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1. धཁਪఆษڧձ ԣ඿ࠃཱେֶ Ֆᖒ෨ 2024 ೥ 1 ݄ 17 ೔ Berry and Haile (2021): Foundations of demand estimation. CONTENTS 1 Introduction 3 1.1 Why estimate demand? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Our focus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 The challenges of demand estimation 3 2.1 The first fundamental challenge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The second fundamental challenge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Demand is not regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 A surprisingly difficult case: exogenous prices . . . . . . . . . . . . . . . . . . . . . . 4 2.5 Many common tools fall short . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.5.1 Controls, including fixed effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.5.2 Control function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5.3 Average treatment effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.6 Balancing flexibility and practicality . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.7 Demand or utilities? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Discrete choice demand 8 3.1 Random utility models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 The canonical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Why random coefficients? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4 Market-level data 11 5 Nonparametric identification: market-level data 12 5.1 Insights from parametric models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.1.1 Multinomial logit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.1.2 Nested logit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.1.3 The BLP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1
2. 5.1.4 Index, inversion, and instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.2 Nonparametric demand model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.2.1 A nonparametric index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.2.2 Inverting demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.3 Identification via instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.4.1 Why 2J instruments? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.4.2 Why BLP instruments? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.4.3 Why the index? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.4.4 Further restrictions and tradeoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6 Micro data, panels, and ranked choices 19 6.1 Micro data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6.2 Consumer panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.3 Ranked choice data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 6.4 Hybrids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Appendix.A ଟ߲ϩδοτϞσϧɿબ୒֬཰ͷಋग़ 24 Appendix.B ଟ߲ϩδοτϞσϧɿIIA 26 2
3. 1 Introduction 1.1 Why estimate demand? ܦࡁओମͷҙࢥܾఆϞσϧΛ׆༻࣮ͨ͠ূ෼ੳͷΞϓϩʔνͰ͋Δߏ଄ਪఆ͸ҎԼͷखॱ͔ΒͳΔɻ 1. ෼ੳ͍ͨ͠ܦࡁࣄ৅ʹؔ͢ΔܦࡁϞσϧΛߏஙʢeconomic modelʣ 2. ܦࡁϞσϧ಺ͷύϥϝλʢe.g., धཁؔ਺ɾඅ༻ؔ਺ʣΛɺσʔλΛ༻͍ͯਪఆʢeconometric modelʣ 3. ਪఆͨ͠ϞσϧΛ༻͍ͯɺ൓࣮Ծ૝ʢcounterfactualʣγϛϡϨʔγϣϯ ߏ଄ਪఆͷڧΈɿܦࡁϞσϧΛ༻͍ͨγϛϡϨʔγϣϯ͕Ͱ͖Δ͜ͱɻ • ྫɿ͋Δ੓ࡦͷ୅ΘΓʹผͷ੓ࡦΛߦ͏ͱͲ͏ͳΔ͔ • ҙࢥܾఆʹ͓͚Δ͋ΔϝΧχζϜɾνϟωϧΛࣺ৅ͨ͠ͱ͖ʹͲ͏ͳΔ͔ • ҼՌޮՌɾ੓ࡦޮՌͷਪఆͷҰͭͷखஈ • ܦࡁֶతʹؔ৺ͷ͋ΔΞ΢τΧϜʢܦࡁްੜͳͲʣΛݟΔ͜ͱ͕Ͱ͖Δ ߏ଄ਪఆͷऑΈɿܦࡁϞσϧʹڧ͘ґڌ͍ͯ͠Δ͜ͱɻڧ͍ԾఆΛஔ͘ඞཁ͕͋Δ&࣮૷ίετ͕ߴ͍ɻ धཁؔ਺ͷਪఆ͸ߏ଄ਪఆʹ͓͍ͯඇৗʹॏཁͳύʔτͷҰͭͰ͋Δɻ 1.2 Our focus ຊษڧձʢChapter 1ʣͰ͸ҎԼͷجຊతͳ໰୊ʹऔΓ૊Ήɻ • धཁਪఆͷࣝผ໰୊ • empirical IO ʹ͓͚ΔجຊతͳϞσϧ • ૢ࡞ม਺ͷબ୒ɾ໾ׂ • ҟͳΔσʔλλΠϓʹ͍ͭͯʢmicro data, panel data, ranked choice data, hybridʣ • nonparametric identification 2 The challenges of demand estimation धཁਪఆͷࣝผ໰୊͸ɺधཁʹӨڹΛ༩͑Δ؍ଌ͞Εͳ͍ཁૉʢunobserved demand shocksʣ͕͋Δ৔߹ ʢi.e., Ձ֨ͷ಺ੜੑʣʹ೉͘͠ͳΔɻ 2.1 The first fundamental challenge ׬શڝ૪Ͱࡒ͕ 1 ͭͷࢢ৔Λߟ͑Δɻधཁͱڙڅ͸ಉ࣌ํఔ͔ࣜΒಛ௃͚ͮΒΕΔɻ Q = D(X, P, U) (2.1) P = C(W, Q, V ) (2.2) ͜͜Ͱɺ 3
4. • Q : quantity • P : price • X : observable demand shifters • W : observable cost shifters • U∈ R : unobserved demand shifters • V ∈ R : unobserved cost shifters Ձ֨ͱྔͷΈ͕಺ੜతͰ͋ΔͱԾఆ͢Δɻi.e., (X, W) ⊥ ⊥(U, V ). ΋͠ɺU ͕ͳ͍ͱͨ͠Βɺ୯ʹ (X, P) ͱ Q ͷؔ܎͕धཁΛද͢͜ͱʹͳΔ͕ U ͸΄ͱΜͲଘࡏ͢ΔͷͰɺࣝ ผͷͨΊͷ޻෉͕ඞཁͰ͋Δɻ ͜ͷ৔߹ͷधཁͷϊϯύϥϝτϦοΫࣝผ͸ɺ಺ੜੑΛ൐͏ճؼϞσϧͷ৔߹ͷࣝผͱಉ͡ૢ࡞ม਺৚͔݅Β ಋ͔ΕΔɻͭ·ΓɺधཁࣝผͷͨΊʹ͸ૢ࡞ม਺͔ΒಘΒΕΔΑ͏ͳ֎ੜతͳมಈ͕ඞཁɻ 2.2 The second fundamental challenge ͲΜͳࡒͷधཁྔ΋ɺ1 ͭҎ্ͷજࡏతͳधཁγϣοΫʢlatent demand shocksʣʹґଘ͓ͯ͠Γɺڵຯͷ ͋Δࡒͷधཁ͸ؔ࿈͢Δ͢΂ͯͷࡒͷՁ֨ͱ੡඼ଐੑʹґଘ͢Δɻྫ͑͹ɺ୅ସ඼΍ิ׬඼ͷՁ֨ʢ·ͨ͸੡ ඼ଐੑʣ͕มΘΕ͹ɺधཁ͸มԽ͢Δɻ͜ΕΒͷཁҼ͸͋Δࡒͷधཁ͔Βআ֎͢Δ͜ͱ͸Ͱ͖ͳ͍ͨΊɺ͋Δ Ձ֨ͷมԽͷΑ͏ͳ ceteris paribus Λఆٛ͢Δࡍʹݻఆ͞ΕΔʮଞͷ͢΂ͯʯͷதʹؚ·ΕΔɻ ͜Ε͸ɺࣝผͷͨΊʹधཁؔ਺ʹؔ͢ΔԾఆΛೖΕͳ͍ݶΓਪఆͰ͖ͳ͍͜ͱΛࣔ͠ɺधཁਪఆ͸ඪ४తͳ ճؼ෼ੳͱͯ͠ѻ͏͜ͱ͸Ͱ͖ͳ͍ɻ 2.3 Demand is not regression Ϛʔέοτʹ J ݸͷࡒ͕͋Δͱ͢Δɻ֤ࡒ j = 1, . . . , J ʹؔ͢Δधཁ͸ Qj = Dj (X, P, U) (2.3) ͜͜ͰɺX = (X1 , . . . , XJ ), P = (P1 , . . . , PJ ), U = (U1 , . . . , UJ ). धཁγϣοΫʢstructural errorsʣU = (U1 , . . . , UJ ) ͸͢΂ͯͷࡒͱ૬ؔ͢ΔͷͰ͜Ε͸ճؼࣜʢregression equationʣͰ͸ͳ͍*1ɻ·ͨɺDj ͷਪఆͷͨΊʹɺJ ݸͷՁ֨ʹؔ͢Δྑ͍ૢ࡞ม਺Λݟ͚͖ͭͯͨͱͯ͠ ΋͜Ε͸े෼Ͱ͸ͳ͍ʢwe will see this formally in later sectionsʣ ɻಉ࣌ํఔࣜϞσϧ͔ΒधཁΛࣝผ͢Δ ͨΊʹ͸ J ݸͷ price ͱ J ݸͷ quantity ʹؔ͢Δૢ࡞ม਺͕ J × J ݸඞཁʹͳΔɻ ಺ੜੑͱ unobserved variables ͕͋Δ৔߹ʹ weighted average responses ʢྫɿLATEʣͷਪఆΛ͢Δ͕ɺ ໌Β͔ʹ͜ΕʹΑͬͯधཁਪఆΛ͢Δ͜ͱ͸Ͱ͖ͳ͍ɻ 2.4 A surprisingly difficult case: exogenous prices ࣜʢ2.3ʣͷӈลͷधཁγϣοΫʢߏ଄ޡࠩʣʹΑΔ໰୊Λཧղ͢ΔͨΊʹҎԼͷ໰୊Λߟ͑Δɻ ଟ͘ͷେ͖͍ࢢ৔ʢmany large marketsʣt ʹ͓͍ͯɺݚڀऀ͕ϥϯμϜʹՁ֨ϕΫτϧ (p1t , . . . , pJt ) Λׂ Γ౰ͯɺधཁྔ (q1t , . . . , qJt ) Λ؍ଌͰ͖Δͱ͢Δ*2ɻ͜ͷ৔߹Ͱ΋धཁਪఆͷͨΊͷࣝผ͸೉͍͜͠ͱΛҎ *1 ճؼࣜ͸ɺY = f(X) + E ͱͯ͠ఆࣜԽ͞ΕΔɻ *2 ؍࡯͞Εͨ਺ྔ͸ɺ֎ੜతʹઃఆ͞ΕͨՁ֨ʹ͓͍ͯڙڅ͞Εͨ਺ྔͰ͸ͳ͘ɺधཁ͞Εͨ਺ྔͰ͋ΔͱԾఆ͍ͯ͠Δ͜ͱʹ஫ҙɻ 4
5. ԼͰݟΔɻ ྫ͑͹ɺधཁ஄ྗੑͰ͸ଞͷ͢΂ͯΛݻఆͨ͠··Ձ֨ΛมԽͤ͞Δඞཁ͕͋Γɺݻఆ͠ͳ͚Ε͹ͳΒͳ͍ ΋ͷͷதʹधཁγϣοΫ͕͋ΔɻՁ͕֨ϥϯμϜʹׂΓ౰ͯΒΕ͍ͯΔͨΊɺՁ֨ͱधཁγϣοΫͷؒͷ૬ؔ ͸ແ͘ͳΔ͕ɺधཁγϣοΫͦͷ΋ͷΛݻఆ͢Δ͜ͱ͸Ͱ͖ͳ͍ɻՁ֨ͷϥϯμϜͳมಈʹ൐͏਺ྔͷ؍ଌ͞ Εͨมಈ͸धཁγϣοΫͷϕΫτϧ ʢU1 , . . . , UJ ʣ ʹ͍ͭͯੵ෼͢Δ͜ͱͰ LATE ΛܭࢉͰ͖Δɻ͔͜͠͠͏ ͨ͠ฏۉ஋͸ɺधཁͷ஄ྗੑΛ໌Β͔ʹ͠ͳ͍ͷͰͦ͜·Ͱخ͘͠ͳ͍ɻՁ֨΁ͷૢ࡞ม਺ͦΕࣗମ͸धཁͷ ࣝผʹ͸े෼Ͱ͸ͳ͍ɻ Αͬͯɺؔ৺ͷ͋Δ൓ࣄ࣮ʢྫ͑͹ɺ߹ซޙͷ݁Ռʣʹର͢Δधཁ൓ԠͷఆྔԽ΍ɺۉߧՁ֨৚݅Λ௨ͯ͡ اۀͷϚʔΫΞοϓΛਪ࿦͢Δ͜ͱ͸Ͱ͖ͳ͍ʢWe return to this issue in Section 2.5.3ʣ ɻ ؔ਺ܗΛԾఆ͢ΔͱՁ֨ͷ಺ੜੑʹରॲ͢Δ͜ͱ͕Ͱ͖Δɻࣜʢ2.3ʣͷधཁؔ਺ΛҎԼͷΑ͏ʹԾఆ͢Δɻ Qj = Dj (X, P, U) = Dj (X, P, Ej (U)) (2.4) ͜͜ͰɺEj (U) ͸ U1 , . . . , UJ ʹ͍ͭͯઢܗͷεΧϥʔͰ͋Γɺ∂Dj /∂Ej (U) > 0 ͱ͢Δɻ ͜ͷͱ͖ɺ Qj |X, P ͷ෼෍ʹ͓͚Δ τ ෼Ґ఺͸ɺ ͦͷ τ ͷ΋ͱͰ Ej (U) Λݻఆͨ͠ͱ͖ͷधཁ Dj (X, P, Ej (U)) Λද͢ (Matzkin (2003))ɻ Ҏ্ΑΓɺधཁؔ਺ͷؔ਺ܗͱͯࣜ͠ʢ2.4ʣΛԾఆͨ͠ͱ͖ɺP ⊥ ⊥ Uɺ΋͘͠͸ P ͕֎ੜม਺ͱߟ͑ΒΕ Δ৔߹ʢ΋͘͠͸ J ݸͷ P ʹ͍ͭͯͷૢ࡞ม਺͕ಘΒΕΔ৔߹ʣʹ͸धཁؔ਺Λࣝผ͢Δ͜ͱ͕Ͱ͖Δɻ͠ ͔͠ɺj ࡒʹର͢ΔधཁγϣοΫ͕୯ௐ૿ՃͰɺઢܗ͔ͭεΧϥʔʢEj (U)ʣͰ༩͑ΒΕΔͱ͢Δͷ͸ͱͯ΋ ڧ͍ԾఆͷͨΊΑΓॊೈͳϞσϧ͕ඞཁɻ 2.5 Many common tools fall short Θ͟Θ͟ෳࡶͳϞσϧΛ࢖͏ඞཁ͕͋Δͷ͔ʁˠࡒ͕ 1 ͭͷࢢ৔Λߟ͑ͨͱͯ͠΋௨ৗͷ empirical tools Ͱ͸धཁΛਪఆͰ͖ͳ͍͜ͱΛ͜ͷ section Ͱ͸ٞ࿦͞Ε͍ͯΔɻ 2.5.1 Controls, including fixed effects ؍࡯Ͱ͖ͳ͍ม਺ͷଘࡏʹΑͬͯੜ͡Δ಺ੜੑͷ໰୊ʹ͸ɺ͜ΕΛऔΓআ͘Α͏ͳʠcontrolsʡΛ୳͢ͷ͕ ࣗવͳΞϓϩʔνͰ͋ΔɻधཁਪఆΛ͢Δʹ͋ͨͬͯ͜ͷΞϓϩʔν্͕ख͍͘͘ʹ͸ controls ʹ͍ͭͯҎ Լͷ 2 ͕ͭຬͨ͞ΕΔඞཁ͕͋Δɻ 1. ؍࡯Ͱ͖ͳ͍ม਺ʹΑΔधཁ΁ͷશͯͷޮՌΛؚΉ 2. controls ͸Ձ֨ʹؔ͢ΔશͯͷมಈΛؚΜͰ͸͍͚ͳ͍ʢআ֎੍໿ͬΆ͍΋ͷʣ ྫ͑͹ɺࣜʢ2.1ʣʹ͓͍ͯશͯͷ demand shifters Λ X ؚ͕Ήͱ͢Δͱɺશͯͷ (X, P) ʹ͓͍ͯ׬શͳध ཁͷ fit/prediction Λ͢Δ͜ͱ͕Մೳ͕ͩɺͦΜͳ͜ͱ͸͋Γಘͳ͍ɻ controls ͷީิͱͯ͠ɺݻఆޮՌ͕ߟ͑ΒΕΔɻݻఆޮՌ͸ʠcontrol for everythingʡͱղऍ͞ΕΔͨΊɻ ݻఆޮՌʹ͓͍ͯ΋ controls ͱͯ͠ 2 ͭͷཁ݅Λຬͨ͢͜ͱඞཁͰ͋Δɻͭ·ΓɺधཁʹӨڹΛ༩͑Δશͯ 5
6. ͷཁҼʹ͍ͭͯݻఆޮՌʹؚ·Ε͓ͯΓɺ͔ͭՁ֨ʹӨڹΛ༩͑Δ΋ͷ͸Ұ੾ؚ·ͳ͍͜ͱɻࡒ͝ͱɺ΋͘͠ ͸ࢢ৔͝ͱͷϨϕϧͰͷજࡏతͳधཁγϣοΫͷଘࡏʹΑΓɺݻఆޮՌʹΑΔΞϓϩʔν͸͜ΕΒͷ؍࡯Ͱ ͖ͳ͍ཁҼʹ͍ͭͯ control ͢Δ΋ͷͱղऍ͞ΕΔɻ͔͠͠ɺproduct-level fixed effects ͸धཁγϣοΫ͕ product ͱ market ͝ͱʹมಈ͢Δ৔߹ʹ͸ controls ͱͯ͠े෼Ͱ͸ͳ͍ɻ given market ʹ͓͍ͯશͯͷফඅऀʹͱͬͯಉ͡ࡒՁ֨Λߟ͑ͯ΋ɺͦΕͧΕͷ product×market ͝ͱͷݻ ఆޮՌ͸Ձ֨ͷมಈ΋ؚΜͰ͠·͏ͨΊɺधཁਪఆ͕Ͱ͖ͳ͘ͳͬͯ͠·͏ɻͭ·Γɺsupply-side ͷՁ֨ม ಈΛݟΔͨΊͷ௥Ճతͳཁૉ͕ඞཁɻ 2.5.2 Control function ʠControl functionʡ Ξϓϩʔν͸ 2SLS ͱࣅ͍ͯͯɺҎԼͷࣜΛߟ͑Δɻ T = F(Z, E1 ) : reduced-form equation (2.5) Y = G(T, E2 ) : outcome equation (2.6) ͜͜Ͱɺ • Tɿ಺ੜม਺ʢtreatmentʣ • E1 , E2 ɿεΧϥʔͱԾఆ͞ΕͨޡࠩͰɺ∂F/∂E1 > 0, ∂G/∂E2 > 0. T ΛՁ֨ͱͯ͠धཁྔ Y Λਪఆͨ͘͠ͳΔɻ׬શڝ૪ͷ single-good economy ΛԾఆͯ͠΋͜ͷܗͰ͸ʢڧ ͍ԾఆΛೖΕͳ͍ݶΓʣधཁਪఆͰ͖ͳ͍ɻधཁਪఆͷͱ͖ʹ͸ɺT Λ P Ͱஔ͖׵͑ͯɺ P = R(X, W, U, V ) (2.7) ͱͯ͠ɺdemand shock U ͱ cost shock V ͷશͯͷߏ଄ޡࠩΛؚΉ΋ͷͱͯ͠ఆࣜԽ͢ΔɻU ͱ V ͕εΧ ϥʔͱͯ͠ P ʹؚ·Ε͍͔ͯͯͭ (X, W) ⊥ ⊥(U, V ) ͳΒ control function ΞϓϩʔνͰधཁਪఆ͕Մೳɻ͠ ͔͠ɺෳ਺ͷࡒ͕͋Δ৔߹Λߟ͑Δͱɺ͋Δࡒ΁ͷधཁ͸ෳ਺ͷ಺ੜతͳՁ֨΍ͦΕͧΕͷ demand, supply shocks ʹґଘ͢ΔͷͰࣝผ͕ࠔ೉ʹͳΔɻ 2.5.3 Average treatment effects LATE ౳ͷฏۉతͳ൓ԠΛࣔ͢ࢦඪ͸धཁਪఆʹ΋࢖͑Δͷ͔ʁˠ In general, the answer is no. ࣜʢ2.1ʣ ɺࣜʢ2.2ʣʹ͓͍ͯɺফඅ੫ τ ͷۉߧՁ֨ P∗(X, W, U, V, τ) ΁ͷޮՌΛΈΔͱ͢Δɻࣜʢ2.2ʣΛ Q = S(W, P, V ) ͱॻ͖׵͑Δɻ ∂P∗(X, W, U, V, τ) ∂τ = ∂S(W,P,V ) ∂P ∂S(W,P,V ) ∂P + ∂D(X,P,U) ∂P (2.8) ఆٛΑΓ LATE ͸જࡏతͳม਺શͯʹؔ͢ΔฏۉͰɺ͜Ε͸ͦΕΒશͯΛݻఆͨ͠΋ͷͰ͸ͳ͍ͨΊ LATE Ͱ͸ͳ͍ɻΑͬͯɺLATE Ξϓϩʔν͸ؔ৺ͷ͋Δม਺ͷྔʹ͍ͭͯͷ ceteris paribus counterfactual Λੜ Έग़ͤͳ͍ɻͭ·Γɺtax change ͷҼՌޮՌ͕Θ͔Βͳ͍ɻ 2.6 Balancing flexibility and practicality धཁਪఆ͕࣋ͭ՝୊ͱͯ͠ɺ 6
7. 1. ݁Ռʹର͢Δڧ͍ priori restriction Λආ͚ΔͨΊʹे෼ʹॊೈͰɺ 2. practical application ΛՄೳʹ͢ΔͨΊʹ؆ུԽ͞Εͨ empirical specifications Λݟ͚ͭΔ͜ͱͰ͋Δɻࢢ৔ʹΑͬͯ͸ɺີ઀ʹؔ࿈͢Δࡒͷ਺͕ଟ͘ͳΔ͜ͱ͕͋ Δɻ͋Δࡒͷधཁ͸ɺؔ࿈͢ΔࡒͷಛੑͱՁ֨ʹґଘ͢ΔͷͰɺJ ݸͷࡒΛ࣋ͭ demand system ͸ɺ֤఺Ͱ J2 ݸͷՁ֨஄ྗੑΛ࣋ͭɻͭ·Γɺݪཧతʹ͸ϊϯύϥϝτϦοΫਪఆ͕ՄೳͰ͋ͬͯ΋ɺ࣮ࡍʹ͸ɺར༻ Մೳͳσʔλʹର࣮ͯ͠༻తͳ empirical model ΛಘΔͨΊʹ੍໿Λ՝͢ඞཁ͕͋Δɻ 2.7 Demand or utilities? ࣮ূ IO ͰधཁਪఆΛϞσϧԽ͢Δ৔߹ͷ࠷΋ҰൠతͳΞϓϩʔν͸ɺফඅऀͷޮ༻Λಛఆ͢Δ͜ͱ͔Β࢝ ·Δɻधཁਪఆͷओͳ໨త͸ɺधཁྔ͕Ձ֨΍ͦͷଞͷ؍ଌม਺ͷ ceteris paribus ͷมಈʹͲͷΑ͏ʹ൓Ԡ ͢Δ͔Λఆྔతʹදݱ͢Δ͜ͱɻফඅऀͷޮ༻࠷େԽΛߟ͑ΔΞϓϩʔνͷར఺ͱͯ͠͸ɺൺֱతগ਺ͷύϥ ϝʔλͰଟ͘ͷࡒʢ͕ͨͬͯ͠ɺଟ͘ͷࣗଞՁ֨஄ྗੑʣͷधཁΛදݱͰ͖Δ͜ͱʢe.g., Berry (1994)ʣ ɻফ අऀϨϕϧͰ͸ɺޮ༻ؔ਺Λ௨ͯ͡ΑΓ༰қʹఆࣜԽͰ͖ΔΑ͏ͳ੍໿΍ରশੑ৚݅Λ՝͍ͨ͠ͱߟ͑Δ͜ͱ ͕ଟ͍ɻ ͦͷΑ͏ͳ੍໿ͷྫͱͯ͠͸ɺ࣍ͷΑ͏ͳ΋ͷ͕͋Δɻ • ༩͑ΒΕͨࡒͷू߹ʹର͢Δબ޷ͷҟ࣭ੑ͸ɺࡒͷಛੑʹର͢Δফඅऀͷᅂ޷ͷҟ࣭ੑ͔Β෦෼తʹੜ ͡Δͱ͍͏Ծఆ • ࡒ j ͷಛੑͷมಈ͸ɺଞͷࡒʹର͢Δ૬ରతͳࡒ j ͷັྗΛม͑Δ͕ɺଞͷࡒͷ૊ͷ૬ରతͳັྗ͸ม ͑ͳ͍ͱ͍͏Ծఆ • ଞͷ͢΂ͯͷ৚͕݅ಉ͡Ͱ͋Ε͹ɺ֤ফඅऀ͸ɺࡒͷ໊শʹؔ܎ͳ͘ɺͲͷΑ͏ͳࡒಛੑͷ૊ʢe.g., Ձ ֨ͱ඼࣭ʣͷؒͰ΋ɺ୯Ұͷݶք୅ସ཰Λ࣋ͭͱ͍͏Ծఆ ͜ͷΑ͏ͳλΠϓͷ੍໿͸ɺωΨςΟϒͳଆ໘΋͋Δ͕ͦͷγϯϓϧ͞ͱॊೈੑʹ͍ͭͯద੾ͳόϥϯεΛఏ ڙ͢ΔՄೳੑ͕͋Δɻ 7
8. 3 Discrete choice demand ཭ࢄબ୒ϞσϧͰ͸Ξ΢ταΠυάοζΛఆٛ͢Δ͜ͱ͕ॏཁͰɺఆٛ͠ͳ͍ͱधཁͷՁ֨஄ྗੑ͕ৗʹ 0 ʹͳͬͯ͠·͏ɻྫ͑͹ɺอݥࢢ৔Ͱ͢΂ͯͷอݥͷՁ͕֨౳͘͠ 2 ഒʹͳͬͨ৔߹Ͱ΋อݥʹܖ໿͢ΔՈܭ ͷ਺ʹ͸Өڹ͕ͳ͍ɻ 3.1 Random utility models • ফඅऀ i ʹͱͬͯར༻Մೳͳʠinside goodsʡΛ j = 1, . . . , Ji . j = 0 ͰΞ΢ταΠυάοζɻ • Xi Λ؍࡯Ͱ͖Δଐੑͷू߹ͱ͠ɺফඅऀ i ʹ͍ͭͯɺར༻Մೳͳࡒʹ͍ͭͯɺࢢ৔ʹ͍ͭͯɺ·ͨ෼ੳ ऀʹͱͬͯ؍࡯Ͱ͖ͳ͍Α͏ͳଐੑΛؚΉ • ফඅऀͷબ୒ू߹͸ Ji , Xi ʹΑܾͬͯఆ͞ΕΔ • ফඅऀ i ͕ࡒ j Λߪೖ͢Δ͜ͱ͔ΒಘΒΕΔؒ઀ޮ༻Λ uij ͱ͢Δ • ফඅऀ i ͸શͯͷࡒʹର͢ΔࣗΒͷޮ༻Λ஌͓ͬͯΓɺ࠷΋ޮ༻͕ߴ͘ͳΔࡒΛબ୒͢ΔͱԾఆʢݦࣔ બ޷ʣ બ୒ू߹ (Ji , Xi ) Λॴ༩ͱͨ͠ͱ͖ɺͦΕͧΕͷফඅऀͷޮ༻ϕΫτϧ (ui0 , ui1 , . . . , uiJi ) ͸ i.i.d. Ͱ Fu (· | Ji , Xi ) ʹै͏ͱ͢Δɻ ޮ༻͸ং਺తͳ֓೦Ͱɺͦͷ஋ࣗମʹ͸ҙຯ͕ͳ͍ɻͦͷͨΊɺҰൠੑΛࣦ͏͜ͱͳͦ͘ΕͧΕͷফඅऀͷޮ ༻ͷ location ͱ scale Λج४Խ͢Δ͜ͱ͕Ͱ͖Δɻ ʢOnly differences in Utility Matterʣ *3 ·ͨɺuij = uik for j ̸= k ͕ͱͳΔ֬཰͸ 0 ͱԾఆ͢Δɻ ফඅऀ i ͷબ୒ϕΫτϧΛ (qi1 , . . . , qiJi ) ͱ͢Δɻ͜͜Ͱɺ qij = 1{uij ≥ uik ∀k ∈ {0, 1, . . . , Ji }} Ͱ͋ΔɻΑͬͯɺফඅऀ i ͕ࡒ j Λબ୒͢Δ֬཰͸ҎԼͷ௨Γɻ sij = E[qij | Ji , Xi ] = Aij dFu (ui0 , ui1 , . . . , uiJi | Ji , Xi ) ͜͜Ͱɺ Aij = (ui0 , ui1 , . . . , uiJi ) ∈ RJi+1 : uij ≥ uik ∀k . ྫ͑͹ɺJi = 2 Ͱߟ͑Δɻࡒ j ͷՁ֨Λ pj ͱ͢Δͱɺ uij = µij − pj for j > 0. ͜͜Ͱɺ(µi1 , µi2 ) ͸෼෍ Fµ (·) ͔Βಠཱʹυϩʔ͞ΕΔͱ͢Δɻ·ͨɺnormalization ͷͨΊʹ ui0 = 0 ͱ͢ ΔʢΞ΢ταΠυάοζͷޮ༻Λ 0ʣ ɻ *3 Train (2009), p19-28. ࢀর 8
9. ফඅऀ i ʹΑΔ֤ࡒͷબ୒֬཰͸ɺFu (·) ʹΑ֤ͬͯ region ʹׂΓ౰ͯΒΕΔ֬཰ʢprobability measureʣ ʹରԠ͢Δͱߟ͑Δ͜ͱ͕Ͱ͖Δɻ 3.2 The canonical model ཭ࢄબ୒ϞσϧͰ͸ɺࢢ৔ t Ͱফඅऀ i ͕ࡒ j Λফඅ͢Δ͜ͱ͔ΒಘΔޮ༻ؔ਺ܗʹ͍ͭͯҎԼͷԾఆΛ ͓͘ɻ uijt = xjt βjt − αit pjt + ξjt + ϵijt for j > 0 (3.1) ͜͜Ͱɺ • j ∈ Jt ɿJt ͸ inside goods ͷू߹ • xjt ∈ RKɿxjt ͸ࡒͷଐੑ K ࣍ϕΫτϧ*4 ·ͨɺ ui0t = ϵi0t (3.2) ͱͯ͠Ξ΢ταΠυάοζ j = 0 ͔Βͷޮ༻Λج४Խ͢Δɻ • ࢢ৔ t = 1, . . . , T ͸ɺ࣌ؒతɾ஍ཧతʹ෼͚ΒΕΔ • ར༻Մೳͳࡒͷ਺Λ Jt = |Jt | ͱ͢Δ • ξjt ͸ফඅऀؒͰڞ௨ͷɺj ʹಛ༗ͳଐੑΛද͍ͯͯ͠ɺྫ͑͹ɺϒϥϯυྗ΍ CM ʹΑΔاۀΠϝʔ δͳͲ • ߴ͍஋ͷ ξjt ͸ࢢ৔ t ͷফඅऀʹͱͬͯฏۉతͳ taste ͕ߴ͍͜ͱΛ͍ࣔͯ͠Δ ҎԼɺxt = (x1t , . . . , xJt,t ), pt = (p1t , . . . , pJt,t ), ξt = (ξ1t , . . . , ξJt,t ), Xt = (xt , pt , ξt ) ͱ͢Δɻ ௨ৗɺpt ͱ ξjt ͸૬ؔ͢Δͱߟ͑ΒΕΔɻͳͥͳΒɺࢢ৔ t ͷࡒ j ͷۉߧՁ֨͸ xt ͱ ξt ͷશͯͷཁૉʹґଘ ͢Δͱߟ͑ΒΕΔͨΊɻ·ͨɺۉߧՁ֨͸જࡏతͳݶքඅ༻΁ͷγϣοΫʹΑͬͯӨڹΛड͚ɺ͜Εʢજࡏత ͳݶքඅ༻΁ͷγϣοΫʣ͸؍࡯Ͱ͖ͳ͍धཁ΁ͷγϣοΫͱ૬ؔ͢Δͱߟ͑ΒΕΔɻ *4 xjt ͷ࠷ॳͷཁૉ͸ 1 Ͱɺξjt ͷฏۉΛଊ͍͑ͯΔͱղऍͰ͖Δɻ 9
10. ࡒͷଐੑ xt ͸֎ੜͩͱԾఆ͞ΕΔ͕ɺ͜Ε͸ essential ͳԾఆͰ͸ͳ͘ɺྑ͍ૢ࡞ม਺͕͋Ε͹͍͍ʢͦΜͳ ʹͳͦ͞͏ʁʣ ɻ ϵijt ʹ͍ͭͯ͸ɺҎԼͷΑ͏ͳԾఆ͕ஔ͔ΕΔɻ *5 • i.i.d. draw from ୈҰछۃ஋෼෍ʢΨϯϕϧ෼෍ʣ *6ɿmixed multinomial logit model Λಋग़Ͱ͖Δ • i.i.d. draw from ਖ਼ن෼෍ɿmixed multinomial probit model Λಋग़Ͱ͖Δ ʠmixedʡ͸ɺ؍࡯͞ΕΔɾ͞Εͳ͍ଐੑͷؒͰͷݶք୅ସ཰Λ൓ө͢Δύϥϝʔλ αit , βit ͷɺফඅऀؒͰͷ ҟ࣭ੑΛදݱ͍ͯ͠Δɻ mixed logit ͷ৔߹ɺબ୒֬཰ʢϚʔέοτγΣΞʣΛҎԼͷࣜͷ௨ΓಘΔɻ sjt = exjtβit−αitpjt+ξjt Jt k=0 exktβit−αitpkt+ξkt dF(αit , βit ; t) (3.3) ͜͜ͰɺF(·; t) ͸ࢢ৔ t ʹ͓͚Δ (αit , βit ) ͷಉ࣌෼෍Ͱɺফඅऀͷબ޷ύϥϝʔλ (αit , βit ) ͸ʠrandom coefficientsʡͱݺ͹ΕΔɻ ϥϯμϜ܎਺ϕΫτϧ βit ͷͦΕͧΕͷཁૉ k ͸ҎԼͷ௨Γɻ β(k) it = β(k) 0 + β(k) v v(k) it + L l=1 β(l,k) d dilt (3.4) ͜͜Ͱɺ • β(k) 0 ɿx(k) jt ʹର͢ΔશͯͷফඅऀͰڞ௨ͷબ޷ύϥϝʔλ • dilt ɿݸਓ i ͷଐੑͰɺ͜Εͷ෼෍͸ط஌ͱ͢Δɻi.e., dilt ∼ Ft (d). • v(k) it ɿpre-specified distributionʢਖ਼ن෼෍ͳͲʣʹै͏֬཰ม਺ɻi.e., vi ∼ G(v). • β(l,k) d and β(k) v ɿҟͳΔଐੑ dit or ҟͳΔબ޷γϣοΫ v(k) it Λ࣋ͭফඅऀؒͰͷɺ੡඼ଐੑ x(k) jt ʹର ͢Δબ޷ͷ౓߹͍ dilt ͱ v(k) it ͷҧ͍͸෼෍͕ط஌Ͱ͋Δ͔Ͳ͏͔ɻྫ͑͹ɺं΁ͷधཁΛߟ͑ͨ࣌ʹɺՈ଒͕ͨ͘͞Μ͍Δਓ͸ ΑΓେ͖͍ं΁ͷબ޷ύϥϝʔλͷ஋͕େ͖͘ͳΔͱߟ͑ΒΕΔɻՈ଒ͷ਺͸ެతσʔλ͔ΒೖखͰ͖Δ৔߹ ΋͋ΔͷͰͦͷͱ͖ʹ͸෼෍͕ط஌ɻ͜͏࣮ͨ͠ࡍͷ෼෍͸ফඅऀؒͷજࡏతͳબ޷ͷҟ࣭ੑΛଊ͑Δ͜ͱ͕ Ͱ͖ΔɻͦͷҰํͰɺंͷ೩අʹؔ͢Δબ޷ύϥϝʔλʢͷ෼෍ʣ͸෼ੳऀʹͱͬͯ͸Θ͔Βͳ͍ɻͦͷͨ Ίɺ͜ͷજࡏతͳબ޷ͷҟ࣭ੑ͸ɺͳΜΒ͔ͷ෼෍ΛԾఆͯ͠ɺv(k) it ʹΑͬͯଊ͑Δͱղऍ͢Δɻ Ձ֨ʹؔ͢Δݸਓ i ͝ͱͷબ޷ύϥϝʔλ αit ͸ҎԼͷ௨ΓఆࣜԽ͞ΕΔɻ ln(αit ) = α0 + αy yit + αv v0 it ͜͜Ͱɺyit ͸ॴಘͳͲͷɺՁ֨൓ԠʹӨڹΛ༩͑ΔͱԾఆ͞ΕΔফඅऀݻ༗ͷई౓ɻyit ͸෦෼తʹɺ΋͘͠ ͸΄ͱΜͲશ͕ͯ dit ͱඃ͍ͬͯΔɻ *5 ϵijt ͷಠཱੑ͸ mixed probit model Ͱ֎͞ΕΔɻ *6 ୈҰछۃ஋෼෍ʹै͏֬཰ม਺ X ͷ෼෍ؔ਺ F ͸ɺF(x) = exp{− exp{−( x−µ η )}} ͱͳΔɻ 10
11. 3.3 Why random coefficients? ϥϯμϜ܎਺ (αit , βit ) Λ༻͍ΔϞνϕʔγϣϯ͸ɺϥϯμϜ܎਺Ͱ͸ͳ͍৔߹Λߟ͑ͯΈΔ͜ͱͰΘ͔Δɻ uijt = xjt β0 − αpjt + ξjt + ϵijt (3.5) ͜͜Ͱɺ δjt = xjt β0 − αpjt + ξjt (3.6) ͱ͢Δͱɺ uijt = δjt + ϵijt (3.7) δjt ͸ʠmean utilityʡͱݺ͹Εɺ *7ݸਓؒͰҟͳΒͳ͍ޮ༻ͷཁૉɻࡒͷબ୒֬཰͸ɺฏۉޮ༻ͷΈʹґଘ͠ ͍ͯΔ͜ͱ͕Θ͔Δɻ·ͨಉ༷ʹɺՁ֨஄ྗੑ͕ฏۉޮ༻ͷΈʹґଘ͍ͯ͠Δɻ ͜ͷϞσϧͰ͸ɺࡒ j ͷࣗݾՁ֨஄ྗੑ͕ࡒ j ͷՁ֨ͱબ୒֬཰ͷΈʹΑܾͬͯ·Γɺࡒ l ͷՁ֨ pl ͕มԽ ͨ࣌͠ͷࡒ j ͷަࠩՁ֨஄ྗੑ͸ࡒ l ͷબ୒֬཰ͱՁ͚֨ͩʹґଘ͠ɺࡒ j ͷଐੑ΍બ୒֬཰ʹ͸Ұ੾ґଘ͠ ͳ͍ɻ·ͨɺಉ͡બ୒֬཰ʢϚʔέοτγΣΞʣͷࡒ͸ಉ͡Ձ֨஄ྗੑͰɺಉ͡ϚʔΫΞοϓ཰ͰɺަࠩՁ֨ ஄ྗੑ͕શͯಉ͡ʹͳΔɻ⇒ ࡒͷؒͷ୅ସੑΛଊ͑ΒΕ͍ͯͳ͍ͨΊɺͱͯ΋໰୊͕͋ΔͷͰ͸ͳ͍͔ɻ ϥϯμϜ܎਺ͷ΋ͱͰ͸ɺϚʔέοτγΣΞ͕ಉ͡ 2 ͭͷࡒʢྫɿτϥοΫͱϛχόϯʣ͕͋ͬͨͱ͖ʹɺ ͦͷࡒͷଐੑΛߟྀ͢Δ͜ͱ͕Ͱ͖ΔͷͰɺୈ 3 ͷࡒʢϑΝϛϦʔ޲͚ͯͷ SUVʣʹର͢ΔՁ֨஄ྗੑͷ༧ ଌ஋΋શ͘ҟͳͬͨ΋ͷʹͳΔɻ͕ͨͬͯ͠ɺϥϯμϜ܎਺͸ࠩผԽ͞Εͨࡒͷ key dimensions ʹԊͬͨফ අऀͷબ޷ͷҟ࣭ੑΛଊ͑Δ͜ͱ͕Ͱ͖Δɻ In practical: xt ʹͲͷม਺ΛؚΊΕ͹͍͍͔໰୊͕͋ΔɻΊͬͪΌଟ͍਺ͷϥϯμϜ܎਺ʹΑͬͯফඅऀͷҟ࣭ੑΛଊ ͑Α͏ͱ͢Δͱ impricise ͳਪఆͷՄೳੑ͕͋ΔɻGillen et atl. (2014) ͱ Gillen et al. (2019) ͸ xt Λ large data set ͔Β data-driven ʹબͿํ๏ΛఏҊ͍ͯ͠ΔɻEconomic model ͔Βߟ͑ΒΕΔɺফඅऀͷ୅ସύ λʔϯΛܾఆ͢Δͷʹॏཁͳม਺ΛબͿ͜ͱ͕Αͦ͞͏Ͱ͋Δɻ΄ͱΜͲͷ৔߹Ͱɺ؍࡯͞ΕΔ (xt , pt ) ͷશ ͯͷཁૉΛೖΕΔͷ͸ద੾Ͱ͸ͳ͍ɻૢ࡞ม਺ͱࣝผʹؔ͢Δٞ࿦Λ͢Δ section Ͱ࠶౓͜ͷ issue Λߟ͑ͯ ͍Δɻ 4 Market-level data *7 E[ϵijt] ̸= 0 ͷ࣌ɺE[uijt] = E[δjt] + E[ϵijt] ̸= E[δjt] ͱͳΔ͜ͱʹ஫ҙɻඞͣ͠΋ޮ༻ͷظ଴஋Λҙຯ͢ΔΘ͚Ͱ͸ͳ͍ 11
12. 5 Nonparametric identification: market-level data Berry and Haile (2014) ͸ɺmarket-level data ͔͠खݩʹͳ͍ͱ͖ʹɺधཁͷϊϯύϥϝτϦοΫϞσϧ ͷࣝผͷͨΊʹඞཁͳ৚݅Λࣔͨ͠ɻ 5.1 Insights from parametric models familiar ͳ parametric models ΛϨϏϡʔ͢Δ͜ͱͰɺधཁͷࣝผͷͨΊͷ intuition ͱ insights Λݟ͍ͨɻ 5.1.1 Multinomial logit Ϛʔέοτ t Ͱফඅऀ i ͕ࡒ j ͔ΒಘΔޮ༻͸ɺҎԼͷΑ͏ʹԾఆ͞ΕΔɻ uijt = xjt β − αpjt + ξjt + ϵijt . ͜͜Ͱɺthe linear index Λ δjt = xjt β − αpjt + ξjt (5.1) ͱ·ͱΊΔͱɺϚʔέοτγΣΞ͸ҎԼͷࣜͰಘΒΕΔɻ sjt = eδjt 1 + k eδkt (5.2) ͜Ε͸ɺδ1t , . . . , δJt ʢthe indicesʣͷඇઢܗͳؔ਺ͱͳ͍ͬͯΔɻॏཁͳͷ͸ɺindices ͔ΒϚʔέοτγΣ Ξ΁ͷม׵͕؆୯ͳ͜ͱͰɺ δjt = ln(sjt ) − ln(s0t ) (5.3) ln(sjt ) − ln(s0t ) = xjt β − αpjt + ξjt (5.4) ΑΓɺઢܗʹਪఆͰ͖ΔࣜΛಘΔ͜ͱ͕Ͱ͖Δ (Berry (1994))ɻ͜ͷࣜͷࣝผͷͨΊʹ͸ɺӈลͷՁ֨ pjt ΁ ͷҰͭͷૢ࡞ม਺Ͱे෼ɻ ׬શͳϊϯύϥϝτϦοΫϞσϧΛݟΔલʹɺ࣍ͷ৚͔݅ΒͳΔ semi-parametric model Λߟ͑ͯΈΔɻ 1. xjt Λ 2 ͭʹ෼͚Δ x(1) jt , x(2) jt ͜ͱ͕Ͱ͖ɺx(1) jt ∈ R ͱ͢Δɻ 2. ଟ߲ϩδοτϞσϧͷߏ଄͸ɺ֎ੜม਺ x(2) t ʹΑͬͯ৚͚݅ͮͨޙʹͷΈอͨΕΔ΋ͷͱ͢Δɻ ͜ͷԾఆʹΑͬͯɺधཁϞσϧͷதͰͷ x(2) t ͷΑΓॊೈͳѻ͍͕ՄೳͱͳΔɻྫ͑͹ɺࡒ j ͷฏۉޮ༻͸ɺ x(2) t ͷϊϯύϥϝτϦοΫؔ਺ʹΑͬͯӨڹΛड͚͏Δ͠ɺ΋͘͠͸ k ̸= j ͷ xkt *8ʹΑͬͯ΋ӨڹΛड͚ ͏Δɻ ͜ͷΑΓॊೈͳधཁϞσϧͷࣝผΛߟ͑Δɻγϯϓϧʹ x(2) t ͷ஋Λݻఆͯ͠ɺϊʔςʔγϣϯ͔Βམͱ͢ɻ ͦ͏͢ΔͱɺϥϯμϜޮ༻͸ uijt = x(1) jt β(1) − αpjt + ξjt + ϵijt j = 1, . . . , J. *8 Handbook of IO p.34 Ͱ͸ʮxkj ʯͱͳ͍ͬͯΔ͕ޡ২ʁ 12
13. ͱͳΔɻมܗ͢Δͱ the inverted demand equation ͸ɺ x(1) jt + ˜ ξjt = 1 β(1) (ln(sjt ) − ln(s0t )) + α β(1) pjt (5.5) x(1) jt = 1 β(1) (ln(sjt ) − ln(s0t )) + α β(1) pjt − ˜ ξjt (5.6) ͱͳΔɻ͜͜Ͱɺ˜ ξjt = ξjt β(1) . ࠨลͷม਺ x(1) jt ͸֎ੜతͳͷͰɺ˜ ξjt ͱฏۉಠཱͱͳ͍ͬͯΔɻ ࣜ (5.6) ͷΠϯϓϦέʔγϣϯͱͯ͠ɺӈลʹ 2 ͭͷ಺ੜม਺ʢ(ln(sjt ) − ln(s0t )) , pjt ʣ͕͋Δʹ΋ؔΘΒͣɺ धཁͷࣝผͷͨΊʹ͸ 1 ͭͷૢ࡞ม਺ zjt ͕͋Ε͹े෼Ͱ͋Δͱ͍͏͜ͱ͕Θ͔Δʢ͔ͩΒ௨ৗͷճؼϞσϧ ͱ͸ҟͳΔʣ ɻಛʹɺE[ξjt | x(1) jt , zjt ] = 0 ͱ͍͏৚݅෇͖ͷϞʔϝϯτ৚͕݅ɺ1 ͭͷૢ࡞ม਺͕͋Δ͜ͱ͔ ΒಘΒΕɺӈลͷ 2 ͭͷύϥϝʔλʔΛࣝผ͢Δ͜ͱ͕Ͱ͖Δɻ 5.1.2 Nested logit ࣍ͷ nested logit model Λߟ͑Δɻ ln(sjt ) − ln(s0t ) = xjt β − αpjt + (1 − λ) ln(sj/g,t ) + ξjt j = 1, . . . , J. ͜͜Ͱɺg ͸ࡒ j ͕ॴଐ͢Δ nest (or group) Λҙຯ͢Δɻଟ߲ϩδοτϞσϧʹɺwithin-group share Ͱ͋ Δ sj/g,t ʹؔ͢Δ܎਺ λ ͕௥Ճ͞Ε͍ͯΔɻΑͬͯɺՁ֨ͷૢ࡞ม਺ʹՃ͑ͯɺ಺ੜม਺ ln(sj/g,t ) ʹؔ͢Δ ૢ࡞ม਺͕௥ՃͰඞཁͱͳΔɻ ಉ༷ʹɺx(2) t Ͱ৚͚݅ͮͯɺϊʔςʔγϣϯ͔Β͜ΕΛམͱ͢ͱɺthe inverted demand system ͸ɺ x(1) jt + ˜ ξjt = 1 β(1) ln(sjt ) − ln(s0t − (1 − λ) ln(sj/g,t ) + α β(1) pjt (5.7) ͱͳΔɻӈล͸ɺଟ߲ϩδοτϞσϧΑΓ΋ෳࡶͳࢢ৔γΣΞͱՁ֨ͷؔ਺ͱͳ͍ͬͯΔɻ͜ΕΛࣜ (5.6) ͷ Α͏ʹมܗͨ͠ͱͯ͠΋ɺࣝผͷͨΊʹඞཁͳૢ࡞ม਺ͷ set ͸มΘΒͳ͍ɻ 5.1.3 The BLP model The BLP mixed logit model Λߟ͑Δɻ͜͜Ͱɺxjt ͷগͳ͘ͱ΋Ұͭͷཁૉ x(1) jt ͸ϥϯμϜ܎਺ͳ͠ͰϞ σϧʹೖ͍ͬͯΔʢthis is a restrictionʣ͜ͱʹ஫ҙ͢Δɻલʹٞ࿦͍ͯ͠Δ௨Γɺ௨ৗ the demand system ͸ inverse Λ͓࣋ͬͯΓɺ͜Ε͸ฏۉޮ༻ϕΫτϧ δt ΋͘͠͸धཁγϣοΫ ξt ʹ͍ͭͯॻ͔ΕΔɻ x(1) jt + ˜ ξjt = 1 β(1) ˜ δj st , pt , x(2) t , θ . (5.8) ӈล͸ࢢ৔γΣΞͷٯؔ਺ ˜ δj ͔ΒͳΔɻ͜ͷؔ਺͸ model ͷ͢΂ͯͷύϥϝʔλʹґଘ͓ͯ͠ΓɺΑΓෳࡶ ͳࢢ৔γΣΞͱՁ֨ͷؔ਺ͱͳ͍ͬͯΔɻ·ͨɺ͜ΕΒ͢΂ͯ͸धཁγϣοΫ ˜ ξjt ͱ૬ؔ͢Δɻ ͢Ͱʹٞ࿦͞Εͨ௨Γɺ֎ੜม਺ xjt ͱՁ֨ pjt ͷૢ࡞ม਺͔ΒͳΔૢ࡞ม਺ zjt ͕͋Δ࣌ɺϞʔϝϯτ৚݅ E[ξjt zjt ] = 0 ͸ࣝผͷͨΊʹे෼Ͱ͸ͳ͍ɻͳͥͳΒɺࣜ (5.8) ͸ɺӈลͷ st ͱ pt ͷे෼ͳมಈΛ࡞Γग़ͤ Δૢ࡞ม਺͕ඞཁͰ͋Δ͜ͱΛҙຯ͍ͯ͠Δ͔Βɻ 5.1.4 Index, inversion, and instruments Ҏ্ͷɺελϯμʔυͳύϥϝτϦοΫϞσϧ͕Ͳ͏ࣝผ͞ΕΔ͔ʹ͍ͭͯͷϨϏϡʔΑΓҎԼ͕Θ͔Δɻ • ͦΕͧΕͷࡒʹ͍ͭͯɺधཁγϣοΫ͸ indicesʢࣜͷࠨลʹ੔ཧͨ͠΋ͷʣΛ௨ͯ͠ޮ༻ʹೖΔ 13
14. • the indices ͱࢢ৔γΣΞͷ 1 ର 1 ͷؔ܎ʹΑͬͯɺdemand system ͷ inversion ͕Ͱ͖Δ • ૢ࡞ม਺͸ɺٯधཁʢthe inverse demandʣ˜ δj ͷཁૉͷࣝผͷͨΊʹ࢖༻͞ΕΔ ҎԼͰ͸ɺ͜ΕΒͷΞΠσΞ͕ɺmarket-level data ͔ΒͷधཁͷϊϯύϥϝτϦοΫͳࣝผΛڐ͢Ծఆ΍৚ ݅Λݟ͍ͯ͘ɻBerry Haile (2014) ʹैͬͯɺҰൠతͳधཁϞσϧʹϊϯύϥϝτϦοΫͳ index ΁ͷ੍໿Λ ՝͢͜ͱͰɺdemand system ͔Β inversion Λ͢Δ͜ͱ͕Ͱ͖ɺҰͭͷࣜʹ͖ͭҰͭͷधཁγϣοΫ͔ΒͳΔ ٯधཁΛੜΈग़͢ɻ͜ͷ࣌ɺඪ४తͳૢ࡞ม਺ͷ৚͔݅Βɺ͜ΕΒͷٯधཁؔ਺Λࣝผ͢Δ͜ͱ͕Ͱ͖Δɻ 5.2 Nonparametric demand model ࢢ৔ t Ͱͷࡒ j ͷधཁ͸ҎԼͷ௨Γɻ sjt = σj (xt , pt , ξt ) j = 1, . . . , J. (5.9) ͜͜Ͱɺ • sjt ɿࢢ৔Ϩϕϧσʔλ͔Β؍࡯͢Δ͜ͱ͕Ͱ͖Δ • xt ɿ؍࡯Ͱ͖Δશͯͷ֎ੜม਺ΛؚΉ • pt ɿશͯͷࡒͷՁ֨ • ξt ɿ؍࡯Ͱ͖ͳ͍ J ࣍ݩͷधཁγϣοΫ ϥϯμϜޮ༻཭ࢄબ୒ϞσϧΑΓɺ ޮ༻ϕΫτϧ (ui1t , . . . , uiJt ) ͸ i.i.d. ͰԿΒ͔ͷಉ࣌෼෍ FU (· | xt , pt , ξt ) ͔ΒಘΒΕΔͱ͢Δɻ͜͜Ͱɺࣜʢ5.9ʣͷ demand system ͸ϥϯμϜޮ༻ͷ specification Λඞཁͱ͍ͯ͠ ͳ͍ɻΑͬͯɺधཁϞσϧ͸Ұൠతͳ΋ͷͰɺ།Ұͷ੍໿͸धཁγϣοΫ͕εΧϥʔͰ J ݸ͋ΔͱԾఆ͍ͯ͠ Δ͜ͱɻ 5.2.1 A nonparametric index xt Λ x(1) t , x(2) t ʹ෼ׂ͠ɺ͜͜Ͱɺx(1) t ∈ RJ ͱ͢ΔɻͦΕͧΕͷࢢ৔ t ʹ͍ͭͯɺindices ͷϕΫτϧΛ δt = (δ1t , . . . , δJt ) ͱ͠ɺ͜͜Ͱɺ δjt = x(1) t βj + ξjt . (5.10) Assumption 5.1 (Index). ∀j, σj (xt , pt , ξt ) = σj x(2) t , δt , pt . Ծఆ 5.1 ͸ɺx(1) t ͱ ξjt ͕ɺindex Ͱ͋Δ δjt Λ௨ͯ͠ͷΈϊϯύϥϝτϦοΫؔ਺ σ ʹೖΔ͜ͱΛཁٻ͠ ͍ͯΔɻ͢Ͱʹٞ࿦ͨ͠ྫͷΑ͏ʹɺधཁਪఆʹ͓͍ͯ͜ͷλΠϓͷ੍໿͸ɺ҉໧తʹύϥϝτϦοΫϞσϧ Ͱ޿͘࢖ΘΕ͍ͯΔɻ͜͜Ͱɺthe indices (δ1t , . . . , δJt ) ͸ͦΕͧΕͷࡒ j ʹ͍ͭͯϊϯύϥϝτϦοΫؔ਺ σ Λ௨ͯ͠धཁΛมԽͤ͞Δ͜ͱΛڐ͞Ε͍ͯΔɻͳͥ͜ͷ index ʹؔ͢ΔԾఆ͕ॏཁͳͷ͔ΛҎԼͰ֬ೝ ͠ɺ ʠmicro dataʡ͕͋Δͱ͖ʹ͸͜ͷԾఆ͕ͲͷΑ͏ʹ؇ΊΒΕΔͷ͔Λ section 7 Ͱ֬ೝ͢Δɻ ύϥϝτϦοΫϞσϧͱಉ༷ɺ؍࡯Ͱ͖ͳ͍धཁγϣοΫ͸ natural ͳ location ΍ scale Λ࣋ͨͳ͍ɻͦͷͨ ΊɺϢχʔΫͳબ޷ͷ representation Λ࣋ͭͨΊʹ͜ΕΒΛඪ४Խ͢Δඞཁ͕͋ΔɻΑͬͯɺҰൠੑΛࣦ͏͜ ͱͳ͘ E[ξjt ] = 0 ͱ |βj | = 1 ∀jʢscale ͷඪ४ԽʣΛԾఆ͢Δ͜ͱ͕Ͱ͖Δɻैͬͯɺindices ͸ δjt = x(1) jt + ξjt . (5.11) ͱͳΔɻ·ͨɺ֎ੜม਺Ͱ͋Δ x(2) jt ͸ࣝผʹ͓͍ͯԿͷ໾ׂ΋Ռͨ͞ͳ͍ͷͰɺx(2) t Λ೚ҙͷ஋ʹݻఆ͠ɺ ϊʔςʔγϣϯ͔Βམͱ͢ɻ 14
15. 5.2.2 Inverting demand ύϥϝτϦοΫϞσϧͷྫͰ͸ɺthe demand system Λ inverting ͢Δͷ͕ॏཁͳ޻ఔͩͬͨɻϊϯύϥϝ τϦοΫϞσϧͰ΋ɺ ʠconnected substitutesʡ৚݅ͷ΋ͱͰ͜ͷઓུΛ࢖͏͜ͱ͕Ͱ͖Δ (Berry, Gandhi, and Haile (2013))ɻ ඪ४తͳ཭ࢄબ୒Ϟσϧ͸ɺશͯͷ୅ସతͳࡒͷؒͷબ୒ΛؚΜͰ͍Δɻྫ͑͹ɺ΋͠ uijt ͕Ձ֨ pjt ʹͭ ͍ͯ୯ௐݮগͩͱ͢Δͱɺpjt ͷݮগ͸ࡒ j ͷࢢ৔γΣΞΛ૿Ճͤ͞ɺଞͷ͢΂ͯͷࡒͷࢢ৔γΣΞΛݮগ͞ ͤΔɻҎԼͷʠconnected substitutesʡԾఆͷ΋ͱͰɺ୅ସతͳࡒͷؒͰͷબ୒ͷ֓೦Λ 2 ͭͷํ๏Ͱ૿ڧ͞ ͤΔ͜ͱ͕Ͱ͖Δɻ·ͣɺindex Ͱ͋Δ δjt ʹʢϚΠφεͷʣՁ֨ͷΑ͏ʹৼΔ෣͏͜ͱΛཁٻ͢Δɻͭ·Γɺ δjt ͷ૿Ճ͸ଞͷ͢΂ͯͷࡒ k ̸= j ͷࢢ৔γΣΞΛݮগͤ͞Δ͜ͱɻ࣍ʹɺࡒͷؒͷɺ࠷খݶͷ strict ͳ୅ସ Λཁٻ͢Δɻ Definition 5.1 (Connected Substitutes). Goods (0, 1, . . . , J) are connected substitutes in δt if both 1. σk (δt , pt ) is nonincreasing in δjt ∀j > 0, k ̸= j, and any (δt , pt ) ∈ R2J ; 2. for each (δt , pt ) ∈ supp(δt , pt ) and any nonempty K ⊆ {1, . . . , J}, there exist k ∈ K and l / ∈ K such that σl (δt , pt ) is strictly decreasing in δkt . Assumption 5.2. Goods (0, 1, . . . , J) are connected substitutes in δt . ఆٛ 5.1 ͷ 1 ͸ɺindices ʹؔͯ͠ࡒ͕ weak substitutes Ͱ͋Δ͜ͱΛཁٻ͍ͯ͠Δɻͭ·Γɺindex Ͱ͋ Δ δjt ͷ૿Ճ͕ଞͷࡒͷधཁΛඞͣ weakly ʹݮগͤ͞Δඞཁ͕͋Δɻ͜Ε͸ɺδjt ͕ࡒ j ͷ࣭ΛมԽͤ͞Δཁ ҼΛද͍ͯ͠Δͱ͢Ε͹ࣗಈతʹຬͨ͞ΕΔ͸ͣɻΑͬͯɺ1 ͸ weak substitution ͷΈΛཁٻ͍ͯ͠Δ͕ɺ2 ͷํ͸গͳ͘ͱ΋ࡒ j = 0, 1, . . . , J ͷதͰͷ strict substitution Λཁٻ͍ͯ͠Δɻ Ծఆ 5.2 ͷͨΊͷे෼ͳ৚݅ͱͯ͠͸ɺͦΕͧΕͷ uijt ͕ δjt ͷݫີͳ૿Ճؔ਺Ͱ͋Δ͜ͱɺ͔ͭ (δkt , pkt ) for k ̸= j ʹӨڹΛड͚ͳ͍͜ͱɻ Berry, Gandhi, and Haile (2013) ͸ɺ෯޿͍धཁϞσϧʹ͓͍ͯ͜ͷ৚͕݅ຬͨ͞ΕΔ͜ͱΛ࣮ূ͠ɺ धཁͷ invertibility ͸ɺधཁγεςϜͷ͋Δ injective transformationʢ1 ର 1 ͷؔ܎ʣʹରͯ͠ connected substitutes ৚͕݅੒ཱ͢Δͱ͖͸͍ͭͰ΋อূ͞ΕΔͨΊɺࡒ͕ิ׬Ͱ͋Δ͍͔ͭ͘ͷέʔεͰ΋ɺ͜ΕΒ ͷ৚݅ʹΑͬͯ invertibility ͕࣮ূ͞ΕΔ͜ͱΛ͍ࣔͯ͠Δɻ Ҏ্ΑΓɺಛʹॏཁͳͷ͸͢΂ͯͷधཁʹରͯ͠ɺϕΫτϧ st such that sjt > 0 ∀j ʹ͍ͭͯɺधཁγες Ϝͷ inverse ͕ଘࡏͯ͠ҎԼͷܗΛͱΔ͜ͱɻ δjt = σ−1 j (st ; pt ) j = 1, . . . , J. (5.12) 5.3 Identification via instruments ࣜ (5.12) ΑΓɺधཁͷࣝผʹඞཁͳૢ࡞ม਺ͷ৚݅ΛݟΔɻ͜ͷࣝผͷͨΊʹඞཁͳ৚݅͸ճؼϞσϧͷ ࣝผͷͨΊʹඞཁͳૢ࡞ม਺ͷ৚݅ͱಉ͡ɻϊϯύϥϝτϦοΫճؼϞσϧͰ͸ɺ࣍ͷܗͷࣜʹڵຯ͕͋Δɻ y = Γ(x) + ϵ (5.13) ͜͜Ͱɺx ∈ RK. Newey and Powell (2003) ͸ɺE[ϵ|z] = 0 Λຬͨ͢ૢ࡞ม਺ z Λॴ༩ͱͨ͠ͱ͖ʹɺ ճؼؔ਺ Γ ͷࣝผͷͨΊʹඞཁे෼ͳ৚݅͸ඪ४తͳʠcompletenessʡcondition Ͱ͋Δ͜ͱΛࣔͨ͠ɻ 15
16. ʠcompletenessʡcondition ͱ͸ɺؔ਺ B(·) ∈ RK ͷΫϥεͷதͰʢE[B(x) | x] < ∞ʣ ɺE[B(x) | z] = 0 Λ ຬͨ͢།Ұͷؔ਺ B ͸ɺͦͷఆٛҬ্ʢdomainʣͰ͸΄΅࣮֬ʹʢ֬཰ 1 Ͱʣ0 ʹࣸ૾͞ΕΔؔ਺ͷ͜ͱɻ Assumption 5.3 (Instruments). . 1. For all j = 1, . . . , J, E ξjt | zt , x(1) t = 0 almost surely. 2. For all functions B(st , pt ) with finite expectation, if E B(st , pt ) | zt , x(1) t = 0 almost surely then B(st , pt ) = 0 almost surely. ͜͜Ͱɺࣜ (5.12) Λมܗ͢Δ͜ͱͰɺ x(1) jt = σ−1 j (st ; pt ) − ξjt (5.14) ΛಘΔɻ Theorem 5.1. Under Assumptions 5.1-5.3, for all j = 1, . . . , J, (i) σ−1 j is identified with probability 1 for all t, and (ii) the function σj (Xt ) is identified on X.*9 Proof. ࣜ (5.14) ͷ྆ลΛ zt , x(1) t Ͱ৚݅෇͚ͨظ଴஋ΛͱΔɻ E ξjt | zt , x(1) t = E σ−1 j (st , pt ) | zt , x(1) t − x(1) jt Assumption 5.3 ΑΓɺ E σ−1 j (st , pt ) | zt , x(1) t − x(1) jt = 0 a.s. ͋Δผͷؔ਺ ˜ σ−1 j ͕͋Δͱͯ͠ɺҎԼΛಉ༷ʹຬͨ͢ͱ͢Δɻ E ˜ σ−1 j (st , pt ) | zt , x(1) t − x(1) jt = 0 a.s. ͜͜ͰɺB(st , pt ) = σ−1 j (st , pt ) − ˜ σ−1 j ͱ͢Δͱɺ E B(st , pt ) | zt , x(1) t = 0 a.s. Assumption 5.3 ΑΓɺ֬཰ 1 Ͱ ˜ σ−1 j = σ−1 j ͱͳΔɻ͜Ε͸ɺσ−1 j ͕ࣝผ͞Εͨ͜ͱΛ͍ࣔͯ͠Δɻ͜ΕΛ ͢΂ͯͷ j ʹ܁Γฦ͢͜ͱͰɺࣜ (5.14) ΑΓ ξjt ͕ϢχʔΫʹ֬཰ 1 Ͱܾఆ͞ΕΔɻ͜ΕʹΑͬͯ (i) ͕ূ໌ Ͱ͖ͨɻબ୒֬཰ʢϚʔέοτγΣΞʣ͸؍࡯͞Εɺधཁؔ਺ σj (Xt ) ͷ͢΂ͯͷཁૉ͸ʢξjt ͕ϢχʔΫʹܾ ·ͬͨ͜ͱͰʣ͢΂ͯΘ͔͍ͬͯΔঢ়ଶͳͷͰɺ(ii) ΋ূ໌͞Εͨɻ 5.4 Discussion 5.4.1 Why 2J instruments? Theorem 5.1 ͸֎ੜม਺Ͱ͋Δ x(1) t ͱআ֎͞Ε͍ͯΔૢ࡞ม਺ zt ʹ͍ͭͯͷૢ࡞ม਺ͱͯ͠ͷ৚݅Λओு ͍ͯ͠Δɻ x(1) jt = σ−1 j (st , pt ) − ξjt (5.15) *9 Xt ͸͜͜Ͱ͸ɺXt = (xt, pt, ξt) ͱͯ͠ఆٛ͞ΕΔɻ͜ͷͱ͖ɺࢢ৔͸ (Jt, Xt) ͷ૊Έ߹ΘͤͰఆٛ͞ΕΔɻ 16
17. Ձ֨ʹ͍ͭͯͷ J ݸͷૢ࡞ม਺ͷΈͰ͸ࣝผʹे෼Ͱ͸ͳ͍ɻՁ֨΁ͷ֎ੜతͳมԽΛى͜͢ࡍʹ͸ଞͷ͢ ΂ͯͷཁҼΛݻఆ͠ͳ͚Ε͹ͳΒͳ͍͕ɺ؍࡯͞Εͳ͍ ξjt ΛؚΉ δt ͸ݻఆͰ͖ͳ͍ɻ͜͜Ͱɺ δjt = σ−1 j (st ; pt ) j = 1, . . . , J. ͱΠϯόʔδϣϯͰ͖Δ͜ͱʹ஫ҙ͢Ε͹ɺ͜͜ͷࢢ৔γΣΞ st ʹؔ͢Δૢ࡞ม਺Λ௥ՃͰ J ݸ༻ҙ͢Δ͜ ͱͰधཁΛࣝผͰ͖ΔɻΑͬͯɺඞཁͳૢ࡞ม਺ͷ਺͸ 2J ݸɻ 5.4.2 Why BLP instruments? J ࣍ݩ͋Δ x(1) t ͷཁૉ͸ BLP ૢ࡞ม਺ͷҰྫɻ͜Ε͸ڙڅۉߧΑΓՁ֨ʹӨڹΛ༩͑Δ͕ɺՁ֨Λॴ༩ͱ ͨ͠ͱ͖ʹधཁʹ΋௚઀ӨڹΛ༩͑ΔɻΑͬͯɺx(1) t ͷཁૉ͸ࢢ৔γΣΞ΁ͷૢ࡞ม਺ͱͯ͠ͷ໾ׂΛՌͨ ͢ɻΑͬͯɺࢢ৔Ϩϕϧͷσʔλ͔͠खݩʹͳ͍৔߹ʹ͸͜ΕΒ͸ॏཁͳૢ࡞ม਺ͱͳΔɻ͜ΕΛݟΔͨΊʹ ྫΛߟ͑ΔɻͦΕͧΕͷࢢ৔ʹ͓͍ͯՁ͕֨֎ੜతʹ༩͑ΒΕ͍ͯΔʢξt ͱฏۉಠཱʣͱ͢Δɻ͜ͷͱ͖Ͱ ΋ɺξt ͸ݻఆ͞Εͳ͍ͷͰधཁࣝผ͸·ͩͰ͖ͳ͍ɻInverse धཁ͸ ξjt = σ−1 j (st ; pt ) − x(1) jt ͱͳΔɻ΋͘͠͸ɺ sjt = hj (s−jt , pt , x(1) jt , ξjt ). hj ͷࣝผʹ͸ s−jt ʹର͢Δ J − 1 ݸͷૢ࡞ม਺͕ඞཁɻࢢ৔γΣΞ΁ͷૢ࡞ม਺ͱͯ͠ɺx(1) jt ʹؚ·Ε͍ͯ ΔΑ͏ͳίετγϑλʔ͸࢖͑ͳ͍ɻͳͥͳΒɺίετγϑλʔ͸Ձ֨Λ௨ͯ͡ͷΈࢢ৔γΣΞʹӨڹΛ༩͑ ͍ͯΔͨΊͰɺ͢ͰʹՁ֨Λ௨ͯ͡௚઀༩͍͑ͯΔӨڹҎ্ʹ͸มಈΛ༩͑ͳ͍ɻx(2) t ΋৚͚݅ͮͯʢݻఆ͠ ͯʣ͋ΔͷͰ֎ੜతͳมಈΛ༩͑ͳ͍ɻΑͬͯɺx(1) −jt ͕ૢ࡞ม਺ͷީิʹͳ͍ͬͯΔɻ Ҏ্ΑΓɺΑΓҰൠతͳϞσϧʹ͓͍ͯ J ݸͷՁ֨ͱࢢ৔γΣΞͦΕͧΕʹର͢Δ 2J ݸͷૢ࡞ม਺͕धཁͷ ࣝผͷͨΊʹඞཁɻBLP ૢ࡞ม਺͸ࢢ৔Ϩϕϧσʔλʹ͓͚Δૢ࡞ม਺ͷީิɻ 5.4.3 Why the index? BLP ૢ࡞ม਺͕ࢢ৔γΣΞʹ֎ੜతͳมಈΛ༩͑Δͷ͸ൺֱత໌Β͔͕ͩɺআ֎੍໿Λຬ͍ͨͯ͠Δͷ͔ Ͳ͏͔Λߟ͍͑ͨɻͲͷΑ͏ͳέʔεͰద੾ʹআ֎͞Ε͍ͯΔͷ͔ʁ x(2) t Ͱ৚͚݅ͮͯɺٯधཁؔ਺͸ ξj t = σ−1 j (st , pt ; x(2) t ) − x(1) jt ͱ͔͚Δɻx(1) t ͸আ֎͞Ε͍ͯΔͷͰɺs−jt ͷࣝผʹ x(1) −jt Λૢ࡞ม਺ͱͯ͠࢖͏͜ͱ͕Ͱ͖Δɻ͜Ε͸ɺࡒ ͷಛੑ x(1) jt ͔Β͸ޮ༻ΛಘΔ͕ɺx(1 −jt ͔Β͸ޮ༻Λಘͳ͍ͱԾఆ͞Ε͍ͯΔͨΊͰɺξjt ʹೖͬͯ͜ͳ͍ͷͰ আ֎͞Ε͍ͯΔɻ 5.4.4 Further restrictions and tradeoffs ௥ՃతͳԾఆ͕ૢ࡞ม਺ͷ৚݅ΛͲ͏؇ΊΔ͔ʹ͍ͭͯΈΔɻδjt ΛҎԼͷΑ͏ʹԾఆ͢Δɻ δjt = ξjt − αpjt ج४Խͱͯ͠ α = 1 ͱ͢Δɻxt Ͱ৚͚݅ͮΔͱ ξjt = σ−1 j (st ) + pjt 17
18. ΋͘͠͸ pjt = −σ−1 j (st ) + ξjt . ͱͳΔɻ͜ͷέʔεͰ͸ J ݸͷࢢ৔γΣΞ st ΁ͷૢ࡞ม਺Ͱࣝผʹे෼ͰɺՁ֨΁ͷૢ࡞ม਺͸͍Βͳ͍ɻ ίετγϑλʔ΍ Hausman ૢ࡞ม਺ɺWaldfogel-Fan (WF) ૢ࡞ม਺ͳͲ͕ީิɻ 2 ͭ໨ͷྫʹ͍ͭͯߟ͑Δɻx(1) t ͱՁ͕֨ index Λ௨ͯ͠ͷΈޮ༻ʹӨڹΛ༩͑Δͱ͠ɺδjt ͸ δjt = ξjt + x(1) jt − αpjt ͱ͢Δɻ͜͜Ͱ α = 1 ͱ͢ΔɻΑͬͯɺ xijt = σ−1 j (st ) − x(1) jt + pjt ΋͘͠͸ pjt = −σ−1 j (st ) + x(1) jt + xijt ͱͳΔɻ͜͜Ͱ΋ࢢ৔γΣΞ st ʹର͢Δૢ࡞ม਺ʢBLP ૢ࡞ม਺ͳͲʣͷΈ͕ඞཁʹͳΔɻ Αͬͯɺؔ਺ܗͷԾఆͱૢ࡞ม਺ͷཁ݅ʹ͸τϨʔυΦϑ͕͋ΔɻखݩͷσʔληοτͰར༻Մೳͳ֎ੜతม ಈͱɺ͢΂ͯͷϊϯύϥϝτϦοΫͳधཁγεςϜΛࣝผ͢ΔͨΊʹඞཁͱͳΔมಈͱͷؒͷΪϟοϓΛຒΊ Δ͜ͱ͕ඞཁͱͳΔɻ 18
19. 6 Micro data, panels, and ranked choices 6.1 Micro data धཁਪఆͷจ຺ʹ͓͍ͯɺ ʠmicro dataʡͱ͸ɺফඅऀ i ͷଐੑ dit ͱফඅऀ i ͷબ୒ qit ΛϚονͰ͖ ΔΑ͏ͳσʔλͷ͜ͱΛࢦ͢ɻࢢ৔Ϩϕϧσʔλ͔Β͸ɺͦΕͧΕͷࢢ৔ʹ͍ͭͯ dit ͱ qit ͷपล෼෍ Fd (dit ), Fq (qit ) ͕Θ͔Δ͕ɺmicro data ͕͋Ε͹͜Εͷಉ࣌෼෍ Fdq (dit , qit ) ͕Θ͔Δɻ͜ͷ͜ͱ͸ɺdit ͕ফඅऀͷબ୒ΛͲ͏ม͑Δ͔ʹ͍ͭͯͷ৘ใΛఏڙͯ͘͠ΕΔɻ΋͠෼ੳऀ͕ࢢ৔Ϩϕϧͷधཁʹ͔͠ ڵຯ͕ͳ͍৔߹Ͱ΋ɺͦΕͧΕͷࢢ৔ͷதͰফඅऀؒͷมಈΛར༻͢Δ͜ͱ͕Ͱ͖ͯɺ͜Ε͸ product × market-level ͷधཁγϣοΫΛݻఆͰ͖Δ఺Ͱخ͍͠ɻ ·ͨɺmicro data Λ࢖͏͜ͱͷخ͠͞ͷҰͭʹɺwithin-market variation ͕͋Δ͜ͱͰૢ࡞ม਺ʹ՝͢৚ ݅Λ؇࿨Ͱ͖Δ͜ͱ͕͋Δɻmicro data ͸धཁਪఆʹ͓͚Δύϥϝʔλͷࣝผʹඞཁͳૢ࡞ม਺ͷ਺ΛݮΒ ͢͜ͱ͕Ͱ͖ɺ৽͍͠ར༻Մೳͳૢ࡞ม਺Λ micro data ͔Β࡞Δ͜ͱ͕Ͱ͖Δɻmixed logit specification Λྫʹߟ͑ͯΈΔɻ uijt = xjt βit − α0 pjt + ξjt + ϵijt ͜͜Ͱɺ β(k) it = β(k) 0 + L l=1 β(l,k) d dilt + β(k) v v(k) it . ผͷදݱͰॻ͖׵͑Δͱɺ uijt = δjt + µijt (vit ; βd , βv ) + ϵijt , (6.1) ͜͜Ͱɺ µijt (vit ; βd , βv ) = K k=1 x(k) jt β(k) v + L l=1 β(l,k) d dilt (6.2) δjt = xjt β0 − α0 pjt + ξjt . (6.3) McFadden, Talvitie, Cosslett, Hasan, Johnson, Reid, and Train (1977) Ͱ͸ɺδjt Λʠalternative-specificʡ constant ͱݺΜͰ͍Δɻྫ͑͹ɺधཁͷՁ֨஄ྗੑΛݟ͍ͨ࣌ʹՁ֨ pjt Ҏ֎ͷཁૉ͸ݻఆ͞Εͳ͚Ε͹ͳΒ ͳ͍͕ɺξjt ͱ૬ؔͯ͠͠·͏৔߹ʹ͸͏·͍͔͘ͳ͍ɻ͜ͷ constant ʹؚ·ΕΔજࡏతͳधཁγϣοΫ ξjt ͕ simultaneity/endogeneity ͷ໰୊ΛҾ͖ى͜͢ɻͨͩ͠ɺmicro data ͕खݩʹ͋Δ৔߹ʹ͸ɺಉ࣌ੑͷ໰ ୊ʹର͢Δ෦෼తͳղܾࡦΛఏڙͰ͖Δɻ ಉ࣌ੑɾ಺ੜੑͷ໰୊͸ɺmarket×product ϨϕϧͷधཁγϣοΫ͕͋Δ͜ͱʹىҼ͢Δɻ͜ͷ಺ੜੑͷ໰୊ ʹରॲ͢ΔͨΊʹ͸ɺࢢ৔ؒͷมಈͱՁ֨ʹؔ͢Δૢ࡞ม਺͸Ҏલͱͯ͠ඞཁɻ ࢢ৔ t ʹ͓͚Δফඅऀ i ͷࡒ j ʹؔ͢Δબ୒֬཰͸ɺ sijt = exp(δjt + µijt (vit ; βd , βv )) Jt k=0 exp(δkt + µikt (vit ; βd , βv )) dFv (vit ) for j = 0, 1, . . . , Jt (6.4) ͜͜Ͱɺফඅऀ i ͕ࡒ j Λબ୒ͨ͜͠ͱΛ j(i) ͱͯ͠ද͢ɻࣜ (6.4) ʹ͓͍ͯ j Λ j(i) Ͱஔ͖׵͑Δͱɺফඅ ऀͷબ୒ʹ͓͚Δ໬౓ؔ਺ʹͳΔʢύϥϝʔλ (δ, αy , βd , βv ) ͷؔ਺ʣ ɻ͜͜Ͱɺδ = {δt }T t=1 ͱ͍ͯ͠Δɻͭ 19
20. ·Γɺ L(δ, βd , βv ) = i,t exp{δj(i)t + µij(i)t (vit ; βd , βv )} Jt k=0 exp{δkt + µikt (vit ; βd , βv )} dFv (vit ). (6.5) ͜ͷ໬౓Λ࠷େʹ͢Δύϥϝʔλ (δ, αy , βd , βv ) ͕ਪఆ஋ɻ͜ͷύϥϝʔλ͚ͩͰ͸ͳ͘ɺࣜ (6.3) ͷ α0 , β0 ΋ٻΊΔ͜ͱ͕ඞཁͰ͋Δɻ͜ͷͨΊͷ࠷΋γϯϓϧͳํ๏͸ɺδjt on xjt , pjt ͷ second-step linear IV regression Λ͢Δ͜ͱɻྫ͑͹ɺBayer, Ferreira, and McMillan (2007) ͱ Bayer and Timmins (2007) Ͱ ͸ɺ͜ͷํ๏Λ࢖ͬͯ residential location ΁ͷधཁΛਪఆͨ͠ɻ ·ͨɺ͜͜Ͱ͸Ձ֨ pjt ʹର͢ΔҰͭͷʢআ֎͞Εͨʣૢ࡞ม਺ͰधཁΛࣝผͰ͖Δɻૢ࡞ม਺ͷྫͱͯ͠ ͸ɺίετγϑλʔɺͦͷ୅ཧม਺ɺBLP ૢ࡞ม਺΍ WF ૢ࡞ม਺ʢ͍ۙ͠ࢢ৔ͷಛੑΛ࢖ͬͨ΋ͷʣ͕͋ ΔɻಛʹɺWF ૢ࡞ม਺ʹ͍ͭͯ͸ࢢ৔Ϩϕϧͷσʔλͷ৔߹ͱ͸ҟͳΓɺmicro data ͷ΋ͱͰ͸ࢢ৔ t ʹ ͓͚Δࡒ j ͷՁ֨ pjt ʹ͍ͭͯɺࢢ৔ t ͷಛੑΛ WF ૢ࡞ม਺ͱͯ͠࢖༻͢Δ͜ͱ͕Ͱ͖Δɻྫ͑͹ɺxjt ͷ ཁૉͰ͸ͳ͘ɺ͔ͭ ξjt ͱ૬ؔ͠ͳ͍Α͏ͳɺdit ͷࢢ৔Ϩϕϧฏۉ஋ͳͲɻ ैͬͯɺmicro data ͷ΋ͱͰͷύϥϝτϦοΫϞσϧͰ͸ɺ͢΂ͯͷύϥϝʔλΛࣝผ͢ΔͨΊʹɺՁ֨ ΁ͷҰͭͷૢ࡞ม਺ͷΈͰे෼Ͱ͋Δ͜ͱ͕Θ͔Γɺ͔ͭ৽͍͠ WF ૢ࡞ม਺ͳͲɺ௥Ճతͳૢ࡞ม਺Λ࡞ ੒͢Δ͜ͱ΋Ͱ͖Δɻ ͜ΕΒͷ two-step Ξϓϩʔν͸༗ӹ͕ͩɺࢢ৔಺มಈʢwithin-market variationʣͱࢢ৔ؒมಈʢcross- market variationʣͷ྆ํΛར༻ͯ͠ɺ͢΂ͯͷύϥϝʔλΛҰ౓ʹਪఆ͢Δํ͕๬·͍͠৔߹͕ଟ͍ɻ࠷௿ Ͱ΋ɺҰ౓ʹ͢΂ͯͷύϥϝʔλΛਪఆ͢Δ͜ͱ͸ޮ཰ੑͷ؍఺͔ΒΑ͘ͳΔɻ One-step ਪఆ͸ɺ༨෼ͳύϥϝʔλ δ Λਪఆ͢Δ͜ͱΛආ͚Δ͜ͱ͕Ͱ͖Δ (?)ɻ micro data ʹ͓͚Δ׬શͳύϥϝτϦοΫϞσϧͰ͸ɺߟ͍͑ͯΔࢢ৔͕Ұ͔ͭ͠ͳ͍৔߹Ͱ΋ɺύ ϥϝʔλ (βd , βv ) Λ within-market variation ͷΈΛ࢖༻ͯ͠ਪఆ͢Δ͜ͱ͕Ͱ͖Δʢ͔΋ʣ ɻ͜ͷՄೳੑ ͸ɺparametric structure ʹґଘ͍ͯ͠Δ͜ͱ͕ϊϯύϥϝτϦοΫϞσϧΛߟ͑Δ͜ͱͰΘ͔ΔʢSection 7ʣ ɻϊϯύϥϝτϦοΫͰߟ͑Δͱɺdit ͷ૬ରతͳޮՌ (βd ) ΛಘΔʹ͸ within-market and cross-market variation ͕ඞཁɻ·ͨɺϊϯύϥϝτϦοΫͳࣝผʹඞཁͳσʔλͷมಈΛ͢΂ͯར༻͢Δ͜ͱͰɺύϥϝτ ϦοΫϞσϧΛΑΓਖ਼֬ʹਪఆͰ͖Δ͜ͱ͕ଟ͍ɻྫ͑͹ɺBerry, Levinsohn, and Pakes (2004) ͸ɺmicro data ʹ͓͍ͯҰͭͷࢢ৔͔ΒϥϯμϜ܎਺཭ࢄબ୒ϞσϧΛਪఆ݁ՌʹϊΠζ͕ଟ͔͕ͬͨɺchoice-set variation ΛՃ͑ͨʢ ʠsecond-choiceʡ data ͷܗΛͨ͠΋ͷɻ͋Δछͷ cross-market dataʣ͜ͱͰɺ݁Ռ͕ ΑΓਖ਼֬ʹͳͬͨ͜ͱΛใࠂ͍ͯ͠Δɻ͜Ε͸ɺcross-market variation ͕ύϥϝʔλ (βd , βv ) Λਪఆ͢Δ্ ͰॏཁͰ͋Δ͜ͱΛࣔࠦ͢Δɻ શͯͷύϥϝʔλΛҰ౓ʹਪఆ͢ΔͨΊʹɺࣜ (6.5) ͷύϥϝʔλ (δ, βd , βv ) ʹؔ͢Δ likelihood score Λ ൓ө͢ΔΑ͏ͳϞʔϝϯτ৚݅Λ࢖͏ɻ E[(δjt − xjt β0 − α0 pjt ) ξjt zjt ] = 0. ͜͜Ͱɺzjt ͸֎ੜม਺ xjt ͱՁ֨΁ͷআ֎͞Εͨૢ࡞ม਺͔ΒͳΔɻ͜͜Ͱ͸ɺࢢ৔Ϩϕϧσʔλͷ࣌ͱಉ ͡Α͏ͳܗΛ͍ͯ͠Δ͕ɺඞཁͳআ֎͞Εͨૢ࡞ม਺͕Ձ֨ pjt ΁ͷ 1 ͚ͭͩͱ͍͏఺Ͱҟͳ͍ͬͯΔɻ 20
21. γϛϡϨʔγϣϯ͞Εͨ໬౓ؔ਺Λ࢖͏ͷ͸͋·Γ޷·͘͠ͳ͍ɻਖ਼֬ʹର਺໬౓ؔ਺Λܭࢉ͢Δͷ͸ɺಛ ʹফඅऀͷ਺͕ଟ͍࣌ɺࢢ৔ͷ਺͕ଟ͍ͱ͖΍ɺ͍͔ͭ͘ͷਅͷબ୒֬཰͕ 0 ʹ͍ۙͱ͖ʹܭࢉ͕େมɻҰ ൠతʹ͸ɺ໬౓ؔ਺Λ࢖༻͢ΔͷΛආ͚ͯɺϞʔϝϯτ৚݅Λ࢖༻ͯ͠ਪఆ͢Δɻྫ͑͹ɺࢢ৔Ϩϕϧσʔ λͷࡍʹར༻͞ΕΔΑ͏ͳࢢ৔γΣΞΛ൓ө͢ΔϞʔϝϯτ৚݅ʹՃ͑ͯɺ ʠmicro momentsʡΛ࢖༻͢Δɻ ʠmicro momentsʡ͸ɺফඅऀ i ͷଐੑͱফඅऀ i ʹΑΔࡒͷબ୒ j(i) ͷಉ࣌෼෍ͷॏཁͳಛ௃Λଊ͑Δ΋ͷ (Berry et al. (2004) ʹৄ͍͠આ໌)ɻ యܕతͳ micro moments ͱͯ͠͸ɺબ͹ΕͨࡒͷಛੑΛॴ༩ͱͨ͠ɺফඅऀͷଐੑͷ৚݅෇͖ظ଴஋΍ڞ ෼ࢄͳͲɻࣗಈंͷྫͰ͸ɺࣗಈंͷΫϥεʢminivan, compact, luxury, pickupʣΛ৚݅ͱͨ͠Ո଒ͷେ͖ ͞ɺ೥ྸɺॴಘͷظ଴஋ͳͲ͕࢖༻͞ΕΔɻ ͜͏ͨ͠ one-step method-of-moments Ξϓϩʔν͸ɺmicro data Λ࣋ͭͱ͖ʹ௚ߦ৚݅΁ͷ੍໿Λ؇࿨ ͢Δ͜ͱ͕Ͱ͖Δ͜ͱΛ͍ࣔͯ͠Δɻͭ·Γɺࢢ৔Ϩϕϧσʔλͷ৔߹ͷඇઢܗύϥϝʔλΛࣝผ͢ΔϞʔϝ ϯτ৚݅͸ɺ(δ, βd , βv ) Λࣝผ͢Δͷʹे෼ͳ micro moments Ͱஔ͖׵͑Δ͜ͱ͕Ͱ͖Δɻ 6.2 Consumer panels धཁਪఆͷจ຺ʹ͓͚Δʠpanel dataʡͱ͸ɺͦΕͧΕͷݸਓʹ͓͍ͯෳ਺ͷબ୒ػձʢchoice occasionsʣ ͷ݁Ռ͕ه࿥͞Ε͍ͯΔͱ͖Λݴ͏ɻ͜͜Ͱ͸͜ΕΛʠConsumer panelʡͱݺͿɻConsumer panel ͷྫͱ ͯ͠͸ҎԼͷ௨Γɻ • ಉ͡ফඅऀ͕ҟͳΔ৯ྉ඼ళͰങ͍෺Λͨ͠ͱ͖ͷσʔλ • ಉ͡Ո଒͕ҟͳΔ೥ʹࣗಈंΛߪೖͨ͠ͱ͖ͷσʔλ • ಉ͡ैۀһ͕ҟͳΔ೥ͷՃೖظؒʹ݈߁อݥΛબ୒ͨ͠ͱ͖ͷσʔλ ಉ͡ফඅऀΛҟͳΔબ୒৔໘Ͱ؍࡯͢Δ͜ͱͰɺ୅ସύλʔϯͷܾఆʹ͓͚Δݸਓಛੑͷ໾ׂʹ͍ͭͯɺ͞Β ʹଟ͘ͷ৘ใΛఏڙͰ͖Δɻ ɺྫ͑͹ɺ͋Δফඅऀ͕ɺࡒͷબ୒ͷ swiching Λ༠ൃ͢Δͷʹे෼ͳ֎ੜతՁ֨ ্ঢʹ൓Ԡͯ͠ɺͲͷ੡඼ʹ୅ସ͢Δ͔Λ௚઀؍࡯͢Δ͜ͱ͕Ͱ͖Δɻ mixed logit model Λߟ͑Δɻ • ࢢ৔Λ஍ཧతͳ΋ͷ m ͱ࣌ؒ t ʹ෼ׂͯ͠ఆٛ • t ∈ {0, 1} • ফඅऀͷબ޷͕࣌ؒΛ௨ͯ͡ҰఆͱԾఆ • ࡒಛ༗ͷγϣοΫ ϵijmt ͸֤ظؒʹ͓͍ͯ৽͘͠ i.i.d. Ͱυϩʔ͞ΕΔͱ͢Δ ࢢ৔ m ʹ͓͍ͯɺফඅऀ i ͕ظؒ 0 ʹࡒ jɺظؒ 1 ʹࡒ k Λબ୒͢Δ֬཰͸ sijkm = eδjm0+µijm0(vim;βd,βv) 1 + l eδlm0+µilm0(vim;βd,βv) eδkm1+µikm1(vim;βd,βv) 1 + l eδlm1+µilm1(vim;βd,βv) dFv (vim ). (6.6) ͱ͔͚Δɻ͜͜Ͱɺj ͱ k ΛͦΕͧΕ j(i), k(i) Ͱஔ͖׵͑ͯɺશͯͷফඅऀͱࢢ৔ m ʹ͓͍ͯੵ࿨ΛऔΕ͹ ໬౓ؔ਺ (ύϥϝʔλ (δ, βd , βv ) ͷؔ਺) ʹͳΔɻ ର਺໬౓͔ΒύϥϝʔλΛܭࢉ͢Δ͜ͱ͸ΑΓγϏΞʹͳΔɻ2 ظؒϞσϧͰ͑͞ɺ(J + 1)2 ͷબ୒֬཰ͷ 21
22. ૊Έ߹Θ͕ͤ͋Γɺ·ͨ 1 ظ໨ʹࡒ jɺ2 ظ໨ʹࡒ k ΛબͿਅͷ֬཰͕ۃ୺ʹগͳ͍૊Έ߹Θͤ΋ग़ͯ͘Δͨ Ίɺܭࢉ͕ۃΊͯ೉͍͠ɻ͜͏ͨ͠໰୊Λආ͚ΔͨΊʹ΋ aggregate moments ͱ micro moments Λ࢖ͬͨ ਪఆ͕޷·ΕΔɻ·ͨɺConsumer panel ͕ఏڙ͢Δ௥Ճతͳ৘ใɺͭ·Γબ୒৔໘ؒͷಉҰফඅऀͷબ୒ؒ ͷؔ܎ΛͱΒ͑ͨϞʔϝϯτΛ௥Ճ͢Δ͜ͱ͕Ͱ͖Δɻ ྫ͑͹ɺ͋Δ঎඼ΧςΰϦʔʹ͓͚Δ৯ྉ඼ͷߪೖʹؔ͢Δσʔλͷ৔߹ɺങ͍෺ؒͰબ୒͞Εͨࡒͷಛੑؒ ͷڞ෼ࢄΛϞʔϝϯτͱؚͯ͠ΊΔ͜ͱ͕Ͱ͖Δʢ͔΋ʣ ɻ 6.3 Ranked choice data ֤ফඅऀͷ঎඼ͷॱҐ෇͚ʹؔ͢ΔσʔλΛఏڙ͢Δ΋ͷ΋͋Δɻྫͱͯ͠͸ɺίϯδϣΠϯτ෼ੳʢϚʔ έςΟϯάͰҰൠతʣ΍ɺϥϯΫ෇͚͞Εֶͨߍબ୒ͳͲ͕͋Δɻݪཧతʹ͸ɺফඅऀͷ͢΂ͯͷ঎඼ʹର ͢ΔϥϯΫ෇͚ΛݟΔ͜ͱ͕Ͱ͖Δ͕ɺ௨ৗ͸্Ґͷબ୒ࢶͷ shortlist ͚ͩΛ؍࡯͢Δ͜ͱʹͳΔɻRanked choice data ͸ consumer panel ͷཧ૝తͳ΋ͷͱͯ͠ߟ͑ΒΕΔɻফඅऀʹରͯ͠ɺ1 ൪ʹ޷Ήࡒɺ2 ൪໨ʹ ޷Ήࡒɺ3 ൪໨ʹ޷Ήࡒ,... ͱฉ͚ΔͷͰɺconsumer panel ͱಉ͡Α͏ͳʢෳ਺ͷબ୒ू߹ʣ݁ՌΛ؍࡯Ͱ͖ ΔɻRanked choice data ͕ consumer panel ΑΓ΋ྑ͍఺͸গͳ͘ͱ΋ 2 ͭ͋Δɻ • ࣌ؒతͳ۠੾Γ͕ͳ͍ͨΊɺϞσϧͷͲͷ֬཰తཁૉΛʠchoice occasionsʡʹΘͨͬͯݻఆతͱΈͳ ͢΂͖͔ʹ͍ͭͯͷٙ໰Λආ͚Δ͜ͱ͕Ͱ͖Δʢconsumer panel Ͱ͸બ޷͕࣌ؒΛ௨ͯ͠ݻఆͱԾఆ ͨ͠ʣ • Ranked choice data ͕ఏڙ͢Δʠvariation in the choice setʡͷ type ͸ɺͲͷࡒ͕࠷΋͍ۙ୅ସࡒͰ ͋Δ͔ΛධՁ͢Δͷʹཧ૝త धཁਪఆͷओཁͳ՝୊ͱͯ͠஫໨͞ΕΔ୅ସύλʔϯ͸ɺୈҰબ୒ͱୈೋબ୒ͷؔ܎ʹີ઀ʹؔ܎͍ͯ͠Δɻ ैͬͯɺୈҰબ୒ࢶͱୈೋબ୒ࢶΛ௚઀؍࡯Ͱ͖Δ͜ͱ͸ඇৗʹڧྗɻ ਪఆͷํ๏͸ consumer panel ͷ࣌ͱಉ༷ɻ໬౓ΞϓϩʔνΑΓ΋Ϟʔϝϯτ৚݅Λ࢖ͬͨਪఆ͕ྑ͍ɻ ranked choice data Ͱ࢖͑ΔϞʔϝϯτ৚݅ͱͯ͠ɺྫ͑͹ɺࢢ৔γΣΞʢୈҰબ୒ͷฏۉબ୒֬཰ʣͱɺফ අऀಛੑͷߏ੒ཁૉɺୈҰબ୒ࡒͷಛੑɺୈҰબ୒ࡒͱୈೋબ୒ࡒͷಛੑͷؒͷڞ෼ࢄΛಛ௃෇͚ΔϞʔϝϯ τΛ૊Έ߹ΘͤΔ͜ͱ͕Ͱ͖Δɻ 6.4 Hybrids Ұൠతʹ͸ɺෳ਺ࢢ৔ͷσʔλͱݶఆతͳ micro data ͔ ranked choice data ͕࢖͑Δ৔߹͕ଟ͍ɻ෼ੳྫ ͱͯ͠͸ҎԼͷ௨Γɻ • Petrin (2002): ৽͍͠ࡒͷಋೖʹΑΔ welfare ͷ૿ՃΛ෼ੳɻࣗಈं΁ͷूܭ͞ΕͨϚʔέοτγΣΞ ͱɺগͳ͍αϯϓϧαΠζͷ micro data (from Consumer Expenditure Survey) Λ૊Έ߹ΘͤͯϞʔ ϝϯτ৚݅Λߏ੒ɻ • Goeree (2008): ޿ࠂͱ PC ΁ͷधཁʹ͍ͭͯ෼ੳɻݸʑͷ PC model ʹ͍ͭͯͷϚʔέοτγΣΞͱɺ ϒϥϯυ͝ͱͷফඅऀʹΑΔߪങʹؔ͢Δಛੑʢfrom ݶΒΕͨ micro dataʣΛ૊Έ߹ΘͤͯϞʔϝϯ τ৚݅Λߏ੒ͨ͠ɻ ͭ·Γɺmicro data ͕͋Δ࣌ͷϞʔϝϯτ৚݅Λ·ͱΊΔͱҎԼͷ௨Γɻ 22
23. 1. BLP-style moments: ूܭ͞ΕͨϚʔέοτγΣΞͱɺૢ࡞ม਺ 2. micro moments: ݸਓͷಛੑͱબ୒͞Εͨࡒͷಛੑͱͷؒͷɺ༧ଌ͞Εͨؔ܎ͱ࣮ࡍͷؔ܎ʢڞ෼ࢄ· ͨ͸৚݅෇͖ฏۉʣͷࠩͱͯ͠ఆٛ͞ΕͨϞʔϝϯτ 23
24. Appendix.A ଟ߲ϩδοτϞσϧɿબ୒֬཰ͷಋग़ ҎԼɺTrain (2009) ʹґڌɻফඅऀ i ͕ࡒ j Λফඅ͢Δ͜ͱ͔ΒಘΔޮ༻Λ uij = Vij + ϵij ͱ͢Δɻͨͩ͠ɺ؆୯ԽͷͨΊʹࢢ৔ t ͷΠϯσοΫε͸མͱ͍ͯ͠ΔɻVij ͕෼ੳऀʹͱͬͯ؍࡯Ͱ͖Δཁ ҼͰɺϵij ͕؍࡯Ͱ͖ͳ͍ཁҼɻ ϵij ͕ i.i.d. ͰୈҰछۃ஋෼෍ʹै͏ͱԾఆ͢Δͱɺ෼෍ؔ਺ͱ֬཰ີ౓ؔ਺͸ F(ϵij ) = e−e−ϵij f(ϵij ) = e−ϵij e−e−ϵij ͱͳΔɻফඅऀ i ͕ࡒ j Λফඅ͢Δͱ͖ɺࡒ j ͔ΒಘΒΕΔޮ༻͕ࡒͷબ୒ू߹ͷதͰ࠷େʹͳ͍ͬͯΔ͸ͣ ʢݦࣔબ޷ͷԾఆʣ ɻΑͬͯɺফඅऀ i ͕ࡒ j Λબ୒͢Δ֬཰͸ҎԼͷ௨Γɻ Pij = Pr(Vij + ϵij > Vik + ϵik ∀k ̸= j) = Pr(ϵij < ϵij + Vij − Vik ∀k ̸= j) ϵij Λॴ༩ͱͯ͠ߟ͑Δͱɺ͜ͷ֬཰͸ ϵij ͷ෼෍ؔ਺ʹͳ͍ͬͯͯɺPij ͸ exp(− exp(−(ϵij + Vij − Vik ))) Λ ∀k ̸= j ʹֻ͍͚ͭͯࢉΛऔͬͨ΋ͷͱͳΔɻϵ ͸ i.i.d. ͳͷͰɺϵij ͕ॴ༩ͷͱ͖ɺҎԼͷ͕ࣜ੒ཱ͢Δɻ Pij | ϵij = J k̸=j exp(− exp(−(ϵij + Vij − Vik ))) ΋ͪΖΜ ϵij ͸ॴ༩Ͱ͸ͳ͍ͷͰɺϵij ͷऔΓ͏Δ஋ʹ͍ͭͯɺີ౓ͰॏΈ෇͚ͨ͠΋ͷΛ଍͠߹ΘͤΔʢੵ෼ ͢Δʣ͜ͱͰɺࡒ j ͷબ୒֬཰Λܭࢉ͢Δ͜ͱ͕Ͱ͖Δɻ *10 Pij = ∞ −∞   J k̸=j exp(− exp(−(ϵij + Vij − Vik )))   f(ϵij )dϵij = ∞ −∞   J k̸=j exp(− exp(−(ϵij + Vij − Vik )))   e−ϵij e−e−ϵij dϵij ҎԼͰ͸ɺදهͷ؆୯ԽͷͨΊʹ s = ϵij ͱ͢Δɻ Pij = ∞ −∞   J k̸=j exp(− exp(−(s + Vij − Vik )))   e−se−e−s ds = ∞ −∞ J k exp(− exp(−(s + Vij − Vik ))) e−sds = ∞ −∞ exp − J k e−(s+Vij −Vik) e−sds = ∞ −∞ exp −e−s J k e−(Vij −Vik) e−sds *10 ͳͥͳΒɺPij | ϵij = P(Pij and ϵij ) f(ϵij ) ͱͳΔͷͰɺ ∫ Pij | ϵijf(ϵij)dϵij = ∫ P(Pij and ϵij)dϵij = Pij ͱͳΔͨΊɻ 24
25. ͜͜Ͱɺt = exp(−s) ͱ͓͘ͱɺ dt ds = −e−s ⇔ dt = −e−sds ͱͳΔɻ ·ͨɺs ͕ −∞ → ∞ ͷͱ͖ t ͸ ∞ → 0 ͱͳΔɻҎ্ΑΓɺ Pij = 0 ∞ exp −t J k e−(Vij −Vik) (−dt) = ∞ 0 exp −t J k e−(Vij −Vik) dt = exp(−t J k e−(Vij −Vik)) − J k e−(Vij −Vik) ∞ 0 = 1 J k e−(Vij −Vik) = eVij J k eVik . ࡒू߹ʹΞ΢ταΠυάοζΛՃ͑ɺશ෦Ͱ J + 1 ݸͷࡒʹ͍ͭͯߟ͑Δɻ͜͜ͰɺJ + 1 ݸͷબ୒ࢶͷ͏ͪɺ J ݸ͕ܾ·Ε͹࢒Γͷ J + 1 ݸ໨͸ࣗવͱܾ·Δɻj = 0 Λج४ʢΞ΢ταΠυάοζʣͱ͠ɺ্ࣜͷ෼฼෼ ࢠΛ eVi0 ͰׂΔͱɺ Pi0 = 1 1 + J k=1 eVik−Vi0 Pij = eVij −Vi0 1 + J k=1 eVik−Vi0 j = 1, . . . , J ͱͳΔɻΞ΢ταΠυάοζ͔Βͷޮ༻͸ޡ߲͔ࠩΒͷΈಘΒΕΔͱԾఆ͍ͯͨ͜͠ͱΛࢥ͍ग़ͤ͹ɺ Vi0 = 0 ͳͷͰɺࡒ j ͷબ୒֬཰͸ҎԼͷ௨ΓʹͳΔɻ Pi0 = 1 1 + J k=1 eVik Pij = eVij 1 + J k=1 eVik j = 1, . . . , J ྫ͑͹ɺVij = xj β0 − αpj + ξj ͱ͢Ε͹ɺফඅऀ i ͕ࡒ j Λߪೖ͢Δ֬཰͸ҎԼͷ௨ΓʹͳΔɻ Pij = exp(xj β0 − αpj + ξj ) 1 + J k=1 exp(xk β0 − αpk + ξj ) ಋग़Ͱ͖ͨʂ 25
26. Appendix.B ଟ߲ϩδοτϞσϧɿIIA આ໌͸ɺ্෢߁྄, ༞ଠԕࢁ, ௚थए৿, ҆ދ౉ล (2021). ʰϓϥΠγϯάͷਅ਷͸୅ସੑʹ͋Γʱ. ࣮ূϏ δωεɾΤίϊϛΫε. ೔ຊධ࿦ࣾ. ʹґڌɻ ࡒ j ͷϚʔέοτγΣΞ sj ͸ɺࡒ j ͷબ୒֬཰ͱ౳͍͠ɻͦ͜ͰɺϚʔέοτγΣΞΛ sj = exp(δj ) J k=1 exp(δk ) ͱදݱ͢Δɻ͜ͷͱ͖ɺ೚ҙͷࡒ j ͱ l ͷϚʔέοτγΣΞͷൺ཰͸ҎԼͷ௨Γɻ sj sl = exp(δj ) exp(δl ) ͜Ε͸ɺࡒ j ͱ l ΛબͿൺ཰͸ɺଞͷୈ 3 ͷࡒ΍ࡒͷଐੑͱ͸ແؔ܎ʹͳΔ͜ͱΛ͓ࣔͯ͠Γɺ͜ΕΛແؔ܎ ͳબ୒ࢶ͔Βͷಠཱੑʢindependence of irrelevant alternatives; IIAʣͱݺͿɻ References Bayer, Patrick, Fernando Ferreira, and Robert McMillan. 2007. “A unified framework for mea- suring preferences for schools and neighborhoods.” Journal of political economy 115 (4): 588–638. Bayer, Patrick, and Christopher Timmins. 2007. “Estimating equilibrium models of sorting across locations.” The Economic Journal 117 (518): 353–374. Berry, Steven, James Levinsohn, and Ariel Pakes. 2004. “Differentiated products demand systems from a combination of micro and macro data: The new car market.” Journal of political Economy 112 (1): 68–105. Berry, Steven T, and Philip A Haile. 2021. “Foundations of demand estimation.” In Handbook of industrial organization, Volume 4. 1–62, Elsevier. Goeree, Michelle Sovinsky. 2008. “Limited information and advertising in the US personal computer industry.” Econometrica 76 (5): 1017–1074. McFadden, Daniel, Antti Talvitie, Stephen Cosslett, Ibrahim Hasan, Michael Johnson, Fred Reid, and Kenneth Train. 1977. “Demand model estimation and validation.” Urban Travel Demand Forecasting Project, Phase 1. Petrin, Amil. 2002. “Quantifying the benefits of new products: The case of the minivan.” Journal of political Economy 110 (4): 705–729. 26