1 ౷ܭతҼՌਪ࿦ͱσʔλղੳ ʙద੾ͳӡ༻Λ໨ࢦͯ͠ʙ தଜɹ஌ൟ 6QEBUFEPO.BZUI 5XJUUFS!UPNPTIJHF@O &NBJMUPNPTIJHFOBLBNVSB!HNBJMDPN

ࠓճͷൃදͷαϚϦʔ ‣ ҼՌਪ࿦ͷେࡶ೺ͳྺ࢙ͱɺ࠷ۙͷ܏޲είΞΛ༻͍ͨҼՌਪ࿦ͷ٧Ίͷ؁ ͞΋આ໌͠·͢ʢQ஋ͷޡ༻ͱࣅͨΑ͏ͳงғؾΛײ͡Δ෦෼͕͋Δʣɻ ‣ ܏޲είΞʹΑΔʮٯ֬཰ॏΈ෇͚ʯͱ͸ຊ࣭తʹԿΛҙຯ͢Δͷ͔ΛվΊ ͯղઆ͢Δʢڞมྔ௼Γ߹͍ͷࢹ఺ʣɻ ‣ ಘΒΕͨҼՌޮՌͷਪఆྔʢ*18ਪఆྔʣΛͲͷΑ͏ʹධՁ͢Ε͹Α͍ͷ͔ ʹ͍ͭͯղઆ͢Δɻ ‣ ҼՌਪ࿦Λద੾ʹӡ༻͢ΔͨΊʹɺղੳऀ͕౿Ή΂͖ϓϩηεʹ͍ͭͯղઆ ͢Δɻ ‣ ʮ͓·͚ʯ ‣ ۙ೥ͷ౷ܭతҼՌਪ࿦ͷൃలʹ͍ͭͯ֓આ͢Δɻ !2 ʮΞΧσϛΞࢹ఺͔ΒݟΔ౷ܭతҼՌਪ࿦ͷద੾ͳӡ༻ͱɺ ࣮σʔλղੳϓϩηεʹ͍ͭͯʯ

໨࣍ ‣ ܏޲είΞ͸ສೳༀͳͷ͔ʂʁ ‣ ϥϯμϜԽൺֱ࣮ݧʢ3BOEPNJ[FE$POUSPM5SJBMʣ ‣ ݹయతճؼϞσϧͷഁ୼ ‣ ೣ΋उࢠ΋܏޲είΞɻ ‣ ҼՌਪ࿦ͷຊ࣭ͱ࣮σʔλͷղੳ ‣ ܏޲είΞͷٯ਺ͦ͜ຊ࣭ʂʢ1SPQFOTJUZ4DPSFʣ ‣ σʔλ͕ͳ͍ͳΒɺ͹Β͖ͭΛߟ͑Δʢ#PPUTUSBQ.FUIPEʣ ‣ ฏۉॲஔޮՌͰ͸ͳ͘ɺ0WFSMBQΛߟ͑Δʢ0WFSMBQ8FJHIUTʣ ‣ ౷ܭతҼՌਪ࿦ͷ࠷ۙͷൃల ‣ ͋ͷਓͷҼՌޮՌ͸Ͳ͏ͩΖ͏͔ʁʢ$BVTBM'PSFTU3-FBSOFSʣ ‣ ·ͱΊ !3

ϥϯμϜԽൺֱ࣮ݧ* ‣ ೥୅͔ΒɺσʔλΛ༻͍ͨޮՌͷଌఆ͸ߦΘΕ͖ͯͨɻ ‣ ༗໊ͳͷ͸ɺϑΟογϟʔ͕ߦͬͨ೶ༀ࣮ݧͰ͋Δɻ !4 ංྉ ͋Γ ංྉ ͳ͠ μ0 μ1 μ1 − μ0 = 0?? ɾ೔౰ͨΓҧ͏ ɾਫ͸͚΋ҧ͏ ࡞෺ͷ੒௕ʹؔ࿈͢Δ ม਺͕ංྉΛ͋Γͳ͠Ͱ ҧ͑͹ग़ͯ͘Δ݁Ռ ɾංྉͷޮՌ ɾ౔৕ͷಛ࣭ ͭΛࠞͥͨޮՌ ංྉ ͋Γ ϥϯμϜʹ ංྉ͋Γ ංྉͳ͠ ഑ஔͯ͠ΈΑ͏ ྫ͑͹ίΠϯτεͰʂ

ϥϯμϜԽൺֱ࣮ݧ** ‣ ϥϯμϜԽൺֱ࣮ݧ͸ɺௐ΂͍ͨӨڹʹରͯ͠ɺ ଞͷม਺͕Өڹ͢ΔͷΛ཈͑ΔͨΊͷख๏ɻ ‣ ྫ͑͹ɺ೔র͕࣌ؒɺ౦ʹߦ͚͹ߦ͘΄Ͳ௕͍ ৔߹ɺͦΕ͕ۮવͰ͋ͬͯ΋࡞෺ͷ੒௕΁ͷӨ ڹ͸ආ͚ΒΕͳ͍ɻ ‣ ϥϯμϜԽൺֱ࣮ݧ͸ɺ݁ՌʹӨڹ͕͋Δม਺ ͕ɺංྉ͋Γɾͳ͠ͷ྆ํͰಉ͡෼෍Λ࣋ͭΑ ͏ʹ͢Δͱ͍͏ࢹ఺Ͱઃܭ͞ΕΔɻ ‣ ͭ·ΓɺॲஔͷޮՌΛਪఆ͢Δࡍʹ஫ҙΛ෷͏ ΂͖͜ͱ͸ʮॲஔ܈ʯͱʮରর܈ʯ͕ɺॲஔม ਺Ҏ֎ͷม਺ʹ͍ͭͯۉ࣭ͳूஂͰ͋Δ͔ͱ͍ ͏͜ͱΛߟ͑Δඞཁ͕͋ΔʢΊͪΌͪ͘Όେ ࣄʣɻ !5 ංྉ ͋Γ ංྉ ͳ͠ ๺ ౦ ೆ ੢ ೔র࣌ؒ ౦ 5ʢॲஔม਺ʣcc9ʢڞมྔʣ

ϥϯμϜԽൺֱ࣮ݧ*** ‣ :Λ݁Ռม਺ɺ8Λංྉͷ༗ແɺ9Λ೔র࣌ؒͱ͢Δɻ ‣ ͜ͷͱ͖ɺӈͷਤͷΑ͏ͳؔ܎͕͋Δɻ ‣ ͜͜Ͱɺ4675"ΛԾఆ͢Δɻ ‣ ͜ͷͱ͖ɺ஌Γ͍ͨ݁Ռ͸ ‣ ࣮ࡍʹܭࢉ͢Δฏۉͷࠩ͸ɺ ‣ !6 X W Y = WY1 + (1 − W)Y0 E[Y1 ] − E[Y0 ] E[Y|W = 1] − E[Y|W = 0] E[Y|W = 1] = ∫ × yf(y, x|w = 1)dydx = ∫ × y1 f(y1 , x|w = 1)dy1 dx = ∫ × y1 f(y1 , x)dy1 dx = E[Y1 ] X W 3BOEPNJ[BUJPO 3$5ͳͷͰ ˎڞมྔ͸ॲஔʹ͸Өڹ͠ͳ͍ ˎॲஔ͸જࡏ݁Ռม਺ʹӨڹ͠ͳ͍ Y0 , Y1 , X ⊥ ⊥ W Y0 , Y1 Y0 , Y1

ϥϯμϜԽൺֱ࣮ݧ*7 ‣ ϥϯμϜԽ͸෺ཧతʹɺ݁Ռม਺ʹӨڹ͢Δม਺9͕ɺॲஔม਺8ʹӨڹΛ ༩͑ͳ͍Α͏ʹ͢ΔͨΊͷํ๏ɻ ‣ ղऍΛม͑Ε͹ɺॲஔΛड͚ͨ܈ͱɺड͚ͳ͔ͬͨ܈Ͱɺڞมྔͷ෼෍ͷࠩ ͕ͳ͘ͳΔΑ͏ʹ͍ͯ͠Δɻ ‣ ҼՌਪ࿦ͱ͸ڀۃతʹɺʮ݁Ռม਺ʹӨڹ͢Δม਺Λ͢΂ͯʯॲஔ܈ͱରর ܈Ͱɺ3BOEPNJ[FͰ͖Ε͹ɺͦΕͰ͢΂ͯࣄ଍ΓΔɻ ‣ Ͱ͸ɺ3$5͕Ͱ͖ͳ͍৔߹ͲͷΑ͏ʹ͢Ε͹͍͍ͩΖ͏ʁ ‣ ॲஔม਺8͕ɺ9ͷӨڹΛड͚͍ͯΔ৔߹ɺͲͷΑ͏ʹͯ͠ௐ੔͢Ε͹͍͍ ͩΖ͏ʁ ‣ ٸʹɺݱ୅తͳ࿩ʹߦ͘લʹɺ΋͏গ͠ྺ࢙తͳ࿩Λ͍ͨ͠ɻ ‣ ͜͜Ͱɺొ৔͢Δͷ͕ݹయతͳճؼϞσϧʢ"/07"ʣͰ͋Δɻ !7

ݹయతճؼϞσϧͷഁ୼* ‣ ೶ༀ͕͋Δ২෺ͷ੒௕ʹ༩͑ΔӨڹΛݟΔͨΊɺ ‣ :Λ২෺ͷ੒௕౓߹͍ʢDNʣ ‣ 8Λ೶ༀͷ༗ແ ‣ 9Λ೔ࣹྔʢL8N?ʣ ‣ ͱ͍͏ม਺͕؍ଌ͞Εͨͱ͠·͢ɻ ‣ ೶ༀͷࢄ෍͸೔ࣹྔ͕ଟ͍৔ॴͷํʹଟ͘ࢃ͍ͨͱ͠·͢ɻ ‣ ͜ͷͱ͖ɺ೶ༀΛ·͍ͨ৔߹ͷ੒௕ͱɺࢃ͔ͳ͔ͬͨ৔߹ͷ੒௕ͷજࡏ݁Ռ ม਺͸ɺҎԼͷΑ͏ʹද͞ΕΔͱ͠·͢ʢЋఆ਺F͸ޡࠩʣɻ ‣ ͜ͷͱ͖ɺ:Λ9ͱ8Ͱճؼͨ͠ͱ͖ͷ8ͷճؼ܎਺͸ɺ೶ༀޮՌͰ͠ΐ͏ ͔ʁ !8 Y1 = Xβ + α + e Y0 = Xβ + e

ݹయతճؼϞσϧͷഁ୼** ‣ ͜ͷ৔߹ɺ݁Ռม਺ʹର͢ΔϞσϧ͸ɺ ‣ :9Ќ8Ћޡ߲ࠩ ‣ ͱද͞ΕΔɻ ‣ ઢܗճؼϞσϧΛ౰ͯ͸Ίͨ৔߹ͷɺ8ͷճؼ܎਺ͷਪఆྔ͸ɺ ‣ ౴͑ɿճؼ܎਺͸ҼՌޮՌʹର͢ΔҰகਪఆྔͰ͋Δɻ ‣ ཧ༝͸ɾɾɾ ‣ 8͔Β:΁ͷӨڹ͸ɺજࡏ݁Ռม਺ͷ͕ࠩ9ʹΑΒͣҰఆͷͱ ͖ɺҼՌޮՌ͸ճؼ܎਺ʹҰக͢Δ͔Βɻ !9 argmin ̂ β, ̂ α (Y − Xβ − Wα)2 ⇔ ̂ α = 1 N1 WT(Y − Xβ) → E[Y − Xβ|W = 1] (N → ∞) = E[Wα + ε|W = 1] = α

ݹయతճؼϞσϧͷഁ୼*** ‣ Ͱ͸ɺ࣍ͷΑ͏ͳ৔߹͸Ͳ͏Ͱ͠ΐ͏͔ʁ ‣ ͜ͷ৔߹͸ɺ͏·͘ਪఆͰ͖Δɻ ‣ ॲஔม਺ʹӨڹ͢Δม਺͕ɺҼՌతޮՌ ʹؚ·Εͳ͍ɻ ‣ ͜ͷ৔߹ɺަབྷ͸ଘࡏ͠ͳ͍ɻ !10 Y1 = X1 + X2 + X3 + e Y0 = X1 + X2 + e W ∼ Bernoulli(p) logit(p) = (X1 + X2 )/2 9ͱ9͕ॲஔʹӨڹ͢Δ 9ͱ9͕྆ํͷજࡏ݁Ռม਺ʹӨڹ͠ 9͕:ͷΈʹؚ·ΕΔ

ݹయతճؼϞσϧͷഁ୼*7 ‣ Ͱ͸ɺ࣍ͷΑ͏ͳ৔߹͸Ͳ͏Ͱ͠ΐ͏͔ʁ ‣ જࡏ݁Ռม਺ͷࠩ͸ɺ9ʹґଘɻ ‣ ॲஔม਺΋9ʹґଘɻ ‣ ͜ͷͱ͖ɺճؼ܎਺ͷ஋͸ɺҼՌతޮՌʹର ͯ͠όΠΞεͷ͋ΔਪఆྔͰ͋Δɻ !11 Y1 = X1 + X2 + X3 + e Y0 = X1 + X2 + e W ∼ Bernoulli(p) logit(p) = (X1 + X2 + X3 )/3 9ͱ9ͱ9͕ॲஔʹӨڹ͢Δ 9ͱ9͕྆ํͷજࡏ݁Ռม਺ʹӨڹ͠ 9͕ҼՌޮՌʹӨڹ͢Δ

ݹయతճؼϞσϧͷഁ୼7 ‣ ʲ͜͜·Ͱͷ·ͱΊͱ௥هʳ ‣ ݹయతճؼϞσϧ͸ɺҼՌతޮՌʢજࡏ݁Ռม਺ͷࠩʣͱɺॲஔม਺ͷ྆ ํʹӨڹΛ༩͑Δม਺͕ଘࡏ͢Δ৔߹ɺഁ୼͢Δɻ ‣ ೶ༀͷྫͰઃఆͨ͠ϞσϧͰ͸ɺ೶ༀΛ༩͑ΒΕͯ΋༩͑ΒΕͳͯ͘ ΋ɺ੒௕཰͸ҰఆͰ͋Γɺ೶ༀͰͦΕ͕͋Δఔ౓৳͹ͤΔͱ͍͏Ϟσϧɻ ‣ ͜ͷϞσϧ͸ɺݱ৅ʹରͯ͠ૉ௚ʹϞσϦϯά͍ͯ͠ΔͷͰɺ໰୊͕͋Δ Θ͚Ͱ͸ͳ͍ɻ ‣ Ͱ͸ɺͦ΋ͦ΋ʮҼՌతޮՌʯͱʮॲஔม਺ʯͷ྆ํʹӨڹΛ༩͑Δڞมྔ ͕ଘࡏ͢Δͷ͸ͲͷΑ͏ͳ৔߹͔ɻ ‣ ʲ޿ࠂʳޮՌ͕͋Γͦ͏ͳਓʹ޿ࠂΛ͏ͪɺޮՌͷେ͖͕͞ɺʮॲஔม਺ʯ ʹӨڹ͢Δڞมྔʹґଘ͢Δɻ ‣ ʲӸֶʳλόίΛٵ͏͔ٵΘͳ͍͔͸ɺͦͷਓͷݸਓಛੑʹґଘ͠ɺͨ͹ ͜ʹΑΔ࣬පൃ঱཰΋ݸਓಛੑʹΑܾͬͯఆ͞ΕΔɻ !12

ݹయతճؼϞσϧͷഁ୼7* ‣ ݹయతͳճؼϞσϧ͸ɺಛʹ݁Ռม਺ʹରͯ͠ม਺ͷӨڹ͕ઢܗͰ͋Δ͜ͱ Λظ଴͢Δ͕ɺ࣮ࡍͷσʔλͷղੳʹ͓͍ͯ͸ɺ݁Ռม਺͸ਖ਼ن෼෍ͱ͸ݶ Βͳ͍ɻ ‣ $7ͷ৔߹\ ^CJOBSZ ‣ ਓ਺ͷ৔߹\ ^EJTDSFUF ‣ ͭ·ΓɺҰൠతʹߟ͑Ε͹ɺજࡏ݁Ռม਺ʹର͢Δظ଴஋ΛϞσϦϯά͠ ͯɺͦͷࠩΛܭࢉ͢Δͱ͍͏͜ͱ͕ඞཁʹͳΔɻ ‣ ͜͏ͳͬͯ͘Δͱɺ͍Α͍Αखଓ͖͕൥ࡶͰ͋Γɺؔ਺HͱПͷઃఆΛ͢Δͳ Μͯ͜ͱ͸΍Γͨ͘ͳ͍ʢͨͩɺۙ೥͸ػցֶशͰ͜ΕΛղফ͢ΔΞϓϩʔ ν΋͋Δʣɻ !13 g1 (E[Y1 |X]) = ϕ1 (X) g0 (E[Y0 |X]) = ϕ0 (X) E[Y1 − Y0 ] = E[g−1 1 (ϕ1 (X))] − E[g−1 0 (ϕ0 (X))]

ೣ΋उࢠ΋܏޲είΞʂʂ* ‣ ͦΜͳͱ͖ɺࢲͨͪ͸ʮ܏޲είΞʯͱ͍͏΋ͷΛݟ͚ͭ·͢ɻ ‣ ʮͲ͏΍Βɺॲஔม਺ʹର͢ΔϩδεςΟοΫճؼϞσϧ͔ɺΛ༧ଌ͢ Δػցֶश͔Βਪఆ͞ΕΔॲஔ֬཰ʯΛ༻͍ͯɺॏΈ෇͚͢Ε͹ҼՌతޮՌ ΛਪఆͰ͖ΔΒ͍͠ʂʂʂ ‣ ŠŠŠŠŠ܏޲είΞΛ༻͍ͨղੳQSPDFEVSFŠŠŠŠŠ ‣ ॲஔม਺8Λɺڞมྔ9ʢ؍ଌ͞Ε͍ͯΔ΋ͷʣͰճؼ͢Δɻ ‣ -PHJTUJDSFHSFTTJPO3'9HCPPTU-1FOBMUZΛ࢖͏ ‣ "6$͕Ҏ্͘Β͍ͳΒɺͱΓ͋͑ͣ0,ɻ ‣ ٯ֬཰ॏΈ෇͚Λ͢Δɻ ‣ ͜ͷਪఆ݁ՌΛϨϙʔτ͓ͯ͠͠·͍ ‣ ŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠ !14 ̂ E[Y1 − Y0 ] = N ∑ i=1 Wi Yi π(Xi ) / N ∑ i=1 Wi π(Xi ) − N ∑ i=1 (1 − Wi )Yi 1 − π(Xi ) / N ∑ i=1 1 − Wi 1 − π(Xi ) Y X W

໨࣍ ‣ ܏޲είΞ͸ສೳༀͳͷ͔ʂʁ ‣ ϥϯμϜԽൺֱ࣮ݧʢ3BOEPNJ[FE$POUSPM5SJBMʣ ‣ ݹయతճؼϞσϧͷഁ୼ ‣ ೣ΋उࢠ΋܏޲είΞɻ ‣ ҼՌਪ࿦ͷຊ࣭ͱ࣮σʔλͷղੳ ‣ ܏޲είΞͷٯ਺ͦ͜ຊ࣭ʂʢ1SPQFOTJUZ4DPSFʣ ‣ σʔλ͕ͳ͍ͳΒɺ͹Β͖ͭΛߟ͑Δʢ#PPUTUSBQ.FUIPEʣ ‣ ฏۉॲஔޮՌͰ͸ͳ͘ɺ0WFSMBQΛߟ͑Δʢ0WFSMBQ8FJHIUTʣ ‣ ౷ܭతҼՌਪ࿦ͷ࠷ۙͷൃల ‣ ͋ͷਓͷҼՌޮՌ͸Ͳ͏ͩΖ͏͔ʁʢ$BVTBM'PSFTU3-FBSOFSʣ ‣ ·ͱΊ !15

16 ॲஔ܈ͱରর܈ͷॏΈ෇͚෼෍ʹண໨ͤΑʂ

#BMBODJOH$PWBSJBUFT* ‣ ͔͜͜Βͷ࿩͸ɺ͓ͦΒ͘ڭՊॻʹ͸ࡌ͍ͬͯͳ͍࿩Ͱ͢ɻ ‣ ͜Ε͕ͦ͜܏޲είΞΛ༻͍ͨ౷ܭతҼՌਪ࿦ͷॏཁͳ֓೦Ͱ͋Δͱʮࢲ ͸ʯࢥ͍ͬͯ·͢ɻ ‣ ͥͻɺ͜ͷߟ͑ํΛϚελʔ͍ͯͩ͘͠͞ʂ !17

#BMBODJOH$PWBSJBUFT** ‣ ۩ମతͳ࿩ʹೖΔલʹɺ6ODPOGPVOEFEOFTTʢڧ͘ແࢹՄೳͳׂΓ෇͚ʣʹ ͍ͭͯઆ໌Λ͓͖ͯ͠·͢ɻ ‣ ͜Ε͸ɺજࡏ݁Ռม਺ͱॲஔม਺ͷ྆ํʹӨڹ͢Δม਺Λɺࢲͨͪ͸؍ଌͰ ͖͍ͯΔͱ͍͏ԾఆͰ͢ɻ ‣ ͜ͷԾఆ͕ͳ͍ͱɺ࿩͕࢝·Γ·ͤΜ͕ɺݱ࣮ͷ໰୊Ͱ͢΂ͯ؍ଌͰ͖͍ͯ Δͱ͍͏ঢ়گ͸ߟ͑ʹ͘͘ɺ੒Γཱͨͳ͍͜ͱ΋ଟ্͍ʹɺνΣοΫͰ͖ͳ ͍ͱ͍͏ͳ͔ͳ͔ո͍͠ԾఆͰ͸͋Γ·͢ɻ ‣ ͨͩɺ੒Γཱͭͱͯ͠ߟ͓͔͑ͯͳ͍ͱ࢝·Βͳ͍ͱ͍͏ͷ΋ࣄ࣮Ͱ͢ɻ ‣ ͔͠͠ɺ͜ͷԾఆ͕੒Γཱ͍ͬͯͳ͍ͱҼՌతޮՌΛਪఆͨ݁͠Ռʹҙຯ͕ ͳ͍͔ͱݴΘΕΔͱͦ͏Ͱ͸͋Γ·ͤΜɻ ‣ ࠓ͔Βͷ࿩ͰͦΕ͕Θ͔Γ·͢ɻ !18 (Y1 , Y0 ) ⊥ ⊥ W X

#BMBODJOH$PWBSJBUFT***ϥϯμϜԽൺֱ࣮ݧͱ ɹɹɹɹɹɹɹɹɹɹɹݹయతϞσϧͷഁ୼͕ڭ͑ͯ͘Εͨ͜ͱ ‣ ϥϯμϜԽ͸ɺॲஔม਺ͱ݁Ռม਺ͷӨڹ͢Δม਺͕ɺͭͷ܈Ͱಉ͡෼෍ʹ ै͏Α͏ʹ෺ཧతʹௐ੔ͨ͠ɻ ‣ ҼՌਪ࿦ͱ͸ڀۃతʹɺʮॲஔͱ݁Ռม਺ʹӨڹ͢Δม਺ͷ෼෍Λ͢΂ͯʯ ॲஔ܈ͱରর܈Ͱɺ౳͘͠Ͱ͖Ε͹Α͍ͱ͍͏͜ͱʹͳΓ·͢ɻ !19 ංྉ ͋Γ ංྉ ͳ͠ ස౓ ೔র࣌ؒ ස౓ ೔র࣌ؒ

#BMBODJOH$PWBSJBUFT*7ʢ਺ֶతͳٞ࿦͕ඞཁʣ ‣ ॲஔ܈ͷ݁Ռม਺ͷฏۉΛɺ͢΂ͯͷඪຊ͕ॲஔΛड͚ͨ৔߹ͷ݁Ռม਺ͷ ฏۉ΁मਖ਼͢Δɻ ‣ ͜Ε͕ɺॲஔ܈Ͱͷ݁Ռม਺ͷฏۉΛॻ͖Լͨ͠৔߹Ͱ͢ɻ͜͜Ͱɺॏཁͳ ͷ͸ੵ෼ͷதʹG YcX ͕͋Δ͜ͱͰ͢ɻ ‣ ͜Ε͸ɺॲஔ܈ͷڞมྔͷ෼෍Ͱ͋Γɺ͜Ε͸ʮॲஔΛड͚ͨ৔߹ͷજࡏ݁ Ռม਺Λɺڞมྔͷ෼෍શମͰ͸ͳ͘ɺॲஔ܈ͷڞมྔ෼෍Ͱੵ෼͍ͯ͠ ΔʯͷͰ࿪Έ͕ੜ͍ͯ͡Δͱ͍͏;͏ʹղऍ͠·͢ɻ ‣ Ͱ͸ɺͦͷ࿪ΈΛ΋ͱʹ໭͠·͠ΐ͏ɻ !20 E[WY] = ∫ w ⋅ y ⋅ f(y, w, x)dydxdw = ∫ y1 ⋅ f(y1 , w = 1,x)dydx = P(W = 1) ∫ y1 ⋅ f(y1 |w = 1,x)f(x|w = 1)dydx = P(W = 1) ∫ y1 ⋅ f(y1 |x)f(x|w = 1)dydx

#BMBODJOH$PWBSJBUFT7ʢ਺ֶతͳٞ࿦ʣ ‣ ͍·ɺ ‣ ͱ͠·͠ΐ͏ɻ͜ΕͰॏΈ෇͚ΒΕͨ8:ͷظ଴஋͸ɺॲஔ܈ͰͷҼՌޮՌ ͷਪఆྔʹͳΓ·͢ɻ ‣ ͱ͜ΖͰɺ͜ͷॏΈͷٯ਺͸ɺ ‣ ͱͳΔͷͰɺ͜Ε͸܏޲είΞͰ͢ɻ !21 w1 (x) = f(x) f(x|w = 1)P(W = 1) E [w1 (x) ⋅ WY] = P(W = 1) ∫ y1 ⋅ f(y1 |x)f(x|w = 1) ⋅ w1 (x)dy1 dx = ∫ y1 ⋅ f(y1 |x)f(x)dy1 dx = E[Y1 ] 1 w1 (x) = f(x|w = 1)P(W = 1) f(x) = P(W = 1|X = x)

#BMBODJOH$PWBSJBUFT7*ڞมྔ෼෍Λൺֱ͢Δ܈Ͱ౳͘͢͠Δɻ ‣ ΑͬͯɺҼՌਪ࿦ʹ͓͍ͯ܏޲είΞ͕ຊ࣭తͳͷͰ͸ͳ͘ɺͦͷٯ਺͕࣋ͭ ҙຯͦ͜ຊ࣭తͰ͋Δ͜ͱ͸ݴ͏·Ͱ΋͋Γ·ͤΜɻ ‣ ͜ͷΑ͏ͳΛ༻͍Δͱɺ෼෍͕ॲஔ܈ͱରর܈Ͱ࿪ΜͰ͍ͯ΋ɺॏΈ෇͚ʹ Αͬͯमਖ਼͕ߦΘΕɺ෼෍͕౳͘͠ͳΔ͜ͱ͕Θ͔Γ·͢ɻ !22 Covariate Density −3 −2 −1 0 1 2 3 0.00 0.10 0.20 0.30 Covariate Density −3 −2 −1 0 1 2 3 0.00 0.10 0.20 0.30 Covariate Density −3 −2 −1 0 1 2 3 0.00 0.05 0.10 0.15 Covariate Density −3 −2 −1 0 1 2 3 0.00 0.05 0.10 0.15 0.20 ॲஔ܈ ରর܈ ௐ੔લ ௐ੔લ

#BMBODJOH$PWBSJBUFT7**ʢຊ࣭తͳ࿩ʣ ‣ ͜Ε͸ɺʮٙࣅతͳϥϯμϜԽʯͱΈͳ͢͜ͱ͕Ͱ͖·͢ɻ ‣ ͭͷ܈ͷ෼෍Λ௼Γ߹Θ͍ͤͯΔʢόϥϯγϯά͍ͯ͠Δʣͱ͍͏ͷ͸ɺॲ ஔม਺ʹӨڹΛ༩͑Δม਺જࡏ݁Ռม਺ʹӨڹΛ༩͑ΔӨڹ͕ɺͭͷ܈ Ͱಉ͡෼෍Λ࣋ͭΑ͏ʹ͢Δͱ͍͏͜ͱʹରԠ͠·͢ɻ ‣ ͜ͷ࿩ͷ݁Ռɺʮ"6$͕Ҏ্͕Ͳ͏ͷ͜͏ͷʯΈ͍ͨͳ࿩͸ʮφϯηϯ εʯͩͱ͍͏͜ͱ͕Θ͔Γ·͢ɻ ‣ ܏޲είΞΛ༻͍ͨҼՌతޮՌͷਪ࿦ʹ͓͍ͯ͸ ‣ ॲஔͱજࡏ݁Ռม਺ʹӨڹΛ༩͑Δม਺͸Կ͔Λߟ͑Δ ‣ ͦͷม਺͕ɺॏΈ෇͚ʹΑͬͯॲஔ܈ରর܈Ͱಉ͡෼෍ʹै͏͔Λ֬ೝ ͢Δ ‣ ͱ͍͏͜ͱΛ֬ೝ͠ͳͯ͘͸ͳΓ·ͤΜɻ"6$΍ଞͷ౷ܭతͳج४ྔ͸ͳΜ ͷࢀߟʹ΋ͳΒͳ͍ͷͰ͢ɻ !23

#BMBODJOH$PWBSJBUFT7*** ‣ ͜ͷ͜ͱ͕Θ͔͍ͬͯΔͱɺʮ܏޲είΞΛ࢖ͬͯॏΈ෇͚͢Δʯ͜ͱ͸ಛʹ ࣍ͷΑ͏ʹݴ͍׵͑ͯ΋ࠩ͠ࢧ͕͑͋Γ·ͤΜɻ ‣ ͭ·ΓɺൺֱͰ͖ͳ͍͸ͣͷ܈Λɺ਺ֶతʹௐ੔͢Δ͜ͱͰʮٖࣅతʹʯൺֱ Մೳʹ͢Δͱ͍͏͜ͱʹͳΓ·͢ɻ ‣ ͦͯ͠ɺ࣮ࡍʮਅͷ܏޲είΞʯ͕Θ͔͍ͬͯΕ͹ɺ͍͍Ͱ͕͢ɺ࣮ࡍͷղੳͰ ͸ਪఆͨ͠܏޲είΞΛ༻͍ΔͷͰɺ ‣ ͱ͍͏࡞ۀ͸ઈରʹඞཁͰ͢ɻ ‣ ͜ΕͰɺࢲ͕ͨͪ܏޲είΞΛ༻͍ͯղੳ͢Δͱ͖ʹ΍Δ΂͖͜ͱ͕ͭΘ͔Γ ·ͨ͠ɻ !24 ൺֱ͢Δ܈Λɺٖࣅతʹۉ࣭ͳू߹ʹ͢Δ͜ͱɻ ࣮ࡍɺਪఆͨ͠܏޲είΞ͕෼෍Λ౳͍ͯ͘͠͠Δ͔֬ೝ͢Δ

#BMBODJOH$PWBSJBUFT*9 ‣ ڧ͘ແࢹՄೳͳׂΓ෇͚͕੒Γཱ͍ͬͯͳ͍ͷʹɺ܏޲είΞʹΑͬͯॏΈ ෇͚ͯ͠ਪఆ͢Δҙຯ͸͋Δͷʁͱ͍͏͜ͱΛݴ͍·ͨ͠ɻ ‣ ͔͠͠ɺ͍·ͳΒʮҙຯ͕͋Δʯͱݴ͑Δ͸ͣͰ͢ɻ ‣ ͳͥͳΒʮ܏޲είΞʹΑΔॏΈ෇͚ʯͱ͸ɺ ‣ ൺֱ͢Δ܈Λʮ౤ೖͨ͠มྔʯʹ͓͍ͯۉ࣭Խ͢Δߦҝ͔ͩΒɻ ‣ ͭ·Γɺগͳ͘ͱ΋ௐ੔Λߦͬͨ΄͏͕ɺ୯७ͳൺֱΛ͢ΔΑΓ΋ɺϚγͳ ݁ՌΛಘ͍ͯΔͱߟ͑͸ࣗવͰ͢ɻ ‣ ʲ݁࿦ʳڧ͘ແࢹՄೳͳׂΓ෇͚͕੒Γཱͭͷʹे෼ͳڞมྔ͕؍ଌ͞Εͯ ͍ͳ͔ͬͨͱͯ͠΋ɺॲஔͷޮՌΛਪఆ͢Δʹ͋ͨͬͯʮӨڹΛ༩͍͑ͯ Δʯͱߟ͑ΒΕΔม਺Λʮௐ੔ͯ͠ʯɺӨڹΛਪఆ͢Δ͜ͱ͸ҙຯ͕͋Δɻ !25

26 ਪఆྔͷ͹Β͖ͭʹ͍ͭͯߟ͑Α͏

͹Β͖ͭΛߟ͑Α͏*ʢܽଌʣ ‣ ౷ܭతҼՌਪ࿦Ͱ͸ɺʮજࡏ݁Ռม਺ͷࠩʯΛղੳ͢ΔΘ͚Ͱ͕͢ɺ͜Ε͸ ͲΜͳ͜ͱΛ͍ͯ͠Δ͔ͱ͍͏ͱɺɺɺ ‣ ӈԼͷਤʹ͓͍ͯɺӈ্ͱࠨԼ͕ʮ׬શʹ؍ଌ͞Εͳ͍ʯͱ͍͏ɺେ෦෼ͷ ܽଌ͕͋Δঢ়ଶͰɺͳΜͱ͔ਪఆ͠Α͏͍ͯ͠Δʮ͔ͳΓ଍ݩͷո͍͠ʯղ ੳͳΘ͚Ͱ͢ɻ !27 ॲஔม਺ 8 જࡏ݁Ռม਺ : જࡏ݁Ռม਺ : ॲஔม਺ 8 ؍ଌ ؍ଌ ܽଌ ܽଌ ؍ଌ ڞมྔ 9

͹Β͖ͭΛߟ͑Α͏** ‣ ҼՌޮՌͷਪఆͱ͸͍͏ͳΕ͹ɺ͜ͷܽଌͷ෦෼͸ ‣ ʮ͜Ε͘Β͍ͩΑʔʋ ʔʆ ϊʯ ‣ ͱ͍͏༧૝Λͯ͠ɺਪఆ͍ͯ͠ΔΘ͚͔ͩΒɺ ‣ ਓؒͰݴ͏ͱ͜Ζͷ ‣ ʮͨͿΜɺɺʓʓ͘Β͍ͩΑͶʯ ‣ ͱ͍͏͜ͱΛɺσʔλͰ΋͖ͪΜͱߟ͑ͳ͍ͱɺ ‣ ͦΜͳʹڧ͍͜ͱݴ͑ͳ͍ͷʹɺ ‣ ʮޮՌ͋ΔͰʂʂʢؔ੢หʣʯ ‣ Έ͍ͨͳ͜ͱΛݴͬͪΌͬͯɺ΍ͬͯΈͨΒޮՌͳ͍͡ΌΜʂʢඪ४ޠʣ ‣ ͱ͍͏ࣄʹͳΓ͔Ͷ·ͤΜɻ ‣ σʔλΛղੳ͢Δ͜ͱ͸ɺʮ͹Β͖ͭʯΛߟ͑Δͷͱಉ͡ ‣ ͱ͍͏͙Β͍େࣄͳ͜ͱͰ͢ 㱼ʆ !28

͹Β͖ͭΛߟ͑Α͏*** ‣ ਪఆͨ͠ͳΒɺͦͷਪఆྔʹ͸ͲΕ͘Β͍ͷ͹Β͖͕ͭ͋Δ͔͸ߟ͑ͳ͍ͱ ͍͚·ͤΜɻ ‣ ࣮ࡍɺσʔλ͔ΒԿ͔Λਪ࿦͢Δ৔߹ʹ͸ɺҎԼͷΑ͏ͳঢ়گΛ૝ఆ͢Δඞ ཁ͕͋Γ·͢ɻͭ·Γɺσʔλ͔Βܭࢉ͞Εͨ஋͸ɺ֬཰తʹಘΒΕͨαϯ ϓϧͷͭͳͷͰɺϒϨ෯Λ࣋ͭͱ͍͏ͷ͕ࣗવͳղऍͰ͢ɻ !29 F /ճαϯϓϧ ΛऔΔ ෼෍ ඪຊ E [ WY π(X) ] ύϥϝʔλ ऩଋ 1 N N ∑ i=1 Wi Yi π(Xi ) ਪఆྔ ࣮ݱ஋ {Xi , Wi , Yi }N i=1 ∼ F σʔλ 9 8 : ࣮ݱ஋ 1 N N ∑ i=1 wi yi π(xi ) ਪఆྔͷ࣮ݱ஋

͹Β͖ͭΛߟ͑Α͏*7 !30 ඪຊ 1 N N ∑ i=1 Wi Yi π(Xi ) ਪఆྔ Կ౓΋ σʔλΛ ൃੜ {Xi , Wi , Yi }N i=1 ∼ F σʔλ 9 8 : ࣮ݱ஋ 1 N N ∑ i=1 wi yi π(xi ) σʔλͰ ࡞ͬͨਪఆ஋ σʔλ 9 8 : σʔλ# 9 8 : ʜ σʔλͰ ࡞ͬͨਪఆ஋ 1 N N ∑ i=1 wi yi π(xi ) σʔλ#Ͱ ࡞ͬͨਪఆ஋ 1 N N ∑ i=1 wi yi π(xi ) ਪఆ஋ͷώετάϥϜ ʢݱ࣮ʹ͸ॻ͚ͳ͍ʣ ಘΒΕ͍ͯΔ σʔλ͔Βܭࢉ͞Εͨ஋͸ ͔͜͜΋ ͠Εͳ͍ ͔͜͜΋ ͠Εͳ͍ ͔͜͜΋ ͠Εͳ͍

͹Β͖ͭΛߟ͑Α͏7 ‣ ࣗ෼͕ಘͨ஋͕ӈ୺ͷ஋ͩͬͨ৔߹ɺͦͷ஋͚ͩͰ݁ՌΛ൑அ͢Δͱɻ ‣ ࣮͸ʮޮՌ͕ͳ͍΋ͷʯ͕ɺʮޮՌ͕͋Δʯͱग़ͯ͠·ͬͨɻ ‣ ࣮͸ʮޮՌ͕͋Δ΋ͷʯ͕ɺʮޮՌ͕ͳ͍ʯͱग़ͯ͠·͏͜ͱ͕͋Δɻ ‣ ͳͷͰɺࢲͨͪ͸ਪఆྔ͕ʮͲΜͳ;͏ʹ͹Βͭ͘ͷ͔ʯΛɺ͖ͪΜͱ֬ೝ ͓ͯ͘͠΄͏͕҆৺Ͱ͖·͢ɻ ‣ ʮͲΜͳ;͏ʹ͹Βͭ͘ͷ͔ʯΛɺ؍ଌ͞Ε͍ͯΔσʔλΛ༻͍ͯɺٖࣅత ʹߦ͏ํ๏͕ʮ#PPUTUSBQ๏ʢ&GSPO ʣʯͰ͢ɻ !31 ͔͜͜΋ ͠Εͳ͍ ͔͜͜΋ ͠Εͳ͍ O

͹Β͖ͭΛߟ͑Α͏7*ʢ#PPUTUSBQ๏ʣ !32 1 N N ∑ i=1 Wi Yi π(Xi ) ਪఆྔ σʔλ {Xi , Wi , Yi }N i=1 ∼ F 9 8 : ࣮ݱ஋ 1 N N ∑ i=1 wi yi π(xi ) σʔλͰ ࡞ͬͨਪఆ஋ ϒʔτετϥοϓαϯϓϧͷ ώετάϥϜʢݱ࣮Ͱॻ͚Δʣ σʔλ ϒʔτετϥοϓ αϯϓϧ σʔλ͔Β Ϧαϯϓϧ ʜ 1 N N ∑ i=1 w(1) i y(1) i π(x(1) i ) ϒʔτετϥοϓαϯϓϧͰ ࡞ͬͨਪఆ஋ αΠζ͸/ ճ਺Λ# ϒʔτετϥοϓ αϯϓϧ# X(1) W(1) Y(1) X(B) W(B) Y(B) 1 N N ∑ i=1 w(B) i y(B) i π(x(B) i ) ʜ ώετάϥϜʹ͢Δ ਪఆ஋ ਪఆ஋

͹Β͖ͭΛߟ͑Α͏7** ‣ ͜ͷ"d'ͷͭͷ஋ʹͭ ͍ͯ͸ɺҼՌޮՌͷਪఆ Λͨ͠ղੳͰ͸ɺઈରʹ ࣔͨ͠΄͏͕͍͍ɻ ‣ ·ͨɺӈͷΑ͏ͳਤ΋߹ Θͤͯࣔ͢͜ͱͰɺͲͷ ఔ౓ɺਪఆ͕͹Βͭ͘ͷ ͔͕Θ͔Γ΍͘͢ͳΔɻ ‣ ͜Ε͸ͥͻ΍ͬͯ΄͍͠ Ͱ͢ʂʂ !33 σʔλͷ ਪఆ஋ #PPUTUSBQ αϯϓϧͷฏۉ ਪఆ஋ ϒʔτετϥοϓα ϯϓϧͷฏۉ ਪఆ෼ࢄ ϒʔτετϥοϓ αϯϓϧͷ෼ࢄ ৴པ۠ؒ ϒʔτετϥοϓ ৴པ۠ؒ " # $ % & ' #PPUTUSBQ $POpEFODF*OUFSWBM "TZNQUPUJD *OUFSWBM

34 ਪఆ͍ͨ͠ޮՌ͸ͳʹ͔ʁ

ಛఆͷੑ࣭ͷ܈ʹண໨͢ΔҼՌతޮՌ* ‣ 8FJHIUFE"WFSBHF5SFBUNFOU&⒎FDU 8"5& ॏΈ෇͚ฏۉॲஔޮՌ ͸ɺ ࣍ͷΑ͏ʹఆٛ͞ΕΔɻ ‣ ͜͜ͰɺI Y ʹ͢ΔͱɺɹɹɹɹɹɹʹͳΔͷͰɺ͜Ε͸"5&ΛؚΉਪ ఆྔͰ͋Δɻ ‣ G Y I Y ͸ɺI Y ʹΑͬͯG Y ʹରͯ͠ॏΈΛ෇͚ͯɺG Y ͷͲͷ෦෼ʹண໨͢ Δ͔Λද͢ɻ ‣ ྫ͑͹ɺI Y FYQ Y ͷॏΈΛ͚ͭΔ৔߹Λߟ͑Δͱɺ9͕େ͖ ͘ͳΔʹ࿈ΕͯI Y ͕େ͖͘ͳΔͷͰɺG Y ͷ෼෍ͷӈଆʹେ͖ͳॏΈ͕ͭ͘ɻ !35 τ(x) = E[Y1 − Y0 |X = x] τh = ∫ τ(dx)f(x)h(x)μ(dx) ∫ f(x)h(x)μ(dx) ͜͜Ͱɺ τ1 = E[Y1 − Y0 ] Y Y Y͕େ͖͍΄Ͳ େ͖ͳॏΈ

ಛఆͷੑ࣭ͷ܈ʹண໨͢ΔҼՌతޮՌ** ‣ I Y F Y F Y ͱ͍͏ܗΛԾఆ͢ΔͱɺG Y I Y ͸ͲͷΑ͏ʹͳΔͩΖ͏ ͔ʁ ‣ ܏޲είΞ͕ͷ෦෼ʢͭ·ΓɺॲஔΛड͚Δ֬཰͕ʣͷ෦෼ʹେ ͖ͳॏΈ͕͔͔Γɺ܏޲είΞ͕খ͍͞େ͖͍෦෼ʹ͸ॏΈ͕͔ͭͳ ͍ɻ ‣ ࠷ॳͷํͷ܏޲είΞͰͷॏΈ෇͚͸ɺॲஔ܈ͰͷฏۉΛɺॲஔΛड͚ͨ৔ ߹ͷજࡏ݁Ռม਺ͷظ଴஋ʹɺิਖ਼͢Δ΋ͷͰͨ͠ɻ ‣ ͜͜ͰٻΊͨॏΈ͸ɺ࣮͸࣍ͷ໰୊ͰɺI Y ͷ৔߹ͷղʹରԠ͠·͢ɻ ‣ ͭ·Γɺॲஔ܈ͱରর܈ͷڞมྔ෼෍Λ΋ͱͷڞมྔ෼෍ʹରͯ͠௼Γ߹Θ ͤΔͨΊͷॏΈͰ͋Δͱߟ͑ΒΕ·͢ʢ#BMBODJOH8FJHIU -JFUBM !36 E [w1 (x) ⋅ WY] = E[Y1 ] f(x|w = 1)w1 (x) = f(x|w = 0)w0 (x) = f(x)h(x)

ಛఆͷੑ࣭ͷ܈ʹண໨͢ΔҼՌతޮՌ*** ‣ σʔλ͸/Ͱɺڞมྔ9͸ / ͔Βੜ੒ͨ͠ɻ ‣ ܏޲είΞ͸MPHJUF Y Yͱ͠ ͨɻ·ͨॲஔม਺8d#FS F Y ‣ ͦͯ͠ɺɹɹɹɹɹɹɹɹɹɹʹ ରͯ͠ɺVOXFJHIUFEPWFSMBQ XFJHIUJOWFSTFQSPCBCJMJUZ XFJHIUΛߦͬͨ৔߹ͷɺ෼෍ͷ௼ Γ߹͍Λܭࢉ͢Δɻ ‣ ͜͜Ͱ࢖༻ͨ͠܏޲είΞ͸ɺਪ ఆ஋Ͱ͸ͳ͘ਅͷ஋Λར༻ͨ͠ɻ ‣ ݁ՌɺPWFSMBQͷ৔߹ͱɺ*18ͷ ৔߹Ͱ෼෍ͷ௼Γ߹͍Ͱҧ͍͕ݟ ΒΕͨɻ !37 f(x):original distribution x Density −4 −2 0 2 4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 f(x)e(x)(1−e(x)): overlap distribution x Density −4 −2 0 2 4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 f(X|T = 0), f(X|T = 1) Unweighted x Density −4 −2 0 2 4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 using overlap weight x Density −4 −2 0 2 4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 using inverse probability weights x Density −4 −2 0 2 4 0.0 0.1 0.2 0.3 0.4 0.5 0.6

ಛఆͷੑ࣭ͷ܈ʹண໨͢ΔҼՌతޮՌ*7 ‣ ࣍ͷ݁Ռ͕ɺ#BMBODJOH8FJHIUͷॏཁੑΛอূ͢Δɻ !38 5IFPSFN -J .PSHBOBOE;BTMBWTLZ 5\ ^ͷ৔߹ɺ#BMBODJOH8FJHIU͔Βߏ੒͞ΕΔਪఆྔɹɹ͸XFJHIUFE BWFSBHFUSFBUNFOUF⒎FDU 8"5&ɺ ʹର͢ΔҰகਪఆྔͰ͋Δɻ ͜͜Ͱɺ͸ɺɹɹɹɹɹɹɹɹɹɹɹɹɹͱ͢Δͱ͖ɺ Ͱఆٛ͞ΕΔɻ ·ͨɺ5\ ^ͷ৔߹ͷ#BMBODJOHXFJHIUͱ͸ҎԼΛຬͨ͢ɺॏΈͰ͋Δɻ ̂ τh = ∑ i w1 (xi )Zi Yi ∑ i w1 (xi )Zi − ∑ i w0 (xi )(1 − Zi )Yi ∑ i w0 (xi )(1 − Zi ) → τh (n → ∞) ̂ τh fX|T=1 (x) × w1 (x) = fX|T=0 (x) × w0 (x) = f(x)h(x) τ(x) = E[Y(1) − Y(0)|X = x] τh τh ≡ ∫ τ(dx)f(x)h(x)μ(dx) ∫ f(x)h(x)μ(dx) τh

ಛఆͷੑ࣭ͷ܈ʹண໨͢ΔҼՌతޮՌ7 ‣ ͜͜ͰɺɹɹɹɹɹɹɹɹɹΛ༻͍ͨ৔߹ͷॏΈΛPWFSMBQXFJHIUͱ͍͏ɻ ‣ ͜ͷΑ͏ͳॏΈ෇͚͸ɺӈԼਤͷΑ͏ͳ෼෍ͷॏΈΛߏ੒͢Δɻ ‣ ͜Ε͸ɺॲஔ܈ͱରর܈Ͱ͔ͿΓ͕େ͖͍෦෼ʹେ͖ͳॏΈ͕͔͔Γɺαϯ ϓϧ͕ଘࡏ͠ͳ͍෦෼ʹ͸ॏΈΛ͔͚ͳ͍ɻ !39 h(x) = e(x){1 − e(x)} fX|T=1 (x) × w1 (x) = fX|T=0 (x) × w0 (x) = f(x)e(x){1 − e(x)} ‣ ͜ͷͱ͖ɺ8"5&͸ɺॲஔΛड͚Δ֬཰͕ ͷαϯϓϧʹରͯ͠࠷΋େ͖ͳॏΈ͕ ͔͔ͬͨҼՌޮՌͷਪఆྔͱͳ͍ͬͯΔɻ ‣ ͜ͷ஋͕े෼େ͖͍͜ͱ͸ɺ޿ࠂΛݟΔՄ ೳੑͷ͋Δਓʹɺ޿ࠂΛଧͭ͜ͱͰɺߪങ ߦಈͷଅਐ͕Ͱ͖Δ͜ͱΛද͢ɻ

ಛఆͷੑ࣭ͷ܈ʹண໨͢ΔҼՌతޮՌ7* ‣ ͜ͷํ๏Ͱɺ༷ʑͳʮಛఆͷ৚݅Λຬͨ͢܈ʯʹର͢ΔҼՌతޮՌΛਪఆ͢ Δ͜ͱ͕Ͱ͖Δɻ !40 #Z-JFUBM #BMBODJOH$PWBSJBUFTWJB1SPQFOTJUZ4DPSF8FJHIUJOH

Original Covariate Density −2 −1 0 1 2 0.0 0.1 0.2 0.3 0.4 0.5 Covariate Density −2 −1 0 1 2 0.00 0.10 0.20 0.30 Covariate Density −2 −1 0 1 2 0.0 0.1 0.2 0.3 0.4 ATE:h(x)=1 Covariate Density −2 −1 0 1 2 0.00 0.10 0.20 0.30 Covariate Density −2 −1 0 1 2 0.00 0.10 0.20 0.30 Covariate Density −2 −1 0 1 2 0.00 0.10 0.20 0.30 ATT:h(x)=e(x) Covariate Density −2 −1 0 1 2 0.0 0.1 0.2 0.3 0.4 0.5 Covariate Density −2 −1 0 1 2 0.0 0.1 0.2 0.3 0.4 0.5 Covariate Density −2 −1 0 1 2 0.0 0.1 0.2 0.3 0.4 0.5 ATU:h(x) = 1−e(x) Covariate Density −2 −1 0 1 2 0.0 0.2 0.4 0.6 Covariate Density −2 −1 0 1 2 0.0 0.1 0.2 0.3 0.4 0.5 Covariate Density −2 −1 0 1 2 0.0 0.1 0.2 0.3 0.4 Overlap:h(x) = e(x)(1−e(x)) Covariate Density −2 −1 0 1 2 0.0 0.1 0.2 0.3 Covariate Density −2 −1 0 1 2 0.0 0.1 0.2 0.3 Covariate Density −2 −1 0 1 2 0.0 0.1 0.2 0.3 0.4 Matching:h(x) = min{e(x),1−e(x)} Covariate Density −2 −1 0 1 2 0.0 0.1 0.2 0.3 0.4 Covariate Density −2 −1 0 1 2 0.0 0.1 0.2 0.3 0.4 Covariate Density −2 −1 0 1 2 0.0 0.1 0.2 0.3 0.4 0.5 ಛఆͷੑ࣭ͷ܈ʹண໨͢ΔҼՌతޮՌ7** !41

ಛఆͷੑ࣭ͷ܈ʹண໨͢ΔҼՌతޮՌ7*** !42 ATE Covariate Density −2 −1 0 1 2 0.00 0.10 0.20 0.30 ATT Covariate Density −2 −1 0 1 2 0.0 0.1 0.2 0.3 0.4 ATU Covariate Density −2 −1 0 1 2 0.0 0.1 0.2 0.3 0.4 0.5 Overlap Covariate Density −2 −1 0 1 2 0.00 0.10 0.20 0.30 "5&"WFSBHF5SFBUNFOU&GGFDU ͢΂ͯͷαϯϓϧʹॲஔΛߦͬͨ৔߹ͷ݁Ռͱ ͢΂ͯͷαϯϓϧʹॲஔΛߦΘͳ͔ͬͨ৔߹ͷ݁Ռͷൺֱ "55"WFSBHF5SFBUNFOU&GGFDUPOUIF5SFBUFE ॲஔΛड͚ͨਓͷूஂʹண໨ͯ͠ɺ ॲஔΛड͚ͨ৔߹ͱɺॲஔΛड͚ͳ͔ͬͨ৔߹ͷ݁ՌΛൺֱ͢Δ "56"WFSBHF5SFBUNFOU&GGFDUPOUIF6OUSFBUFE ॲஔΛड͚ͳ͔ͬͨਓͷूஂʹண໨ͯ͠ɺ ॲஔΛड͚ͨ৔߹ͱɺॲஔΛड͚ͳ͔ͬͨ৔߹ͷ݁ՌΛൺֱ͢Δ "50"WFSBHF5SFBUNFOU&GGFDUPOUIF0WFSMBQ ॲஔΛड͚Δ֬཰ͱɺॲஔΛड͚ͳ͍֬཰͕͋Δఔ౓͋Δूஂʹண໨ͯ͠ɺ ॲஔΛड͚ͨ৔߹ͱɺॲஔΛड͚ͳ͔ͬͨ৔߹ͷ݁ՌΛൺֱ͢Δ

໨࣍ ‣ ܏޲είΞ͸ສೳༀͳͷ͔ʂʁ ‣ ϥϯμϜԽൺֱ࣮ݧʢ3BOEPNJ[FE$POUSPM5SJBMʣ ‣ ݹయతճؼϞσϧͷഁ୼ ‣ ೣ΋उࢠ΋܏޲είΞɻ ‣ ҼՌਪ࿦ͷຊ࣭ͱ࣮σʔλͷղੳ ‣ ܏޲είΞͷٯ਺ͦ͜ຊ࣭ʂʢ1SPQFOTJUZ4DPSFʣ ‣ σʔλ͕ͳ͍ͳΒɺ͹Β͖ͭΛߟ͑Δʢ#PPUTUSBQ.FUIPEʣ ‣ ฏۉॲஔޮՌͰ͸ͳ͘ɺ0WFSMBQΛߟ͑Δʢ0WFSMBQ8FJHIUTʣ ‣ ౷ܭతҼՌਪ࿦ͷ࠷ۙͷൃల ‣ ͋ͷਓͷҼՌޮՌ͸Ͳ͏ͩΖ͏͔ʁʢ$BVTBM'PSFTU3-FBSOFSʣ ‣ ·ͱΊ !43

ݸਓʹର͢ΔҼՌޮՌͷਪఆ* .-º$BVTBMJUZʣ ‣ ۙ೥ɼ4UBOGPSEେֶʹ͓͍ͯػցֶशΛ༻͍ͨ)FUFSPHFOFPVT5SFBUNFOU &⒎FDUͷਪఆ͕׆ൃԽ͍ͯ͠Δɽ ‣ ػցֶशΛ༻͍Α͏ͱ͍͏ྲྀΕ͸ɼҎલ͔Β΋͕͋ͬͨɼ൴Βͷ DPOUSJCVUJPOͷଟ͘͸ʮಘΒΕΔਪఆྔͷ઴ۙతੑ࣭ʯΛ໌Β͔ʹͨ͠ͱ ͍͏఺Ͱ͋Δɽ ‣ ͭ·Γɼ౷ܭతਪ࿦ԾઆݕఆͳͲ͕ՄೳʹͳΔΑ͏ͳ'SBNFXPSLΛ࡞ Γग़ͨ͠ɽ ‣ Ϟνϕʔγϣϯ͸ʮ1FSTPOBMJ[FE.FEJDJOFʯʢݸผԽҩྍɿͲΜͳਓʹ ͲΜͳॲஔΛ͢Ε͹ɼޮՌ͕࠷େʹͳΔͷ͔Λ஌Γ͍ͨʣ ‣ େ͖͘෼͚Δͱɼ̎ͭͷΞϓϩʔν͕औΒΕ͍ͯΔ ‣ 3BOEPN'PSFTUΛԠ༻ͨ͠ख๏ʂ ‣ ݁Ռม਺܏޲είΞΛ.-3FHSFTTJPOͰਪఆ͔ͯ͠Βɼਪఆͨ͠஋Λ ༻͍ͯճؼͷ໰୊Λ.-Ͱղ͘ํ๏ !44

ݸਓʹର͢ΔҼՌޮՌͷਪఆ** 1SPQFOTJUZ5SFF !45 ॲஔม਺8ͱڞมྔ9ʹ3FHSFTTJPO5SFFΛ౰ͯ͸ ΊΔͱ͍͏໰୊Λߟ͑ΔʢTQMJUDSJUFSJPO͸4- MPHMJLFMJIPPEʣͳͲɽ ͜ͷͱ͖ɼ-FBGͷதʹೖ͍ͬͯΔඪຊ͸ɼ͍ۙ͠ڞ มྔΛ࣋ͭʢͭ·ΓҰछͷ૚ผԽͰ͋Δʣɽ Αͬͯɼ-FBGΛݻఆͨ͠৔߹ʹ͸ɼ-FBGͷதͰ͸ɼ ڧ͘ແࢹͰ͖ΔׂΓ෇͚͕ۙࣅతʹ੒Γཱͭͱߟ͑ Δ͜ͱ͕Ͱ͖Δɽ ·ͨɼ೚ҙͷ-FBG- 9 ͸઴ۙతʹEJBN - 9 ˠ Ͱ͋Δ͔Βɼ઴ۙతʹ͸9Λݻఆͨ͠΋ͱͰɼҼՌ తޮՌΛਪఆ͍ͯ͠Δ͜ͱʹͳΔɽ 9 9 0 ֤-FBGͷ਺ࣈ͸ɼͦͷ-FBG಺ʹ͋Δαϯϓϧͷ ฏۉͰܭࢉ͞ΕΔॲஔ֬཰Ͱ͋Δɽ E[Y(1)|X ∈ L(X)] − E[Y(1)|X ∈ L(X)] ≈ E[Y(1)|W = 1,X ∈ L] − E[Y(0)|W = 0,X ∈ L] 1SPQFOTJUZUSFF

ݸਓʹର͢ΔҼՌޮՌͷਪఆ*** $BVTBM'PSFTUʣ ‣ 1SPQFOTJUZ5SFFΛCBTFMFBSOFSͱ͢Δ3BOEPN'PSFTUΛʮ$BVTBM'PSFTU 8BHFSBOE"UIFZ ʯͱ͍͏ɽ ‣ <8BHFSBOE"UIFZ >Ͱ͸ɼ3BOEPN'PSFTUͷ઴ۙਖ਼نੑʹର͢Δূ ໌Λ༩͍͑ͯΔʢ਺஋తʹܭࢉՄೳͳਪఆྔ΋ʣɽ ‣ ͜ͷख๏ͷDPOUSJCVUJPO͸େ͖ͭ͘ ‣ ʮ܏޲είΞͷਪఆʯ͕ඞཁͳ͍ʢඞཁͳڞมྔΛಛఆ͢Ε͹ྑ͍ʣɽ ‣ ਪఆ͞ΕΔҼՌతޮՌ͸ʮ)FUFSPHFOFPVTʯͰ͋Γɼڞมྔ͕༩͑ΒΕ ͨݩͰͷҼՌతޮՌͷਪఆ͕Մೳ ‣ ͲΜͳਓʹରͯ͠ɼͲΕ͘Β͍ͷޮՌ͕͋Δ͔͕Θ͔Δʂ ‣ ࿦จʹ͸ɼݟࣄͳγϛϡϨʔγϣϯ݁Ռ͕ࡌ͍ͬͯΔɽ ‣ ߹Θͤͯɼศརͳख๏ʹ͸ɼ૬Ԡͷܽ఺͕ଘࡏ͢ΔͷͰɼQBDLBHF\HSG^Ͱݕ ূͯ͠΄͍͠ɽ !46 E[Y(1) − Y(0)|X = x]

ݸਓʹର͢ΔҼՌޮՌͷਪఆ*7 QBDLBHF\HSG^ʣ ‣ 4BNQMFTJ[F/ ‣ /VNCFSPGGFBUVSFTQ ‣ ҼՌޮՌͷؔ਺͸ɼ͕ΑΓେ͖͍৔߹ʹͷΈଘࡏ͠ɼͦͷӨڹ͸ઢܕత Ͱ͋ΔΑ͏ͳઃఆͰ͋Δɽ !47 Xj ∼ N(0,1) j = 1,2,...,p W ∼ Bernoulli (0.4 + 0.2 × I(X1 > 0)) Y = W × max(X1 ,0) + X[,2] + min(X3 ,0) + N(0,1) X1 $BVTBMF⒎FDU X1 / &TUJNBUFE 5SVF&⒎FDU $*

ݸਓʹର͢ΔҼՌޮՌͷਪఆ7 3-FBSOFSʣ ‣ 3-FBSOFS͸΋ͱ΋ͱʢ/JFBOE8BHFS ʣͰ̎஋ͷॲஔʹରͯ͠ఏҊ ͞Εͨख๏ɽ ‣ &<:c9>ͱ&<8c9>ΛͦΕͧΕɼద౰ͳճؼϞσϧͰਪఆͨ͠ޙͰɼ࣍ͷํఔ ࣜͷղΛ-PDBM-JOFBS'PSFTU 'SJFECFSH FUBM Ͱղ͘ɽ ‣ 9͕༩͑ΒΕͨ΋ͱͰͷɼඇઢܗͳҼՌޮՌ͕ਪఆՄೳͱͳΔɽ ‣ ࠷ۙͷ3BOEPN'PSFTUपΓͷൃల͸͍͢͝ʂ ‣ (FOFSBMJ[FE3BOEPN'PSFTU "UIFZ 5JCTIJSBOJ BOE8BHFS ͸ 'PSFTUͰํఔࣜΛղ͘͜ͱ͕Ͱ͖Δʢ*OTUSVNFOUBMWBSJBCMF΍ɼRVBOUJMF SFHSFTTJPOͳͲ΋Ͱ͖Δɽʣɽ !48 Y − ̂ E[Y|X] = (W − ̂ E[W|X])τ(X) + ε

ݸਓʹର͢ΔҼՌޮՌͷਪఆ7*ʢͦͷଞͷ࿩୊ʣ ‣ $BVTBMJUZº5SBOTGFS-FBSOJOH͕࢝·ͬͨɽ ‣ ,VO[FMFUBM 5SBOTGFS-FBSOJOHGPS&TUJNBUJOH$BVTBM &⒎FDUTVTJOH/FVSBM/FUXPSLT BS9JWW ‣ ҼՌਪ࿦ʹసҠֶशΛద༻͢Δͱ͍͏ϑϨʔϜϫʔΫ͸ɼ͜Ε͔ΒͷτϨ ϯυʹͳΔՄೳੑ͕େ͖͍ɽ ‣ ·ͣ͸ɼ%PNBJO"EBQUJPO͔Β࢝·ΔͱࢥΘΕ·͢ɽ !49

໨࣍ ‣ ܏޲είΞ͸ສೳༀͳͷ͔ʂʁ ‣ ϥϯμϜԽൺֱ࣮ݧʢ3BOEPNJ[FE$POUSPM5SJBMʣ ‣ ݹయతճؼϞσϧͷഁ୼ ‣ ೣ΋उࢠ΋܏޲είΞɻ ‣ ҼՌਪ࿦ͷຊ࣭ͱ࣮σʔλͷղੳ ‣ ܏޲είΞͷٯ਺ͦ͜ຊ࣭ʂʢ1SPQFOTJUZ4DPSFʣ ‣ σʔλ͕ͳ͍ͳΒɺ͹Β͖ͭΛߟ͑Δʢ#PPUTUSBQ.FUIPEʣ ‣ ฏۉॲஔޮՌͰ͸ͳ͘ɺ0WFSMBQΛߟ͑Δʢ0WFSMBQ8FJHIUTʣ ‣ ౷ܭతҼՌਪ࿦ͷ࠷ۙͷൃల ‣ ͋ͷਓͷҼՌޮՌ͸Ͳ͏ͩΖ͏͔ʁʢ$BVTBM'PSFTU3-FBSOFSʣ ‣ ·ͱΊ !50

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