Instrumental Variables
Brady Nealcausalcourse.com

Unobserved Confounding<latexit sha1_base64="kM2Kw9peVD0/n13t7Mo4wEYSnXc=">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</latexit>

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Unobserved Confounding
Week 5:<latexit sha1_base64="kM2Kw9peVD0/n13t7Mo4wEYSnXc=">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</latexit>

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Unobserved Confounding
Week 5:• Frontdoor adjustment<latexit sha1_base64="kM2Kw9peVD0/n13t7Mo4wEYSnXc=">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</latexit>

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Unobserved Confounding
Week 5:• Frontdoor adjustment• Unconfounded children criterion<latexit sha1_base64="kM2Kw9peVD0/n13t7Mo4wEYSnXc=">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</latexit>

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Unobserved Confounding
Week 5:• Frontdoor adjustment• Unconfounded children criterion• Some other fancy application of
do-calculus<latexit sha1_base64="kM2Kw9peVD0/n13t7Mo4wEYSnXc=">AAAWq3icrVhZb9tGEN7YTZu4ba4CfukLG9tAAsiq7LZIgMBAACdBUTRFgjpHYxkBj5VEiFfJpRVH0E/oD+xT/0pnvl2KpC7KQSSIXA5nvpmda3flJIGfqU7n3ysbm19c/fKra9e3vv7m2xs3b92+8zqL89SVr9w4iNO3jp3JwI/kK+WrQL5NUmmHTiDfOMNjfv/mXKaZH0cn6iKRZ6Hdj/ye79qKSO9vb3zoOrLvR2PlDz8mvqvyVE62LPPZs/6IPZlNn7sRPZ5mylay5dnZQHpn1r1uHsVOJtNz6blx1IvzyCOF9y1bWfc6rc59a7y7mGV38mgRMkMqmoMKZaTuW6eODOKRFcieso7inrUY64y1TKVWIMe5cuNQTnFTvz9oBDZCVdi9GVw/ylSaG5NtJz6XFZOnhgGtZC0AKw5/6vWrDk9sNVjuYknMNW89uqRk4Y15uUoElvLuFczVyS+xqSsjr5Zk72/tdNodfKz5wYEZ7AjzeRHfvv6P6ApPxMIVuQiFFJFQNA6ELTL6nooD0REJ0c7EmGgpjXy8l2Iitkg2Jy5JHDZRh3Tt09OpoUb0zJgZpF3SEtAvJUlL7Bkej8Y9UPWd9VsV3mU6xsBmGy/o7hjMkKhKDIjaJFdwri/n0DdsmLWiOTzEbH2aSQIK+8GtzblH94CeFc2QrxfEKWnkkVRKI5doAVE1hXWkdNeeZ98MEAkbfJJGbPUqu1dZze9j+g4Jy6ZxBkvZVks8M3GJoFnCVuYJENPliB9ohtq+VVi96Rx8xF3PgnlPiDIUH2lUR16ls5pdy7mUQZ4s0MX8ChwRcWdEiZH5PvnPJ456rrqEyTpsRLKPuSTIoLZB/g3vi2xOKJ77sCdD1Czkmw89iams0u7CzoDuDrBTkh/TuwH0sC9aZI8EdorsLbzYAndKTyM8ubDRnaG3UZkczxbNibOrRbgxvfGnWNoT2p+FRdq6MdEs890Xv0O/pLi34BXO6xbJxrBQZ3EGXZGJ/RH1FfZLSFemsmcLSgs6QtgxQLYMiVbq00is70eaRYb6CQzqmO4BvJYApQXt7J8RxiHmxBHNoXtEYw96WNqiXtcWD4wdk5pW9o2P7jSv1UL+ce3qzjirleUiVKplsvIImjriZzNjjgDbXfWaa7qh7skZZjeZscqB/WXX4Byt28dSZXzqsfKQjQNI1XHZi4vnuWguh6iTFmLJvu8TxxGkeoaeoTYTo2mrUn/HpkpixMPFPTQ5oeudtY1m3oyJHhpPBcSn65zXJH7zN2HayKkuvFd9vwrxqZEvVgPu+GNQV8u9RmXVZVMajadvVsv7GEn4R4844hovga8TzOWH6c+qvFkX+/PgRXOInLHKIK2L7iE3Fs/8ROxQL9ihrKr7kyPNlBS7k4TGu1POXcJl3iG0REThCh9P308aY/CEKBPjsR7eFXVY2uCj83ng7JrM0xy6otQMB1f1aq2cl4vyRtNXy3rTfK/LavpqWaacIzd90w8yYL1tkFOIA0uElSw9aZCKsSZourb5rzXsi7DW5iYvUmPhmwbJED1Gzyo2Vfm80T6FLpgan7DMu0/w4Dn63wQ7oMv6sZRVl/JmKXfxST4t5UeX9GwpGV7Sv6XkxwZJibXBn9JY5mVjTTHXi0YunU9Na4Kk1aTYv+hqa+HqkBd5zfZMZ7LMesNZoVfHI/SA1lr9eX7d4Uo+aKzjHN3XQR9KodlbUTuvPhtaGcF8jfllyPfyfLlOfS2Sm82bcg9R7P2zyuriA7s8F4yxhxuZdajYze0jfjoCv1Ine047JB4fi59od3ZA12e1NWtdVD4l5abbVXEPCfkBrTZPsCpdHjcxZ5n92omnruMpaSh+E3M2LLkDs4NfhnQZvYuRdc54cyeJWU16rzggLn1+umhALLNwMd7q+C5G1v9zxNi/l6eoz4FcPSGuh1fmzPLzfNO/DMvPvgnde3QN0I8zw/sEWRegL0vqEPp8rvc+x3P/NHD+HlJVcO6O796Z0VfHnz279nH2zImj5NFnEAVpTasiLjsFJ5BRZq+X4cRYnovbtXOuxpu1LavkVWjwjkwvkqbXb72/tXMw+9/Z/OD1Yfvgl3bn5eHO44fmf7Vr4ntxV9wjPz0Qj6kyX5Bf3Y3/Nq9u3ti8ub2//ef2u+2uZt24YmS+E7XPtvwfPIi3+Q==</latexit>

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Unobserved Confounding
Week 5:• Frontdoor adjustment• Unconfounded children criterion• Some other fancy application of
do-calculus

Week 7:

Brady Neal<latexit sha1_base64="kM2Kw9peVD0/n13t7Mo4wEYSnXc=">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</latexit>

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Unobserved Confounding
Week 5:• Frontdoor adjustment• Unconfounded children criterion• Some other fancy application of
do-calculus

Week 7:<latexit sha1_base64="kM2Kw9peVD0/n13t7Mo4wEYSnXc=">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</latexit>

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• Set identification (bounds)

Brady Neal

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Unobserved Confounding
Week 5:• Frontdoor adjustment• Unconfounded children criterion• Some other fancy application of
do-calculus

Week 7:<latexit sha1_base64="kM2Kw9peVD0/n13t7Mo4wEYSnXc=">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</latexit>

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• Set identification (bounds)• Sensitivity analysis
Brady Neal

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Unobserved Confounding
Week 5:• Frontdoor adjustment• Unconfounded children criterion• Some other fancy application of
do-calculus

Week 7:This week:
Instrumental variables<latexit sha1_base64="kM2Kw9peVD0/n13t7Mo4wEYSnXc=">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</latexit>

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• Set identification (bounds)• Sensitivity analysis
Brady Neal

2 / 33

Unobserved Confounding
Week 5:• Frontdoor adjustment• Unconfounded children criterion• Some other fancy application of
do-calculus

Week 7:This week:
Instrumental variables<latexit sha1_base64="uSnWmzenLZoB/622kgJqyZD0I1U=">AAAWqXicrVjbbttGEN3YvSRu0iYp4Je+sLEDJKisym6LBAgMBHASFEVTJICTuLEMg5eVRIi3kku7jqBP6Af2of/SmbNLkdSNUhAJIpfDmTOzc9tdOUngZ6rT+ffaxuZnn3/x5fUbW1/dvPX1N7fv3H2bxXnqyjduHMTpiWNnMvAj+Ub5KpAnSSrt0AnkO2d4xO/fXcg08+PoWF0l8iy0+5Hf811bEen8zsZF15F9Pxopf/gh8V2Vp3K8ZZnPfeuP2JPZ5Lkb0eNppmwlW56dDaR3Zj3o5lHsZDK9kJ4bR704jzxS+NCylfWg0+o8tEa781l2x0/mITOkojmoUEbqoXXqyCC+tALZU9Zh3LPmY52xlonUEuQ4V24cyglu6vcHjcBGaAmsH2UqzY3FthNfyIrFE7sAVrIWeBV/P/f6VX8nthos9rAk5pqznqwpWThjVq4SgEbe6twXmNSVkVdLsfPbO512Bx9rdrBvBjvCfF7Fd278I7rCE7FwRS5CIUUkFI0DYYuMvqdiX3REQrQzMSJaSiMf76UYiy2SzYlLEodN1CFd+/R0aqgRPTNmBmmXtAT0S0nSEvcNj0fjHqj6zvqtCu8iHSNgs41XdHcMZkhUJQZEbZIrOFeXc+gbNsxa0RweY7Y+zSQBhf3g1ubco3tAz4pmyNcr4pQ08kgqpZFLtIComsI6Urprz7NvBoiEDT5JI7Z6md3LrOb3MX2HhGXTOIOlbKslXpi4RNAsYSvzBIjpYsS/aYbavmVYvckcfMRdz4J5j4kyFB9oVEdeprOaXYu5lEEez9HF/AocEXFnRImR+T75zyeOeq66hMk6bESyj7kkyKC2Qf4N74tsTiiee7AnQ9Qs5JsPPYmprNLuws6A7g6wU5If0bsB9LAvWmSPBHaK7C282AJ3Sk+XeHJhoztFb6MyOZ4tmhNnV4twY3rjT7C0J7Q/C4u0dSOiWea7J36Hfklxb8ErnNctko1hoc7iDLoiE/tD6ivsl5CuTGXPFpQWdISwY4BsGRKt1KeRWN+PNIsM9RMY1BHdA3gtAUoL2tk/lxiHmBNHNIfuSxp70MPSFvW6tnhk7BjXtLJvfHSnWa0W8o9rV3fGaa0sF6FSLZOVh9DUET+bGXME2O6q11zTDXVPzjC78ZRVDuwvuwbnaN0+lirjU4+Vh2wcQKqOy16cP895czlAnbQQS/Z9nzgOIdUz9Ay1mRhNW5X6OzJVEiMeLu6hyQld76ztcurNiOih8VRAfLrOeU3iN38Rpo2c6sJ71ffLEJ8b+WI14I4/AnW53FtUVl02pdFo8ma5vI+RhH/0iCOu8RL4OsFcvp/8rMqbVbE/DV40g8gZqwzSqugecmP+zI/FDvWCHcqquj850kxJsTtJaLw74dwlXOYdQktEFK7w0eT9uDEGz4gyNh7r4V1Rh6UNPjqfB86uyTzNoStKTXFwVS/Xynk5L280fbmsN8n3uqymL5dlygVy0zf9IAPWSYOcQhxYIqxk6XGDVIw1QdO1zX+uYF+EtTY3eZEaC981SIboMXpWsanKl432KXTB1PiEZd5/hAcv0P/G2AGt68dSVq3lzVLu6qN8WspfrunZUjJc07+l5IcGSYm1wZ/QWOZ1Y00x16tGLp1PTWuCpNWk2L/oamvh6pAXec32TGeyzHrDWaFXx0P0gNZK/Xl23eFK3m+s4xzd10EfSqHZW1I7bz4ZWhnBfIX5Zcj38ny5Sn3Nk5vOm3IPUez9s8rq4gO7PBeMsIe7NOtQsZvbQ/x0BH6lTvaSdkg8PhI/0e5sn64vamvWqqh8SspNt6viHhDyI1ptnmFVWh83MWeZvdqJp67jOWkofmNzNiy5A7ODX4S0jt75yDpnvJmTxLQmvVccEJc+P101IJZZOB9veXznI+v/OWLs38tT1KdArp4QV8Mrc2bxeb7pX4bFZ9+E7j26BujHmeF9hqwL0JcldQh9Ptd7n6OZfxo4fw+oKjh3R/fuTumr40+fXfs4e+bEUfLoM4iCtKZVERedghPIKLPXy3BiLM/F7do5V+NN25ZV8io0eIemF0nT67fOb+/sT/93Njt4e9De/6XdeX2w8/Sx+V/tuvhO3BMPyE+PxFOqzFfkV3fjv83NzZubt7Z/2H69fbL9XrNuXDMy34raZ9v9H7Int5s=</latexit>

Z

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Y

• Set identification (bounds)• Sensitivity analysis
Brady Neal

2 / 33

What is an Instrument?
No Nonparametric Identification of the ATE
Warm-Up: Linear Setting
Nonparametric Identification of Local ATE
More General Settings for the ATE
Brady Neal3 / 33

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

What is an Instrument?

4 / 33Assumption 1: RelevanceZ has a causal effect on T<latexit sha1_base64="/hLeD4FcMEV8euIIoCVN+BKXDP0=">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</latexit>

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Brady Neal

What is an Instrument?

Y

5 / 33Assumption 1: RelevanceZ has a causal effect on T<latexit sha1_base64="/hLeD4FcMEV8euIIoCVN+BKXDP0=">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</latexit>

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Brady Neal

What is an Instrument?

Y

5 / 33Assumption 2: Exclusion Restriction
The causal effect of Z on Y is fully mediated by T<latexit sha1_base64="/hLeD4FcMEV8euIIoCVN+BKXDP0=">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</latexit>

U
Z

T

Brady Neal

What is an Instrument?

Y

6 / 33Assumption 2: Exclusion Restriction
The causal effect of Z on Y is fully mediated by T<latexit sha1_base64="/hLeD4FcMEV8euIIoCVN+BKXDP0=">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</latexit>

U
Z

T

Brady Neal

What is an Instrument?

Y

6 / 33Assumption 2: Exclusion Restriction
The causal effect of Z on Y is fully mediated by T<latexit sha1_base64="SXUJchWKlejMrSU7143taV/9zHo=">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</latexit>

U
Z

T

Brady Neal

What is an Instrument?

Y

6 / 33Assumption 2: Exclusion Restriction
The causal effect of Z on Y is fully mediated by T<latexit sha1_base64="SXUJchWKlejMrSU7143taV/9zHo=">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</latexit>

U
Z

T

Y

Recall:Removing edges corresponds to adding assumptions
Brady Neal

What is an Instrument?

6 / 33Assumption 3: Instrumental Unconfoundedness
Z is unconfounded (no unblockable backdoor paths to Y)<latexit sha1_base64="SXUJchWKlejMrSU7143taV/9zHo=">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</latexit>

U
Z

T

Brady Neal

What is an Instrument?

Y

7 / 33Assumption 3: Instrumental Unconfoundedness
Z is unconfounded (no unblockable backdoor paths to Y)<latexit sha1_base64="SXUJchWKlejMrSU7143taV/9zHo=">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</latexit>

U
Z

T

Brady Neal

What is an Instrument?

Y

7 / 33Assumption 3: Instrumental Unconfoundedness
Z is unconfounded (no unblockable backdoor paths to Y)<latexit sha1_base64="K53ughT9jty+/kkw1QKV+eOgpqA=">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</latexit>

U
Z

T

Brady Neal

What is an Instrument?

Y

7 / 33

Conditional Instruments<latexit sha1_base64="0iqNsrzTQ+P1Sl711NXqyPFhwHs=">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</latexit>

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Z

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Brady Neal

What is an Instrument?

Y
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Conditional Instruments<latexit sha1_base64="TgeuzHqWqDHtSNbnWDpV8XNDN/o=">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</latexit>

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Brady Neal

What is an Instrument?

Y
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Conditional Instruments
Slightly weaker version of Assumption 3:
Unconfoundedness after conditioning
on observed variables<latexit sha1_base64="TgeuzHqWqDHtSNbnWDpV8XNDN/o=">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</latexit>

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Brady Neal

What is an Instrument?

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Question:
What are the 3 assumptions we need to saythat a given variable is an instrument, andwhat do they correspond to graphically?

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

No Nonparametric Identification of the ATE

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No Nonparametric Identification of the ATE

Brady Neal

No Nonparametric Identification of the ATE

11 / 33

No Nonparametric Identification of the ATE
Why didn’t we see instruments in Week 5 Identification?

Brady Neal

No Nonparametric Identification of the ATE

11 / 33

No Nonparametric Identification of the ATE
Why didn’t we see instruments in Week 5 Identification?
Week 5 was about nonparametric identification

Brady Neal

No Nonparametric Identification of the ATE

11 / 33

No Nonparametric Identification of the ATE
Why didn’t we see instruments in Week 5 Identification?
Week 5 was about nonparametric identification
Recall necessary condition for nonparametric identification:
For each backdoor path from T to any child that is an ancestor of Y, it ispossible to block that path<latexit sha1_base64="uSnWmzenLZoB/622kgJqyZD0I1U=">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</latexit>

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Brady Neal

No Nonparametric Identification of the ATE

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

Warm-Up: Linear Setting

12 / 33

Another Assumption: Linear Outcome
Y := T + ↵u U
<latexit sha1_base64="UeVu8hQ8nYOpqwWbbqcmxjVPKxk=">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</latexit>

Brady Neal

Warm-Up: Linear Setting

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Another Assumption: Linear Outcome
Y := T + ↵u U
<latexit sha1_base64="UeVu8hQ8nYOpqwWbbqcmxjVPKxk=">AAAUiXicrVjbbttGEN2kt8RpGyf1W1/YKAUKVFYlN4WDoAIC2AmKogEcwLc2CgyKXFmEeCsvVh1Bn9BP6of0ua/tP3Tm7FIkdSGlwBIkrmZnzszObXfVD10nTtrtv2/d/uDDjz7+5M7drXuffvb5/e0HD0/jII0seWIFbhCd981Yuo4vTxInceV5GEnT67vyrD864PmzKxnFTuAfJ9ehfOuZl74zcCwzIdLF9lkvSBMr8KTxrGv0bOkmptFLCCHxpJ8Y3xo90w2H5kUv9YN+LKMraVuBPwhS3ybUK9M1ls9cbDfarTZexuKgowcNoV9HwYO7f4qesEUgLJEKT0jhi4TGrjBFTO83oiPaIiTaWzEhWkQjB/NSTMUWyabEJYnDJOqIvi/p1xtN9ek3Y8aQtkiLS5+IJA3xteaxaTwAVT1Zv1HgXaVjAmy28ZqefY3pETURQ6LWyWWc68v16e3VrDqhNTzFah1aSQgK+8EqrXlAT5d+J7RC/r4mTkkjm6QiGllEc4mqKKwjoqfyPPtmiEiY4JM0Yqur7K6ymucDeo8Iy6RxDEvZVkO81HHxoVnCVuZxEdPViH/QCpV9VViD2RocxF2tgnmPiTIS72hURq7SWcyu1VyJRp4u0cX8CTh84o6JEiDzHfKfQxzlXLUIk3WYiOQl1hIig1oa+WfMZ9kcUjx3YU+MqBnINwd6Ql1Zud2ZnS49+8COSH5Cc0PoYV80yR4J7AjZm3mxCe6Ifo3xy4KN1hy9hcrkeDZpTZxdTcINaMaZYSlPKH9mFinrJkQz9HtX/AL9kuLehFc4r5skG8BClcUxdPk69l3qK+wXj76Zyp7NKE3o8GDHENkyIlquTyGxvu9oFTHqx9WoE3q68FoIlCa0s3/GGHtYE0c0he4xjW3oYWmDel1L7Gs7piWt7BsH3WlRq4H849pVnXFeK8v5qFRDZ2UXmtriiV4xR4DtLnrN0t1Q9eQYq5vOWdWH/XnX4Bwt28dSeXzKsbKRjUNIlXHZi8vXuWwte6iTJmLJvr8kji6kBpoeozZDrWmrUH8HukoCxMPC09M5oeqdtY3nZiZE97SnXOJTdc57Es/8TpgmcqoH7xXnqxBfaPlsN+COPwG1Wu4UlVWWjWg0mc1UyzsYSfhHjTjiCi+Er0Os5avZxyjMrIt9M3j+AiJnbKKR1kW3kRvLV34sGtQLGpRVZX9ypJkS4XQS0vjxjPMx4TLvCFp8onCFT2bz09oYHBJlqj02wFxWh7kNDjqfDc6ezjzFoSoqmePgqq7Wynm5LG8UvVrWnuV7WVbRq2WZcoXcdHQ/iIF1XiOXIA4s4RWy9LhGKsCeoOjK5l/XsM/HXpvqvIi0hWc1kh56jFpVoKvyVa19CbpgpH3CMr+9hwev0P+mOAFt6sdcNtnIm7nc9Xv5NJcfb+jZXNLb0L+55LsaSYm9wZnRWOZ1bU0x11Etl8qnuj1B0m6SnV9UtTXx3Scv8p5t685k6P2Gs0Ltjl30gOZa/Xlx3+FK7tTWcYru20cfiqDZrqidkxtDyyOYrrG+GPme3y/Xqa9lcvN5k58hsrN/XNhdHGDn94IJznBjvQ9lp7ldxE9F4CfqZK/ohMTjA/E9nc469P2ytGeti8q3pFR3uyLuHiHv025ziF1pc9xQ32V2Szeeso4XpCH7TPXdMOd29Ql+FdImepcjq5yxF24S85rUWXFIXOr+dF2DmGfhcrzq+C5HVv9zBDi/57eom0Au3hDXw8tzZvV9vu5fhtV335CeA/p20Y9jzXuIrHPRlyV1CHU/V2efg4V/Gjh/96gqOHcnjx7O6Svjz99dL3H3TIkj51F3kATSilZEXHULDiGT6LNejBtjfi9ule65Cm/etriQV57G6+peJHWv37rYbnTm/ztbHJzutTo/tNqvnzSe/6j/V7sjvhSPxDfkp33xnCrziPxqib/EP+Jf8d/OvZ3OztOdZ4r19i0t84UovXYO/gd1egmg</latexit>

Z doesn’t appear in this structural
equation because of the exclusion
restriction assumption<latexit sha1_base64="uSnWmzenLZoB/622kgJqyZD0I1U=">AAAWqXicrVjbbttGEN3YvSRu0iYp4Je+sLEDJKisym6LBAgMBHASFEVTJICTuLEMg5eVRIi3kku7jqBP6Af2of/SmbNLkdSNUhAJIpfDmTOzc9tdOUngZ6rT+ffaxuZnn3/x5fUbW1/dvPX1N7fv3H2bxXnqyjduHMTpiWNnMvAj+Ub5KpAnSSrt0AnkO2d4xO/fXcg08+PoWF0l8iy0+5Hf811bEen8zsZF15F9Pxopf/gh8V2Vp3K8ZZnPfeuP2JPZ5Lkb0eNppmwlW56dDaR3Zj3o5lHsZDK9kJ4bR704jzxS+NCylfWg0+o8tEa781l2x0/mITOkojmoUEbqoXXqyCC+tALZU9Zh3LPmY52xlonUEuQ4V24cyglu6vcHjcBGaAmsH2UqzY3FthNfyIrFE7sAVrIWeBV/P/f6VX8nthos9rAk5pqznqwpWThjVq4SgEbe6twXmNSVkVdLsfPbO512Bx9rdrBvBjvCfF7Fd278I7rCE7FwRS5CIUUkFI0DYYuMvqdiX3REQrQzMSJaSiMf76UYiy2SzYlLEodN1CFd+/R0aqgRPTNmBmmXtAT0S0nSEvcNj0fjHqj6zvqtCu8iHSNgs41XdHcMZkhUJQZEbZIrOFeXc+gbNsxa0RweY7Y+zSQBhf3g1ubco3tAz4pmyNcr4pQ08kgqpZFLtIComsI6Urprz7NvBoiEDT5JI7Z6md3LrOb3MX2HhGXTOIOlbKslXpi4RNAsYSvzBIjpYsS/aYbavmVYvckcfMRdz4J5j4kyFB9oVEdeprOaXYu5lEEez9HF/AocEXFnRImR+T75zyeOeq66hMk6bESyj7kkyKC2Qf4N74tsTiiee7AnQ9Qs5JsPPYmprNLuws6A7g6wU5If0bsB9LAvWmSPBHaK7C282AJ3Sk+XeHJhoztFb6MyOZ4tmhNnV4twY3rjT7C0J7Q/C4u0dSOiWea7J36Hfklxb8ErnNctko1hoc7iDLoiE/tD6ivsl5CuTGXPFpQWdISwY4BsGRKt1KeRWN+PNIsM9RMY1BHdA3gtAUoL2tk/lxiHmBNHNIfuSxp70MPSFvW6tnhk7BjXtLJvfHSnWa0W8o9rV3fGaa0sF6FSLZOVh9DUET+bGXME2O6q11zTDXVPzjC78ZRVDuwvuwbnaN0+lirjU4+Vh2wcQKqOy16cP895czlAnbQQS/Z9nzgOIdUz9Ay1mRhNW5X6OzJVEiMeLu6hyQld76ztcurNiOih8VRAfLrOeU3iN38Rpo2c6sJ71ffLEJ8b+WI14I4/AnW53FtUVl02pdFo8ma5vI+RhH/0iCOu8RL4OsFcvp/8rMqbVbE/DV40g8gZqwzSqugecmP+zI/FDvWCHcqquj850kxJsTtJaLw74dwlXOYdQktEFK7w0eT9uDEGz4gyNh7r4V1Rh6UNPjqfB86uyTzNoStKTXFwVS/Xynk5L280fbmsN8n3uqymL5dlygVy0zf9IAPWSYOcQhxYIqxk6XGDVIw1QdO1zX+uYF+EtTY3eZEaC981SIboMXpWsanKl432KXTB1PiEZd5/hAcv0P/G2AGt68dSVq3lzVLu6qN8WspfrunZUjJc07+l5IcGSYm1wZ/QWOZ1Y00x16tGLp1PTWuCpNWk2L/oamvh6pAXec32TGeyzHrDWaFXx0P0gNZK/Xl23eFK3m+s4xzd10EfSqHZW1I7bz4ZWhnBfIX5Zcj38ny5Sn3Nk5vOm3IPUez9s8rq4gO7PBeMsIe7NOtQsZvbQ/x0BH6lTvaSdkg8PhI/0e5sn64vamvWqqh8SspNt6viHhDyI1ptnmFVWh83MWeZvdqJp67jOWkofmNzNiy5A7ODX4S0jt75yDpnvJmTxLQmvVccEJc+P101IJZZOB9veXznI+v/OWLs38tT1KdArp4QV8Mrc2bxeb7pX4bFZ9+E7j26BujHmeF9hqwL0JcldQh9Ptd7n6OZfxo4fw+oKjh3R/fuTumr40+fXfs4e+bEUfLoM4iCtKZVERedghPIKLPXy3BiLM/F7do5V+NN25ZV8io0eIemF0nT67fOb+/sT/93Njt4e9De/6XdeX2w8/Sx+V/tuvhO3BMPyE+PxFOqzFfkV3fjv83NzZubt7Z/2H69fbL9XrNuXDMy34raZ9v9H7Int5s=</latexit>

Z

U

T

Brady Neal

Warm-Up: Linear SettingY

13 / 33

Another Assumption: Linear Outcome
Y := T + ↵u U
<latexit sha1_base64="UeVu8hQ8nYOpqwWbbqcmxjVPKxk=">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</latexit>

Z doesn’t appear in this structural
equation because of the exclusion
restriction assumption<latexit sha1_base64="uSnWmzenLZoB/622kgJqyZD0I1U=">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</latexit>

Z

U

T

Y

LinearBrady Neal

Warm-Up: Linear Setting

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Warm-Up: Binary Linear Setting<latexit sha1_base64="hm9P+ROKGa+XIlenBfnjjn0ePYA=">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</latexit>

Z

U

T

Y

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">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</latexit>

Brady Neal

Warm-Up: Linear Setting

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Warm-Up: Binary Linear Setting<latexit sha1_base64="hm9P+ROKGa+XIlenBfnjjn0ePYA=">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</latexit>

Z

U

T

Y

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">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</latexit>

Brady Neal

Warm-Up: Linear Setting

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Warm-Up: Binary Linear Setting<latexit sha1_base64="hm9P+ROKGa+XIlenBfnjjn0ePYA=">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</latexit>

Z

U

T

Y

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">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</latexit>

Brady Neal

Warm-Up: Linear Setting

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Warm-Up: Binary Linear Setting<latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit>

E[Y | Z = 1]

E[Y | Z = 0]

= E[ T + ↵u U | Z = 1]

E[ T + ↵u U | Z = 0]

= (E[T | Z = 1]

E[T | Z = 0]) + ↵u (E[U | Z = 1]

= (E[T | Z = 1]

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

E[T | Z = 0]) + ↵u (E[U ]E[U ])

E[U | Z = 0])<latexit sha1_base64="hm9P+ROKGa+XIlenBfnjjn0ePYA=">AAAWqXicrVjbbttGEN3YvSRu0iYp4Je+sLEDJKisym6KFAgMBHASFEVTJICTuLEMg5eVRIi3kku7jqBP6Af2of/SmbNLkdSNUhAJIpfDmTOzc9tdOUngZ6rT+ffaxuZnn3/x5fUbW1/dvPX1N7fv3H2bxXnqyjduHMTpiWNnMvAj+Ub5KpAnSSrt0AnkO2d4xO/fXcg08+PoWF0l8iy0+5Hf811bEen8zsZF15F9Pxopf/gh8V2Vp3K8ZZnPfeuP2JPZ5Lkb0eNppmwlW56dDaR3Zj3o5lHsZDK9kJ4bR704jzxS+NCylfWg0+o8tEa781l2x0/mITOkojmoUEbqoXXqyCC+tALZU9Zh3LPmY52xlonUEuQ4V24cyglu6vcHjcBGaAmsH2UqzY3FthNfyIrFE7sAVrIWeBV/P/f6VX8nthos9rAk5pqznqwpWThjVq4SgEbe6twXmNSVkVdLsfPbO512Bx9rdrBvBjvCfF7Fd278I7rCE7FwRS5CIUUkFI0DYYuMvqdiX3REQrQzMSJaSiMf76UYiy2SzYlLEodN1CFd+/R0aqgRPTNmBmmXtAT0S0nSEvcNj0fjHqj6zvqtCu8iHSNgs41XdHcMZkhUJQZEbZIrOFeXc+gbNsxa0Rx+wWx9mkkCCvvBrc25R/eAnhXNkK9XxClp5JFUSiOXaAFRNYV1pHTXnmffDBAJG3ySRmz1MruXWc3vY/oOCcumcQZL2VZLvDBxiaBZwlbmCRDTxYh/0wy1fcuwepM5+Ii7ngXzHhNlKD7QqI68TGc1uxZzKYM8nqOL+RU4IuLOiBIj833yn08c9Vx1CZN12IhkH3NJkEFtg/wb3hfZnFA892BPhqhZyDcfehJTWaXdhZ0B3R1gpyQ/oncD6GFftMgeCewU2Vt4sQXulJ4u8eTCRneK3kZlcjxbNCfOrhbhxvTGn2BpT2h/FhZp60ZEs8x3T/wO/ZLi3oJXOK9bJBvDQp3FGXRFJvaH1FfYLyFdmcqeLSgt6AhhxwDZMiRaqU8jsb4faRYZ6icwqCO6B/BaApQWtLN/LjEOMSeOaA7dlzT2oIelLep1bfHY2DGuaWXf+OhOs1ot5B/Xru6M01pZLkKlWiYrD6GpIx6ZGXME2O6q11zTDXVPzjC78ZRVDuwvuwbnaN0+lirjU4+Vh2wcQKqOy16cP895czlAnbQQS/Z9nzgOIdUz9Ay1mRhNW5X6OzJVEiMeLu6hyQld76ztcurNiOih8VRAfLrOeU3iN38Rpo2c6sJ71ffLEJ8b+WI14I4/AnW53FtUVl02pdFo8ma5vI+RhH/0iCOu8RL4OsFcvp/8rMqbVbE/DV40g8gZqwzSqugecmP+zI/FDvWCHcqquj850kxJsTtJaLw74dwlXOYdQktEFK7w0eT9uDEGz4gyNh7r4V1Rh6UNPjqfB86uyTzNoStKTXFwVS/Xynk5L280fbmsN8n3uqymL5dlygVy0zf9IAPWSYOcQhxYIqxk6XGDVIw1QdO1zX+uYF+EtTY3eZEaC981SIboMXpWsanKl432KXTB1PiEZd5/hAcv0P/G2AGt68dSVq3lzVLu6qN8WspfrunZUjJc07+l5IcGSYm1wZ/QWOZ1Y00x16tGLp1PTWuCpNWk2L/oamvh6pAXec32TGeyzHrDWaFXx0P0gNZK/Xl23eFK3m+s4xzd10EfSqHZW1I7bz4ZWhnBfIX5Zcj38ny5Sn3Nk5vOm3IPUez9s8rq4gO7PBeMsIe7NOtQsZvbQ/x0BH6lTvaSdkg8PhI/0e5sn64vamvWqqh8SspNt6viHhDyY1ptnmFVWh83MWeZvdqJp67jOWkofmNzNiy5A7ODX4S0jt75yDpnvJmTxLQmvVccEJc+P101IJZZOB9veXznI+v/OWLs38tT1KdArp4QV8Mrc2bxeb7pX4bFZ9+E7j26BujHmeF9hqwL0JcldQh9Ptd7n6OZfxo4fw+oKjh3R/fuTumr40+fXfs4e+bEUfLoM4iCtKZVERedghPIKLPXy3BiLM/F7do5V+NN25ZV8io0eIemF0nT67fOb+/sT/93Njt4e9De/7ndeX2w8/SR+V/tuvhO3BMPyE+PxVOqzFfkV3fjv83NzZubt7Z/2H69fbL9XrNuXDMy34raZ9v9H7Dzt5c=</latexit>

Z

U

T

Y

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">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</latexit>

Brady Neal

Warm-Up: Linear Setting

14 / 33

Warm-Up: Binary Linear Setting<latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit><latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit>

E[Y | Z = 1]

E[Y | Z = 0]

= E[ T + ↵u U | Z = 1]
= (E[T | Z = 1]
= (E[T | Z = 1]
= (E[T | Z = 1]

E[ T + ↵u U | Z = 0]

(exclusion restriction and linear outcome assumptions)

E[T | Z = 0]) + ↵u (E[U | Z = 1]
E[T | Z = 0]) + ↵u (E[U ] E[U ])E[U | Z = 0])

E[T | Z = 0])<latexit sha1_base64="hm9P+ROKGa+XIlenBfnjjn0ePYA=">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</latexit>

Z

U

T

Y

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">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</latexit>

Brady Neal

Warm-Up: Linear Setting

14 / 33

Warm-Up: Binary Linear Setting<latexit 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E[Y | Z = 1]

E[Y | Z = 0]

= E[ T + ↵u U | Z = 1]
= (E[T | Z = 1]
= (E[T | Z = 1]
= (E[T | Z = 1]

E[ T + ↵u U | Z = 0]

(exclusion restriction and linear outcome assumptions)

E[T | Z = 0]) + ↵u (E[U | Z = 1]
E[T | Z = 0]) + ↵u (E[U ] E[U ])E[T | Z = 0])

E[U | Z = 0])<latexit sha1_base64="hm9P+ROKGa+XIlenBfnjjn0ePYA=">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</latexit>

Z

U

T

Y

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">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</latexit>

Brady Neal

Warm-Up: Linear Setting

14 / 33

Warm-Up: Binary Linear Setting<latexit 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E[Y | Z = 1]

E[Y | Z = 0]

= E[ T + ↵u U | Z = 1]
= (E[T | Z = 1]
= (E[T | Z = 1]
= (E[T | Z = 1]

E[ T + ↵u U | Z = 0]

(exclusion restriction and linear outcome assumptions)

E[T | Z = 0]) + ↵u (E[U | Z = 1]
E[T | Z = 0]) + ↵u (E[U ] E[U ])E[T | Z = 0])

E[U | Z = 0])(instrumental unconfoundedness assumption)<latexit sha1_base64="hm9P+ROKGa+XIlenBfnjjn0ePYA=">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</latexit>

Z

U

T

Y

Y := T + ↵u U
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Brady Neal

Warm-Up: Linear Setting

14 / 33

Warm-Up: Binary Linear Setting<latexit 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E[Y | Z = 1] E[Y | Z = 0]
= E[ T + ↵u U | Z = 1] E[ T + ↵u U | Z = 0]
= (E[T | Z = 1]
= (E[T | Z = 1]
= (E[T | Z = 1]

(exclusion restriction and linear outcome assumptions)

E[T | Z = 0]) + ↵u (E[U | Z = 1]
E[T | Z = 0]) + ↵u (E[U ] E[U ])E[T | Z = 0])

E[U | Z = 0])(instrumental unconfoundedness assumption)<latexit sha1_base64="hm9P+ROKGa+XIlenBfnjjn0ePYA=">AAAWqXicrVjbbttGEN3YvSRu0iYp4Je+sLEDJKisym6KFAgMBHASFEVTJICTuLEMg5eVRIi3kku7jqBP6Af2of/SmbNLkdSNUhAJIpfDmTOzc9tdOUngZ6rT+ffaxuZnn3/x5fUbW1/dvPX1N7fv3H2bxXnqyjduHMTpiWNnMvAj+Ub5KpAnSSrt0AnkO2d4xO/fXcg08+PoWF0l8iy0+5Hf811bEen8zsZF15F9Pxopf/gh8V2Vp3K8ZZnPfeuP2JPZ5Lkb0eNppmwlW56dDaR3Zj3o5lHsZDK9kJ4bR704jzxS+NCylfWg0+o8tEa781l2x0/mITOkojmoUEbqoXXqyCC+tALZU9Zh3LPmY52xlonUEuQ4V24cyglu6vcHjcBGaAmsH2UqzY3FthNfyIrFE7sAVrIWeBV/P/f6VX8nthos9rAk5pqznqwpWThjVq4SgEbe6twXmNSVkVdLsfPbO512Bx9rdrBvBjvCfF7Fd278I7rCE7FwRS5CIUUkFI0DYYuMvqdiX3REQrQzMSJaSiMf76UYiy2SzYlLEodN1CFd+/R0aqgRPTNmBmmXtAT0S0nSEvcNj0fjHqj6zvqtCu8iHSNgs41XdHcMZkhUJQZEbZIrOFeXc+gbNsxa0Rx+wWx9mkkCCvvBrc25R/eAnhXNkK9XxClp5JFUSiOXaAFRNYV1pHTXnmffDBAJG3ySRmz1MruXWc3vY/oOCcumcQZL2VZLvDBxiaBZwlbmCRDTxYh/0wy1fcuwepM5+Ii7ngXzHhNlKD7QqI68TGc1uxZzKYM8nqOL+RU4IuLOiBIj833yn08c9Vx1CZN12IhkH3NJkEFtg/wb3hfZnFA892BPhqhZyDcfehJTWaXdhZ0B3R1gpyQ/oncD6GFftMgeCewU2Vt4sQXulJ4u8eTCRneK3kZlcjxbNCfOrhbhxvTGn2BpT2h/FhZp60ZEs8x3T/wO/ZLi3oJXOK9bJBvDQp3FGXRFJvaH1FfYLyFdmcqeLSgt6AhhxwDZMiRaqU8jsb4faRYZ6icwqCO6B/BaApQWtLN/LjEOMSeOaA7dlzT2oIelLep1bfHY2DGuaWXf+OhOs1ot5B/Xru6M01pZLkKlWiYrD6GpIx6ZGXME2O6q11zTDXVPzjC78ZRVDuwvuwbnaN0+lirjU4+Vh2wcQKqOy16cP895czlAnbQQS/Z9nzgOIdUz9Ay1mRhNW5X6OzJVEiMeLu6hyQld76ztcurNiOih8VRAfLrOeU3iN38Rpo2c6sJ71ffLEJ8b+WI14I4/AnW53FtUVl02pdFo8ma5vI+RhH/0iCOu8RL4OsFcvp/8rMqbVbE/DV40g8gZqwzSqugecmP+zI/FDvWCHcqquj850kxJsTtJaLw74dwlXOYdQktEFK7w0eT9uDEGz4gyNh7r4V1Rh6UNPjqfB86uyTzNoStKTXFwVS/Xynk5L280fbmsN8n3uqymL5dlygVy0zf9IAPWSYOcQhxYIqxk6XGDVIw1QdO1zX+uYF+EtTY3eZEaC981SIboMXpWsanKl432KXTB1PiEZd5/hAcv0P/G2AGt68dSVq3lzVLu6qN8WspfrunZUjJc07+l5IcGSYm1wZ/QWOZ1Y00x16tGLp1PTWuCpNWk2L/oamvh6pAXec32TGeyzHrDWaFXx0P0gNZK/Xl23eFK3m+s4xzd10EfSqHZW1I7bz4ZWhnBfIX5Zcj38ny5Sn3Nk5vOm3IPUez9s8rq4gO7PBeMsIe7NOtQsZvbQ/x0BH6lTvaSdkg8PhI/0e5sn64vamvWqqh8SspNt6viHhDyY1ptnmFVWh83MWeZvdqJp67jOWkofmNzNiy5A7ODX4S0jt75yDpnvJmTxLQmvVccEJc+P101IJZZOB9veXznI+v/OWLs38tT1KdArp4QV8Mrc2bxeb7pX4bFZ9+E7j26BujHmeF9hqwL0JcldQh9Ptd7n6OZfxo4fw+oKjh3R/fuTumr40+fXfs4e+bEUfLoM4iCtKZVERedghPIKLPXy3BiLM/F7do5V+NN25ZV8io0eIemF0nT67fOb+/sT/93Njt4e9De/7ndeX2w8/SR+V/tuvhO3BMPyE+PxVOqzFfkV3fjv83NzZubt7Z/2H69fbL9XrNuXDMy34raZ9v9H7Dzt5c=</latexit>

Z

U

T

Y

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">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</latexit>

Brady Neal

Warm-Up: Linear Setting

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Warm-Up: Binary Linear Setting<latexit 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E[Y | Z = 1] E[Y | Z = 0]
= E[ T + ↵u U | Z = 1] E[ T + ↵u U | Z = 0]
= (E[T | Z = 1]
= (E[T | Z = 1]
= (E[T | Z = 1]

(exclusion restriction and linear outcome assumptions)

E[T | Z = 0]) + ↵u (E[U | Z = 1]
E[T | Z = 0]) + ↵u (E[U ] E[U ])E[T | Z = 0])

E[U | Z = 0])(instrumental unconfoundedness assumption)<latexit sha1_base64="hm9P+ROKGa+XIlenBfnjjn0ePYA=">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</latexit><latexit sha1_base64="3cuWur6hLCfODW+HoFEKOFHnbPs=">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</latexit>

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

E[Y | Z = 0]
E[T | Z = 0]

Z

U

T

Y

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">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</latexit>

Brady Neal

Warm-Up: Linear Setting

14 / 33

Warm-Up: Binary Linear Setting<latexit 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E[Y | Z = 1] E[Y | Z = 0]
= E[ T + ↵u U | Z = 1] E[ T + ↵u U | Z = 0]
= (E[T | Z = 1]
= (E[T | Z = 1]
= (E[T | Z = 1]

(exclusion restriction and linear outcome assumptions)

E[T | Z = 0]) + ↵u (E[U | Z = 1]
E[T | Z = 0]) + ↵u (E[U ] E[U ])E[T | Z = 0])

E[U | Z = 0])(instrumental unconfoundedness assumption)<latexit sha1_base64="hm9P+ROKGa+XIlenBfnjjn0ePYA=">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</latexit><latexit sha1_base64="3cuWur6hLCfODW+HoFEKOFHnbPs=">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</latexit>

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

E[Y | Z = 0]
E[T | Z = 0]Z(non-zero due to relevance assumption)

U

T

Y

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">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</latexit>

Brady Neal

Warm-Up: Linear Setting

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Multiplying Path Coefficients in Linear Setting<latexit sha1_base64="NIoWxOqWvkAXO6A/wEjPiAFLHZo=">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</latexit>

Z

U

↵

z

T
<latexit sha1_base64="fyZ1p99ikpkndGU282pak6zlpWw=">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</latexit>

Brady NealE[Y | Z = 1]=
E[T | Z = 1]

Y
E[Y | Z = 0]
E[T | Z = 0]

Warm-Up: Linear Setting

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Brady NealE[Y | Z = 1]=
E[T | Z = 1]

Y
E[Y | Z = 0]
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Warm-Up: Linear Setting

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Brady NealY↵z=
E[T | Z = 1] E[T | Z = 0]
Warm-Up: Linear Setting

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Brady NealY↵z=
E[T | Z = 1] E[T | Z = 0]
Warm-Up: Linear Setting

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Multiplying Path Coefficients in Linear Setting<latexit sha1_base64="NIoWxOqWvkAXO6A/wEjPiAFLHZo=">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</latexit>

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Brady NealY↵z=
E[T | Z = 1] E[T | Z = 0]
Warm-Up: Linear Setting

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Multiplying Path Coefficients in Linear Setting<latexit sha1_base64="NIoWxOqWvkAXO6A/wEjPiAFLHZo=">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</latexit>

Z

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z

T

Y
<latexit sha1_base64="8fxaAO836FB6EaO7JISLrsyoqlM=">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</latexit>

Brady Neal

↵z
=
↵z

Warm-Up: Linear Setting

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Wald Estimator
Wald estimand:<latexit sha1_base64="3cuWur6hLCfODW+HoFEKOFHnbPs=">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</latexit>

Brady NealE[Y | Z = 1]=
E[T | Z = 1]

E[Y | Z = 0]
E[T | Z = 0]

Warm-Up: Linear Setting

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Wald Estimator
Wald estimand:<latexit sha1_base64="3cuWur6hLCfODW+HoFEKOFHnbPs=">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</latexit>

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

E[Y | Z = 0]
E[T | Z = 0]

Wald estimator:<latexit sha1_base64="BCiNb02P+Gj7F9mYtsGPolA0Ja8=">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</latexit>ˆ=Brady NealP

1
i:zi =1 Yi
n1P1i:zi =1 Ti
n1P1i:zi =0 Yi
n0P1i:zi =0 Tin0Warm-Up: Linear Setting

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Continuous Linear Setting<latexit sha1_base64="3cuWur6hLCfODW+HoFEKOFHnbPs=">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</latexit>

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

E[Y | Z = 0]
E[T | Z = 0]<latexit sha1_base64="hm9P+ROKGa+XIlenBfnjjn0ePYA=">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</latexit>

Z

U

T

Y

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">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</latexit>

Brady Neal

Warm-Up: Linear Setting

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Continuous Linear Setting<latexit sha1_base64="3cuWur6hLCfODW+HoFEKOFHnbPs=">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</latexit>

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

E[Y | Z = 0]
E[T | Z = 0]

What if T and Z are continuous?
<latexit sha1_base64="hm9P+ROKGa+XIlenBfnjjn0ePYA=">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</latexit>

Z

U

T

Y

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">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</latexit>

Brady Neal

Warm-Up: Linear Setting

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Continuous Linear Setting<latexit sha1_base64="3cuWur6hLCfODW+HoFEKOFHnbPs=">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</latexit>

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

E[Y | Z = 0]
E[T | Z = 0]<latexit sha1_base64="+eCRfgBW3aclHm3N1hl5+O3MQK8=">AAAUvnicrVhZb9tGEN6kV+JeTuq3vrBRCqQArUpuCudFgAE7QRE0gAP4aqPAkEjKIsSr5MqKQ+gvFf01BfrU/pTOfLsUSR2kFFiCyNXszDezcy2X/chzE9lq/XPn7kcff/LpZ/fub33+xZdffb394OFZEo5jyzm1Qi+ML/q9xPHcwDmVrvSciyh2en7fc877o0OeP7924sQNgxN5Ezlv/d5V4A5cqyeJdLn9sms7nuwZHaM7iHtW2j0Mr590w7G0Qt8xja4bJDIe+04gf5jqSUn4kilz05fbjVazhY+xOGjrQUPoz3H44P6foitsEQpLjIUvHBEISWNP9ERC3zeiLVoiItpbkRItppGLeUdMxRbJjonLIY4eUUd0vaJ/bzQ1oP+MmUDaIi0e/WKSNMT3msem8QBUdWf9RoF3lY4U2GzjDd37GtMnqhRDotbJZZzry/Xp69esWtIanmG1Lq0kAoX9YJXWPKC7R/8lrZCvN8Tp0MgmqZhGFtE8oioK64jprjzPvhkiEj3wOTRiq6vsrrKa50P6jgirR+MElrKthnih4xJAswNbmcdDTFcjvqMVKvuqsAazNbiIu1oF854QZSTe06iMXKWzmF2ruaRGni7RxfwSHAFxJ0QJkfku+c8ljnKuWoTJOnqI5BXWEiGDmhr5JeazbI4onruwJ0HUDOSbCz2Rrqzc7sxOj+59YMckn9LcEHrYFybZ4wA7RvZmXjTBHdO/Cf5ZsNGaozdRmRxPk9bE2WUSbkgz7gxLeUL5M7NIWZcSzdDfXfEr9DsUdxNe4bw2STaEhSqLE+gKdOw71FfYLz5dmcqezSgmdPiwY4hsGREt16eQWN+PtIoE9eNp1JTuHrwWAcWEdvbPBGMfa+KIjqF7QmMbeljaoF7XFPvajmlJK/vGRXda1Gog/7h2VWec18pyASrV0FnZgaaWeKpXzBFgu4tes3Q3VD05weqmc1b1YX/eNThHy/axVB6fcqxsZOMQUmVc9uLydS5byx7qxEQs2fdXxNGB1EDTE9RmpDVtFervUFdJiHhYuPs6J1S9s7bJ3ExKdF97yiM+Vee8J/HMH4TZQ0514b3ifBXicy2f7Qbc8VNQq+XOUFll2ZhG6WymWt7FyIF/1IgjrvAi+DrCWr6b/YzCzLrYt4MXLCByxkqNtC66jdxYvvIT0aBe0KCsKvuTI82UGE8nEY0fzzgfEy7zjqAlIApXeDqbn9bG4IgoU+2xAeayOsxtcNH5bHB2deYpDlVRco6Dq7paK+flsrxR9GpZe5bvZVlFr5ZlyjVy09X9IAHWRY2cRBxYwi9k6UmNVIg9QdGVzb+tYV+AvXas8yLWFp7XSProMWpVoa7KV7X2SXTBWPuEZX7/AA9eo/9N8QS0qR9zWbmRN3O5mw/yaS4/2dCzuaS/oX9zyfc1kg72BndGY5nXtTXFXMe1XCqf6vYEh3aT7PlFVZuJa5+8yHu2rTuTofcbzgq1O3bQA8y1+vPivsOV3K6t4zG6bx99KIZmu6J2Tm8NLY/geI31Jcj3/Hy5Tn0tk1s/bw4Rh2XdMZtRTyD5U0h2ekgK+5ML/PxkkeIpcKJ3sux5cBcZoGL4C/XCV/SMNYWmn+j5rk3XF6Vdb11UPmeNdb8s4u4R8j7tV0fY1zbHjfRpaLd0ZirreE4ast9Uny5zbk+fAVYhbaJ3ObLKOnvhLDKvST1tDolLncBuahDzPF6OVx3f5cjqTUmIE0B+DrsN5OIZcz28PGdWvxGoe0+x+vQc0X1AVw8dPdG8R8g6D53doR6jTvjq6elw4V0F5+8eVQXnbvro4Zy+Mv786fcKp9cxceQ86hQjIa1oRcRV5+gIMlI/LSY4c+Yn62bppKzw5m1LCnnla7yO7keO3i22Lrcb7fm3b4uDs71m++dm6/XTxsEz/WbunvhWPBJPyE/74oAq85j8aom/xN/iX/HfzsHOYMffCRXr3Tta5htR+uy8+x+w2xHE</latexit>

Cov(Y, Z)=
Cov(T, Z)

What if T and Z are continuous?
<latexit sha1_base64="hm9P+ROKGa+XIlenBfnjjn0ePYA=">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</latexit>

Z

U

T

Y

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">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</latexit>

Brady Neal

Warm-Up: Linear Setting

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Continuous Linear Setting<latexit sha1_base64="hm9P+ROKGa+XIlenBfnjjn0ePYA=">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</latexit>

Z

U

T

Y

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">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</latexit>

Brady Neal

Warm-Up: Linear Setting

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Continuous Linear Setting<latexit sha1_base64="hm9P+ROKGa+XIlenBfnjjn0ePYA=">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</latexit>

Z

U

T

Y

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">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</latexit>

Brady Neal

Warm-Up: Linear Setting

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Continuous Linear Setting
Cov(Y, Z) = E[Y Z]<latexit sha1_base64="BGCvaNfK+MP2Ge6hHXBQZJcFZPM=">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</latexit>

E[Y ]E[Z]= E[( T + ↵u U )Z]

E[ T + ↵u U ]E[Z]= E[T Z] + ↵u E[U Z]

E[T ]E[Z]= (E[T Z]

E[T ]E[Z]) + ↵u (E[U Z]↵u E[U ]E[Z]E[U ]E[Z])= Cov(T, Z) + ↵u Cov(U, Z)= Cov(T, Z)<latexit sha1_base64="hm9P+ROKGa+XIlenBfnjjn0ePYA=">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</latexit>

Z

U

T

Y

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">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</latexit>

Brady Neal

Warm-Up: Linear Setting

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Continuous Linear Setting
Cov(Y, Z) = E[Y Z] E[Y ]E[Z]= E[( T + ↵u U )Z] E[ T + ↵u U ]E[Z]E[T ]E[Z] ↵u E[U ]E[Z]= E[T Z] + ↵u E[U Z]= (E[T Z] E[T ]E[Z]) + ↵u (E[U Z] E[U ]E[Z])= Cov(T, Z) + ↵u Cov(U, Z)= Cov(T, Z)<latexit 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Z

U

T

Y

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">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</latexit>

Brady Neal

Warm-Up: Linear Setting

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Continuous Linear Setting
Cov(Y, Z) = E[Y Z] E[Y ]E[Z](exclusion restriction and linear outcome assumptions)= E[( T + ↵u U )Z] E[ T + ↵u U ]E[Z]E[T ]E[Z] ↵u E[U ]E[Z]= E[T Z] + ↵u E[U Z]= (E[T Z] E[T ]E[Z]) + ↵u (E[U Z] E[U ]E[Z])= Cov(T, Z) + ↵u Cov(U, Z)= Cov(T, Z)<latexit 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Z

U

T

Y

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">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</latexit>

Brady Neal

Warm-Up: Linear Setting

17 / 33

Continuous Linear Setting
Cov(Y, Z) = E[Y Z] E[Y ]E[Z](exclusion restriction and linear outcome assumptions)= E[( T + ↵u U )Z] E[ T + ↵u U ]E[Z]E[T ]E[Z] ↵u E[U ]E[Z]= E[T Z] + ↵u E[U Z]= (E[T Z] E[T ]E[Z]) + ↵u (E[U Z] E[U ]E[Z])= Cov(T, Z) + ↵u Cov(U, Z)= Cov(T, Z)<latexit 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Z

U

T

Y

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">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</latexit>

Brady Neal

Warm-Up: Linear Setting

17 / 33

Continuous Linear Setting
Cov(Y, Z) = E[Y Z] E[Y ]E[Z](exclusion restriction and linear outcome assumptions)= E[( T + ↵u U )Z] E[ T + ↵u U ]E[Z]E[T ]E[Z] ↵u E[U ]E[Z]= E[T Z] + ↵u E[U Z]= (E[T Z] E[T ]E[Z]) + ↵u (E[U Z] E[U ]E[Z])= Cov(T, Z) + ↵u Cov(U, Z)= Cov(T, Z)<latexit 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Z

U

T

Y

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">AAAUiXicrVhZb9tGEN6kV+L0cFK/9YWNUqBAZVVyU9gIICCAnaAoGsABfLVRYFDkyiLEqzysOoJ+Qn9Sf0if+9r+h858uxRJHaQUWILE1ezMN7Nz7a76oevESbv99527H3z40cef3Lu/9eDTzz7/Yvvho7M4SCNLnlqBG0QXfTOWruPL08RJXHkRRtL0+q48748Oef78WkaxE/gnyU0o33rmle8MHMtMiHS5fd4L0sQKPGk86xo9W7qJafQSQkg86SfGd0bPdMOhedlL/aAfy+ha2lbgD4LUtwn12nSN5TOX2412q42XsTjo6EFD6Ndx8PD+n6InbBEIS6TCE1L4IqGxK0wR0/uN6Ii2CIn2VkyIFtHIwbwUU7FFsilxSeIwiTqi7yv69UZTffrNmDGkLdLi0iciSUN8o3lsGg9AVU/WbxR4V+mYAJttvKFnX2N6RE3EkKh1chnn+nJ9ens1q05oDQdYrUMrCUFhP1ilNQ/o6dLvhFbI3zfEKWlkk1REI4toLlEVhXVE9FSeZ98MEQkTfJJGbHWV3VVW83xA7xFhmTSOYSnbaoiXOi4+NEvYyjwuYroa8Q9aobKvCmswW4ODuKtVMO8JUUbiHY3KyFU6i9m1mivRyNMlupg/AYdP3DFRAmS+Q/5ziKOcqxZhsg4TkbzCWkJkUEsj/4z5LJtDiucu7IkRNQP55kBPqCsrtzuz06VnH9gRyU9obgg97Ism2SOBHSF7My82wR3RrzF+WbDRmqO3UJkczyatibOrSbgBzTgzLOUJ5c/MImXdhGiGfu+KX6BfUtyb8ArndZNkA1iosjiGLl/Hvkt9hf3i0TdT2bMZpQkdHuwYIltGRMv1KSTW9z2tIkb9uBp1Qk8XXguB0oR29s8YYw9r4oim0D2msQ09LG1Qr2uJfW3HtKSVfeOgOy1qNZB/XLuqM85rZTkflWrorOxCU1s81SvmCLDdRa9ZuhuqnhxjddM5q/qwP+8anKNl+1gqj085VjaycQipMi57cfk6l61lD3XSRCzZ91fE0YXUQNNj1GaoNW0V6u9QV0mAeFh4ejonVL2ztvHczITonvaUS3yqznlP4pnfCdNETvXgveJ8FeILLZ/tBtzxJ6BWy52hssqyEY0ms5lqeQcjCf+oEUdc4YXwdYi1fD37GIWZdbFvB89fQOSMTTTSuug2cmP5yk9Eg3pBg7Kq7E+ONFMinE5CGj+ZcT4hXOYdQYtPFK7wyWx+WhuDI6JMtccGmMvqMLfBQeezwdnTmac4VEUlcxxc1dVaOS+X5Y2iV8vas3wvyyp6tSxTrpGbju4HMbAuauQSxIElvEKWntRIBdgTFF3Z/Osa9vnYa1OdF5G28LxG0kOPUasKdFW+qrUvQReMtE9Y5rf38OA1+t8UJ6BN/ZjLJht5M5e7eS+f5vLjDT2bS3ob+jeXfFcjKbE3ODMay7yurSnmOq7lUvlUtydI2k2y84uqtia+++RF3rNt3ZkMvd9wVqjdsYse0FyrPy/uO1zJndo6TtF9++hDETTbFbVzemtoeQTTNdYXI9/z++U69bVMbj5v8jNEdvaPC7uLA+z8XjDBGW6s96HsNLeL+KkI/ESd7BWdkHh8KH6g01mHvl+W9qx1UfmWlOpuV8TdI+R92m2OsCttjhvqu8xu6cZT1vGCNGSfqb4b5tyuPsGvQtpE73JklTP2wk1iXpM6Kw6JS92fbmoQ8yxcjlcd3+XI6n+OAOf3/BZ1G8jFG+J6eHnOrL7P1/3LsPruG9JzQN8u+nGseY+QdS76sqQOoe7n6uxzuPBPA+fvHlUF5+7k8aM5fWX8+bvrFe6eKXHkPOoOkkBa0YqIq27BIWQSfdaLcWPM78Wt0j1X4c3bFhfyytN4Xd2LpO71W5fbjc78f2eLg7O9VufHVvv108bzA/2/2j3xlXgsviU/7YvnVJnH5FdL/CX+Ef+K/3Ye7HR2DnaeKda7d7TMl6L02jn8H3RGCZw=</latexit>

Brady Neal

Warm-Up: Linear Setting

17 / 33

Continuous Linear Setting
Cov(Y, Z) = E[Y Z] E[Y ]E[Z](exclusion restriction and linear outcome assumptions)= E[( T + ↵u U )Z] E[ T + ↵u U ]E[Z]E[T ]E[Z] ↵u E[U ]E[Z]= E[T Z] + ↵u E[U Z]= (E[T Z] E[T ]E[Z]) + ↵u (E[U Z] E[U ]E[Z])= Cov(T, Z) + ↵u Cov(U, Z)= Cov(T, Z)<latexit 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Z

U

T

Y

Y := T + ↵u U
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Brady Neal

Warm-Up: Linear Setting

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Continuous Linear Setting
Cov(Y, Z) = E[Y Z] E[Y ]E[Z](exclusion restriction and linear outcome assumptions)= E[( T + ↵u U )Z] E[ T + ↵u U ]E[Z]E[T ]E[Z] ↵u E[U ]E[Z]= E[T Z] + ↵u E[U Z]= (E[T Z] E[T ]E[Z]) + ↵u (E[U Z] E[U ]E[Z])= Cov(T, Z) + ↵u Cov(U, Z)= Cov(T, Z) (instrumental unconfoundedness assumption)<latexit 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Z

U

T

Y

Y := T + ↵u U
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Brady Neal

Warm-Up: Linear Setting

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Continuous Linear Setting
Cov(Y, Z) = E[Y Z] E[Y ]E[Z](exclusion restriction and linear outcome assumptions)= E[( T + ↵u U )Z] E[ T + ↵u U ]E[Z]E[T ]E[Z] ↵u E[U ]E[Z]= E[T Z] + ↵u E[U Z]= (E[T Z] E[T ]E[Z]) + ↵u (E[U Z] E[U ]E[Z])= Cov(T, Z) + ↵u Cov(U, Z)= Cov(T, Z) (instrumental unconfoundedness assumption)<latexit 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Z

U<latexit sha1_base64="+eCRfgBW3aclHm3N1hl5+O3MQK8=">AAAUvnicrVhZb9tGEN6kV+JeTuq3vrBRCqQArUpuCudFgAE7QRE0gAP4aqPAkEjKIsSr5MqKQ+gvFf01BfrU/pTOfLsUSR2kFFiCyNXszDezcy2X/chzE9lq/XPn7kcff/LpZ/fub33+xZdffb394OFZEo5jyzm1Qi+ML/q9xPHcwDmVrvSciyh2en7fc877o0OeP7924sQNgxN5Ezlv/d5V4A5cqyeJdLn9sms7nuwZHaM7iHtW2j0Mr590w7G0Qt8xja4bJDIe+04gf5jqSUn4kilz05fbjVazhY+xOGjrQUPoz3H44P6foitsEQpLjIUvHBEISWNP9ERC3zeiLVoiItpbkRItppGLeUdMxRbJjonLIY4eUUd0vaJ/bzQ1oP+MmUDaIi0e/WKSNMT3msem8QBUdWf9RoF3lY4U2GzjDd37GtMnqhRDotbJZZzry/Xp69esWtIanmG1Lq0kAoX9YJXWPKC7R/8lrZCvN8Tp0MgmqZhGFtE8oioK64jprjzPvhkiEj3wOTRiq6vsrrKa50P6jgirR+MElrKthnih4xJAswNbmcdDTFcjvqMVKvuqsAazNbiIu1oF854QZSTe06iMXKWzmF2ruaRGni7RxfwSHAFxJ0QJkfku+c8ljnKuWoTJOnqI5BXWEiGDmhr5JeazbI4onruwJ0HUDOSbCz2Rrqzc7sxOj+59YMckn9LcEHrYFybZ4wA7RvZmXjTBHdO/Cf5ZsNGaozdRmRxPk9bE2WUSbkgz7gxLeUL5M7NIWZcSzdDfXfEr9DsUdxNe4bw2STaEhSqLE+gKdOw71FfYLz5dmcqezSgmdPiwY4hsGREt16eQWN+PtIoE9eNp1JTuHrwWAcWEdvbPBGMfa+KIjqF7QmMbeljaoF7XFPvajmlJK/vGRXda1Gog/7h2VWec18pyASrV0FnZgaaWeKpXzBFgu4tes3Q3VD05weqmc1b1YX/eNThHy/axVB6fcqxsZOMQUmVc9uLydS5byx7qxEQs2fdXxNGB1EDTE9RmpDVtFervUFdJiHhYuPs6J1S9s7bJ3ExKdF97yiM+Vee8J/HMH4TZQ0514b3ifBXicy2f7Qbc8VNQq+XOUFll2ZhG6WymWt7FyIF/1IgjrvAi+DrCWr6b/YzCzLrYt4MXLCByxkqNtC66jdxYvvIT0aBe0KCsKvuTI82UGE8nEY0fzzgfEy7zjqAlIApXeDqbn9bG4IgoU+2xAeayOsxtcNH5bHB2deYpDlVRco6Dq7paK+flsrxR9GpZe5bvZVlFr5ZlyjVy09X9IAHWRY2cRBxYwi9k6UmNVIg9QdGVzb+tYV+AvXas8yLWFp7XSProMWpVoa7KV7X2SXTBWPuEZX7/AA9eo/9N8QS0qR9zWbmRN3O5mw/yaS4/2dCzuaS/oX9zyfc1kg72BndGY5nXtTXFXMe1XCqf6vYEh3aT7PlFVZuJa5+8yHu2rTuTofcbzgq1O3bQA8y1+vPivsOV3K6t4zG6bx99KIZmu6J2Tm8NLY/geI31Jcj3/Hy5Tn0tk1s/bw4Rh2XdMZtRTyD5U0h2ekgK+5ML/PxkkeIpcKJ3sux5cBcZoGL4C/XCV/SMNYWmn+j5rk3XF6Vdb11UPmeNdb8s4u4R8j7tV0fY1zbHjfRpaLd0ZirreE4ast9Uny5zbk+fAVYhbaJ3ObLKOnvhLDKvST1tDolLncBuahDzPF6OVx3f5cjqTUmIE0B+DrsN5OIZcz28PGdWvxGoe0+x+vQc0X1AVw8dPdG8R8g6D53doR6jTvjq6elw4V0F5+8eVQXnbvro4Zy+Mv786fcKp9cxceQ86hQjIa1oRcRV5+gIMlI/LSY4c+Yn62bppKzw5m1LCnnla7yO7keO3i22Lrcb7fm3b4uDs71m++dm6/XTxsEz/WbunvhWPBJPyE/74oAq85j8aom/xN/iX/HfzsHOYMffCRXr3Tta5htR+uy8+x+w2xHE</latexit>

Cov(Y, Z)=
Cov(T, Z)TY

Y := T + ↵u U
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Brady Neal

Warm-Up: Linear Setting

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Continuous Linear Setting
Cov(Y, Z) = E[Y Z] E[Y ]E[Z](exclusion restriction and linear outcome assumptions)= E[( T + ↵u U )Z] E[ T + ↵u U ]E[Z]E[T ]E[Z] ↵u E[U ]E[Z]= E[T Z] + ↵u E[U Z]= (E[T Z] E[T ]E[Z]) + ↵u (E[U Z] E[U ]E[Z])= Cov(T, Z) + ↵u Cov(U, Z)= Cov(T, Z) (instrumental unconfoundedness assumption)<latexit 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Z

U<latexit sha1_base64="+eCRfgBW3aclHm3N1hl5+O3MQK8=">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</latexit>

Cov(Y, Z)=
Cov(T, Z)T(non-zero due to relevance assumption)
Brady Neal

Warm-Up: Linear SettingY

Y := T + ↵u U
<latexit sha1_base64="HWMiX08wbFAVaft4aHfc0jXX83w=">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</latexit>

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Continuous Linear Setting: Estimator 1<latexit sha1_base64="oEV3IjO/P2oPp2MoXoeqsf1CQ7s=">AAAUvnicrVhZb9tGEN6kV+IecVK/9YWNUiAFaFVyU7gvAgzYCYqgARzAVxsFhkSuLEK8SlJWHEJ/qeivKdCn9qd05tulSOogpcASRK5mZ76ZnWu57IeuEyet1j937n708Seffnbv/tbnX3z51YPth4/O4mAcWfLUCtwguuj3Yuk6vjxNnMSVF2Eke17flef90SHPn1/LKHYC/yS5CeVbr3flOwPH6iVEutx+2bWlm/SMjtEdRD0r7R4G10+7wTixAk+aRtfx4yQae9JPvp/qyYTwE6bMTV9uN1rNFj7G4qCtBw2hP8fBw/t/iq6wRSAsMRaekMIXCY1d0RMxfd+ItmiJkGhvRUq0iEYO5qWYii2SHROXJI4eUUd0vaJ/bzTVp/+MGUPaIi0u/SKSNMR3msem8QBUdWf9RoF3lY4U2GzjDd37GtMjaiKGRK2TyzjXl+vT16tZdUJr+BmrdWglISjsB6u05gHdXfqf0Ar5ekOckkY2SUU0sojmElVRWEdEd+V59s0QkeiBT9KIra6yu8pqng/oOyKsHo1jWMq2GuKFjosPzRK2Mo+LmK5GfEcrVPZVYQ1ma3AQd7UK5j0hyki8p1EZuUpnMbtWcyUaebpEF/Mn4PCJOyZKgMx3yH8OcZRz1SJM1tFDJK+wlhAZ1NTILzGfZXNI8dyFPTGiZiDfHOgJdWXldmd2unTvAzsi+ZTmhtDDvjDJHgnsCNmbedEEd0T/JvhnwUZrjt5EZXI8TVoTZ5dJuAHNODMs5Qnlz8wiZV1KNEN/d8Wv0C8p7ia8wnltkmwAC1UWx9Dl69h3qK+wXzy6MpU9m1FM6PBgxxDZMiJark8hsb4faBUx6sfVqCndXXgtBIoJ7eyfCcYe1sQRHUP3hMY29LC0Qb2uKfa1HdOSVvaNg+60qNVA/nHtqs44r5XlfFSqobOyA00t8UyvmCPAdhe9ZuluqHpyjNVN56zqw/68a3COlu1jqTw+5VjZyMYhpMq47MXl61y2lj3UiYlYsu+viKMDqYGmx6jNUGvaKtTfoa6SAPGwcPd0Tqh6Z22TuZmU6J72lEt8qs55T+KZPwizh5zqwnvF+SrE51o+2w2446egVsudobLKshGN0tlMtbyDkYR/1IgjrvBC+DrEWr6d/YzCzLrYt4PnLyByxiYaaV10G7mxfOUnokG9oEFZVfYnR5opEZ5OQho/mXE+IVzmHUGLTxSu8HQ2P62NwRFRptpjA8xldZjb4KDz2eDs6sxTHKqikjkOrupqrZyXy/JG0atl7Vm+l2UVvVqWKdfITUf3gxhYFzVyCeLAEl4hS09qpALsCYqubP5tDft87LVjnReRtvC8RtJDj1GrCnRVvqq1L0EXjLRPWOb3D/DgNfrfFE9Am/oxl0028mYud/NBPs3lJxt6Npf0NvRvLvm+RlJib3BmNJZ5XVtTzHVcy6XyqW5PkLSbZM8vqtpMXPvkRd6zbd2ZDL3fcFao3bGDHmCu1Z8X9x2u5HZtHY/RffvoQxE02xW1c3praHkEx2usL0a+5+fLdeprmdz6eXOIOCzrjtmMegLJn0Ky00Nc2J8c4OcnixRPgRO9k2XPg7vIABXDX6gXvqJnrCk0/UjPd226vijteuui8jlrrPtlEXePkPdpvzrCvrY5bqhPQ7ulM1NZx3PSkP2m+nSZc7v6DLAKaRO9y5FV1tkLZ5F5Teppc0hc6gR2U4OY5/FyvOr4LkdWb0oCnADyc9htIBfPmOvh5Tmz+o1A3XuK1afnkO4Durro6LHmPULWuejsknqMOuGrp6fDhXcVnL97VBWcu+njR3P6yvjzp98rnF7HxJHzqFNMAmlFKyKuOkeHkEn002KMM2d+sm6WTsoKb962uJBXnsbr6H4k9W6xdbndaM+/fVscnO012z81W6+fNQ4O9Ju5e+Ib8Vg8JT/tiwOqzGPyqyX+En+Lf8V/Owc7gx1vJ1Csd+9oma9F6bPz7n+zQxHM</latexit>

Cov(Y, Z)=
Cov(T, Z)

Brady Neal

Warm-Up: Linear Setting

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Continuous Linear Setting: Estimator 1<latexit sha1_base64="1UfPzxDyErL8XNDr0YdqGu72nvA=">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</latexit>d
Cov(Y,Z)ˆ=d
Cov(T,
Z)

Brady Neal

Warm-Up: Linear Setting

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Two-Stage Least Squares Estimator<latexit sha1_base64="uSnWmzenLZoB/622kgJqyZD0I1U=">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</latexit>

Z

U

T

Brady Neal

Warm-Up: Linear SettingY

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Two-Stage Least Squares Estimator
1. Linearly regress T on Z to estimate E[T | Z]. This gives us the
projection of T onto Z: T̂<latexit 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Z

U

T

Brady Neal

Warm-Up: Linear SettingY

19 / 33

Two-Stage Least Squares Estimator
1. Linearly regress T on Z to estimate E[T | Z]. This gives us the
projection of T onto Z: T̂
2. Linearly regress Y on T̂ to estimate E[Y | T̂ ] . Obtain our estimate ˆ as
the fitted coefficient in front of T̂ .
<latexit 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Z

U

T

Brady Neal

Warm-Up: Linear SettingY

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Two-Stage Least Squares Estimator
1. Linearly regress T on Z to estimate E[T | Z]. This gives us the
projection of T onto Z: T̂
2. Linearly regress Y on T̂ to estimate E[Y | T̂ ] . Obtain our estimate ˆ as
the fitted coefficient in front of T̂ .
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Z

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T̂ = Ê[T | Z]
Brady Neal

T

Warm-Up: Linear SettingY

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Two-Stage Least Squares Estimator
1. Linearly regress T on Z to estimate E[T | Z]. This gives us the
projection of T onto Z: T̂
2. Linearly regress Y on T̂ to estimate E[Y | T̂ ] . Obtain our estimate ˆ as
the fitted coefficient in front of T̂ .
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Z

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T̂ = Ê[T | Z]
Brady NealT̂
Warm-Up: Linear SettingY

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Two-Stage Least Squares Estimator
1. Linearly regress T on Z to estimate E[T | Z]. This gives us the
projection of T onto Z: T̂
2. Linearly regress Y on T̂ to estimate E[Y | T̂ ] . Obtain our estimate ˆ as
the fitted coefficient in front of T̂ .
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Z

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T̂ = Ê[T | Z]
Brady NealT̂
Warm-Up: Linear SettingY

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Two-Stage Least Squares Estimator
1. Linearly regress T on Z to estimate E[T | Z]. This gives us the
projection of T onto Z: T̂
2. Linearly regress Y on T̂ to estimate E[Y | T̂ ] . Obtain our estimate ˆ as
the fitted coefficient in front of T̂ .
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Z

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T̂ = Ê[T | Z]
Brady NealT̂
Warm-Up: Linear SettingY

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Two-Stage Least Squares Estimator
1. Linearly regress T on Z to estimate E[T | Z]. This gives us the
projection of T onto Z: T̂
2. Linearly regress Y on T̂ to estimate E[Y | T̂ ] . Obtain our estimate ˆ as
the fitted coefficient in front of T̂ .
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Also works as an
estimator in thebinary setting<latexit sha1_base64="MITRZaN0AvA2gq0L++sFNIgCr/E=">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</latexit>

Z

U<latexit sha1_base64="7hJNLSbvPbbIVSt2QBaKnRXtOqY=">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</latexit>

T̂ = Ê[T | Z]
Brady NealT̂
Warm-Up: Linear SettingY

19 / 33

Question:
In the binary linear, setting where is each
assumption used in the proof below?
<latexit sha1_base64="dRKcYo4A4lEwHJrBAYMuOLPrWZA=">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</latexit>

E[Y | Z = 1]

E[Y | Z = 0]

= E[ T + ↵u U | Z = 1]

E[ T + ↵u U | Z = 0]

= (E[T | Z = 1]

E[T | Z = 0]) + ↵u (E[U | Z = 1]

= (E[T | Z = 1]

E[T | Z = 0])= (E[T | Z = 1]
<latexit sha1_base64="8qrl2ZQIvUvYlZZfeqMm9sVinJ8=">AAAU0nicrVhLb9tGEF6lbZq4Lyf1rRc2SoEeaFVynboXAwFsB0XRAA7gV2sZBkVSFiG+SlJWbUKHotf2P/Rv9d905tulSOpBSoElSFzNznwzO6/dVS90nThpt/9rPPrgw48ef/zk6cYnn372+Rebz56fxcEoMu1TM3CD6KJnxLbr+PZp4iSufRFGtuH1XPu8Nzzg+fNbO4qdwD9J7kL7yjNufKfvmEZCpOvNf7uW7SaGtq91+5Fhpt2jy24wSszAs7Wu51ha1/HjJBp5tp8QU+dK29ZqeNpXE8AkZEcCWhVQJRdBXW822602Xtr8oKMGTaFex8Gzp/+IrrBEIEwxEp6whS8SGrvCEDG9L0VHtEVItCuREi2ikYN5W0zEBsmOiMsmDoOoQ/q+oV+XiurTb8aMIW2SFpc+EUlq4hvFY9G4D6p8sn6twLtMRwpstvGOnj2F6RE1EQOi1sllnKvL9ejt1aw6oTX8iNU6tJIQFPaDWVpzn54u/U5ohfx9R5w2jSySimhkEs0lqqSwjoie0vPsmwEiYYDPphFbXWV3ldU8H9B7SFgGjWNYyrZq4o2Kiw/NNmxlHhcxXY74B61Q2leF1Z+uwUHc5SqY94QoQ3FPozJylc5idi3nShTyZIEu5k/A4RN3TJQAme+Q/xziKOeqSZisw0Akb7CWEBnUUsg/Yz7L5pDiuQ17YkRNQ7450BOqysrtzux06dkDdkTyKc0NoId9oZM9NrAjZG/mRR3cEf0a45cJG80ZeguVyfHUaU2cXTrhBjTjTLGkJ6Q/M4ukdSnRNPXeFr9Av01x1+EVzmudZANYKLM4hi5fxX6f+gr7xaNvprJnM4oOHR7sGCBbhkTL9Ukk1vcdrSJG/bgKNaWnC6+FQNGhnf0zxtjDmjiiI+ge09iCHpbWqNe1xJ6yY1LSyr5x0J3mtWrIP65d2RlntbKcj0rVVFbuQ1Nb7KoVcwTY7qLXTNUNZU+OsbrJjFU92J93Dc7Rsn0slcenHCsL2TiAVBmXvbh4nYvWsoM60RFL9v0NcexDqq/oMWozVJo2CvV3oKokQDxMPD2VE7LeWdt4ZiYluqc85RKfrHPek3jmd8I0kFNdeK84X4V4pOSz3YA7fgpqtdwZKqssG9Eonc5UyzsY2fCPHHHEJV4IX4dYy9fTj1aYWRX7YfD8OUTO2EQhrYpuITcWr/xENKkXNCmryv7kSDMlwukkpPHLKedLwmXeIbT4ROEKT6fzk9oYHBJlojzWx1xWh7kNDjqfBc6uyjzJISsqmeHgqq7Wynm5KG8kvVrWmuZ7WVbSq2WZcovcdFQ/iIF1USOXIA4s4RWy9KRGKsCeIOnS5l9XsM/HXjtSeREpC89rJD30GLmqQFXl21r7EnTBSPmEZX57Dw/eov9NcAJa14+5bLKWN3O5u/fyaS4/XtOzuaS3pn9zyfsaSRt7gzOlscy72ppiruNaLplPdXuCTbtJdn6R1abju0de5D3bUp1JU/sNZ4XcHffRA/SV+vP8vsOV3Kmt4xG6bw99KIJmq6J2Th8MLY/gaIX1xcj3/H65Sn0tkpvNm/wMkZ3948Lu4gA7vxekOMON1T6Unea2ET8ZgZ+ok72lExKPD8T3dDrr0Peb0p61Kirfkkaq2xVxdwh5j3abQ+xK6+OG6i6zXbrxlHUckYbsM1F3w5zbVSf4ZUjr6F2MLHPGmrtJzGqSZ8UBccn7010NYp6Fi/Gq47sYWf7PEeD8nt+iHgK5eENcDS/PmeX3+bp/GZbffUN69unbRT+OFe8hss5FX7apQ8j7uTz7HMz908D5u0NVwbmbvng+o6+MP3t3vcHdc0QcOY+8gySQlrQi4rJbcAiZRJ31YtwY83txq3TPlXiztsWFvPIU3r7qRbbq9RvXm83O7H9n84OznVbnVav9brf5elf9r/ZEfCVeiG/JT3viNVXmMfnVbDxu6I1XjR+2Trbut/7c+kuyPmoomS9F6bX19//FEx+f</latexit>=

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

E[T | Z = 0]) + ↵u (E[U ]E[Y | Z = 0]
E[T | Z = 0]

E[U ])

E[U | Z = 0])

Question:
In the continuous linear setting, prove thefollowing:<latexit sha1_base64="WEcbnDFOAVNR4xJgWQS71CyS6k0=">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</latexit>

Cov(Y, Z)=
Cov(T, Z)

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

Nonparametric Identification of Local ATE

22 / 33

Linear Outcome Assumption as Homogeneity
Linear outcome assumption: Y := T + ↵u U
<latexit sha1_base64="UeVu8hQ8nYOpqwWbbqcmxjVPKxk=">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</latexit>

Brady Neal

Nonparametric Identification of Local ATE

23 / 33

Linear Outcome Assumption as Homogeneity
Linear outcome assumption: Y := T + ↵u U
<latexit sha1_base64="UeVu8hQ8nYOpqwWbbqcmxjVPKxk=">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</latexit>

There are other variants of the linear outcome assumption that all require
the treatment effect to be homogeneous (the same for all units) in someway (see, e.g., Section 16.3 of Hernán & Robins (2020))

Brady Neal

Nonparametric Identification of Local ATE

23 / 33

Linear Outcome Assumption as Homogeneity
Linear outcome assumption: Y := T + ↵u U
<latexit sha1_base64="UeVu8hQ8nYOpqwWbbqcmxjVPKxk=">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</latexit>

There are other variants of the linear outcome assumption that all require
the treatment effect to be homogeneous (the same for all units) in someway (see, e.g., Section 16.3 of Hernán & Robins (2020))
Very restricting!

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Nonparametric Identification of Local ATE

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Can we get identification withoutparametric assumptions?

Potential Outcomes Notation with Instruments<latexit sha1_base64="uSnWmzenLZoB/622kgJqyZD0I1U=">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</latexit>

Z

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Y

Nonparametric Identification of Local ATE

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Potential Outcomes Notation with Instruments<latexit sha1_base64="M63Vb4/ai+tR0Jrq/f++NR10cms=">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</latexit>

Y (1) and Y (0) are short for Y (T = 1) and Y (T = 0)<latexit sha1_base64="uSnWmzenLZoB/622kgJqyZD0I1U=">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</latexit>

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Nonparametric Identification of Local ATE

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Potential Outcomes Notation with Instruments<latexit sha1_base64="M63Vb4/ai+tR0Jrq/f++NR10cms=">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</latexit>

Y (1) and Y (0) are short for Y (T = 1) and Y (T = 0)<latexit sha1_base64="d6IaN9EZSd+xwhLp8Bud/kYpVUg=">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</latexit>

We now have T (Z = 1) and T (Z = 0) or T (1) and T (0) for short<latexit sha1_base64="uSnWmzenLZoB/622kgJqyZD0I1U=">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</latexit>

Z

U

T
Brady Neal

Y

Nonparametric Identification of Local ATE

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Potential Outcomes Notation with Instruments<latexit sha1_base64="M63Vb4/ai+tR0Jrq/f++NR10cms=">AAAUrHicrVjbbttGEF2nbZq4lzip3/rCRimQALIiuSnchwoIYCcoigZwAN/ayDAkcmUR4q3kyo4t6BP6gX3rp3Tm7FIkdSGlwBIkrmZnzszObXfVizw3Uc3mvxv3Pvv8i/tfPni4+dXX33z7aOvxk5MkHMW2PLZDL4zPet1Eem4gj5WrPHkWxbLr9zx52hvu8/zplYwTNwyO1E0kz/3uZeD2XburiHSxddsJR8oOffm89cLqKPlRja1u4FgTazrRzCZiaSWDMFZWP4zzLB1FOpUvA2W1raVABa7mi4utWrPRxMuaH7TMoCbM6zB8/PAf0RGOCIUtRsIXUgRC0dgTXZHQ+4NoiaaIiHYuxkSLaeRiXoqJ2CTZEXFJ4ugSdUjfl/Trg6EG9JsxE0jbpMWjT0ySlvjR8Dg07oOqn6zfyvEu0zEGNtt4Q8+ewfSJqsSAqFVyKefqcj16+xWrVrSGX7Bal1YSgcJ+sAtr7tPTo9+KVsjfN8QpaeSQVEwjm2geUTWFdcT01J5n3wwQiS74JI3Y6jK7y6zm+ZDeQ8Lq0jiBpWyrJd6auATQLGEr83iI6XLEj7RCbV8ZVn+6Bhdx16tg3iOiDMUtjYrIZTrz2bWcSxnkyQJdzK/AERB3QpQQme+S/1ziKOaqTZiso4tIXmItETKoYZB/x3yazRHFcwf2JIiahXxzoScylZXZndrp0bMH7JjkxzQ3gB72RZ3skcCOkb2pF+vgjunXNX7ZsNGeoTdQmRzPOq2Js6tOuCHNuFMs7Qntz9Qibd2YaJZ574g/oF9S3OvwCud1nWRDWKizOIGuwMS+TX2F/eLTN1PZsymlDh0+7BggW4ZEy/RpJNb3klaRoH48gzqmpwevRUCpQzv75xpjH2viiI6g+5rGDvSwtEW9riH2jB2Tglb2jYvuNK/VQv5x7erOOKuV5QJUqmWysg1NTfHKrJgjwHbnvWabbqh7coLVTWas6sH+rGtwjhbtY6ksPsVYOcjGAaSKuOzFxetctJZd1EkdsWTfXxJHG1J9Q09Qm5HRtJmrv31TJSHiYePpm5zQ9c7armdmxkT3jac84tN1znsSz/xNmF3kVAfey8+XIb4x8uluwB1/DGq53Akqqygb02g8nSmXdzGS8I8eccQ1XgRfR1jLD9OPlZtZFftu8II5RM5YZZBWRXeQG4tXfiRq1AtqlFVFf3KkmRLjdBLR+NmU8xnhMu8QWgKicIWPp/OTyhgcEGViPNbHXFqHmQ0uOp8Dzo7JPM2hK0rNcHBVl2vlvFyUN5peLutM870oq+nlsky5Qm66ph8kwDqrkFOIA0v4uSw9qpAKsSdourb5zxXsC7DXjkxexMbC0wpJHz1Gryo0Vfmu0j6FLhgbn7DMX5/gwSv0vwlOQOv6MZNVa3kzk7v5JJ9m8tdrejaT9Nf0byZ5WyEpsTe4UxrLvK+sKeY6rOTS+VS1J0jaTdLzi662Or575EXesx3TmSyz33BW6N2xjR5QX6k/z+87XMmtyjoeofv20IdiaHZKauf4ztCyCI5WWF+CfM/ul6vU1yK52bzJzhDp2T/J7S4usLN7wRhnuGuzD6WnuR3ET0fgN+pk7+iExON98ROdzlr0/bawZ62Kyrekkel2edxdQt6j3eYAu9L6uJG5y+wUbjxFHW9IQ/qZmLthxu2ZE/wypHX0LkbWOePM3SRmNemz4oC49P3ppgIxy8LFeOXxXYys/+cIcX7PblF3gZy/Ia6Gl+XM8vt81b8My+++ET379O2hHyeG9wBZ56EvS+oQ+n6uzz77c/80cP7uUlVw7o6fPpnRV8Sfvbte4u45Io6MR99BFKQ1LY+47BYcQUaZs16CG2N2L24U7rkab9a2JJdXvsFrm14kTa/fvNiqtWb/O5sfnOw2Wj83mu9f1V7/av5XeyC+F0/Fc/LTnnhNlXlIfrXFfxv3Nx5tbG03to+2P2yfa9Z7G0bmO1F4bff/B7YxEQ8=</latexit>

Y (1) and Y (0) are short for Y (T = 1) and Y (T = 0)
We also have Y (Z = 1) and Y (Z = 0)<latexit sha1_base64="def7f/hUKSNd3e3nkF4vCLcuX5A=">AAAUh3icrVjbbttGEN2kt9i9Oanf+sLGKZAAsiK5aZ2HCkhhJyiKBnAA39ooMChyZRHirSRlxSH0Cf2ifkkf+9r+RGfOLkVSF1IKLEHianbmzOzcdle90HXipNX6+9btDz786ONP7mxsfvrZ5198uXX33mkcjCJLnliBG0TnPTOWruPLk8RJXHkeRtL0eq486w0PeP7sSkaxE/jHyXUo33jmpe/0HctMiHSxddJN5NskPZOG6caBMTCvpDExusEosQJPPuw6fpxEI0/6idEx2o8MxW6Yvr2crfXoYmun1WzhZcwP2nqwI/TrKLi78afoClsEwhIj4QkpfJHQ2BWmiOn9WrRFS4REeyNSokU0cjAvxURskuyIuCRxmEQd0vcl/XqtqT79ZswY0hZpcekTkaQhvtU8No37oKon6zcKvMt0pMBmG6/p2dOYHlETMSBqnVzGubpcj95ezaoTWsNTrNahlYSgsB+s0pr79HTpd0Ir5O9r4pQ0skkqopFFNJeoisI6Inoqz7NvBoiECT5JI7a6yu4qq3k+oPeQsEwax7CUbTXECx0XH5olbGUeFzFdjviWVqjsq8LqT9fgIO5qFcx7TJSheEejMnKVzmJ2LedKNPJkgS7mT8DhE3dMlACZ75D/HOIo56pFmKzDRCQvsZYQGdTUyL9gPsvmkOK5C3tiRM1AvjnQE+rKyu3O7HTp2QN2RPIpzQ2gh33RIHsksCNkb+bFBrgj+jXGLws2WjP0JiqT49mgNXF2NQg3oBlniqU8ofyZWaSsS4lm6Peu+BX6JcW9Aa9wXjdINoCFKotj6PJ17DvUV9gvHn0zlT2bURrQ4cGOAbJlSLRcn0JifY9pFTHqx9WoKT1deC0ESgPa2T9jjD2siSM6gu4xjW3oYWmDel1T7Gs7JiWt7BsH3Wleq4H849pVnXFWK8v5qFRDZ2UHmlriiV4xR4DtLnrN0t1Q9eQYq5vMWNWD/XnX4Bwt28dSeXzKsbKRjQNIlXHZi4vXuWgte6iTBmLJvr8kjg6k+poeozZDrWmzUH8HukoCxMPC09M5oeqdtY1nZlKie9pTLvGpOuc9iWf+IEwTOdWF94rzVYjPtXy2G3DHT0GtljtFZZVlIxql05lqeQcjCf+oEUdc4YXwdYi1fDP9GIWZVbFvBs+fQ+SMTTTSqug2cmPxyo/FDvWCHcqqsj850kyJcDoJafxgyvmAcJl3CC0+UbjC0+n8pDYGh0SZaI/1MZfVYW6Dg85ng7OrM09xqIpKZji4qqu1cl4uyhtFr5a1p/lellX0almmXCE3Hd0PYmCd18gliANLeIUsPa6RCrAnKLqy+bcV7POx1450XkTawrMaSQ89Rq0q0FX5sta+BF0w0j5hmd/fw4NX6H8TnIDW9WMum6zlzVzu+r18msuP1/RsLumt6d9c8l2NpMTe4ExpLPOqtqaY66iWS+VT3Z4gaTfJzi+q2hr47pEXec+2dWcy9H7DWaF2xw56QGOl/jy/73Alt2vreITu20MfiqDZrqidkxtDyyM4WmF9MfI9v1+uUl+L5GbzJj9DZGf/uLC7OMDO7wUpznBjvQ9lp7ldxE9F4GfqZC/phMTjA/Ednc7a9P2itGetisq3pJHudkXcPULep93mELvS+rihvsvslm48ZR3PSUP2mei7Yc7t6hP8MqR19C5GVjljz90kZjWps+KAuNT96boGMc/CxXjV8V2MrP7nCHB+z29RN4FcvCGuhpfnzPL7fN2/DMvvviE9+/Ttoh/HmvcQWeeiL0vqEOp+rs4+B3P/NHD+7lFVcO6m9+/N6Cvjz95dL3H3HBFHzqPuIAmkFa2IuOwWHEIm0We9GDfG/F7cLN1zFd6sbXEhrzyN19G9SOpev3mxtdOe/e9sfnC612x/32y9erLz7Ef9v9od8bW4Lx6Sn/bFM6rMI/KrJf4S/4h/xX/bG9uPt3/YfqpYb9/SMl+J0mv7p/8BxhAGRQ==</latexit><latexit sha1_base64="d6IaN9EZSd+xwhLp8Bud/kYpVUg=">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</latexit>

We now have T (Z = 1) and T (Z = 0) or T (1) and T (0) for short<latexit sha1_base64="uSnWmzenLZoB/622kgJqyZD0I1U=">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</latexit>

Z

U

T
Brady Neal

Y

Nonparametric Identification of Local ATE

25 / 33

Principal Strata
Break data into 4 strata (groups) based on how the instrument
affects the treatment they takeZ
U<latexit sha1_base64="7Wpmtk4gGKaYaQv+qqBHDDO4V28=">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</latexit>

T

Brady Neal

Nonparametric Identification of Local ATE

Y

26 / 33

Principal Strata
Break data into 4 strata (groups) based on how the instrument
affects the treatment they takeZ
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T

Brady Neal

Nonparametric Identification of Local ATE

Y

26 / 33

Principal Strata
Break data into 4 strata (groups) based on how the instrument
affects the treatment they takeZ
U<latexit sha1_base64="7Wpmtk4gGKaYaQv+qqBHDDO4V28=">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</latexit>• Compliers:

Brady Neal<latexit sha1_base64="oeupem7J3nlOID/wbAq7V7swhcs=">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</latexit>

T (Z = 1) = 1 ,T (Z = 0) = 0

Nonparametric Identification of Local ATE

TY

26 / 33

Principal Strata
Break data into 4 strata (groups) based on how the instrument
affects the treatment they takeZ
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Brady Neal<latexit sha1_base64="oeupem7J3nlOID/wbAq7V7swhcs=">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</latexit>

T (Z = 1) = 1 ,T (Z = 0) = 0

TY<latexit sha1_base64="xD8zMCKqXrs71+Hc7nLNY0XPcuc=">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</latexit>

T (Z = 1) = 0 ,T (Z = 0) = 1

Nonparametric Identification of Local ATE

26 / 33

Principal Strata
Break data into 4 strata (groups) based on how the instrument
affects the treatment they takeZ
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T (Z = 1) = 1 , T (Z = 0) = 0• Compliers:• Defiers:
T (Z = 1) = 0 , T (Z = 0) = 1• Always-takers: T (Z = 1) = 1 , T (Z = 0) = 1<latexit sha1_base64="oeupem7J3nlOID/wbAq7V7swhcs=">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</latexit>

TY<latexit sha1_base64="xD8zMCKqXrs71+Hc7nLNY0XPcuc=">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</latexit><latexit sha1_base64="lpzpnPZaGkQ0k+6ItoP5qSHQ4oQ=">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</latexit>

Brady Neal

Nonparametric Identification of Local ATE

26 / 33

Principal Strata
Break data into 4 strata (groups) based on how the instrument
affects the treatment they takeZ
U<latexit sha1_base64="7Wpmtk4gGKaYaQv+qqBHDDO4V28=">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</latexit>• Compliers:• Defiers:• Always-takers:• Never-takers:
Brady Neal<latexit sha1_base64="oeupem7J3nlOID/wbAq7V7swhcs=">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</latexit>

T (Z = 1) = 1 ,T (Z = 0) = 0

TY<latexit sha1_base64="xD8zMCKqXrs71+Hc7nLNY0XPcuc=">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</latexit>

T (Z = 1) = 0 ,T (Z = 0) = 1<latexit sha1_base64="lpzpnPZaGkQ0k+6ItoP5qSHQ4oQ=">AAAUfHicrVjbbttGEN2kt8S9OYnf+sJGLpCitCq5ad2HCghgJyiCBnAA39ooMCSKsgjzFpKy6wj6hP5Lv6ZAn9ovKXrm7EoUdaMc2ILJ1ezMmdm57a7ase+lWa32963b773/wYcf3bm79vEnn372+fq9+0dp1E8c99CJ/Cg5abdS1/dC9zDzMt89iRO3FbR997h9vivzxxduknpReJBdxe7roHUWel3PaWUgna4/b2ZgzwI3zB41vTDNkr6MrYZV/1oeVtO2bKv5pt/qWAtZa2Q9Xa/UqjX+WbODuhlUlPnbj+7d/UM1VUdFylF9FShXhSrD2FctleLzStVVTcWgvVYD0BKMPM67aqjWINsHlwuOFqjneJ7h2ytDDfFdMFNKO9Di4z+BpKW+MjwdjLuk6rfotyZ4F+kYEFtsvMK7bTADUDPVA7VMbsS5ulwbn6Bk1RnW8CNX62ElMSniB6ew5i7ePr5nWKE8r8DpYtSBVIKRA5oPqqaIjgRv7XnxTY+RaJHPxUisXmb3MqtlPsLnHFgtjFNaKrZa6pmJS0jNLm0VHp8xXYz4O1ao7VuG1R2vwWPc9SqE9wCUc/UWoyLyMp2T2bWYKzPIwzm6hD8jRwjuFJSIme/Bfx44irnqAFN0tBjJM64lZgZVDfJzzo+yOUY8t2hPyqhZzDePemJTWbndIzt9vNvETiA/wFyPesQXNuxxiZ0we0detMmd4Nslvzm00ZmiV1mZEk8ba5LssoEbYcYbY2lPaH+OLNLWDUCzzGdL/UL9LuJu0yuS1zZkI1qoszilrtDEvoG+In4J8BSqeHZEsakjoB09Zss5aLk+jST6vsUqUtaPb1AHePv0WkwUm9rFP5ccB1yTRLRP3ZcYd6hHpC30uqraMXYMC1rFNx6706xWi/kntas747RWkQtZqZbJygY11dRjs2KJgNg96TXHdEPdk1OubjhlVZv2511DcrRon0jl8SnGqsNs7FGqiCtenL/OeWvZZp3YjKX4/gwcDUp1DT1lbcZG09pE/e2aKokYD4fvwOSErnfRdjk1MwA9MJ7ywafrXPYkmXkDzBZzqknvTc4vQ3xq5Ee7gXT8AanL5Y5YWUXZBKPBeGa5vMeRS//okURc48X0dcy1fDn+tyZmVsW+GbxwBlEyNjNIq6J3mBvzV36gKugFFWRV0Z8SaaEkPJ3EGG+OOTeBK7zn1BKCIhU+GM8PS2OwB8rQeKzLuVEd5jZ47HwdcjZN5mkOXVHZFIdU9XKtkpfz8kbTl8t2xvlelNX05bJCuWBueqYfpMQ6KZHLGAeRCCay9KBEKuKeoOna5l9XsC/kXts3eZEYC49LJAP2GL2qyFTli1L7MnbBxPhEZH57Bw9esP8NeQK6rh9z2exa3szlrt7Jp7n85TU9m0sG1/RvLvm2RNLl3uCNaSLzsrSmhGu/lEvnU9me4GI3GZ1fdLXZfLbhRdmzO6YzWWa/kazQu2ODPcBeqT/P7jtSyfXSOu6z+7bZhxJq7iypncMbQ8sj2F9hfSnzPb9frlJf8+Sm8yY/Q4zO/unE7uIRO78XDHiGuzT70Og0t8X46Qj8jE72AickGe+q73A6q+P5rLBnrYoqt6S+6XaTuNtA3sFus8dd6fq4sbnLbBVuPEUdT6Fh9D80d8Oc2zcn+EVI19E7H1nnTGfmJjGtSZ8Ve+DS96erEsQ8C+fjLY/vfGT9O0fE83t+i7oJ5Mkb4mp4ec4svs+X/cqw+O4b493F02c/Tg3vHrPOZ1920SH0/VyffXZnfmmQ/N1GVUjuDh7en9JXxJ++u57x7tkHR86j7yAZpTVtEnHRLTimTGbOeilvjPm9uFq452q8advSibwKDF7D9CLX9Pq10/VKffq3s9nB0Xa1/n219vJx5clP5ne1O+oL9VA9gp921BNU5j786qg/1V/qH/Xvg/82Nje+2djSrLdvGZkHqvC38cP/tpEAqQ==</latexit>

T (Z = 1) = 1 ,T (Z = 0) = 1

T (Z = 1) = 0 ,T (Z = 0) = 0<latexit sha1_base64="NqOPrwwRK/pv6KXEPBk7WPSbRkQ=">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</latexit>

Nonparametric Identification of Local ATE

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Principal Strata
Break data into 4 strata (groups) based on how the instrument
affects the treatment they takeZ
U<latexit sha1_base64="7Wpmtk4gGKaYaQv+qqBHDDO4V28=">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</latexit>• Compliers:• Defiers:• Always-takers:• Never-takers:<latexit sha1_base64="oeupem7J3nlOID/wbAq7V7swhcs=">AAAUfHicrVjbbttGEN2kt8S9OYnf+sJGLpCitCq5ad2HCghgJyiCBnAA39ooMCSKsgjzFpKy6wj6hP5Lv6ZAn9ovKXrm7EoUdaMc2ILJ1ezMmdm57a7ase+lWa32963b773/wYcf3bm79vEnn372+fq9+0dp1E8c99CJ/Cg5abdS1/dC9zDzMt89iRO3FbR997h9vivzxxduknpReJBdxe7roHUWel3PaWUgna4/b2ZgzwI3zB41vTDNkr6MrYZV/1oeVtO2bKv5pt/qWAtZa8JaO12v1Ko1/lmzg7oZVJT524/u3f1DNVVHRcpRfRUoV4Uqw9hXLZXi80rVVU3FoL1WA9ASjDzOu2qo1iDbB5cLjhao53ie4dsrQw3xXTBTSjvQ4uM/gaSlvjI8HYy7pOq36LcmeBfpGBBbbLzCu20wA1Az1QO1TG7EubpcG5+gZNUZ1vAjV+thJTEp4gensOYu3j6+Z1ihPK/A6WLUgVSCkQOaD6qmiI4Eb+158U2PkWiRz8VIrF5m9zKrZT7C5xxYLYxTWiq2WuqZiUtIzS5tFR6fMV2M+DtWqO1bhtUdr8Fj3PUqhPcAlHP1FqMi8jKdk9m1mCszyMM5uoQ/I0cI7hSUiJnvwX8eOIq56gBTdLQYyTOuJWYGVQ3yc86PsjlGPLdoT8qoWcw3j3piU1m53SM7fbzbxE4gP8Bcj3rEFzbscYmdMHtHXrTJneDbJb85tNGZoldZmRJPG2uS7LKBG2HGG2NpT2h/jizS1g1As8xnS/1C/S7ibtMrktc2ZCNaqLM4pa7QxL6BviJ+CfAUqnh2RLGpI6AdPWbLOWi5Po0k+r7FKlLWj29QB3j79FpMFJvaxT+XHAdck0S0T92XGHeoR6Qt9Lqq2jF2DAtaxTceu9OsVov5J7WrO+O0VpELWamWycoGNdXUY7NiiYDYPek1x3RD3ZNTrm44ZVWb9uddQ3K0aJ9I5fEpxqrDbOxRqogrXpy/znlr2Wad2Iyl+P4MHA1KdQ09ZW3GRtPaRP3tmiqJGA+H78DkhK530XY5NTMAPTCe8sGn61z2JJl5A8wWc6pJ703OL0N8auRHu4F0/AGpy+WOWFlF2QSjwXhmubzHkUv/6JFEXOPF9HXMtXw5/rcmZlbFvhm8cAZRMjYzSKuid5gb81d+oCroBRVkVdGfEmmhJDydxBhvjjk3gSu859QSgiIVPhjPD0tjsAfK0Hisy7lRHeY2eOx8HXI2TeZpDl1R2RSHVPVyrZKX8/JG05fLdsb5XpTV9OWyQrlgbnqmH6TEOimRyxgHkQgmsvSgRCrinqDp2uZfV7Av5F7bN3mRGAuPSyQD9hi9qshU5YtS+zJ2wcT4RGR+ewcPXrD/DXkCuq4fc9nsWt7M5a7eyae5/OU1PZtLBtf0by75tkTS5d7gjWki87K0poRrv5RL51PZnuBiNxmdX3S12Xy24UXZszumM1lmv5Gs0Ltjgz3AXqk/z+47Usn10jrus/u22YcSau4sqZ3DG0PLI9hfYX0p8z2/X65SX/PkpvMmP0OMzv7pxO7iETu/Fwx4hrs0+9DoNLfF+OkI/IxO9gInJBnvqu9wOqvj+aywZ62KKrekvul2k7jbQN7BbrPHXen6uLG5y2wVbjxFHU+hYfQ/NHfDnNs3J/hFSNfROx9Z50xn5iYxrUmfFXvg0venqxLEPAvn4y2P73xk/TtHxPN7fou6CeTJG+JqeHnOLL7Pl/3KsPjuG+PdxdNnP04N7x6zzmdfdtEh9P1cn312Z35pkPzdRlVI7g4e3p/SV8Sfvrue8e7ZB0fOo+8gGaU1bRJx0S04pkxmznopb4z5vbhauOdqvGnb0om8Cgxew/Qi1/T6tdP1Sn36t7PZwdF2tf59tfbyceXJT+Z3tTvqC/VQPYKfdtQTVOY+/OqoP9Vf6h/174P/NjY3vtnY0qy3bxmZB6rwt/HD/6LEAKg=</latexit>

T (Z = 1) = 1 ,T (Z = 0) = 0

T<latexit sha1_base64="xD8zMCKqXrs71+Hc7nLNY0XPcuc=">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</latexit>

T (Z = 1) = 0 ,T (Z = 0) = 1<latexit sha1_base64="lpzpnPZaGkQ0k+6ItoP5qSHQ4oQ=">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</latexit>

T (Z = 1) = 1 ,T (Z = 0) = 1

T (Z = 1) = 0 ,T (Z = 0) = 0<latexit sha1_base64="uJ8AEZwBJ4xhnUEPOEHwwHt79XA=">AAAWsHicrVhZb9tGEN7YPRL3ylH4pS9s7AAJIKuymyIFAgMBnARF0RQJ4CRuLMOgyJXEilfJpV1H0E/oD+x7f0hnvl2KpA5SCiJB5HI4883sXLurXux7qep0/r22sfnJp599fv3G1hdffvX1Nzdv3X6TRlniyNdO5EfJSc9Ope+F8rXylC9P4kTaQc+Xb3ujI37/9kImqReFx+oqlmeBPQi9vufYikjntzYm3Z4ceOFYeaP3seeoLJGTLct87lm/R65Mp8/dkB5PU2Ur2XLtdCjdM+t+NwujXiqTC+k6UdiPstAlhQ8sW1n3O63OA2u8u5hld/J4ETJDKpqDCmSoHlinPelHl5Yv+8o6jPrWYqwz1jKVqkGOMuVEgZziJt5g2AhshGpgvTBVSWYsXsXWQiBHLXn9mTsoez221XC5nyUxV1z2eE3J3CXzcqUwLOW9lzOXXbDEpq4M3Uqmnd/c6bQ7+Fjzg30z2BHm8zK6deMf0RWuiIQjMhEIKUKhaOwLW6T0PRX7oiNiop2JMdESGnl4L8VEbJFsRlySOGyijug6oKdTQw3pmTFTSDukxadfQpKWuGd4XBr3QdV31m+VeJfpGAObbbyie89gBkRVYkjUJrmcc3W5Hn2DhlkrmsPPmK1HM4lBYT84lTn36e7Ts6IZ8vWKOCWNXJJKaOQQzSeqprCOhO7a8+ybISJhg0/SiK2us7vOan4f0XdEWDaNU1jKtlriuYlLCM0StjKPj5guR/ybZqjtq8PqT+fgIe56Fsx7TJSReE+jKnKdznJ2LedSBnmyQBfzK3CExJ0SJULme+Q/jziqueoQJuuwEckB5hIjg9oG+Ve8z7M5pnjuwZ4UUbOQbx70xKayCrtzO32694CdkPyY3g2hh33RInsksBNkb+7FFrgTerrEkwMbnRl6G5XJ8WzRnDi7WoQb0RtviqU9of2ZW6StGxPNMt898Rv0S4p7C17hvG6RbAQLdRan0BWa2B9SX2G/BHRlKns2p7SgI4AdQ2TLiGiFPo3E+n6gWaSoH9+gjunuw2sxUFrQzv65xDjAnDiiGXRf0tiFHpa2qNe1xSNjx6SilX3joTvNa7WQf1y7ujPOamW5EJVqmaw8hKaOeGhmzBFgu8tec0w31D05xewmM1b1YH/RNThHq/axVBGfaqxcZOMQUlVc9uLieS6aywHqpIVYsu8HxHEIqb6hp6jN2GjaKtXfkamSCPFwcA9MTuh6Z22XM2/GRA+Mp3zi03XOaxK/+YswbeRUF94rv69DfGbk89WAO/4Y1Hq5N6isqmxCo/H0Tb28h5GEf/SII67xYvg6xly+n/6s0ptVsT8OXjiHyBmrDNKq6C5yY/HMj8UO9YIdyqqqPznSTEmwO4lpvDvl3CVc5h1BS0gUrvDx9P2kMQZPiTIxHuvjXV6HhQ0eOp8Lzq7JPM2hK0rNcHBV12vlvFyUN5peL+tO870qq+n1sky5QG56ph+kwDppkFOIA0sEpSw9bpCKsCZourb5jxXsC7HWZiYvEmPh2wbJAD1GzyoyVfmi0T6FLpgYn7DMuw/w4AX63wQ7oHX9WMiqtbxZyF19kE8L+cs1PVtIBmv6t5B83yApsTZ4UxrLvGqsKeZ62cil86lpTZC0muT7F11tLVx75EVes13TmSyz3nBW6NXxED2gtVJ/nl93uJL3G+s4Q/ftoQ8l0OzW1M7rj4ZWRDBbYX4p8r04X65SX4vkZvOm2EPke/+0tLp4wC7OBWPs4S7NOpTv5vYQPx2BX6iTvaAdEo+PxI+0O9un6/PKmrUqKp+SMtPtyrgHhPyIVpunWJXWx43NWWavcuKp6nhGGvLfxJwNC27f7OCXIa2jdzGyzhl37iQxq0nvFYfEpc9PVw2IRRYuxquP72Jk/T9HhP17cYr6GMjlE+JqeEXOLD/PN/3LsPzsG9O9T1cf/Tg1vE+RdT76sqQOoc/neu9zNPdPA+fvAVUF5+747u0ZfVX82bPrAGfPjDgKHn0GUZDWtDLislNwDBll9nopTozFubhdOedqvFnb0lJeBQbv0PQiaXr91vnNnf3Z/87mB28O2vs/tTuvDnaePDT/q10X34m74j756ZF4QpX5kvzqbPy3ubV5Z/Pb7YPtk+3zbVuzblwzMndE5bP95//+Ibrk</latexit>

Z

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T

Brady Neal

Y

Nonparametric Identification of Local ATE

Y

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Principal Strata
Break data into 4 strata (groups) based on how the instrument
affects the treatment they takeZ
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T (Z = 1) = 1 ,T (Z = 0) = 0

T<latexit sha1_base64="xD8zMCKqXrs71+Hc7nLNY0XPcuc=">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</latexit>

T (Z = 1) = 0 ,T (Z = 0) = 1<latexit sha1_base64="lpzpnPZaGkQ0k+6ItoP5qSHQ4oQ=">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</latexit>

T (Z = 1) = 1 ,T (Z = 0) = 1

T (Z = 1) = 0 ,T (Z = 0) = 0<latexit sha1_base64="uJ8AEZwBJ4xhnUEPOEHwwHt79XA=">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</latexit>

Z

U<latexit sha1_base64="NqOPrwwRK/pv6KXEPBk7WPSbRkQ=">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</latexit>

T

Brady Neal

Y

Nonparametric Identification of Local ATE

Y

26 / 33

Monotonicity Assumption (No Defiers)<latexit sha1_base64="uzQcI2iod6GX+k3WLvmwihepPrs=">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</latexit>

Brady Neal

8i,

Ti (Z = 1)

Ti (Z = 0)

Nonparametric Identification of Local ATE

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Monotonicity Assumption (No Defiers)<latexit sha1_base64="uzQcI2iod6GX+k3WLvmwihepPrs=">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</latexit>

8i,

Ti (Z = 1)

Ti (Z = 0)<latexit sha1_base64="7Wpmtk4gGKaYaQv+qqBHDDO4V28=">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</latexit>

Z

T

Brady Neal

Nonparametric Identification of Local ATE

UY

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Monotonicity Assumption (No Defiers)<latexit sha1_base64="uzQcI2iod6GX+k3WLvmwihepPrs=">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</latexit>

8i,

Ti (Z = 1)

Ti (Z = 0)<latexit sha1_base64="7Wpmtk4gGKaYaQv+qqBHDDO4V28=">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</latexit>

Z• Compliers:• Defiers:• Always-takers:• Never-takers:
Brady NealU<latexit sha1_base64="oeupem7J3nlOID/wbAq7V7swhcs=">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</latexit>

T (Z = 1) = 1 ,T (Z = 0) = 0

TY<latexit sha1_base64="xD8zMCKqXrs71+Hc7nLNY0XPcuc=">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</latexit>

T (Z = 1) = 0 ,T (Z = 0) = 1<latexit sha1_base64="lpzpnPZaGkQ0k+6ItoP5qSHQ4oQ=">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</latexit>

T (Z = 1) = 1 ,T (Z = 0) = 1

T (Z = 1) = 0 ,T (Z = 0) = 0<latexit sha1_base64="NqOPrwwRK/pv6KXEPBk7WPSbRkQ=">AAAUfHicrVjbbttGEN2kt8S9OYnf+sJGLpCitCq5ad2HCghgJyiCBnAA39ooMCSKsgjzFpKy6wj6hP5Lv6ZAn9ovKXrm7EoUdaMc2ILJ1ezMmdm57a7ase+lWa32963b773/wYcf3bm79vEnn372+fq9+0dp1E8c99CJ/Cg5abdS1/dC9zDzMt89iRO3FbR997h9vivzxxduknpReJBdxe7roHUWel3PaWUgna4/b2ZgzwI3zB41vTDNkr6MrYZV/xqPmtW0Ldtqvum3OtZC1hpZT9crtWqNf9bsoG4GFWX+9qN7d/9QTdVRkXJUXwXKVaHKMPZVS6X4vFJ1VVMxaK/VALQEI4/zrhqqNcj2weWCowXqOZ5n+PbKUEN8F8yU0g60+PhPIGmprwxPB+Muqfot+q0J3kU6BsQWG6/wbhvMANRM9UAtkxtxri7XxicoWXWGNfzI1XpYSUyK+MEprLmLt4/vGVYozytwuhh1IJVg5IDmg6opoiPBW3tefNNjJFrkczESq5fZvcxqmY/wOQdWC+OUloqtlnpm4hJSs0tbhcdnTBcj/o4VavuWYXXHa/AYd70K4T0A5Vy9xaiIvEznZHYt5soM8nCOLuHPyBGCOwUlYuZ78J8HjmKuOsAUHS1G8oxriZlBVYP8nPOjbI4Rzy3akzJqFvPNo57YVFZu98hOH+82sRPIDzDXox7xhQ17XGInzN6RF21yJ/h2yW8ObXSm6FVWpsTTxpoku2zgRpjxxljaE9qfI4u0dQPQLPPZUr9Qv4u42/SK5LUN2YgW6ixOqSs0sW+gr4hfAjyFKp4dUWzqCGhHj9lyDlquTyOJvm+xipT14xvUAd4+vRYTxaZ28c8lxwHXJBHtU/clxh3qEWkLva6qdowdw4JW8Y3H7jSr1WL+Se3qzjitVeRCVqplsrJBTTX12KxYIiB2T3rNMd1Q9+SUqxtOWdWm/XnXkBwt2idSeXyKseowG3uUKuKKF+evc95atlknNmMpvj8DR4NSXUNPWZux0bQ2UX+7pkoixsPhOzA5oetdtF1OzQxAD4ynfPDpOpc9SWbeALPFnGrSe5PzyxCfGvnRbiAdf0DqcrkjVlZRNsFoMJ5ZLu9x5NI/eiQR13gxfR1zLV+O/62JmVWxbwYvnEGUjM0M0qroHebG/JUfqAp6QQVZVfSnRFooCU8nMcabY85N4ArvObWEoEiFD8bzw9IY7IEyNB7rcm5Uh7kNHjtfh5xNk3maQ1dUNsUhVb1cq+TlvLzR9OWynXG+F2U1fbmsUC6Ym57pBymxTkrkMsZBJIKJLD0okYq4J2i6tvnXFewLudf2TV4kxsLjEsmAPUavKjJV+aLUvoxdMDE+EZnf3sGDF+x/Q56AruvHXDa7ljdzuat38mkuf3lNz+aSwTX9m0u+LZF0uTd4Y5rIvCytKeHaL+XS+VS2J7jYTUbnF11tNp9teFH27I7pTJbZbyQr9O7YYA+wV+rPs/uOVHK9tI777L5t9qGEmjtLaufwxtDyCPZXWF/KfM/vl6vU1zy56bzJzxCjs386sbt4xM7vBQOe4S7NPjQ6zW0xfjoCP6OTvcAJSca76juczup4PivsWauiyi2pb7rdJO42kHew2+xxV7o+bmzuMluFG09Rx1NoGP0Pzd0w5/bNCX4R0nX0zkfWOdOZuUlMa9JnxR649P3pqgQxz8L5eMvjOx9Z/84R8fye36JuAnnyhrgaXp4zi+/zZb8yLL77xnh38fTZj1PDu8es89mXXXQIfT/XZ5/dmV8aJH+3URWSu4OH96f0FfGn765nvHv2wZHz6DtIRmlNm0RcdAuOKZOZs17KG2N+L64W7rkab9q2dCKvAoPXML3INb1+7XS9Up/+7Wx2cLRdrX9frb18XHnyk/ld7Y76Qj1Uj+CnHfUElbkPvzrqT/WX+kf9++C/jc2Nbza2NOvtW0bmgSr8bfzwP47MAKc=</latexit>

Nonparametric Identification of Local ATE

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Monotonicity Assumption (No Defiers)<latexit sha1_base64="uzQcI2iod6GX+k3WLvmwihepPrs=">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</latexit>

8i,

Ti (Z = 1)

Ti (Z = 0)<latexit sha1_base64="7Wpmtk4gGKaYaQv+qqBHDDO4V28=">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</latexit>

Z• Compliers:• Defiers:• Always-takers:• Never-takers:
Brady NealU<latexit sha1_base64="oeupem7J3nlOID/wbAq7V7swhcs=">AAAUfHicrVjbbttGEN2kt8S9OYnf+sJGLpCitCq5ad2HCghgJyiCBnAA39ooMCSKsgjzFpKy6wj6hP5Lv6ZAn9ovKXrm7EoUdaMc2ILJ1ezMmdm57a7ase+lWa32963b773/wYcf3bm79vEnn372+fq9+0dp1E8c99CJ/Cg5abdS1/dC9zDzMt89iRO3FbR997h9vivzxxduknpReJBdxe7roHUWel3PaWUgna4/b2ZgzwI3zB41vTDNkr6MrYZV/1oeVtO2bKv5pt/qWAtZa8JaO12v1Ko1/lmzg7oZVJT524/u3f1DNVVHRcpRfRUoV4Uqw9hXLZXi80rVVU3FoL1WA9ASjDzOu2qo1iDbB5cLjhao53ie4dsrQw3xXTBTSjvQ4uM/gaSlvjI8HYy7pOq36LcmeBfpGBBbbLzCu20wA1Az1QO1TG7EubpcG5+gZNUZ1vAjV+thJTEp4gensOYu3j6+Z1ihPK/A6WLUgVSCkQOaD6qmiI4Eb+158U2PkWiRz8VIrF5m9zKrZT7C5xxYLYxTWiq2WuqZiUtIzS5tFR6fMV2M+DtWqO1bhtUdr8Fj3PUqhPcAlHP1FqMi8jKdk9m1mCszyMM5uoQ/I0cI7hSUiJnvwX8eOIq56gBTdLQYyTOuJWYGVQ3yc86PsjlGPLdoT8qoWcw3j3piU1m53SM7fbzbxE4gP8Bcj3rEFzbscYmdMHtHXrTJneDbJb85tNGZoldZmRJPG2uS7LKBG2HGG2NpT2h/jizS1g1As8xnS/1C/S7ibtMrktc2ZCNaqLM4pa7QxL6BviJ+CfAUqnh2RLGpI6AdPWbLOWi5Po0k+r7FKlLWj29QB3j79FpMFJvaxT+XHAdck0S0T92XGHeoR6Qt9Lqq2jF2DAtaxTceu9OsVov5J7WrO+O0VpELWamWycoGNdXUY7NiiYDYPek1x3RD3ZNTrm44ZVWb9uddQ3K0aJ9I5fEpxqrDbOxRqogrXpy/znlr2Wad2Iyl+P4MHA1KdQ09ZW3GRtPaRP3tmiqJGA+H78DkhK530XY5NTMAPTCe8sGn61z2JJl5A8wWc6pJ703OL0N8auRHu4F0/AGpy+WOWFlF2QSjwXhmubzHkUv/6JFEXOPF9HXMtXw5/rcmZlbFvhm8cAZRMjYzSKuid5gb81d+oCroBRVkVdGfEmmhJDydxBhvjjk3gSu859QSgiIVPhjPD0tjsAfK0Hisy7lRHeY2eOx8HXI2TeZpDl1R2RSHVPVyrZKX8/JG05fLdsb5XpTV9OWyQrlgbnqmH6TEOimRyxgHkQgmsvSgRCrinqDp2uZfV7Av5F7bN3mRGAuPSyQD9hi9qshU5YtS+zJ2wcT4RGR+ewcPXrD/DXkCuq4fc9nsWt7M5a7eyae5/OU1PZtLBtf0by75tkTS5d7gjWki87K0poRrv5RL51PZnuBiNxmdX3S12Xy24UXZszumM1lmv5Gs0Ltjgz3AXqk/z+47Usn10jrus/u22YcSau4sqZ3DG0PLI9hfYX0p8z2/X65SX/PkpvMmP0OMzv7pxO7iETu/Fwx4hrs0+9DoNLfF+OkI/IxO9gInJBnvqu9wOqvj+aywZ62KKrekvul2k7jbQN7BbrPHXen6uLG5y2wVbjxFHU+hYfQ/NHfDnNs3J/hFSNfROx9Z50xn5iYxrUmfFXvg0venqxLEPAvn4y2P73xk/TtHxPN7fou6CeTJG+JqeHnOLL7Pl/3KsPjuG+PdxdNnP04N7x6zzmdfdtEh9P1cn312Z35pkPzdRlVI7g4e3p/SV8Sfvrue8e7ZB0fOo+8gGaU1bRJx0S04pkxmznopb4z5vbhauOdqvGnb0om8Cgxew/Qi1/T6tdP1Sn36t7PZwdF2tf59tfbyceXJT+Z3tTvqC/VQPYKfdtQTVOY+/OqoP9Vf6h/174P/NjY3vtnY0qy3bxmZB6rwt/HD/6LEAKg=</latexit>

T (Z = 1) = 1 ,T (Z = 0) = 0

TY<latexit sha1_base64="xD8zMCKqXrs71+Hc7nLNY0XPcuc=">AAAUfHicrVjbbttGEN2kt8S9OYnf+sJGLpCitCq5ad2HCghgJyiCBnAA39ooMCSKsgjzFpKy6wj6hP5Lv6ZAn9ovKXrm7EoUdaMc2ILJ1ezMmdm57a7ase+lWa32963b773/wYcf3bm79vEnn372+fq9+0dp1E8c99CJ/Cg5abdS1/dC9zDzMt89iRO3FbR997h9vivzxxduknpReJBdxe7roHUWel3PaWUgna4/b2ZgzwI3zB41vTDNkr6MrYZV/xqPmtW0Ldtqvum3OtZC1pqw1k/XK7VqjX/W7KBuBhVl/vaje3f/UE3VUZFyVF8FylWhyjD2VUul+LxSdVVTMWiv1QC0BCOP864aqjXI9sHlgqMF6jmeZ/j2ylBDfBfMlNIOtPj4TyBpqa8MTwfjLqn6LfqtCd5FOgbEFhuv8G4bzADUTPVALZMbca4u18YnKFl1hjX8yNV6WElMivjBKay5i7eP7xlWKM8rcLoYdSCVYOSA5oOqKaIjwVt7XnzTYyRa5HMxEquX2b3MapmP8DkHVgvjlJaKrZZ6ZuISUrNLW4XHZ0wXI/6OFWr7lmF1x2vwGHe9CuE9AOVcvcWoiLxM52R2LebKDPJwji7hz8gRgjsFJWLme/CfB45irjrAFB0tRvKMa4mZQVWD/Jzzo2yOEc8t2pMyahbzzaOe2FRWbvfITh/vNrETyA8w16Me8YUNe1xiJ8zekRdtcif4dslvDm10puhVVqbE08aaJLts4EaY8cZY2hPanyOLtHUD0Czz2VK/UL+LuNv0iuS1DdmIFuosTqkrNLFvoK+IXwI8hSqeHVFs6ghoR4/Zcg5ark8jib5vsYqU9eMb1AHePr0WE8WmdvHPJccB1yQR7VP3JcYd6hFpC72uqnaMHcOCVvGNx+40q9Vi/knt6s44rVXkQlaqZbKyQU019disWCIgdk96zTHdUPfklKsbTlnVpv1515AcLdonUnl8irHqMBt7lCriihfnr3PeWrZZJzZjKb4/A0eDUl1DT1mbsdG0NlF/u6ZKIsbD4TswOaHrXbRdTs0MQA+Mp3zw6TqXPUlm3gCzxZxq0nuT88sQnxr50W4gHX9A6nK5I1ZWUTbBaDCeWS7vceTSP3okEdd4MX0dcy1fjv+tiZlVsW8GL5xBlIzNDNKq6B3mxvyVH6gKekEFWVX0p0RaKAlPJzHGm2POTeAK7zm1hKBIhQ/G88PSGOyBMjQe63JuVIe5DR47X4ecTZN5mkNXVDbFIVW9XKvk5by80fTlsp1xvhdlNX25rFAumJue6QcpsU5K5DLGQSSCiSw9KJGKuCdourb51xXsC7nX9k1eJMbC4xLJgD1GryoyVfmi1L6MXTAxPhGZ397Bgxfsf0OegK7rx1w2u5Y3c7mrd/JpLn95Tc/mksE1/ZtLvi2RdLk3eGOayLwsrSnh2i/l0vlUtie42E1G5xddbTafbXhR9uyO6UyW2W8kK/Tu2GAPsFfqz7P7jlRyvbSO++y+bfahhJo7S2rn8MbQ8gj2V1hfynzP75er1Nc8uem8yc8Qo7N/OrG7eMTO7wUDnuEuzT40Os1tMX46Aj+jk73ACUnGu+o7nM7qeD4r7FmrosotqW+63STuNpB3sNvscVe6Pm5s7jJbhRtPUcdTaBj9D83dMOf2zQl+EdJ19M5H1jnTmblJTGvSZ8UeuPT96aoEMc/C+XjL4zsfWf/OEfH8nt+ibgJ58oa4Gl6eM4vv82W/Miy++8Z4d/H02Y9Tw7vHrPPZl110CH0/12ef3ZlfGiR/t1EVkruDh/en9BXxp++uZ7x79sGR8+g7SEZpTZtEXHQLjimTmbNeyhtjfi+uFu65Gm/atnQirwKD1zC9yDW9fu10vVKf/u1sdnC0Xa1/X629fFx58pP5Xe2O+kI9VI/gpx31BJW5D7866k/1l/pH/fvgv43NjW82tjTr7VtG5oEq/G388D+imQCo</latexit>

T (Z = 1) = 0 ,T (Z = 0) = 1<latexit sha1_base64="lpzpnPZaGkQ0k+6ItoP5qSHQ4oQ=">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</latexit>

T (Z = 1) = 1 ,T (Z = 0) = 1

T (Z = 1) = 0 ,T (Z = 0) = 0<latexit sha1_base64="NqOPrwwRK/pv6KXEPBk7WPSbRkQ=">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</latexit>

Nonparametric Identification of Local ATE

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Monotonicity Assumption (No Defiers)<latexit sha1_base64="uzQcI2iod6GX+k3WLvmwihepPrs=">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</latexit>

8i,

Ti (Z = 1)

Ti (Z = 0)<latexit sha1_base64="7Wpmtk4gGKaYaQv+qqBHDDO4V28=">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</latexit>

Z• Compliers:• Defiers:• Always-takers:• Never-takers:
Brady NealU<latexit sha1_base64="oeupem7J3nlOID/wbAq7V7swhcs=">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</latexit>

T (Z = 1) = 1 ,T (Z = 0) = 0

TY<latexit sha1_base64="xD8zMCKqXrs71+Hc7nLNY0XPcuc=">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</latexit>

T (Z = 1) = 0 ,T (Z = 0) = 1<latexit sha1_base64="lpzpnPZaGkQ0k+6ItoP5qSHQ4oQ=">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</latexit>

T (Z = 1) = 1 ,T (Z = 0) = 1

T (Z = 1) = 0 ,T (Z = 0) = 0<latexit sha1_base64="NqOPrwwRK/pv6KXEPBk7WPSbRkQ=">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</latexit>

Nonparametric Identification of Local ATE

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Deriving Local ATE Identification

with Monotonicity Assumption

Brady Neal

Nonparametric Identification of Local ATE

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Deriving Local ATE Identification<latexit sha1_base64="Bng2qDf2uUOF27+TuP3gaxDy3tQ=">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</latexit>

E[Y (Z = 1)

Y (Z = 0)]= E[Y (Z = 1)
+ E[Y (Z = 1)
+ E[Y (Z = 1)
+ E[Y (Z = 1)
= E[Y (Z = 1)

Brady Nealwith Monotonicity Assumption

Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)

Y (Z = 0) | T (1) = 0, T (0) = 1] P (T (1) = 0, T (0) = 1)
Y (Z = 0) | T (1) = 1, T (0) = 1] P (T (1) = 1, T (0) = 1)
Y (Z = 0) | T (1) = 0, T (0) = 0] P (T (1) = 0, T (0) = 0)

Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)

Nonparametric Identification of Local ATE

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Deriving Local ATE Identification<latexit sha1_base64="Bng2qDf2uUOF27+TuP3gaxDy3tQ=">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</latexit><latexit sha1_base64="Bng2qDf2uUOF27+TuP3gaxDy3tQ=">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</latexit>

with Monotonicity Assumption

E[Y (Z = 1) Y (Z = 0)]= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)
+ E[Y (Z = 1)
+ E[Y (Z = 1)

Y (Z = 0) | T (1) = 0, T (0) = 1] P (T (1) = 0, T (0) = 1)
Y (Z = 0) | T (1) = 1, T (0) = 1] P (T (1) = 1, T (0) = 1)

+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 0] P (T (1) = 0, T (0) = 0)
= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)

Brady Neal

Nonparametric Identification of Local ATE(Compliers)(Defiers)(Always-takers)(Never-takers)

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Deriving Local ATE Identification<latexit sha1_base64="Bng2qDf2uUOF27+TuP3gaxDy3tQ=">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</latexit><latexit sha1_base64="Bng2qDf2uUOF27+TuP3gaxDy3tQ=">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</latexit>

with Monotonicity Assumption

E[Y (Z = 1) Y (Z = 0)]= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)
+ E[Y (Z = 1)
+ E[Y (Z = 1)

Y (Z = 0) | T (1) = 0, T (0) = 1] P (T (1) = 0, T (0) = 1)
Y (Z = 0) | T (1) = 1, T (0) = 1] P (T (1) = 1, T (0) = 1)

+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 0] P (T (1) = 0, T (0) = 0)
= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)

Brady Neal

Nonparametric Identification of Local ATE(Compliers)(Defiers)(Always-takers)(Never-takers)

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Deriving Local ATE Identification<latexit sha1_base64="Bng2qDf2uUOF27+TuP3gaxDy3tQ=">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</latexit><latexit sha1_base64="Bng2qDf2uUOF27+TuP3gaxDy3tQ=">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</latexit>

with Monotonicity Assumption

E[Y (Z = 1) Y (Z = 0)]= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)
+ E[Y (Z = 1)
+ E[Y (Z = 1)

Y (Z = 0) | T (1) = 0, T (0) = 1] P (T (1) = 0, T (0) = 1)
Y (Z = 0) | T (1) = 1, T (0) = 1] P (T (1) = 1, T (0) = 1)

+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 0] P (T (1) = 0, T (0) = 0)
= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)

Brady Neal

Nonparametric Identification of Local ATE(Compliers)(Defiers)(Always-takers)(Never-takers)

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Deriving Local ATE Identification<latexit sha1_base64="Bng2qDf2uUOF27+TuP3gaxDy3tQ=">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</latexit><latexit sha1_base64="Bng2qDf2uUOF27+TuP3gaxDy3tQ=">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</latexit>

with Monotonicity Assumption

E[Y (Z = 1) Y (Z = 0)]= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)
+ E[Y (Z = 1)
+ E[Y (Z = 1)

Y (Z = 0) | T (1) = 0, T (0) = 1] P (T (1) = 0, T (0) = 1)
Y (Z = 0) | T (1) = 1, T (0) = 1] P (T (1) = 1, T (0) = 1)

+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 0] P (T (1) = 0, T (0) = 0)
= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)

Brady Neal

Nonparametric Identification of Local ATE(Compliers)(Defiers)(Always-takers)(Never-takers)

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Deriving Local ATE Identification<latexit sha1_base64="Bng2qDf2uUOF27+TuP3gaxDy3tQ=">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</latexit><latexit sha1_base64="Bng2qDf2uUOF27+TuP3gaxDy3tQ=">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</latexit>

with Monotonicity Assumption

E[Y (Z = 1) Y (Z = 0)]= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)
+ E[Y (Z = 1)
+ E[Y (Z = 1)

Y (Z = 0) | T (1) = 0, T (0) = 1] P (T (1) = 0, T (0) = 1)
Y (Z = 0) | T (1) = 1, T (0) = 1] P (T (1) = 1, T (0) = 1)

+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 0] P (T (1) = 0, T (0) = 0)
= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)

Brady Neal

Nonparametric Identification of Local ATE(Compliers)(Defiers)(Always-takers)(Never-takers)

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Deriving Local ATE Identification<latexit sha1_base64="Bng2qDf2uUOF27+TuP3gaxDy3tQ=">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</latexit><latexit sha1_base64="Bng2qDf2uUOF27+TuP3gaxDy3tQ=">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</latexit>

with Monotonicity Assumption

E[Y (Z = 1) Y (Z = 0)]= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)
+ E[Y (Z = 1)
+ E[Y (Z = 1)

Y (Z = 0) | T (1) = 0, T (0) = 1] P (T (1) = 0, T (0) = 1)
Y (Z = 0) | T (1) = 1, T (0) = 1] P (T (1) = 1, T (0) = 1)

+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 0] P (T (1) = 0, T (0) = 0)
= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)

Brady Neal

Nonparametric Identification of Local ATE(Compliers)(Defiers)(Always-takers)(Never-takers)

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Deriving Local ATE Identification<latexit 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with Monotonicity Assumption

E[Y (Z = 1) Y (Z = 0)]= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 1] P (T (1) = 0, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 1] P (T (1) = 1, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 0] P (T (1) = 0, T (0) = 0)
= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)

Brady Neal

Nonparametric Identification of Local ATE(Compliers)(Defiers)(Always-takers)(Never-takers)

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Deriving Local ATE Identification<latexit 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with Monotonicity Assumption

E[Y (Z = 1) Y (Z = 0)]= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 1] P (T (1) = 0, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 1] P (T (1) = 1, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 0] P (T (1) = 0, T (0) = 0)
= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)<latexit sha1_base64="6/C9JDS05OcmoeWaG0G3AfSB11c=">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</latexit><latexit sha1_base64="Ll7TKa5O//2N2eH3L3ezgzOQweM=">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</latexit>

E[Y (Z(T = 1)

Brady Neal

(Compliers)(Defiers)(Always-takers)(Never-takers)

E[Y (Z = 1) Y (Z = 0)]
Y (Z(T = 0) | T (1) = 1, T (0) = 0] =P (T (1) = 1, T (0) = 0)
E[Y | Z = 1] E[Y | Z = 0]=
P (T (1) = 1, T (0) = 0)
E[Y | Z = 1] E[Y | Z = 0]=
E[T | Z = 1] E[T | Z = 0]

Nonparametric Identification of Local ATE

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Deriving Local ATE Identification<latexit 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with Monotonicity Assumption

E[Y (Z = 1) Y (Z = 0)]= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 1] P (T (1) = 0, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 1] P (T (1) = 1, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 0] P (T (1) = 0, T (0) = 0)
= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)<latexit sha1_base64="Ll7TKa5O//2N2eH3L3ezgzOQweM=">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</latexit><latexit sha1_base64="Ll7TKa5O//2N2eH3L3ezgzOQweM=">AAAWJnicrVjZbttGFJ24W+IuWQo/9YWN08IGZFVyU6QvAgI4CYqiARLAWZrICCSKsglzK0nFdQQBfW2B/ke/pm9F0bf+Q3+g554ZiqQWUgosQeLozr3n3rnbzKgfeW6Stlr/XNp459333v/g8pXNDz/6+JOr167feJqEo9h2ntihF8bP+73E8dzAeZK6qec8j2Kn5/c951n/9EDmn7124sQNg8P0PHKO/N5x4A5du5eC9Or6pf+69192w1Fqh76z000hm/pOkFodq71r7VnZlFWea+1aXd8dWDlxB+yQaRRJLSG1jjYtvL7sWN1h3LPHJX1ukKTxaJHCmbnW7tFk3I12VlG4O7G63aVKjeFlzUeiuZqnta4BF64cLHkMqoAquQD16tp2q9niy5oftM1gW5nXo/D6ld9VVw1UqGw1Ur5yVKBSjD3VUwneL1VbtVQE2pEagxZj5HLeURO1CdkRuBxw9EA9xfcxfr001AC/BTOhtA0tHj4xJC31heEZYDwkVT9Fv1XgXaZjTGyx8RzPvsH0QU3VCah1chnn6nJ9vP2aVadYw7dcrYuVRKSIH+zSmod4evidYoXyfQ5OB6MBpGKMbNA8UDVFdMR4as+Lb04YiR75HIzE6iq7q6yW+RDvU2D1ME5oqdhqqQcmLgE1O7RVeDzGdDniz1ihtq8Kazhdg8u461UI7yEop+oNRmXkKp3F7FrOlRrkyQJdwp+SIwB3AkrIzHfhPxcc5Vy1gSk6eozkMdcSMYOaBvl7zmfZHCGee7QnYdQs5ptLPZGprNzuzE4Pzz6xY8iPMXdCPeKLBuxxiB0zezMvNsgd49cZf9m00Z6hN1mZEs8G1iTZ1QBuiBl3iqU9of2ZWaStG4Nmmfee+oH6HcS9Qa9IXjcgG9JCncUJdQUm9h30FfGLj2+himczSoM6fNpxwmw5BS3Xp5FE31dYRcL68QzqGE+PXouI0qB28c8Zxz7XJBEdUfcZxgPqEWkLva6p7hg7JiWt4huX3Wleq8X8k9rVnXFWq8gFrFTLZGWHmlrqtlmxREDsLnrNNt1Q9+SEq5vMWNWn/XnXkBwt2ydSeXzKsRowG08oVcYVLy5e56K17LNOGoyl+P4YHB1KDQ09YW1GRtNmof4OTJWEjIfNp29yQte7aDubmRmD7htPeeDTdS57ksz8BMwec6pL7xXnqxDvG/lsN5COPya1Wu4pK6ssG2M0ns5Uy7scOfSPHknENV5EX0dcy+fTj1WYWRX7YvCCOUTJ2NQgrYo+YG4sXvmh2kYv2EZWlf0pkRZKzNNJhPGtKect4ArvKbUEoEiFj6fzk9oY3ANlYjw25FxWh7kNLjvfgJxdk3maQ1dUOsMhVV2tVfJyUd5oerXsYJrvZVlNr5YVymvmpmv6QUKs5zVyKeMgEn4hSw9rpELuCZqubf5xBfsC7rUjkxexsfBZjaTPHqNXFZqqfFhrX8ouGBufiMyLt/Dga/a/CU9A6/oxl03X8mYud/5WPs3lz9b0bC7pr+nfXPJNjaTDvcGd0kTmcW1NCdejWi6dT3V7goPdJDu/6Gpr8LsPL8qePTCdyTL7jWSF3h077AGNlfrz/L4jldyureMRu2+ffSim5kFF7Ty5MLQ8gqMV1pcw3/P75Sr1tUhuNm/yM0R29k8Ku4tL7PxeMOYZ7szsQ9lpbo/x0xH4Dp3sIU5IMj5QX+N01sb3g9KetSqq3JJGptsVcfeBfAe7zT3uSuvjRuYus1e68ZR13IeG7DMxd8Oc2zMn+GVI6+hdjKxzZjB3k5jVpM+KJ+DS96fzGsQ8CxfjVcd3MbL+nyPk+T2/RV0EcvGGuBpenjPL7/N1/zIsv/tGeA7x7bEfJ4b3HrPOY1920CH0/VyffQ7m/mmQ/N1HVUjujm/emNFXxp+9ux7z7jkCR86j7yAppTWtiLjsFhxRJjVnvYQ3xvxe3CzdczXerG1JIa98g9cxvcgxvX7z1bXt9ux/Z/ODp/vN9jfN1uPb23dvm//VLqvP1E21Az/dUXdRmY/gV3vjxcYvG79u/Lb1x9afW39t/a1ZNy4ZmU9V6bX17/8ffpBz</latexit>

E[Y (T = 1)

Brady Neal

(Compliers)(Defiers)(Always-takers)(Never-takers)

E[Y (Z = 1) Y (Z = 0)]
Y (T = 0) | T (1) = 1, T (0) = 0] =P (T (1) = 1, T (0) = 0)
E[Y | Z = 1] E[Y | Z = 0]=
P (T (1) = 1, T (0) = 0)
E[Y | Z = 1] E[Y | Z = 0]=
E[T | Z = 1] E[T | Z = 0]

Nonparametric Identification of Local ATE

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Deriving Local ATE Identification<latexit 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with Monotonicity Assumption

E[Y (Z = 1) Y (Z = 0)]= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 1] P (T (1) = 0, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 1] P (T (1) = 1, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 0] P (T (1) = 0, T (0) = 0)
= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)

Local ATE (LATE) or Complieraverage causal effect (CACE): E[Y (T = 1)
<latexit sha1_base64="Ll7TKa5O//2N2eH3L3ezgzOQweM=">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</latexit><latexit sha1_base64="Ll7TKa5O//2N2eH3L3ezgzOQweM=">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</latexit>

Brady Neal

(Compliers)(Defiers)(Always-takers)(Never-takers)

E[Y (Z = 1) Y (Z = 0)]
Y (T = 0) | T (1) = 1, T (0) = 0] =P (T (1) = 1, T (0) = 0)
E[Y | Z = 1] E[Y | Z = 0]=
P (T (1) = 1, T (0) = 0)
E[Y | Z = 1] E[Y | Z = 0]=
E[T | Z = 1] E[T | Z = 0]

Nonparametric Identification of Local ATE

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Deriving Local ATE Identification<latexit 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with Monotonicity Assumption

E[Y (Z = 1) Y (Z = 0)]= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 1] P (T (1) = 0, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 1] P (T (1) = 1, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 0] P (T (1) = 0, T (0) = 0)
= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)

Local ATE (LATE) or Complieraverage causal effect (CACE): E[Y (T = 1)
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Brady Neal

(Compliers)(Defiers)(Always-takers)(Never-takers)

E[Y (Z = 1) Y (Z = 0)]
Y (T = 0) | T (1) = 1, T (0) = 0] =P (T (1) = 1, T (0) = 0)
E[Y | Z = 1] E[Y | Z = 0]=
P (T (1) = 1, T (0) = 0)
E[Y | Z = 1] E[Y | Z = 0]=
E[T | Z = 1] E[T | Z = 0]

Nonparametric Identification of Local ATE

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Deriving Local ATE Identification<latexit 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with Monotonicity Assumption

E[Y (Z = 1) Y (Z = 0)]= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 1] P (T (1) = 0, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 1] P (T (1) = 1, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 0] P (T (1) = 0, T (0) = 0)
= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)(Compliers)(Defiers)(Always-takers)(Never-takers)

Local ATE (LATE) or Complieraverage causal effect (CACE): E[Y (T = 1)

E[Y (Z = 1) Y (Z = 0)]
Y (T = 0) | T (1) = 1, T (0) = 0] =P (T (1) = 1, T (0) = 0)
E[Y | Z = 1] E[Y | Z = 0]
P (T (1) = 1, T (0) = 0) = 1 P (T = 0 | Z = 1) P (T = 1 | Z = 0)=
P (T (1) = 1, T (0) = 0)
= 1 (1 P (T = 1 | Z = 1)) P (T = 1 | Z = 0)
E[Y | Z = 1] E[Y | Z = 0]
= P (T = 1 | Z = 1) P (T = 1 | Z = 0)=
E[T | Z = 1] E[T | Z = 0]
= E[T | Z = 1] E[T | Z = 0]<latexit 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Brady Neal

Nonparametric Identification of Local ATE

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Deriving Local ATE Identification<latexit 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with Monotonicity Assumption

E[Y (Z = 1) Y (Z = 0)]= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 1] P (T (1) = 0, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 1] P (T (1) = 1, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 0] P (T (1) = 0, T (0) = 0)
= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)(Compliers)(Defiers)(Always-takers)(Never-takers)

Local ATE (LATE) or Complieraverage causal effect (CACE): E[Y (T = 1)

E[Y (Z = 1) Y (Z = 0)]
Y (T = 0) | T (1) = 1, T (0) = 0] =P (T (1) = 1, T (0) = 0)
E[Y | Z = 1] E[Y | Z = 0]
P (T (1) = 1, T (0) = 0) = 1 P (T = 0 | Z = 1) P (T = 1 | Z = 0)=
P (T (1) = 1, T (0) = 0)
= 1 (1 P (T = 1 | Z = 1)) P (T = 1 | Z = 0)
E[Y | Z = 1] E[Y | Z = 0]
= P (T = 1 | Z = 1) P (T = 1 | Z = 0)=
E[T | Z = 1] E[T | Z = 0]
= E[T | Z = 1] E[T | Z = 0]<latexit 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Brady Neal

Nonparametric Identification of Local ATE

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Deriving Local ATE Identification<latexit 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with Monotonicity Assumption

E[Y (Z = 1) Y (Z = 0)]= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 1] P (T (1) = 0, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 1] P (T (1) = 1, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 0] P (T (1) = 0, T (0) = 0)
= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)(Compliers)(Defiers)(Always-takers)(Never-takers)

Local ATE (LATE) or Complieraverage causal effect (CACE): E[Y (T = 1)

E[Y (Z = 1) Y (Z = 0)]
Y (T = 0) | T (1) = 1, T (0) = 0] =P (T (1) = 1, T (0) = 0)
E[Y | Z = 1] E[Y | Z = 0]
P (T (1) = 1, T (0) = 0) = 1 P (T = 0 | Z = 1) P (T = 1 | Z = 0)=
P (T (1) = 1, T (0) = 0)
= 1 (1 P (T = 1 | Z = 1)) P (T = 1 | Z = 0)
E[Y | Z = 1] E[Y | Z = 0]
= P (T = 1 | Z = 1) P (T = 1 | Z = 0)=
E[T | Z = 1] E[T | Z = 0]
= E[T | Z = 1] E[T | Z = 0]<latexit 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Brady Neal

Nonparametric Identification of Local ATE

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Deriving Local ATE Identification<latexit 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with Monotonicity Assumption

E[Y (Z = 1) Y (Z = 0)]= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 1] P (T (1) = 0, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 1] P (T (1) = 1, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 0] P (T (1) = 0, T (0) = 0)
= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)(Compliers)(Defiers)(Always-takers)(Never-takers)

Local ATE (LATE) or Complieraverage causal effect (CACE): E[Y (T = 1)

E[Y (Z = 1) Y (Z = 0)]
Y (T = 0) | T (1) = 1, T (0) = 0] =P (T (1) = 1, T (0) = 0)
E[Y | Z = 1] E[Y | Z = 0]
P (T (1) = 1, T (0) = 0) = 1 P (T = 0 | Z = 1) P (T = 1 | Z = 0)=
P (T (1) = 1, T (0) = 0)
= 1 (1 P (T = 1 | Z = 1)) P (T = 1 | Z = 0)
E[Y | Z = 1] E[Y | Z = 0]
= P (T = 1 | Z = 1) P (T = 1 | Z = 0)=
E[T | Z = 1] E[T | Z = 0]
= E[T | Z = 1] E[T | Z = 0]<latexit 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Brady Neal

Nonparametric Identification of Local ATE

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Deriving Local ATE Identification<latexit 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with Monotonicity Assumption

E[Y (Z = 1) Y (Z = 0)]= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 1] P (T (1) = 0, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 1] P (T (1) = 1, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 0] P (T (1) = 0, T (0) = 0)
= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)(Compliers)(Defiers)(Always-takers)(Never-takers)

Local ATE (LATE) or Complieraverage causal effect (CACE): E[Y (T = 1)

E[Y (Z = 1) Y (Z = 0)]
Y (T = 0) | T (1) = 1, T (0) = 0] =P (T (1) = 1, T (0) = 0)
E[Y | Z = 1] E[Y | Z = 0]
P (T (1) = 1, T (0) = 0) = 1 P (T = 0 | Z = 1) P (T = 1 | Z = 0)=
P (T (1) = 1, T (0) = 0)
= 1 (1 P (T = 1 | Z = 1)) P (T = 1 | Z = 0)
E[Y | Z = 1] E[Y | Z = 0]
= P (T = 1 | Z = 1) P (T = 1 | Z = 0)=
E[T | Z = 1] E[T | Z = 0]
= E[T | Z = 1] E[T | Z = 0]<latexit 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Brady Neal

Nonparametric Identification of Local ATE

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Deriving Local ATE Identification<latexit 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with Monotonicity Assumption

E[Y (Z = 1) Y (Z = 0)]= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 1] P (T (1) = 0, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 1] P (T (1) = 1, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 0] P (T (1) = 0, T (0) = 0)
= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)(Compliers)(Defiers)(Always-takers)(Never-takers)

Local ATE (LATE) or Complieraverage causal effect (CACE): E[Y (T = 1)

E[Y (Z = 1) Y (Z = 0)]
Y (T = 0) | T (1) = 1, T (0) = 0] =P (T (1) = 1, T (0) = 0)
E[Y | Z = 1] E[Y | Z = 0]
P (T (1) = 1, T (0) = 0) = 1 P (T = 0 | Z = 1) P (T = 1 | Z = 0)=
P (T (1) = 1, T (0) = 0)
= 1 (1 P (T = 1 | Z = 1)) P (T = 1 | Z = 0)
E[Y | Z = 1] E[Y | Z = 0]
= P (T = 1 | Z = 1) P (T = 1 | Z = 0)=
E[T | Z = 1] E[T | Z = 0]
= E[T | Z = 1] E[T | Z = 0]<latexit 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Brady Neal

Nonparametric Identification of Local ATE

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Deriving Local ATE Identification<latexit 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with Monotonicity Assumption

E[Y (Z = 1) Y (Z = 0)]= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 1] P (T (1) = 0, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 1] P (T (1) = 1, T (0) = 1)
+ E[Y (Z = 1) Y (Z = 0) | T (1) = 0, T (0) = 0] P (T (1) = 0, T (0) = 0)
= E[Y (Z = 1) Y (Z = 0) | T (1) = 1, T (0) = 0] P (T (1) = 1, T (0) = 0)(Compliers)(Defiers)(Always-takers)(Never-takers)

Local ATE (LATE) or Complieraverage causal effect (CACE): E[Y (T = 1)

E[Y (Z = 1) Y (Z = 0)]
Y (T = 0) | T (1) = 1, T (0) = 0] =P (T (1) = 1, T (0) = 0)
E[Y | Z = 1] E[Y | Z = 0]
P (T (1) = 1, T (0) = 0) = 1 P (T = 0 | Z = 1) P (T = 1 | Z = 0)=
P (T (1) = 1, T (0) = 0)
= 1 (1 P (T = 1 | Z = 1)) P (T = 1 | Z = 0)
E[Y | Z = 1] E[Y | Z = 0]
= P (T = 1 | Z = 1) P (T = 1 | Z = 0)=
E[T | Z = 1] E[T | Z = 0]
= E[T | Z = 1] E[T | Z = 0]<latexit 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sha1_base64="Bbr3Wp37P7BYg47sjFTGHhhGVGA=">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</latexit>

Brady Neal

Nonparametric Identification of Local ATE

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Nonparametric Identification of LATE
Under Monotonicity Assumption
Local ATE (LATE) or complier average causal effect (CACE):<latexit sha1_base64="6+dePFDORB6ijcHK8mqFYEDqM/k=">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</latexit>

E[Y (T = 1)

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

Nonparametric Identification of Local ATE

E[Y | Z = 0]
E[T | Z = 0]

29 / 33

Nonparametric Identification of LATE
Under Monotonicity Assumption
Local ATE (LATE) or complier average causal effect (CACE):<latexit sha1_base64="6+dePFDORB6ijcHK8mqFYEDqM/k=">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</latexit>

E[Y (T = 1)
<latexit sha1_base64="xc6uTgr35ljdnf3+YuNhkMh3+C4=">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</latexit>

E[Y (T = 1)

Brady NealE[Y | Z = 1]
Y (T = 0) | T (1) = 1, T (0) = 0] =E[T | Z = 1]
Y (T = 0)] (ATE, for contrast)

Nonparametric Identification of Local ATE

E[Y | Z = 0]
E[T | Z = 0]

29 / 33

Nonparametric Identification of LATE
Under Monotonicity Assumption
Local ATE (LATE) or complier average causal effect (CACE):<latexit sha1_base64="6+dePFDORB6ijcHK8mqFYEDqM/k=">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</latexit>

E[Y (T = 1)
<latexit sha1_base64="xc6uTgr35ljdnf3+YuNhkMh3+C4=">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</latexit>

E[Y (T = 1)

Brady NealE[Y | Z = 1]
Y (T = 0) | T (1) = 1, T (0) = 0] =E[T | Z = 1]
Y (T = 0)] (ATE, for contrast)

Nonparametric Identification of Local ATE

E[Y | Z = 0]
E[T | Z = 0]

29 / 33

Nonparametric Identification of LATE
Under Monotonicity Assumption
Local ATE (LATE) or complier average causal effect (CACE):<latexit sha1_base64="6+dePFDORB6ijcHK8mqFYEDqM/k=">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</latexit>

E[Y (T = 1)
<latexit sha1_base64="xc6uTgr35ljdnf3+YuNhkMh3+C4=">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</latexit>

E[Y (T = 1)

E[Y | Z = 1]
Y (T = 0) | T (1) = 1, T (0) = 0] =E[T | Z = 1]
Y (T = 0)] (ATE, for contrast)

E[Y | Z = 0]
E[T | Z = 0]This is the Wald estimand!

Brady Neal

Nonparametric Identification of Local ATE

29 / 33

Nonparametric Identification of LATE
Under Monotonicity Assumption
Local ATE (LATE) or complier average causal effect (CACE):<latexit sha1_base64="6+dePFDORB6ijcHK8mqFYEDqM/k=">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</latexit>

E[Y (T = 1)
<latexit sha1_base64="xc6uTgr35ljdnf3+YuNhkMh3+C4=">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</latexit>

E[Y (T = 1)

Problems:

Brady NealE[Y | Z = 1]
Y (T = 0) | T (1) = 1, T (0) = 0] =E[T | Z = 1]
Y (T = 0)] (ATE, for contrast)

E[Y | Z = 0]
E[T | Z = 0]This is the Wald estimand!

Nonparametric Identification of Local ATE

29 / 33

Nonparametric Identification of LATE
Under Monotonicity Assumption
Local ATE (LATE) or complier average causal effect (CACE):<latexit sha1_base64="6+dePFDORB6ijcHK8mqFYEDqM/k=">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</latexit>

E[Y (T = 1)
<latexit sha1_base64="xc6uTgr35ljdnf3+YuNhkMh3+C4=">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</latexit>

E[Y (T = 1)

E[Y | Z = 1]
Y (T = 0) | T (1) = 1, T (0) = 0] =E[T | Z = 1]
Y (T = 0)] (ATE, for contrast)

Problems:• Monotonicity isn’t always satisfied

Brady NealE[Y | Z = 0]
E[T | Z = 0]This is the Wald estimand!

Nonparametric Identification of Local ATE

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Nonparametric Identification of LATE
Under Monotonicity Assumption
Local ATE (LATE) or complier average causal effect (CACE):<latexit sha1_base64="6+dePFDORB6ijcHK8mqFYEDqM/k=">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</latexit>

E[Y (T = 1)
<latexit sha1_base64="xc6uTgr35ljdnf3+YuNhkMh3+C4=">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</latexit>

E[Y (T = 1)

E[Y | Z = 1]
Y (T = 0) | T (1) = 1, T (0) = 0] =E[T | Z = 1]
Y (T = 0)] (ATE, for contrast)

E[Y | Z = 0]
E[T | Z = 0]This is the Wald estimand!

Problems:• Monotonicity isn’t always satisfied• Even if it is, we are usually interested in the average effect across thewhole population (ATE), rather than just the compliers (CACE)
Brady Neal

Nonparametric Identification of Local ATE

29 / 33

Question:
What causal estimand can we nonparametrically
identify with an instrument and themonotonicity assumption?

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

More General Settings for the ATE

31 / 33

Nonparametric Outcome with Additive Noise<latexit sha1_base64="va00cDgPgeSGB8x25gx1KuU8QT0=">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</latexit>

Y := f (T, W ) + U

Brady Neal

More General Settings for the ATE

32 / 33

Nonparametric Outcome with Additive Noise<latexit sha1_base64="va00cDgPgeSGB8x25gx1KuU8QT0=">AAAUcnicrVjpbttGEN6kV+JeTuJf7R8mSoEUpVXJTeGigAADdoKiaAAH8NVEgSGRK4sQr/Kw6gh6hL5Gn6Z/W6Dv0QfozLdLkdRByoElSFzNznwzO9fuqh+6Tpy0Wv/euv3e+x98+NGduxsff/LpZ59v3rt/EgdpZMljK3CD6Kzfi6Xr+PI4cRJXnoWR7Hl9V572R/s8f3opo9gJ/KPkKpRvvN6F7wwcq5cQ6XxzrxukiRV40vixYwyedBMSTjzpJ6bRtQJ/EKS+TeJfG98Y3dQP+rGMLqVdmDnfbLSaLbyMxUFbDxpCvw6De3f/EF1hi0BYIhWekMIXCY1d0RMxvV+LtmiJkGhvxIRoEY0czEsxFRskmxKXJI4eUUf0fUG/XmuqT78ZM4a0RVpc+kQkaYivNI9N4wGo6sn6jQLvKh0TYLONV/Tsa0yPqIkYErVOLuNcX65Pb69m1Qmt4Qes1qGVhKCwH6zSmgf0dOl3Qivk7yvilDSySSqikUU0l6iKwjoieirPs2+GiEQPfJJGbHWV3VVW83xA7xFh9Wgcw1K21RDPdVx8aJawlXlcxHQ14u+0QmVfFdZgtgYHcVerYN4joozEWxqVkat0FrNrNVeikadLdDF/Ag6fuGOiBMh8h/znEEc5Vy3CZB09RPICawmRQU2N/DPms2wOKZ7bsCdG1AzkmwM9oa6s3O7MTpeefWBHJD+huSH0sC9MskcCO0L2Zl40wR3RrzF+WbDRmqM3UZkcT5PWxNllEm5AM84MS3lC+TOzSFk3IZqh39viF+iXFHcTXuG8Nkk2gIUqi2Po8nXsO9RX2C8efTOVPZtRTOjwYMcQ2TIiWq5PIbG+b2kVMerH1agTerrwWggUE9rZP2OMPayJI5pC95jGNvSwtEG9ril2tR3Tklb2jYPutKjVQP5x7arOOK+V5XxUqqGzsgNNLfFUr5gjwHYXvWbpbqh6cozVTees6sP+vGtwjpbtY6k8PuVY2cjGIaTKuOzF5etctpYd1ImJWLLvL4ijA6mBpseozVBr2ijU376ukgDxsPD0dE6oemdt47mZCdE97SmX+FSd857EM78RZg851YX3ivNViM+0fLYbcMefgFotd4LKKstGNJrMZqrlHYwk/KNGHHGFF8LXIdbycPYxCjPrYt8Mnr+AyBmbaKR10W3kxvKVH4kG9YIGZVXZnxxppkQ4nYQ0fjzjfEy4zDuCFp8oXOGT2fy0NgYHRJlqjw0wl9VhboODzmeDs6szT3GoikrmOLiqq7VyXi7LG0WvlrVn+V6WVfRqWaZcIjcd3Q9iYJ3VyCWIA0t4hSw9qpEKsCcourL51zXs87HXpjovIm3haY2khx6jVhXoqnxRa1+CLhhpn7DMq3fw4CX63xQnoOv6MZdNruXNXO7qnXyay4+v6dlc0rumf3PJtzWSEnuDM6OxzMvammKuw1oulU91e4Kk3SQ7v6hqM/HdJy/ynm3rzmTo/YazQu2OHfQAc63+vLjvcCW3a+s4Rfftow9F0GxX1M7xjaHlEUzXWF+MfM/vl+vU1zK5+bzJzxDZ2T8u7C4OsPN7wQRnuLHeh7LT3DbipyLwE3WyF3RC4vG++I5OZ236fl7as9ZF5VtSqrtdEXeHkHdptznArnR93FDfZbZLN56yjmekIftM9d0w53b1CX4V0nX0LkdWOWMv3CTmNamz4pC41P3pqgYxz8LleNXxXY6s/ucIcH7Pb1E3gVy8Ia6Hl+fM6vt83b8Mq+++IT0H9O2iH8ea9wBZ56IvS+oQ6n6uzj77C/80cP7uUFVw7k4e3Z/TV8afv7te4O6ZEkfOo+4gCaQVrYi46hYcQibRZ70YN8b8Xtws3XMV3rxtcSGvPI3X0b1I6l6/cb7ZaM//d7Y4ONlptr9vtl4+bezt6f/V7ogvxSPxhPy0K/aoMg/Jr5b4U/wl/hb/PPhv64uth1v6T7jbt7TMA1F6bZn/AxTm/8o=</latexit>

Y := f (T, W ) + U
where f can be some very flexible model such as a deep neural network(see, e.g., Hartford et al. (2017), Xu et al. (2020), and references therein)<latexit sha1_base64="MLErlg+XMWPJ1veUnjomczgsjfY=">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</latexit>

Brady Neal

More General Settings for the ATE

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Nonparametric Outcome with Additive Noise(Semi-parametric)<latexit sha1_base64="va00cDgPgeSGB8x25gx1KuU8QT0=">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</latexit>

Y := f (T, W ) + U
where f can be some very flexible model such as a deep neural network(see, e.g., Hartford et al. (2017), Xu et al. (2020), and references therein)<latexit sha1_base64="MLErlg+XMWPJ1veUnjomczgsjfY=">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</latexit>

Brady Neal

More General Settings for the ATE

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Set Identification of ATE(rather than point identification)

Brady Neal

More General Settings for the ATE

33 / 33

Set Identification of ATE(rather than point identification)
Although, we can’t point identify the ATE, we can bound it.
See Section 8.2 of Pearl’s Causality book

Brady Neal

More General Settings for the ATE

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Set Identification of ATE(rather than point identification)
Although, we can’t point identify the ATE, we can bound it.
See Section 8.2 of Pearl’s Causality book
Similarly, we can relax the additive noise assumption if we are content with
set identification, rather than point identification.
Previous slide (minus W):

Brady Neal<latexit sha1_base64="l2lw2Z3j8fSQk3HKMLnZixUCxAM=">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</latexit>

Y := f (T ) + U

More General Settings for the ATE

33 / 33

Set Identification of ATE(rather than point identification)
Although, we can’t point identify the ATE, we can bound it.
See Section 8.2 of Pearl’s Causality book
Similarly, we can relax the additive noise assumption if we are content with
set identification, rather than point identification.
Previous slide (minus W):<latexit sha1_base64="l2lw2Z3j8fSQk3HKMLnZixUCxAM=">AAAUZHicrVjrbhtFFJ62XNpQIG3CLyS01EUqwjF2WpQKyVKlpBVCVEql3EpdRfbuOF55b+wlJrX8CPznafgLr8AL8Byc882sd9eXXbuKLXvHZ875zplzmxn3AseO4mbz3xs3b33w4Ucf376z8cndTz/7fPPe/ZPIT0JTHpu+44dnvW4kHduTx7EdO/IsCGXX7TnytDfc5/nTSxlGtu8dxVeBfOt2Lzy7b5vdmEjnm7sdP4lN35XGj22j/6gTk3DsSi/+1vjO6CSe34tkeCkt0/f6fuJZBHW+WWs2mngZ84OWHtSEfh369+78ITrCEr4wRSJcIYUnYho7oisier8RLdEUAdHeijHRQhrZmJdiIjZINiEuSRxdog7p+4J+vdFUj34zZgRpk7Q49AlJ0hDfaB6Lxn1Q1ZP1GzneZTrGwGYbr+jZ05guUWMxIGqVXMq5ulyP3m7FqmNaw1Os1qaVBKCwH8zCmvv0dOh3TCvk7yvilDSySCqkkUk0h6iKwjpCeirPs28GiEQXfJJGbHWZ3WVW87xP7yFhdWkcwVK21RAvdFw8aJawlXkcxHQ54u+0QmVfGVZ/ugYbcVerYN4jogzFOxoVkct05rNrOVeskScLdDF/DA6PuCOi+Mh8m/xnE0cxV03CZB1dRPICawmQQQ2N/DPm02wOKJ47sCdC1Azkmw09ga6szO7UToeePWCHJD+muQH0sC/qZI8EdojsTb1YB3dIv0b4ZcJGc4beQGVyPOu0Js6uOuH6NGNPsZQnlD9Ti5R1Y6IZ+r0jfoF+SXGvwyuc13WS9WGhyuIIujwd+zb1FfaLS99MZc+mlDp0uLBjgGwZEi3Tp5BY3/e0igj142jUMT0deC0ASh3a2T8jjF2siSOaQPeIxhb0sLRBva4h9rQdk4JW9o2N7jSv1UD+ce2qzjirleU8VKqhs7INTU3xRK+YI8B2571m6m6oenKE1U1mrOrB/qxrcI4W7WOpLD7FWFnIxgGkirjsxcXrXLSWXdRJHbFk318QRxtSfU2PUJuB1rSRq799XSU+4mHi6eqcUPXO2kYzM2Oiu9pTDvGpOuc9iWd+I8wucqoD7+XnyxCfa/l0N+COPwa1XO4ElVWUDWk0ns6Uy9sYSfhHjTjiCi+ArwOs5evpx8jNrIp9PXjeHCJnbKyRVkW3kBuLV34katQLapRVRX9ypJkS4nQS0PjhlPMh4TLvEFo8onCFj6fzk8oYHBBloj3Wx1xah5kNNjqfBc6OzjzFoSoqnuHgqi7Xynm5KG8UvVzWmuZ7UVbRy2WZconctHU/iIB1ViEXIw4s4eay9KhCyseeoOjK5tcr2Odhr010XoTawtMKSRc9Rq3K11X5stK+GF0w1D5hmV/fw4OX6H8TnIDW9WMmG6/lzUzu6r18msmP1vRsJumu6d9M8l2FpMTeYE9pLPOqsqaY67CSS+VT1Z4gaTdJzy+q2ur47pEXec+2dGcy9H7DWaF2xzZ6QH2l/jy/73AltyrrOEH37aEPhdBsldTO8bWhZRFMVlhfhHzP7per1Nciudm8yc4Q6dk/yu0uNrCze8EYZ7iR3ofS09wO4qci8BN1spd0QuLxvnhMp7MWfb8o7FmrovItKdHdLo+7S8h7tNscYFdaHzfQd5mdwo2nqOM5aUg/E303zLgdfYJfhrSO3sXIKmesuZvErCZ1VhwQl7o/XVUgZlm4GK88vouR1f8cPs7v2S3qOpDzN8TV8LKcWX6fr/qXYfndN6Bnn74d9ONI8x4g6xz0ZUkdQt3P1dlnf+6fBs7fXaoKzt3xg/sz+or4s3fXC9w9E+LIeNQdJIa0ouURl92CA8jE+qwX4caY3YsbhXuuwpu1Lcrllavx2roXSd3rN843a63Z/87mBye7jdYPjearJ7VnT/X/arfFl+KBeER+2hPPqDIPya+m+FP8Jf4W/2z9t313e2v7C8V684aW2RKF1/ZX/wPSiPno</latexit>

Y := f (T ) + U

Relaxing additive noise assumption: Y := f (T, U )<latexit sha1_base64="y7n/LNP3rKTpEU+seY82qrvZ5eM=">AAAUY3icrVjpbttGEN6kV+K0jeP4X1GAjVIgBWhVclM4KCAggJ2gKBrAAXy1UWBI5MoixKs8rDqCHqG/+zT92z5DH6Dv0ZlvlyKpg5QDS5C4mp35Znau3VU/dJ04abX+vXX7gw8/+viTO3c37n362ef3Nx9sncRBGlny2ArcIDrr92LpOr48TpzElWdhJHte35Wn/dE+z59eyih2Av8ouQrlW6934TsDx+olRDrfbHeDNLECTxo/dIzBk25Cwokn/cQ0uqkf9GMZXUrbCvxBkPo2AX1zvtloNVt4GYuDth40hH4dBg/u/iG6whaBsEQqPCGFLxIau6InYnq/EW3REiHR3ooJ0SIaOZiXYio2SDYlLkkcPaKO6PuCfr3RVJ9+M2YMaYu0uPSJSNIQX2sem8YDUNWT9RsF3lU6JsBmG6/o2deYHlETMSRqnVzGub5cn95ezaoTWsMzrNahlYSgsB+s0poH9HTpd0Ir5O8r4pQ0skkqopFFNJeoisI6Inoqz7NvhohED3ySRmx1ld1VVvN8QO8RYfVoHMNSttUQL3VcfGiWsJV5XMR0NeLvtEJlXxXWYLYGB3FXq2DeI6KMxDsalZGrdBazazVXopGnS3QxfwIOn7hjogTIfIf85xBHOVctwmQdPUTyAmsJkUFNjfwT5rNsDimeO7AnRtQM5JsDPaGurNzuzE6Xnn1gRyQ/obkh9LAvTLJHAjtC9mZeNMEd0a8xflmw0ZqjN1GZHE+T1sTZZRJuQDPODEt5Qvkzs0hZNyGaod874mfolxR3E17hvDZJNoCFKotj6PJ17DvUV9gvHn0zlT2bUUzo8GDHENkyIlquTyGxvm9pFTHqx9WoE3q68FoIFBPa2T9jjD2siSOaQveYxjb0sLRBva4p9rQd05JW9o2D7rSo1UD+ce2qzjivleV8VKqhs7IDTS3xVK+YI8B2F71m6W6oenKM1U3nrOrD/rxrcI6W7WOpPD7lWNnIxiGkyrjsxeXrXLaWXdSJiViy7y+IowOpgabHqM1Qa9oo1N++rpIA8bDw9HROqHpnbeO5mQnRPe0pl/hUnfOexDO/EWYPOdWF94rzVYgvtHy2G3DHn4BaLXeCyirLRjSazGaq5R2MJPyjRhxxhRfC1yHW8tXsYxRm1sW+GTx/AZEzNtFI66LbyI3lKz8SDeoFDcqqsj850kyJcDoJafx4xvmYcJl3BC0+UbjCJ7P5aW0MDogy1R4bYC6rw9wGB53PBmdXZ57iUBWVzHFwVVdr5bxcljeKXi1rz/K9LKvo1bJMuURuOrofxMA6q5FLEAeW8ApZelQjFWBPUHRl8y9r2Odjr011XkTawtMaSQ89Rq0q0FX5qta+BF0w0j5hmV/fw4OX6H9TnICu68dcNrmWN3O5q/fyaS4/vqZnc0nvmv7NJd/VSErsDc6MxjKva2uKuQ5ruVQ+1e0JknaT7Pyiqs3Ed5+8yHu2rTuTofcbzgq1O3bQA8y1+vPivsOV3K6t4xTdt48+FEGzXVE7xzeGlkcwXWN9MfI9v1+uU1/L5ObzJj9DZGf/uLC7OMDO7wUTnOHGeh/KTnM7iJ+KwI/UyV7RCYnH++I7Op216ftlac9aF5VvSanudkXcXULeo93mALvS9XFDfZfZKd14yjpekIbsM9V3w5zb1Sf4VUjX0bscWeWMvXCTmNekzopD4lL3p6saxDwLl+NVx3c5svqfI8D5Pb9F3QRy8Ya4Hl6eM6vv83X/Mqy++4b0HNC3i34ca94DZJ2LviypQ6j7uTr77C/808D5u0tVwbk7ebQ1p6+MP393vcDdMyWOnEfdQRJIK1oRcdUtOIRMos96MW6M+b24WbrnKrx52+JCXnkar6N7kdS9fuN8s9Ge/+9scXCy22x/32y9ftp4/kz/r3ZHfCEeiSfkpz3xnCrzkPxqiT/FX+Jv8c/D/7bvbW9tbyvW27e0zENRem1/+T8uv/m/</latexit>

Brady Neal

More General Settings for the ATE

33 / 33

Set Identification of ATE(rather than point identification)
Although, we can’t point identify the ATE, we can bound it.
See Section 8.2 of Pearl’s Causality book
Similarly, we can relax the additive noise assumption if we are content with
set identification, rather than point identification.
Previous slide (minus W):<latexit sha1_base64="l2lw2Z3j8fSQk3HKMLnZixUCxAM=">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</latexit>

Y := f (T ) + U

Relaxing additive noise assumption: Y := f (T, U )<latexit sha1_base64="y7n/LNP3rKTpEU+seY82qrvZ5eM=">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</latexit>

See Kilbertus et al. (2020) and references therein
Brady Neal

More General Settings for the ATE

33 / 33