Amanothi ezembalo

Kulesi sithasiselo, ngizofingqa imibono ethile mayelana nokwenza ucwaningo oluvela emininingwaneni engeyona yokuhlola kufomu elithile lembalo. Kunezimboni ezimbili eziyinhloko: uhlaka lwesigcafu se-causal, oluhambisana kakhulu neJearl Pearl kanye nosebenza nabo, kanye nohlaka olungenzeka lwemiphumela, oluhlobene kakhulu noDonald Rubin kanye nozakwethu. Ngizokwethula uhlaka lwezinzuzo ezikhona ngoba luhlangene kakhulu nemibono emanothi ematheksthi ekupheleni kwesahluko 3 no-4. Ngolunye ulwazi mayelana nohlaka lwesigcawu se-causal, ngincoma Pearl, Glymour, and Jewell (2016) (isingeniso ) Pearl (2009) (ethuthukile). Ukuze ukwelashwa ubude bebhuku le-causal inference ehlanganisa uhlaka lwamazinga okungenzeka kanye nohlaka lwe-causal graph, ngincoma Morgan and Winship (2014) .

Umgomo walesi sithasiselo kukusiza ukuthi ukhululeke ngokukwaziswa nesitayela sekhono lemiphumela engase ibe khona ukuze ukwazi ukuguqulwa kwezinye izinto zobuchwepheshe ezibhaliwe ngalesi sihloko. Okokuqala, ngizochaza ukuhlelwa kokuhlelwa kwemiphumela. Khona-ke, ngizoyisebenzisa ukuze ngiqhubeke ngibheka ukuhlolwa kwemvelo okunjengeyena ka- Angrist (1990) ngomphumela wemisebenzi yezempi ngemali engenayo. Lesi sihlomelo sisondela kakhulu ku- Imbens and Rubin (2015) .

Uhlaka lokubheka imiphumela

Umhlahlandlela weziphumo ongenzeka unamakhemikhali amathathu okuyinhloko: amayunithi , ukwelashwa , kanye nemiphumela engenzeka . Ukuze sibone lezi zakhi, ake sicabangele inguqulo esetshenzisiwe yombuzo obhekiswe ku- Angrist (1990) : Iyini imiphumela yomsebenzi wezempi kumholo? Kule ndaba, singachaza amayunithi ukuba abe abantu abafanelekile ukulungiswa kwe-1970 e-United States, futhi singabhala laba bantu nge- \(i = 1, \ldots, N\) . Imithi yokwelapha kuleli cala ingaba "ukukhonza emasosheni" noma "ukukhonza emasosheni." Ngizobiza lezi zimo zokwelashwa nokulawula, futhi ngizobhala \(W_i = 1\) uma umuntu \(i\) usesimweni sokwelashwa futhi \(W_i = 0\) uma umuntu \(i\) esesimweni sokulawula. Okokugcina, imiphumela engase ibe yinto ebonakala yinkimbinkimbi ngoba ihlanganisa imiphumela "engaba khona"; izinto okungenzeka zenzeke. Ngomuntu ngamunye ovumelekile ukwenza umyalo we-1970, singacabangela inani abazobe bawuthola ngo-1978 uma bekhonza empini, engizoyibiza ngokuthi \(Y_i(1)\) , nemali ababeyoyithola 1978 uma bengakhonzanga empini, engizokubiza ngokuthi \(Y_i(0)\) . Esikhathini sohlaka lokusebenza okungenzeka, \(Y_i(1)\) kanye \(Y_i(0)\) kuthathwa njengamanani alinganiselwe, kuyilapho \(W_i\) engahleliwe.

Ukukhethwa kwamayunithi, ukwelashwa, nemiphumela kubalulekile ngoba kuchaza ukuthi yini-futhi ayikwazi ukufundwa kulolu cwaningo. Ukukhethwa kwamayunithi-abantu abafanelekile ekubhalweni kuka-1970-akubandakanyi abesifazane, ngakho-ke ngaphandle kokucabanga okungeziwe, lolu cwaningo ngeke lusitshele lutho ngomphumela wemisebenzi yempi yabesifazane. Izinqumo mayelana nendlela yokuchaza ukwelashwa nemiphumela zibalulekile futhi. Isibonelo, uma ukwelashwa kwesithakazelo kugxile ekukhonzeni empini noma ukulwa nokulwa? Ingabe umphumela wenzalo kufanele ube yimali engenayo noma ukwaneliseka komsebenzi? Ekugcineni, ukukhethwa kwamayunithi, ukwelashwa, nemiphumela kufanele kuqhutshwe yizinhloso zesayensi nezenqubomgomo zesifundo.

Njengoba kunikezwe ukukhethwa kwamayunithi, ukwelashwa, kanye nemiphumela engaba khona, umphumela we-causal wezokwelapha kumuntu \(i\) , \(\tau_i\) ,

\[ \tau_i = Y_i(1) - Y_i(0) \qquad(2.1)\]

Ngamanye amazwi, siqhathanisa ukuthi umuntu othile \(i\) uzothola kanjani ngemuva kokukhonza ukuthi umuntu \(i\) angakanani ayothola ngaphandle kokukhonza. Kuye, eq. 2.1 yindlela ecacile yokuchaza umphumela we-causal, kanti nakuba ulula kakhulu, loluhlaka (Imbens and Rubin 2015) ezindleleni eziningi ezibalulekile nezithakazelisayo (Imbens and Rubin 2015) .

Uma usebenzisa uhlaka lwezinto ezikhona, ngivame ukukuthola kuwusizo ukubhala itafula ekhombisa imiphumela engaba khona kanye nemiphumela yokwelapha yazo zonke izingxenye (ithebula 2.5). Uma ungeke ucabange itafula elinjengalesi sifundo sakho, khona-ke kungase kudingeke ukuba uqonde ngokucacile kuncazelo yakho yamayunithi akho, ukwelashwa, nemiphumela engenzeka.

Lapho kuchaza umphumela we-causal ngale ndlela, noma kunjalo, sibhekene nenkinga. Cishe zonke izimo, asikwazi ukugcina imiphumela emibili. Okusho ukuthi umuntu othize ukhonze noma akazange akhonze. Ngakho-ke, sibona omunye wemiphumela \(Y_i(1)\) noma \(Y_i(0)\) -kodwa hhayi kokubili. Ukuhluleka ukugcina imiphumela emibili ingaba inkinga enkulu Holland (1986) eyayibiza ngokuthi yiNkinga Eyisisekelo Ye-Causal Inference .

Ngenhlanhla, uma senza ucwaningo, asikho nje umuntu oyedwa; Kunalokho, sinabantu abaningi, futhi lokhu kunikeza indlela ehambelana nenkinga ebalulekile ye-Causal Inference. Esikhundleni sokuzama ukulinganisa umphumela womuthi ngamunye, singalinganisa umphumela wokwelapha isilinganiso kuwo wonke amayunithi:

\[ \text{ATE} = \bar{\tau} = \frac{1}{N} \sum_{i=1}^N \tau_i \qquad(2.2)\]

Lesi sibalo sisabonakaliswa ngokwemibandela ye- \(\tau_i\) , \(\tau_i\) , kodwa ngezinye i-algebra (eq 2.8 ye- Gerber and Green (2012) ), sithola

\[ \text{ATE} = \frac{1}{N} \sum_{i=1}^N Y_i(1) - \frac{1}{N} \sum_{i=1}^N Y_i(0) \qquad(2.3)\]

Lokhu kubonisa ukuthi uma singakwazi ukulinganisa umphumela wesilinganiso sabantu ngaphansi kokwelashwa ( \(N^{-1} \sum_{i=1}^N Y_i(1)\) ) nomphumela wesilinganiso sabantu ngaphansi kokulawulwa ( \(N^{-1} \sum_{i=1}^N Y_i(1)\) ), khona-ke singakwazi ukulinganisa umphumela wokwelapha isilinganiso, ngisho ngaphandle kokulinganisa umphumela wokwelapha komuntu othile.

Manje njengoba ngichaze ukulinganisela kwethu-into esizama ukuyiqhathanisa-ngizophendukela ukuthi singayilinganisa kanjani ngempela nedatha. Futhi lapha sigijima ngokuqondile enkingeni esiyibona kuphela eminye yemiphumela engaba khona kumuntu ngamunye; sibona \(Y_i(0)\) noma \(Y_i(1)\) (ithebula 2.6). Singakwazi ukulinganisa umphumela wokwelashwa ojwayelekile ngokuqhathanisa ukuhola kwabantu abaye bakhonza kwimali yabantu abangakhonzanga:

\[ \widehat{\text{ATE}} = \underbrace{\frac{1}{N_t} \sum_{i:W_i=1} Y_i(1)}_{\text{average earnings, treatment}} - \underbrace{\frac{1}{N_c} \sum_{i:W_i=0} Y_i(0)}_{\text{average earnings, control}} \qquad(2.4)\]

lapho \(N_t\) ne \(N_c\) yizinombolo zabantu \(N_c\) kanye nezimo zokulawula. Le ndlela izokusebenza kahle uma isabelo sokwelashwa sizimele ngaphandle kwemiphumela engenzeka, isimo esibizwa ngezinye izikhathi esibizwa ngokuthi singazi . Ngeshwa, uma kungekho ukuhlolwa, ukungazi kangako kunelisekile, okusho ukuthi umlinganisi ku-eq. 2.4 angeke kwenzeke ukulinganisa okuhle. Enye indlela yokucabanga ngayo ukuthi lapho kungekho khona isabelo esiphuthumayo sokwelashwa, eq. 2.4 alifani nokufana nokuthanda; kuqhathanisa ukuhola kwezihlobo ezahlukene zabantu. Noma evezwe ngendlela ehlukile, ngaphandle kwesabelo esiphuthumayo sokwelashwa, isabelo sokwelapha cishe sihlobene nemiphumela engenzeka.

Esahlukweni 4, ngizochaza ukuthi ukuhlolwa okulawulwa ngokungahleliwe kungasiza kanjani abacwaningi ukuba benze izilinganiso ze-causal, futhi ngizochaza ukuthi abacwaningi bangasebenzisa kanjani izivivinyo zemvelo, njenge-lottery ehleliwe.

Ukuhlolwa kwemvelo

Enye indlela yokwenza izilinganiso ze-causal ngaphandle kokusebenzisa ukuhlola ukuhlola into eyenzekayo emhlabeni eye yanikezela ukwelashwa ngezikhathi ezithile. Le ndlela ibizwa ngokuthi ukuhlolwa kwemvelo . Ezimweni eziningi, ngeshwa, imvelo ayinikezi ngezikhathi ezithile ukwelashwa okufunayo kubantu abanentshisekelo. Kodwa ngezinye izikhathi, imvelo ngokungahleliwe idlulisa ukwelashwa okuhlobene. Ngokuyinhloko, ngizocabangela icala lapho kukhona ukwelashwa okwesikhashana okukhuthaza abantu ukuthi bathole ukwelashwa okuyinhloko . Isibonelo, lo mbhalo ungacatshangwa ukuthi ukwelashwa okwesibili okwenziwe ngokungahleliwe okwakhuthaza abanye abantu ukuba bathathe ukwelashwa okuyinhloko, okwakungamasevisi. Ngezinye izikhathi lokhu kuklanywa kubizwa ngokuthi ukuklama isikhuthazo . Futhi indlela yokuhlaziya engizokuchaza ngayo ukusingatha lesi simo ngezinye izikhathi kuthiwa yiziguquguquko ze-instrumental . Kulesi silungiselelo, ngezinye izicabangela, abacwaningi bangasebenzisa isikhuthazo sokufunda ngomphumela wezokwelapha eziyinhloko ze-subset ethile yamayunithi.

Ukuze sibheke ukwelashwa okubili okuhlukene-isikhuthazo kanye nokwelapha okuyinhloko-sidinga ukwaziswa okusha. Ake sithi abanye abantu \(Z_i = 1\) ngokungahleliwe ( \(Z_i = 1\) ) noma abhalwa ( \(Z_i = 0\) ); kulesi simo, \(Z_i\) ngezinye izikhathi kuthiwa iyisitsha .

Phakathi kwalabo ababhalwa phansi, abanye bakhonza ( \(Z_i = 1, W_i = 1\) ) kanti abanye abazange ( \(Z_i = 1, W_i = 0\) ). Ngokufanayo, phakathi kwalabo abangabhalwanga, abanye bakhonze ( \(Z_i = 0, W_i = 1\) ) kanti abanye abazange ( \(Z_i = 0, W_i = 0\) ). Imiphumela emihle yomuntu ngamunye manje ingakwazi ukwandiswa ukuze ibonise isimo sayo kokubili isikhuthazo kanye nokwelashwa. Isibonelo, vumela \(Y(1, W_i(1))\) abe yimali yomuntu \(i\) uma \(W_i(1)\) , lapho \(W_i(1)\) isimo sakhe sesevisi uma kuhlelwa. Ngaphezu kwalokho, singakwazi ukwahlukanisa abantu ngamaqembu amane: amakhamphani, abaqashi, abahlukumezi, kanye nabahlala njalo (ithebula 2.7).

Ngaphambi kokuba sixoxe ngokulinganisa umphumela wokwelashwa (okungukuthi, inkonzo yezempi), singakwazi ukuchaza kuqala imiphumela emibili yesikhuthazo (ie, ukubhala). Okokuqala, singachaza umphumela wesikhuthazo ekwelapheni okuyinhloko. Okwesibili, singachaza umphumela wesikhuthazo emphumela. Kuzovela ukuthi le mibi emibili ingahlanganiswa ukuze inikeze ukulinganisa umphumela wokwelapha eqenjini elithile labantu.

Okokuqala, umphumela wesikhuthazo sokwelashwa ungachazwa ngomuntu \(i\) njenge

\[ \text{ITT}_{W,i} = W_i(1) - W_i(0) \qquad(2.5)\]

Ngaphezu kwalokho, le ningi ingachazwa ngaphezu kwendawo yonke njenge

\[ \text{ITT}_{W} = \frac{1}{N} \sum_{i=1}^N [W_i(1) - W_i(0)] \qquad(2.6)\]

Okokugcina, singalinganisa \(\text{ITT} _{W}\) besebenzisa idatha:

\[ \widehat{\text{ITT}_{W}} = \bar{W}^{\text{obs}}_1 - \bar{W}^{\text{obs}}_0 \qquad(2.7)\]

lapho \(\bar{W}^{\text{obs}}_1\) yizinga elitholwayo yokwelashwa kulabo abakhuthazwayo futhi \(\bar{W}^{\text{obs}}_0\) izinga elibhekele ukwelashwa kulabo abangazange bakhuthazwe. \(\text{ITT}_W\) ngezinye izikhathi ebizwa ngokuthi izinga abatholako.

Okulandelayo, umphumela wesikhuthazo kumphumela ungachazwa ngomuntu \(i\) njenge:

\[ \text{ITT}_{Y,i} = Y_i(1, W_i(1)) - Y_i(0, W_i(0)) \qquad(2.8)\]

Ngaphezu kwalokho, le ningi ingachazwa ngaphezu kwendawo yonke njenge

\[ \text{ITT}_{Y} = \frac{1}{N} \sum_{i=1}^N [Y_i(1, W_i(1)) - Y_i(0, W_i(0))] \qquad(2.9)\]

Okokugcina, singalinganisa \(\text{ITT}_{Y}\) besebenzisa idatha:

\[ \widehat{\text{ITT}_{Y}} = \bar{Y}^{\text{obs}}_1 - \bar{Y}^{\text{obs}}_0 \qquad(2.10)\]

lapho \(\bar{Y}^{\text{obs}}_1\) ngumphumela obonisiwe (isib., ukuhola) kulabo abakhuthazwayo (isib, ukuhlelwa) kanye \(\bar{W}^{\text{obs}}_0\) ngumphumela obhekiwe kulabo abangazange bakhuthazwe.

Ekugcineni, sibheka umphumela wesithakazelo: umphumela wokuphathwa okuyisisekelo (isb., Umsebenzi wezempi) emphumela (isib., Ukuhola). Ngeshwa, kuvela ukuthi umuntu akakwazi, ngokujwayelekile, ukulinganisa lo mphumela kuwo wonke amayunithi. Kodwa-ke, ngokucabanga okunye, abacwaningi bangalinganisa umphumela wokwelashwa kwabathengisi (okusho ukuthi, abantu abazokhonza uma behlelwa kanye nabantu abangayikukhonza uma kungabhalwanga, ithebula 2.7). Ngizobiza lokhu kulinganiselwa ukuthi umphumela we-causal isilinganiso we- complier (i-CACE) (okubuye futhi ngezinye izikhathi kuthiwa umphumela wendawo yokwelapha isilinganiso , LATE):

\[ \text{CACE} = \frac{1}{N_{\text{co}}} \sum_{i:G_i=\text{co}} [Y(1, W_i(1)) - Y(0, W_i(0))] \qquad(2.11)\]

lapho \(G_i\) iqembu lomuntu \(i\) (bona ithebula 2.7) kanye \(N_{\text{co}}\) yinombolo yamakhamphani. Ngamanye amazwi, eq. 2.11 uqhathanisa amaholo wezinkampani ezibhalwa \(Y_i(1, W_i(1))\) futhi engabhalwanga \(Y_i(0, W_i(0))\) . Ukulinganisela ku-eq. 2.11 kubonakala kunzima ukulinganisa idatha ekhonjisiwe ngoba akunakwenzeka ukubona abakwa-compress besebenzisa imininingwane egcinwe kuphela (ukuthi uma umuntu ehlanganisa nawe kuzodingeka uqaphele ukuthi usebenze yini lapho ehlelwa futhi ngabe ukhonza lapho engabhalwanga).

Kuvela-mhlawumbe ngokumangalisa-ukuthi uma kunamakhomishini, bese kunikezwa eyodwa kwenza izinkombiso ezintathu ezengeziwe, kungenzeka ukuba ulinganise i-CACE kusuka kumininingwane ehlonziwe. Okokuqala, umuntu kufanele acabange ukuthi isabelo sokwelashwa siyiphutha. Endabeni yeloti ehleliwe lokhu kunengqondo. Kodwa-ke, kwezinye izilungiselelo lapho ukuhlolwa kwemvelo kungathembeki ekuziphatheni komzimba, lokhu kucabanga kungase kube nzima kakhulu. Okwesibili, umuntu kufanele acabange ukuthi awusizo ukungcola (lokhu kucabanga kuthiwa ngezinye izikhathi kuthiwa yi-monotonicity assumption). Ngokwemibandela yalolu daba kubonakala kunengqondo ukucabanga ukuthi kunabantu abambalwa kakhulu abangeke bakhonze uma kubhalwe phansi futhi kuzosebenza uma kungabhalwanga. Okwesithathu, futhi ekugcineni, kuza ukucabanga okubaluleke kunazo zonke okubizwa ngokuthi umkhawulo wokukhipha . Ngaphansi komkhawulo wokukhishwa, umuntu kufanele acabange ukuthi wonke umphumela wesabelo sokwelashwa udluliselwa ngelashwa ngokwayo. Ngamanye amazwi, umuntu kufanele acabange ukuthi akukho mphumela oqondile wesikhuthazo kwimiphumela. Uma kwenzeka i-lottery ehleliwe, isibonelo, umuntu kudingeka acabange ukuthi isimo sesimiso esingenalutho asinomthelela kumholo ngaphandle kwezempi (isibalo 2.11). Umkhawulo wokuqedwa ungaphula uma, ngokwesibonelo, abantu ababhaliwe basebenzise isikhathi esiningi esikoleni ukuze bagweme isevisi noma uma abaqashi bengenakubalwa amathuba okuqasha abantu ababhaliwe.

Uma lezi zimo ezintathu (isabelo okungahleliwe zokwelapha, akukho okungahambisani nomkhawulo wokukhushulwa) zihlangene, ke

\[ \text{CACE} = \frac{\text{ITT}_Y}{\text{ITT}_W} \qquad(2.12)\]

ngakho-ke singalinganisa i-CACE:

\[ \widehat{\text{CACE}} = \frac{\widehat{\text{ITT}_Y}}{\widehat{\text{ITT}_W}} \qquad(2.13)\]

Enye indlela yokucabanga nge-CACE yukuthi umehluko emiphumeleni phakathi kwalabo abakhuthazwayo nalabo abangakhuthazwa, abahlonywe isilinganiso sokubanjwa.

Kukhona ama-caveats amabili abalulekile okufanele agcine engqondweni. Okokuqala, umkhawulo wokuqedwa ukucabanga okuqinile, futhi kudinga ukulungiswa esimweni sokwethenjelwa, okuyinto evame ukufuna ubuchwepheshe bendawo. Umkhawulo wokukhishwa awukwazi ukulungiswa ngokungahleliwe kwesikhuthazo. Okwesibili, inselele evamile yokusebenza kanye nokuhlaziywa okuguquguqukayo kwezinto ezizayo lapho kufika isikhuthazo sinomthelela omncane ekutholeni ukwelashwa (uma \(\text{ITT}_W\) encane). Lokhu kubizwa ngokuthi ithuluzi elibuthakathaka , futhi liholela ezinkingeni ezihlukahlukene (Imbens and Rosenbaum 2005; Murray 2006) . Enye indlela yokucabanga ngenkinga ngezinsimbi ezibuthakathaka yilezi \(\widehat{\text{CACE}}\) ingaba nokuzwela \(\widehat{\text{ITT}_Y}\) amancane ku- \(\widehat{\text{ITT}_Y}\) -ngenxa yokuthi ukwephula komkhawulo wokukhishwa-ngoba lezi \(\widehat{\text{ITT}_W}\) (bheka iq. 2.13). Kakhulu, uma ukwelashwa okwenzelwa imvelo akusho umthelela omkhulu empilweni oyinakekelayo, uzobe uthola isikhathi esinzima ufunda mayelana nokwelashwa okukhathalelayo.

Bheka isahluko 23 no-24 se- Imbens and Rubin (2015) ukuze uthole inguqulo ehlelekile yale ngxoxo. Indlela yokwehlisa ngokwezomnotho ezenzakalweni zezinsimbi ngokuvamile iboniswa ngokulinganisa ukulingana, hhayi imiphumela engenzeka. Ukuze uthole isingeniso kule ngenye indlela, bheka i- Angrist and Pischke (2009) , futhi uma uqhathanisa phakathi kwalezi zindlela ezimbili, bheka isigaba 24.6 se- Imbens and Rubin (2015) . Okunye okunye, isethulo esincane esingavumelekile sendlela yokuguquguquka yezinto ezihlinzekwayo kuhlinzekwa esahlukweni 6 se- Gerber and Green (2012) . Ukuze uthole okwengeziwe emkhawulweni wokuqedwa, bheka D. Jones (2015) . Aronow and Carnegie (2013) bachaza isethi eyengeziwe yokucabanga okungasetshenziswa ukulinganisa i-ATE esikhundleni se-CACE. Ukuze uthole ukuthi ukuhlolwa kwezemvelo kungaba yinkimbinkimbi kangakanani ukuhumusha, bheka Sekhon and Titiunik (2012) . Ukuze uthole isingeniso esivamile ngokwengeziwe ekuhlolweni kwemvelo-okuhamba okungaphezu kwezinto eziguqukayo zokusebenza ukuze kufaka phakathi imiklamo efana nokuyeka ukuyeka-ubone Dunning (2012) .