## Amanqaku eMathematika

Kule sihlomelo, ndiya kufingqa ezinye iingcamango malunga nokwenza uthayibhile kwi-data engeyiyo yokuhlola kwifom ye-mathematical. Kukho iindlela ezimbini eziphambili: isakhelo segrafus causal, ininzi ehambelana neJearl Pearl kunye noogxa, kunye nezikhokelo zeziphumo, ezininzi ezinxulumene noDonald Rubin kunye noogxa nabo. Ndiza kuzisa isicwangciso-sikhokelo sesiphumo kuba sisondelelene kakhulu kwiingcamango kumanqaku eemathematika ekupheleni kwesahluko sesi-3 nakwi-4. Pearl, Glymour, and Jewell (2016) (ekuqaleni ) kunye Pearl (2009) (phambili). Ngonyango lobungakanani besikhokelo senkcazo ebandakanya i-framework framework kunye nesakhelo segrafus ye causal, ndincoma Morgan and Winship (2014) .

Injongo yale sihlomelo kukukunceda ukhululeke ngokubaluleka kunye nesitayela sesithethe seziphumo ezingenako ukuze uguqule kwezinye izinto ezibhaliweyo ezibhalwe kwesi sihloko. Okokuqala, ndiza kuchaza izikhokelo zeziphumo. Emva koko, ndiya kuyisebenzisa ukuqhubela phambili ukuxoxa ngezilingo zemvelo ezifana ne- Angrist (1990) kwimpumelelo yenkonzo yempi kwimbuyekezo. Esi sihlomelo sisondela kakhulu kwi- Imbens and Rubin (2015) .

Isikhokelo sesiphumo

Isikhokelo sesiphumo esinokubakho sinezinto ezintathu eziphambili: iiyunithi , unyango kunye neziphumo ezinokwenzeka . Ukuze sifanekise ezi zinto, makhe sicinge ngenguqu ebhalwe phantsi kwi- Angrist (1990) : Yintoni eyimpembelelo kwinkonzo yempi kwimali? Kule meko, singachaza iinqununu ukuba zibe ngabantu abanelungelo lokuqulunqwa kwee-1970 kwi-United States, kwaye sinokubhenketisa aba bantu ngokuthi $$i = 1, \ldots, N$$ . Unyango lwalolu hlobo lunokuba "ukukhonza emkhosini" okanye "ukukhonza emkhosini." Ndiza kubiza le meko kunye nokulawula, kwaye ndiza kubhala $$W_i = 1$$ ukuba umntu $$i$$ kwimeko yonyango kwaye $$W_i = 0$$ ukuba umntu $$i$$ $$W_i = 0$$ lolawulo. Ekugqibeleni, iziphumo ezinokuthi zilukhuni ngakumbi ngenxa yokuba zibandakanya iziphumo "ezinokwenzeka"; izinto eza kwenzeka. Ngomntu ngamnye ofanelekileyo kwisiqulatho se-1970, sinokucinga ngomlinganiselo ababeza kuwufumana ngo-1978 ukuba bekhonza emkhosini, endiya kuthiwa $$Y_i(1)$$ , kunye nesixa abaya kuzuza Ngowe-1978 ukuba abazange bakhonze emkhosini, endiza kubiza $$Y_i(0)$$ . Kukhokelo $$Y_i(1)$$ esinokuthi, $$Y_i(1)$$ kunye $$Y_i(0)$$ kuthathwa njengamanani alinganisiweyo, ngelixa $$W_i$$ iyinto eguquguqukayo.

Ukukhethwa kweeyunithi, unyango kunye neziphumo zibalulekileyo kuba kuchaza oko-kwaye akunakukwazi ukufundwa kwisifundo. Ukukhethwa kweeyunithi-abantu abanelungelo lokuqulunqwa kwee-1970-ababandakanyi kwabasetyhini, ngoko ke ngaphandle kokucinga okungeziwe, esi sifundo asiyi kusitshela nantoni na ngemiphumo yomsebenzi wempi yabasetyhini. Izigqibo malunga nendlela yokuchaza unyango kunye neziphumo zibalulekile ngokunjalo. Ngokomzekelo, ukuba unyango lomdla lujolise ekukhonzeni umkhosi okanye ukulwa nokulwa? Ngaba isiphumo somdla siya kuba ngumvuzo okanye ukwaneliseka komsebenzi? Ekugqibeleni, ukhetho lweeyunithi, unyango, kunye neziphumo kufuneka ziqhutywe ziinjongo zesayensi kunye nenqubomgomo yesifundo.

Ngenxa yokukhetha kweeyunithi, unyango kunye neziphumo ezinokwenzeka, umphumo wonyango kumntu $$i$$ , $$\tau_i$$ ,

$\tau_i = Y_i(1) - Y_i(0) \qquad(2.1)$

Ngamanye amagama, sithelekisa ukuba umntu $$i$$ uza kuzuza emva kokukhonza ukuba umntu $$i$$ uza kuzuza ngaphandle kokukhonza. Kum, eq. 2.1 yindlela ecacileyo yokuchaza umphumo we-causal, kwaye nangona ilula kakhulu, esi sikhokelo siphendulela ukuba senzeke kwiindlela ezininzi ezibalulekileyo (Imbens and Rubin 2015) .

Xa usebenzisa izikhokelo zeziphumo, ndandifumana kukunceda ukubhala itafile ezibonisa iziphumo ezinokwenzeka kunye nemiphumo yonyango kuwo onke amacandelo (itafile 2.5). Ukuba awukwazi ukucinga itafile enjengaleyo ekufundeni kwakho, ngoko unokufuna ukuba uchaneke ngakumbi kwiinkcazelo zakho zeeyunithi zakho, unyango kunye neziphumo ezinokwenzeka.

Itheyibhile 2.5: Itheyibhile yeziPhumo eziPhambili
Umntu Iintlawulo kwimeko yokunyanga Imali ekulawuleni imeko Impembelelo yonyango
1 $$Y_1(1)$$ $$Y_1(0)$$ $$\tau_1$$
2 $$Y_2(1)$$ $$Y_2(0)$$ $$\tau_2$$
$$\vdots$$ $$\vdots$$ $$\vdots$$ $$\vdots$$
$$N$$ $$Y_N(1)$$ $$Y_N(0)$$ $$\tau_N$$
Kuthetha $$\bar{Y}(1)$$ $$\bar{Y}(0)$$ $$\bar{\tau}$$

Xa kuchazwa umphumo we-causal ngale ndlela, kunjalo, siba yingxaki. Phantse kuzo zonke iimeko, asikwazi ukugcina iziphumo zombini. Okokuthi, umntu othile oye wakhonza okanye akazange akhonze. Ngoko ke, sigcina esinye seziphumo ezinokwenzeka- $$Y_i(1)$$ okanye $$Y_i(0)$$ -kodwa asikho zombini. Ukungakwazi ukugcina iziphumo eziphambili ziyiyona ngxaki enkulu apho i- Holland (1986) ibibiza ngokuba yiNgxaki ebalulekileyo ye-Causal Inference .

Ngethamsanqa, xa senza uphando, asikho nje umntu omnye; Kunoko, sinabantu abaninzi, kwaye oku kunika indlela ejikeleze ingxaki ebalulekileyo ye-Causal Inference. Esikhundleni sokuzama ukuqikelela umgangatho wonyango lomntu ngamnye, sinokuqikelela ukuba umgangatho wonyango oqhelekileyo kuwo onke amacandelo:

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

Olu lingelo lusetyenziswa ngokwemigaqo ye $$\tau_i$$ , engenakukwazi ukuyenza, kodwa kunye ne-algebra (eq 2.8 Gerber and Green (2012) ), siyafumana

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

Oku kubonisa ukuba xa singakwazi ukuqikelela labemi avareji isiphumo phantsi unyango ( $$N^{-1} \sum_{i=1}^N Y_i(1)$$ ) kunye sabantu avareji isiphumo phantsi kolawulo ( $$N^{-1} \sum_{i=1}^N Y_i(1)$$ ), ngoko sinako ukuqikelela umgangatho wonyango, nangaphandle kokuqikelela umphumo wonyango kunoma yimuphi umntu othile.

Ngoku ukuba ndichazile ukuqikelela kwethu-into esizama ukuyiqikelela-ndiya kutshintsha indlela esinokuyiqikelela ngayo ngedatha. Kwaye silapha sihamba ngqo kwingxaki esiyibona kuphela enye yeziphumo ezinokwenzeka kumntu ngamnye; sibona $$Y_i(0)$$ okanye $$Y_i(1)$$ (itheyibhile 2.6). Singaqikelela ukuba umlinganiselo wonyango oqhelekileyo ngokuthelekisa umvuzo wabantu abaye bakhonza kumvuzo wabantu abangazange bakhonze:

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

apho $$N_t$$ kunye $$N_c$$ ngamanani abantu $$N_c$$ kunye nolawulo lweemeko. Le ndlela iza kusebenza kakuhle ukuba isabelo sonyango sizimeleyo kwiziphumo ezinokuthi, imeko ebizwa ngezinye iinkcukacha ngokungaziwa . Ngelishwa, ekungabikho kovavanyo, ukungazi kudla ngokuqinileyo, oko kuthetha ukuba uqikelelo kwi-eq. 2.4 akakwazi ukuvelisa uqikelelo olulungileyo. Enye indlela yokucinga ngayo kukuba ekungabikho kwesabelo sonyango esingahleliyo, eq. 2.4 akafani noko kufana; kuthelekiswa nokufumana iintlobo ezahlukeneyo zabantu. Okanye kuboniswe ngokucacileyo, ngaphandle kwesabelo sonyango, ukunikezelwa kwonyango mhlawumbi kuhambelana neziphumo ezinokwenzeka.

Kwisahluko 4, ndiza kuchaza indlela ukuhlolwa okungalawulwa ngandlela-thile kunokunceda abaphandi benze ukuqikelela kwe-causal, kwaye ndiza kuchaza indlela abaphandi abanokuyisebenzisa ngayo ukuhlolwa kwemvelo, njenge-lottery.

Itheyibhile 2.6: Itheyibhile yeziphumo eziqwalaselwe
Umntu Iintlawulo kwimeko yokunyanga Imali ekulawuleni imeko Impembelelo yonyango
1 ? $$Y_1(0)$$ ?
2 $$Y_2(1)$$ ? ?
$$\vdots$$ $$\vdots$$ $$\vdots$$ $$\vdots$$
$$N$$ $$Y_N(1)$$ ? ?
Kuthetha ? ? ?

Uvavanyo lwendalo

Enye indlela yokwenza uqikelelo lwe-causal ngaphandle kokuqhuba uvavanyo kukujonga into eyenzekayo kwihlabathi eliye lalinikezela ngonyango unyango. Le ndlela ibizwa ngokuba yizilingo zemvelo . Kwiimeko ezininzi, ngelanga, ubunjani abukhiphi ngokukhawuleza unyango olufunayo kubemi banomdla. Kodwa ngamanye amaxesha, uhlobo lwendalo ludlulisa unyango olunxulumene. Ngokukodwa, ndiya kuqwalasela imeko apho kukho unyango lwesibini olukhuthaza abantu ukuba bafumane unyango lokuqala . Ngokomzekelo, uxwebhu olubhaliweyo lunokuthathwa njengonyango olwenziwe ngenyango oluthile olukhuthaze abanye abantu ukuba bathathe unyango oluphambili, olusebenza emkhosini. Ngezinye ixesha lo mhlaba ubizwa ngokuba yi- design design . Kwaye indlela yokuhlalutya endiya kuchaza ngayo ukujongana nale meko ngamanye amaxesha ibizwa ngokuba yizixhobo eziguqukayo . Kule ndlela, kunye nokucinga, abaphandi bangasebenzisa isikhuthazo sokuba bafunde malunga nomphumo wonyango oluphambili kwi-subset ethile yeeyunithi.

Ukuze sikwazi ukuphatha unyango olwahlukileyo-ukhuthazo kunye nophando oluphambili-sifuna ukuphawula olutsha. Masithi abanye abantu baqulunqwa ngokukhawuleza ( $$Z_i = 1$$ ) okanye $$Z_i = 0$$ ( $$Z_i = 0$$ ); Kule meko, $$Z_i$$ ngamanye amaxesha kuthiwa isixhobo .

Phakathi kwabo babhalwa, abanye bakhonza ( $$Z_i = 1, W_i = 1$$ ) kwaye ezinye azizange ( $$Z_i = 1, W_i = 0$$ ). Ngokufanayo, phakathi kwabo babengabhalwanga, abanye bakhonza ( $$Z_i = 0, W_i = 1$$ ) kwaye abanye abazange ( $$Z_i = 0, W_i = 0$$ ). Iziphumo ezingenakho kumntu ngamnye ngoku ziyakwandiswa ukuze zibonise isimo sazo zombini isikhuthazo kunye nonyango. Ngokomzekelo, vumela $$Y(1, W_i(1))$$ ibe yimali yomntu $$i$$ ukuba $$W_i(1)$$ , apho $$W_i(1)$$ yinkonzo yenkonzo yakhe xa ibhalwa. Ukuqhubela phambili, sinokuhlukanisa uluntu zibe ngamaqela amane: abaxhamli, abazenzi-nto, abahlambululi, kunye nabahlala besithatha (itafile 2.7).

Itheyibhile 2.7: Iintlobo ezine zabantu
Uhlobo Inkonzo ukuba yenziwe Inkonzo ukuba ingaqulunqwa
Iincindi Ewe, $$W_i(Z_i=1) = 1$$ Hayi, $$W_i(Z_i=0) = 0$$
Abazange bathathe Hayi, $$W_i(Z_i=1) = 0$$ Hayi, $$W_i(Z_i=0) = 0$$
Abahlaziyi Hayi, $$W_i(Z_i=1) = 0$$ Ewe, $$W_i(Z_i=0) = 1$$
Abathathi-njalo Ewe, $$W_i(Z_i=1) = 1$$ Ewe, $$W_i(Z_i=0) = 1$$

Ngaphambi kokuba sixoxe ngokulinganisa umphumo wonyango (oko kukuthi, inkonzo yempi), sinokuqala ukuchaza iziphumo ezimbini zokhuthazo (oko kukuthi, kubhalwe). Okokuqala, sinokuchaza intsingiselo yokukhuthazwa kwonyango lokuqala. Okwesibini, sinokuchaza intsingiselo yokhuthazo kwisiphumo. Kuya kubakho ukuba ezi zimbini zinokudibaniswa ukubonelela umlinganiselo wonyango kwiqela elithile labantu.

Okokuqala, umphumo wokukhuthazwa kwonyango unokuchazwa kumntu $$i$$ njengoko

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

Ngaphezulu, lo buninzi lunokuchazwa ngaphezu kwabantu bonke

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

Ekugqibeleni, sinokulinganisela $$\text{ITT} _{W}$$ usebenzisa idatha:

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

apho $$\bar{W}^{\text{obs}}_1$$ yinqanaba elichongiweyo lonyango kulabo bakhuthazwa kwaye $$\bar{W}^{\text{obs}}_0$$ izinga lokufumana unyango kulabo abangazange bakhuthazwe. $$\text{ITT}_W$$ ngamanye amaxesha ubizwa ngokuba yizinga lokunyusa .

Okulandelayo, umphumo wokukhuthazwa kwisiphumo ungachazwa kumntu $$i$$ njenge:

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

Ngaphezulu, lo buninzi lunokuchazwa ngaphezu kwabantu bonke

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

Ekugqibeleni, sinokulinganisela $$\text{ITT}_{Y}$$ usebenzisa idatha:

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

apho $$\bar{Y}^{\text{obs}}_1$$ $$\bar{W}^{\text{obs}}_0$$ .

Ekugqibeleni, sijonga ingqalelo kwimpembelelo yempatho: umphumo wonyango oluphambili (umz., Inkonzo yempi) kwisiphumo (umzekelo, umvuzo). Ngelishwa, kuvela ukuba umntu akakwazi, ngokuqhelekileyo, ukuqikelela le mpawu kuzo zonke iiyunithi. Nangona kunjalo, ngezinye iingcamango, abaphandi banokulinganisela impembelelo yonyango kwiinkampani (oko kukuthi, abantu abaza kukhonza xa bequlunqwe kunye nabantu abangayi kukhonza xa bengabhalwanga, itafile 2.7). Ndiza kubiza oku estimand lo complier avareji isiphumo imbangela (CACE) (oko kukuthi kwakhona ngamanye amaxesha kuthiwa yi unyango avareji yasekuhlaleni, KADE):

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

apho $$G_i$$ unikela iqela lomntu $$i$$ (bona itheyibhile 2.7) kunye $$N_{\text{co}}$$ liyinombolo yamacompress. Ngamanye amagama, eq. 2.11 uqhathanisa umvuzo weenkampani eziqulunqwe $$Y_i(1, W_i(1))$$ kwaye engabhalwanga $$Y_i(0, W_i(0))$$ . Uqikelelo kwi-eq. 2.11 kubonakala kunzima ukuqikelela kwi-data ephawulweyo kuba akunakwenzeka ukuchonga abaxhamli basebenzise idatha egcinwe kuphela (ukuba ukwazi ukuba umntu uhambelana nawe uya kufuneka ukhangele ukuba usebenze xa wayilungiswa kwaye ingaba wayekhonza xa engaqulunqwa).

Kuye kwenzeka-mhlawumbi ngenye indlela-ukuba ukuba kukho nawaphi na ama-compress, kwaye unikwe enye yenza izizathu ezintathu ezongezelelweyo, kunokwenzeka ukuba uqikelele i-CACE kwiinkcukacha ezibonwayo. Okokuqala, umntu kufuneka acinge ukuba isabelo sonyango sihleli. Kwimeko yoluhlu lwebhanti oku kuqikelele. Nangona kunjalo, kwezinye izicwangciso apho iimvavanyo zendalo zingathembeki kwizinto ezizenzekelayo, ukucinga oku kunokuba nzima kakhulu. Okwesibini, umntu kufuneka acinge ukuba azikho iinqambi (oku kuthethwa ngamanye amaxesha kuthiwa yi-monotonicity assumption). Kwimeko yolu qulunqo kubonakala kunengqiqo ukucinga ukuba banabantu abambalwa abangeke bakhonze ukuba baqulunqwe kwaye baya kukhonza xa bengabhalwanga. Okwesithathu, kwaye ekugqibeleni, kuza ukuchithwa okubaluleke kakhulu okubizwa ngokuba ngumngcipheko wokukhutshwa . Ngaphantsi komda wokukhutshwa, umntu kufuneka acinge ukuba yonke impembelelo yesabelo sonyango idluliselwa ngonyango ngokwawo. Ngamanye amagama, umntu kufuneka acinge ukuba akukho miphumo ngqo yokukhuthazwa kwiziphumo. Kwimeko yolu qulunqo lweloti, umzekelo, umntu kufuneka acinge ukuba imeko yeprogram ayinasiphumo kwimali ngaphandle kweenkonzo zempi (umzobo 2.11). Umngcipheko wokukhutshwa ungaphula umthetho ukuba, ngokomzekelo, abantu ababhaliweyo basebenzisa ixesha elide kwisikolo ukuze baphephe inkonzo okanye ukuba abaqeshi babengenakukwazi ukuqesha abantu ababhaliweyo.

Ukuba le miqathango emithathu (isabelo esingenanto kwonyango, akukho nongcoliseko kunye nokuthintela ukukhutshwa), ke ngoko

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

ngoko sinokulinganisela i-CACE:

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

Enye indlela yokucinga nge-CACE kukuba kukuba umehluko kwiziphumo phakathi kwabo bakhuthazwayo nabangakhuthazwayo, bahlonywe ngumlinganiselo wokufumana.

Kukho ezibini ezibalulekileyo zokugcina izikhumbuzo. Okokuqala, umda wokunciphisa ukukhutshwa kukubakho ukungqinelana, kwaye kufuneka ukuba ulungelelwe kwimeko-yimeko, efuna ukuba nolwazi lwengingqi. Umngcipheko wokubanjelwa awukwazi ukulungiswa ngokukhawuleza kokukhuthaza. Okwesibini, umngeni oqhelekileyo osebenzayo kunye nohlalutyo oluguquguqukayo lwezahlulo luza xa isikhuthazo singenakuchukumisa kakhulu ukufumana unyango (xa $$\text{ITT}_W$$ encinci). Oku kuthiwa yinto ebuthathaka , kwaye ikhokelela kwiintlobo ezahlukeneyo zeengxaki (Imbens and Rosenbaum 2005; Murray 2006) . Enye indlela yokucinga ngeengxaki ngezixhobo ezinobuthakathaka kukuba $$\widehat{\text{CACE}}$$ ukukhathazeka ezincinci kwi $$\widehat{\text{ITT}_Y}$$ ngenxa ukuphulwa komda wokukhutshwa-ngenxa yokuba ezi zinto ziphazamisekile ziphakanyiswe yincinci $$\widehat{\text{ITT}_W}$$ (bona iq. 2.13). Ngokukhawuleza, ukuba unyango oluthileyo lunegalelo elikhulu kunyango olukhathalelayo, ngoko uya kuba nzima ukufunda ngonyango onokukhathalela.

Jonga isahluko 23 no-24 se- Imbens and Rubin (2015) ngencwadana Imbens and Rubin (2015) yale ngxoxo. Indlela yokwenza uqoqosho ngokwemveli kwizinto eziguquguqukayo ezisetyenziswayo ngokuqhelekileyo zibonakaliswe ngokwemilinganiselo yokuqikelela ukulingana, kungekhona iziphumo ezinokwenzeka. Ukwenza isingeniso kule ngenye indlela, khangela i- Angrist and Pischke (2009) , kunye nokuthelekisa phakathi kweendlela ezimbini, jonga kwicandelo 24.6 lika- Imbens and Rubin (2015) . Ngenye indlela, isethulo esincinane esingekho phantsi kwesimo sendlela yokuguquguquka esisetyenziswayo sinikwe kwisahluko 6 Gerber and Green (2012) . Ukufumana okungakumbi ngokuthintela ukukhutshwa, jonga D. Jones (2015) . Aronow and Carnegie (2013) ichaza isethi eyongezelelweyo yeengcinga ezingasetyenziselwa ukuqikelela i-ATE kunokuba i-CACE. Ngezinye iinkcukacha malunga nokuba iimvavanyo zendalo Sekhon and Titiunik (2012) ukutolika, khangela Sekhon and Titiunik (2012) . Ukungeniswa ngokubanzi kwimizingo yemvelo-enye ehamba ngaphaya kweendlela zokuguquguquka ezisetyenziswayo ukuba zibandakanye iiplani ezifana nokuguquka kwe-regression-yabona i- Dunning (2012) .