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❲✐❧❞ ❇♦♦tstr❛♣ ■♥❢❡r❡♥❝❡ ❢♦r ❲✐❧❞❧② ❉✐✛❡r❡♥t ❈❧✉st❡r ❙✐③❡s ▼❛tt❤❡✇ ❉✳ ❲❡❜❜ ❖❝t♦❜❡r ✾✱ ✷✵✶✸

■♥tr♦❞✉❝t✐♦♥ ❚❤✐s ♣❛♣❡r ✐s ❥♦✐♥t ✇✐t❤✿ ❏❛♠❡s ●✳ ▼❛❝❑✐♥♥♦♥ ❉❡♣❛rt♠❡♥t ♦❢ ❊❝♦♥♦♠✐❝s ◗✉❡❡♥✬s ❯♥✐✈❡rs✐t② ❑✐♥❣st♦♥✱ ❖♥t❛r✐♦✱ ❈❛♥❛❞❛ ❑✼▲ ✸◆✻ ❈♦♥tr✐❜✉t✐♦♥s✿ Pr❡s❡♥ts ▼♦♥t❡ ❈❛r❧♦ ❡✈✐❞❡♥❝❡ t❤❛t ♦✈❡rt✉r♥s t❤❡ ❵r✉❧❡ ♦❢ ✹✷✬ ❙❤♦✇s t❤❛t ❉✐❉ ❛♥❞ ❈❘❱❊ ❡st✐♠❛t✐♦♥ ✇♦r❦s ♣♦♦r❧② ✇❤❡♥ t❤❡ ♣r♦♣♦rt✐♦♥ ♦❢ ❝❧✉st❡rs tr❡❛t❡❞ ✐s ✈❡r② s♠❛❧❧ ♦r ✈❡r② ❧❛r❣❡

❙t❛♥❞❛r❞ ▼❡t❤♦❞ ❢♦r ❉✐✛❡r❡♥❝❡✲✐♥✲❉✐✛❡r❡♥❝❡s ❙✉♣♣♦s❡ ②♦✉ ✇❛♥t t♦ ❡st✐♠❛t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♥❡❛r ❞✐✛❡r❡♥❝❡✲✐♥✲❞✐✛❡r❡♥❝❡s ❡q✉❛t✐♦♥ Y igt = β ✵ + β ✶ ∗ treat g + β ✷ ∗ year t + β ✸ ∗ treat g ∗ year t + X igt γ + ǫ igt ✭✶✮ Y igt ♦❜s❡r✈❛t✐♦♥ ❢♦r ♣❡rs♦♥ ✐ ✐♥ ❣r♦✉♣ ❣ ❛♥❞ t✐♠❡ t treat g ❞✉♠♠② ❢♦r ✐❢ t❤❡ ♣❡rs♦♥ ✐s ✐♥ t❤❡ tr❡❛t♠❡♥t ❣r♦✉♣ year t ❞✉♠♠② ✐❢ ✐♥ t✐♠❡ ♣❡r✐♦❞ ❛❢t❡r tr❡❛t♠❡♥t X igt ♦t❤❡r ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s treat g ∗ year t ✐s t❤❡ ❉✐❉ t❡r♠

❊st✐♠❛t✐♥❣ β ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ✐♥❢❡r❡♥❝❡ ❢♦r t❤❡ ❖▲❙ ❡st✐♠❛t❡ ♦❢ ˆ β ✸ ❲✐t❤ t❤❡ ❛ss✉♠♣t✐♦♥s t❤❛t ❞❛t❛ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦✈❡r ❣✱ ❜✉t ❡rr♦rs ❛r❡ ❝♦rr❡❧❛t❡❞ ✇✐t❤✐♥ ❝❧✉st❡r E [ u g ] = ✵ ′ E [ u g u g ] = Σ g ✱ ✈✐♦❧❛t❡s t❤❡ ✐✳✐✳❞✳ ❛ss✉♠♣t✐♦♥ ′ E [ u g u h ] = ✵ ❢♦r ❝❧✉st❡r h � = g ❲❤❡♥ t❤❡♥ ❤❛✈❡ √ N (ˆ β − β ) ∼ ℵ [ ✵ , NV [ˆ β ]]

❈❧✉st❡r ❘♦❜✉st ❱❛r✐❛♥❝❡ ❊st✐♠❛t♦r ✭❈❘❱❊✮   G V CR [ˆ ˆ ′ X ) − ✶ � ′ ′  ( X ′ X ) − ✶ β ] = ( X X g ˜ u g ˜ u g X  g g = ✶ ✐♥ t❤❡ s✐♠♣❧❡st ❝❛s❡ t❤❡ ❖▲❙ r❡s✐❞✉❛❧s ❛r❡ ✉s❡❞ ˜ u g = ˆ u g = y g − X g β g = ✶ ˜ U g ˜ ✐♥ ♦t❤❡r ❝❛s❡s � G ′ g ✱ ✐s r❡♣❧❛❝❡❞ ❜② � G ′ ′ g = ✶ X g ˆ u g ˆ u g X U g ❙t❛t❛ ✉s❡s✿ � ✶ N g � ˜ � U g = ˆ u ig , X g i = ✶

❆s②♠♣t♦t✐❝s ❯♥❞❡r❧②✐♥❣ ❈❘❱❊ ●❡♥❡r❛❧ r❡s✉❧ts ♦♥ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ❡st✐♠❛t✐♦♥ ✐♥ ❲❤✐t❡ ✭✶✾✽✹✮ ✐♠♣❧② ♦❢ t❤❡ ❈❘❱❊ ✐s ❝♦♥s✐st❡♥t ✉♥❞❡r t❤r❡❡ ❦❡② ❛ss✉♠♣t✐♦♥s✿ ❆✶✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ❝❧✉st❡rs ❣♦❡s t♦ ✐♥✜♥✐t②✳ ❆✷✳ ❚❤❡ ✇✐t❤✐♥✲❝❧✉st❡r ❝♦rr❡❧❛t✐♦♥ ✐s ❝♦♥st❛♥t ❛❝r♦ss ❝❧✉st❡rs✳ ❆✸✳ ❚❤❡ ✐♥❞✐✈✐❞✉❛❧ ❝❧✉st❡rs ❝♦♥t❛✐♥ ❛♥ ❡q✉❛❧ ♥✉♠❜❡r ♦❢ ♦❜s❡r✈❛t✐♦♥s✳ ❈❛rt❡r✱ ❙❝❤♥❡♣❡❧ ❛♥❞ ❙t❡✐❣❡r✇❛❧❞ ✭✷✵✶✷✮ r❡❧❛① ❆✶ ❛♥❞ ❆✷✳ ❚❤✐s t❛❧❦ ❝♦♥❝❡r♥s ❆✸✳

❘❡❧❛t❡❞ ▲✐t❡r❛t✉r❡ ❈❧✉st❡r❡❞ ❊rr♦rs ❑❧♦❡❦ ✭✶✾✽✶✮ ▼♦✉❧t♦♥ ✭✶✾✾✵✮ ■♥❢❡r❡♥❝❡ ✐♥ ❉✐✛❡r❡♥❝❡✲✐♥✲❉✐✛❡r❡♥❝❡ ❈♦♥❧❡② ❛♥❞ ❚❛❜❡r ✭✷✵✶✶✮ ❉♦♥❛❧❞ ❛♥❞ ▲❛♥❣ ✭✷✵✵✼✮ ❇♦♦tstr❛♣ ■♥❢❡r❡♥❝❡ ✐♥ ❉✐✛❡r❡♥❝❡✲✐♥✲❉✐✛❡r❡♥❝❡s ❇❡rtr❛♥❞✱ ❉✉✢♦ ❛♥❞ ▼✉❧❧❛✐♥❛t❤❛♥ ✭✷✵✵✹✮ ❈❛♠❡r♦♥✱ ●❡❧❜❛❝❤ ❛♥❞ ▼✐❧❧❡r ✭✷✵✵✽✮ ❲❡❜❜ ✭✷✵✶✸✮

❘❡❥❡❝t✐♦♥ ❋r❡q✉❡♥❝✐❡s ❜② ◆✉♠❜❡r ♦❢ ❈❧✉st❡rs ◆✉♠❜❡r ♦❢ ●r♦✉♣s ✭●✮ ✺ ✶✵ ✶✺ ✷✵ ✷✺ ✸✵ ❖▲❙ ∼ N ( ✵ , ✶ ) ✵✳✹✻✽ ✵✳✹✽✻ ✵✳✹✾✸ ✵✳✹✾✹ ✵✳✹✽✾ ✵✳✹✾✾ ❈❘❱❊ ∼ N ( ✵ , ✶ ) ✵✳✷✶✶ ✵✳✶✸✸ ✵✳✶✵✽ ✵✳✵✾✹ ✵✳✵✽✹ ✵✳✵✽✵ ❈❘❱❊ ∼ T ( G − ✶ ) ✵✳✶✵✵ ✵✳✵✾✵ ✵✳✵✽✶ ✵✳✵✼✺ ✵✳✵✼✵ ✵✳✵✻✾ ◆♦t❡s✿ ❘❡♣❧✐❝❛t✐♦♥ ♦❢ s✐♠✉❧❛t✐♦♥s ♣❡r❢♦r♠❡❞ ❜② ❈●▼✳ ❘❡❥❡❝t✐♦♥ ❢r❡q✉❡♥❝✐❡s ❡st✐♠❛t❡❞ ✇✐t❤ ✺✵✱✵✵✵ r❡♣❧✐❝❛t✐♦♥s✳

❚❤❡ ✏❘✉❧❡ ♦❢ ✹✷✑ ❈❧❛✐♠✿ ✏■♥ ❛ ❉❉ s❝❡♥❛r✐♦ ✇❤❡r❡ ②♦✉✬❞ ❧✐❦❡ t♦ ❝❧✉st❡r ♦♥ st❛t❡ ♦r s♦♠❡ ♦t❤❡r ❝r♦ss✲s❡❝t✐♦♥❛❧ ❞✐♠❡♥s✐♦♥✱ t❤❡ r❡❧❡✈❛♥t ❞✐♠❡♥s✐♦♥ ❢♦r ❝♦✉♥t✐♥❣ ❝❧✉st❡rs ✐s t❤❡ ♥✉♠❜❡r ♦❢ st❛t❡s ♦r ❝r♦ss✲s❡❝t✐♦♥❛❧ ❣r♦✉♣s✳ ❚❤❡r❡❢♦r❡✱ ❢♦❧❧♦✇✐♥❣ ❉♦✉❣❧❛s ❆❞❛♠✬s ❞✐❝t✉♠ t❤❛t t❤❡ ❛♥s✇❡r t♦ ❧✐❢❡✱ t❤❡ ✉♥✐✈❡rs❡✱ ❛♥❞ ❡✈❡r②t❤✐♥❣ ✐s ✹✷✱ ✇❡ ❜❡❧✐❡✈❡ t❤❡ q✉❡st✐♦♥ ✐s✿ ❍♦✇ ♠❛♥② ❝❧✉st❡rs ❛r❡ ❡♥♦✉❣❤ ❢♦r r❡❧✐❛❜❧❡ ✐♥❢❡r❡♥❝❡ ✉s✐♥❣ t❤❡ st❛♥❞❛r❞ ❝❧✉st❡r ❛❞❥✉st♠❡♥t❄✑ ❆♥❣r✐st ❛♥❞ P✐s❝❤❦❡✱ ▼♦st❧② ❍❛r♠❧❡ss ❊❝♦♥♦♠❡tr✐❝s ✱ ♣❛❣❡ ✸✶✾✳ ❘❡s♣♦♥s❡✿ ❚r✉❡ ✐❢ ❝❧✉st❡rs ❛r❡ ♦❢ ❡q✉❛❧ s✐③❡✱ ❢❛❧s❡ ♦t❤❡r✇✐s❡✳

❚❤❡ ✏❘✉❧❡ ♦❢ ✹✷✑ ✻ ✶✵ ✷✵ ✺✵ ❖▲❙ ✵✳✸✽✸ ✵✳✹✹✸ ✵✳✸✾✵ ✵✳✹✾✵ ❈❘❱❊ ✵✳✶✺✸ ✵✳✶✵✺ ✵✳✵✽✵ ✵✳✵✺✺ ◆♦t❡s✿ ❇❡rtr❛♥❞✱ ❉✉✢♦ ❛♥❞ ▼✉❧❧❛✐♥❛t❤❛♥ ✭✷✵✵✹✮ ▼♦♥t❡ ❈❛r❧♦ ❙✐♠✉❧❛t✐♦♥s ✉s✐♥❣ ❈P❙ ❛❣❣r❡❣❛t❡ ❞❛t❛✳ ❙✐♠✉❧❛t✐♦♥s s✉❝❤ ❛s t❤❡s❡✱ ❛♥❞ t❤♦s❡ ❜② ❈❛♠❡r♦♥✱ ●❡❧❜❛❝❤ ❛♥❞ ▼✐❧❧❡r ✭✷✵✵✽✮ ❤❛✈❡ ❧❡❞ t♦ ❛ s❤♦rt❤❛♥❞ ❵r✉❧❡ ♦❢ ✹✷✬✱ ✇❤❡♥ ❆✶ ✐s ❛♣♣r♦①✐♠❛t❡❧② s❛t✐s✜❡❞✳ ✏❈✉rr❡♥t ❝♦♥s❡♥s✉s ❛♣♣❡❛rs t♦ ❜❡ t❤❛t G = ✺✵ ✐s ❡♥♦✉❣❤ ❢♦r st❛t❡✲②❡❛r ♣❛♥❡❧ ❞❛t❛✳✑ ❈❛♠❡r♦♥ ❛♥❞ ▼✐❧❧❡r ✭✷✵✶✸✮

Pr♦❝❡❞✉r❡ ❢♦r ❲✐❧❞ ❈❧✉st❡r ❇♦♦tstr❛♣✲t ✶✳ ❡st✐♠❛t❡ ❡q✉❛t✐♦♥ ✶ ❛♥❞ ♦❜t❛✐♥ ❡st✐♠❛t❡s ♦❢ ˆ β ✱ ˆ γ ❛♥❞ ˆ ǫ igt ✶❛✳ ❡st✐♠❛t❡ ❛ r❡str✐❝t❡❞ ✈❡rs✐♦♥ ♦❢ ❡q✉❛t✐♦♥ ✶ ✇❤✐❝❤ ✐♠♣♦s❡s t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s✱ ♦❜t❛✐♥ ˜ ǫ igt ❛♥❞ ❡q✉✐✈❛❧❡♥t ✷✳ ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ t❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ ˆ β ✸ s♦ ❝❛❧❝✉❧❛t❡ t❤❡ t✲st❛t✐st✐❝✱ ˆ t ✱ ✉s✐♥❣ ❝❧✉st❡r r♦❜✉st st❛♥❞❛r❞ ❡rr♦rs ✸✳ ❝❤♦♦s❡ ❛ ♥✉♠❜❡r ♦❢ ❜♦♦tstr❛♣s✱ B ✱ ❛♥❞ ❢♦r ❡❛❝❤ ✐t❡r❛t✐♦♥ ❣❡♥❡r❛t❡ ❛ ♥❡✇ ❜♦♦tstr❛♣ s❛♠♣❧❡ ❢r♦♠ t❤❡ ❜♦♦tstr❛♣ ❉●P✿ igt = ˜ β ✵ + ˜ β ✶ ∗ treat g + ˜ y ∗ β ✷ ∗ year t + ✭✷✮ ˜ β ✸ ∗ treat g ∗ year t + X igt ˜ γ + f (˜ u igt ) v ∗ g , u igt ) tr❛♥s❢♦r♠s t❤❡ i th r❡s✐❞✉❛❧ ✐♥ t✐♠❡ t ❢r♦♠ ❣r♦✉♣ g ✱ ˜ ✇❤❡r❡ f (˜ u igt ✱ ❛♥❞ v g ✐s ❛ ❜♦♦tstr❛♣ ✇❡✐❣❤t✳ ■♠♣♦s❡ t❤❡ ♥✉❧❧ ❜② s❡tt✐♥❣ ˜ β ✸ = ✵