Difference-in-Differences
Brady Nealcausalcourse.com

Motivation and Preliminaries
Difference-in-Differences Overview
Assumptions and Proof
Problems with Difference-in-Differences

Brady Neal

2 / 25

Motivation and Preliminaries
Difference-in-Differences Overview
Assumptions and Proof
Problems with Difference-in-Differences

Brady Neal

Motivation and Preliminaries

3 / 25

Motivation
Context: Treatment and control group both before and after treatment administered
Advantage: Use time dimension to help with identification
Treatment
administeredoutcome

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Brady Neal

Motivation and Preliminaries

4 / 25

Example from Card & Krueger (1994)
Minimum wage
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employment

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time
Brady Neal

Motivation and Preliminaries

5 / 25Average Treatment Effect on the Treated (ATT)

Brady Neal

Motivation and Preliminaries

6 / 25Average Treatment Effect on the Treated (ATT)
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E[Y (1)

Brady Neal

Y (0)] = E[Y | T = 1]

E[Y | T = 0]

Motivation and Preliminaries

6 / 25Average Treatment Effect on the Treated (ATT)
Unconfoundedness:

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E[Y (1)

Brady Neal

Y (0)] = E[Y | T = 1]

E[Y | T = 0]

Motivation and Preliminaries(Y (0), Y (1)) ?? T<latexit sha1_base64="D5f7B1rjye1XD6q4mhbO9Bf8J+Y=">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</latexit>

6 / 25Average Treatment Effect on the Treated (ATT)
Unconfoundedness:

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E[Y (1)

Brady Neal

Y (0)] = E[Y | T = 1]

E[Y | T = 0]

Motivation and Preliminaries(Y (0), Y (1)) ?? T<latexit sha1_base64="D5f7B1rjye1XD6q4mhbO9Bf8J+Y=">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</latexit>

6 / 25Average Treatment Effect on the Treated (ATT)
Unconfoundedness:

ATE:<latexit sha1_base64="vFl/7vZyIfYMc4akmWOEzBEsI7E=">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</latexit><latexit sha1_base64="vFl/7vZyIfYMc4akmWOEzBEsI7E=">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</latexit>

E[Y (1)

Y (0)] = E[Y | T = 1]

E[Y | T = 0]

(Y (0), Y (1)) ?? T<latexit sha1_base64="D5f7B1rjye1XD6q4mhbO9Bf8J+Y=">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</latexit>

ATT:
E[Y (1)<latexit sha1_base64="9/ClvRHvi/8lFMggdK7KC4h2YSM=">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</latexit>

Y (0) | T = 1] = E[Y (1) | T = 1]
= E[Y | T = 1]

E[Y (0) | T = 1]

= E[Y | T = 1]

E[Y | T = 0]

= E[Y | T = 1]
Brady NealE[Y (0) | T = 1]

E[Y (0) | T = 0]

Motivation and Preliminaries

6 / 25Average Treatment Effect on the Treated (ATT)
Unconfoundedness:

ATE:<latexit sha1_base64="vFl/7vZyIfYMc4akmWOEzBEsI7E=">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</latexit><latexit sha1_base64="vFl/7vZyIfYMc4akmWOEzBEsI7E=">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</latexit>

E[Y (1)

Y (0)] = E[Y | T = 1]

E[Y | T = 0]

(Y (0), Y (1)) ?? T<latexit sha1_base64="D5f7B1rjye1XD6q4mhbO9Bf8J+Y=">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</latexit>

ATT:
E[Y (1)<latexit sha1_base64="9/ClvRHvi/8lFMggdK7KC4h2YSM=">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</latexit>

Y (0) | T = 1] = E[Y (1) | T = 1]
= E[Y | T = 1]

E[Y (0) | T = 1]

= E[Y | T = 1]

E[Y | T = 0]

= E[Y | T = 1]
Brady NealE[Y (0) | T = 1]

E[Y (0) | T = 0]

Motivation and Preliminaries

6 / 25Average Treatment Effect on the Treated (ATT)
Unconfoundedness:

ATE:<latexit sha1_base64="vFl/7vZyIfYMc4akmWOEzBEsI7E=">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</latexit><latexit sha1_base64="vFl/7vZyIfYMc4akmWOEzBEsI7E=">AAAUu3icrVjbbttGEN2kt8S9Oanf+sJGKeAAsiq5KZwXAQFsp0XRAA7ga2PDkCjKZs1bScquI+iT2p/pU4H2X3rm7FIUdaMUWILI1ezMmdm5LZftyHOTtF7/5979Dz786ONPHjxc+fSzz7/4cvXR48Mk7MW2c2CHXhgft1uJ47mBc5C6qeccR7HT8tuec9S+2pb5o2snTtww2E9vI+fMb10Ebte1WylI56s/nu6+PVlvPLM2rJP1+rMzq2kJxTr13Y51mgIq9Z0gBblxBp4Zc/Wz89VKvVbnx5ocNMygosxnL3z08C91qjoqVLbqKV85KlApxp5qqQTft6qh6ioC7Uz1QYsxcjnvqIFagWwPXA44WqBe4XqBf28NNcB/wUwobUOLh18MSUt9a3g6GHdJ1XfRb43wztLRJ7bYeIt722D6oKbqEtQyuYxzcbk2vn7JqlOs4QVX62IlESniB7uw5i7uHv6nWKFcb8HpYNSBVIyRDZoHqqaIjhh37XnxzSUj0SKfg5FYPc/ueVbLfIjvFbBaGCe0VGy11CsTl4CaHdoqPB5jOhvxD6xQ2zcPqztcg8u461UI7z4oV+odRkXkeTpHs2s2V2qQB1N0CX9KjgDcCSghM9+F/1xwFHPVBqboaDGSF1xLxAyqGeSfOZ9lc4R4btCehFGzmG8u9USmsnK7Mzs93NvEjiHfx9wl9YgvqrDHIXbM7M28WCV3jH83/GfTRnuMXmNlSjyrWJNkVxW4IWbcIZb2hPZnZpG2rg+aZb4b6hfqdxD3Kr0ieV2FbEgLdRYn1BWY2DfRV8QvPq5CFc9mlCp1+LTjktlyBVquTyOJvu+wioT14xnUPu4evRYRpUrt4p8bjn2uSSLao+4bjDvUI9IWel1NbRk7BgWt4huX3WlSq8X8k9rVnXFcq8gFrFTLZGWTmurquVmxREDsHvWabbqh7skJVzcYs6pN+/OuITlatE+k8vgUY9VhNl5SqogrXpy+zmlr2WSdVBlL8f0FOJqU6hp6wtqMjKaVkfrbNlUSMh42777JCV3vou1mbKYPum885YFP17nsSTLzOzBbzKlTem90fh7irpHPdgPp+H1S58sdsrKKsjFG/eHMfHmXI4f+0SOJuMaL6OuIa/lm+LNGZhbFvhu8YAJRMjY1SIuid5gb01e+ryroBRVkVdGfEmmhxHw6iTB+OuR8ClzhvaKWABSp8P5wflAagx1QBsZjXc5ldZjb4LLzdch5ajJPc+iKSsc4pKrna5W8nJY3mj5ftjPM96Ksps+XFco1c9M1/SAh1nGJXMo4iIQ/kqX7JVIh9wRN1zafLGBfwL22Z/IiNhYelUj67DF6VaGpytel9qXsgrHxicj8+h4evGb/G/AJaFk/5rLpUt7M5W7fy6e5/M2Sns0l/SX9m0u+K5F0uDe4Q5rIvCmtKeHaK+XS+VS2JzjYTbLnF11tVV7b8KLs2R3TmSyz30hW6N2xyR5QXag/T+47UsmN0jrusfu22Ydiau7MqZ2DO0PLI9hbYH0J8z0/Xy5SX9PkFs+bbcZhWnfMZvQTSP4Ukp0ekpH9ySV+frLo8ynwxuxk2fPgBjNAx/An9MLXeMYaUNP3eL5r4PqqsOstiirnrJ7pl6O4m0Dewn61w31tedzInIY2Cmemoo5daMh+A3O6zLk9cwaYhbSM3unIOus6E2eRcU36afMSXPoEdluCmOfxdLz58Z2OrN+UhDwB5Oewu0AePWMuhpfnzOw3AmXvKWafniPcu7h67OiJ4d1h1nns7A56jD7h66en7Yl3FZK/m6gKyd3+k8dj+or446ffC55ee+DIefQpJqW0po0izjpHR5RJzdNiwjNnfrKuFU7KGm/ctmQkr3yD1zT9yDG7xcr5aqUx/vZtcnC4WWv8UKu/eV55+cK8mXugvlZP1Dr8tKVeojL34Fdb/an+Vv+q/9aaa/bab2ueZr1/z8h8pQqftd7/QaAK1w==</latexit>

E[Y (1)

Y (0)] = E[Y | T = 1]

E[Y | T = 0]

(Y (0), Y (1)) ?? T

WeakerUnconfoundedness:

ATT:
E[Y (1)<latexit sha1_base64="9/ClvRHvi/8lFMggdK7KC4h2YSM=">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</latexit>

Y (0) | T = 1] = E[Y (1) | T = 1]

E[Y (0) | T = 1]

= E[Y | T = 1]

E[Y (0) | T = 1]

= E[Y | T = 1]

E[Y | T = 0]

= E[Y | T = 1]
Brady Neal<latexit sha1_base64="D5f7B1rjye1XD6q4mhbO9Bf8J+Y=">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</latexit>

Y (0) ?? T<latexit sha1_base64="gWO2eWBKWJyi4omMXcdiRNAO2sw=">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</latexit>

E[Y (0) | T = 0]

Motivation and Preliminaries

6 / 25Average Treatment Effect on the Treated (ATT)
Unconfoundedness:

ATE:<latexit sha1_base64="vFl/7vZyIfYMc4akmWOEzBEsI7E=">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</latexit><latexit sha1_base64="vFl/7vZyIfYMc4akmWOEzBEsI7E=">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</latexit>

E[Y (1)

Y (0)] = E[Y | T = 1]

E[Y | T = 0]

(Y (0), Y (1)) ?? T

WeakerUnconfoundedness:

ATT:
E[Y (1)<latexit sha1_base64="9/ClvRHvi/8lFMggdK7KC4h2YSM=">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</latexit><latexit sha1_base64="ma9be3dzNMNuvQupnevxKnxLegM=">AAAUz3icrVjbbttGEN2kt8S9xEn91hc2TgEHkFTJTeG8GAhgOyiKBrABX2sbhkRRNmHeSlJWHUFFX/sz/Yd+Rr+gr+0f9MzZpSjqRjmIBJGr2Zkzs3NbLluR5yZpvf73vfsffPjRx588eLj06Weff/Fo+fGTwyTsxrZzYIdeGB+3monjuYFzkLqp5xxHsdP0W55z1LrekvmjGydO3DDYT28j59xvXgZux7WbKUgXy6dnO6cna43nVtU6Was/t858t22dpYBIfSdIrU2rcY6LcE2fq86aq59fLK/Wa3V+rMlBwwxWlfnsho8f/qnOVFuFylZd5StHBSrF2FNNleB7qhqqriLQzlUftBgjl/OOGqglyHbB5YCjCeo1rpf4d2qoAf4LZkJpG1o8/GJIWuobw9PGuEOqvot+a4R3lo4+scXGW9xbBtMHNVVXoJbJZZyLy7Xw9UtWnWINL7laFyuJSBE/2IU1d3D38D/FCuV6C04HozakYoxs0DxQNUV0xLhrz4tvrhiJJvkcjMTqeXbPs1rmQ3yvgdXEOKGlYqulXpu4BNTs0Fbh8RjT2Yi/YoXavnlYneEaXMZdr0J490G5Vm8xKiLP0zmaXbO5UoM8mKJL+FNyBOBOQAmZ+S7854KjmKs2MEVHk5G85FoiZlDNIP/I+SybI8SzSnsSRs1ivrnUE5nKyu3O7PRwbxE7hnwfc1fUI76owB6H2DGzN/Nihdwx/vX4z6aN9hi9xsqUeFawJsmuCnBDzLhDLO0J7c/MIm1dHzTLfKvqJ+p3EPcKvSJ5XYFsSAt1FifUFZjYb6KviF98XIUqns0oFerwaccVs+UatFyfRhJ932IVCevHM6h93D16LSJKhdrFPz2Ofa5JItql7h7GbeoRaQu9rqY2jB2DglbxjcvuNKnVYv5J7erOOK5V5AJWqmWycpOa6uqFWbFEQOwe9ZptuqHuyQlXNxizqkX7864hOVq0T6Ty+BRj1WY2XlGqiCtenL7OaWtZZ51UGEvx/SU4NinVMfSEtRkZTUsj9bdlqiRkPGzefZMTut5FW29spg+6bzzlgU/XuexJMvMLMJvMqTN6b3R+HuKOkc92A+n4fVLnyx2ysoqyMUb94cx8eZcjh/7RI4m4xovo64hr+Xr4s0ZmFsV+P3jBBKJkbGqQFkVvMzemr3xfraIXrCKriv6USAsl5tNJhPGzIecz4ArvNbUEoEiF94fzg9IYbIMyMB7rcC6rw9wGl52vTc4zk3maQ1dUOsYhVT1fq+TltLzR9Pmy7WG+F2U1fb6sUG6Ym67pBwmxjkvkUsZBJPyRLN0vkQq5J2i6tvlkAfsC7rVdkxexsfCoRNJnj9GrCk1Vvim1L2UXjI1PRObnd/DgDfvfgE9Ad/VjLpveyZu53O07+TSX793Rs7mkf0f/5pJvSyQd7g3ukCYye6U1JVy7pVw6n8r2BAe7Sfb8oqutwmsLXpQ9u206k2X2G8kKvTtusgdUFurPk/uOVHKjtI677L4t9qGYmttzaufgvaHlEewusL6E+Z6fLxepr2lyi+fNFuMwrTtmM/oJJH8KyU4Pycj+5BI/P1n0+RTYMztZ9jxYZQboGP6AXvgGz1gDavoOz3cNXF8Xdr1FUeWc1TX9chR3Hcgb2K+2ua/dHTcyp6Fq4cxU1LEDDdlvYE6XObdnzgCzkO6idzqyzrr2xFlkXJN+2rwClz6B3ZYg5nk8HW9+fKcj6zclIU8A+TnsfSCPnjEXw8tzZvYbgbL3FLNPzxHuHVw9dvTE8G4z6zx2dgc9Rp/w9dPT1sS7CsnfdVSF5G7/6ZMxfUX88dPvJU+vXXDkPPoUk1Ja00YRZ52jI8qk5mkx4ZkzP1nXCidljTduWzKSV77B2zT9yDG7xdLF8mpj/O3b5OBwvdb4vlbfe7H66qV5M/dAfaWeqjX4aUO9QmXuwq+2+kv9o/5V/63srfRWflv5XbPev2dkvlSFz8of/wMg5hJT</latexit>

Y (0) | T = 1] = E[Y (1)| T |=T 1]| T(0)= |0]T = 1]
= 1]E[YE[Y= E[Y | T = 1]

E[Y (0) | T = 1]

= E[Y | T = 1]

E[Y | T = 0]

= E[Y | T = 1]
Brady Neal<latexit sha1_base64="D5f7B1rjye1XD6q4mhbO9Bf8J+Y=">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</latexit>

Y (0) ?? T<latexit sha1_base64="gWO2eWBKWJyi4omMXcdiRNAO2sw=">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</latexit>

E[Y (0) | T = 0]

Motivation and Preliminaries

6 / 25Average Treatment Effect on the Treated (ATT)
Unconfoundedness:

ATE:<latexit sha1_base64="vFl/7vZyIfYMc4akmWOEzBEsI7E=">AAAUu3icrVjbbttGEN2kt8S9Oanf+sJGKeAAsiq5KZwXAQFsp0XRAA7ga2PDkCjKZs1bScquI+iT2p/pU4H2X3rm7FIUdaMUWILI1ezMmdm5LZftyHOTtF7/5979Dz786ONPHjxc+fSzz7/4cvXR48Mk7MW2c2CHXhgft1uJ47mBc5C6qeccR7HT8tuec9S+2pb5o2snTtww2E9vI+fMb10Ebte1WylI56s/nu6+PVlvPLM2rJP1+rMzq2kJxTr13Y51mgIq9Z0gBblxBp4Zc/Wz89VKvVbnx5ocNMygosxnL3z08C91qjoqVLbqKV85KlApxp5qqQTft6qh6ioC7Uz1QYsxcjnvqIFagWwPXA44WqBe4XqBf28NNcB/wUwobUOLh18MSUt9a3g6GHdJ1XfRb43wztLRJ7bYeIt722D6oKbqEtQyuYxzcbk2vn7JqlOs4QVX62IlESniB7uw5i7uHv6nWKFcb8HpYNSBVIyRDZoHqqaIjhh37XnxzSUj0SKfg5FYPc/ueVbLfIjvFbBaGCe0VGy11CsTl4CaHdoqPB5jOhvxD6xQ2zcPqztcg8u461UI7z4oV+odRkXkeTpHs2s2V2qQB1N0CX9KjgDcCSghM9+F/1xwFHPVBqboaDGSF1xLxAyqGeSfOZ9lc4R4btCehFGzmG8u9USmsnK7Mzs93NvEjiHfx9wl9YgvqrDHIXbM7M28WCV3jH83/GfTRnuMXmNlSjyrWJNkVxW4IWbcIZb2hPZnZpG2rg+aZb4b6hfqdxD3Kr0ieV2FbEgLdRYn1BWY2DfRV8QvPq5CFc9mlCp1+LTjktlyBVquTyOJvu+wioT14xnUPu4evRYRpUrt4p8bjn2uSSLao+4bjDvUI9IWel1NbRk7BgWt4huX3WlSq8X8k9rVnXFcq8gFrFTLZGWTmurquVmxREDsHvWabbqh7skJVzcYs6pN+/OuITlatE+k8vgUY9VhNl5SqogrXpy+zmlr2WSdVBlL8f0FOJqU6hp6wtqMjKaVkfrbNlUSMh42777JCV3vou1mbKYPum885YFP17nsSTLzOzBbzKlTem90fh7irpHPdgPp+H1S58sdsrKKsjFG/eHMfHmXI4f+0SOJuMaL6OuIa/lm+LNGZhbFvhu8YAJRMjY1SIuid5gb01e+ryroBRVkVdGfEmmhxHw6iTB+OuR8ClzhvaKWABSp8P5wflAagx1QBsZjXc5ldZjb4LLzdch5ajJPc+iKSsc4pKrna5W8nJY3mj5ftjPM96Ksps+XFco1c9M1/SAh1nGJXMo4iIQ/kqX7JVIh9wRN1zafLGBfwL22Z/IiNhYelUj67DF6VaGpytel9qXsgrHxicj8+h4evGb/G/AJaFk/5rLpUt7M5W7fy6e5/M2Sns0l/SX9m0u+K5F0uDe4Q5rIvCmtKeHaK+XS+VS2JzjYTbLnF11tVV7b8KLs2R3TmSyz30hW6N2xyR5QXag/T+47UsmN0jrusfu22Ydiau7MqZ2DO0PLI9hbYH0J8z0/Xy5SX9PkFs+bbcZhWnfMZvQTSP4Ukp0ekpH9ySV+frLo8ynwxuxk2fPgBjNAx/An9MLXeMYaUNP3eL5r4PqqsOstiirnrJ7pl6O4m0Dewn61w31tedzInIY2Cmemoo5daMh+A3O6zLk9cwaYhbSM3unIOus6E2eRcU36afMSXPoEdluCmOfxdLz58Z2OrN+UhDwB5Oewu0AePWMuhpfnzOw3AmXvKWafniPcu7h67OiJ4d1h1nns7A56jD7h66en7Yl3FZK/m6gKyd3+k8dj+or446ffC55ee+DIefQpJqW0po0izjpHR5RJzdNiwjNnfrKuFU7KGm/ctmQkr3yD1zT9yDG7xcr5aqUx/vZtcnC4WWv8UKu/eV55+cK8mXugvlZP1Dr8tKVeojL34Fdb/an+Vv+q/9aaa/bab2ueZr1/z8h8pQqftd7/QaAK1w==</latexit><latexit sha1_base64="vFl/7vZyIfYMc4akmWOEzBEsI7E=">AAAUu3icrVjbbttGEN2kt8S9Oanf+sJGKeAAsiq5KZwXAQFsp0XRAA7ga2PDkCjKZs1bScquI+iT2p/pU4H2X3rm7FIUdaMUWILI1ezMmdm5LZftyHOTtF7/5979Dz786ONPHjxc+fSzz7/4cvXR48Mk7MW2c2CHXhgft1uJ47mBc5C6qeccR7HT8tuec9S+2pb5o2snTtww2E9vI+fMb10Ebte1WylI56s/nu6+PVlvPLM2rJP1+rMzq2kJxTr13Y51mgIq9Z0gBblxBp4Zc/Wz89VKvVbnx5ocNMygosxnL3z08C91qjoqVLbqKV85KlApxp5qqQTft6qh6ioC7Uz1QYsxcjnvqIFagWwPXA44WqBe4XqBf28NNcB/wUwobUOLh18MSUt9a3g6GHdJ1XfRb43wztLRJ7bYeIt722D6oKbqEtQyuYxzcbk2vn7JqlOs4QVX62IlESniB7uw5i7uHv6nWKFcb8HpYNSBVIyRDZoHqqaIjhh37XnxzSUj0SKfg5FYPc/ueVbLfIjvFbBaGCe0VGy11CsTl4CaHdoqPB5jOhvxD6xQ2zcPqztcg8u461UI7z4oV+odRkXkeTpHs2s2V2qQB1N0CX9KjgDcCSghM9+F/1xwFHPVBqboaDGSF1xLxAyqGeSfOZ9lc4R4btCehFGzmG8u9USmsnK7Mzs93NvEjiHfx9wl9YgvqrDHIXbM7M28WCV3jH83/GfTRnuMXmNlSjyrWJNkVxW4IWbcIZb2hPZnZpG2rg+aZb4b6hfqdxD3Kr0ieV2FbEgLdRYn1BWY2DfRV8QvPq5CFc9mlCp1+LTjktlyBVquTyOJvu+wioT14xnUPu4evRYRpUrt4p8bjn2uSSLao+4bjDvUI9IWel1NbRk7BgWt4huX3WlSq8X8k9rVnXFcq8gFrFTLZGWTmurquVmxREDsHvWabbqh7skJVzcYs6pN+/OuITlatE+k8vgUY9VhNl5SqogrXpy+zmlr2WSdVBlL8f0FOJqU6hp6wtqMjKaVkfrbNlUSMh42777JCV3vou1mbKYPum885YFP17nsSTLzOzBbzKlTem90fh7irpHPdgPp+H1S58sdsrKKsjFG/eHMfHmXI4f+0SOJuMaL6OuIa/lm+LNGZhbFvhu8YAJRMjY1SIuid5gb01e+ryroBRVkVdGfEmmhxHw6iTB+OuR8ClzhvaKWABSp8P5wflAagx1QBsZjXc5ldZjb4LLzdch5ajJPc+iKSsc4pKrna5W8nJY3mj5ftjPM96Ksps+XFco1c9M1/SAh1nGJXMo4iIQ/kqX7JVIh9wRN1zafLGBfwL22Z/IiNhYelUj67DF6VaGpytel9qXsgrHxicj8+h4evGb/G/AJaFk/5rLpUt7M5W7fy6e5/M2Sns0l/SX9m0u+K5F0uDe4Q5rIvCmtKeHaK+XS+VS2JzjYTbLnF11tVV7b8KLs2R3TmSyz30hW6N2xyR5QXag/T+47UsmN0jrusfu22Ydiau7MqZ2DO0PLI9hbYH0J8z0/Xy5SX9PkFs+bbcZhWnfMZvQTSP4Ukp0ekpH9ySV+frLo8ynwxuxk2fPgBjNAx/An9MLXeMYaUNP3eL5r4PqqsOstiirnrJ7pl6O4m0Dewn61w31tedzInIY2Cmemoo5daMh+A3O6zLk9cwaYhbSM3unIOus6E2eRcU36afMSXPoEdluCmOfxdLz58Z2OrN+UhDwB5Oewu0AePWMuhpfnzOw3AmXvKWafniPcu7h67OiJ4d1h1nns7A56jD7h66en7Yl3FZK/m6gKyd3+k8dj+or446ffC55ee+DIefQpJqW0po0izjpHR5RJzdNiwjNnfrKuFU7KGm/ctmQkr3yD1zT9yDG7xcr5aqUx/vZtcnC4WWv8UKu/eV55+cK8mXugvlZP1Dr8tKVeojL34Fdb/an+Vv+q/9aaa/bab2ueZr1/z8h8pQqftd7/QaAK1w==</latexit>

E[Y (1)

Y (0)] = E[Y | T = 1]

E[Y | T = 0]

(Y (0), Y (1)) ?? T

WeakerUnconfoundedness:

ATT:
E[Y (1)<latexit sha1_base64="9/ClvRHvi/8lFMggdK7KC4h2YSM=">AAAVkXicvVhLb9tGEF6lr8R92Ql8ag9slBYOIKmSm8LpQYBR20HRNoAD2LEdyzAkkrII8VWSsuoIOvXX9NJD+2v6bzrz7VIkJYqUA6MSRK5mZ76Znddy2fNtK4yazX8r9957/4MPP7r/YO3jTz797PP1jYevQ28U6Oax7tlecNrrhqZtueZxZEW2eeoHZtfp2eZJb7jH8yfXZhBannsU3fjmhdO9cq2+pXcjIl1uVL7sHJyfbbWeanXtbKv5VOs4lqF1IsKIHNONtLbWutC+aWsxW+58XU4vE+901hTC/y7eXFU8V/ZyvdpsNPHRFgctNagK9Tn0Nh78JTrCEJ7QxUg4whSuiGhsi64I6XsuWqIpfKJdiAnRAhpZmDfFVKyR7Ii4TOLoEnVI1yv6d66oLv1nzBDSOmmx6ReQpCa+VjwGjfugyjvr11K8y3RMgM023tC9pzAdokZiQNQyuZhzdbkefZ2SVUe0hudYrUUr8UFhP+iZNffpbtP/iFbI1xviNGlkkFRAI51oNlElhXUEdJeeZ98MEIku+EwasdVFdhdZzfMefYeE1aVxCEvZVk28UHFxodmErcxjI6bLEX+nFUr7irD6szVYiLtcBfMeEWUo3tIoi1ykM51dy7kihTzN0cX8EThc4g6J4iHzLfKfRRzZXNUJk3V0EckrrMVHBjUU8s+Yj7PZp3jWYU+IqGnINwt6fFVZid2xnTbde8AOSH5CcwPoYV/UyB4T2AGyN/ZiDdwB/Rvjnw4b9Tl6A5XJ8azRmji7aoTr0Yw1w5KekP6MLZLWTYimqW9d/Ar9JsW9Bq9wXtdI1oOFMotD6HJV7NvUV9gvDl2Zyp6NKTXocGDHANkyJFqiTyKxvm9pFSHqx1aoE7rb8JoPlBq0s3/GGDtYE0d0BN1jGhvQw9Ia9bqG2FF2TDNa2TcWutOiVg35x7UrO+O8VpZzUamayso2NDXFM7VijgDbnfaarrqh7MkhVjeds6oH+5OuwTmatY+lkvhkY2UgGweQyuKyF/PXmbeWbdRJDbFk318RRxtSfUUPUZu+0rSWqr89VSUe4qHj7qickPXO2sZzMxOiO8pTNvHJOuc9iWd+I8wucqoD76XnixAPlHy8G3DHn4BaLPcalZWVDWg0mc0Uy1sYmfCPHHHEJZ4PX/tYy1ezn5aaWRX7bvDcBUTO2EghrYpuIDfyV34kqtQLqpRVWX9ypJkS4OnEp/GTGecTwmXeIbS4ROEKn8zmp6Ux2CfKVHmsj7m4DhMbLHQ+A5wdlXmSQ1ZUNMfBVV2slfMyL28kvVjWmOV7VlbSi2WZco3ctFQ/CIF1WiIXIQ4s4aSy9KhEysOeIOnS5rMV7HOx145UXgTKwpMSSQc9Rq7KU1X5stS+CF0wUD5hmTfv4MFr9L8pnoBu68dENrqVNxO5m3fyaSI/vqVnE0nnlv5NJN+WSJrYG6wZjWVeldYUcx2Wcsl8KtsTTNpN4ucXWW01XHvkRd6zDdWZNLXfcFbI3bGNHlBbqT8v7jtcya3SOh6h+/bQhwJoNgpq5/jO0JIIjlZYX4h8T86Xq9RXntzqebOHOOR1x3hGPoEkTyHx6SFM7U8W8JOTxQRPgWO1k8XPg3VkgIzhT9QLX9Iz1hSavqPnuxZdX2R2vVVR+Zw1Uv0yjbtNyDu0X+1jX7s9rq9OQ/XMmSmr44A0xL+pOl0m3LY6AyxDuo3efGSZdcbCWWRek3zaHBCXPIHdlCAmeZyPVxzffGT5psTDCSA5h90FcvqMuRpekjPL3wiUvadYfnr26d6nq42OHirefWSdjc5uUo+RJ3z59LS38K6C83ebqoJzd/L44Zy+LP786fcKp9cRcSQ88hQTQVrS0ojLztE+ZCL1tBjizJmcrBuZk7LEm7ctTOWVo/Daqh+ZardYu1yvtubfvi0OXm83Wt83mq+eVXefqzdz98UX4rHYIj/tiF2qzEPyq175o/Jn5e/KP5uPNn/Y3N38UbLeqyiZRyLz2fzlP/RjSfU=</latexit><latexit sha1_base64="9/ClvRHvi/8lFMggdK7KC4h2YSM=">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</latexit>

Y (0) | T = 1] = E[Y (1) | T = 1] E[Y (0) | T = 1]
= E[Y | T = 1] E[Y (0) | T = 1]
= E[Y | T = 1]
= E[Y | T = 1]

Brady Neal<latexit sha1_base64="D5f7B1rjye1XD6q4mhbO9Bf8J+Y=">AAAUj3icrVhZb9tGEN6kV+IesVO/5YWNUsAGaFVyUzgvLgzYCdqiARzAZ+PAkMiVRYhXScqqIuihv6ZvRftz+m868+1SJHWQUmAJIlezM9/MzrVctkPXiZNG47979z/6+JNPP3vwcO3zL7786tH6xuOzOOhHljy1AjeILtqtWLqOL08TJ3HlRRjJltd25Xm7d8jz57cyip3AP0mGoXzntW58p+NYrYRI1+tPti63GtumcbnV3N42rhzflqGki58YJ9frtUa9gY8xO2jqQU3oz3Gw8fBvcSVsEQhL9IUnpPBFQmNXtERM37eiKRoiJNo7MSJaRCMH81KMxRrJ9olLEkeLqD263tC/t5rq03/GjCFtkRaXfhFJGuJbzWPTuAOqurN+I8e7SMcI2GzjkO5tjekRNRFdolbJpZzLy7Xp61WsOqE1vMBqHVpJCAr7wSqsuUN3l/4ntEK+DolT0sgmqYhGFtFcoioK64jorjzPvukiEi3wSRqx1WV2l1nN8wF9e4TVonEMS9lWQ7zScfGhWcJW5nER08WIf9AKlX1lWJ3JGhzEXa2CeU+I0hPvaVRELtOZz67FXIlGHs/RxfwJOHzijokSIPMd8p9DHMVctQiTdbQQyRusJUQG1TXyL5hPszmkeO7AnhhRM5BvDvSEurIyu1M7Xbq3gR2R/IjmutDDvjDJHgnsCNmbetEEd0T/BvhnwUZril5HZXI8TVoTZ5dJuAHNOBMs5Qnlz9QiZd2IaIb+7ohfoV9S3E14hfPaJNkAFqosjqHL17Hfp77CfvHoylT2bEoxocODHV1kS49omT6FxPq+o1XEqB9Xo47o7sJrIVBMaGf/DDD2sCaOaB+6BzS2oYelDep1dbGn7RgXtLJvHHSnWa0G8o9rV3XGaa0s56NSDZ2V+9DUEM/1ijkCbHfea5buhqonx1jdeMqqNuzPugbnaNE+lsriU4yVjWzsQqqIy16cv855a9lFnZiIJfv+hjj2IdXR9Bi1GWpNa7n6O9RVEiAeFu6ezglV76xtMDUzIrqnPeUSn6pz3pN45nfCbCGnruC9/HwZ4kstn+4G3PFHoJbLnaGyirIRjUaTmXJ5ByMJ/6gRR1zhhfB1iLV8M/kZuZllse8Gz59B5IxNNNKy6DZyY/7KT0SNekGNsqroT440UyI8nYQ0fjbhfEa4zNuDFp8oXOGjyfy4MgZHRBlrj3Uwl9ZhZoODzmeD80pnnuJQFZVMcXBVl2vlvJyXN4peLmtP8r0oq+jlsky5RW46uh/EwLqokEsQB5bwcll6UiEVYE9QdGXz5RL2+dhr+zovIm3heYWkhx6jVhXoqnxdaV+CLhhpn7DMbx/gwVv0vzGegFb1YyabrOTNTG74QT7N5AcrejaT9Fb0byb5vkJSYm9wJjSWeVNZU8x1XMml8qlqT5C0m6TPL6raTFzb5EXes23dmQy933BWqN1xHz3AXKo/z+47XMnNyjruo/u20YciaLZLauf0ztCyCPaXWF+MfM/Ol8vU1zy55fPmEHGY1x3TGfUEkj2FpKeHOLc/OcDPThYjPAUO9E6WPg/uIANUDH+iXvianrHG0PQ9Pd816fqqsOsti8rnrL7ul3ncXULeo/3qCPva6rihPg3tFM5MRR0vSUP6G+vTZcbt6jPAIqRV9M5HVllnz5xFpjWpp80ucakT2LACMcvj+Xjl8Z2PrN6UBDgBZOewu0DOnzGXw8tyZvEbgar3FItPzyHdO3R10dFjzXuErHPR2SX1GHXCV09PhzPvKjh/d6kqOHdHTx9P6SviT59+b3B67RNHxqNOMQmkFS2PuOgcHUIm0U+LMc6c2cm6XjgpK7xp2+JcXnkab1/3I6l3i7Xr9Vpz+u3b7OBst978od5487x28EK/mXsgnoinYov8tCcOqDKPya+W+FP8Jf4R/25ubO5t/rh5oFjv39MyX4vCZ/Pn/wHmHvuw</latexit>

Y (0) ?? T<latexit sha1_base64="gWO2eWBKWJyi4omMXcdiRNAO2sw=">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</latexit>

E[Y (0) | T = 0]
E[Y | T = 0]

Motivation and Preliminaries

6 / 25Average Treatment Effect on the Treated (ATT)
Unconfoundedness:

ATE:<latexit sha1_base64="vFl/7vZyIfYMc4akmWOEzBEsI7E=">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</latexit><latexit sha1_base64="vFl/7vZyIfYMc4akmWOEzBEsI7E=">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</latexit>

E[Y (1)

Y (0)] = E[Y | T = 1]

E[Y | T = 0]

(Y (0), Y (1)) ?? T<latexit sha1_base64="D5f7B1rjye1XD6q4mhbO9Bf8J+Y=">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</latexit>

ATT:
E[Y (1)<latexit sha1_base64="9/ClvRHvi/8lFMggdK7KC4h2YSM=">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</latexit>

Y (0) | T = 1] = E[Y (1) | T = 1]
= E[Y | T = 1]

E[Y (0) | T = 1]

= E[Y | T = 1]

E[Y | T = 0]

= E[Y | T = 1]
Brady NealE[Y (0) | T = 1]

E[Y (0) | T = 0]

Motivation and Preliminaries

6 / 25Average Treatment Effect on the Treated (ATT)
Unconfoundedness:

ATE:<latexit sha1_base64="vFl/7vZyIfYMc4akmWOEzBEsI7E=">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</latexit><latexit sha1_base64="vFl/7vZyIfYMc4akmWOEzBEsI7E=">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</latexit>

E[Y (1)

Y (0)] = E[Y | T = 1]

E[Y | T = 0]

(Y (0), Y (1)) ?? T<latexit sha1_base64="D5f7B1rjye1XD6q4mhbO9Bf8J+Y=">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</latexit>

ATT:
E[Y (1)<latexit sha1_base64="9/ClvRHvi/8lFMggdK7KC4h2YSM=">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</latexit><latexit sha1_base64="9/ClvRHvi/8lFMggdK7KC4h2YSM=">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</latexit>

Brady Neal

Y (0) | T = 1] = E[Y (1) | T = 1] E[Y (0) | T = 1]
= E[Y | T = 1] E[Y (0) | T = 1]
= E[Y | T = 1] E[Y (0) | T = 0]
= E[Y | T = 1] E[Y | T = 0]
Motivation and Preliminaries

6 / 25Average Treatment Effect on the Treated (ATT)
Unconfoundedness:

ATE:<latexit sha1_base64="vFl/7vZyIfYMc4akmWOEzBEsI7E=">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</latexit><latexit sha1_base64="vFl/7vZyIfYMc4akmWOEzBEsI7E=">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</latexit>

E[Y (1)

Y (0)] = E[Y | T = 1]

E[Y | T = 0]

(Y (0), Y (1)) ?? T<latexit sha1_base64="D5f7B1rjye1XD6q4mhbO9Bf8J+Y=">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</latexit>

ATT:
E[Y (1)<latexit 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Brady Neal

Y (0) | T = 1] = E[Y (1) | T = 1] E[Y (0) | T = 1]
= E[Y | T = 1] E[Y (0) | T = 1]
= E[Y | T = 1] E[Y (0) | T = 0]
= E[Y | T = 1] E[Y | T = 0]
Motivation and Preliminaries

6 / 25Average Treatment Effect on the Treated (ATT)
Unconfoundedness:

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E[Y (1)

Y (0)] = E[Y | T = 1]

E[Y | T = 0]

(Y (0), Y (1)) ?? T<latexit sha1_base64="D5f7B1rjye1XD6q4mhbO9Bf8J+Y=">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</latexit>

ATT:
E[Y (1)<latexit 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sha1_base64="9/ClvRHvi/8lFMggdK7KC4h2YSM=">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</latexit><latexit sha1_base64="9/ClvRHvi/8lFMggdK7KC4h2YSM=">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</latexit><latexit sha1_base64="9/ClvRHvi/8lFMggdK7KC4h2YSM=">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</latexit>

Y (0) | T = 1] = E[Y (1) | T = 1] E[Y (0) | T = 1]
= E[Y | T = 1] E[Y (0) | T = 1]
= E[Y | T = 1] E[Y (0) | T = 0]
(Y (0) ?? T )= E[Y | T = 1] E[Y | T = 0]<latexit sha1_base64="3hYDtUrzXv25gamI80APurq8u+4=">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</latexit>

Brady Neal

Motivation and Preliminaries

6 / 25Average Treatment Effect on the Treated (ATT)
Unconfoundedness:

ATE:<latexit sha1_base64="vFl/7vZyIfYMc4akmWOEzBEsI7E=">AAAUu3icrVjbbttGEN2kt8S9Oanf+sJGKeAAsiq5KZwXAQFsp0XRAA7ga2PDkCjKZs1bScquI+iT2p/pU4H2X3rm7FIUdaMUWILI1ezMmdm5LZftyHOTtF7/5979Dz786ONPHjxc+fSzz7/4cvXR48Mk7MW2c2CHXhgft1uJ47mBc5C6qeccR7HT8tuec9S+2pb5o2snTtww2E9vI+fMb10Ebte1WylI56s/nu6+PVlvPLM2rJP1+rMzq2kJxTr13Y51mgIq9Z0gBblxBp4Zc/Wz89VKvVbnx5ocNMygosxnL3z08C91qjoqVLbqKV85KlApxp5qqQTft6qh6ioC7Uz1QYsxcjnvqIFagWwPXA44WqBe4XqBf28NNcB/wUwobUOLh18MSUt9a3g6GHdJ1XfRb43wztLRJ7bYeIt722D6oKbqEtQyuYxzcbk2vn7JqlOs4QVX62IlESniB7uw5i7uHv6nWKFcb8HpYNSBVIyRDZoHqqaIjhh37XnxzSUj0SKfg5FYPc/ueVbLfIjvFbBaGCe0VGy11CsTl4CaHdoqPB5jOhvxD6xQ2zcPqztcg8u461UI7z4oV+odRkXkeTpHs2s2V2qQB1N0CX9KjgDcCSghM9+F/1xwFHPVBqboaDGSF1xLxAyqGeSfOZ9lc4R4btCehFGzmG8u9USmsnK7Mzs93NvEjiHfx9wl9YgvqrDHIXbM7M28WCV3jH83/GfTRnuMXmNlSjyrWJNkVxW4IWbcIZb2hPZnZpG2rg+aZb4b6hfqdxD3Kr0ieV2FbEgLdRYn1BWY2DfRV8QvPq5CFc9mlCp1+LTjktlyBVquTyOJvu+wioT14xnUPu4evRYRpUrt4p8bjn2uSSLao+4bjDvUI9IWel1NbRk7BgWt4huX3WlSq8X8k9rVnXFcq8gFrFTLZGWTmurquVmxREDsHvWabbqh7skJVzcYs6pN+/OuITlatE+k8vgUY9VhNl5SqogrXpy+zmlr2WSdVBlL8f0FOJqU6hp6wtqMjKaVkfrbNlUSMh42777JCV3vou1mbKYPum885YFP17nsSTLzOzBbzKlTem90fh7irpHPdgPp+H1S58sdsrKKsjFG/eHMfHmXI4f+0SOJuMaL6OuIa/lm+LNGZhbFvhu8YAJRMjY1SIuid5gb01e+ryroBRVkVdGfEmmhxHw6iTB+OuR8ClzhvaKWABSp8P5wflAagx1QBsZjXc5ldZjb4LLzdch5ajJPc+iKSsc4pKrna5W8nJY3mj5ftjPM96Ksps+XFco1c9M1/SAh1nGJXMo4iIQ/kqX7JVIh9wRN1zafLGBfwL22Z/IiNhYelUj67DF6VaGpytel9qXsgrHxicj8+h4evGb/G/AJaFk/5rLpUt7M5W7fy6e5/M2Sns0l/SX9m0u+K5F0uDe4Q5rIvCmtKeHaK+XS+VS2JzjYTbLnF11tVV7b8KLs2R3TmSyz30hW6N2xyR5QXag/T+47UsmN0jrusfu22Ydiau7MqZ2DO0PLI9hbYH0J8z0/Xy5SX9PkFs+bbcZhWnfMZvQTSP4Ukp0ekpH9ySV+frLo8ynwxuxk2fPgBjNAx/An9MLXeMYaUNP3eL5r4PqqsOstiirnrJ7pl6O4m0Dewn61w31tedzInIY2Cmemoo5daMh+A3O6zLk9cwaYhbSM3unIOus6E2eRcU36afMSXPoEdluCmOfxdLz58Z2OrN+UhDwB5Oewu0AePWMuhpfnzOw3AmXvKWafniPcu7h67OiJ4d1h1nns7A56jD7h66en7Yl3FZK/m6gKyd3+k8dj+or446ffC55ee+DIefQpJqW0po0izjpHR5RJzdNiwjNnfrKuFU7KGm/ctmQkr3yD1zT9yDG7xcr5aqUx/vZtcnC4WWv8UKu/eV55+cK8mXugvlZP1Dr8tKVeojL34Fdb/an+Vv+q/9aaa/bab2ueZr1/z8h8pQqftd7/QaAK1w==</latexit><latexit sha1_base64="vFl/7vZyIfYMc4akmWOEzBEsI7E=">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</latexit>

E[Y (1)

Y (0)] = E[Y | T = 1]

E[Y | T = 0]

(Y (0), Y (1)) ?? T<latexit sha1_base64="D5f7B1rjye1XD6q4mhbO9Bf8J+Y=">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</latexit>

ATT:
E[Y (1)<latexit 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Y (0) | T = 1] = E[Y (1) | T = 1] E[Y (0) | T = 1]
= E[Y | T = 1] E[Y (0) | T = 1]
= E[Y | T = 1] E[Y (0) | T = 0]
(Y (0) ?? T )= E[Y | T = 1] E[Y | T = 0]<latexit sha1_base64="3hYDtUrzXv25gamI80APurq8u+4=">AAAUiXicrVjbbttGEN2kt8TpxUndp76wcQo4gKxKbgobBQwEsBMURQM4gG9pHBgSRVmEeCtJWXUEfUxfW6D/07/pmbNLUdSNUmAJIlezM2dm57ZcNiPPTdJa7b87dz/6+JNPP7t3f+3B5198+dX6w0enSdiLbefEDr0wPm82EsdzA+ckdVPPOY9ip+E3Pees2T2Q+bNrJ07cMDhObyLnnd+4Cty2azdSkC7Xv9l6s1V7al24QcuJHFyC1Dp+erm+WavW+LGmB3Uz2FTmcxQ+vP+vulAtFSpb9ZSvHBWoFGNPNVSC71tVVzUVgfZODUCLMXI576ihWoNsD1wOOBqgdnG9wr+3hhrgv2AmlLahxcMvhqSlvjc8LYzbpOq76LfGeOfpGBBbbLzBvWkwfVBT1QG1TC7jXF6uia9fsuoUa9jjal2sJCJF/GAX1tzG3cP/FCuU6w04HYxakIoxskHzQNUU0RHjrj0vvukwEg3yORiJ1YvsXmS1zIf4doHVwDihpWKrpV6auATU7NBW4fEY0/mIf2KF2r5FWO3RGlzGXa9CeI9B6ar3GBWRF+kcz675XKlBHs7QJfwpOQJwJ6CEzHwX/nPBUcxVG5iio8FIXnEtETOoapB/5XyWzRHiuU17EkbNYr651BOZysrtzuz0cG8SO4b8AHMd6hFfVGCPQ+yY2Zt5sULuGP/6/GfTRnuCXmVlSjwrWJNkVwW4IWbcEZb2hPZnZpG2bgCaZb7b6jfqdxD3Cr0ieV2BbEgLdRYn1BWY2O+jr4hffFyFKp7NKBXq8GlHh9nSBS3Xp5FE3w9YRcL68QzqAHePXouIUqF28U+fY59rkoj2qLuPcYt6RNpCr6uqXWPHsKBVfOOyO01rtZh/Uru6M05qFbmAlWqZrNynppp6ZlYsERC7x71mm26oe3LC1Q0nrGrS/rxrSI4W7ROpPD7FWLWYjR1KFXHFi7PXOWstO6yTCmMpvr8Cxz6l2oaesDYjo2ltrP4OTJWEjIfNu29yQte7aOtPzAxA942nPPDpOpc9SWb+AGaDOXVB743PL0J8YeSz3UA6/oDUxXKnrKyibIzRYDSzWN7lyKF/9EgirvEi+jriWr4b/ayxmWWxbwcvmEKUjE0N0rLoLebG7JUfq030gk1kVdGfEmmhxHw6iTB+MuJ8Alzh7VJLAIpU+GA0PyyNwSEoQ+OxNueyOsxtcNn5WuS8MJmnOXRFpRMcUtWLtUpezsobTV8s2xrle1FW0xfLCuWauemafpAQ67xELmUcRMIfy9LjEqmQe4Kma5vfLGFfwL22Z/IiNhaelUj67DF6VaGpylel9qXsgrHxicj8/gEevGb/G/IJaFU/5rLpSt7M5W4+yKe5fH9Fz+aS/or+zSXfl0g63BvcEU1kXpfWlHAdlXLpfCrbExzsJtnzi662Cq9NeFH27JbpTJbZbyQr9O64zx5QWao/T+87Usn10jrusfs22Ydiam4tqJ2TW0PLI9hbYn0J8z0/Xy5TX7Pkls+bA8ZhVnfMZvQTSP4Ukp0ekrH9ySV+frIY8Cmwb3ay7HlwmxmgY/gLeuErPGMNqelHPN/VcX1Z2PWWRZVzVs/0y3HcHSDvYr865L62Om5kTkPbhTNTUccLaMh+Q3O6zLk9cwaYh7SK3tnIOutaU2eRSU36abMDLn0CuylBzPN4Nt7i+M5G1m9KQp4A8nPYbSCPnzGXw8tzZv4bgbL3FPNPzxHubVw9dvTE8B4y6zx2dgc9Rp/w9dPTwdS7CsnfHVSF5O7g8aMJfUX8ydPvFU+vPXDkPPoUk1Ja08YR552jI8qk5mkx4ZkzP1lXCydljTdpWzKWV77B2zf9yDG7xdrl+mZ98u3b9OB0p1r/qVp7/Wzz+Z55M3dPfaseqy34aVc9R2Uewa82tP2l/lb/bDzYqG/sbfysWe/eMTJfq8Jn4+B/YL76TQ==</latexit>

Brady Neal

Motivation and Preliminaries

6 / 25

Question:
What is the difference between the ATE
and the ATT?

Motivation and Preliminaries
Difference-in-Differences Overview
Assumptions and Proof
Problems with Difference-in-Differences

Brady Neal

Difference-in-Differences Overview

8 / 25

Introducing Time

Brady Neal

Difference-in-Differences Overview

9 / 25

Introducing Time<latexit sha1_base64="gPeVt40VbbIgq17ra5BRvR27Apw=">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</latexit><latexit sha1_base64="rv0f3xFXVtgAmN42lVmvw7DNjRY=">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</latexit>

Brady NealE[Y | T = 1]

E[Y | T = 0]

Difference-in-Differences Overview

9 / 25

Introducing Time<latexit sha1_base64="gPeVt40VbbIgq17ra5BRvR27Apw=">AAAUjXicrVjbbttGEN2kt8S9xEn9UvSFjVOgD7IquQ7ch6oIYDstigZwAN9SyzAkirIJ81aSsuoIQr+mr+n39G965uxSFHWjHFiCyNXszJnZuS2X7chzk7RW++/e/Q8+/OjjTx48XPn0s8+/eLT6+MlREvZi2zm0Qy+MT9qtxPHcwDlM3dRzTqLYafltzzluX+3I/PG1EyduGBykN5Fz5rcuArfr2q0UpPPVr5p7p2+spu92rGYKwdR3gtRqWPWz89X1WrXGjzU9qJvBujKf/fDxw3eqqToqVLbqKV85KlApxp5qqQTfU1VXNRWBdqYGoMUYuZx31FCtQLYHLgccLVCvcL3Av1NDDfBfMBNK29Di4RdD0lLfGp4Oxl1S9V30W2O883QMiC023uDeNpg+qKm6BLVMLuNcXq6Nr1+y6hRr+JGrdbGSiBTxg11Ycxd3D/9TrFCuN+B0MOpAKsbIBs0DVVNER4y79rz45pKRaJHPwUisXmT3IqtlPsT3ClgtjBNaKrZa6qWJS0DNDm0VHo8xnY/4F1ao7VuE1R2twWXc9SqE9wCUK/UWoyLyIp3j2TWfKzXIwxm6hD8lRwDuBJSQme/Cfy44irlqA1N0tBjJC64lYgZVDfJvnM+yOUI8N2hPwqhZzDeXeiJTWbndmZ0e7m1ix5AfYO6SesQXFdjjEDtm9mZerJA7xr8+/9m00Z6gV1mZEs8K1iTZVQFuiBl3hKU9of2ZWaStG4Bmme+G+p36HcS9Qq9IXlcgG9JCncUJdQUm9g30FfGLj6tQxbMZpUIdPu24ZLZcgZbr00ii73usImH9eAZ1gLtHr0VEqVC7+KfPsc81SUR71N3HuEM9Im2h11XVtrFjWNAqvnHZnaa1Wsw/qV3dGSe1ilzASrVMVjaoqaa2zIolAmL3uNds0w11T064uuGEVW3an3cNydGifSKVx6cYqw6z8ZJSRVzx4ux1zlrLJuukwliK7y/A0aBU19AT1mZkNK2M1d+OqZKQ8bB5901O6HoXbf2JmQHovvGUBz5d57InycyfwGwxp5r03vj8IsQ9I5/tBtLxB6QuljtiZRVlY4wGo5nF8i5HDv2jRxJxjRfR1xHX8s3oZ43NLIt9N3jBFKJkbGqQlkXvMDdmr/xAraMXrCOriv6USAsl5tNJhPGzEecz4ArvFbUEoEiFD0bzw9IY7IIyNB7rci6rw9wGl52vQ86myTzNoSsqneCQql6sVfJyVt5o+mLZzijfi7KavlhWKNfMTdf0g4RYJyVyKeMgEv5Ylh6USIXcEzRd2/xmCfsC7rU9kxexsfC4RNJnj9GrCk1Vviq1L2UXjI1PROaP9/DgNfvfkE9At/VjLpveypu53M17+TSX79/Ss7mkf0v/5pJvSyQd7g3uiCYyr0trSrj2S7l0PpXtCQ52k+z5RVdbhdc2vCh7dsd0JsvsN5IVendssAdUlurP0/uOVHK9tI577L5t9qGYmjsLaufwztDyCPaWWF/CfM/Pl8vU1yy55fNmh3GY1R2zGf0Ekj+FZKeHZGx/comfnywGfArsm50sex7cYAboGP6KXvgKz1hDavoBz3d1XF8Wdr1lUeWc1TP9chx3E8jb2K92ua/dHjcyp6GNwpmpqGMPGrLf0Jwuc27PnAHmId1G72xknXWdqbPIpCb9tHkJLn0CuylBzPN4Nt7i+M5G1m9KQp4A8nPYXSCPnzGXw8tzZv4bgbL3FPNPzxHuXVw9dvTE8O4y6zx2dgc9Rp/w9dPTztS7CsnfTVSF5O7g6ZMJfUX8ydPvBU+vPXDkPPoUk1Ja08YR552jI8qk5mkx4ZkzP1lXCydljTdpWzKWV77Ba5h+5JjdYuV8db0++fZtenC0Wa0/r9Zeb62/2DJv5h6or9VT9R38tK1eoDL34Vdb/a3+Ue/Uv2uP1p6v/bT2s2a9f8/IfKkKn7Vf/gdHyfwu</latexit>

E[Y | T = 1]

<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">AAAUl3icrVjbbttGEN2kt8S9Oalfir6wcQo4gKxKrgv3xWgAXxAUDWADtmPXNgKJomxCvJWk7DqCPqA/0762n9K/6ZmzS1HUjVJgCSJXszNnZue2XDYjz03SWu2/Bw8/+PCjjz959Hjp088+/+LL5SdPT5KwG9vOsR16YXzabCSO5wbOceqmnnMaxU7Db3rOm2ZnR+bf3Dhx4obBUXoXOZd+4ypw267dSEF6u7x6sXd+tlZ/Ya1bZ2u1F9aF77asixQQqe8EqbVt1S/BVavW+LHGB3UzWFXmcxA+efyXulAtFSpbdZWvHBWoFGNPNVSC77mqq5qKQLtUPdBijFzOO6qvliDbBZcDjgaoHVyv8O/cUAP8F8yE0ja0ePjFkLTUd4anhXGbVH0X/dYQ7zQdPWKLjXe4Nw2mD2qqrkEtk8s455dr4uuXrDrFGn7ial2sJCJF/GAX1tzG3cP/FCuU6x04HYxakIoxskHzQNUU0RHjrj0vvrlmJBrkczASq2fZPctqmQ/x7QCrgXFCS8VWS+2buATU7NBW4fEY0+mIf2CF2r5ZWO3BGlzGXa9CeI9A6ah3GBWRZ+kczq7pXKlB7k/QJfwpOQJwJ6CEzHwX/nPBUcxVG5iio8FIXnEtETOoapB/4XyWzRHiuU57EkbNYr651BOZysrtzuz0cG8SO4Z8D3PX1CO+qMAeh9gxszfzYoXcMf7d8p9NG+0RepWVKfGsYE2SXRXghphxB1jaE9qfmUXauh5olvmuq1+p30HcK/SK5HUFsiEt1FmcUFdgYr+NviJ+8XEVqng2o1Sow6cd18yWDmi5Po0k+r7HKhLWj2dQe7h79FpElAq1i39uOfa5Jolol7pvMW5Rj0hb6HVVtWXs6Be0im9cdqdxrRbzT2pXd8ZRrSIXsFItk5Xb1FRTm2bFEgGxe9hrtumGuicnXF1/xKom7c+7huRo0T6RyuNTjFWL2XhNqSKueHHyOietZYN1UmEsxfdX4NimVNvQE9ZmZDQtDdXfjqmSkPGwefdNTuh6F223IzM90H3jKQ98us5lT5KZ34HZYE5d0HvD87MQ94x8thtIx++ROlvuhJVVlI0x6g1mZsu7HDn0jx5JxDVeRF9HXMu3g581NDMv9v3gBWOIkrGpQZoXvcXcmLzyI7WKXrCKrCr6UyItlJhPJxHGzwecz4ErvB1qCUCRCu8N5vulMdgFpW881uZcVoe5DS47X4ucFybzNIeuqHSEQ6p6tlbJy0l5o+mzZVuDfC/KavpsWaHcMDdd0w8SYp2WyKWMg0j4Q1l6VCIVck/QdG3z2Rz2BdxruyYvYmPhmxJJnz1Gryo0Vfm61L6UXTA2PhGZ397Dgzfsf30+AS3qx1w2Xcibudzde/k0l79d0LO5pL+gf3PJdyWSDvcGd0ATmcPSmhKug1IunU9le4KD3SR7ftHVVuG1CS/Knt0yncky+41khd4dt9kDKnP15/F9Ryq5XlrHXXbfJvtQTM2tGbVzfG9oeQS7c6wvYb7n58t56muS3Px5s8M4TOqO2Yx+AsmfQrLTQzK0P7nEz08WPT4F3pqdLHseXGcG6Bi+Qi98jWesPjX9gOe7Oq77hV1vXlQ5Z3VNvxzG3QDyFvarXe5ri+NG5jS0XjgzFXXsQUP265vTZc7tmTPANKRF9E5G1lnXGjuLjGrST5vX4NInsLsSxDyPJ+PNju9kZP2mJOQJID+H3Qfy8BlzPrw8Z6a/ESh7TzH99Bzh3sbVY0dPDO8us85jZ3fQY/QJXz897Yy9q5D83UBVSO72nj0d0VfEHz39XvH02gVHzqNPMSmlNW0Ycdo5OqJMap4WE54585N1tXBS1nijtiVDeeUbvG3TjxyzWyy9XV6tj759Gx+cbFTrP1Zrh5urLzfNm7lH6hv1TK3BT1vqJSrzAH611Z/qb/WP+nfl65WfV/ZXXmnWhw+MzFeq8Fk5/B+BGv5b</latexit><latexit sha1_base64="rv0f3xFXVtgAmN42lVmvw7DNjRY=">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</latexit>

E[Y (1)

Y (0) | T = 1]

E[Y | T = 0]

Unconfoundedness:<latexit sha1_base64="YOwbpWQvSRpWPa9I0mIbb0kj6tU=">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</latexit>

Brady Neal

Y (0) ?? T

Difference-in-Differences Overview

9 / 25

Introducing Time<latexit sha1_base64="gPeVt40VbbIgq17ra5BRvR27Apw=">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</latexit>

E[Y | T = 1]

<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">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</latexit><latexit sha1_base64="rv0f3xFXVtgAmN42lVmvw7DNjRY=">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</latexit>

E[Y (1)

Y (0) | T = 1]

E[Y | T = 0]

Unconfoundedness:<latexit sha1_base64="YOwbpWQvSRpWPa9I0mIbb0kj6tU=">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</latexit>

ATT estimand without time: E[Y (1)<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">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</latexit>

Brady Neal

Y (0) ?? TY (0) | T = 1]

Difference-in-Differences Overview

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Introducing Time<latexit sha1_base64="gPeVt40VbbIgq17ra5BRvR27Apw=">AAAUjXicrVjbbttGEN2kt8S9xEn9UvSFjVOgD7IquQ7ch6oIYDstigZwAN9SyzAkirIJ81aSsuoIQr+mr+n39G965uxSFHWjHFiCyNXszJnZuS2X7chzk7RW++/e/Q8+/OjjTx48XPn0s8+/eLT6+MlREvZi2zm0Qy+MT9qtxPHcwDlM3dRzTqLYafltzzluX+3I/PG1EyduGBykN5Fz5rcuArfr2q0UpPPVr5p7p2+spu92rGYKwdR3gtRqWPWz89X1WrXGjzU9qJvBujKf/fDxw3eqqToqVLbqKV85KlApxp5qqQTfU1VXNRWBdqYGoMUYuZx31FCtQLYHLgccLVCvcL3Av1NDDfBfMBNK29Di4RdD0lLfGp4Oxl1S9V30W2O883QMiC023uDeNpg+qKm6BLVMLuNcXq6Nr1+y6hRr+JGrdbGSiBTxg11Ycxd3D/9TrFCuN+B0MOpAKsbIBs0DVVNER4y79rz45pKRaJHPwUisXmT3IqtlPsT3ClgtjBNaKrZa6qWJS0DNDm0VHo8xnY/4F1ao7VuE1R2twWXc9SqE9wCUK/UWoyLyIp3j2TWfKzXIwxm6hD8lRwDuBJSQme/Cfy44irlqA1N0tBjJC64lYgZVDfJvnM+yOUI8N2hPwqhZzDeXeiJTWbndmZ0e7m1ix5AfYO6SesQXFdjjEDtm9mZerJA7xr8+/9m00Z6gV1mZEs8K1iTZVQFuiBl3hKU9of2ZWaStG4Bmme+G+p36HcS9Qq9IXlcgG9JCncUJdQUm9g30FfGLj6tQxbMZpUIdPu24ZLZcgZbr00ii73usImH9eAZ1gLtHr0VEqVC7+KfPsc81SUR71N3HuEM9Im2h11XVtrFjWNAqvnHZnaa1Wsw/qV3dGSe1ilzASrVMVjaoqaa2zIolAmL3uNds0w11T064uuGEVW3an3cNydGifSKVx6cYqw6z8ZJSRVzx4ux1zlrLJuukwliK7y/A0aBU19AT1mZkNK2M1d+OqZKQ8bB5901O6HoXbf2JmQHovvGUBz5d57InycyfwGwxp5r03vj8IsQ9I5/tBtLxB6QuljtiZRVlY4wGo5nF8i5HDv2jRxJxjRfR1xHX8s3oZ43NLIt9N3jBFKJkbGqQlkXvMDdmr/xAraMXrCOriv6USAsl5tNJhPGzEecz4ArvFbUEoEiFD0bzw9IY7IIyNB7rci6rw9wGl52vQ86myTzNoSsqneCQql6sVfJyVt5o+mLZzijfi7KavlhWKNfMTdf0g4RYJyVyKeMgEv5Ylh6USIXcEzRd2/xmCfsC7rU9kxexsfC4RNJnj9GrCk1Vviq1L2UXjI1PROaP9/DgNfvfkE9At/VjLpveypu53M17+TSX79/Ss7mkf0v/5pJvSyQd7g3uiCYyr0trSrj2S7l0PpXtCQ52k+z5RVdbhdc2vCh7dsd0JsvsN5IVendssAdUlurP0/uOVHK9tI577L5t9qGYmjsLaufwztDyCPaWWF/CfM/Pl8vU1yy55fNmh3GY1R2zGf0Ekj+FZKeHZGx/comfnywGfArsm50sex7cYAboGP6KXvgKz1hDavoBz3d1XF8Wdr1lUeWc1TP9chx3E8jb2K92ua/dHjcyp6GNwpmpqGMPGrLf0Jwuc27PnAHmId1G72xknXWdqbPIpCb9tHkJLn0CuylBzPN4Nt7i+M5G1m9KQp4A8nPYXSCPnzGXw8tzZv4bgbL3FPNPzxHuXVw9dvTE8O4y6zx2dgc9Rp/w9dPTztS7CsnfTVSF5O7g6ZMJfUX8ydPvBU+vPXDkPPoUk1Ja08YR552jI8qk5mkx4ZkzP1lXCydljTdpWzKWV77Ba5h+5JjdYuV8db0++fZtenC0Wa0/r9Zeb62/2DJv5h6or9VT9R38tK1eoDL34Vdb/a3+Ue/Uv2uP1p6v/bT2s2a9f8/IfKkKn7Vf/gdHyfwu</latexit>

E[Y | T = 1]

<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">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</latexit><latexit sha1_base64="rv0f3xFXVtgAmN42lVmvw7DNjRY=">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</latexit>

E[Y (1)

Y (0) | T = 1]

E[Y | T = 0]

Unconfoundedness:<latexit sha1_base64="YOwbpWQvSRpWPa9I0mIbb0kj6tU=">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</latexit>

ATT estimand without time: E[Y (1)<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">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</latexit>

Brady Neal

Y (0) ?? TY (0) | T = 1]

Difference-in-Differences Overview

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Introducing Time<latexit sha1_base64="gPeVt40VbbIgq17ra5BRvR27Apw=">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</latexit><latexit sha1_base64="rv0f3xFXVtgAmN42lVmvw7DNjRY=">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</latexit>

ATT estimand without time: E[Y (1)<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">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</latexit>

Brady NealE[Y | T = 1]

E[Y | T = 0]

Y (0) | T = 1]

Difference-in-Differences Overview

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Introducing Time<latexit sha1_base64="gPeVt40VbbIgq17ra5BRvR27Apw=">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</latexit><latexit sha1_base64="rv0f3xFXVtgAmN42lVmvw7DNjRY=">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</latexit>

E[Y | T = 1]

E[Y | T = 0]time
ATT estimand without time: E[Y (1)<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">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</latexit>

Brady Neal

Y (0) | T = 1]

Difference-in-Differences Overview

9 / 25

Introducing TimeE[Y1 | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit>

E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>time
ATT estimand without time: E[Y (1)<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">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</latexit>

Brady Neal

Y (0) | T = 1]

Difference-in-Differences Overview

9 / 25

Introducing TimeE[Y1 | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit>

E[Y0 | T = 1]
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E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>time

ATT estimand without time: E[Y (1)<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">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</latexit>

Brady Neal

Y (0) | T = 1]

Difference-in-Differences Overview

9 / 25

Introducing TimeE[Y1 | T = 1]
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Treatment
Group
E[Y0 | T = 1]
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ControlGroup

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E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>time

ATT estimand without time: E[Y (1)<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">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</latexit>

Brady Neal

Y (0) | T = 1]

Difference-in-Differences Overview

9 / 25

Introducing Time

Treatment
administered

E[Y1 | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit>

Treatment
Group
E[Y0 | T = 1]
<latexit sha1_base64="GWKSlxQAH7OAOhoCpjA48XtUgBE=">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</latexit>

ControlGroup

E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>time

ATT estimand without time: E[Y (1)<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">AAAUl3icrVjbbttGEN2kt8S9Oalfir6wcQo4gKxKrgv3xWgAXxAUDWADtmPXNgKJomxCvJWk7DqCPqA/0762n9K/6ZmzS1HUjVJgCSJXszNnZue2XDYjz03SWu2/Bw8/+PCjjz959Hjp088+/+LL5SdPT5KwG9vOsR16YXzabCSO5wbOceqmnnMaxU7Db3rOm2ZnR+bf3Dhx4obBUXoXOZd+4ypw267dSEF6u7x6sXd+tlZ/Ya1bZ2u1F9aF77asixQQqe8EqbVt1S/BVavW+LHGB3UzWFXmcxA+efyXulAtFSpbdZWvHBWoFGNPNVSC77mqq5qKQLtUPdBijFzOO6qvliDbBZcDjgaoHVyv8O/cUAP8F8yE0ja0ePjFkLTUd4anhXGbVH0X/dYQ7zQdPWKLjXe4Nw2mD2qqrkEtk8s455dr4uuXrDrFGn7ial2sJCJF/GAX1tzG3cP/FCuU6x04HYxakIoxskHzQNUU0RHjrj0vvrlmJBrkczASq2fZPctqmQ/x7QCrgXFCS8VWS+2buATU7NBW4fEY0+mIf2CF2r5ZWO3BGlzGXa9CeI9A6ah3GBWRZ+kczq7pXKlB7k/QJfwpOQJwJ6CEzHwX/nPBUcxVG5iio8FIXnEtETOoapB/4XyWzRHiuU57EkbNYr651BOZysrtzuz0cG8SO4Z8D3PX1CO+qMAeh9gxszfzYoXcMf7d8p9NG+0RepWVKfGsYE2SXRXghphxB1jaE9qfmUXauh5olvmuq1+p30HcK/SK5HUFsiEt1FmcUFdgYr+NviJ+8XEVqng2o1Sow6cd18yWDmi5Po0k+r7HKhLWj2dQe7h79FpElAq1i39uOfa5Jolol7pvMW5Rj0hb6HVVtWXs6Be0im9cdqdxrRbzT2pXd8ZRrSIXsFItk5Xb1FRTm2bFEgGxe9hrtumGuicnXF1/xKom7c+7huRo0T6RyuNTjFWL2XhNqSKueHHyOietZYN1UmEsxfdX4NimVNvQE9ZmZDQtDdXfjqmSkPGwefdNTuh6F223IzM90H3jKQ98us5lT5KZ34HZYE5d0HvD87MQ94x8thtIx++ROlvuhJVVlI0x6g1mZsu7HDn0jx5JxDVeRF9HXMu3g581NDMv9v3gBWOIkrGpQZoXvcXcmLzyI7WKXrCKrCr6UyItlJhPJxHGzwecz4ErvB1qCUCRCu8N5vulMdgFpW881uZcVoe5DS47X4ucFybzNIeuqHSEQ6p6tlbJy0l5o+mzZVuDfC/KavpsWaHcMDdd0w8SYp2WyKWMg0j4Q1l6VCIVck/QdG3z2Rz2BdxruyYvYmPhmxJJnz1Gryo0Vfm61L6UXTA2PhGZ397Dgzfsf30+AS3qx1w2Xcibudzde/k0l79d0LO5pL+gf3PJdyWSDvcGd0ATmcPSmhKug1IunU9le4KD3SR7ftHVVuG1CS/Knt0yncky+41khd4dt9kDKnP15/F9Ryq5XlrHXXbfJvtQTM2tGbVzfG9oeQS7c6wvYb7n58t56muS3Px5s8M4TOqO2Yx+AsmfQrLTQzK0P7nEz08WPT4F3pqdLHseXGcG6Bi+Qi98jWesPjX9gOe7Oq77hV1vXlQ5Z3VNvxzG3QDyFvarXe5ri+NG5jS0XjgzFXXsQUP265vTZc7tmTPANKRF9E5G1lnXGjuLjGrST5vX4NInsLsSxDyPJ+PNju9kZP2mJOQJID+H3Qfy8BlzPrw8Z6a/ESh7TzH99Bzh3sbVY0dPDO8us85jZ3fQY/QJXz897Yy9q5D83UBVSO72nj0d0VfEHz39XvH02gVHzqNPMSmlNW0Ycdo5OqJMap4WE54585N1tXBS1nijtiVDeeUbvG3TjxyzWyy9XV6tj759Gx+cbFTrP1Zrh5urLzfNm7lH6hv1TK3BT1vqJSrzAH611Z/qb/WP+nfl65WfV/ZXXmnWhw+MzFeq8Fk5/B+BGv5b</latexit>

Brady Neal

Y (0) | T = 1]

Difference-in-Differences Overview

9 / 25

Introducing Time

Treatment
administered

E[Y1 | T = 1]
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Only group that’s
received treatment

Treatment
Group
E[Y0 | T = 1]
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ControlGroup

E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">AAAUj3icrVjbbttGEN2kt8S92anf8sJGKdAHWZFdF+6LCwO2g7ZoAAfwLY0NQ6IoixBvJSm7jqCHfk3fivZz+jc9c3YpirpRCixB5Gp25szs3JbLZuS5SVqv//fg4QcffvTxJ48er3z62edffLm69uQ0CXux7ZzYoRfG581G4nhu4Jykbuo551HsNPym55w1u/syf3bjxIkbBsfpXeRc+o3rwG27diMF6Wr16cXh2zdXdevCd1vWRQrR1HeC1Nq16pdXq5V6rc6PNTnYNIOKMp+jcO3x3+pCtVSobNVTvnJUoFKMPdVQCb5v1aaqqwi0S9UHLcbI5byjBmoFsj1wOeBogNrF9Rr/3hpqgP+CmVDahhYPvxiSlvrG8LQwbpOq76LfGuGdpaNPbLHxDvemwfRBTVUH1DK5jHNxuSa+fsmqU6zhB67WxUoiUsQPdmHNbdw9/E+xQrnegdPBqAWpGCMbNA9UTREdMe7a8+KbDiPRIJ+DkVg9z+55Vst8iG8XWA2ME1oqtlrqpYlLQM0ObRUejzGdjfgHVqjtm4fVHq7BZdz1KoT3GJSueodREXmeztHsms2VGuTBFF3Cn5IjAHcCSsjMd+E/FxzFXLWBKToajOQ11xIxg2oG+RfOZ9kcIZ4btCdh1Czmm0s9kams3O7MTg/3JrFjyPcx16Ee8UUV9jjEjpm9mRer5I7x75b/bNpoj9FrrEyJZxVrkuyqAjfEjDvE0p7Q/sws0tb1QbPMd0P9Sv0O4l6lVySvq5ANaaHO4oS6AhP7XfQV8YuPq1DFsxmlSh0+7egwW7qg5fo0kuh7gVUkrB/PoPZx9+i1iChVahf/3HLsc00S0R5132Lcoh6RttDramrH2DEoaBXfuOxOk1ot5p/Uru6M41pFLmClWiYrd6mprrbNiiUCYveo12zTDXVPTri6wZhVTdqfdw3J0aJ9IpXHpxirFrOxQ6kirnhx+jqnrWWLdVJlLMX31+DYpVTb0BPWZmQ0rYzU376pkpDxsHn3TU7oehdtt2MzfdB94ykPfLrOZU+Smd+B2WBOXdB7o/PzEA+NfLYbSMfvkzpf7pSVVZSNMeoPZ+bLuxw59I8eScQ1XkRfR1zL18OfNTKzKPb94AUTiJKxqUFaFL3F3Ji+8mNVQS+oIKuK/pRICyXm00mE8fMh53PgCm+XWgJQpML7w/lBaQwOQBkYj7U5l9VhboPLztci54XJPM2hKyod45Cqnq9V8nJa3mj6fNnWMN+Lspo+X1YoN8xN1/SDhFjnJXIp4yAS/kiWHpdIhdwTNF3b/GYB+wLutT2TF7Gx8KxE0meP0asKTVW+KrUvZReMjU9E5rf38OAN+9+AT0DL+jGXTZfyZi53914+zeVvl/RsLukv6d9c8l2JpMO9wR3SROZ1aU0J11Epl86nsj3BwW6SPb/oaqvy2oQXZc9umc5kmf1GskLvjrvsAdWF+vPkviOVvFlaxz123yb7UEzNrTm1c3JvaHkEewusL2G+5+fLReprmtziebPPOEzrjtmMfgLJn0Ky00Mysj+5xM9PFn0+Bd6anSx7HtxgBugY/oRe+ArPWANq+g7Pd5u4vizseouiyjmrZ/rlKO4WkHewXx1wX1seNzKnoY3Cmamo4xAast/AnC5zbs+cAWYhLaN3OrLOutbEWWRck37a7IBLn8DuShDzPJ6ONz++05H1m5KQJ4D8HHYfyKNnzMXw8pyZ/Uag7D3F7NNzhHsbV48dPTG8B8w6j53dQY/RJ3z99LQ/8a5C8ncLVSG523/2ZExfEX/89HvN02sPHDmPPsWklNa0UcRZ5+iIMql5Wkx45sxP1rXCSVnjjduWjOSVb/B2TT9yzG6xcrVa2Rx/+zY5ON2qbX5fq7/eruxtmzdzj9RT9Ux9Cz/tqD1U5hH8aqs/1V/qH/Xv+tr6zvqP63ua9eEDI/OVKnzWf/4f5wH80A==</latexit>time

ATT estimand without time: E[Y (1)<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">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</latexit>

Brady Neal

Y (0) | T = 1]

Difference-in-Differences Overview

9 / 25

Introducing Time

Treatment
administered

E[Y1 | T = 1]
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Only group that’s
received treatment

Treatment
Group
E[Y0 | T = 1]
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ControlGroup

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E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>time

ATT estimand without time: E[Y (1)<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">AAAUl3icrVjbbttGEN2kt8S9Oalfir6wcQo4gKxKrgv3xWgAXxAUDWADtmPXNgKJomxCvJWk7DqCPqA/0762n9K/6ZmzS1HUjVJgCSJXszNnZue2XDYjz03SWu2/Bw8/+PCjjz959Hjp088+/+LL5SdPT5KwG9vOsR16YXzabCSO5wbOceqmnnMaxU7Db3rOm2ZnR+bf3Dhx4obBUXoXOZd+4ypw267dSEF6u7x6sXd+tlZ/Ya1bZ2u1F9aF77asixQQqe8EqbVt1S/BVavW+LHGB3UzWFXmcxA+efyXulAtFSpbdZWvHBWoFGNPNVSC77mqq5qKQLtUPdBijFzOO6qvliDbBZcDjgaoHVyv8O/cUAP8F8yE0ja0ePjFkLTUd4anhXGbVH0X/dYQ7zQdPWKLjXe4Nw2mD2qqrkEtk8s455dr4uuXrDrFGn7ial2sJCJF/GAX1tzG3cP/FCuU6x04HYxakIoxskHzQNUU0RHjrj0vvrlmJBrkczASq2fZPctqmQ/x7QCrgXFCS8VWS+2buATU7NBW4fEY0+mIf2CF2r5ZWO3BGlzGXa9CeI9A6ah3GBWRZ+kczq7pXKlB7k/QJfwpOQJwJ6CEzHwX/nPBUcxVG5iio8FIXnEtETOoapB/4XyWzRHiuU57EkbNYr651BOZysrtzuz0cG8SO4Z8D3PX1CO+qMAeh9gxszfzYoXcMf7d8p9NG+0RepWVKfGsYE2SXRXghphxB1jaE9qfmUXauh5olvmuq1+p30HcK/SK5HUFsiEt1FmcUFdgYr+NviJ+8XEVqng2o1Sow6cd18yWDmi5Po0k+r7HKhLWj2dQe7h79FpElAq1i39uOfa5Jolol7pvMW5Rj0hb6HVVtWXs6Be0im9cdqdxrRbzT2pXd8ZRrSIXsFItk5Xb1FRTm2bFEgGxe9hrtumGuicnXF1/xKom7c+7huRo0T6RyuNTjFWL2XhNqSKueHHyOietZYN1UmEsxfdX4NimVNvQE9ZmZDQtDdXfjqmSkPGwefdNTuh6F223IzM90H3jKQ98us5lT5KZ34HZYE5d0HvD87MQ94x8thtIx++ROlvuhJVVlI0x6g1mZsu7HDn0jx5JxDVeRF9HXMu3g581NDMv9v3gBWOIkrGpQZoXvcXcmLzyI7WKXrCKrCr6UyItlJhPJxHGzwecz4ErvB1qCUCRCu8N5vulMdgFpW881uZcVoe5DS47X4ucFybzNIeuqHSEQ6p6tlbJy0l5o+mzZVuDfC/KavpsWaHcMDdd0w8SYp2WyKWMg0j4Q1l6VCIVck/QdG3z2Rz2BdxruyYvYmPhmxJJnz1Gryo0Vfm61L6UXTA2PhGZ397Dgzfsf30+AS3qx1w2Xcibudzde/k0l79d0LO5pL+gf3PJdyWSDvcGd0ATmcPSmhKug1IunU9le4KD3SR7ftHVVuG1CS/Knt0yncky+41khd4dt9kDKnP15/F9Ryq5XlrHXXbfJvtQTM2tGbVzfG9oeQS7c6wvYb7n58t56muS3Px5s8M4TOqO2Yx+AsmfQrLTQzK0P7nEz08WPT4F3pqdLHseXGcG6Bi+Qi98jWesPjX9gOe7Oq77hV1vXlQ5Z3VNvxzG3QDyFvarXe5ri+NG5jS0XjgzFXXsQUP265vTZc7tmTPANKRF9E5G1lnXGjuLjGrST5vX4NInsLsSxDyPJ+PNju9kZP2mJOQJID+H3Qfy8BlzPrw8Z6a/ESh7TzH99Bzh3sbVY0dPDO8us85jZ3fQY/QJXz897Yy9q5D83UBVSO72nj0d0VfEHz39XvH02gVHzqNPMSmlNW0Ycdo5OqJMap4WE54585N1tXBS1nijtiVDeeUbvG3TjxyzWyy9XV6tj759Gx+cbFTrP1Zrh5urLzfNm7lH6hv1TK3BT1vqJSrzAH611Z/qb/WP+nfl65WfV/ZXXmnWhw+MzFeq8Fk5/B+BGv5b</latexit>

ATT estimand with time: E[Y1 (1)<latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">AAAUm3icrVjbbttGEN2kt8S9xEn9VhRg4xRNAFmVXBfui4EAtoMiSAAH8CWpZRgSRdmEeCtJWXUEfUJ/pS99bT+kf9MzZ5eiqBvlwBJErmZnzszObblsRZ6bpLXaf3fufvTxJ59+du/+yudffPnVg9WHj46TsBfbzpEdemH8ttVMHM8NnKPUTT3nbRQ7Tb/lOSet7q7Mn1w5ceKGwWF6HTlnfvMicDuu3UxBOl/9obF/+u68/rT+zNqwZFB7ZjV8t201UsCkvhOk1o5VPztfXa9Va/xY04O6Gawr8zkIH97/SzVUW4XKVj3lK0cFKsXYU02V4Huq6qqmItDO1AC0GCOX844aqhXI9sDlgKMJahfXC/w7NdQA/wUzobQNLR5+MSQt9b3haWPcIVXfRb81xjtPx4DYYuM17i2D6YOaqktQy+QyzuXlWvj6JatOsYZfuFoXK4lIET/YhTV3cPfwP8UK5XoNTgejNqRijGzQPFA1RXTEuGvPi28uGYkm+RyMxOpFdi+yWuZDfLvAamKc0FKx1VIvTFwCanZoq/B4jOl8xD+wQm3fIqzOaA0u465XIbyHoHTVe4yKyIt0jmfXfK7UIA9n6BL+lBwBuBNQQma+C/+54Cjmqg1M0dFkJC+4logZVDXILzmfZXOEeG7QnoRRs5hvLvVEprJyuzM7PdxbxI4hP8DcJfWILyqwxyF2zOzNvFghd4x/ff6zaaM9Qa+yMiWeFaxJsqsC3BAz7ghLe0L7M7NIWzcAzTLfDfWK+h3EvUKvSF5XIBvSQp3FCXUFJvY76CviFx9XoYpnM0qFOnzaccls6YKW69NIou9HrCJh/XgGdYC7R69FRKlQu/inz7HPNUlEe9Tdx7hNPSJtoddV1baxY1jQKr5x2Z2mtVrMP6ld3RkntYpcwEq1TFbuUFNNbZkVSwTE7nGv2aYb6p6ccHXDCatatD/vGpKjRftEKo9PMVZtZuMlpYq44sXZ65y1lk3WSYWxFN9fgGOHUh1DT1ibkdG0MlZ/u6ZKQsbD5t03OaHrXbT1J2YGoPvGUx74dJ3LniQzvwOzyZxq0Hvj84sQ9418thtIxx+QuljumJVVlI0xGoxmFsu7HDn0jx5JxDVeRF9HXMt3o581NrMs9u3gBVOIkrGpQVoWvc3cmL3yQ7WOXrCOrCr6UyItlJhPJxHGT0acT4ArvF1qCUCRCh+M5oelMdgDZWg81uFcVoe5DS47X5ucDZN5mkNXVDrBIVW9WKvk5ay80fTFsu1RvhdlNX2xrFCumJuu6QcJsd6WyKWMg0j4Y1l6WCIVck/QdG3zuyXsC7jX9kxexMbCkxJJnz1Gryo0Vfm61L6UXTA2PhGZ3z7Ag1fsf0M+Ad3Uj7lseiNv5nLXH+TTXL5/Q8/mkv4N/ZtLvi+RdLg3uCOayLwprSnhOijl0vlUtic42E2y5xddbRVeW/Ci7Nlt05kss99IVujdcYc9oLJUf57ed6SS66V13GP3bbEPxdTcXlA7R7eGlkewt8T6EuZ7fr5cpr5myS2fN7uMw6zumM3oJ5D8KSQ7PSRj+5NL/PxkMeBTYN/sZNnz4AYzQMfwV/TC13jGGlLTT3i+q+P6orDrLYsq56ye6ZfjuJtA3sZ+tcd97ea4kTkNbRTOTEUd+9CQ/YbmdJlze+YMMA/pJnpnI+usa0+dRSY16afNS3DpE9h1CWKex7PxFsd3NrJ+UxLyBJCfw24DefyMuRxenjPz3wiUvaeYf3qOcO/g6rGjJ4Z3j1nnsbM76DH6hK+fnnan3lVI/m6iKiR3B48fTegr4k+efi94eu2BI+fRp5iU0po2jjjvHB1RJjVPiwnPnPnJulo4KWu8SduSsbzyDd6O6UeO2S1WzlfX65Nv36YHx5vV+s/V2put9edb5s3cPfWNeqyewk/b6jkq8wB+tdWf6m/1j/p37du13bWXa6806907RuZrVfisHf0PG2j/ow==</latexit>

Brady Neal

Y (0) | T = 1]

Y1 (0) | T = 1]

Difference-in-Differences Overview

9 / 25

Introducing Time

Treatment
administered

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit><latexit sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">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</latexit>

Treatment
Group
E[Y0 | T = 1]
<latexit sha1_base64="GWKSlxQAH7OAOhoCpjA48XtUgBE=">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</latexit>

ControlGroup

E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>time

ATT estimand without time: E[Y (1)<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">AAAUl3icrVjbbttGEN2kt8S9Oalfir6wcQo4gKxKrgv3xWgAXxAUDWADtmPXNgKJomxCvJWk7DqCPqA/0762n9K/6ZmzS1HUjVJgCSJXszNnZue2XDYjz03SWu2/Bw8/+PCjjz959Hjp088+/+LL5SdPT5KwG9vOsR16YXzabCSO5wbOceqmnnMaxU7Db3rOm2ZnR+bf3Dhx4obBUXoXOZd+4ypw267dSEF6u7x6sXd+tlZ/Ya1bZ2u1F9aF77asixQQqe8EqbVt1S/BVavW+LHGB3UzWFXmcxA+efyXulAtFSpbdZWvHBWoFGNPNVSC77mqq5qKQLtUPdBijFzOO6qvliDbBZcDjgaoHVyv8O/cUAP8F8yE0ja0ePjFkLTUd4anhXGbVH0X/dYQ7zQdPWKLjXe4Nw2mD2qqrkEtk8s455dr4uuXrDrFGn7ial2sJCJF/GAX1tzG3cP/FCuU6x04HYxakIoxskHzQNUU0RHjrj0vvrlmJBrkczASq2fZPctqmQ/x7QCrgXFCS8VWS+2buATU7NBW4fEY0+mIf2CF2r5ZWO3BGlzGXa9CeI9A6ah3GBWRZ+kczq7pXKlB7k/QJfwpOQJwJ6CEzHwX/nPBUcxVG5iio8FIXnEtETOoapB/4XyWzRHiuU57EkbNYr651BOZysrtzuz0cG8SO4Z8D3PX1CO+qMAeh9gxszfzYoXcMf7d8p9NG+0RepWVKfGsYE2SXRXghphxB1jaE9qfmUXauh5olvmuq1+p30HcK/SK5HUFsiEt1FmcUFdgYr+NviJ+8XEVqng2o1Sow6cd18yWDmi5Po0k+r7HKhLWj2dQe7h79FpElAq1i39uOfa5Jolol7pvMW5Rj0hb6HVVtWXs6Be0im9cdqdxrRbzT2pXd8ZRrSIXsFItk5Xb1FRTm2bFEgGxe9hrtumGuicnXF1/xKom7c+7huRo0T6RyuNTjFWL2XhNqSKueHHyOietZYN1UmEsxfdX4NimVNvQE9ZmZDQtDdXfjqmSkPGwefdNTuh6F223IzM90H3jKQ98us5lT5KZ34HZYE5d0HvD87MQ94x8thtIx++ROlvuhJVVlI0x6g1mZsu7HDn0jx5JxDVeRF9HXMu3g581NDMv9v3gBWOIkrGpQZoXvcXcmLzyI7WKXrCKrCr6UyItlJhPJxHGzwecz4ErvB1qCUCRCu8N5vulMdgFpW881uZcVoe5DS47X4ucFybzNIeuqHSEQ6p6tlbJy0l5o+mzZVuDfC/KavpsWaHcMDdd0w8SYp2WyKWMg0j4Q1l6VCIVck/QdG3z2Rz2BdxruyYvYmPhmxJJnz1Gryo0Vfm61L6UXTA2PhGZ397Dgzfsf30+AS3qx1w2Xcibudzde/k0l79d0LO5pL+gf3PJdyWSDvcGd0ATmcPSmhKug1IunU9le4KD3SR7ftHVVuG1CS/Knt0yncky+41khd4dt9kDKnP15/F9Ryq5XlrHXXbfJvtQTM2tGbVzfG9oeQS7c6wvYb7n58t56muS3Px5s8M4TOqO2Yx+AsmfQrLTQzK0P7nEz08WPT4F3pqdLHseXGcG6Bi+Qi98jWesPjX9gOe7Oq77hV1vXlQ5Z3VNvxzG3QDyFvarXe5ri+NG5jS0XjgzFXXsQUP265vTZc7tmTPANKRF9E5G1lnXGjuLjGrST5vX4NInsLsSxDyPJ+PNju9kZP2mJOQJID+H3Qfy8BlzPrw8Z6a/ESh7TzH99Bzh3sbVY0dPDO8us85jZ3fQY/QJXz897Yy9q5D83UBVSO72nj0d0VfEHz39XvH02gVHzqNPMSmlNW0Ycdo5OqJMap4WE54585N1tXBS1nijtiVDeeUbvG3TjxyzWyy9XV6tj759Gx+cbFTrP1Zrh5urLzfNm7lH6hv1TK3BT1vqJSrzAH611Z/qb/WP+nfl65WfV/ZXXmnWhw+MzFeq8Fk5/B+BGv5b</latexit>

ATT estimand with time: E[Y1 (1)<latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

Brady Neal

Y (0) | T = 1]

Y1 (0) | T = 1]

Difference-in-Differences Overview

9 / 25

Introducing Time

Treatment
administered

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit><latexit sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">AAAUsHicrVjbbttGEN2kt8S9Oanf+sJGKZACsiq5LtyHGghgOyiKBnAA31LbMCSKslnxVpKy6gj6mX5Av6Ov7Vv/pmfOLkVRN0pBJIhczc6cmZ3bctmKPDdJ6/X/7t1/7/0PPvzowcO1jz/59LPP1x89PknCXmw7x3bohfFZq5k4nhs4x6mbes5ZFDtNv+U5p63unsyf3jpx4obBUXoXOZd+8zpwO67dTEG6Wv/x4uD89VXDuvDdtnWRQjT1nSC1dq3GJS569lnjm5kMV+uVeq3OjzU9aJhBRZnPYfjo4V/qQrVVqGzVU75yVKBSjD3VVAm+56qh6ioC7VINQIsxcjnvqKFag2wPXA44mqB2cb3Gv3NDDfBfMBNK29Di4RdD0lJfG542xh1S9V30W2O883QMiC023uHeMpg+qKm6AbVMLuNcXq6Fr1+y6hRr+IGrdbGSiBTxg11Ycwd3D/9TrFCud+B0MGpDKsbIBs0DVVNER4y79rz45oaRaJLPwUisXmT3IqtlPsS3C6wmxgktFVst9cLEJaBmh7YKj8eYzkf8AyvU9i3C6ozW4DLuehXCewRKV73BqIi8SOd4ds3nSg3ycIYu4U/JEYA7ASVk5rvwnwuOYq7awBQdTUbymmuJmEE1g/wz57NsjhDPTdqTMGoW882lnshUVm53ZqeHe4vYMeQHmLuhHvFFFfY4xI6ZvZkXq+SO8a/PfzZttCfoNVamxLOKNUl2VYEbYsYdYWlPaH9mFmnrBqBZ5rupfqF+B3Gv0iuS11XIhrRQZ3FCXYGJ/S76ivjFx1Wo4tmMUqUOn3bcMFu6oOX6NJLo+xarSFg/nkEd4O7RaxFRqtQu/ulz7HNNEtEedfcxblOPSFvodTW1Y+wYFrSKb1x2p2mtFvNPald3xkmtIhewUi2TlbvUVFfbZsUSAbF73Gu26Ya6Jydc3XDCqhbtz7uG5GjRPpHK41OMVZvZeEOpIq54cfY6Z61li3VSZSzF99fg2KVUx9AT1mZkNK2N1d+eqZKQ8bB5901O6HoXbf2JmQHovvGUBz5d57InyczvwGwypy7ovfH5RYgHRj7bDaTjD0hdLHfCyirKxhgNRjOL5V2OHPpHjyTiGi+iryOu5avRzxqbWRb73eAFU4iSsalBWha9zdyYvfIjVUEvqCCriv6USAsl5tNJhPHTEedT4Apvl1oCUKTCB6P5YWkM9kEZGo91OJfVYW6Dy87XJueFyTzNoSsqneCQql6sVfJyVt5o+mLZ9ijfi7KavlhWKLfMTdf0g4RYZyVyKeMgEv5Ylh6VSIXcEzRd2/x6CfsC7rU9kxexsfC0RNJnj9GrCk1Vviy1L2UXjI1PRObXt/DgLfvfkE9Aq/oxl01X8mYud/dWPs3l+yt6Npf0V/RvLvmmRNLh3uCOaCLzqrSmhOuwlEvnU9me4GA3yZ5fdLVVeW3Bi7Jnt01nssx+I1mhd8dd9oDqUv15et+RSm6U1nGP3bfFPhRTc3tB7Ry/M7Q8gr0l1pcw3/Pz5TL1NUtu+bzZYxxmdcdsRj+B5E8h2ekhGdufXOLnJ4sBnwL7ZifLngc3mQE6hj+hF77EM9aQmr7D810D1xeFXW9ZVDln9Uy/HMfdAvIO9qt97mur40bmNLRZODMVdRxAQ/YbmtNlzu2ZM8A8pFX0zkbWWdeeOotMatJPmzfg0iewuxLEPI9n4y2O72xk/aYk5AkgP4e9C+TxM+ZyeHnOzH8jUPaeYv7pOcK9g6vHjp4Y3n1mncfO7qDH6BO+fnram3pXIfm7haqQ3B08eTyhr4g/efq95um1B46cR59iUkpr2jjivHN0RJnUPC0mPHPmJ+ta4aSs8SZtS8byyjd4u6YfOWa3WLtarzQm375ND062ao3va/VX25Xn2+bN3AP1pXqinsFPO+o5KvMQfrXVn+pv9Y/6d2Nr42zjaqOpWe/fMzJfqMJn47f/AcaqCCA=</latexit>

Treatment
Group
E[Y0 | T = 1]
<latexit sha1_base64="GWKSlxQAH7OAOhoCpjA48XtUgBE=">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</latexit>

ControlGroup

E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>time

ATT estimand without time: E[Y (1)<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">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</latexit>

ATT estimand with time: E[Y1 (1)<latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

Brady Neal

Y (0) | T = 1]

Y1 (0) | T = 1]

Difference-in-Differences Overview

9 / 25

Introducing Time

Treatment
administered

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
E[Y1 (0) | T = 1] ?
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Treatment
Group
E[Y0 | T = 1]
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ControlGroup

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E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>time

ATT estimand without time: E[Y (1)<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">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</latexit>

ATT estimand with time: E[Y1 (1)<latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

Brady Neal

Y (0) | T = 1]

Y1 (0) | T = 1]

Difference-in-Differences Overview

9 / 25

Introducing Time

Treatment
administered

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
E[Y1 (0) | T = 1] ?
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Treatment
Group
E[Y0 | T = 1]
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ControlGroup

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E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>time

ATT estimand without time: E[Y (1)<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">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</latexit>

ATT estimand with time: E[Y1 (1)<latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

Brady Neal

Y (0) | T = 1]

Y1 (0) | T = 1]

Difference-in-Differences Overview

9 / 25

Introducing Time

Treatment
administered

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
E[Y1 (0) | T = 1] ?
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Treatment
Group
E[Y0 | T = 1]
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ControlGroup

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E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>time

ATT estimand without time: E[Y (1)<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">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</latexit>

ATT estimand with time: E[Y1 (1)<latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">AAAUm3icrVjbbttGEN2kt8S9xEn9VhRg4xRNAFmVXBfui4EAtoMiSAAH8CWpZRgSRdmEeCtJWXUEfUJ/pS99bT+kf9MzZ5eiqBvlwBJErmZnzszObblsRZ6bpLXaf3fufvTxJ59+du/+yudffPnVg9WHj46TsBfbzpEdemH8ttVMHM8NnKPUTT3nbRQ7Tb/lOSet7q7Mn1w5ceKGwWF6HTlnfvMicDuu3UxBOl/9obF/+u68/rT+zNqwZFB7ZjV8t201UsCkvhOk1o5VPztfXa9Va/xY04O6Gawr8zkIH97/SzVUW4XKVj3lK0cFKsXYU02V4Huq6qqmItDO1AC0GCOX844aqhXI9sDlgKMJahfXC/w7NdQA/wUzobQNLR5+MSQt9b3haWPcIVXfRb81xjtPx4DYYuM17i2D6YOaqktQy+QyzuXlWvj6JatOsYZfuFoXK4lIET/YhTV3cPfwP8UK5XoNTgejNqRijGzQPFA1RXTEuGvPi28uGYkm+RyMxOpFdi+yWuZDfLvAamKc0FKx1VIvTFwCanZoq/B4jOl8xD+wQm3fIqzOaA0u465XIbyHoHTVe4yKyIt0jmfXfK7UIA9n6BL+lBwBuBNQQma+C/+54Cjmqg1M0dFkJC+4logZVDXILzmfZXOEeG7QnoRRs5hvLvVEprJyuzM7PdxbxI4hP8DcJfWILyqwxyF2zOzNvFghd4x/ff6zaaM9Qa+yMiWeFaxJsqsC3BAz7ghLe0L7M7NIWzcAzTLfDfWK+h3EvUKvSF5XIBvSQp3FCXUFJvY76CviFx9XoYpnM0qFOnzaccls6YKW69NIou9HrCJh/XgGdYC7R69FRKlQu/inz7HPNUlEe9Tdx7hNPSJtoddV1baxY1jQKr5x2Z2mtVrMP6ld3RkntYpcwEq1TFbuUFNNbZkVSwTE7nGv2aYb6p6ccHXDCatatD/vGpKjRftEKo9PMVZtZuMlpYq44sXZ65y1lk3WSYWxFN9fgGOHUh1DT1ibkdG0MlZ/u6ZKQsbD5t03OaHrXbT1J2YGoPvGUx74dJ3LniQzvwOzyZxq0Hvj84sQ9418thtIxx+QuljumJVVlI0xGoxmFsu7HDn0jx5JxDVeRF9HXMt3o581NrMs9u3gBVOIkrGpQVoWvc3cmL3yQ7WOXrCOrCr6UyItlJhPJxHGT0acT4ArvF1qCUCRCh+M5oelMdgDZWg81uFcVoe5DS47X5ucDZN5mkNXVDrBIVW9WKvk5ay80fTFsu1RvhdlNX2xrFCumJuu6QcJsd6WyKWMg0j4Y1l6WCIVck/QdG3zuyXsC7jX9kxexMbCkxJJnz1Gryo0Vfm61L6UXTA2PhGZ3z7Ag1fsf0M+Ad3Uj7lseiNv5nLXH+TTXL5/Q8/mkv4N/ZtLvi+RdLg3uCOayLwprSnhOijl0vlUtic42E2y5xddbRVeW/Ci7Nlt05kss99IVujdcYc9oLJUf57ed6SS66V13GP3bbEPxdTcXlA7R7eGlkewt8T6EuZ7fr5cpr5myS2fN7uMw6zumM3oJ5D8KSQ7PSRj+5NL/PxkMeBTYN/sZNnz4AYzQMfwV/TC13jGGlLTT3i+q+P6orDrLYsq56ye6ZfjuJtA3sZ+tcd97ea4kTkNbRTOTEUd+9CQ/YbmdJlze+YMMA/pJnpnI+usa0+dRSY16afNS3DpE9h1CWKex7PxFsd3NrJ+UxLyBJCfw24DefyMuRxenjPz3wiUvaeYf3qOcO/g6rGjJ4Z3j1nnsbM76DH6hK+fnnan3lVI/m6iKiR3B48fTegr4k+efi94eu2BI+fRp5iU0po2jjjvHB1RJjVPiwnPnPnJulo4KWu8SduSsbzyDd6O6UeO2S1WzlfX65Nv36YHx5vV+s/V2put9edb5s3cPfWNeqyewk/b6jkq8wB+tdWf6m/1j/p37du13bWXa6806907RuZrVfisHf0PG2j/ow==</latexit>

Brady Neal

Y (0) | T = 1]

Y1 (0) | T = 1]

Difference-in-Differences Overview

9 / 25

Introducing Time

Treatment
administered

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit><latexit sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">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</latexit>

Treatment
Group
E[Y1 (0) | T = 1] ?
<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit><latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit>

E[Y0 | T = 1]
<latexit sha1_base64="GWKSlxQAH7OAOhoCpjA48XtUgBE=">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</latexit>

ControlGroup

E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>time

ATT estimand without time: E[Y (1)<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">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</latexit>

ATT estimand with time: E[Y1 (1)<latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

Brady Neal

Y (0) | T = 1]

Y1 (0) | T = 1]

Difference-in-Differences Overview

9 / 25

Introducing Time

Treatment
administered

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit><latexit sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">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</latexit>

TreatmentGroup<latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

E[Y1 (1)Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
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E[Y0 | T = 1]
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ControlGroup

E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>time

ATT estimand without time: E[Y (1)<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">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</latexit>

ATT estimand with time: E[Y1 (1)<latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

Brady Neal

Y (0) | T = 1]

Y1 (0) | T = 1]

Difference-in-Differences Overview

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Introducing Time

Treatment
administered

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit><latexit sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">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</latexit>

TreatmentGroup<latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

E[Y1 (1)Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
Assume away<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit><latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit>

E[Y0 | T = 1]
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ControlGroup

E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>time

ATT estimand without time: E[Y (1)<latexit sha1_base64="gm5CZCXThNe1/cBAzQ2rU87AxUQ=">AAAUl3icrVjbbttGEN2kt8S9Oalfir6wcQo4gKxKrgv3xWgAXxAUDWADtmPXNgKJomxCvJWk7DqCPqA/0762n9K/6ZmzS1HUjVJgCSJXszNnZue2XDYjz03SWu2/Bw8/+PCjjz959Hjp088+/+LL5SdPT5KwG9vOsR16YXzabCSO5wbOceqmnnMaxU7Db3rOm2ZnR+bf3Dhx4obBUXoXOZd+4ypw267dSEF6u7x6sXd+tlZ/Ya1bZ2u1F9aF77asixQQqe8EqbVt1S/BVavW+LHGB3UzWFXmcxA+efyXulAtFSpbdZWvHBWoFGNPNVSC77mqq5qKQLtUPdBijFzOO6qvliDbBZcDjgaoHVyv8O/cUAP8F8yE0ja0ePjFkLTUd4anhXGbVH0X/dYQ7zQdPWKLjXe4Nw2mD2qqrkEtk8s455dr4uuXrDrFGn7ial2sJCJF/GAX1tzG3cP/FCuU6x04HYxakIoxskHzQNUU0RHjrj0vvrlmJBrkczASq2fZPctqmQ/x7QCrgXFCS8VWS+2buATU7NBW4fEY0+mIf2CF2r5ZWO3BGlzGXa9CeI9A6ah3GBWRZ+kczq7pXKlB7k/QJfwpOQJwJ6CEzHwX/nPBUcxVG5iio8FIXnEtETOoapB/4XyWzRHiuU57EkbNYr651BOZysrtzuz0cG8SO4Z8D3PX1CO+qMAeh9gxszfzYoXcMf7d8p9NG+0RepWVKfGsYE2SXRXghphxB1jaE9qfmUXauh5olvmuq1+p30HcK/SK5HUFsiEt1FmcUFdgYr+NviJ+8XEVqng2o1Sow6cd18yWDmi5Po0k+r7HKhLWj2dQe7h79FpElAq1i39uOfa5Jolol7pvMW5Rj0hb6HVVtWXs6Be0im9cdqdxrRbzT2pXd8ZRrSIXsFItk5Xb1FRTm2bFEgGxe9hrtumGuicnXF1/xKom7c+7huRo0T6RyuNTjFWL2XhNqSKueHHyOietZYN1UmEsxfdX4NimVNvQE9ZmZDQtDdXfjqmSkPGwefdNTuh6F223IzM90H3jKQ98us5lT5KZ34HZYE5d0HvD87MQ94x8thtIx++ROlvuhJVVlI0x6g1mZsu7HDn0jx5JxDVeRF9HXMu3g581NDMv9v3gBWOIkrGpQZoXvcXcmLzyI7WKXrCKrCr6UyItlJhPJxHGzwecz4ErvB1qCUCRCu8N5vulMdgFpW881uZcVoe5DS47X4ucFybzNIeuqHSEQ6p6tlbJy0l5o+mzZVuDfC/KavpsWaHcMDdd0w8SYp2WyKWMg0j4Q1l6VCIVck/QdG3z2Rz2BdxruyYvYmPhmxJJnz1Gryo0Vfm61L6UXTA2PhGZ397Dgzfsf30+AS3qx1w2Xcibudzde/k0l79d0LO5pL+gf3PJdyWSDvcGd0ATmcPSmhKug1IunU9le4KD3SR7ftHVVuG1CS/Knt0yncky+41khd4dt9kDKnP15/F9Ryq5XlrHXXbfJvtQTM2tGbVzfG9oeQS7c6wvYb7n58t56muS3Px5s8M4TOqO2Yx+AsmfQrLTQzK0P7nEz08WPT4F3pqdLHseXGcG6Bi+Qi98jWesPjX9gOe7Oq77hV1vXlQ5Z3VNvxzG3QDyFvarXe5ri+NG5jS0XjgzFXXsQUP265vTZc7tmTPANKRF9E5G1lnXGjuLjGrST5vX4NInsLsSxDyPJ+PNju9kZP2mJOQJID+H3Qfy8BlzPrw8Z6a/ESh7TzH99Bzh3sbVY0dPDO8us85jZ3fQY/QJXz897Yy9q5D83UBVSO72nj0d0VfEHz39XvH02gVHzqNPMSmlNW0Ycdo5OqJMap4WE54585N1tXBS1nijtiVDeeUbvG3TjxyzWyy9XV6tj759Gx+cbFTrP1Zrh5urLzfNm7lH6hv1TK3BT1vqJSrzAH611Z/qb/WP+nfl65WfV/ZXXmnWhw+MzFeq8Fk5/B+BGv5b</latexit>

ATT estimand with time: E[Y1 (1)<latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

Brady Neal

Y (0) | T = 1]

Y1 (0) | T = 1]

Difference-in-Differences Overview

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Difference-in-Differences
Treatment
administered

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">AAAUj3icrVjbbttGEN2kt8S92anf8sLGKdAHWZFcF+6LCgO2g7ZoAAfwLY0MQ6IomzBvJSmrjqCHfk3fivZz+jc9c3YpirpRDixB5Gp25szs3JbLduS5SVqr/ffg4QcffvTxJ48er3z62edffLm69uQkCXux7RzboRfGZ+1W4nhu4Bynbuo5Z1HstPy255y2r/dk/vTGiRM3DI7S28g591uXgdt17VYK0sXq0+bB2zcXdavpux2rmUI09Z0gtRpW/fxidaNWrfFjTQ/qZrChzOcwXHv8t2qqjgqVrXrKV44KVIqxp1oqwfetqquaikA7VwPQYoxczjtqqFYg2wOXA44WqNe4XuLfW0MN8F8wE0rb0OLhF0PSUt8Yng7GXVL1XfRbY7zzdAyILTbe4t42mD6oqboCtUwu41xero2vX7LqFGv4gat1sZKIFPGDXVhzF3cP/1OsUK634HQw6kAqxsgGzQNVU0RHjLv2vPjmipFokc/BSKxeZPciq2U+xPcaWC2ME1oqtlrqpYlLQM0ObRUejzGdj/gHVqjtW4TVHa3BZdz1KoT3CJRr9Q6jIvIinePZNZ8rNcjDGbqEPyVHAO4ElJCZ78J/LjiKuWoDU3S0GMlLriViBlUN8i+cz7I5Qjw3aU/CqFnMN5d6IlNZud2ZnR7ubWLHkB9g7op6xBcV2OMQO2b2Zl6skDvGvz7/2bTRnqBXWZkSzwrWJNlVAW6IGXeEpT2h/ZlZpK0bgGaZ76b6lfodxL1Cr0heVyAb0kKdxQl1BSb2DfQV8YuPq1DFsxmlQh0+7bhitlyDluvTSKLvBVaRsH48gzrA3aPXIqJUqF380+fY55okoj3q7mPcoR6RttDrqmrH2DEsaBXfuOxO01ot5p/Uru6Mk1pFLmClWiYrG9RUU9tmxRIBsXvca7bphronJ1zdcMKqNu3Pu4bkaNE+kcrjU4xVh9l4Rakirnhx9jpnrWWLdVJhLMX3l+BoUKpr6AlrMzKaVsbqb89USch42Lz7Jid0vYu2/sTMAHTfeMoDn65z2ZNk5ndgtphTTXpvfH4R4oGRz3YD6fgDUhfLnbCyirIxRoPRzGJ5lyOH/tEjibjGi+jriGv5evSzxmaWxb4fvGAKUTI2NUjLoneYG7NXfqQ20As2kFVFf0qkhRLz6STC+PmI8zlwhfeaWgJQpMIHo/lhaQz2QRkaj3U5l9VhboPLztchZ9NknubQFZVOcEhVL9YqeTkrbzR9sWxnlO9FWU1fLCuUG+ama/pBQqyzErmUcRAJfyxLj0qkQu4Jmq5tfrOEfQH32p7Ji9hYeFoi6bPH6FWFpipfldqXsgvGxici89t7ePCG/W/IJ6C7+jGXTe/kzVzu9r18msv37+jZXNK/o39zyXclkg73BndEE5nXpTUlXIelXDqfyvYEB7tJ9vyiq63CaxtelD27YzqTZfYbyQq9OzbYAypL9efpfUcquV5axz123zb7UEzNnQW1c3xvaHkEe0usL2G+5+fLZeprltzyebPHOMzqjtmMfgLJn0Ky00Mytj+5xM9PFgM+BfbNTpY9D24yA3QMf0IvfIVnrCE1fYfnuzquLwu73rKocs7qmX45jrsF5B3sV/vc1+6OG5nT0GbhzFTUcQAN2W9oTpc5t2fOAPOQ7qJ3NrLOus7UWWRSk37avAKXPoHdliDmeTwbb3F8ZyPrNyUhTwD5Oew+kMfPmMvh5Tkz/41A2XuK+afnCPcurh47emJ495l1Hju7gx6jT/j66Wlv6l2F5O8WqkJyd/DsyYS+Iv7k6feSp9ceOHIefYpJKa1p44jzztERZVLztJjwzJmfrKuFk7LGm7QtGcsr3+A1TD9yzG6xcrG6UZ98+zY9ONmq1r+v1l5vb+xumzdzj9RT9Ux9Cz/tqF1U5iH8aqs/1V/qH/Xv+tr6zvqP67ua9eEDI/OVKnzWf/4fD0T80g==</latexit><latexit sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">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</latexit>

Treatment
Group

E[Y1 (1)<latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">AAAUm3icrVjbbttGEN2kt8S9xEn9VhRg4xRNAFmVXBfui4EAtoMiSAAH8CWpZRgSRdmEeCtJWXUEfUJ/pS99bT+kf9MzZ5eiqBvlwBJErmZnzszObblsRZ6bpLXaf3fufvTxJ59+du/+yudffPnVg9WHj46TsBfbzpEdemH8ttVMHM8NnKPUTT3nbRQ7Tb/lOSet7q7Mn1w5ceKGwWF6HTlnfvMicDuu3UxBOl/9obF/+u68/rT+zNqwZFB7ZjV8t201UsCkvhOk1o5VPztfXa9Va/xY04O6Gawr8zkIH97/SzVUW4XKVj3lK0cFKsXYU02V4Huq6qqmItDO1AC0GCOX844aqhXI9sDlgKMJahfXC/w7NdQA/wUzobQNLR5+MSQt9b3haWPcIVXfRb81xjtPx4DYYuM17i2D6YOaqktQy+QyzuXlWvj6JatOsYZfuFoXK4lIET/YhTV3cPfwP8UK5XoNTgejNqRijGzQPFA1RXTEuGvPi28uGYkm+RyMxOpFdi+yWuZDfLvAamKc0FKx1VIvTFwCanZoq/B4jOl8xD+wQm3fIqzOaA0u465XIbyHoHTVe4yKyIt0jmfXfK7UIA9n6BL+lBwBuBNQQma+C/+54Cjmqg1M0dFkJC+4logZVDXILzmfZXOEeG7QnoRRs5hvLvVEprJyuzM7PdxbxI4hP8DcJfWILyqwxyF2zOzNvFghd4x/ff6zaaM9Qa+yMiWeFaxJsqsC3BAz7ghLe0L7M7NIWzcAzTLfDfWK+h3EvUKvSF5XIBvSQp3FCXUFJvY76CviFx9XoYpnM0qFOnzaccls6YKW69NIou9HrCJh/XgGdYC7R69FRKlQu/inz7HPNUlEe9Tdx7hNPSJtoddV1baxY1jQKr5x2Z2mtVrMP6ld3RkntYpcwEq1TFbuUFNNbZkVSwTE7nGv2aYb6p6ccHXDCatatD/vGpKjRftEKo9PMVZtZuMlpYq44sXZ65y1lk3WSYWxFN9fgGOHUh1DT1ibkdG0MlZ/u6ZKQsbD5t03OaHrXbT1J2YGoPvGUx74dJ3LniQzvwOzyZxq0Hvj84sQ9418thtIxx+QuljumJVVlI0xGoxmFsu7HDn0jx5JxDVeRF9HXMt3o581NrMs9u3gBVOIkrGpQVoWvc3cmL3yQ7WOXrCOrCr6UyItlJhPJxHGT0acT4ArvF1qCUCRCh+M5oelMdgDZWg81uFcVoe5DS47X5ucDZN5mkNXVDrBIVW9WKvk5ay80fTFsu1RvhdlNX2xrFCumJuu6QcJsd6WyKWMg0j4Y1l6WCIVck/QdG3zuyXsC7jX9kxexMbCkxJJnz1Gryo0Vfm61L6UXTA2PhGZ3z7Ag1fsf0M+Ad3Uj7lseiNv5nLXH+TTXL5/Q8/mkv4N/ZtLvi+RdLg3uCOayLwprSnhOijl0vlUtic42E2y5xddbRVeW/Ci7Nlt05kss99IVujdcYc9oLJUf57ed6SS66V13GP3bbEPxdTcXlA7R7eGlkewt8T6EuZ7fr5cpr5myS2fN7uMw6zumM3oJ5D8KSQ7PSRj+5NL/PxkMeBTYN/sZNnz4AYzQMfwV/TC13jGGlLTT3i+q+P6orDrLYsq56ye6ZfjuJtA3sZ+tcd97ea4kTkNbRTOTEUd+9CQ/YbmdJlze+YMMA/pJnpnI+usa0+dRSY16afNS3DpE9h1CWKex7PxFsd3NrJ+UxLyBJCfw24DefyMuRxenjPz3wiUvaeYf3qOcO/g6rGjJ4Z3j1nnsbM76DH6hK+fnnan3lVI/m6iKiR3B48fTegr4k+efi94eu2BI+fRp5iU0po2jjjvHB1RJjVPiwnPnPnJulo4KWu8SduSsbzyDd6O6UeO2S1WzlfX65Nv36YHx5vV+s/V2put9edb5s3cPfWNeqyewk/b6jkq8wB+tdWf6m/1j/p37du13bWXa6806907RuZrVfisHf0PG2j/ow==</latexit>

Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">AAAUl3icrVjZbttGFJ2kW+JuTuqXoi9snAIpIKuS68J9MRLAC4KiAWzAW2oZhkRRFmFuJSm7jqAP6M+0r+2n9G967pmhKGqjHFiCyNGde8+9c7fhsBV5bpLWav89ePjBhx99/Mmjx0uffvb5F18uP3l6nIS92HaO7NAL49NWM3E8N3COUjf1nNModpp+y3NOWlfbMn9y7cSJGwaH6W3knPvNy8DtuHYzBeliebWxe/b2ov6i9r3V8N221UghnfpOkFpbVv3calSsl+CqVWv8WJODuhmsKvPZD588/ks1VFuFylY95StHBSrF2FNNleB7puqqpiLQzlUftBgjl/OOGqglyPbA5YCjCeoVrpf4d2aoAf4LZkJpG1o8/GJIWuo7w9PGuEOqvot+a4R3lo4+scXGW9xbBtMHNVVdUMvkMs7F5Vr4+iWrTrGGn7laFyuJSBE/2IU1d3D38D/FCuV6C04HozakYoxs0DxQNUV0xLhrz4tvuoxEk3wORmL1PLvnWS3zIb5XwGpinNBSsdVSeyYuATU7tFV4PMZ0NuIfWKG2bx5WZ7gGl3HXqxDeQ1Cu1DuMisjzdI5m12yu1CAPpugS/pQcAbgTUEJmvgv/ueAo5qoNTNHRZCQvuZaIGVQ1yL9wPsvmCPFcoz0Jo2Yx31zqiUxl5XZndnq4t4gdQ76PuS71iC8qsMchdszszbxYIXeMfzf8Z9NGe4xeZWVKPCtYk2RXBbghZtwhlvaE9mdmkbauD5plvmvqV+p3EPcKvSJ5XYFsSAt1FifUFZjYb6GviF98XIUqns0oFerwaUeX2XIFWq5PI4m+H7CKhPXjGdQ+7h69FhGlQu3inxuOfa5JItqj7huM29Qj0hZ6XVVtGjsGBa3iG5fdaVKrxfyT2tWdcVyryAWsVMtk5RY11dSGWbFEQOwe9ZptuqHuyQlXNxizqkX7864hOVq0T6Ty+BRj1WY2dilVxBUvTl/ntLWss04qjKX4/hIcW5TqGHrC2oyMpqWR+ts2VRIyHjbvvskJXe+i7WZspg+6bzzlgU/XuexJMvM7MJvMqQa9Nzo/D3HXyGe7gXT8Pqnz5Y5ZWUXZGKP+cGa+vMuRQ//okURc40X0dcS1fDv8WSMzi2LfD14wgSgZmxqkRdHbzI3pKz9Uq+gFq8iqoj8l0kKJ+XQSYfx8yPkcuMJ7RS0BKFLh/eH8oDQGO6AMjMc6nMvqMLfBZedrk7NhMk9z6IpKxzikqudrlbycljeaPl+2Pcz3oqymz5cVyjVz0zX9ICHWaYlcyjiIhD+SpYclUiH3BE3XNr9dwL6Ae23P5EVsLDwpkfTZY/SqQlOVb0rtS9kFY+MTkfntPTx4zf434BPQXf2Yy6Z38mYud/tePs3lb+7o2VzSv6N/c8l3JZIO9wZ3SBOZg9KaEq79Ui6dT2V7goPdJHt+0dVW4bUFL8qe3TadyTL7jWSF3h232AMqC/XnyX1HKrleWsc9dt8W+1BMze05tXN0b2h5BHsLrC9hvufny0Xqa5rc4nmzzThM647ZjH4CyZ9CstNDMrI/ucTPTxZ9PgXemJ0sex5cYwboGL5GL3yDZ6wBNf2I57s6rnuFXW9RVDln9Uy/HMVdB/Im9qsd7mt3x43MaWitcGYq6tiFhuw3MKfLnNszZ4BZSHfROx1ZZ1174iwyrkk/bXbBpU9gtyWIeR5Px5sf3+nI+k1JyBNAfg67D+TRM+ZieHnOzH4jUPaeYvbpOcK9g6vHjp4Y3h1mncfO7qDH6BO+fnrannhXIfm7jqqQ3O0/ezqmr4g/fvq95Om1B46cR59iUkpr2ijirHN0RJnUPC0mPHPmJ+tq4aSs8cZtS0byyjd4W6YfOWa3WLpYXq2Pv32bHByvV+s/VWsHG6uvNsybuUfqG/VMvYCfNtUrVOY+/GqrP9Xf6h/178rXKy9X9lZea9aHD4zMV6rwWTn4H8iH/qo=</latexit>

E[Y0 | T = 1]
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E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit><latexit sha1_base64="4Gwi+xAwrn/BaW60eKvMSBAkJZk=">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</latexit>

E[Y1 (1)Brady Nealtime

Y1 (0) | T = 1] = (E[Y1 | T = 1]

E[Y0 | T = 1])(E[Y1 | T = 0]

Difference-in-Differences Overview

E[Y0 | T = 0])

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Difference-in-Differences
Treatment
administered

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit><latexit sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">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</latexit>

Treatment
Group

E[Y1 (1)<latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit>

E[Y0 | T = 1]
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E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit><latexit sha1_base64="4Gwi+xAwrn/BaW60eKvMSBAkJZk=">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</latexit>

E[Y1 (1)Brady Nealtime

Y1 (0) | T = 1] = (E[Y1 | T = 1]

E[Y0 | T = 1])(E[Y1 | T = 0]

Difference-in-Differences Overview

E[Y0 | T = 0])

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Difference-in-Differences
Treatment
administered

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit><latexit sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">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</latexit>

Treatment
Group

E[Y1 (1)<latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit>

E[Y0 | T = 1]
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E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit><latexit sha1_base64="4Gwi+xAwrn/BaW60eKvMSBAkJZk=">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</latexit>

E[Y1 (1)Brady Nealtime

Y1 (0) | T = 1] = (E[Y1 | T = 1]

E[Y0 | T = 1])(E[Y1 | T = 0]

Difference-in-Differences Overview

E[Y0 | T = 0])

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Difference-in-Differences
Treatment
administered

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit><latexit sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">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</latexit>

Treatment
Group

E[Y1 (1)<latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit>

E[Y0 | T = 1]
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E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">AAAUj3icrVjbbttGEN2kt8S92anf8sJGKdAHWZFdF+6LCwO2g7ZoAAfwLY0NQ6IoixBvJSm7jqCHfk3fivZz+jc9c3YpirpRCixB5Gp25szs3JbLZuS5SVqv//fg4QcffvTxJ48er3z62edffLm69uQ0CXux7ZzYoRfG581G4nhu4Jykbuo551HsNPym55w1u/syf3bjxIkbBsfpXeRc+o3rwG27diMF6Wr16cXh2zdXdevCd1vWRQrR1HeC1Nq16pdXq5V6rc6PNTnYNIOKMp+jcO3x3+pCtVSobNVTvnJUoFKMPdVQCb5v1aaqqwi0S9UHLcbI5byjBmoFsj1wOeBogNrF9Rr/3hpqgP+CmVDahhYPvxiSlvrG8LQwbpOq76LfGuGdpaNPbLHxDvemwfRBTVUH1DK5jHNxuSa+fsmqU6zhB67WxUoiUsQPdmHNbdw9/E+xQrnegdPBqAWpGCMbNA9UTREdMe7a8+KbDiPRIJ+DkVg9z+55Vst8iG8XWA2ME1oqtlrqpYlLQM0ObRUejzGdjfgHVqjtm4fVHq7BZdz1KoT3GJSueodREXmeztHsms2VGuTBFF3Cn5IjAHcCSsjMd+E/FxzFXLWBKToajOQ11xIxg2oG+RfOZ9kcIZ4btCdh1Czmm0s9kams3O7MTg/3JrFjyPcx16Ee8UUV9jjEjpm9mRer5I7x75b/bNpoj9FrrEyJZxVrkuyqAjfEjDvE0p7Q/sws0tb1QbPMd0P9Sv0O4l6lVySvq5ANaaHO4oS6AhP7XfQV8YuPq1DFsxmlSh0+7egwW7qg5fo0kuh7gVUkrB/PoPZx9+i1iChVahf/3HLsc00S0R5132Lcoh6RttDramrH2DEoaBXfuOxOk1ot5p/Uru6M41pFLmClWiYrd6mprrbNiiUCYveo12zTDXVPTri6wZhVTdqfdw3J0aJ9IpXHpxirFrOxQ6kirnhx+jqnrWWLdVJlLMX31+DYpVTb0BPWZmQ0rYzU376pkpDxsHn3TU7oehdtt2MzfdB94ykPfLrOZU+Smd+B2WBOXdB7o/PzEA+NfLYbSMfvkzpf7pSVVZSNMeoPZ+bLuxw59I8eScQ1XkRfR1zL18OfNTKzKPb94AUTiJKxqUFaFL3F3Ji+8mNVQS+oIKuK/pRICyXm00mE8fMh53PgCm+XWgJQpML7w/lBaQwOQBkYj7U5l9VhboPLztci54XJPM2hKyod45Cqnq9V8nJa3mj6fNnWMN+Lspo+X1YoN8xN1/SDhFjnJXIp4yAS/kiWHpdIhdwTNF3b/GYB+wLutT2TF7Gx8KxE0meP0asKTVW+KrUvZReMjU9E5rf38OAN+9+AT0DL+jGXTZfyZi53914+zeVvl/RsLukv6d9c8l2JpMO9wR3SROZ1aU0J11Epl86nsj3BwW6SPb/oaqvy2oQXZc9umc5kmf1GskLvjrvsAdWF+vPkviOVvFlaxz123yb7UEzNrTm1c3JvaHkEewusL2G+5+fLReprmtziebPPOEzrjtmMfgLJn0Ky00Mysj+5xM9PFn0+Bd6anSx7HtxgBugY/oRe+ArPWANq+g7Pd5u4vizseouiyjmrZ/rlKO4WkHewXx1wX1seNzKnoY3Cmamo4xAast/AnC5zbs+cAWYhLaN3OrLOutbEWWRck37a7IBLn8DuShDzPJ6ONz++05H1m5KQJ4D8HHYfyKNnzMXw8pyZ/Uag7D3F7NNzhHsbV48dPTG8B8w6j53dQY/RJ3z99LQ/8a5C8ncLVSG523/2ZExfEX/89HvN02sPHDmPPsWklNa0UcRZ5+iIMql5Wkx45sxP1rXCSVnjjduWjOSVb/B2TT9yzG6xcrVa2Rx/+zY5ON2qbX5fq7/eruxtmzdzj9RT9Ux9Cz/tqD1U5hH8aqs/1V/qH/Xv+tr6zvqP63ua9eEDI/OVKnzWf/4f5wH80A==</latexit><latexit sha1_base64="4Gwi+xAwrn/BaW60eKvMSBAkJZk=">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</latexit>

E[Y1 (1)Brady Nealtime

Y1 (0) | T = 1] = (E[Y1 | T = 1]

E[Y0 | T = 1])(E[Y1 | T = 0]

Difference-in-Differences Overview

E[Y0 | T = 0])

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Difference-in-Differences
Treatment
administered

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit><latexit sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">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</latexit>

Treatment
Group

E[Y1 (1)<latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">AAAUm3icrVjbbttGEN2kt8S9xEn9VhRg4xRNAFmVXBfui4EAtoMiSAAH8CWpZRgSRdmEeCtJWXUEfUJ/pS99bT+kf9MzZ5eiqBvlwBJErmZnzszObblsRZ6bpLXaf3fufvTxJ59+du/+yudffPnVg9WHj46TsBfbzpEdemH8ttVMHM8NnKPUTT3nbRQ7Tb/lOSet7q7Mn1w5ceKGwWF6HTlnfvMicDuu3UxBOl/9obF/+u68/rT+zNqwZFB7ZjV8t201UsCkvhOk1o5VPztfXa9Va/xY04O6Gawr8zkIH97/SzVUW4XKVj3lK0cFKsXYU02V4Huq6qqmItDO1AC0GCOX844aqhXI9sDlgKMJahfXC/w7NdQA/wUzobQNLR5+MSQt9b3haWPcIVXfRb81xjtPx4DYYuM17i2D6YOaqktQy+QyzuXlWvj6JatOsYZfuFoXK4lIET/YhTV3cPfwP8UK5XoNTgejNqRijGzQPFA1RXTEuGvPi28uGYkm+RyMxOpFdi+yWuZDfLvAamKc0FKx1VIvTFwCanZoq/B4jOl8xD+wQm3fIqzOaA0u465XIbyHoHTVe4yKyIt0jmfXfK7UIA9n6BL+lBwBuBNQQma+C/+54Cjmqg1M0dFkJC+4logZVDXILzmfZXOEeG7QnoRRs5hvLvVEprJyuzM7PdxbxI4hP8DcJfWILyqwxyF2zOzNvFghd4x/ff6zaaM9Qa+yMiWeFaxJsqsC3BAz7ghLe0L7M7NIWzcAzTLfDfWK+h3EvUKvSF5XIBvSQp3FCXUFJvY76CviFx9XoYpnM0qFOnzaccls6YKW69NIou9HrCJh/XgGdYC7R69FRKlQu/inz7HPNUlEe9Tdx7hNPSJtoddV1baxY1jQKr5x2Z2mtVrMP6ld3RkntYpcwEq1TFbuUFNNbZkVSwTE7nGv2aYb6p6ccHXDCatatD/vGpKjRftEKo9PMVZtZuMlpYq44sXZ65y1lk3WSYWxFN9fgGOHUh1DT1ibkdG0MlZ/u6ZKQsbD5t03OaHrXbT1J2YGoPvGUx74dJ3LniQzvwOzyZxq0Hvj84sQ9418thtIxx+QuljumJVVlI0xGoxmFsu7HDn0jx5JxDVeRF9HXMt3o581NrMs9u3gBVOIkrGpQVoWvc3cmL3yQ7WOXrCOrCr6UyItlJhPJxHGT0acT4ArvF1qCUCRCh+M5oelMdgDZWg81uFcVoe5DS47X5ucDZN5mkNXVDrBIVW9WKvk5ay80fTFsu1RvhdlNX2xrFCumJuu6QcJsd6WyKWMg0j4Y1l6WCIVck/QdG3zuyXsC7jX9kxexMbCkxJJnz1Gryo0Vfm61L6UXTA2PhGZ3z7Ag1fsf0M+Ad3Uj7lseiNv5nLXH+TTXL5/Q8/mkv4N/ZtLvi+RdLg3uCOayLwprSnhOijl0vlUtic42E2y5xddbRVeW/Ci7Nlt05kss99IVujdcYc9oLJUf57ed6SS66V13GP3bbEPxdTcXlA7R7eGlkewt8T6EuZ7fr5cpr5myS2fN7uMw6zumM3oJ5D8KSQ7PSRj+5NL/PxkMeBTYN/sZNnz4AYzQMfwV/TC13jGGlLTT3i+q+P6orDrLYsq56ye6ZfjuJtA3sZ+tcd97ea4kTkNbRTOTEUd+9CQ/YbmdJlze+YMMA/pJnpnI+usa0+dRSY16afNS3DpE9h1CWKex7PxFsd3NrJ+UxLyBJCfw24DefyMuRxenjPz3wiUvaeYf3qOcO/g6rGjJ4Z3j1nnsbM76DH6hK+fnnan3lVI/m6iKiR3B48fTegr4k+efi94eu2BI+fRp5iU0po2jjjvHB1RJjVPiwnPnPnJulo4KWu8SduSsbzyDd6O6UeO2S1WzlfX65Nv36YHx5vV+s/V2put9edb5s3cPfWNeqyewk/b6jkq8wB+tdWf6m/1j/p37du13bWXa6806907RuZrVfisHf0PG2j/ow==</latexit>

Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit>

E[Y0 | T = 1]
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E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit><latexit sha1_base64="4Gwi+xAwrn/BaW60eKvMSBAkJZk=">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</latexit>

E[Y1 (1)Brady Nealtime

Y1 (0) | T = 1] = (E[Y1 | T = 1]

E[Y0 | T = 1])(E[Y1 | T = 0]

Difference-in-Differences Overview

E[Y0 | T = 0])

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Difference-in-Differences
Treatment
administered

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit><latexit sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">AAAUsHicrVjbbttGEN2kt8S9Oanf+sJGKZACsiq5LtyHGghgOyiKBnAA31LbMCSKslnxVpKy6gj6mX5Av6Ov7Vv/pmfOLkVRN0pBJIhczc6cmZ3bctmKPDdJ6/X/7t1/7/0PPvzowcO1jz/59LPP1x89PknCXmw7x3bohfFZq5k4nhs4x6mbes5ZFDtNv+U5p63unsyf3jpx4obBUXoXOZd+8zpwO67dTEG6Wv/x4uD89VXDuvDdtnWRQjT1nSC1dq3GJS569lnjm5kMV+uVeq3OjzU9aJhBRZnPYfjo4V/qQrVVqGzVU75yVKBSjD3VVAm+56qh6ioC7VINQIsxcjnvqKFag2wPXA44mqB2cb3Gv3NDDfBfMBNK29Di4RdD0lJfG542xh1S9V30W2O883QMiC023uHeMpg+qKm6AbVMLuNcXq6Fr1+y6hRr+IGrdbGSiBTxg11Ycwd3D/9TrFCud+B0MGpDKsbIBs0DVVNER4y79rz45oaRaJLPwUisXmT3IqtlPsS3C6wmxgktFVst9cLEJaBmh7YKj8eYzkf8AyvU9i3C6ozW4DLuehXCewRKV73BqIi8SOd4ds3nSg3ycIYu4U/JEYA7ASVk5rvwnwuOYq7awBQdTUbymmuJmEE1g/wz57NsjhDPTdqTMGoW882lnshUVm53ZqeHe4vYMeQHmLuhHvFFFfY4xI6ZvZkXq+SO8a/PfzZttCfoNVamxLOKNUl2VYEbYsYdYWlPaH9mFmnrBqBZ5rupfqF+B3Gv0iuS11XIhrRQZ3FCXYGJ/S76ivjFx1Wo4tmMUqUOn3bcMFu6oOX6NJLo+xarSFg/nkEd4O7RaxFRqtQu/ulz7HNNEtEedfcxblOPSFvodTW1Y+wYFrSKb1x2p2mtFvNPald3xkmtIhewUi2TlbvUVFfbZsUSAbF73Gu26Ya6Jydc3XDCqhbtz7uG5GjRPpHK41OMVZvZeEOpIq54cfY6Z61li3VSZSzF99fg2KVUx9AT1mZkNK2N1d+eqZKQ8bB5901O6HoXbf2JmQHovvGUBz5d57InyczvwGwypy7ovfH5RYgHRj7bDaTjD0hdLHfCyirKxhgNRjOL5V2OHPpHjyTiGi+iryOu5avRzxqbWRb73eAFU4iSsalBWha9zdyYvfIjVUEvqCCriv6USAsl5tNJhPHTEedT4Apvl1oCUKTCB6P5YWkM9kEZGo91OJfVYW6Dy87XJueFyTzNoSsqneCQql6sVfJyVt5o+mLZ9ijfi7KavlhWKLfMTdf0g4RYZyVyKeMgEv5Ylh6VSIXcEzRd2/x6CfsC7rU9kxexsfC0RNJnj9GrCk1Vviy1L2UXjI1PRObXt/DgLfvfkE9Aq/oxl01X8mYud/dWPs3l+yt6Npf0V/RvLvmmRNLh3uCOaCLzqrSmhOuwlEvnU9me4GA3yZ5fdLVVeW3Bi7Jnt01nssx+I1mhd8dd9oDqUv15et+RSm6U1nGP3bfFPhRTc3tB7Ry/M7Q8gr0l1pcw3/Pz5TL1NUtu+bzZYxxmdcdsRj+B5E8h2ekhGdufXOLnJ4sBnwL7ZifLngc3mQE6hj+hF77EM9aQmr7D810D1xeFXW9ZVDln9Uy/HMfdAvIO9qt97mur40bmNLRZODMVdRxAQ/YbmtNlzu2ZM8A8pFX0zkbWWdeeOotMatJPmzfg0iewuxLEPI9n4y2O72xk/aYk5AkgP4e9C+TxM+ZyeHnOzH8jUPaeYv7pOcK9g6vHjp4Y3n1mncfO7qDH6BO+fnram3pXIfm7haqQ3B08eTyhr4g/efq95um1B46cR59iUkpr2jjivHN0RJnUPC0mPHPmJ+ta4aSs8SZtS8byyjd4u6YfOWa3WLtarzQm375ND062ao3va/VX25Xn2+bN3AP1pXqinsFPO+o5KvMQfrXVn+pv9Y/6d2Nr42zjaqOpWe/fMzJfqMJn47f/AcaqCCA=</latexit>

Treatment
Group

E[Y1 (1)<latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit>

E[Y0 | T = 1]
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E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit><latexit sha1_base64="4Gwi+xAwrn/BaW60eKvMSBAkJZk=">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</latexit>

E[Y1 (1)Brady Nealtime

Y1 (0) | T = 1] = (E[Y1 | T = 1]

E[Y0 | T = 1])(E[Y1 | T = 0]

Difference-in-Differences Overview

E[Y0 | T = 0])

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Difference-in-Differences
Treatment
administered

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit><latexit sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">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</latexit>

Treatment
Group

E[Y1 (1)<latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">AAAUl3icrVjZbttGFJ2kW+JuTuqXoi9snAIpIKuS68J9MRLAC4KiAWzAW2oZhkRRFmFuJSm7jqAP6M+0r+2n9G967pmhKGqjHFiCyNGde8+9c7fhsBV5bpLWav89ePjBhx99/Mmjx0uffvb5F18uP3l6nIS92HaO7NAL49NWM3E8N3COUjf1nNModpp+y3NOWlfbMn9y7cSJGwaH6W3knPvNy8DtuHYzBeliebWxe/b2ov6i9r3V8N221UghnfpOkFpbVv3calSsl+CqVWv8WJODuhmsKvPZD588/ks1VFuFylY95StHBSrF2FNNleB7puqqpiLQzlUftBgjl/OOGqglyPbA5YCjCeoVrpf4d2aoAf4LZkJpG1o8/GJIWuo7w9PGuEOqvot+a4R3lo4+scXGW9xbBtMHNVVdUMvkMs7F5Vr4+iWrTrGGn7laFyuJSBE/2IU1d3D38D/FCuV6C04HozakYoxs0DxQNUV0xLhrz4tvuoxEk3wORmL1PLvnWS3zIb5XwGpinNBSsdVSeyYuATU7tFV4PMZ0NuIfWKG2bx5WZ7gGl3HXqxDeQ1Cu1DuMisjzdI5m12yu1CAPpugS/pQcAbgTUEJmvgv/ueAo5qoNTNHRZCQvuZaIGVQ1yL9wPsvmCPFcoz0Jo2Yx31zqiUxl5XZndnq4t4gdQ76PuS71iC8qsMchdszszbxYIXeMfzf8Z9NGe4xeZWVKPCtYk2RXBbghZtwhlvaE9mdmkbauD5plvmvqV+p3EPcKvSJ5XYFsSAt1FifUFZjYb6GviF98XIUqns0oFerwaUeX2XIFWq5PI4m+H7CKhPXjGdQ+7h69FhGlQu3inxuOfa5JItqj7huM29Qj0hZ6XVVtGjsGBa3iG5fdaVKrxfyT2tWdcVyryAWsVMtk5RY11dSGWbFEQOwe9ZptuqHuyQlXNxizqkX7864hOVq0T6Ty+BRj1WY2dilVxBUvTl/ntLWss04qjKX4/hIcW5TqGHrC2oyMpqWR+ts2VRIyHjbvvskJXe+i7WZspg+6bzzlgU/XuexJMvM7MJvMqQa9Nzo/D3HXyGe7gXT8Pqnz5Y5ZWUXZGKP+cGa+vMuRQ//okURc40X0dcS1fDv8WSMzi2LfD14wgSgZmxqkRdHbzI3pKz9Uq+gFq8iqoj8l0kKJ+XQSYfx8yPkcuMJ7RS0BKFLh/eH8oDQGO6AMjMc6nMvqMLfBZedrk7NhMk9z6IpKxzikqudrlbycljeaPl+2Pcz3oqymz5cVyjVz0zX9ICHWaYlcyjiIhD+SpYclUiH3BE3XNr9dwL6Ae23P5EVsLDwpkfTZY/SqQlOVb0rtS9kFY+MTkfntPTx4zf434BPQXf2Yy6Z38mYud/tePs3lb+7o2VzSv6N/c8l3JZIO9wZ3SBOZg9KaEq79Ui6dT2V7goPdJHt+0dVW4bUFL8qe3TadyTL7jWSF3h232AMqC/XnyX1HKrleWsc9dt8W+1BMze05tXN0b2h5BHsLrC9hvufny0Xqa5rc4nmzzThM647ZjH4CyZ9CstNDMrI/ucTPTxZ9PgXemJ0sex5cYwboGL5GL3yDZ6wBNf2I57s6rnuFXW9RVDln9Uy/HMVdB/Im9qsd7mt3x43MaWitcGYq6tiFhuw3MKfLnNszZ4BZSHfROx1ZZ1174iwyrkk/bXbBpU9gtyWIeR5Px5sf3+nI+k1JyBNAfg67D+TRM+ZieHnOzH4jUPaeYvbpOcK9g6vHjp4Y3h1mncfO7qDH6BO+fnrannhXIfm7jqqQ3O0/ezqmr4g/fvq95Om1B46cR59iUkpr2ijirHN0RJnUPC0mPHPmJ+tq4aSs8cZtS0byyjd4W6YfOWa3WLpYXq2Pv32bHByvV+s/VWsHG6uvNsybuUfqG/VMvYCfNtUrVOY+/GqrP9Xf6h/178rXKy9X9lZea9aHD4zMV6rwWTn4H8iH/qo=</latexit>

E[Y0 | T = 1]
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E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit><latexit sha1_base64="4Gwi+xAwrn/BaW60eKvMSBAkJZk=">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</latexit>

E[Y1 (1)Brady Nealtime

Y1 (0) | T = 1] = (E[Y1 | T = 1]

E[Y0 | T = 1])(E[Y1 | T = 0]

Difference-in-Differences Overview

E[Y0 | T = 0])

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Difference-in-Differences
Treatment
administered

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit><latexit sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">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</latexit>

Treatment
Group

E[Y1 (1)<latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">AAAUl3icrVjZbttGFJ2kW+JuTuqXoi9snAIpIKuS68J9MRLAC4KiAWzAW2oZhkRRFmFuJSm7jqAP6M+0r+2n9G967pmhKGqjHFiCyNGde8+9c7fhsBV5bpLWav89ePjBhx99/Mmjx0uffvb5F18uP3l6nIS92HaO7NAL49NWM3E8N3COUjf1nNModpp+y3NOWlfbMn9y7cSJGwaH6W3knPvNy8DtuHYzBeliebWxe/b2ov6i9r3V8N221UghnfpOkFpbVv3calSsl+CqVWv8WJODuhmsKvPZD588/ks1VFuFylY95StHBSrF2FNNleB7puqqpiLQzlUftBgjl/OOGqglyPbA5YCjCeoVrpf4d2aoAf4LZkJpG1o8/GJIWuo7w9PGuEOqvot+a4R3lo4+scXGW9xbBtMHNVVdUMvkMs7F5Vr4+iWrTrGGn7laFyuJSBE/2IU1d3D38D/FCuV6C04HozakYoxs0DxQNUV0xLhrz4tvuoxEk3wORmL1PLvnWS3zIb5XwGpinNBSsdVSeyYuATU7tFV4PMZ0NuIfWKG2bx5WZ7gGl3HXqxDeQ1Cu1DuMisjzdI5m12yu1CAPpugS/pQcAbgTUEJmvgv/ueAo5qoNTNHRZCQvuZaIGVQ1yL9wPsvmCPFcoz0Jo2Yx31zqiUxl5XZndnq4t4gdQ76PuS71iC8qsMchdszszbxYIXeMfzf8Z9NGe4xeZWVKPCtYk2RXBbghZtwhlvaE9mdmkbauD5plvmvqV+p3EPcKvSJ5XYFsSAt1FifUFZjYb6GviF98XIUqns0oFerwaUeX2XIFWq5PI4m+H7CKhPXjGdQ+7h69FhGlQu3inxuOfa5JItqj7huM29Qj0hZ6XVVtGjsGBa3iG5fdaVKrxfyT2tWdcVyryAWsVMtk5RY11dSGWbFEQOwe9ZptuqHuyQlXNxizqkX7864hOVq0T6Ty+BRj1WY2dilVxBUvTl/ntLWss04qjKX4/hIcW5TqGHrC2oyMpqWR+ts2VRIyHjbvvskJXe+i7WZspg+6bzzlgU/XuexJMvM7MJvMqQa9Nzo/D3HXyGe7gXT8Pqnz5Y5ZWUXZGKP+cGa+vMuRQ//okURc40X0dcS1fDv8WSMzi2LfD14wgSgZmxqkRdHbzI3pKz9Uq+gFq8iqoj8l0kKJ+XQSYfx8yPkcuMJ7RS0BKFLh/eH8oDQGO6AMjMc6nMvqMLfBZedrk7NhMk9z6IpKxzikqudrlbycljeaPl+2Pcz3oqymz5cVyjVz0zX9ICHWaYlcyjiIhD+SpYclUiH3BE3XNr9dwL6Ae23P5EVsLDwpkfTZY/SqQlOVb0rtS9kFY+MTkfntPTx4zf434BPQXf2Yy6Z38mYud/tePs3lb+7o2VzSv6N/c8l3JZIO9wZ3SBOZg9KaEq79Ui6dT2V7goPdJHt+0dVW4bUFL8qe3TadyTL7jWSF3h232AMqC/XnyX1HKrleWsc9dt8W+1BMze05tXN0b2h5BHsLrC9hvufny0Xqa5rc4nmzzThM647ZjH4CyZ9CstNDMrI/ucTPTxZ9PgXemJ0sex5cYwboGL5GL3yDZ6wBNf2I57s6rnuFXW9RVDln9Uy/HMVdB/Im9qsd7mt3x43MaWitcGYq6tiFhuw3MKfLnNszZ4BZSHfROx1ZZ1174iwyrkk/bXbBpU9gtyWIeR5Px5sf3+nI+k1JyBNAfg67D+TRM+ZieHnOzH4jUPaeYvbpOcK9g6vHjp4Y3h1mncfO7qDH6BO+fnrannhXIfm7jqqQ3O0/ezqmr4g/fvq95Om1B46cR59iUkpr2ijirHN0RJnUPC0mPHPmJ+tq4aSs8cZtS0byyjd4W6YfOWa3WLpYXq2Pv32bHByvV+s/VWsHG6uvNsybuUfqG/VMvYCfNtUrVOY+/GqrP9Xf6h/178rXKy9X9lZea9aHD4zMV6rwWTn4H8iH/qo=</latexit>

E[Y0 | T = 1]
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ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit><latexit sha1_base64="4Gwi+xAwrn/BaW60eKvMSBAkJZk=">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</latexit>

E[Y1 (1)Brady Nealtime

Y1 (0) | T = 1] = (E[Y1 | T = 1]

E[Y0 | T = 1])(E[Y1 | T = 0]

Difference-in-Differences Overview

E[Y0 | T = 0])

10 / 25

Tolerates Time-Invariant Unobserved Confounding
Unobserved confounders that are constant with time are no problem,since they’ll cancel out in the time differences<latexit sha1_base64="4Gwi+xAwrn/BaW60eKvMSBAkJZk=">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</latexit>

E[Y1 (1)Y1 (0) | T = 1] = (E[Y1 | T = 1]

E[Y0 | T = 1])(E[Y1 | T = 0]

Time difference intreatment group

Brady Neal

Difference-in-Differences Overview

E[Y0 | T = 0])

Time difference incontrol group

11 / 25

Question:
How would you estimate the terms on theright-hand side of the difference-indifferences equation?
<latexit sha1_base64="4Gwi+xAwrn/BaW60eKvMSBAkJZk=">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</latexit>

E[Y1 (1)Y1 (0) | T = 1] = (E[Y1 | T = 1]

E[Y0 | T = 1])(E[Y1 | T = 0]

E[Y0 | T = 0])

Motivation and Preliminaries
Difference-in-Differences Overview
Assumptions and Proof
Problems with Difference-in-Differences

Brady Neal

Assumptions and Proof

13 / 25

Consistency Assumption Extended to Time<latexit sha1_base64="W5glzZkbZx6BlP4KlW/DX8dUdSA=">AAAU2XicrVjLbttGFJ30mbgvJ/WuGzZOgRSQVdlNkWwEBHASFEUDJIDzaKPAoKiRTZivkJRdR9Ciu6Lb/kd/pF/Qz+i23fTcM0OR1ItyEAkih3fuPffOfQ1H/STws7zT+fvSO+++9/4HH16+svHRx598+tnm1WtPs3iUevqJFwdx+rzvZjrwI/0k9/NAP09S7Yb9QD/rn+zL/LNTnWZ+HB3k54l+GbpHkT/0PTcH6XBT94Zx6gaB08vdkdNyeq9G7gAPwMhDHeVOt/Jw6oLPD2GXzpxePMq9ONSHlOzWn2/WhL4+3NzutDv8OPODXTvYVvbzKL565U/VUwMVK0+NVKi0ilSOcaBcleH7Qu2qjkpAe6nGoKUY+ZzXaqI2IDsClwaHC+oJrkd4emGpEZ4FM6O0By0BfikkHfWV5RlgPCTV3EW/U+FdpmNMbLHxHPe+xQxBzdUxqE1yBef6cn18w4ZV51jDHa7Wx0oSUsQPXm3NQ9wDPOdYoVzPwakxGkAqxcgDLQDVUERHirvxvPjmmJFwyacxEqtX2b3KapmP8T0BlotxRkvFVkc9sHGJqFnTVuEJGNPliL9ghca+VVjD6Rp8xt2sQngPQDlRrzGqI6/SWc2u5Vy5RZ4s0CX8OTkicGegxMx8H/7zwVHPVQ+YosNlJI+4loQZ1LbIP3C+yOYE8dyhPRmj5jDffOpJbGWVdhd2Brj3iZ1Cfoy5Y+oRX7RgjyZ2yuwtvNgid4qnMz55tNGbobdZmRLPFtYk2dUCbowZf4plPGH8WVhkrBuD5tjvjvqR+jXi3qJXJK9bkI1pocnijLoiG/su+or4JcRVqOLZgtKijpB2HDNbTkAr9Rkk0fcNVpGxfgKLOsY9oNcSorSoXfxzxnHINUlER9R9hvGAekTaQa9rq9vWjklNq/jGZ3ea1+ow/6R2TWec1SpyESvVsVnZpaaOumVXLBEQu6te82w3ND054+omM1b1aX/ZNSRH6/aJVBmfeqwGzMZjStVxxYuL17loLXuskxZjKb4/AkeXUkNLz1ibidW0Uam/fVslMePh8R7anDD1LtrOZmbGoIfWUwH4TJ3LniQzr4DpMqd69F51fhXifStf7AbS8cekrpZ7ysqqy6YYjaczq+V9jjT9Y0YScYOX0NcJ1/Ll9OdUZtbFfjt40RyiZGxukdZFHzA3Fq/8QG2jF2wjq+r+lEgLJeXbSYLxjSnnDeAK7wm1RKBIhY+n85PGGNwDZWI9NuRcUYelDT4734CcPZt5hsNUVD7DIVW9Wqvk5aK8MfTVsoNpvtdlDX21rFBOmZu+7QcZsZ43yOWMg0iElSw9aJCKuScYurH5pzXsi7jXjmxepNbCZw2SIXuMWVVsq/Jho305u2BqfSIyP7+BB0/Z/yZ8A7qoH0vZ/ELeLOXO38inpfzZBT1bSoYX9G8p+bpBUnNv8Kc0kXncWFPC9aiRy+RT056gsZsU7y+m2lq89uFF2bMHtjM5dr+RrDC7Y5c9oLVWf57fd6SSdxvreMTu22cfSql5sKJ2nrw1tDKCozXWlzHfy/PlOvW1SG79vNlnHBZ1x2LGvIGUbyHF6SGr7E8+8cuTxZhvgWd2JyveB3eYASaG36MXPsQ71oSavsX73S6uD2q73rqocs4a2X5Zxd0D8m3sV/e4r10cN7GnoZ3amamu4z40FL+JPV2W3IE9AyxDuojexcgm6wZzZ5FZTeZt8xhc5gR23oBY5vFivNXxXYxs/imJeQIoz2FvA7l6xlwPr8yZ5f8INP1Psfz0nOA+xDVgR88s7z1mXcDOrtFjzAnfvD3tz/1XIfm7h6qQ3B1fvzajr44/e/o94ul1BI6Sx5xickobWhVx2Tk6oUxu3xYznjnLk3W7dlI2eLO2ZZW8Ci1e1/YjbXeLjcPN7d3Zf9/mB0/32rvftTuPb23fvWP/mbusvlDX1U346ba6i8p8BL966i/1j/pX/bf1YuvXrd+2fjes71yyMp+r2mfrj/8BHL0csg==</latexit>

Brady Neal

8⌧,

T = t =) Y⌧ = Y⌧ (t)Assumptions and Proof

14 / 25

Consistency Assumption Extended to Time<latexit sha1_base64="W5glzZkbZx6BlP4KlW/DX8dUdSA=">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</latexit>

8⌧,

Causal estimand

T = t =) Y⌧ = Y⌧ (t)Statistical estimand

Examples: E[Y⌧ (1) | T = 1] = E[Y⌧ | T = 1]
<latexit sha1_base64="hVJaTtcP+g3Uus53bVIvUj4YKwU=">AAAUxHicrVhZb9tGEN6kV+IecVK/9YWNUiAFZFVyUzgvBgL4QFE0gAP4aiPDkCjKIsyrJGXXEdSf1Mf+lwJ9bX9Hv/l2KYq6KAWWIHI1O/PN7FzLZTvy3CSt1/++d/+DDz/6+JMHD9c+/ezzLx6tP35ykoT92HaO7dAL47N2K3E8N3COUzf1nLModlp+23NO21e7Mn967cSJGwZH6W3knPuty8DtunYrBeli/bC5/7YZ9lM79J2LZtrqP298azV9t2M1U+CkvhOk1o7VOMdlknUm38V6pV6r82NNDxpmUFHmcxg+fvinaqqOCpWt+spXjgpUirGnWirB961qqLqKQDtXA9BijFzOO2qo1iDbB5cDjhaoV7he4t9bQw3wXzATStvQ4uEXQ9JS3xieDsZdUvVd9FtjvPN0DIgtNt7i3jaYPqip6oFaJpdxLi/XxtcvWXWKNbzkal2sJCJF/GAX1tzF3cP/FCuU6y04HYw6kIoxskHzQNUU0RHjrj0vvukxEi3yORiJ1YvsXmS1zIf4XgGrhXFCS8VWSx2YuATU7NBW4fEY0/mIv2OF2r5FWN3RGlzGXa9CeI9AuVLvMCoiL9I5nl3zuVKDPJyhS/hTcgTgTkAJmfku/OeCo5irNjBFR4uRvORaImZQzSD/xPksmyPEc5P2JIyaxXxzqScylZXbndnp4d4mdgz5AeZ61CO+qMIeh9gxszfzYpXcMf7d8J9NG+0Jeo2VKfGsYk2SXVXghphxR1jaE9qfmUXaugFolvluqp+p30Hcq/SK5HUVsiEt1FmcUFdgYr+DviJ+8XEVqng2o1Spw6cdPWbLFWi5Po0k+r7DKhLWj2dQB7h79FpElCq1i39uOPa5Jolon7pvMO5Qj0hb6HU1tW3sGBa0im9cdqdprRbzT2pXd8ZJrSIXsFItk5U71FRXL8yKJQJi97jXbNMNdU9OuLrhhFVt2p93DcnRon0ilcenGKsOs7FHqSKueHH2OmetZYt1UmUsxfeX4NihVNfQE9ZmZDStjdXfrqmSkPGwefdNTuh6F203EzMD0H3jKQ98us5lT5KZ34DZYk416b3x+UWI+0Y+2w2k4w9IXSx3wsoqysYYDUYzi+Vdjhz6R48k4hovoq8jruXr0c8am1kW+27wgilEydjUIC2L3mFuzF75kaqgF1SQVUV/SqSFEvPpJML42YjzGXCF94paAlCkwgej+WFpDPZAGRqPdTmX1WFug8vO1yFn02Se5tAVlU5wSFUv1ip5OStvNH2xbGeU70VZTV8sK5Rr5qZr+kFCrLMSuZRxEAl/LEuPSqRC7gmarm3+ZQn7Au61fZMXsbHwtETSZ4/RqwpNVb4utS9lF4yNT0Tm1/fw4DX735BPQKv6MZdNV/JmLnf7Xj7N5W9W9Gwu6a/o31zyXYmkw73BHdFE5k1pTQnXYSmXzqeyPcHBbpI9v+hqq/Lahhdlz+6YzmSZ/UayQu+OO+wB1aX68/S+I5XcKK3jPrtvm30opubOgto5vjO0PIL9JdaXMN/z8+Uy9TVLbvm82WUcZnXHbEY/geRPIdnpIRnbn1zi5yeLAZ8Cb8xOlj0PbjIDdAx/RC98jWesITV9j+e7Bq4HhV1vWVQ5Z/VNvxzH3QLyNvarPe5rq+NG5jS0WTgzFXXsQ0P2G5rTZc7tmTPAPKRV9M5G1lnXmTqLTGrST5s9cOkT2G0JYp7Hs/EWx3c2sn5TEvIEkJ/D7gJ5/Iy5HF6eM/PfCJS9p5h/eo5w7+LqsaMnhnePWeexszvoMfqEr5+edqfeVUj+bqEqJHcHT59M6CviT55+L3l67YMj59GnmJTSmjaOOO8cHVEmNU+LCc+c+cm6Vjgpa7xJ25KxvPIN3o7pR47ZLdYu1iuNybdv04OTrVrjh1r9zYvKq5fmzdwD9ZV6qp7DT9vqFSrzEH611V/qH/Wv+m/jYMPbSDb6mvX+PSPzpSp8Nv74H/0YEdQ=</latexit>

Brady Neal

Assumptions and Proof

14 / 25

Consistency Assumption Extended to Time<latexit sha1_base64="W5glzZkbZx6BlP4KlW/DX8dUdSA=">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</latexit>

8⌧,

Causal estimand

T = t =) Y⌧ = Y⌧ (t)Causal estimand

Statistical estimand

Statistical estimand

Examples: E[Y⌧ (1) | T = 1] = E[Y⌧ | T = 1] E[Y⌧ (0) | T = 0] = E[Y⌧ | T = 0]<latexit sha1_base64="hVJaTtcP+g3Uus53bVIvUj4YKwU=">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</latexit>

Brady Neal<latexit sha1_base64="IImBzr+nHAAMCAaFYKYJ0rbRWb0=">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</latexit>

Assumptions and Proof

14 / 25

Consistency Assumption Extended to Time<latexit sha1_base64="W5glzZkbZx6BlP4KlW/DX8dUdSA=">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</latexit>

8⌧,

Causal estimand

T = t =) Y⌧ = Y⌧ (t)Causal estimand

Statistical estimand

Statistical estimand

Examples: E[Y⌧ (1) | T = 1] = E[Y⌧ | T = 1] E[Y⌧ (0) | T = 0] = E[Y⌧ | T = 0]<latexit sha1_base64="hVJaTtcP+g3Uus53bVIvUj4YKwU=">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</latexit>

Counterfactual
Quantities:

Brady Neal<latexit sha1_base64="IImBzr+nHAAMCAaFYKYJ0rbRWb0=">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</latexit>

E[Y⌧ (1) | T = 0]<latexit sha1_base64="bd1NBiLd1X1wiRxlrSSVUrwunBo=">AAAUnHicrVjbbttGEN2kt8S9OanfChRslAIuIKuSm8J5ERDAdlC0NeAAvrWWYUjkyiLEW0nKqkPoF/opfepr+x/9m86cXYqkLqQUWILI1ezMmdm5LZe9wLGjuNn878HD997/4MOPHj3e+PiTTz/7fPPJ07PIH4WmPDV9xw8vet1IOrYnT2M7duRFEMqu23PkeW+4z/PntzKMbN87ie8CeeV2bzy7b5vdmEjXm9udw8uOP4pN35XXnbg72m59a3Rc2zI6MeHErvRio200r643a81GEx9jftDSg5rQn2P/yeO/REdYwhemGAlXSOGJmMaO6IqIvpeiJZoiINqVSIgW0sjGvBQTsUGyI+KSxNEl6pCuN/TvUlM9+s+YEaRN0uLQLyRJQ3yjeSwa90FVd9Zv5HiX6UiAzTbe0b2nMV2ixmJA1Cq5lHN1uR593YpVx7SGl1itTSsJQGE/mIU19+nu0P+YVsjXO+KUNLJIKqSRSTSHqIrCOkK6K8+zbwaIRBd8kkZsdZndZVbzvE/fIWF1aRzBUrbVEK91XDxolrCVeRzEdDniH7RCZV8ZVn+6BhtxV6tg3hOiDMVbGhWRy3Tms2s5V6yRJwt0MX8MDo+4I6L4yHyb/GcTRzFXTcJkHV1E8gZrCZBBDY38E+bTbA4onjuwJ0LUDOSbDT2BrqzM7tROh+49YIckn9DcAHrYF3WyRwI7RPamXqyDO6R/Y/wzYaM5Q2+gMjmedVoTZ1edcH2asadYyhPKn6lFyrqEaIb+7ohfoF9S3OvwCud1nWR9WKiyOIIuT8e+TX2F/eLSlans2ZRShw4XdgyQLUOiZfoUEuv7jlYRoX4cjZrQ3YHXAqDUoZ39M8bYxZo4oiPoHtPYgh6WNqjXNcSetmNS0Mq+sdGd5rUayD+uXdUZZ7WynIdKNXRWtqGpKV7oFXME2O6810zdDVVPjrC6yYxVPdifdQ3O0aJ9LJXFpxgrC9k4gFQRl724eJ2L1rKLOqkjluz7G+JoQ6qv6RFqM9CaNnL1t6+rxEc8TNxdnROq3lnbeGYmIbqrPeUQn6pz3pN45nfC7CKnOvBefr4M8VDLp7sBd/wE1HK5M1RWUTakUTKdKZe3MZLwjxpxxBVeAF8HWMvX05+Rm1kV+37wvDlEzthYI62KbiE3Fq/8RNSoF9Qoq4r+5EgzJcTTSUDj51PO54TLvENo8YjCFZ5M5yeVMTggykR7rI+5tA4zG2x0PgucHZ15ikNVVDzDwVVdrpXzclHeKHq5rDXN96KsopfLMuUWuWnrfhAB66JCLkYcWMLNZelJhZSPPUHRlc2/rmCfh712pPMi1BaeV0i66DFqVb6uyqNK+2J0wVD7hGV+ewcP3qL/TfAEtK4fM9l4LW9mcnfv5NNMfrymZzNJd03/ZpJvKyQl9gZ7SmOZN5U1xVzHlVwqn6r2BEm7Sfr8oqqtjmuPvMh7tqU7k6H3G84KtTu20QPqK/Xn+X2HK7lVWccjdN8e+lAIzVZJ7ZzeG1oWwdEK64uQ79n5cpX6WiS3et7sIw6LumM6o55AsqeQ9PQQ5fYnG/jZySLBU+BY72Tp8+AOMkDF8EfqhUf0jDWBpu/p+a5F19eFXW9VVD5njXS/zOPuEvIe7VcH2NfWxw30aWincGYq6jgkDelvok+XGbejzwDLkNbRuxhZZZ01dxaZ1aSeNgfEpU5gdxWIWR4vxiuP72Jk9abExwkgO4fdB3L+jLkaXpYzy98IVL2nWH56Dujep6uDjh5p3gNknYPOLqnHqBO+enran3tXwfm7S1XBuZs8ezqjr4g/e/q9wel1RBwZjzrFxJBWtDzisnN0AJlYPy1GOHNmJ+tG4aSs8GZti3J55Wq8tu5HUu8WG9ebtdbs27f5wdluo/VDo/nmRe3VS/1m7pH4UjwT2+SnPfGKKvOY/GqKP8Xf4h/x79ZXWwdbP28dKdaHD7TMF6Lw2Tr7H/7/Alw=</latexit>

Assumptions and Proof

14 / 25

Consistency Assumption Extended to Time<latexit sha1_base64="W5glzZkbZx6BlP4KlW/DX8dUdSA=">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</latexit>

8⌧,

Causal estimand

T = t =) Y⌧ = Y⌧ (t)Causal estimand

Statistical estimand

Statistical estimand

Examples: E[Y⌧ (1) | T = 1] = E[Y⌧ | T = 1] E[Y⌧ (0) | T = 0] = E[Y⌧ | T = 0]<latexit sha1_base64="hVJaTtcP+g3Uus53bVIvUj4YKwU=">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</latexit>

Counterfactual
Quantities:

Brady Neal

E[Y⌧ (1) | T = 0]<latexit sha1_base64="bd1NBiLd1X1wiRxlrSSVUrwunBo=">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</latexit><latexit sha1_base64="IImBzr+nHAAMCAaFYKYJ0rbRWb0=">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</latexit>

andE[Y⌧ (0) | T = 1]
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Assumptions and Proof

14 / 25

Parallel Trends Assumption
Treatment
administered

Treatment
Group

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
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E[Y1 (1)Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
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E[Y0 | T = 1]
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ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>

Brady Nealtime
Assumptions and Proof

15 / 25

Parallel Trends Assumption
Treatment
administered

Treatment
Group

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit 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sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

E[Y1 (1)Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
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E[Y0 | T = 1]
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ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>

Brady Nealtime
Assumptions and Proof

15 / 25

Parallel Trends Assumption
Treatment
administered

Treatment
Group

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit><latexit sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">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</latexit><latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

E[Y1 (1)Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
Assumptions<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit><latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit>

E[Y0 | T = 1]
<latexit sha1_base64="GWKSlxQAH7OAOhoCpjA48XtUgBE=">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</latexit>

E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>

Brady Nealtime
Assumptions and Proof

15 / 25

Parallel Trends Assumption
Treatment
administered

Treatment
Group

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit><latexit sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">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</latexit><latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

E[Y1 (1)Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
Assumptions<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit><latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit>

E[Y0 | T = 1]
<latexit sha1_base64="GWKSlxQAH7OAOhoCpjA48XtUgBE=">AAAUj3icrVjbbttGEN2kt8S92anf8sLGKdAHWZFcF+6LCgO2g7ZoAAfwLY0MQ6IomzBvJSmrjqCHfk3fivZz+jc9c3YpirpRDixB5Gp25szs3JbLduS5SVqr/ffg4QcffvTxJ48er3z62edffLm69uQkCXux7RzboRfGZ+1W4nhu4Bynbuo5Z1HstPy255y2r/dk/vTGiRM3DI7S28g591uXgdt17VYK0sXq0+bB2zcXNavpux2rmUI09Z0gtRpW/fxidaNWrfFjTQ/qZrChzOcwXHv8t2qqjgqVrXrKV44KVIqxp1oqwfetqquaikA7VwPQYoxczjtqqFYg2wOXA44WqNe4XuLfW0MN8F8wE0rb0OLhF0PSUt8Yng7GXVL1XfRbY7zzdAyILTbe4t42mD6oqboCtUwu41xero2vX7LqFGv4gat1sZKIFPGDXVhzF3cP/1OsUK634HQw6kAqxsgGzQNVU0RHjLv2vPjmipFokc/BSKxeZPciq2U+xPcaWC2ME1oqtlrqpYlLQM0ObRUejzGdj/gHVqjtW4TVHa3BZdz1KoT3CJRr9Q6jIvIinePZNZ8rNcjDGbqEPyVHAO4ElJCZ78J/LjiKuWoDU3S0GMlLriViBlUN8i+cz7I5Qjw3aU/CqFnMN5d6IlNZud2ZnR7ubWLHkB9g7op6xBcV2OMQO2b2Zl6skDvGvz7/2bTRnqBXWZkSzwrWJNlVAW6IGXeEpT2h/ZlZpK0bgGaZ76b6lfodxL1Cr0heVyAb0kKdxQl1BSb2DfQV8YuPq1DFsxmlQh0+7bhitlyDluvTSKLvBVaRsH48gzrA3aPXIqJUqF380+fY55okoj3q7mPcoR6RttDrqmrH2DEsaBXfuOxO01ot5p/Uru6Mk1pFLmClWiYrG9RUU9tmxRIBsXvca7bphronJ1zdcMKqNu3Pu4bkaNE+kcrjU4xVh9l4Rakirnhx9jpnrWWLdVJhLMX3l+BoUKpr6AlrMzKaVsbqb89USch42Lz7Jid0vYu2/sTMAHTfeMoDn65z2ZNk5ndgtphTTXpvfH4R4oGRz3YD6fgDUhfLnbCyirIxRoPRzGJ5lyOH/tEjibjGi+jriGv5evSzxmaWxb4fvGAKUTI2NUjLoneYG7NXfqQ20As2kFVFf0qkhRLz6STC+PmI8zlwhfeaWgJQpMIHo/lhaQz2QRkaj3U5l9VhboPLztchZ9NknubQFZVOcEhVL9YqeTkrbzR9sWxnlO9FWU1fLCuUG+ama/pBQqyzErmUcRAJfyxLj0qkQu4Jmq5tfrOEfQH32p7Ji9hYeFoi6bPH6FWFpipfldqXsgvGxici89t7ePCG/W/IJ6C7+jGXTe/kzVzu9r18msv37+jZXNK/o39zyXclkg73BndEE5nXpTUlXIelXDqfyvYEB7tJ9vyiq63CaxtelD27YzqTZfYbyQq9OzbYAypL9efpfUcquV5axz123zb7UEzNnQW1c3xvaHkEe0usL2G+5+fLZeprltzyebPHOMzqjtmMfgLJn0Ky00Mytj+5xM9PFgM+BfbNTpY9D24yA3QMf0IvfIVnrCE1fYfnuzquLwu73rKocs7qmX45jrsF5B3sV/vc1+6OG5nT0GbhzFTUcQAN2W9oTpc5t2fOAPOQ7qJ3NrLOus7UWWRSk37avAKXPoHdliDmeTwbb3F8ZyPrNyUhTwD5Oew+kMfPmMvh5Tkz/41A2XuK+afnCPcurh47emJ495l1Hju7gx6jT/j66Wlv6l2F5O8WqkJyd/DsyYS+Iv7k6feSp9ceOHIefYpJKa1p44jzztERZVLztJjwzJmfrKuFk7LGm7QtGcsr3+A1TD9yzG6xcrG6UZ98+zY9ONmq1r+v1l5vb+xumzdzj9RT9Ux9Cz/tqF1U5iH8aqs/1V/qH/Xv+tr6zvqP67ua9eEDI/OVKnzWf/4f+xH80Q==</latexit>

E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>

Brady Nealtime
Assumptions and Proof

15 / 25

Parallel Trends Assumption
Treatment
administered

Treatment
Group

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit><latexit sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">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</latexit><latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

E[Y1 (1)Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
Assumptions<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit><latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit>

E[Y0 | T = 1]
<latexit sha1_base64="GWKSlxQAH7OAOhoCpjA48XtUgBE=">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</latexit>

E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">AAAUj3icrVjbbttGEN2kt8S92anf8sLGKdAHWZFcF+6LCgO2g7ZoAAfwLY0MQ6IomzBvJSmrjqCHfk3fivZz+jc9c3YpirpRDixB5Gp25szs3JbLduS5SVqr/ffg4QcffvTxJ48er3z62edffLm69uQkCXux7RzboRfGZ+1W4nhu4Bynbuo5Z1HstPy255y2r/dk/vTGiRM3DI7S28g591uXgdt17VYK0sXq0+bB2zcXdavpux2rmUI09Z0gtRpW7fxidaNWrfFjTQ/qZrChzOcwXHv8t2qqjgqVrXrKV44KVIqxp1oqwfetqquaikA7VwPQYoxczjtqqFYg2wOXA44WqNe4XuLfW0MN8F8wE0rb0OLhF0PSUt8Yng7GXVL1XfRbY7zzdAyILTbe4t42mD6oqboCtUwu41xero2vX7LqFGv4gat1sZKIFPGDXVhzF3cP/1OsUK634HQw6kAqxsgGzQNVU0RHjLv2vPjmipFokc/BSKxeZPciq2U+xPcaWC2ME1oqtlrqpYlLQM0ObRUejzGdj/gHVqjtW4TVHa3BZdz1KoT3CJRr9Q6jIvIinePZNZ8rNcjDGbqEPyVHAO4ElJCZ78J/LjiKuWoDU3S0GMlLriViBlUN8i+cz7I5Qjw3aU/CqFnMN5d6IlNZud2ZnR7ubWLHkB9g7op6xBcV2OMQO2b2Zl6skDvGvz7/2bTRnqBXWZkSzwrWJNlVAW6IGXeEpT2h/ZlZpK0bgGaZ76b6lfodxL1Cr0heVyAb0kKdxQl1BSb2DfQV8YuPq1DFsxmlQh0+7bhitlyDluvTSKLvBVaRsH48gzrA3aPXIqJUqF380+fY55okoj3q7mPcoR6RttDrqmrH2DEsaBXfuOxO01ot5p/Uru6Mk1pFLmClWiYrG9RUU9tmxRIBsXvca7bphronJ1zdcMKqNu3Pu4bkaNE+kcrjU4xVh9l4Rakirnhx9jpnrWWLdVJhLMX3l+BoUKpr6AlrMzKaVsbqb89USch42Lz7Jid0vYu2/sTMAHTfeMoDn65z2ZNk5ndgtphTTXpvfH4R4oGRz3YD6fgDUhfLnbCyirIxRoPRzGJ5lyOH/tEjibjGi+jriGv5evSzxmaWxb4fvGAKUTI2NUjLoneYG7NXfqQ20As2kFVFf0qkhRLz6STC+PmI8zlwhfeaWgJQpMIHo/lhaQz2QRkaj3U5l9VhboPLztchZ9NknubQFZVOcEhVL9YqeTkrbzR9sWxnlO9FWU1fLCuUG+ama/pBQqyzErmUcRAJfyxLj0qkQu4Jmq5tfrOEfQH32p7Ji9hYeFoi6bPH6FWFpipfldqXsgvGxici89t7ePCG/W/IJ6C7+jGXTe/kzVzu9r18msv37+jZXNK/o39zyXclkg73BndEE5nXpTUlXIelXDqfyvYEB7tJ9vyiq63CaxtelD27YzqTZfYbyQq9OzbYAypL9efpfUcquV5axz123zb7UEzNnQW1c3xvaHkEe0usL2G+5+fLZeprltzyebPHOMzqjtmMfgLJn0Ky00Mytj+5xM9PFgM+BfbNTpY9D24yA3QMf0IvfIVnrCE1fYfnuzquLwu73rKocs7qmX45jrsF5B3sV/vc1+6OG5nT0GbhzFTUcQAN2W9oTpc5t2fOAPOQ7qJ3NrLOus7UWWRSk37avAKXPoHdliDmeTwbb3F8ZyPrNyUhTwD5Oew+kMfPmMvh5Tkz/41A2XuK+afnCPcurh47emJ495l1Hju7gx6jT/j66Wlv6l2F5O8WqkJyd/DsyYS+Iv7k6feSp9ceOHIefYpJKa1p44jzztERZVLztJjwzJmfrKuFk7LGm7QtGcsr3+A1TD9yzG6xcrG6UZ98+zY9ONmq1r+v1l5vb+xumzdzj9RT9Ux9Cz/tqF1U5iH8aqs/1V/qH/Xv+tr6zvqP67ua9eEDI/OVKnzWf/4f+yX80Q==</latexit>

ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>

Brady Nealtime
Assumptions and Proof

15 / 25

Parallel Trends Assumption
Treatment
administered

Treatment
Group

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit><latexit sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">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</latexit><latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

E[Y1 (1)Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
Assumptions<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit><latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit>

E[Y0 | T = 1]
<latexit sha1_base64="GWKSlxQAH7OAOhoCpjA48XtUgBE=">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</latexit>

E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>

Brady Nealtime
Assumptions and Proof

15 / 25

Parallel Trends Assumption
Treatment
administered

Treatment
Group

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit><latexit sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">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</latexit><latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

E[Y1 (1)Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
Assumptions<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit><latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit>

E[Y0 | T = 1]
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ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>

time

Parallel Trends Assumption: E[Y1 (0)<latexit sha1_base64="obCRQ1Ebk8Xd9dME4TFe8x/wgMo=">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</latexit>

Brady Neal

Y0 (0) | T = 1] = E[Y1 (0)Assumptions and Proof

Y0 (0) | T = 0]

15 / 25

No Pretreatment EffectTreatment
administered

Treatment
Group

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit 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sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">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</latexit><latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">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</latexit>

E[Y1 (1)Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
Assumptions<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">AAAUl3icrVjZbttGFJ2kW+JuTuqXoi9snAIpIKuS68J9MRLAC4KiAWzAW2oZhkRRFmFuJSm7jqAP6M+0r+2n9G967pmhKGqjHFiCyNGde8+9c7fhsBV5bpLWav89ePjBhx99/Mmjx0uffvb5F18uP3l6nIS92HaO7NAL49NWM3E8N3COUjf1nNModpp+y3NOWlfbMn9y7cSJGwaH6W3knPvNy8DtuHYzBeliebWxe/b2ov6i9r3V8N221UghnfpOkFpbVv3calSsl+CqVWv8WJODuhmsKvPZD588/ks1VFuFylY95StHBSrF2FNNleB7puqqpiLQzlUftBgjl/OOGqglyPbA5YCjCeoVrpf4d2aoAf4LZkJpG1o8/GJIWuo7w9PGuEOqvot+a4R3lo4+scXGW9xbBtMHNVVdUMvkMs7F5Vr4+iWrTrGGn7laFyuJSBE/2IU1d3D38D/FCuV6C04HozakYoxs0DxQNUV0xLhrz4tvuoxEk3wORmL1PLvnWS3zIb5XwGpinNBSsdVSeyYuATU7tFV4PMZ0NuIfWKG2bx5WZ7gGl3HXqxDeQ1Cu1DuMisjzdI5m12yu1CAPpugS/pQcAbgTUEJmvgv/ueAo5qoNTNHRZCQvuZaIGVQ1yL9wPsvmCPFcoz0Jo2Yx31zqiUxl5XZndnq4t4gdQ76PuS71iC8qsMchdszszbxYIXeMfzf8Z9NGe4xeZWVKPCtYk2RXBbghZtwhlvaE9mdmkbauD5plvmvqV+p3EPcKvSJ5XYFsSAt1FifUFZjYb6GviF98XIUqns0oFerwaUeX2XIFWq5PI4m+H7CKhPXjGdQ+7h69FhGlQu3inxuOfa5JItqj7huM29Qj0hZ6XVVtGjsGBa3iG5fdaVKrxfyT2tWdcVyryAWsVMtk5RY11dSGWbFEQOwe9ZptuqHuyQlXNxizqkX7864hOVq0T6Ty+BRj1WY2dilVxBUvTl/ntLWss04qjKX4/hIcW5TqGHrC2oyMpqWR+ts2VRIyHjbvvskJXe+i7WZspg+6bzzlgU/XuexJMvM7MJvMqQa9Nzo/D3HXyGe7gXT8Pqnz5Y5ZWUXZGKP+cGa+vMuRQ//okURc40X0dcS1fDv8WSMzi2LfD14wgSgZmxqkRdHbzI3pKz9Uq+gFq8iqoj8l0kKJ+XQSYfx8yPkcuMJ7RS0BKFLh/eH8oDQGO6AMjMc6nMvqMLfBZedrk7NhMk9z6IpKxzikqudrlbycljeaPl+2Pcz3oqymz5cVyjVz0zX9ICHWaYlcyjiIhD+SpYclUiH3BE3XNr9dwL6Ae23P5EVsLDwpkfTZY/SqQlOVb0rtS9kFY+MTkfntPTx4zf434BPQXf2Yy6Z38mYud/tePs3lb+7o2VzSv6N/c8l3JZIO9wZ3SBOZg9KaEq79Ui6dT2V7goPdJHt+0dVW4bUFL8qe3TadyTL7jWSF3h232AMqC/XnyX1HKrleWsc9dt8W+1BMze05tXN0b2h5BHsLrC9hvufny0Xqa5rc4nmzzThM647ZjH4CyZ9CstNDMrI/ucTPTxZ9PgXemJ0sex5cYwboGL5GL3yDZ6wBNf2I57s6rnuFXW9RVDln9Uy/HMVdB/Im9qsd7mt3x43MaWitcGYq6tiFhuw3MKfLnNszZ4BZSHfROx1ZZ1174iwyrkk/bXbBpU9gtyWIeR5Px5sf3+nI+k1JyBNAfg67D+TRM+ZieHnOzH4jUPaeYvbpOcK9g6vHjp4Y3h1mncfO7qDH6BO+fnrannhXIfm7jqqQ3O0/ezqmr4g/fvq95Om1B46cR59iUkpr2ijirHN0RJnUPC0mPHPmJ+tq4aSs8cZtS0byyjd4W6YfOWa3WLpYXq2Pv32bHByvV+s/VWsHG6uvNsybuUfqG/VMvYCfNtUrVOY+/GqrP9Xf6h/178rXKy9X9lZea9aHD4zMV6rwWTn4H8iH/qo=</latexit><latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit>

E[Y0 | T = 1]
<latexit sha1_base64="GWKSlxQAH7OAOhoCpjA48XtUgBE=">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</latexit>

E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">AAAUj3icrVjbbttGEN2kt8S92anf8sJGKdAHWZFdF+6LCwO2g7ZoAAfwLY0NQ6IoixBvJSm7jqCHfk3fivZz+jc9c3YpirpRCixB5Gp25szs3JbLZuS5SVqv//fg4QcffvTxJ48er3z62edffLm69uQ0CXux7ZzYoRfG581G4nhu4Jykbuo551HsNPym55w1u/syf3bjxIkbBsfpXeRc+o3rwG27diMF6Wr16cXh2zdXdevCd1vWRQrR1HeC1Nq16pdXq5V6rc6PNTnYNIOKMp+jcO3x3+pCtVSobNVTvnJUoFKMPdVQCb5v1aaqqwi0S9UHLcbI5byjBmoFsj1wOeBogNrF9Rr/3hpqgP+CmVDahhYPvxiSlvrG8LQwbpOq76LfGuGdpaNPbLHxDvemwfRBTVUH1DK5jHNxuSa+fsmqU6zhB67WxUoiUsQPdmHNbdw9/E+xQrnegdPBqAWpGCMbNA9UTREdMe7a8+KbDiPRIJ+DkVg9z+55Vst8iG8XWA2ME1oqtlrqpYlLQM0ObRUejzGdjfgHVqjtm4fVHq7BZdz1KoT3GJSueodREXmeztHsms2VGuTBFF3Cn5IjAHcCSsjMd+E/FxzFXLWBKToajOQ11xIxg2oG+RfOZ9kcIZ4btCdh1Czmm0s9kams3O7MTg/3JrFjyPcx16Ee8UUV9jjEjpm9mRer5I7x75b/bNpoj9FrrEyJZxVrkuyqAjfEjDvE0p7Q/sws0tb1QbPMd0P9Sv0O4l6lVySvq5ANaaHO4oS6AhP7XfQV8YuPq1DFsxmlSh0+7egwW7qg5fo0kuh7gVUkrB/PoPZx9+i1iChVahf/3HLsc00S0R5132Lcoh6RttDramrH2DEoaBXfuOxOk1ot5p/Uru6M41pFLmClWiYrd6mprrbNiiUCYveo12zTDXVPTri6wZhVTdqfdw3J0aJ9IpXHpxirFrOxQ6kirnhx+jqnrWWLdVJlLMX31+DYpVTb0BPWZmQ0rYzU376pkpDxsHn3TU7oehdtt2MzfdB94ykPfLrOZU+Smd+B2WBOXdB7o/PzEA+NfLYbSMfvkzpf7pSVVZSNMeoPZ+bLuxw59I8eScQ1XkRfR1zL18OfNTKzKPb94AUTiJKxqUFaFL3F3Ji+8mNVQS+oIKuK/pRICyXm00mE8fMh53PgCm+XWgJQpML7w/lBaQwOQBkYj7U5l9VhboPLztci54XJPM2hKyod45Cqnq9V8nJa3mj6fNnWMN+Lspo+X1YoN8xN1/SDhFjnJXIp4yAS/kiWHpdIhdwTNF3b/GYB+wLutT2TF7Gx8KxE0meP0asKTVW+KrUvZReMjU9E5rf38OAN+9+AT0DL+jGXTZfyZi53914+zeVvl/RsLukv6d9c8l2JpMO9wR3SROZ1aU0J11Epl86nsj3BwW6SPb/oaqvy2oQXZc9umc5kmf1GskLvjrvsAdWF+vPkviOVvFlaxz123yb7UEzNrTm1c3JvaHkEewusL2G+5+fLReprmtziebPPOEzrjtmMfgLJn0Ky00Mysj+5xM9PFn0+Bd6anSx7HtxgBugY/oRe+ArPWANq+g7Pd5u4vizseouiyjmrZ/rlKO4WkHewXx1wX1seNzKnoY3Cmamo4xAast/AnC5zbs+cAWYhLaN3OrLOutbEWWRck37a7IBLn8DuShDzPJ6ONz++05H1m5KQJ4D8HHYfyKNnzMXw8pyZ/Uag7D3F7NNzhHsbV48dPTG8B8w6j53dQY/RJ3z99LQ/8a5C8ncLVSG523/2ZExfEX/89HvN02sPHDmPPsWklNa0UcRZ5+iIMql5Wkx45sxP1rXCSVnjjduWjOSVb/B2TT9yzG6xcrVa2Rx/+zY5ON2qbX5fq7/eruxtmzdzj9RT9Ux9Cz/tqD1U5hH8aqs/1V/qH/Xv+tr6zvqP63ua9eEDI/OVKnzWf/4f5wH80A==</latexit>

Brady Nealtime
Assumptions and Proof

16 / 25

No Pretreatment EffectTreatment
administered

Treatment
Group

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit sha1_base64="iX6414k+YcZ/RuyOxcJbvdpUIjQ=">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</latexit><latexit sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">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</latexit><latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">AAAUm3icrVjbbttGEN2kt8S9xEn9VhRg4xRNAFmVXBfui4EAtoMiSAAH8CWpZRgSRdmEeCtJWXUEfUJ/pS99bT+kf9MzZ5eiqBvlwBJErmZnzszObblsRZ6bpLXaf3fufvTxJ59+du/+yudffPnVg9WHj46TsBfbzpEdemH8ttVMHM8NnKPUTT3nbRQ7Tb/lOSet7q7Mn1w5ceKGwWF6HTlnfvMicDuu3UxBOl/9obF/+u68/rT+zNqwZFB7ZjV8t201UsCkvhOk1o5VPztfXa9Va/xY04O6Gawr8zkIH97/SzVUW4XKVj3lK0cFKsXYU02V4Huq6qqmItDO1AC0GCOX844aqhXI9sDlgKMJahfXC/w7NdQA/wUzobQNLR5+MSQt9b3haWPcIVXfRb81xjtPx4DYYuM17i2D6YOaqktQy+QyzuXlWvj6JatOsYZfuFoXK4lIET/YhTV3cPfwP8UK5XoNTgejNqRijGzQPFA1RXTEuGvPi28uGYkm+RyMxOpFdi+yWuZDfLvAamKc0FKx1VIvTFwCanZoq/B4jOl8xD+wQm3fIqzOaA0u465XIbyHoHTVe4yKyIt0jmfXfK7UIA9n6BL+lBwBuBNQQma+C/+54Cjmqg1M0dFkJC+4logZVDXILzmfZXOEeG7QnoRRs5hvLvVEprJyuzM7PdxbxI4hP8DcJfWILyqwxyF2zOzNvFghd4x/ff6zaaM9Qa+yMiWeFaxJsqsC3BAz7ghLe0L7M7NIWzcAzTLfDfWK+h3EvUKvSF5XIBvSQp3FCXUFJvY76CviFx9XoYpnM0qFOnzaccls6YKW69NIou9HrCJh/XgGdYC7R69FRKlQu/inz7HPNUlEe9Tdx7hNPSJtoddV1baxY1jQKr5x2Z2mtVrMP6ld3RkntYpcwEq1TFbuUFNNbZkVSwTE7nGv2aYb6p6ccHXDCatatD/vGpKjRftEKo9PMVZtZuMlpYq44sXZ65y1lk3WSYWxFN9fgGOHUh1DT1ibkdG0MlZ/u6ZKQsbD5t03OaHrXbT1J2YGoPvGUx74dJ3LniQzvwOzyZxq0Hvj84sQ9418thtIxx+QuljumJVVlI0xGoxmFsu7HDn0jx5JxDVeRF9HXMt3o581NrMs9u3gBVOIkrGpQVoWvc3cmL3yQ7WOXrCOrCr6UyItlJhPJxHGT0acT4ArvF1qCUCRCh+M5oelMdgDZWg81uFcVoe5DS47X5ucDZN5mkNXVDrBIVW9WKvk5ay80fTFsu1RvhdlNX2xrFCumJuu6QcJsd6WyKWMg0j4Y1l6WCIVck/QdG3zuyXsC7jX9kxexMbCkxJJnz1Gryo0Vfm61L6UXTA2PhGZ3z7Ag1fsf0M+Ad3Uj7lseiNv5nLXH+TTXL5/Q8/mkv4N/ZtLvi+RdLg3uCOayLwprSnhOijl0vlUtic42E2y5xddbRVeW/Ci7Nlt05kss99IVujdcYc9oLJUf57ed6SS66V13GP3bbEPxdTcXlA7R7eGlkewt8T6EuZ7fr5cpr5myS2fN7uMw6zumM3oJ5D8KSQ7PSRj+5NL/PxkMeBTYN/sZNnz4AYzQMfwV/TC13jGGlLTT3i+q+P6orDrLYsq56ye6ZfjuJtA3sZ+tcd97ea4kTkNbRTOTEUd+9CQ/YbmdJlze+YMMA/pJnpnI+usa0+dRSY16afNS3DpE9h1CWKex7PxFsd3NrJ+UxLyBJCfw24DefyMuRxenjPz3wiUvaeYf3qOcO/g6rGjJ4Z3j1nnsbM76DH6hK+fnnan3lVI/m6iKiR3B48fTegr4k+efi94eu2BI+fRp5iU0po2jjjvHB1RJjVPiwnPnPnJulo4KWu8SduSsbzyDd6O6UeO2S1WzlfX65Nv36YHx5vV+s/V2put9edb5s3cPfWNeqyewk/b6jkq8wB+tdWf6m/1j/p37du13bWXa6806907RuZrVfisHf0PG2j/ow==</latexit>

E[Y1 (1)Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
Assumptions<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit><latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit>

E[Y0 | T = 1]
<latexit sha1_base64="GWKSlxQAH7OAOhoCpjA48XtUgBE=">AAAUj3icrVjbbttGEN2kt8S92anf8sLGKdAHWZFcF+6LCgO2g7ZoAAfwLY0MQ6IomzBvJSmrjqCHfk3fivZz+jc9c3YpirpRDixB5Gp25szs3JbLduS5SVqr/ffg4QcffvTxJ48er3z62edffLm69uQkCXux7RzboRfGZ+1W4nhu4Bynbuo5Z1HstPy255y2r/dk/vTGiRM3DI7S28g591uXgdt17VYK0sXq0+bB2zcXNavpux2rmUI09Z0gtRpW/fxidaNWrfFjTQ/qZrChzOcwXHv8t2qqjgqVrXrKV44KVIqxp1oqwfetqquaikA7VwPQYoxczjtqqFYg2wOXA44WqNe4XuLfW0MN8F8wE0rb0OLhF0PSUt8Yng7GXVL1XfRbY7zzdAyILTbe4t42mD6oqboCtUwu41xero2vX7LqFGv4gat1sZKIFPGDXVhzF3cP/1OsUK634HQw6kAqxsgGzQNVU0RHjLv2vPjmipFokc/BSKxeZPciq2U+xPcaWC2ME1oqtlrqpYlLQM0ObRUejzGdj/gHVqjtW4TVHa3BZdz1KoT3CJRr9Q6jIvIinePZNZ8rNcjDGbqEPyVHAO4ElJCZ78J/LjiKuWoDU3S0GMlLriViBlUN8i+cz7I5Qjw3aU/CqFnMN5d6IlNZud2ZnR7ubWLHkB9g7op6xBcV2OMQO2b2Zl6skDvGvz7/2bTRnqBXWZkSzwrWJNlVAW6IGXeEpT2h/ZlZpK0bgGaZ76b6lfodxL1Cr0heVyAb0kKdxQl1BSb2DfQV8YuPq1DFsxmlQh0+7bhitlyDluvTSKLvBVaRsH48gzrA3aPXIqJUqF380+fY55okoj3q7mPcoR6RttDrqmrH2DEsaBXfuOxO01ot5p/Uru6Mk1pFLmClWiYrG9RUU9tmxRIBsXvca7bphronJ1zdcMKqNu3Pu4bkaNE+kcrjU4xVh9l4Rakirnhx9jpnrWWLdVJhLMX3l+BoUKpr6AlrMzKaVsbqb89USch42Lz7Jid0vYu2/sTMAHTfeMoDn65z2ZNk5ndgtphTTXpvfH4R4oGRz3YD6fgDUhfLnbCyirIxRoPRzGJ5lyOH/tEjibjGi+jriGv5evSzxmaWxb4fvGAKUTI2NUjLoneYG7NXfqQ20As2kFVFf0qkhRLz6STC+PmI8zlwhfeaWgJQpMIHo/lhaQz2QRkaj3U5l9VhboPLztchZ9NknubQFZVOcEhVL9YqeTkrbzR9sWxnlO9FWU1fLCuUG+ama/pBQqyzErmUcRAJfyxLj0qkQu4Jmq5tfrOEfQH32p7Ji9hYeFoi6bPH6FWFpipfldqXsgvGxici89t7ePCG/W/IJ6C7+jGXTe/kzVzu9r18msv37+jZXNK/o39zyXclkg73BndEE5nXpTUlXIelXDqfyvYEB7tJ9vyiq63CaxtelD27YzqTZfYbyQq9OzbYAypL9efpfUcquV5axz123zb7UEzNnQW1c3xvaHkEe0usL2G+5+fLZeprltzyebPHOMzqjtmMfgLJn0Ky00Mytj+5xM9PFgM+BfbNTpY9D24yA3QMf0IvfIVnrCE1fYfnuzquLwu73rKocs7qmX45jrsF5B3sV/vc1+6OG5nT0GbhzFTUcQAN2W9oTpc5t2fOAPOQ7qJ3NrLOus7UWWRSk37avAKXPoHdliDmeTwbb3F8ZyPrNyUhTwD5Oew+kMfPmMvh5Tkz/41A2XuK+afnCPcurh47emJ495l1Hju7gx6jT/j66Wlv6l2F5O8WqkJyd/DsyYS+Iv7k6feSp9ceOHIefYpJKa1p44jzztERZVLztJjwzJmfrKuFk7LGm7QtGcsr3+A1TD9yzG6xcrG6UZ98+zY9ONmq1r+v1l5vb+xumzdzj9RT9Ux9Cz/tqF1U5iH8aqs/1V/qH/Xv+tr6zvqP67ua9eEDI/OVKnzWf/4f+xH80Q==</latexit>

E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>time

No Pretreatment Effect Assumption: E[Y0 (1) | T = 1]
<latexit sha1_base64="ARXjfm2ubtVJbzl1ahVXcGFeP08=">AAAUxXicrVhZb9tGEN6kV+IecVK/9YWNUiAFZFVyXbgvAgLYTouiAWzAV2sZhkRSFiFeJVdWHULoP+prf0sf+tr+jc58uxRJHaQUWILI1ezMN7NzLZe90HVi2Wz+/eDhe+9/8OFHjx5vfPzJp5892Xz67CwORpFpn5qBG0QXvW5su45vn0pHuvZFGNldr+fa573hPs+f39pR7AT+ibwL7Suve+M7fcfsSiJdbx53Di87wUiagWdfN1+2vjY6nmMZHUkg0rN9abSN1pWxbRT5mkv42kbzerPWbDTxMeYHLT2oCf05Cp4+/lN0hCUCYYqR8IQtfCFp7IquiOl7KVqiKUKiXYmEaBGNHMzbYiI2SHZEXDZxdIk6pOsN/bvUVJ/+M2YMaZO0uPSLSNIQX2kei8Z9UNWd9Rs53mU6EmCzjXd072lMj6hSDIhaJZdyri7Xo69XsWpJa/geq3VoJSEo7AezsOY+3V36L2mFfL0jTptGFklFNDKJ5hJVUVhHRHflefbNAJHogs+mEVtdZneZ1Twf0HdIWF0ax7CUbTXEax0XH5pt2Mo8LmK6HPF3WqGyrwyrP12Dg7irVTDvCVGG4i2NishlOvPZtZxLauTJAl3ML8HhE3dMlACZ75D/HOIo5qpJmKyji0jeYC0hMqihkX/CfJrNIcVzG/bEiJqBfHOgJ9SVldmd2unSvQfsiOQTmhtAD/uiTvbYwI6QvakX6+CO6N8Y/0zYaM7QG6hMjmed1sTZVSfcgGacKZbyhPJnapGyLiGaob/b4mfotynudXiF87pOsgEsVFkcQ5evY9+mvsJ+8ejKVPZsSqlDhwc7BsiWIdEyfQqJ9X1Dq4hRP65GTejuwmshUOrQzv4ZY+xhTRzREXSPaWxBD0sb1OsaYk/bMSloZd846E7zWg3kH9eu6oyzWlnOR6UaOivb0NQUu3rFHAG2O+81U3dD1ZNjrG4yY1UP9mddg3O0aB9LZfEpxspCNg4gVcRlLy5e56K17KBO6ogl+/6GONqQ6mt6jNoMtaaNXP3t6yoJEA8Td0/nhKp31jaemUmI7mlPucSn6pz3JJ75jTC7yKkOvJefL0M81PLpbsAdPwG1XO4MlVWUjWiUTGfK5R2MbPhHjTjiCi+Er0Os5cvpz8jNrIp9P3j+HCJnrNRIq6JbyI3FKz8RNeoFNcqqoj850kyJ8HQS0vjFlPMF4TLvEFp8onCFJ9P5SWUMDogy0R7rYy6tw8wGB53PAmdHZ57iUBUlZzi4qsu1cl4uyhtFL5e1pvlelFX0clmm3CI3Hd0PYmBdVMhJxIElvFyWnlRIBdgTFF3Z/MsK9vnYa0c6LyJt4XmFpIceo1YV6Kp8U2mfRBeMtE9Y5td38OAt+t8ET0Dr+jGTlWt5M5O7eyefZvLjNT2bSXpr+jeTfFshaWNvcKY0ljmurCnmOqrkUvlUtSfYtJukzy+q2uq49siLvGdbujMZer/hrFC7Yxs9oL5Sf57fd7iSW5V1PEL37aEPRdBsldTO6b2hZREcrbC+GPmenS9Xqa9FcqvnzT7isKg7pjPqCSR7CklPD3Fuf3KAn50sEjwFjvVOlj4PbiMDVAx/pF74hp6xJtD0LT3ftej6urDrrYrK56yR7pd53B1C3qP96gD72vq4oT4NbRfOTEUdh6Qh/U306TLjdvUZYBnSOnoXI6uss+bOIrOa1NPmgLjUCeyuAjHL48V45fFdjKzelAQ4AWTnsPtAzp8xV8PLcmb5G4Gq9xTLT88h3ft0ddHRY817gKxz0dlt6jHqhK+envbn3lVw/u5QVXDuJs+fzegr4s+efm9weh0RR8ajTjES0oqWR1x2jg4hI/XTYowzZ3aybhROygpv1rY4l1eexmvrfmTr3WLjerPWmn37Nj8422m0vms0j3drr3b1m7lH4gvxXLwkP+2JV1SZR+RXU/wl/hH/iv+2ftjytuTWrWJ9+EDLfC4Kn60//gffgBAM</latexit>

Brady Neal

Assumptions and Proof

E[Y0 (0) | T = 1] = 0

16 / 25

No Pretreatment EffectTreatment
administered

Treatment
Group

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit 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E[Y1 (1)Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
Assumptions<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">AAAUl3icrVjZbttGFJ2kW+JuTuqXoi9snAIpIKuS68J9MRLAC4KiAWzAW2oZhkRRFmFuJSm7jqAP6M+0r+2n9G967pmhKGqjHFiCyNGde8+9c7fhsBV5bpLWav89ePjBhx99/Mmjx0uffvb5F18uP3l6nIS92HaO7NAL49NWM3E8N3COUjf1nNModpp+y3NOWlfbMn9y7cSJGwaH6W3knPvNy8DtuHYzBeliebWxe/b2ov6i9r3V8N221UghnfpOkFpbVv3calSsl+CqVWv8WJODuhmsKvPZD588/ks1VFuFylY95StHBSrF2FNNleB7puqqpiLQzlUftBgjl/OOGqglyPbA5YCjCeoVrpf4d2aoAf4LZkJpG1o8/GJIWuo7w9PGuEOqvot+a4R3lo4+scXGW9xbBtMHNVVdUMvkMs7F5Vr4+iWrTrGGn7laFyuJSBE/2IU1d3D38D/FCuV6C04HozakYoxs0DxQNUV0xLhrz4tvuoxEk3wORmL1PLvnWS3zIb5XwGpinNBSsdVSeyYuATU7tFV4PMZ0NuIfWKG2bx5WZ7gGl3HXqxDeQ1Cu1DuMisjzdI5m12yu1CAPpugS/pQcAbgTUEJmvgv/ueAo5qoNTNHRZCQvuZaIGVQ1yL9wPsvmCPFcoz0Jo2Yx31zqiUxl5XZndnq4t4gdQ76PuS71iC8qsMchdszszbxYIXeMfzf8Z9NGe4xeZWVKPCtYk2RXBbghZtwhlvaE9mdmkbauD5plvmvqV+p3EPcKvSJ5XYFsSAt1FifUFZjYb6GviF98XIUqns0oFerwaUeX2XIFWq5PI4m+H7CKhPXjGdQ+7h69FhGlQu3inxuOfa5JItqj7huM29Qj0hZ6XVVtGjsGBa3iG5fdaVKrxfyT2tWdcVyryAWsVMtk5RY11dSGWbFEQOwe9ZptuqHuyQlXNxizqkX7864hOVq0T6Ty+BRj1WY2dilVxBUvTl/ntLWss04qjKX4/hIcW5TqGHrC2oyMpqWR+ts2VRIyHjbvvskJXe+i7WZspg+6bzzlgU/XuexJMvM7MJvMqQa9Nzo/D3HXyGe7gXT8Pqnz5Y5ZWUXZGKP+cGa+vMuRQ//okURc40X0dcS1fDv8WSMzi2LfD14wgSgZmxqkRdHbzI3pKz9Uq+gFq8iqoj8l0kKJ+XQSYfx8yPkcuMJ7RS0BKFLh/eH8oDQGO6AMjMc6nMvqMLfBZedrk7NhMk9z6IpKxzikqudrlbycljeaPl+2Pcz3oqymz5cVyjVz0zX9ICHWaYlcyjiIhD+SpYclUiH3BE3XNr9dwL6Ae23P5EVsLDwpkfTZY/SqQlOVb0rtS9kFY+MTkfntPTx4zf434BPQXf2Yy6Z38mYud/tePs3lb+7o2VzSv6N/c8l3JZIO9wZ3SBOZg9KaEq79Ui6dT2V7goPdJHt+0dVW4bUFL8qe3TadyTL7jWSF3h232AMqC/XnyX1HKrleWsc9dt8W+1BMze05tXN0b2h5BHsLrC9hvufny0Xqa5rc4nmzzThM647ZjH4CyZ9CstNDMrI/ucTPTxZ9PgXemJ0sex5cYwboGL5GL3yDZ6wBNf2I57s6rnuFXW9RVDln9Uy/HMVdB/Im9qsd7mt3x43MaWitcGYq6tiFhuw3MKfLnNszZ4BZSHfROx1ZZ1174iwyrkk/bXbBpU9gtyWIeR5Px5sf3+nI+k1JyBNAfg67D+TRM+ZieHnOzH4jUPaeYvbpOcK9g6vHjp4Y3h1mncfO7qDH6BO+fnrannhXIfm7jqqQ3O0/ezqmr4g/fvq95Om1B46cR59iUkpr2ijirHN0RJnUPC0mPHPmJ+tq4aSs8cZtS0byyjd4W6YfOWa3WLpYXq2Pv32bHByvV+s/VWsHG6uvNsybuUfqG/VMvYCfNtUrVOY+/GqrP9Xf6h/178rXKy9X9lZea9aHD4zMV6rwWTn4H8iH/qo=</latexit><latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit>

E[Y0 | T = 1]
<latexit sha1_base64="GWKSlxQAH7OAOhoCpjA48XtUgBE=">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</latexit>

E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>time

No Pretreatment Effect Assumption: E[Y0 (1) | T = 1]
<latexit sha1_base64="ARXjfm2ubtVJbzl1ahVXcGFeP08=">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</latexit>

Brady Neal

Assumptions and Proof

E[Y0 (0) | T = 1] = 0

16 / 25

No Pretreatment EffectTreatment
administered

Treatment
Group

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit 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E[Y1 (1)Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
Assumptions<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit><latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit>

E[Y0 (1) | T = 1] = E[Y0 | T = 1]
<latexit sha1_base64="GWKSlxQAH7OAOhoCpjA48XtUgBE=">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</latexit><latexit sha1_base64="DOnBydaaldsxt1aTVeuV+tbbfkQ=">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</latexit>

E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>time

No Pretreatment Effect Assumption: E[Y0 (1) | T = 1]
<latexit sha1_base64="ARXjfm2ubtVJbzl1ahVXcGFeP08=">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</latexit>

Brady Neal

Assumptions and Proof

E[Y0 (0) | T = 1] = 0

16 / 25

No Pretreatment EffectTreatment
administered

Treatment
Group

E[Y1 | T = 1] = E[Y1 (1) | T = 1]
<latexit 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sha1_base64="xdaVoclE+kDAngvrlih8MIn+Sfo=">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</latexit><latexit sha1_base64="Z0sDXAq916i29khVZqg5KMF66Es=">AAAUm3icrVjbbttGEN2kt8S9xEn9VhRg4xRNAFmVXBfui4EAtoMiSAAH8CWpZRgSRdmEeCtJWXUEfUJ/pS99bT+kf9MzZ5eiqBvlwBJErmZnzszObblsRZ6bpLXaf3fufvTxJ59+du/+yudffPnVg9WHj46TsBfbzpEdemH8ttVMHM8NnKPUTT3nbRQ7Tb/lOSet7q7Mn1w5ceKGwWF6HTlnfvMicDuu3UxBOl/9obF/+u68/rT+zNqwZFB7ZjV8t201UsCkvhOk1o5VPztfXa9Va/xY04O6Gawr8zkIH97/SzVUW4XKVj3lK0cFKsXYU02V4Huq6qqmItDO1AC0GCOX844aqhXI9sDlgKMJahfXC/w7NdQA/wUzobQNLR5+MSQt9b3haWPcIVXfRb81xjtPx4DYYuM17i2D6YOaqktQy+QyzuXlWvj6JatOsYZfuFoXK4lIET/YhTV3cPfwP8UK5XoNTgejNqRijGzQPFA1RXTEuGvPi28uGYkm+RyMxOpFdi+yWuZDfLvAamKc0FKx1VIvTFwCanZoq/B4jOl8xD+wQm3fIqzOaA0u465XIbyHoHTVe4yKyIt0jmfXfK7UIA9n6BL+lBwBuBNQQma+C/+54Cjmqg1M0dFkJC+4logZVDXILzmfZXOEeG7QnoRRs5hvLvVEprJyuzM7PdxbxI4hP8DcJfWILyqwxyF2zOzNvFghd4x/ff6zaaM9Qa+yMiWeFaxJsqsC3BAz7ghLe0L7M7NIWzcAzTLfDfWK+h3EvUKvSF5XIBvSQp3FCXUFJvY76CviFx9XoYpnM0qFOnzaccls6YKW69NIou9HrCJh/XgGdYC7R69FRKlQu/inz7HPNUlEe9Tdx7hNPSJtoddV1baxY1jQKr5x2Z2mtVrMP6ld3RkntYpcwEq1TFbuUFNNbZkVSwTE7nGv2aYb6p6ccHXDCatatD/vGpKjRftEKo9PMVZtZuMlpYq44sXZ65y1lk3WSYWxFN9fgGOHUh1DT1ibkdG0MlZ/u6ZKQsbD5t03OaHrXbT1J2YGoPvGUx74dJ3LniQzvwOzyZxq0Hvj84sQ9418thtIxx+QuljumJVVlI0xGoxmFsu7HDn0jx5JxDVeRF9HXMt3o581NrMs9u3gBVOIkrGpQVoWvc3cmL3yQ7WOXrCOrCr6UyItlJhPJxHGT0acT4ArvF1qCUCRCh+M5oelMdgDZWg81uFcVoe5DS47X5ucDZN5mkNXVDrBIVW9WKvk5ay80fTFsu1RvhdlNX2xrFCumJuu6QcJsd6WyKWMg0j4Y1l6WCIVck/QdG3zuyXsC7jX9kxexMbCkxJJnz1Gryo0Vfm61L6UXTA2PhGZ3z7Ag1fsf0M+Ad3Uj7lseiNv5nLXH+TTXL5/Q8/mkv4N/ZtLvi+RdLg3uCOayLwprSnhOijl0vlUtic42E2y5xddbRVeW/Ci7Nlt05kss99IVujdcYc9oLJUf57ed6SS66V13GP3bbEPxdTcXlA7R7eGlkewt8T6EuZ7fr5cpr5myS2fN7uMw6zumM3oJ5D8KSQ7PSRj+5NL/PxkMeBTYN/sZNnz4AYzQMfwV/TC13jGGlLTT3i+q+P6orDrLYsq56ye6ZfjuJtA3sZ+tcd97ea4kTkNbRTOTEUd+9CQ/YbmdJlze+YMMA/pJnpnI+usa0+dRSY16afNS3DpE9h1CWKex7PxFsd3NrJ+UxLyBJCfw24DefyMuRxenjPz3wiUvaeYf3qOcO/g6rGjJ4Z3j1nnsbM76DH6hK+fnnan3lVI/m6iKiR3B48fTegr4k+efi94eu2BI+fRp5iU0po2jjjvHB1RJjVPiwnPnPnJulo4KWu8SduSsbzyDd6O6UeO2S1WzlfX65Nv36YHx5vV+s/V2put9edb5s3cPfWNeqyewk/b6jkq8wB+tdWf6m/1j/p37du13bWXa6806907RuZrVfisHf0PG2j/ow==</latexit>

E[Y1 (1)Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
Assumptions<latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit><latexit sha1_base64="QZ2+Iln12IwJpBbo7OBMeBVfEOU=">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</latexit>

E[Y0 (1) | T = 1] = E[Y0 | T = 1]
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E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit>time

No Pretreatment Effect Assumption: E[Y0 (1) | T = 1]
<latexit sha1_base64="ARXjfm2ubtVJbzl1ahVXcGFeP08=">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</latexit>

Brady Neal

Assumptions and Proof

E[Y0 (0) | T = 1] = 0

16 / 25

Assumptions Change

Brady Neal

Assumptions and Proof

17 / 25

Assumptions Change
Causal Estimand

Brady Neal

Assumptions

Assumptions and Proof

17 / 25

Assumptions Change
Causal Estimand
E[Y (1)<latexit sha1_base64="9/ClvRHvi/8lFMggdK7KC4h2YSM=">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</latexit>

Assumptions

Y (0) | T = 1] = E[Y (1) | T = 1]
= E[Y | T = 1]

E[Y (0) | T = 1]

= E[Y | T = 1]

E[Y | T = 0]

= E[Y | T = 1]

Brady NealE[Y (0) | T = 1]

E[Y (0) | T = 0]

Assumptions and Proof

17 / 25

Assumptions Change
Causal Estimand
E[Y (1)<latexit sha1_base64="9/ClvRHvi/8lFMggdK7KC4h2YSM=">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</latexit>

Assumptions= E[Y | T = 1]
= E[Y | T = 1]
= E[Y | T = 1]

Brady Neal

Unconfoundedness:
E[Y (0) | T = 1]
Y (0) ?? T
E[Y (0) | T = 1]
E[Y (0) | T = 0]
E[Y | T = 0]

Y (0) | T = 1] = E[Y (1) | T = 1]

Assumptions and Proof<latexit sha1_base64="gWO2eWBKWJyi4omMXcdiRNAO2sw=">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</latexit>

17 / 25

Assumptions Change
Causal Estimand
E[Y (1)<latexit sha1_base64="9/ClvRHvi/8lFMggdK7KC4h2YSM=">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</latexit>

Assumptions= E[Y | T = 1]
= E[Y | T = 1]

<latexit sha1_base64="YaJa8u54w+KJbbrWwdJ+E6Mxch0=">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</latexit>

E[Y1 (1)Brady Neal

Unconfoundedness:
E[Y (0) | T = 1]
Y (0) ?? T
E[Y (0) | T = 1]
E[Y (0) | T = 0]
E[Y | T = 0]

Y (0) | T = 1] = E[Y (1) | T = 1]

Y1 (0) | T = 1]= E[Y | T = 1]

Assumptions and Proof<latexit sha1_base64="gWO2eWBKWJyi4omMXcdiRNAO2sw=">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</latexit>

17 / 25

Assumptions Change
Causal Estimand
E[Y (1)<latexit sha1_base64="9/ClvRHvi/8lFMggdK7KC4h2YSM=">AAAVkXicvVhLb9tGEF6lr8R92Ql8ag9slBYOIKmSm8LpQYBR20HRNoAD2LEdyzAkkrII8VWSsuoIOvXX9NJD+2v6bzrz7VIkJYqUA6MSRK5mZ76Znddy2fNtK4yazX8r9957/4MPP7r/YO3jTz797PP1jYevQ28U6Oax7tlecNrrhqZtueZxZEW2eeoHZtfp2eZJb7jH8yfXZhBannsU3fjmhdO9cq2+pXcjIl1uVL7sHJyfbbWeanXtbKv5VOs4lqF1IsKIHNONtLbWutC+aWsxW+58XU4vE+901hTC/y7eXFU8V/ZyvdpsNPHRFgctNagK9Tn0Nh78JTrCEJ7QxUg4whSuiGhsi64I6XsuWqIpfKJdiAnRAhpZmDfFVKyR7Ii4TOLoEnVI1yv6d66oLv1nzBDSOmmx6ReQpCa+VjwGjfugyjvr11K8y3RMgM023tC9pzAdokZiQNQyuZhzdbkefZ2SVUe0hudYrUUr8UFhP+iZNffpbtP/iFbI1xviNGlkkFRAI51oNlElhXUEdJeeZ98MEIku+EwasdVFdhdZzfMefYeE1aVxCEvZVk28UHFxodmErcxjI6bLEX+nFUr7irD6szVYiLtcBfMeEWUo3tIoi1ykM51dy7kihTzN0cX8EThc4g6J4iHzLfKfRRzZXNUJk3V0EckrrMVHBjUU8s+Yj7PZp3jWYU+IqGnINwt6fFVZid2xnTbde8AOSH5CcwPoYV/UyB4T2AGyN/ZiDdwB/Rvjnw4b9Tl6A5XJ8azRmji7aoTr0Yw1w5KekP6MLZLWTYimqW9d/Ar9JsW9Bq9wXtdI1oOFMotD6HJV7NvUV9gvDl2Zyp6NKTXocGDHANkyJFqiTyKxvm9pFSHqx1aoE7rb8JoPlBq0s3/GGDtYE0d0BN1jGhvQw9Ia9bqG2FF2TDNa2TcWutOiVg35x7UrO+O8VpZzUamayso2NDXFM7VijgDbnfaarrqh7MkhVjeds6oH+5OuwTmatY+lkvhkY2UgGweQyuKyF/PXmbeWbdRJDbFk318RRxtSfUUPUZu+0rSWqr89VSUe4qHj7qickPXO2sZzMxOiO8pTNvHJOuc9iWd+I8wucqoD76XnixAPlHy8G3DHn4BaLPcalZWVDWg0mc0Uy1sYmfCPHHHEJZ4PX/tYy1ezn5aaWRX7bvDcBUTO2EghrYpuIDfyV34kqtQLqpRVWX9ypJkS4OnEp/GTGecTwmXeIbS4ROEKn8zmp6Ux2CfKVHmsj7m4DhMbLHQ+A5wdlXmSQ1ZUNMfBVV2slfMyL28kvVjWmOV7VlbSi2WZco3ctFQ/CIF1WiIXIQ4s4aSy9KhEysOeIOnS5rMV7HOx145UXgTKwpMSSQc9Rq7KU1X5stS+CF0wUD5hmTfv4MFr9L8pnoBu68dENrqVNxO5m3fyaSI/vqVnE0nnlv5NJN+WSJrYG6wZjWVeldYUcx2Wcsl8KtsTTNpN4ucXWW01XHvkRd6zDdWZNLXfcFbI3bGNHlBbqT8v7jtcya3SOh6h+/bQhwJoNgpq5/jO0JIIjlZYX4h8T86Xq9RXntzqebOHOOR1x3hGPoEkTyHx6SFM7U8W8JOTxQRPgWO1k8XPg3VkgIzhT9QLX9Iz1hSavqPnuxZdX2R2vVVR+Zw1Uv0yjbtNyDu0X+1jX7s9rq9OQ/XMmSmr44A0xL+pOl0m3LY6AyxDuo3efGSZdcbCWWRek3zaHBCXPIHdlCAmeZyPVxzffGT5psTDCSA5h90FcvqMuRpekjPL3wiUvadYfnr26d6nq42OHirefWSdjc5uUo+RJ3z59LS38K6C83ebqoJzd/L44Zy+LP786fcKp9cRcSQ88hQTQVrS0ojLztE+ZCL1tBjizJmcrBuZk7LEm7ctTOWVo/Daqh+ZardYu1yvtubfvi0OXm83Wt83mq+eVXefqzdz98UX4rHYIj/tiF2qzEPyq175o/Jn5e/KP5uPNn/Y3N38UbLeqyiZRyLz2fzlP/RjSfU=</latexit><latexit sha1_base64="YaJa8u54w+KJbbrWwdJ+E6Mxch0=">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</latexit>

E[Y1 (1)Brady Neal

Assumptions

Unconfoundedness:
E[Y (0) | T = 1]
Y (0) ?? T
= E[Y | T = 1] E[Y (0) | T = 1]
= E[Y | T = 1] E[Y (0) | T = 0]
Parallel Trends:
T =1] Y0E[Y(0) | |TT==
1] 0]= E[Y1 (0) Y0 (0) | T = 0]
Y (0) | T = 1]= E[Y | E[Y
1 (0)Y (0) | T = 1] = E[Y (1) | T = 1]

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1

Assumptions and Proof

17 / 25

Assumptions Change
Causal Estimand
E[Y (1)<latexit sha1_base64="9/ClvRHvi/8lFMggdK7KC4h2YSM=">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</latexit><latexit sha1_base64="YaJa8u54w+KJbbrWwdJ+E6Mxch0=">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</latexit>

E[Y1 (1)Assumptions

Unconfoundedness:
E[Y (0) | T = 1]
Y (0) ?? T
= E[Y | T = 1] E[Y (0) | T = 1]
= E[Y | T = 1] E[Y (0) | T = 0]
Parallel Trends:
T =1] Y0E[Y(0) | |TT==
1] 0]= E[Y1 (0) Y0 (0) | T = 0]
Y (0) | T = 1]= E[Y | E[Y
1 (0)Y (0) | T = 1] = E[Y (1) | T = 1]

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1<latexit sha1_base64="VgmJf4DWSb/OcMDpwKhv+LWuJ8c=">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</latexit>(Y1 (0)Brady Neal

Assumptions and Proof

Y0 (0)) ?? T

17 / 25

Difference-in-Differences
Treatment
administered

E[Y1 | T = 1]
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Treatment
Group

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Y1 (0) | T = 1]

E[Y1 (0) | T = 1] ?
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E[Y0 | T = 1]
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E[Y1 | T = 0]<latexit sha1_base64="PKjZjI0b5fW+DynfluNHHXheNFs=">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</latexit>

ControlGroup
E[Y0 | T = 0]<latexit sha1_base64="USWa5QqyvB4KOUQNoIvUK7pg4+k=">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</latexit><latexit sha1_base64="4Gwi+xAwrn/BaW60eKvMSBAkJZk=">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</latexit>

E[Y1 (1)Brady Nealtime

Y1 (0) | T = 1] = (E[Y1 | T = 1]

E[Y0 | T = 1])

Assumptions and Proof(E[Y1 | T = 0]

E[Y0 | T = 0])

18 / 25

ProofE[Y1 (1)<latexit sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit>

Y1 (0) | T = 1] = E[Y1 (1) | T = 1]
= E[Y1 | T = 1]

<latexit sha1_base64="tXDizUAisV+QBMe0tFN+A62tD7w=">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</latexit>

E[Y1 (1)Y1 (0) | T = 1] = E[Y1 | T = 1]

= (E[Y1 | T = 1]

Brady NealE[Y1 (0) | T = 1]

E[Y1 (0) | T = 1]

(E[Y0 | T = 1] + E[Y1 | T = 0]
E[Y0 | T = 1])

Assumptions and Proof(E[Y1 | T = 0]

E[Y0 | T = 0])

E[Y0 | T = 0])

19 / 25

ProofE[Y1 (1)<latexit sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit><latexit sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit>

Brady Neal

Y1 (0) | T = 1] = E[Y1 (1) | T = 1] E[Y1 (0) | T = 1]
= E[Y1 | T = 1] E[Y1 (0) | T = 1]

Assumptions and Proof

19 / 25

ProofE[Y1 (1)<latexit sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit><latexit sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit><latexit sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit>

Brady Neal

Y1 (0) | T = 1] = E[Y1 (1) | T = 1] E[Y1 (0) | T = 1]
= E[Y1 | T = 1] E[Y1 (0) | T = 1]

Assumptions and Proof

19 / 25

ProofE[Y1 (1)<latexit sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit><latexit sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit><latexit sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit>

Y1 (0) | T = 1] = E[Y1 (1) | T = 1] E[Y1 (0) | T = 1]
= E[Y1 | T = 1] E[Y1 (0) | T = 1]

Common trends:
E[Y1 (0) | T = 1]
<latexit sha1_base64="aWToTBSDADsG+ACQ2xjp/AM5ekI=">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</latexit>

Brady Neal

E[Y0 (0) | T = 1] = E[Y1 (0) | T = 0]

Assumptions and Proof

E[Y0 (0) | T = 0]

19 / 25

ProofE[Y1 (1)<latexit sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">AAAVT3icrVhZb9tGEF6lV+xeTuo89YWN0iIBJFVyXbgvAlLYDoqiARzAR1rLMCSSkgjxCklZdQT9pf6XPhb9IX0rOvPtUjwkkVJqCSJXszPfzM61XPZ82wqjZvOvyr333v/gw4/ub21//Mmnn32+8+DheeiNA9080z3bC173uqFpW655FlmRbb72A7Pr9Gzzojc65PmLGzMILc89jW5988rpDlyrb+ndiEjXO393ji873jjSPce8bj1tPdPqWup/85nWcSxD60QEGjmmG2ltrXWlfdPWFgSXMtZzfKsAO51tjT553P8Fer1TbTaa+GiLg5YaVIX6nHgPtv4QHWEIT+hiLBxhCldENLZFV4T0vRQt0RQ+0a7ElGgBjSzMm2Imtkl2TFwmcXSJOqLrgP5dKqpL/xkzhLROWmz6BSSpia8Vj0HjPqjyzvq1FO8qHVNgs423dO8pTIeokRgStUwu5lxfrkdfp2TVEa3hB6zWopX4oLAf9Mya+3S36X9EK+TrLXGaNDJIKqCRTjSbqJLCOgK6S8+zb4aIRBd8Jo3Y6iK7i6zmeY++I8Lq0jiEpWyrJl6ouLjQbMJW5rER09WIv9MKpX1FWP35GizEXa6CeU+JMhJvaZRFLtKZzq7VXJFCni3RxfwROFziDoniIfMt8p9FHNlc1QmTdXQRyQHW4iODGgr5Z8zH2exTPOuwJ0TUNOSbBT2+qqzE7thOm+49YAckP6W5IfSwL2pkjwnsANkbe7EG7oD+TfBPh416jt5AZXI8a7Qmzq4a4Xo0Y82xpCekP2OLpHVTomnqWxe/QL9Jca/BK5zXNZL1YKHM4hC6XBX7NvUV9otDV6ayZ2NKDToc2DFEtoyIluiTSKzvW1pFiPqxFeqU7ja85gOlBu3snwnGDtbEER1D94TGBvSwtEa9riEOlB2zjFb2jYXutKhVQ/5x7crOmNfKci4qVVNZ2YampthXK+YIsN1pr+mqG8qeHGJ1s5xVPdifdA3O0ax9LJXEJxsrA9k4hFQWl724fJ3L1rKHOqkhluz7AXG0IdVX9BC16StN26n6O1RV4iEeOu6OyglZ76xtkpuZEt1RnrKJT9Y570k884Ywu8ipDryXni9CPFby8W7AHX8KarHcOSorKxvQaDqfKZa3MDLhHzniiEs8H772sZav5j8tNbMu9t3guQuInLGRQloX3UBuLF/5qahSL6hSVmX9yZFmSoCnE5/GT+acTwiXeUfQ4hKFK3w6n5+VxuCIKDPlsT7m4jpMbLDQ+QxwdlTmSQ5ZUVGOg6u6WCvn5bK8kfRiWWOe71lZSS+WZcoNctNS/SAE1usSuQhxYAknlaWnJVIe9gRJlzb/uoZ9LvbascqLQFl4USLpoMfIVXmqKl+W2hehCwbKJyzz2zt48Ab9b4YnoE39mMhGG3kzkbt9J58m8pMNPZtIOhv6N5F8WyJpYm+w5jSWeVVaU8x1Usol86lsTzBpN4mfX2S11XDtkRd5zzZUZ9LUfsNZIXfHNnpAba3+vLjvcCW3Sut4jO7bQx8KoNkoqJ2zO0NLIjheY30h8j05X65TX8vk1s+bQ8RhWXeMZ+QTSPIUEp8ewtT+ZAE/OVlM8RQ4UTtZ/DxYRwbIGP5EvfAlPWPNoOk7er5r0fVFZtdbF5XPWWPVL9O4e4R8QPvVEfa1zXF9dRqqZ85MWR3HpCH+zdTpMuG21RlgFdImepcjy6wzFs4ieU3yaXNIXPIEdluCmOTxcrzi+C5Hlm9KPJwAknPYXSCnz5jr4SU5s/qNQNl7itWnZ5/ufbra6Oih4j1C1tno7Cb1GHnCl09PhwvvKjh/96gqOHenjx/m9GXx86ffAU6vY+JIeOQpJoK0pKURV52jfchE6mkxxJkzOVk3MidliZe3LUzllaPw2qofmWq32L7eqbbyb98WB+d7jdb3jear/erzffVm7r74UjwWT8lPB+I5VeYJ+VWv/FgZVPzKm90/d//Z/feRYr1XUYMvRObzaOs/Ovk7eA==</latexit><latexit 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sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit>

Y1 (0) | T = 1] = E[Y1 (1) | T = 1] E[Y1 (0) | T = 1]
= E[Y1 | T = 1] E[Y1 (0) | T = 1]

Common trends:
E[Y1 (0) | T = 1]
<latexit sha1_base64="aWToTBSDADsG+ACQ2xjp/AM5ekI=">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</latexit>

E[Y1 (0) | T = 1] = E[Y0 (0) | T = 1] + E[Y1 (0) | T = 0]<latexit sha1_base64="15OBylrrrzFIjKZ30zRScQC0Sgg=">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</latexit>= E[Y0 (0) | T = 1] + E[Y1 | T = 0]
= E[Y0 (1) | T = 1] + E[Y1 | T = 0]
= E[Y0 | T = 1] + E[Y1 | T = 0]

Brady Neal

E[Y0 (0) | T = 1] = E[Y1 (0) | T = 0]

E[Y0 (0) | T = 0]

E[Y0 (0) | T = 0]

E[Y0 | T = 0]
E[Y0 | T = 0]

E[Y0 | T = 0]

Assumptions and Proof

19 / 25

ProofE[Y1 (1)<latexit 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sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit><latexit sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit>

Y1 (0) | T = 1] = E[Y1 (1) | T = 1] E[Y1 (0) | T = 1]
= E[Y1 | T = 1] E[Y1 (0) | T = 1]

Common trends:
E[Y1 (0) | T = 1]
<latexit sha1_base64="aWToTBSDADsG+ACQ2xjp/AM5ekI=">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</latexit>

E[Y0 (0) | T = 1] = E[Y1 (0) | T = 0]

E[Y0 (0) | T = 0]

E[Y1 (0) | T = 1] = E[Y0 (0) | T = 1] + E[Y1 (0) | T = 0] E[Y0 (0) | T = 0]
= E[Y0 (0) | T = 1] + E[Y1 | T = 0] E[Y0 | T = 0]<latexit sha1_base64="15OBylrrrzFIjKZ30zRScQC0Sgg=">AAAWe3icvVhbb+NEFJ4NsOyWW7uIvvBiNgtawA1J6aq8RFqp7QohVupKvezSVFViO40V37Cdlm6UX8Er/AX+Dz8GiXO+Gcd2LrZTFRLFnpw55ztnzm087gWOHcXN5t/3au+8+9799x88XPvgw48+/mR949FJ5I9Cwzo2fMcPX/e6keXYnnUc27FjvQ5Cq+v2HOu0N9zj+dMrK4xs3zuKbwLr3O1eenbfNroxkS42ave1zsFZxx/Fhu9aF62nza+1jmubWicmmNi1vFhra61z7at2jrG5lPHbKoDNc22rCh7xdTprGn1uq7+S8qWa57S2/g+ti9b7H6u9WK83G018tPlBSw3qQn0O/Y2Hf4mOMIUvDDESrrCEJ2IaO6IrIvqeiZZoioBo52JMtJBGNuYtMRFrJDsiLos4ukQd0vWS/p0pqkf/GTOCtEFaHPqFJKmJLxWPSeM+qPLO+rUM7zIdY2CzjTd07ylMl6ixGBC1TC7hrC7Xo69bsuqY1vADVmvTSgJQ2A9Gbs19ujv0P6YV8vWGOC0amSQV0sggmkNUSWEdId2l59k3A0SiCz6LRmx1kd1FVvO8T98hYXVpHMFStlUTL1RcPGi2YCvzOIjpcsTfaIXSviKs/nQNNuIuV8G8R0QZirc0yiMX6cxm13KuWCFPFuhi/hgcHnFHRPGR+Tb5zyaOfK4ahMk6uojkJdYSIIMaCvknzCfZHFA8t2BPhKhpyDcbegJVWandiZ0O3XvADkl+THMD6GFf6GSPBewQ2Zt4UQd3SP+u8c+AjcYMvYHK5HjqtCbOLp1wfZqxp1jSE9KfiUXSujHRNPXdEj9Dv0Vx1+EVzmudZH1YKLM4gi5Pxb5NfYX94tKVqezZhKJDhws7BsiWIdFSfRKJ9X1Hq4hQP45CHdPdgdcCoOjQzv65xtjFmjiiI+i+prEJPSytUa9riF1lxySnlX1jozvNa9WQf1y7sjPOamU5D5WqqaxsQ1NT7KgVcwTY7qzXDNUNZU+OsLrJjFU92J92Dc7RvH0slcYnHysT2TiAVB6Xvbh4nYvWso060RFL9v0lcbQh1Vf0CLUZKE1rmfrbU1XiIx4G7q7KCVnvrO16ZmZMdFd5yiE+Wee8J/HMr4TZRU514L3sfBHigZJPdgPu+GNQi+VOUFl52ZBG4+lMsbyNkQX/yBFHXOIF8HWAtXwx/WmZmarYd4PnzSFyxsYKqSq6idxYvPIjUadeUKesyvuTI82UEE8nAY2fTDmfEC7zDqHFIwpX+Hg6PymNwT5RJspjfcwldZjaYKPzmeDsqMyTHLKi4hkOrupirZyXi/JG0otlzWm+52UlvViWKVfITVv1gwhYr0vkYsSBJdxMlh6VSPnYEyRd2vymgn0e9tqRyotQWXhaIumix8hV+aoqX5baF6MLhsonLPPLLTx4hf43wRPQqn5MZeOVvJnK3dzKp6n89YqeTSXdFf2bSr4tkbSwN9hTGsu8Kq0p5jos5ZL5VLYnWLSbJM8vstp0XHvkRd6zTdWZNLXfcFbI3bGNHqBX6s/z+w5Xcqu0jkfovj30oRCazYLaOb4ztDSCowrri5Dv6fmySn0tkqueN3uIw6LumMzIJ5D0KSQ5PUSZ/ckGfnqyGOMp8FrtZMnz4BYyQMbwR+qFL+kZawJN39PzXYuuL3K7XlVUPmeNVL/M4m4T8i7tV/vY11bHDdRpaCt3ZsrrOCANyW+iTpcpt6POAMuQVtG7GFlmnTl3FpnVJJ82B8QlT2A3JYhpHi/GK47vYmT5psTHCSA9h90FcvaMWQ0vzZnlbwTK3lMsPz0HdO/T1UFHjxTvPrLOQWe3qMfIE758etqbe1fB+btNVcG5O378aEZfHn/29HuJ0+uIOFIeeYqJIS1pWcRl5+gAMrF6Woxw5kxP1o3cSVnizdoWZfLKVXht1Y8stVusXazXW7Nv3+YHJ9uN1rNG89VO/fmOejP3QHwuHoun5Kdd8Zwq85D8atTc2u+1P2p/fvbPZn3zm01dstbuKZlPRe6z+exfyKKb8A==</latexit><latexit sha1_base64="15OBylrrrzFIjKZ30zRScQC0Sgg=">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</latexit>= E[Y0 (1) | T = 1] + E[Y1 | T = 0] E[Y0 | T = 0]
= E[Y0 | T = 1] + E[Y1 | T = 0] E[Y0 | T = 0]

Brady Neal

Assumptions and Proof

19 / 25

ProofE[Y1 (1)<latexit sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit><latexit sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit><latexit sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit>

Y1 (0) | T = 1] = E[Y1 (1) | T = 1] E[Y1 (0) | T = 1]
= E[Y1 | T = 1] E[Y1 (0) | T = 1]

Common trends:
E[Y1 (0) | T = 1]
<latexit sha1_base64="aWToTBSDADsG+ACQ2xjp/AM5ekI=">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</latexit>

E[Y0 (0) | T = 1] = E[Y1 (0) | T = 0]

E[Y1 (0) | T = 1] = E[Y0 (0) | T = 1] + E[Y1 (0) | T = 0] E[Y0 (0) | T = 0]
= E[Y0 (0) | T = 1] + E[Y1 | T = 0] E[Y0 | T = 0]

E[Y0 (0) | T = 0]<latexit sha1_base64="15OBylrrrzFIjKZ30zRScQC0Sgg=">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</latexit><latexit sha1_base64="15OBylrrrzFIjKZ30zRScQC0Sgg=">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</latexit>= E[Y0 (1) | T = 1] + E[Y1 | T = 0] E[Y0 | T = 0]
= E[Y0 | T = 1] + E[Y1 | T = 0] E[Y0 | T = 0]

Brady Neal

Assumptions and Proof<latexit sha1_base64="0+5Yt0nkNb49qYW6PsKTBTbXzsI=">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</latexit>

No pretreatment effect:

E[Y0 (1) | T = 1]

E[Y0 (0) | T = 1] = 0

19 / 25

ProofE[Y1 (1)<latexit 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sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit><latexit sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit>

Y1 (0) | T = 1] = E[Y1 (1) | T = 1] E[Y1 (0) | T = 1]
= E[Y1 | T = 1] E[Y1 (0) | T = 1]

Common trends:
E[Y1 (0) | T = 1]
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E[Y0 (0) | T = 1] = E[Y1 (0) | T = 0]

E[Y1 (0) | T = 1] = E[Y0 (0) | T = 1] + E[Y1 (0) | T = 0] E[Y0 (0) | T = 0]
= E[Y0 (0) | T = 1] + E[Y1 | T = 0] E[Y0 | T = 0]

E[Y0 (0) | T = 0]<latexit 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sha1_base64="15OBylrrrzFIjKZ30zRScQC0Sgg=">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</latexit>= E[Y0 (1) | T = 1] + E[Y1 | T = 0] E[Y0 | T = 0]
= E[Y0 | T = 1] + E[Y1 | T = 0] E[Y0 | T = 0]

Brady Neal

Assumptions and Proof<latexit sha1_base64="0+5Yt0nkNb49qYW6PsKTBTbXzsI=">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</latexit>

No pretreatment effect:

E[Y0 (1) | T = 1]

E[Y0 (0) | T = 1] = 0

19 / 25

ProofE[Y1 (1)<latexit 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sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit><latexit sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit>

Y1 (0) | T = 1] = E[Y1 (1) | T = 1] E[Y1 (0) | T = 1]
= E[Y1 | T = 1] E[Y1 (0) | T = 1]

Common trends:
E[Y1 (0) | T = 1]
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E[Y0 (0) | T = 1] = E[Y1 (0) | T = 0]

E[Y1 (0) | T = 1] = E[Y0 (0) | T = 1] + E[Y1 (0) | T = 0] E[Y0 (0) | T = 0]
= E[Y0 (0) | T = 1] + E[Y1 | T = 0] E[Y0 | T = 0]

E[Y0 (0) | T = 0]<latexit 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sha1_base64="15OBylrrrzFIjKZ30zRScQC0Sgg=">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</latexit>= E[Y0 (1) | T = 1] + E[Y1 | T = 0] E[Y0 | T = 0]
= E[Y0 | T = 1] + E[Y1 | T = 0] E[Y0 | T = 0]

Brady Neal

Assumptions and Proof<latexit sha1_base64="0+5Yt0nkNb49qYW6PsKTBTbXzsI=">AAAUxnicrVjbbttGEN2kt8S9Oanf+sLGKZACsiq5KdwXAQFsB0HRAE7hW2sZhkRRFmHeSlJWHUFAv6jP/ZW+9LX9jJ45uxRF3SgHliByNTtzZnZuy2U78twkrdX+vnf/vfc/+PCjBw/XPv7k088+X3/0+DgJ+7HtHNmhF8an7VbieG7gHKVu6jmnUey0/LbnnLSvdmX+5NqJEzcMDtObyDn3W5eB23XtVgrSxfrPzf2zZthP7dB3LmrP6t9YTd/tWM0UIKnvBKnVsOrnlrVlFRlrCxgbVu1ifbNWrfFjzQ7qZrCpzOcgfPTwT9VUHRUqW/WVrxwVqBRjT7VUgu+ZqquaikA7V0PQYoxczjtqpNYg2weXA44WqFe4XuLfmaEG+C+YCaVtaPHwiyFpqa8NTwfjLqn6LvqtCd5FOobEFhtvcG8bTB/UVPVALZPLOFeXa+Prl6w6xRp+4GpdrCQiRfxgF9bcxd3D/xQrlOsNOB2MOpCKMbJB80DVFNER4649L77pMRIt8jkYidXL7F5mtcyH+F4Bq4VxQkvFVku9NHEJqNmhrcLjMaaLEX/HCrV9y7C64zW4jLtehfAegnKl3mJURF6mczK7FnOlBnk0R5fwp+QIwJ2AEjLzXfjPBUcxV21gio4WI3nJtUTMoKpB/pHzWTZHiOcW7UkYNYv55lJPZCortzuz08O9TewY8kPM9ahHfFGBPQ6xY2Zv5sUKuWP8G/CfTRvtKXqVlSnxrGBNkl0V4IaYccdY2hPan5lF2rohaJb5bqmfqN9B3Cv0iuR1BbIhLdRZnFBXYGLfQF8Rv/i4ClU8m1Eq1OHTjh6z5Qq0XJ9GEn3fYhUJ68czqEPcPXotIkqF2sU/A459rkki2qfuAcYd6hFpC72uqnaMHaOCVvGNy+40q9Vi/knt6s44rVXkAlaqZbKyQU019dysWCIgdk96zTbdUPfkhKsbTVnVpv1515AcLdonUnl8irHqMBt7lCriihfnr3PeWrZZJxXGUnx/CY4GpbqGnrA2I6NpbaL+dk2VhIyHzbtvckLXu2gbTM0MQfeNpzzw6TqXPUlmfgNmiznVpPcm55ch7hv5bDeQjj8kdbncMSurKBtjNBzPLJd3OXLoHz2SiGu8iL6OuJavxj9rYmZV7LvBC2YQJWNTg7Qqeoe5MX/lh2oTvWATWVX0p0RaKDGfTiKMn445nwJXeK+oJQBFKnw4nh+VxmAPlJHxWJdzWR3mNrjsfB1yNk3maQ5dUekUh1T1cq2Sl/PyRtOXy3bG+V6U1fTlskK5Zm66ph8kxDotkUsZB5HwJ7L0sEQq5J6g6drmX1awL+Be2zd5ERsLT0okffYYvarQVOXrUvtSdsHY+ERkfn0HD16z/434BHRbP+ay6a28mcvdvJNPc/nBLT2bS/q39G8u+bZE0uHe4I5pIvOmtKaE66CUS+dT2Z7gYDfJnl90tVV4bcOLsmd3TGeyzH4jWaF3xwZ7QGWl/jy770gl10vruM/u22Yfiqm5s6R2ju4MLY9gf4X1Jcz3/Hy5Sn3Nk1s9b3YZh3ndMZvRTyD5U0h2ekgm9ieX+PnJYsinwIHZybLnwS1mgI7hK/TC13jGGlHTd3i+q+P6srDrrYoq56y+6ZeTuNtA3sF+tcd97fa4kTkNbRXOTEUd+9CQ/UbmdJlze+YMsAjpNnrnI+us68ycRaY16afNHrj0CeymBDHP4/l4y+M7H1m/KQl5AsjPYXeBPHnGXA0vz5nFbwTK3lMsPj1HuHdx9djRE8O7x6zz2Nkd9Bh9wtdPT7sz7yokf7dRFZK7wyePp/QV8adPv5c8vfbBkfPoU0xKaU2bRFx0jo4ok5qnxYRnzvxkXS2clDXetG3JRF75Bq9h+pFjdou1i/XN+vTbt9nB8Xa1/n219ub55ouaeTP3QH2pnqhn8NOOeoHKPIBfbfWX+kf9q/7beLURbPQ3Bpr1/j0j84UqfDb++B+qfRAy</latexit>

No pretreatment effect:

E[Y0 (1) | T = 1]

E[Y0 (0) | T = 1] = 0

19 / 25

ProofE[Y1 (1)<latexit 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sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit><latexit sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit>

Y1 (0) | T = 1] = E[Y1 (1) | T = 1] E[Y1 (0) | T = 1]
= E[Y1 | T = 1] E[Y1 (0) | T = 1]

Common trends:
E[Y1 (0) | T = 1]
<latexit sha1_base64="aWToTBSDADsG+ACQ2xjp/AM5ekI=">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</latexit>

E[Y0 (0) | T = 1] = E[Y1 (0) | T = 0]

E[Y1 (0) | T = 1] = E[Y0 (0) | T = 1] + E[Y1 (0) | T = 0] E[Y0 (0) | T = 0]
= E[Y0 (0) | T = 1] + E[Y1 | T = 0] E[Y0 | T = 0]

E[Y0 (0) | T = 0]<latexit 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sha1_base64="15OBylrrrzFIjKZ30zRScQC0Sgg=">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</latexit><latexit sha1_base64="15OBylrrrzFIjKZ30zRScQC0Sgg=">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</latexit><latexit sha1_base64="15OBylrrrzFIjKZ30zRScQC0Sgg=">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</latexit>= E[Y0 (1) | T = 1] + E[Y1 | T = 0] E[Y0 | T = 0]
= E[Y0 | T = 1] + E[Y1 | T = 0] E[Y0 | T = 0]<latexit sha1_base64="tXDizUAisV+QBMe0tFN+A62tD7w=">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</latexit>

E[Y1 (1)Y1 (0) | T = 1] = E[Y1 | T = 1]

= (E[Y1 | T = 1]

Brady Neal<latexit sha1_base64="0+5Yt0nkNb49qYW6PsKTBTbXzsI=">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</latexit>

No pretreatment effect:

E[Y0 (1) | T = 1]

(E[Y0 | T = 1] + E[Y1 | T = 0]
E[Y0 | T = 1])

Assumptions and Proof(E[Y1 | T = 0]

E[Y0 (0) | T = 1] = 0

E[Y0 | T = 0])

E[Y0 | T = 0])

19 / 25

ProofE[Y1 (1)<latexit 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sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit><latexit sha1_base64="LoMV5Tq/oC9OXR5/2YkSBL62prs=">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</latexit>

Y1 (0) | T = 1] = E[Y1 (1) | T = 1] E[Y1 (0) | T = 1]
= E[Y1 | T = 1] E[Y1 (0) | T = 1]

Common trends:
E[Y1 (0) | T = 1]
<latexit sha1_base64="aWToTBSDADsG+ACQ2xjp/AM5ekI=">AAAVFXicrVhLb9tGEF6ljyTuy0l964WN0iIFJJVy07oXAQFsB0XRAA7gV2sZhkRSFiG+SlJWHUG/o+f+gf6D3opee+6/6cy3S5GUKFIKLEHkanbmm9l5LZf9wLGjWNf/q91759333r//4OHWBx9+9PEn248en0b+ODSsE8N3/PC834ssx/ask9iOHes8CK2e23ess/5on+fPbqwwsn3vOL4NrEu3d+3ZA9voxUS62v6je3ihfal1/XFs+K511X6mf6VpXdc2tW5MQLFrebHW0dqXWlMj3jmjzoyFfJ0cX7uYT18TT7+82q7rLR0fbXnQVoO6UJ8j/9HDP0VXmMIXhhgLV1jCEzGNHdETEX0vRFvoIiDapZgSLaSRjXlLzMQWyY6JyyKOHlFHdL2mfxeK6tF/xowgbZAWh34hSWriC8Vj0ngAqryzfi3Du0rHFNhs4y3d+wrTJWoshkStkks415fr09etWHVMa/geq7VpJQEo7Acjt+YB3R36H9MK+XpLnBaNTJIKaWQQzSGqpLCOkO7S8+ybISLRA59FI7a6zO4yq3nep++IsHo0jmAp26qJlyouHjRbsJV5HMR0NeJvtEJpXxnWYL4GG3GXq2DeY6KMxBsa5ZHLdGazazVXrJBnBbqYPwaHR9wRUXxkvk3+s4kjn6sGYbKOHiJ5jbUEyKCWQv4R80k2BxTPJuyJEDUN+WZDT6AqK7U7sdOhex/YIclPaW4IPeyLBtljATtE9iZebIA7pH8T/DNgo7FAb6EyOZ4NWhNnV4NwfZqx51jSE9KfiUXSuinRNPVtip+g36K4N+AVzusGyfqwUGZxBF2ein2H+gr7xaUrU9mzCaUBHS7sGCJbRkRL9Ukk1vc1rSJC/TgKdUp3B14LgNKAdvbPBGMXa+KIjqF7QmMTelhao17XEnvKjllOK/vGRnda1qoh/7h2ZWdc1MpyHipVU1nZgSZdPFcr5giw3VmvGaobyp4cYXWzBav6sD/tGpyjeftYKo1PPlYmsnEIqTwue7F4nUVr2UWdNBBL9v01cXQgNVD0CLUZKE1bmfrbV1XiIx4G7q7KCVnvrG2yMDMluqs85RCfrHPek3jmV8LsIae68F52vgzxUMknuwF3/Cmo5XKnqKy8bEij6XymXN7GyIJ/5IgjLvEC+DrAWj6f/7TMzLrYd4PnLSFyxsYKaV10E7lRvPJjUadeUKesyvuTI82UEE8nAY2fzjmfEi7zjqDFIwpX+HQ+P6uMwQFRZspjA8wldZjaYKPzmeDsqsyTHLKi4gUOrupyrZyXRXkj6eWy5jzf87KSXi7LlBvkpq36QQSs8wq5GHFgCTeTpccVUj72BEmXNv+8hn0e9tqxyotQWXhWIemix8hV+aoqX1XaF6MLhsonLPPLW3jwBv1vhiegTf2YysYbeTOVu30rn6bykw09m0q6G/o3lXxTIWlhb7DnNJZ5XVlTzHVUySXzqWpPsGg3SZ5fZLU1cO2TF3nPNlVn0tR+w1khd8cOekBjrf68vO9wJbcr63iM7ttHHwqh2SypnZM7Q0sjOF5jfRHyPT1frlNfRXLr580+4lDUHZMZ+QSSPoUkp4cosz/ZwE9PFlM8BU7UTpY8DzaRATKGP1AvfEXPWDNo+oae79p0fZnb9dZF5XPWWPXLLO4uIe/RfnWAfW1z3ECdhpq5M1NexyFpSH4zdbpMuR11BliFtIneYmSZdebSWWRRk3zaHBKXPIHdViCmeVyMVx7fYmT5psTHCSA9h90FcvaMuR5emjOr3whUvadYfXoO6D6gq4OOHineA2Sdg85uUY+RJ3z59LS/9K6C83eXqoJzd/rk8YK+PP7i6fcap9cxcaQ88hQTQ1rSsoirztEBZGL1tBjhzJmerFu5k7LEW7QtyuSVq/A6qh9ZarfYutqutxffvi0PTndb7W9b+uvn9Re6ejP3QHwmnohn5Kc98YIq84j8atTu15q172p7O7/v/LXz984/kvVeTcl8KnKfnX//B7s9KNg=</latexit>

E[Y0 (0) | T = 1] = E[Y1 (0) | T = 0]

E[Y1 (0) | T = 1] = E[Y0 (0) | T = 1] + E[Y1 (0) | T = 0] E[Y0 (0) | T = 0]
= E[Y0 (0) | T = 1] + E[Y1 | T = 0] E[Y0 | T = 0]

E[Y0 (0) | T = 0]<latexit 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sha1_base64="15OBylrrrzFIjKZ30zRScQC0Sgg=">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</latexit><latexit sha1_base64="15OBylrrrzFIjKZ30zRScQC0Sgg=">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</latexit><latexit sha1_base64="15OBylrrrzFIjKZ30zRScQC0Sgg=">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</latexit>= E[Y0 (1) | T = 1] + E[Y1 | T = 0] E[Y0 | T = 0]
= E[Y0 | T = 1] + E[Y1 | T = 0] E[Y0 | T = 0]<latexit sha1_base64="tXDizUAisV+QBMe0tFN+A62tD7w=">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</latexit><latexit sha1_base64="tXDizUAisV+QBMe0tFN+A62tD7w=">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</latexit>

E[Y1 (1)Brady Neal<latexit sha1_base64="0+5Yt0nkNb49qYW6PsKTBTbXzsI=">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</latexit>

No pretreatment effect:

E[Y0 (1) | T = 1]

E[Y0 (0) | T = 1] = 0

Y1 (0) | T = 1] = E[Y1 | T = 1] (E[Y0 | T = 1] + E[Y1 | T = 0] E[Y0 | T = 0])= (E[Y1 | T = 1] E[Y0 | T = 1]) (E[Y1 | T = 0] E[Y0 | T = 0])

Assumptions and Proof

19 / 25

Question:
What is the graphical intuition behind thethe various parts of the proof andassumptions on the previous slide?

Motivation and Preliminaries
Difference-in-Differences Overview
Assumptions and Proof
Problems with Difference-in-Differences

Brady Neal

Problems with Difference-in-Differences

21 / 25

Violations of Parallel Trends
Violation:

Brady NealE[Y1 (0)<latexit sha1_base64="PkdyDPogTXkD8DqjZCXyMqczYMc=">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</latexit>

Y0 (0) | T = 1] 6= E[Y1 (0)Y0 (0) | T = 0]

Problems with Difference-in-Differences

22 / 25

Violations of Parallel Trends
Violation:

E[Y1 (0)<latexit sha1_base64="PkdyDPogTXkD8DqjZCXyMqczYMc=">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</latexit>

Y0 (0) | T = 1] 6= E[Y1 (0)Y0 (0) | T = 0]

Control for relevant confounders:<latexit sha1_base64="cOpk+jcNlCXlxl5iFGpUGdVNEeE=">AAAU6nicrVjbbttGEF27t8RJGyf1W1/YOC1SQFYl14X7YiCA7aAoGsABfGstw5AoyibMW0nKriPoJ/pW9LXP/Z8C/ZieObsURd0oBZYgcjk7c2Z2bstVK/LcJK3V/l1a/uDDjz7+5MHDlUePP/3syerTZ8dJ2I1t58gOvTA+bTUTx3MD5yh1U885jWKn6bc856R1vSvzJzdOnLhhcJjeRc6537wM3I5rN1OQLlZvGvtn1tdWI+ymdug7F/WXtW+sjfy5Js8N321bjRS4qe8EqbVj1SvWyTnukF5UtCaiF6vrtWqNH2t8UDeDdWU+B+HTh/+ohmqrUNmqq3zlqEClGHuqqRJ8z1Rd1VQE2rnqgRZj5HLeUX21AtkuuBxwNEG9xvUST2eGGuBZMBNK29Di4RdD0lJfGZ42xh1S9V30W0O803T0iC023uHeMpg+qKm6ArVMLuOcX66Fr1+y6hRr+IGrdbGSiBTxg11Ycwd3D88pVijXO3A6GLUhFWNkg+aBqimiI8Zde158c8VINMnnYCRWz7J7ltUyH+J7DawmxgktFVst9drEJaBmh7YKj8eYTkf8HSvU9s3C6gzW4DLuehXCewjKtXqHURF5ls7h7JrOlRrk/gRdwp+SIwB3AkrIzHfhPxccxVy1gSk6mozkJdcSMYOqBvknzmfZHCGeG7QnYdQs5ptLPZGprNzuzE4P9xaxY8j3MHdFPeKLCuxxiB0zezMvVsgd4+mWTzZttEfoVVamxLOCNUl2VYAbYsYdYGlPaH9mFmnreqBZ5ruhfqZ+B3Gv0CuS1xXIhrRQZ3FCXYGJ/Q76ivjFx1Wo4tmMUqEOn3ZcMVuuQcv1aSTR9y1WkbB+PIPaw92j1yKiVKhd/HPLsc81SUS71H2LcZt6RNpCr6uqbWNHv6BVfOOyO41rtZh/Uru6M45qFbmAlWqZrNyhppraMiuWCIjdw16zTTfUPTnh6vojVrVof941JEeL9olUHp9irNrMxitKFXHFi5PXOWktm6yTCmMpvr8Exw6lOoaesDYjo2llqP52TZWEjIfNu29yQte7aLsdmemB7htPeeDTdS57ksz8Bswmc6pB7w3Pz0LcN/LZbiAdv0fqbLljVlZRNsaoN5iZLe9y5NA/eiQR13gRfR1xLV8OftbQzLzY94MXjCFKxqYGaV70NnNj8soP1Tp6wTqyquhPibRQYr6dRBi/GHC+AK7wXlNLAIpUeG8w3y+NwR4ofeOxDueyOsxtcNn52uRsmMzTHLqi0hEOqerZWiUvJ+WNps+WbQ/yvSir6bNlhXLD3HRNP0iIdVoilzIOIuEPZelhiVTIPUHTtc2/zGFfwL22a/IiNhaelEj67DF6VaGpyjel9qXsgrHxicj8+h4evGH/6/MNaFE/5rLpQt7M5e7ey6e5/O2Cns0l/QX9m0u+K5F0uDe4A5rIvC2tKeE6KOXS+VS2JzjYTbL3F11tFV5b8KLs2W3TmSyz30hW6N1xhz2gMld/Ht93pJLrpXXcZfdtsQ/F1NyeUTtH94aWR7A7x/oS5nt+vpynvibJzZ83u4zDpO6Yzeg3kPwtJDs9JEP7k0v8/GTR41vgrdnJsvfBDWaAjuGP6IVv8I7Vp6bv8H5Xx/V1YdebF1XOWV3TL4dxN4G8jf1qj/va4riROQ1tFM5MRR370JD9+uZ0mXN75gwwDWkRvZORdda1x84io5r02+YVuPQJ7K4EMc/jyXiz4zsZWf9TEvIEkJ/D7gN5+Iw5H16eM9P/ESj7n2L66TnCvYOrx46eGN49Zp3Hzu6gx+gTvn572h37r0LydxNVIbnbe/5sRF8Rf/T0e8nTaxccOY8+xaSU1rRhxGnn6IgyqXlbTHjmzE/W1cJJWeON2pYM5ZVv8HZMP3LMbrFysbpeH/33bXxwvFmtf1+tvd1af7Vl/pl7oL5Qz9VL+GlbvUJlHsCvtvpvaXnp0dLjNW/tj7U/1/7SrMtLRuZzVfis/f0/0u8ZOg==</latexit>

Brady NealE[Y1 (0)Y0 (0) | T = 1, W ] = E[Y1 (0)Y0 (0) | T = 0, W ]Problems with Difference-in-Differences

22 / 25

Violations of Parallel Trends
Violation:

E[Y1 (0)Y0 (0) | T = 1] 6= E[Y1 (0)<latexit sha1_base64="PkdyDPogTXkD8DqjZCXyMqczYMc=">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</latexit>

Y0 (0) | T = 0]

Control for relevant confounders:<latexit sha1_base64="cOpk+jcNlCXlxl5iFGpUGdVNEeE=">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</latexit>

E[Y1 (0)Y0 (0) | T = 1, W ] = E[Y1 (0)Y0 (0) | T = 0, W ]Violation whenever there is an interaction term between treatment andtime in the structural equation for the outcome Y:
Y := . . . + T ⌧<latexit sha1_base64="XTf0CQVi5+D2Xi15NG+LYryLjmc=">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</latexit>

Brady Neal=)

Parallel trends violation

Problems with Difference-in-Differences

22 / 25

Question:
When we condition on W to get parallel
trends (below), what additional assumption
do we need to satisfy?
<latexit sha1_base64="cOpk+jcNlCXlxl5iFGpUGdVNEeE=">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</latexit>

E[Y1 (0)Y0 (0) | T = 1, W ] = E[Y1 (0)Y0 (0) | T = 0, W ]Parallel Trends is Scale-SpecificE[Y1 (0) | T = 1]
<latexit sha1_base64="Q5+24PPusECcyrZpINZEYYkS9QA=">AAAVFXicrVhLb9tGEF6ljyTuy0l964WN0iIFKFVy3boXAQFsB0XRAA7gV2sZhkRSFiG+SlJWHUG/o+f+gf6D3opee+6/6cy3S5GUKFIKIkHkanbmm9l5LZf9wLGjuNX6r3bvnXffe//+g4dbH3z40cefbD96fBb549CwTg3f8cOLfi+yHNuzTmM7dqyLILR6bt+xzvujA54/v7XCyPa9k/gusK7c3o1nD2yjFxPpevuP7tGl9qXW9cex4bvWdftZ6ytN67q2qXVjAopdy4u1jta+0hoa8c4ZW8xYyNfJ8bWL+Vpr4rWurrfrrWYLH2150FaDulCfY//Rwz9FV5jCF4YYC1dYwhMxjR3RExF9L0VbtERAtCsxJVpIIxvzlpiJLZIdE5dFHD2ijuh6Q/8uFdWj/4wZQdogLQ79QpLUxBeKx6TxAFR5Z/1ahneVjimw2cY7uvcVpkvUWAyJWiWXcK4v16evW7HqmNbwPVZr00oCUNgPRm7NA7o79D+mFfL1jjgtGpkkFdLIIJpDVElhHSHdpefZN0NEogc+i0ZsdZndZVbzvE/fEWH1aBzBUrZVEy9UXDxotmAr8ziI6WrE32iF0r4yrMF8DTbiLlfBvCdEGYnXNMojl+nMZtdqrlghzwp0MX8MDo+4I6L4yHyb/GcTRz5XDcJkHT1E8gZrCZBBTYX8I+aTbA4ong3YEyFqGvLNhp5AVVZqd2KnQ/c+sEOSn9LcEHrYFzrZYwE7RPYmXtTBHdK/Cf4ZsNFYoDdRmRxPndbE2aUTrk8z9hxLekL6M7FIWjclmqa+DfET9FsUdx1e4bzWSdaHhTKLI+jyVOw71FfYLy5dmcqeTSg6dLiwY4hsGREt1SeRWN/XtIoI9eMo1CndHXgtAIoO7eyfCcYu1sQRHUP3hMYm9LC0Rr2uKfaVHbOcVvaNje60rFVD/nHtys64qJXlPFSqprKyA00tsadWzBFgu7NeM1Q3lD05wupmC1b1YX/aNThH8/axVBqffKxMZOMQUnlc9mLxOovWsos60RFL9v0NcXQgNVD0CLUZKE1bmfo7UFXiIx4G7q7KCVnvrG2yMDMluqs85RCfrHPek3jmV8LsIae68F52vgzxSMknuwF3/Cmo5XJnqKy8bEij6XymXN7GyIJ/5IgjLvEC+DrAWj6f/7TMzLrYbwfPW0LkjI0V0rroJnKjeOUnok69oE5ZlfcnR5opIZ5OAho/nXM+JVzmHUGLRxSu8Ol8flYZg0OizJTHBphL6jC1wUbnM8HZVZknOWRFxQscXNXlWjkvi/JG0stlzXm+52UlvVyWKbfITVv1gwhYFxVyMeLAEm4mS08qpHzsCZIubf55Dfs87LVjlRehsvC8QtJFj5Gr8lVVvqy0L0YXDJVPWOaXN/DgLfrfDE9Am/oxlY038mYqd/dGPk3lJxt6NpV0N/RvKvm6QtLC3mDPaSzzqrKmmOu4kkvmU9WeYNFukjy/yGrTce2TF3nPNlVn0tR+w1khd8cOeoC+Vn9e3ne4ktuVdTxG9+2jD4XQbJbUzulbQ0sjOF5jfRHyPT1frlNfRXLr580B4lDUHZMZ+QSSPoUkp4cosz/ZwE9PFlM8BU7UTpY8DzaQATKGP1AvfEnPWDNo+oae79p0fZHb9dZF5XPWWPXLLO4uIe/TfnWIfW1z3ECdhhq5M1NexxFpSH4zdbpMuR11BliFtIneYmSZdebSWWRRk3zaHBKXPIHdVSCmeVyMVx7fYmT5psTHCSA9h70N5OwZcz28NGdWvxGoek+x+vQc0H1AVwcdPVK8h8g6B53doh4jT/jy6elg6V0F5+8uVQXn7vTJ4wV9efzF0+8NTq9j4kh55CkmhrSkZRFXnaMDyMTqaTHCmTM9WTdzJ2WJt2hblMkrV+F1VD+y1G6xdb1dby++fVsenO022982W6/26s/31Ju5B+Iz8UQ8Iz/ti+dUmcfkV6N2v9aofVfb3/l956+dv3f+kaz3akrmU5H77Pz7P7xxKNw=</latexit>

Brady Neal

E[Y0 (0) | T = 1] = E[Y1 (0) | T = 0]

E[Y0 (0) | T = 0]

Problems with Difference-in-Differences

24 / 25

Parallel Trends is Scale-SpecificE[Y1 (0) | T = 1]
<latexit sha1_base64="Q5+24PPusECcyrZpINZEYYkS9QA=">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</latexit>

E[Y0 (0) | T = 1] = E[Y1 (0) | T = 0]

E[Y0 (0) | T = 0]

Does not imply (and is not implied by)<latexit sha1_base64="45lRqwcfJPZl/cB+EweNnDhCzRQ=">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</latexit>E[ log Y1 (0) | T = 1]

Brady NealE[log Y0 (0) | T = 1] = E[log Y1 (0) | T = 0]

Problems with Difference-in-Differences

E[log Y0 (0) | T = 0]

24 / 25

Parallel Trends is Scale-SpecificE[Y1 (0) | T = 1]
<latexit sha1_base64="Q5+24PPusECcyrZpINZEYYkS9QA=">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</latexit>

E[Y0 (0) | T = 1] = E[Y1 (0) | T = 0]

E[Y0 (0) | T = 0]

Does not imply (and is not implied by)<latexit sha1_base64="45lRqwcfJPZl/cB+EweNnDhCzRQ=">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</latexit>E[ log Y1 (0) | T = 1]

E[log Y0 (0) | T = 1] = E[log Y1 (0) | T = 0]E[log Y0 (0) | T = 0]This means that the parallel trends assumptions isn’t nonparametric

Brady Neal

Problems with Difference-in-Differences

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Question:
1. Is parallel trends satisfied if time and treatment
interact in producing the outcome?
2. If parallel trends is satisfied, is it also satisfied forarbitrary transformations of the outcome variable?