPp+jwnNuElZKcQBtKSGxCSu4giE8KATcKYImoNS0gWIWer5NGZaGqkzaYyxTEtbezTmKeaiz0uCfUJYL6Wn3TkOlMeCJMtenRd6iwkOp1Lw1QVtHixv6eVGnSuR8NSCt8/Xzap56qpSHMaQ1FtSZmn+rbDqMGVdyZPoKhiQHZR0Ftav7e0ZXYjV1pqYoHs3MK2VfLBT5INymZMk3ymKpm+f+9rdV4q5/zSpOAI4cA7NrhTPqxSzuHMKdBYJNB/4PQT6J3/5P8V9jc7va3O4zdba0+f1L+D5eh2dDe6F/Wi7ehp9CJ6He1FIvoQfYq+RF9bH1ufW9a3+fQxYU65lbUOK0fvwGprAu</latexit> <latexit sha1_base64="8HLHS2O4Ueq6yvnfYygFxEsT8no=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="6UZUlUoNAMm3rBUqYGrWy8+y6ow=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? 2 1 1 1 ? 3 1 0 0 ? 4 0 0 0 ? 5 0 1 1 ? 6 1 1 1 ? / 2 / / − 1 / − 1 / 3 = 3 3 − 3 3 17 / 41 Brady Neal
<latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">AFE3icbVRLaxRBEJ4kq8b1lejRS2sIiEzW3RiIguCREQ8KCRyIbQ01O72y/6K5xsxnmP3jxr3jxoIhXDx79N9bMTjQT7WGgHl+9vq6ZxCkZsNv9tbC41Lpw8dLy5faVq9eu31hZvbkfbOYF7AmrH+X8ABKGthDiQreOQ9cJwreJpNnpf/te/BWrOLMweHmo+MHErBkUxHK/cHYei5yDeL/GHBNlit9iq131CPVta6nW512L9CrxbWovq8Plpd+jlIrcg0GBSKh3DQ6zo8zLlHKRQU7UEWwHEx4SM4INFwDeEwr4Yq2DpZUja0nl6DrLKejci5DmGmE0JqjuNw3lca/+c7yHD46DCXxmUIRswLDTPF0LKSIZKDwLVjAQuvKRemRhzogGJx3ajTKLpm7tBHkSCsbW2XNq3UgBjCwKmv0dl7Q2UaWlvBMqsc525eSE1eBm5JydhgkJXPyJCoAozSgw61BqeVKTKHgWuGIjz904dAj8Mgslq2624XhA6nIsA5X0WOUucyqZeO5neRhzByFOQVhfrU2Iufd2GmLBlajljgbk8VBi7GyQJYqaoD7LRJQubzM6G684wnHM7SxsWnJdUBuaPp+jwnNuElZKcQBtKSGxCSu4giE8KATcKYImoNS0gWIWer5NGZaGqkzaYyxTEtbezTmKeaiz0uCfUJYL6Wn3TkOlMeCJMtenRd6iwkOp1Lw1QVtHixv6eVGnSuR8NSCt8/Xzap56qpSHMaQ1FtSZmn+rbDqMGVdyZPoKhiQHZR0Ftav7e0ZXYjV1pqYoHs3MK2VfLBT5INymZMk3ymKpm+f+9rdV4q5/zSpOAI4cA7NrhTPqxSzuHMKdBYJNB/4PQT6J3/5P8V9jc7va3O4zdba0+f1L+D5eh2dDe6F/Wi7ehp9CJ6He1FIvoQfYq+RF9bH1ufW9a3+fQxYU65lbUOK0fvwGprAu</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="8HLHS2O4Ueq6yvnfYygFxEsT8no=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="6UZUlUoNAMm3rBUqYGrWy8+y6ow=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">AFE3icbVRLaxRBEJ4kq8b1lejRS2sIiEzW3RiIguCREQ8KCRyIbQ01O72y/6K5xsxnmP3jxr3jxoIhXDx79N9bMTjQT7WGgHl+9vq6ZxCkZsNv9tbC41Lpw8dLy5faVq9eu31hZvbkfbOYF7AmrH+X8ABKGthDiQreOQ9cJwreJpNnpf/te/BWrOLMweHmo+MHErBkUxHK/cHYei5yDeL/GHBNlit9iq131CPVta6nW512L9CrxbWovq8Plpd+jlIrcg0GBSKh3DQ6zo8zLlHKRQU7UEWwHEx4SM4INFwDeEwr4Yq2DpZUja0nl6DrLKejci5DmGmE0JqjuNw3lca/+c7yHD46DCXxmUIRswLDTPF0LKSIZKDwLVjAQuvKRemRhzogGJx3ajTKLpm7tBHkSCsbW2XNq3UgBjCwKmv0dl7Q2UaWlvBMqsc525eSE1eBm5JydhgkJXPyJCoAozSgw61BqeVKTKHgWuGIjz904dAj8Mgslq2624XhA6nIsA5X0WOUucyqZeO5neRhzByFOQVhfrU2Iufd2GmLBlajljgbk8VBi7GyQJYqaoD7LRJQubzM6G684wnHM7SxsWnJdUBuaPp+jwnNuElZKcQBtKSGxCSu4giE8KATcKYImoNS0gWIWer5NGZaGqkzaYyxTEtbezTmKeaiz0uCfUJYL6Wn3TkOlMeCJMtenRd6iwkOp1Lw1QVtHixv6eVGnSuR8NSCt8/Xzap56qpSHMaQ1FtSZmn+rbDqMGVdyZPoKhiQHZR0Ftav7e0ZXYjV1pqYoHs3MK2VfLBT5INymZMk3ymKpm+f+9rdV4q5/zSpOAI4cA7NrhTPqxSzuHMKdBYJNB/4PQT6J3/5P8V9jc7va3O4zdba0+f1L+D5eh2dDe6F/Wi7ehp9CJ6He1FIvoQfYq+RF9bH1ufW9a3+fQxYU65lbUOK0fvwGprAu</latexit> Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? 2 1 1 1 ? 3 1 0 0 ? 4 0 0 0 ? 5 0 1 1 ? 6 1 1 1 ? / 2 / / − 1 / − 1 / 3 = 3 3 − 3 3 17 / 41 Brady Neal
<latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="6UZUlUoNAMm3rBUqYGrWy8+y6ow=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="8HLHS2O4Ueq6yvnfYygFxEsT8no=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="uGwb7/40FtxyqwMkvjDesmjJa3g=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? X 2 1 1 1 ? 3 1 0 0 ? 4 0 0 0 ? 5 0 1 1 ? T Y 6 1 1 1 ? / 2 / / − 1 / − 1 / 3 = 3 3 − 3 3 17 / 41 Brady Neal
<latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">AF83icfVRfbxtFED+XGoqBksIDSLwsuEFuhfFKkgYalSFYQD0VK0lY5K9rbm7NX3n/anSN1T/fC1+AN8coH4hvwMZg9X5tcWrGWrdmZ3/zmt7PjLZySAefzf0a3rk9fve9O+9Pvjwo7sf79375CzY2gs4FVZ/6zgAZQ0cIoSFTxzHrguFDwtNo9j/Olv4IO05gS3Dpar4yspOBIrou93/Pj8+cPsm+W7IB15pzMrxesd7Ncy5KdsAXLriGuvPMly5GvGrky1vNCKonbluX5hNHqad7CMSC42JvOZ/NusTeNrDemSb+eXNy7/Twvrag1GBSKh3CezR0uG+5RCgXtJK8DOC42fAXnZBquISybrlkt2ydPySr6WuQd7rGQ3XIWx1QUjNcR1uxqLzbHzGqvlo0rkYwYleoqhVDy2LnWSk9CFRbMrjwkrQyseaeC6T7mQzKFLod7q3dIC9Cy9g+5GkGymAkUfBUN+LiviGqOiJd0l9tmJ3LxkPXiYuevOwIUEbl9nBUCUZhWYdSi1fNk3UfA6cMVWnrt1mBH45zrErtgeMBSeVaBirpseOnEoWnvtE9bcQUhLEDQ7ncSUe28vQyq4Er0904A8rSmzgYZUSCdEYiomu6STv4hSO8SHmNjW2jL0OyA2dfpExoRk3JYtGkBLEiQ2aZdHIRvZwG3iqANKCVdgJSVnl+mTEsjda3ZpSxHWd19pA42l2qs9Lg61TWCOlp9l6lSmPAU8vcgb5iApXUqmdNEFTR4MbFk3bUxVyNxpQ9nyLpjtPf6qShzWUPRbUtZpXVQ4dpowr+hsuFRkB2UdJU26+3tMV2I1VdaOkH3buCy3zT5cdvkcZiLojlu2HsjPs+6nUTNzfi0pTgCOHAO5Z/GT+s29zAmVdAY5FA/w/fsQL9GCRqb5F604lwXNEQwpXzCnSq4gKS6iuR6bZ9LA/BKgm94q75n503m9zvQFvDnXdxH1LyfQYZTefnjeNs8NZdjT7/tej6aMf+mfpTvJF8lXyIMmSh8mj5KfkSXKaiOTf0d3RZ6Px/X4j/Gf47920FujPufTZLDGf/8HwkH2Ow=</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="6UZUlUoNAMm3rBUqYGrWy8+y6ow=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="8HLHS2O4Ueq6yvnfYygFxEsT8no=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="uGwb7/40FtxyqwMkvjDesmjJa3g=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? X 2 1 1 1 ? 3 1 0 0 ? 4 0 0 0 ? 5 0 1 1 ? T Y 6 1 1 1 ? / 2 / / − 1 / − 1 / 3 = 3 3 − 3 3 17 / 41 Brady Neal
<latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="6UZUlUoNAMm3rBUqYGrWy8+y6ow=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="8HLHS2O4Ueq6yvnfYygFxEsT8no=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> <latexit sha1_base64="wt2lCE4iu6jp6hQfvYVw6vDRF60=">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</latexit> <latexit sha1_base64="3+xP5vsR1L6LDmNEnY+okJIMjY=">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</latexit> Ignorability: ( Y (1) , Y (0)) ⊥ ⊥ T E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Y (1) Y (0) Y (1) − Y (0) i T Y 1 0 0 0 ? X 2 1 1 1 ? 3 1 0 0 ? 4 0 0 0 ? 5 0 1 1 ? T Y 6 1 1 1 ? / 2 / / − 1 / − 1 / 3 = 3 3 − 3 3 17 / 41 Brady Neal
<latexit sha1_base64="hsz8RjcljbDBaGAYNVc09xwW9nw=">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</latexit> <latexit sha1_base64="+csBynRxFr6koeweOUGm1ZEASRU=">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</latexit> <latexit sha1_base64="u9JVBphx7MmuBVj2PHmsIsdN7Y=">AFh3icfVTbjhNHEB0uDmCSsBCJl7x0MCvlYTD2asVFiUitFEU5YFIuwtox1r19NTYLfdN3TUspmfyWvyQ/mbVI8Hdr1EacujupyqOlVd6tIpGXAy+efK1WvXB1/duHlrePvrb769s3P3nGwjRdwJKy/k3JAyhp4AglKnjPHBdKnhdrl4m/+t34IO05hDXDuaL4yspeBIptOd+8XByVtWaFmxQzZjkzl91qeT053RZDzpDvtSmPbCKOvPq9O7198WlRWNBoNC8RBOphOH8g9SqGgHRZNAMfFi/ghETDNYR57Bpo2S5ZKlZbT3+DrLNejIhch7DWJSE1x2W47EvG/KdNFg/m0dpXINgxKZQ3SiGlqVpsEp6EKjWJHDhJXFlYsk9F0gzG26VKXW7rVu7Ql6GlrFd9gtRN1IAI4uCbX7va8q3jUqWNH8qscsO5eoD68HbkZvpbJmQwO3nqACI0iwCsw6lh/6IQreBK7YwnO3DGMC/9aENFW3fuR4QGK5lIFKeuxyp5xKlp7dQxL7iDkFQjruxUJOfenoVcCV6eawBeV5LzJ0NMqGIBPFMiShdHDI6j37nCO9z3qDNja3SrANyQ93Ppkxoxk3FkpAH0JIiVXexREI4fE4FoRNIJS0gXIWeX5Wc60NFI3mp3JCpdpYcdPKUe7CXVWGvwcyqKQnbvU6g0BjyNzM1okfepcC2V2lATtHW0uGEW2z5VKTerAVWfbxa7fvquKh6WUPVYUBdqnlfZc5gzruTCzBTUJAdlHQUNu/t7SVdiNVXWmiZB927grFdicdDGIi1zWcaDt32HXPfe72OSbnkl6YCRwgH3rHih/RjnXIJZz4BjUC/T98kxXoY5BSe4s0m46E4qWEM6N58jDnkViWEF90TOajvb6JkDFwivu4sNkfNgWegXe7OkmJr2lYHqMpefni+F473xdH/8/I/90Yuf+mfpZvZ9iD7MZtmT7MX2a/Zq+woE9nH7M/sr+zvwa3B48GTwbMN9OqVPua7bOsMfv4XHAzX0g=</latexit> <latexit sha1_base64="B6SDsM+n6ge+raWORAaVSuAj6XE=">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</latexit> Another perspective: exchangeability T = 0 T = 1 E [ Y | T = 1] = y 1 E [ Y | T = 0] = y 0 18 / 41 Brady Neal
<latexit sha1_base64="hsz8RjcljbDBaGAYNVc09xwW9nw=">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</latexit> <latexit sha1_base64="+csBynRxFr6koeweOUGm1ZEASRU=">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</latexit> <latexit sha1_base64="u9JVBphx7MmuBVj2PHmsIsdN7Y=">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</latexit> <latexit sha1_base64="B6SDsM+n6ge+raWORAaVSuAj6XE=">AFh3icfVTbjhtFEJ1cDIm5ZBMkXnhpcFbiYeJ4VisSJCwFRYsQ4iFIu5tEO9aqp6fGbrlv6q5h4zTzM7zCD/E3VI8n2fUG0ZHdTlVdaq61JVTMuBs9s+Nm7dujz76+M7d8Sefvb5vb37D06Db2AE2GV9a8qHkBJAycoUcEr54HrSsHLav08+V/+Dj5Ia45x42Ch+dLIRgqOZDrf+7I8OnvNSi1rdszmrFjQZ3NenO9NZtNZf9iHQjEIk2w4L87v35d1la0GgwKxUM4K2YOF5F7lEJBNy7bAI6LNV/CGYmGawiL2DfQsX2y1Kyxnv4GW+9GhG5DmGjK0Jqjqtw3ZeM/+U7a7F5uojSuBbBiG2hplUMLUvTYLX0IFBtSODCS+LKxIp7LpBmNt4pU+luV7d2jbwKHWP7CeibqQARhYFu/zeNJRvF5Usaf5UYp8dy/VbNoB3I7fT2TEhgbv3UQEQpVkGZh1KLd8OQxS8DVyxpeduFaYE/qUNapu8jxgMRyJQOV9NjnTjmVrDz3mxhW3EHIaxDW9ysScu69vQi54EoM8lQD8ryRmDsbZEIRCeKZElG6OGZ0Hv3KEd7kvEWbG1unWQfkhrqfF0xoxk3NkpAH0JIiXexEI4fE04EYRNIJS0gXIWe35Rc60NFK3ml3IGle0q7PpE8rRbUOdlQbfh7IopKfdexcqjQFPI3NzWuRDKtxIpbUBG0dLW6Yx25IVcntakA95JvHvp+hq5qHFdQDFtSVmpdVDhzmjCu5NHMFDclBWUdB4/7+ntOVWE2VtaZJ0L0buBiUWB51sUzLXFXxqOt2fafcD16vY1Ku+aWpwRHCgXes/Dr9WK9cw5l3QGORQP8P32YF+hik1N4izaYn4biJYRL4yXyeGCRGNbQXPVMisnB0ASoWHrFXyYjA+7Uq/BmwPdxqR3FEyPUXH96flQOD2YFofT7387nDz7YXiW7mRfZd9k32ZF9iR7lv2cvchOMpH9kf2Z/ZX9Pbo7ejz6bvR0C715Y4j5Its5ox/BSYr19Q=</latexit> Another perspective: exchangeability T = 0 T = 1 Group A Group B Group A E [ Y | T = 1] = y 1 E [ Y | T = 0] = y 0 18 / 41 Brady Neal
<latexit sha1_base64="hsz8RjcljbDBaGAYNVc09xwW9nw=">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</latexit> <latexit sha1_base64="+csBynRxFr6koeweOUGm1ZEASRU=">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</latexit> <latexit sha1_base64="u9JVBphx7MmuBVj2PHmsIsdN7Y=">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</latexit> <latexit sha1_base64="B6SDsM+n6ge+raWORAaVSuAj6XE=">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</latexit> Another perspective: exchangeability T = 0 T = 1 Group B Group A Group A E [ Y | T = 1] = y 1 E [ Y | T = 0] = y 0 18 / 41 Brady Neal
<latexit sha1_base64="hsz8RjcljbDBaGAYNVc09xwW9nw=">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</latexit> <latexit sha1_base64="+csBynRxFr6koeweOUGm1ZEASRU=">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</latexit> <latexit sha1_base64="u9JVBphx7MmuBVj2PHmsIsdN7Y=">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</latexit> <latexit sha1_base64="B6SDsM+n6ge+raWORAaVSuAj6XE=">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</latexit> Another perspective: exchangeability T = 0 T = 1 Group B Group A Group A E [ Y | T = 1] = y 1 E [ Y | T = 0] = y 0 18 / 41 Brady Neal
<latexit sha1_base64="u9JVBphx7MmuBVj2PHmsIsdN7Y=">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</latexit> <latexit sha1_base64="B6SDsM+n6ge+raWORAaVSuAj6XE=">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</latexit> <latexit sha1_base64="hsz8RjcljbDBaGAYNVc09xwW9nw=">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</latexit> <latexit sha1_base64="+csBynRxFr6koeweOUGm1ZEASRU=">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</latexit> <latexit sha1_base64="gTr4smGTrhcKjcTg6MmgOZTfxMo=">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</latexit> Another perspective: exchangeability T = 0 T = 1 Group B Group A Group A E [ Y | T = 1] = y 1 E [ Y | T = 0] = y 0 Before switch After switch E [ Y (1) | T = 1] = E [ Y (1) | T = 0] 18 / 41 Brady Neal
<latexit sha1_base64="hsz8RjcljbDBaGAYNVc09xwW9nw=">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</latexit> <latexit sha1_base64="+csBynRxFr6koeweOUGm1ZEASRU=">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</latexit> <latexit sha1_base64="gTr4smGTrhcKjcTg6MmgOZTfxMo=">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</latexit> <latexit sha1_base64="u9JVBphx7MmuBVj2PHmsIsdN7Y=">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</latexit> <latexit sha1_base64="gTr4smGTrhcKjcTg6MmgOZTfxMo=">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</latexit> <latexit sha1_base64="B6SDsM+n6ge+raWORAaVSuAj6XE=">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</latexit> Another perspective: exchangeability T = 0 T = 1 Group B Group A Group A E [ Y | T = 1] = y 1 E [ Y | T = 0] = y 0 Before switch After switch E [ Y (1) | T = 1] = E [ Y (1) | T = 0] 0] = E [ Y (1)] 18 / 41 Brady Neal
<latexit sha1_base64="hsz8RjcljbDBaGAYNVc09xwW9nw=">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</latexit> <latexit sha1_base64="+csBynRxFr6koeweOUGm1ZEASRU=">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</latexit> <latexit sha1_base64="gTr4smGTrhcKjcTg6MmgOZTfxMo=">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</latexit> <latexit sha1_base64="u9JVBphx7MmuBVj2PHmsIsdN7Y=">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</latexit> <latexit sha1_base64="gTr4smGTrhcKjcTg6MmgOZTfxMo=">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</latexit> <latexit sha1_base64="B6SDsM+n6ge+raWORAaVSuAj6XE=">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</latexit> <latexit sha1_base64="WFjqyGlwAO5sBmjhLTYUe7gk=">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</latexit> <latexit sha1_base64="WFjqyGlwAO5sBmjhLTYUe7gk=">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</latexit> Another perspective: exchangeability T = 0 T = 1 Group B Group A Group A E [ Y | T = 1] = y 1 E [ Y | T = 0] = y 0 Before switch After switch E [ Y (1) | T = 1] = E [ Y (1) | T = 0] 0] = E [ Y (1)] E [ Y (0) | T = 0] = E [ Y (0) | T = 1] 1] = E [ Y (0)] 18 / 41 Brady Neal
<latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> Aside: identifiability E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] 19 / 41 Brady Neal
<latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> Aside: identifiability E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Causal quantities 19 / 41 Brady Neal
<latexit sha1_base64="UoUCDRHX5dtoKTKNpV9W/EGob3o=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">AF83icfVRfbxtFED+XGoqBksIDSLwsuEFuhfFKkgYalSFYQD0VK0lY5K9rbm7NX3n/anSN1T/fC1+AN8coH4hvwMZg9X5tcWrGWrdmZ3/zmt7PjLZySAefzf0a3rk9fve9O+9Pvjwo7sf79375CzY2gs4FVZ/6zgAZQ0cIoSFTxzHrguFDwtNo9j/Olv4IO05gS3Dpar4yspOBIrou93/Pj8+cPsm+W7IB15pzMrxesd7Ncy5KdsAXLriGuvPMly5GvGrky1vNCKonbluX5hNHqad7CMSC42JvOZ/NusTeNrDemSb+eXNy7/Twvrag1GBSKh3CezR0uG+5RCgXtJK8DOC42fAXnZBquISybrlkt2ydPySr6WuQd7rGQ3XIWx1QUjNcR1uxqLzbHzGqvlo0rkYwYleoqhVDy2LnWSk9CFRbMrjwkrQyseaeC6T7mQzKFLod7q3dIC9Cy9g+5GkGymAkUfBUN+LiviGqOiJd0l9tmJ3LxkPXiYuevOwIUEbl9nBUCUZhWYdSi1fNk3UfA6cMVWnrt1mBH45zrErtgeMBSeVaBirpseOnEoWnvtE9bcQUhLEDQ7ncSUe28vQyq4Er0904A8rSmzgYZUSCdEYiomu6STv4hSO8SHmNjW2jL0OyA2dfpExoRk3JYtGkBLEiQ2aZdHIRvZwG3iqANKCVdgJSVnl+mTEsjda3ZpSxHWd19pA42l2qs9Lg61TWCOlp9l6lSmPAU8vcgb5iApXUqmdNEFTR4MbFk3bUxVyNxpQ9nyLpjtPf6qShzWUPRbUtZpXVQ4dpowr+hsuFRkB2UdJU26+3tMV2I1VdaOkH3buCy3zT5cdvkcZiLojlu2HsjPs+6nUTNzfi0pTgCOHAO5Z/GT+s29zAmVdAY5FA/w/fsQL9GCRqb5F604lwXNEQwpXzCnSq4gKS6iuR6bZ9LA/BKgm94q75n503m9zvQFvDnXdxH1LyfQYZTefnjeNs8NZdjT7/tej6aMf+mfpTvJF8lXyIMmSh8mj5KfkSXKaiOTf0d3RZ6Px/X4j/Gf47920FujPufTZLDGf/8HwkH2Ow=</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> Aside: identifiability E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Causal quantities (accessible, since we Statistical quantities have ) P ( x, t, y ) 19 / 41 Brady Neal
<latexit sha1_base64="5auZG6jeoPm4ozoeNTO2B0vpb7k=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="GpiC0li5hK0qCF5QjyXzo8lM54=">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</latexit> <latexit sha1_base64="UoUCDRHX5dtoKTKNpV9W/EGob3o=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> <latexit sha1_base64="o5h8fSm3fvImYsQ6hSHyW6nG6A=">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</latexit> Aside: identifiability E [ Y (1)] − E [ Y (0)] = E [ Y (1) | T = 1] − E [ Y (0) | T = 0] | | (ignorability) − − = E [ Y | T = 1] − E [ Y | T = 0] Causal quantities (accessible, since we Statistical quantities have ) P ( x, t, y ) A causal quantity (e.g. ) is identifiable if we can E [ Y ( t )] compute it from a purely statistical quantity (e.g. ) E [ Y | t ] 19 / 41 Brady Neal
<latexit sha1_base64="5Vsktr4ZwYPyL47pIvpl3RGjrY=">AFcHicfVTbjhNHEB3IOoBzAZIXJB5oYlaKosHYq5VIHiwhcVEU5QGk3YVovUI1PTV2y31Td0WM5ov4BU+jt/IF6R6PLDrJUpbHtXlVNWp6lIXqtIk8nHS5e/2hl8feXqteE373/fUbN384iq4OEg+l0y68KiCiVhYPSZHGVz4gmELjy2L1OPlf/o0hKmcPaO3xMDCqkpJIDa9OHh9YzQZT7ojvhSmvTDK+vP89c2dv+alk7VBS1JDjMfTiaeTBgIpqbEdzuIHuQKFnjMogWD8aTpmLZily2lqFzgvyXRWc9HNGBiXJuCkQZoGS/6kvG/fMc1Vb+eNMr6mtDKTaGq1oKcSG2LUgWUpNcsgAyKuQq5hACSeDjDrTKFabd151YERWyF2BXPmLpVEgVbNG7ze1Nxvm1UsqRBc4ldcaBWb0UP3o7cTGfLRAxuP0dFJFJ2EYXzpIx62w9RQh1Bi0UAv4xjBv9RxzRVv7vIRKzXKrIJQN1uVNOrYoAYd3EJXiMeYnShW4XYg4huNOYS9Cyl8cGCfJKUe5dVAnFJhnSsTpmqHgc/9PIHyTQ0ut65Ms4ElrufTYU0AmwpkpBHNIoJyVXexTGI8ME40loztEGtlY+YizLAaS6MsrURpyqkpZixjv5kHO0m1DvlKXPoaKRKvDufQpV1mLgkfkZL/I+F6U1htqkreOFzfOmrZPVajNamDZ5s1XT9VyXEJZY9FvW5mdV9jzlArRa2JnGiuWonegYXd/j/lKnOHKxvAk+N4tnvZKM3/aNvO0zEXRPG3bd8RhN4bTJOUC35lS/SM8Bi8mN9NP9EpF3D2E9A6YtD/wzdZkT+WOHVwxLPpSHjQvIR4ZjxDHvQsEsMSq/Oe0XS01zeBupkHDb65l4z32rlZYbB7pm6S3nIwP0bTi0/Pl8LR3ni6P/7txf7o0ZP+Wbqa3c5+yn7OptnD7FH2e/Y8O8xkhtm7H32Yefwa3BncHdDfTypT7mx2zrDH75F+DE0hw=</latexit> Randomized control trial (RCT) T 20 / 41 Brady Neal
<latexit sha1_base64="5Vsktr4ZwYPyL47pIvpl3RGjrY=">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</latexit> <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> Randomized control trial (RCT) T Went to sleep with shoes on Went to sleep without shoes on ( T = 1) ( T = 0) drunk sober sober sober drunk sober drunk drunk sober sober drunk sober drunk drunk sober sober drunk sober sober sober drunk drunk sober drunk sober drunk sober sober 20 / 41 Brady Neal
<latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> <latexit sha1_base64="5Vsktr4ZwYPyL47pIvpl3RGjrY=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> Randomized control trial (RCT) T Slept with shoes on Went to sleep without shoes on ( T = 1) ( T = 0) drunk sober sober sober drunk sober drunk drunk sober sober drunk sober drunk drunk sober sober drunk sober sober sober drunk drunk sober drunk sober drunk sober sober 20 / 41 Brady Neal
<latexit sha1_base64="5Vsktr4ZwYPyL47pIvpl3RGjrY=">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</latexit> <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> Randomized control trial (RCT) T Slept with shoes on Slept without shoes on ( T = 1) ( T = 0) drunk sober sober sober drunk sober drunk drunk sober sober drunk sober drunk drunk sober sober drunk sober sober sober drunk drunk sober drunk sober drunk sober sober 20 / 41 Brady Neal
<latexit sha1_base64="5Vsktr4ZwYPyL47pIvpl3RGjrY=">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</latexit> <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> Randomized control trial (RCT) T Slept with shoes on Slept without shoes on ( T = 1) ( T = 0) drunk drunk sober sober sober sober sober sober drunk drunk drunk drunk drunk drunk sober sober sober sober drunk drunk 20 / 41 Brady Neal
Graphical interpretation of RCT 21 / 41 Brady Neal
<latexit sha1_base64="uGwb7/40FtxyqwMkvjDesmjJa3g=">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</latexit> Graphical interpretation of RCT X T Y 21 / 41 Brady Neal
<latexit sha1_base64="5Vsktr4ZwYPyL47pIvpl3RGjrY=">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</latexit> <latexit sha1_base64="uGwb7/40FtxyqwMkvjDesmjJa3g=">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</latexit> Graphical interpretation of RCT T X T Y 21 / 41 Brady Neal
<latexit sha1_base64="5Vsktr4ZwYPyL47pIvpl3RGjrY=">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</latexit> <latexit sha1_base64="wt2lCE4iu6jp6hQfvYVw6vDRF60=">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</latexit> Graphical interpretation of RCT T X T Y 21 / 41 Brady Neal
Question: What important property does an RCT give us?
<latexit sha1_base64="vdhnOVYCs5M1CLTr9/rvuY12bRc=">AFhnicfVRtbxNHED4oLsG0JVDxqV+2NZES6TB2FJQiYSkSAqGqH6iUBKI4ivb25uyV9027cw1mdT+mX9tfxL9h9nyQOKCuZWt25pmZ58db+GUDgafbx87tbve9vb9zp3/3hx5/ubd5/cBxs7QUcCausf1fwAEoaOEKJCt45D1wXCt4Wixcp/vZv8EFac4hLB2eaz4yspOBIrvPNh9sn2+OdnJ1sj3Z2FSaEhw7PN8cjIajdrGvjXFnDLJuvTm/f+tkWlpRazAoFA/hdDxyeBa5RykUNP1pHcBxseAzOCXTcA3hLb8G7ZFnpJV1tPXIGu9VzMi1yEsdUFIzXEerseS81ux0xqr38+iNK5GMGLVqKoVQ8uSGKyUHgSqJRlceElcmZhzwWSZP21NoVu1vfWLpAXoWFsi70i6kYKYORsM7vfUX1lHJk+SnFlvsUC4+sA68nrlSZ82FBG6+ZAVAlGYWmHUotfzQiSh4HbhiM8/dPAwJ/Ecdkqpu+djxgMRyLgO19NjWTjWVLDz3yxjm3EHISxDWtxMScu69vQi54Ep09lAD8rySmDsbZEIRCeKZClG52Ge0Hv/JEd7nvEabG1smrQNyQ6efjJnQjJuSJSMPoCUREou8zSMQwpNhwKUiaASlpAuQs9Lzi5xpaSuNbuQJc7ZhGZyn2o0q1RnpcEvqSwK6Wn2PqdKY8CTZG5Cg7xHjSup1IqaoKmjwQ2T2HSlCrkaDSi7epPYnqc7VcnDHMoOC+pKz8suw5zxpWcmYmCiuygrKOkfnt/L+hKrKbOWpMSdO8GLrpNnL5s4jQNc1HEl02zHjvmvot6HdPmWrz9BxPCgXds+mv6sHZzDWc+A41FAv0/fFUV6McglfYWSZuWhOKhAunZfIw45FYlhCdTUyGA92u0OAilOvuIuPkvNRM9UL8GZX1zHtG0qmx2h8/en52jeHY73hs/+2hscPO+epY3sl+y3bDsbZ/vZQfY6e5MdZSKL2T/Zv9l/vY3esPe0t7+C3rzR5fycra3ewScpENcM</latexit> Conditional exchangeability Exchangeability: ( Y (1) , Y (0)) ⊥ ⊥ T 23 / 41 Brady Neal
<latexit sha1_base64="vdhnOVYCs5M1CLTr9/rvuY12bRc=">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</latexit> <latexit sha1_base64="DuN46LKiETzH5PvG/Zwp4JQMaEA=">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</latexit> Conditional exchangeability Exchangeability: X ( Y (1) , Y (0)) ⊥ ⊥ T T Y 23 / 41 Brady Neal
<latexit sha1_base64="DuN46LKiETzH5PvG/Zwp4JQMaEA=">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</latexit> <latexit sha1_base64="nx+h6pLGXoiKdS8RQWDjcXaAg8c=">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</latexit> <latexit sha1_base64="vdhnOVYCs5M1CLTr9/rvuY12bRc=">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</latexit> Conditional exchangeability Exchangeability: X ( Y (1) , Y (0)) ⊥ ⊥ T T Y X T Y 23 / 41 Brady Neal
<latexit sha1_base64="0tMjNEcGwa0vylA0lax0f94I2P0=">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</latexit> <latexit sha1_base64="DuN46LKiETzH5PvG/Zwp4JQMaEA=">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</latexit> <latexit sha1_base64="nx+h6pLGXoiKdS8RQWDjcXaAg8c=">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</latexit> <latexit sha1_base64="vdhnOVYCs5M1CLTr9/rvuY12bRc=">AFhnicfVRtbxNHED4oLsG0JVDxqV+2NZES6TB2FJQiYSkSAqGqH6iUBKI4ivb25uyV9027cw1mdT+mX9tfxL9h9nyQOKCuZWt25pmZ58db+GUDgafbx87tbve9vb9zp3/3hx5/ubd5/cBxs7QUcCausf1fwAEoaOEKJCt45D1wXCt4Wixcp/vZv8EFac4hLB2eaz4yspOBIrvPNh9sn2+OdnJ1sj3Z2FSaEhw7PN8cjIajdrGvjXFnDLJuvTm/f+tkWlpRazAoFA/hdDxyeBa5RykUNP1pHcBxseAzOCXTcA3hLb8G7ZFnpJV1tPXIGu9VzMi1yEsdUFIzXEerseS81ux0xqr38+iNK5GMGLVqKoVQ8uSGKyUHgSqJRlceElcmZhzwWSZP21NoVu1vfWLpAXoWFsi70i6kYKYORsM7vfUX1lHJk+SnFlvsUC4+sA68nrlSZ82FBG6+ZAVAlGYWmHUotfzQiSh4HbhiM8/dPAwJ/Ecdkqpu+djxgMRyLgO19NjWTjWVLDz3yxjm3EHISxDWtxMScu69vQi54Ep09lAD8rySmDsbZEIRCeKZClG52Ge0Hv/JEd7nvEabG1smrQNyQ6efjJnQjJuSJSMPoCUREou8zSMQwpNhwKUiaASlpAuQs9Lzi5xpaSuNbuQJc7ZhGZyn2o0q1RnpcEvqSwK6Wn2PqdKY8CTZG5Cg7xHjSup1IqaoKmjwQ2T2HSlCrkaDSi7epPYnqc7VcnDHMoOC+pKz8suw5zxpWcmYmCiuygrKOkfnt/L+hKrKbOWpMSdO8GLrpNnL5s4jQNc1HEl02zHjvmvot6HdPmWrz9BxPCgXds+mv6sHZzDWc+A41FAv0/fFUV6McglfYWSZuWhOKhAunZfIw45FYlhCdTUyGA92u0OAilOvuIuPkvNRM9UL8GZX1zHtG0qmx2h8/en52jeHY73hs/+2hscPO+epY3sl+y3bDsbZ/vZQfY6e5MdZSKL2T/Zv9l/vY3esPe0t7+C3rzR5fycra3ewScpENcM</latexit> Conditional exchangeability Exchangeability: X ( Y (1) , Y (0)) ⊥ ⊥ T T Y Conditional exchangeability: X ( Y (1) , Y (0)) ⊥ ⊥ T | X T Y 23 / 41 Brady Neal
<latexit sha1_base64="DuN46LKiETzH5PvG/Zwp4JQMaEA=">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</latexit> <latexit sha1_base64="0tMjNEcGwa0vylA0lax0f94I2P0=">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</latexit> <latexit sha1_base64="vdhnOVYCs5M1CLTr9/rvuY12bRc=">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</latexit> <latexit sha1_base64="cNh0Io/0qaIFOZ30xwVEGc7NFQg=">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</latexit> Conditional exchangeability Exchangeability: X ( Y (1) , Y (0)) ⊥ ⊥ T T Y Conditional exchangeability: X ( Y (1) , Y (0)) ⊥ ⊥ T | X T Y 23 / 41 Brady Neal
<latexit sha1_base64="0tMjNEcGwa0vylA0lax0f94I2P0=">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</latexit> Identification of conditional average treatment effect Conditional exchangeability: ( Y (1) , Y (0)) ⊥ ⊥ T | X 24 / 41 Brady Neal
<latexit sha1_base64="0tMjNEcGwa0vylA0lax0f94I2P0=">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</latexit> <latexit sha1_base64="LhuXP2DgGQYEdijlu1KsvMEUwFw=">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</latexit> Identification of conditional average treatment effect Conditional exchangeability: ( Y (1) , Y (0)) ⊥ ⊥ T | X E [ Y (1) − Y (0) | X ] = E [ Y (1) | X ] − E [ Y (0) | X ] 24 / 41 Brady Neal
<latexit sha1_base64="LhuXP2DgGQYEdijlu1KsvMEUwFw=">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</latexit> <latexit sha1_base64="LhuXP2DgGQYEdijlu1KsvMEUwFw=">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</latexit> <latexit sha1_base64="0tMjNEcGwa0vylA0lax0f94I2P0=">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</latexit> Identification of conditional average treatment effect Conditional exchangeability: ( Y (1) , Y (0)) ⊥ ⊥ T | X E [ Y (1) − Y (0) | X ] = E [ Y (1) | X ] − E [ Y (0) | X ] | | | − − = E [ Y (1) | T = 1 , X ] − E [ Y (0) | T = 0 , X ] 24 / 41 Brady Neal
<latexit sha1_base64="LhuXP2DgGQYEdijlu1KsvMEUwFw=">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</latexit> <latexit sha1_base64="LhuXP2DgGQYEdijlu1KsvMEUwFw=">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</latexit> <latexit sha1_base64="LhuXP2DgGQYEdijlu1KsvMEUwFw=">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</latexit> <latexit sha1_base64="0tMjNEcGwa0vylA0lax0f94I2P0=">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</latexit> Identification of conditional average treatment effect Conditional exchangeability: ( Y (1) , Y (0)) ⊥ ⊥ T | X E [ Y (1) − Y (0) | X ] = E [ Y (1) | X ] − E [ Y (0) | X ] | | | − − = E [ Y (1) | T = 1 , X ] − E [ Y (0) | T = 0 , X ] | | − = E [ Y | T = 1 , X ] − E [ Y | T = 0 , X ] 24 / 41 Brady Neal
<latexit sha1_base64="LhuXP2DgGQYEdijlu1KsvMEUwFw=">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</latexit> <latexit sha1_base64="0tMjNEcGwa0vylA0lax0f94I2P0=">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</latexit> <latexit sha1_base64="IEVzZeIvoVhRVqcfD9j+mZoxK7U=">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</latexit> <latexit sha1_base64="LhuXP2DgGQYEdijlu1KsvMEUwFw=">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</latexit> <latexit sha1_base64="LhuXP2DgGQYEdijlu1KsvMEUwFw=">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</latexit> Identification of conditional average treatment effect Conditional exchangeability: ( Y (1) , Y (0)) ⊥ ⊥ T | X E [ Y (1) − Y (0) | X ] = E [ Y (1) | X ] − E [ Y (0) | X ] | | | − − = E [ Y (1) | T = 1 , X ] − E [ Y (0) | T = 0 , X ] | | − = E [ Y | T = 1 , X ] − E [ Y | T = 0 , X ] What about the ATE? E [ Y (1) − Y (0)] 24 / 41 Brady Neal
<latexit sha1_base64="QBQ/SlmBKmhrRqHJ9awVHyCWcFA=">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</latexit> The Adjustment Formula (identification of ATE) E [ Y (1) − Y (0)] = E X E [ Y (1) − Y (0) | X ] = E X [ E [ Y | T = 1 , X ] − E [ Y | T = 0 , X ]] 25 / 41 Brady Neal
<latexit sha1_base64="nx+h6pLGXoiKdS8RQWDjcXaAg8c=">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</latexit> <latexit sha1_base64="QBQ/SlmBKmhrRqHJ9awVHyCWcFA=">AF7XicfVRLbyNFEB4vawjmscly5FLgDcpKE2NHkXb3YCnSKghDouUZBN5rKinp2y3C9195D1DvMzuCGu/Cau/BKqx5OHvYi2bNXjq6qvq8qdWyl8GA7/7jz6H34092Pu19vkXz7Z3Xt64U3pOJ5zI427zJlHKTSeBxEkXlqHTOUS3+bL19H/9ld0Xh9FlYWp4rNtZgJzgKZrnd/y04nVwej53AIVwfD51P4bgzZ6fUlbNohU6KAylkWQ/o3KEkzsKkAa8hZzCGUQpZGsGHW47hrSNzYr4I0+vd/nAwbA58KIxaoZ+058313uOrDC8VKgDl8z7yWhow7RiLguse5lpUfL+JLNcUKiZgr9tGraVM+WQqYGUdfHaCxPoyomPJ+pXJCKhYWftsXjf/lm5Rh9nJaCW3LgJqvC81KCcFA7DkUwiEPckUC404QV+AL5hgPNJneRplc1Zu6McvAcl8D7MPRF0LjkAWiZv83s0o3yYqWuKUqcQ+nInle2jBm5Hr7myYAoHruyiPIQg92BsEq8b5vIWemZhLljduEHBP6p9LGrdnVomQ/EciE8lXShyR1zSpE75laVXzCLPi2QG9csok+Zc+bGp5xJ3soDhYGlMxFSa7yIKCJBPGMiSlc1i3j4Mwv4LmVlMKk2Rey1D0zT7cj4AqYLiAKqUcliBfpk0cgQJ+P/BhJQlaoZTCekyhcOwmBSW0UKWCG1GERVzcwQvKUa9DrRE63IVCxYWj3bsNFVqjo5bZMS3yMRWeCSnX1DhtHS2uH1d1myoX69XAos03rpr7tLcqmF9g0WJRPqh5X+XIhSYFHM9jv/FLw0loJ6zfxe0iMospKUSdo7hpvWqXKTusqi8uc59VpXW/6LphrvU5VUdnyC12gJYRFZyH7Jn6gUbZw+haoTSDQ/8PXWZF+dKDUzgTqTUPCMklLiPfGe+RZyIyLHD20NMf9Y/aS6CsMieZrZ5F47M6U0t0+kiVdRrCqbHaLT9HwoXBwNRseDV78c909O2mdpJ/k6+TY5SEbJi+Qk+TF5k5wnPmns9PZ6ztmu7v3T+6f6hjzptzFfJxun+9S/8uvL0</latexit> The Adjustment Formula (identification of ATE) E [ Y (1) − Y (0)] = E X E [ Y (1) − Y (0) | X ] = E X [ E [ Y | T = 1 , X ] − E [ Y | T = 0 , X ]] X T Y 25 / 41 Brady Neal
<latexit sha1_base64="nx+h6pLGXoiKdS8RQWDjcXaAg8c=">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</latexit> <latexit sha1_base64="QBQ/SlmBKmhrRqHJ9awVHyCWcFA=">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</latexit> The Adjustment Formula (identification of ATE) E [ Y (1) − Y (0)] = E X E [ Y (1) − Y (0) | X ] = E X [ E [ Y | T = 1 , X ] − E [ Y | T = 0 , X ]] X T Y 25 / 41 Brady Neal
<latexit sha1_base64="cNh0Io/0qaIFOZ30xwVEGc7NFQg=">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</latexit> <latexit sha1_base64="QBQ/SlmBKmhrRqHJ9awVHyCWcFA=">AF7XicfVRLbyNFEB4vawjmscly5FLgDcpKE2NHkXb3YCnSKghDouUZBN5rKinp2y3C9195D1DvMzuCGu/Cau/BKqx5OHvYi2bNXjq6qvq8qdWyl8GA7/7jz6H34092Pu19vkXz7Z3Xt64U3pOJ5zI427zJlHKTSeBxEkXlqHTOUS3+bL19H/9ld0Xh9FlYWp4rNtZgJzgKZrnd/y04nVwej53AIVwfD51P4bgzZ6fUlbNohU6KAylkWQ/o3KEkzsKkAa8hZzCGUQpZGsGHW47hrSNzYr4I0+vd/nAwbA58KIxaoZ+058313uOrDC8VKgDl8z7yWhow7RiLguse5lpUfL+JLNcUKiZgr9tGraVM+WQqYGUdfHaCxPoyomPJ+pXJCKhYWftsXjf/lm5Rh9nJaCW3LgJqvC81KCcFA7DkUwiEPckUC404QV+AL5hgPNJneRplc1Zu6McvAcl8D7MPRF0LjkAWiZv83s0o3yYqWuKUqcQ+nInle2jBm5Hr7myYAoHruyiPIQg92BsEq8b5vIWemZhLljduEHBP6p9LGrdnVomQ/EciE8lXShyR1zSpE75laVXzCLPi2QG9csok+Zc+bGp5xJ3soDhYGlMxFSa7yIKCJBPGMiSlc1i3j4Mwv4LmVlMKk2Rey1D0zT7cj4AqYLiAKqUcliBfpk0cgQJ+P/BhJQlaoZTCekyhcOwmBSW0UKWCG1GERVzcwQvKUa9DrRE63IVCxYWj3bsNFVqjo5bZMS3yMRWeCSnX1DhtHS2uH1d1myoX69XAos03rpr7tLcqmF9g0WJRPqh5X+XIhSYFHM9jv/FLw0loJ6zfxe0iMospKUSdo7hpvWqXKTusqi8uc59VpXW/6LphrvU5VUdnyC12gJYRFZyH7Jn6gUbZw+haoTSDQ/8PXWZF+dKDUzgTqTUPCMklLiPfGe+RZyIyLHD20NMf9Y/aS6CsMieZrZ5F47M6U0t0+kiVdRrCqbHaLT9HwoXBwNRseDV78c909O2mdpJ/k6+TY5SEbJi+Qk+TF5k5wnPmns9PZ6ztmu7v3T+6f6hjzptzFfJxun+9S/8uvL0</latexit> The Adjustment Formula (identification of ATE) E [ Y (1) − Y (0)] = E X E [ Y (1) − Y (0) | X ] = E X [ E [ Y | T = 1 , X ] − E [ Y | T = 0 , X ]] X T Y 25 / 41 Brady Neal
Recommend
More recommend