Mathematical notes

Muchidimbu ichi, ndichapfupisa dzimwe pfungwa pamusoro pekuita zvinyorwa kubva kune zvisiri izvo zvidzidzo mune imwe fomu zvishoma yemasvomhu. Kune nzira mbiri huru: iyo causal graph framework, inowanzobatanidzwa neJudhiya Pearl uye vashandi pamwe chete, uye zvigaro zvingagadziriswa, izvo zvinowanzobatanidzwa naDonald Rubin nevamwe vashandi. Ini ndichatsanangura zvingangogadziriswa nemigumisiro nokuti inowirirana zvakanyanya nemafungiro ari munyaya dzemasvomhu kumagumo echitsauko 3 ne4. Nokuda Pearl, Glymour, and Jewell (2016) ndinokurudzira Pearl, Glymour, and Jewell (2016) (kutanga ) uye Pearl (2009) (yakatangira). Nokuda kwebhuku-kurera kurapwa kwechikonzero chinosanganisa zvigadziriswa zvinogadzirisa zvirongwa uye causal graph musimboti, ndinorumbidza Morgan and Winship (2014) .

Chinangwa chechidimbu ichi ndechekukubatsira kuti ugare wakasununguka nemashoko uye maitiro emigumisiro yemigumisiro kuitira kuti iwe ugone kushanduka pane zvimwe zvinyorwa zvekugadzira zvakanyorwa mubhuku iri. Chokutanga, ini ndicharondedzera zvingangogadziriswa. Zvadaro, ndichazvishandisa kuti ndirambe ndichikurukurirana kuedza kwepanyama kwakafanana Angrist (1990) pamusoro pemigumisiro yebasa rechiuto pamubhadharo. Izvi zvinyorwa zvinonyanya kuitika kuna Imbens and Rubin (2015) .

Mikana inogona kuitika

Izvo zvinogona kugadzirisa zvirongwa zvine zvinhu zvitatu zvikuru: zvikamu , marapirwo , nemigumisiro inogona kuitika . Kuti tifananidze zvinhu izvi, ngationgororei shanduro yakashandurwa yemubvunzo Angrist (1990) muna Angrist (1990) : Chii chinokonzerwa nebasa rechiuto pamubhadharo? Muchiitiko ichi, tinogona kutsanangura mauniti kuti vave vanhu vakakodzerwa ne1970 kutungamirirwa muUnited States, uye tinogona kunyora vanhu ava ne \(i = 1, \ldots, N\) . Mishonga iyi munyaya iyi inogona kuva "kushanda muhondo" kana "kusashumira muuto." Ndichati idzi idzi kurapwa uye kutonga mamiriro ezvinhu, uye ini ndichanyora \(W_i = 1\) kana munhu \(i\) iri mumamiriro ekurapa uye \(W_i = 0\) kana munhu \(i\) ari mumamiriro ekudzora. Pakupedzisira, zvingaguma zvingangove zvakanyanya kugadzikana zvakaoma nokuti zvinosanganisira "zvingave" zvibereko; zvinhu zvingadai zvakaitika. Kune munhu mumwe nomumwe akakodzera kubasa ra1970, tinogona kufungidzira muwandu wavakange vagamuchira muna 1978 kana vakashanda muuto rehondo, randichazoti \(Y_i(1)\) , nemari yavakange vawana 1978 kana vasina kushumira muuto, izvo zvandichadana \(Y_i(0)\) . Muchikamu chinogona kugadziriswa, \(Y_i(1)\) uye \(Y_i(0)\) inofungidzirwa kuti yakawanda, asi \(W_i\) kushanduka.

Kusarudzwa kwemasuniti, kurapwa, nemigumisiro inokosha nokuti inotsanangura zvinogona-uye zvisingagoni-kudzidziswa kubva pachidzidzo. Chisarudzo chemauniti-vanhu vanofanirwa kuita basa ra1970-havanosanganisi vakadzi, uye saka pasina mafungiro anowedzera, chidzidzo ichi hachatiudza chimwe chinhu pamusoro pemigumisiro yebasa rechiuto kuvakadzi. Zvisarudzo pamusoro pekutsanangura nzira dzokurapa nemigumisiro zvinokosha zvakare. Semuenzaniso, inofanirwa kurapwa kwekufarira inofanira kumira pakushumira muchiuto kana kurwisana here? Ko mhedzisiro yekufarira inobatsira here kana kuti kugutsikana kwebasa? Pakupedzisira, kusarudzwa kwemasuniti, marapirwo, nemigumisiro kunofanirwa kutungamirirwa nechinangwa chesayenzi nechinangwa chekudzidza.

Zvichipa kusarudzwa kwemasuniti, marapirwo, nemigumisiro inogona kuitika, zvinokonzerwa nehutano kune munhu \(i\) , \(\tau_i\) ,

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

Mune mamwe mazwi, tinofananidza kuti munhu \(i\) angadai akawana sei mushure mekushanda kune munhu \(i\) angave awana pasina kushumira. Kwandiri, eq. 2.1 ndiyo nzira yakanakisisa yekutsanangura chinokonzera kukanganisa, uye kunyange zvazvo iri nyore kwazvo, urongwa uhwu (Imbens and Rubin 2015) munzira dzakawanda dzinokosha uye dzinofadza (Imbens and Rubin 2015) .

Paunenge uchishandisa zvigadziriro zvingagadziriswa, ndinowanzoona zvichibatsira kunyora tafura inoratidza migumisiro inogona kuitika uye mishonga yekurapa yezvikamu zvose (tafura 2,5). Kana iwe usingakwanisi kufungidzira tafura yakafanana neyikudzidza kwako, iwe unogona kunge uchida kunyatsotsanangurira pane zvaunofunga nezvemauniti yako, marapirwo, nemigumisiro inogona.

Pakutsanangura zvinokonzera kukanganisa nenzira iyi, zvakadaro, tinomhanya mune dambudziko. Munenge mamiriro ezvinhu ose, hatiiti kuti tione zvose zvinogona kuitika. Ndiko kuti, munhu chaiye wakashanda kana asina kushumira. Saka, tinocherechedza imwe yemigumisiro inokwanisa- \(Y_i(1)\) kana \(Y_i(0)\) asi kwete zvose. Kukundikana kwekuona zvose zvinogona kuitika ndeye dambudziko guru iyo Holland (1986) yakadana iyo Inokosha Dambudziko re Causal Inference .

Nenzira yakanaka, patinenge tichitsvakurudza, hatisi munhu mumwe chete; pane kudaro, tine vanhu vazhinji, uye izvi zvinopa nzira yakapoteredza Chinetso Chinokonzera Causal Inference. Panzvimbo yekuedza kutarisa munhu-wehutano hwehutano hwehutano, tinogona kuenzana nehuwandu hwehutano hwehutano hwezvikwata zvose:

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

Iyi kuenzanisa ichiri kuratidzwa maererano ne \(\tau_i\) , izvo zvisingakwanisi, asi nedzimwe algebra (eq 2.8 Gerber and Green (2012) ), tinowana

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

Izvi zvinoratidza kuti kana tinogona kufungidzira vanhu avhareji mugumisiro pasi kurapwa ( \(N^{-1} \sum_{i=1}^N Y_i(1)\) ) uye vanhu avhareji mugumisiro kudzora ( \(N^{-1} \sum_{i=1}^N Y_i(1)\) ), saka tinogona kuenzana nehuwandu hwehutano hwehutano, kunyange pasina kuenzanisa kushandiswa kwemishonga kune mumwe munhu.

Iye zvino zvandakatsanangura maonero edu-chinhu chatinenge tichiedza kufungidzira-ndichazochinja kuti tingakwanisa sei kuenzana ne data. Uye pano isu tinomhanyira zvakananga mumatambudziko atinongotarisa imwe yemigumisiro inogona kumunhu wese; tinoona \(Y_i(0)\) kana \(Y_i(1)\) (tafura 2.6). Tinogona kuenzana nemavhareji ekurapa kwekuenzanisa nekuenzanisa kuwanikwa kwevanhu vakashandira kumari yevanhu vasina kushumira:

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

apo \(N_t\) uye \(N_c\) ndiwo nhamba dzevanhu vari mukurapa uye kutonga mamiriro. Iyi nzira ichashanda zvakanaka kana kurapwa kwekuita kwakasununguka pane zvingaguma, mamiriro ezvinhu dzimwe nguva anonzi kusaziva . Zvinosuruvarisa, musipo kwekuedza, kusaziva hazviwanzogutsikana, izvo zvinoreva kuti muongorori muq. 2.4 haingazoiti kuti tive nekufungidzira kwakanaka. Imwe nzira yekufunga nayo ndeyokuti pakushaikwa kwekusaita basa rekurapa, eq. 2.4 haisi kuenzanisa nezvakafanana; iri kuenzanisa zviwanikwa zvemhando dzakasiyana dzevanhu. Kana kuti inoratidzika zvakasiyana zvakasiyana, pasina basa risiri rekurapa, kurapa kurapwa kunogona kunge kwakabatana nemigumisiro inogona kuitika.

Muchitsauko 4, ini ndichatsanangura kuti maitiro ekuongororwa akaitwa sei anogona kubatsira vatsvakurudzi kuti vaongorore, uye ini ndicharondedzera kuti vatsvakurudzi vangashandisa sei zvidzidzo zvepanyama, zvakadai sokunyora bhoraji.

Kuongorora kwepanyama

Imwe nzira yekuita zvinokonzera kusarura pasina kukwanisa kuedza ndeyokutsvaga chimwe chinhu chiri kuitika munyika iyo yakarongedza kurapa kwako. Iyi nzira inonzi maitiro ezvisikwa . Mune mamiriro ezvinhu mazhinji, zvinosuruvarisa, zvisikwa hazviwanzoendesa kurapa kwaunoda kune vanhu vanofarira. Asi dzimwe nguva, zvisikwa zvinowanzopa chirwere chakafanana. Kunyanya, ndichafunga nyaya iyo pane imwe nzira yechipiri inokurudzira vanhu kuti vagamuchire kurapwa kwekutanga . Semuenzaniso, kunyora kunogona kuonekwa sokusaitwa kwechipiri kwechirwere ichi chakakurudzira vamwe vanhu kutora kurapwa kwekutanga, kwaive kushanda muchiuto. Izvi zvinodanwa dzimwe nguva zvinokurudzirwa kugadzirwa . Uye iyo nzira yekuongorora iyo yandicharondedzera kubata nayo mamiriro aya dzimwe nguva inonzi instrumental variables . Mumamiriro ezvinhu aya, nemamwe mafungiro, vatsvakurudzi vanogona kushandisa kukurudzirwa kuti vadzidze pamusoro pemagumisiro ekutanga kurapwa kwechinyorwa chiduku chezvikwata.

Kuti tigone kubata mishonga miviri yakasiyana-iyo kukurudzirwa uye kunyanya kurapwa-tinoda humwe hutsva hutsva. Ngatitii vamwe vanhu vanonyora zvinyorwa ( \(Z_i = 1\) ) kana kuti vasina kunyorwa ( \(Z_i = 0\) ); mumamiriro ezvinhu aya, \(Z_i\) dzimwe nguva anonzi chiridzwa .

Pakati peavo vakanyorwa, vamwe vakashumira ( \(Z_i = 1, W_i = 1\) ) uye vamwe havana ( \(Z_i = 1, W_i = 0\) ). Saizvozvowo, pakati peavo vasina kunyorwa, vamwe vakashumira ( \(Z_i = 0, W_i = 1\) ) uye vamwe havana ( \(Z_i = 0, W_i = 0\) ). Izvo zvinogona kuitika kune munhu mumwe nomumwe zvino zvinogona kuwedzerwa kuratidzira mamiriro avo ekukurudzira uye kurapwa. Semuenzaniso, rega \(Y(1, W_i(1))\) ave mubairo wevanhu \(i\) kana akanyorwa, apo \(W_i(1)\) inzvimbo yake yebasa kana yakanyorwa. Uyezve, tinokwanisa kuparadzanisa huwandu mumapoka mana: makambani, vasati vatora, vanosvibisa, uye nguva dzose-vanotora (tafura 2.7).

Tisati tataura nezvekufungidzira kuitika kwekurapa (kureva, basa rechiuto), tinogona kutanga kutsanangura miviri miviri yekurudziro (kureva, kuverengwa). Chokutanga, tinogona kutsanangura kushanda kwekukurudzirwa kwekutanga kurapwa. Chechipiri, tinogona kutsanangura kushanda kwekukurudzirwa pamusoro pemigumisiro. Zvichazoitika kuti izvi zviviri zvinogona kugadziriswa kugovera kuongororwa kwemigumisiro yekurapa pane rimwe boka revanhu.

Chokutanga, chiitiko chekukurudzirwa kwemishonga chinogona kutsanangurwa nemunhu \(i\) se

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

Uyezve, iyi yakawanda inogona kurondedzerwa pamusoro pevanhu vose se

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

Pakupedzisira, tinogona kuenzana \(\text{ITT} _{W}\) uchishandisa data:

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

apo \(\bar{W}^{\text{obs}}_1\) ndiyo yakatarwa muzinga wekurapa kune avo vakakurudzirwa uye \(\bar{W}^{\text{obs}}_0\) is iyo yakacherechedza chiyero cherapa kune avo vasina kukurudzirwa. \(\text{ITT}_W\) inowanzonziwo kuwedzerwa kwezinga .

Zvadaro, zvakakonzerwa nekukurudzirwa pamigumisiro zvinogona kutsanangurwa nemunhu \(i\) se:

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

Uyezve, iyi yakawanda inogona kurondedzerwa pamusoro pevanhu vose se

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

Pakupedzisira, tinogona kuenzanisa \(\text{ITT}_{Y}\) uchishandisa data:

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

apo \(\bar{Y}^{\text{obs}}_1\) ndiyo yakagumuchirwa (semuenzaniso, mubayiro) kune avo vakakurudzirwa (semuenzaniso, yakanyorwa) uye \(\bar{W}^{\text{obs}}_0\) ndizvo zvakaguma zvaitika kune avo vasina kukurudzirwa.

Pakupedzisira, tinotarisa kuguma kwekufarira: mhedzisiro yekutanga kurapwa (semuenzaniso, basa remauto) pamigumisiro (semuenzaniso, kubhadhara). Zvinosuruvarisa, zvinowanzoitika kuti munhu haagoni, kazhinji, anofungidzira izvi zvichiitika pazvikamu zvose. Zvisinei, nedzimwe mafungiro, vatsvakurudzi vanogona kufungidzira kushanda kwekurapa kune vatengesi (kureva, vanhu vanozoshanda kana vakanyorwa uye vanhu vasingazoshandisi kana vasina kunyorwa, tafura 2.7). Ini ndichati iyi inofungidzira iyo inobatanidza chiyero chechiusal effect (CACE) (iyo inowanzonziwo dzimwe nzvimbo inowanikwa mukati mekurapa , EATE):

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

apo \(G_i\) anopa boka revanhu \(i\) (ona tafura 2.7) uye \(N_{\text{co}}\) ndiyo nhamba yemakambani. Mune mamwe mazwi, eq. 2.11 inofananidzwa nemari yevashandi vanobudiswa \(Y_i(1, W_i(1))\) uye vasina kunyora \(Y_i(0, W_i(0))\) . The estimation in eq. 2.11 inoratidzika zvakaoma kuenzanisa kubva pane zvakachengetedzwa nokuti hazvibviri kuziva vatengesi vachishandisa data yakarongeka (kuziva kana mumwe munhu arikubatanidza iwe unofanirwa kuona kana akashanda paakanyorwa uye kana akashanda kana asina kunyora).

Icho chinosvika-zvimwe zvinoshamisa-kuti kana paine chero mabhizimisi, zvinozogoverwa zvinogadzira zvitatu zvekufungidzira, zvinokwanisika kuenzanisa CACE kubva pane zvakachengetedzwa. Kutanga, mumwe anofanirwa kufungidzira kuti basa rekurapa rinongoerekana raitika. Munyaya yekunyorwa kwejota iyi inonzwisisika. Zvisinei, mune dzimwe nzvimbo apo maitiro ezvisikwa haafaniri kuvimba nenyama, izvi zvinogona kunge zvakanyanya kuoma. Chechipiri, mumwe anofanirwa kufunga kuti avo havasi vanosvibisa (izvi zvinonzi dzimwe nguva zvinonzi monotonicity assumption). Muchirevo chezvinyorwa izvi zvinoratidzika zviri nyore kufunga kuti kune vanhu vashomanana vasingazoshandi kana vakanyorwa uye vachashanda kana vasina kunyora. Chechitatu, uye pakupedzisira, kunouya kugadziriswa kunonyanya kukosha kunonzi kuregererwa kwekuregererwa . Pasi pemutemo wekuregererwa, munhu anofunga kufunga kuti zvose zvinogadziriswa zvekurapa zvinopedzwa kuburikidza nemushonga wacho pachako. Mune mamwe mazwi, munhu anofungidzira kuti hapana zvakakonzerwa zvakananga zvekurudziro pamigumisiro. Munyaya yekunyora bhoraji, somuenzaniso, munhu anofunga kufunga kuti hurumende yepamusoro haina simba kune mamwe maitiro kunze kwekushandira basa rechiuto (chirevo 2.11). Kuregererwa kwekuregererwa kunogona kutyorwa kana, somuenzaniso, vanhu vakanyorwa vakashandisa nguva yakawanda kuchikoro kuitira kuti vasaita basa kana kuti vashandi vakanga vasingakwanisi kubhadhara vanhu vakanyorwa.

Kana izvi zvikamu zvitatu (basa rega roga kurapwa, kusina tsvina, uye kubviswa kwekuregererwa) zvinosangana, ipapo

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

saka tinogona kuenzanisa CACE:

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

Imwe nzira yekufunga pamusoro peCCA ndeyokuti mutsauko muzviitiko pakati pevaya vakakurudzirwa uye avo vasina kukurudzirwa, vanofungidzirwa nehuwandu hwekuwedzerwa.

Iko kune mabhuku maviri anokosha ekuchengetedza mupfungwa. Kutanga, kuregererwa kwekuregererwa kusimba kwakasimba, uye kunofanirwa kuve yakarurama pamhosva-ne-kesi, iyo inowanzodikanwa neunyanzvi hwemunharaunda. Kuregererwa kwekuregererwa hakugone kururamiswa nekusarudzwa kwekukurudzira. Chechipiri, chinetso chinowanzoitwa nekugadzirisa kushandiswa kwakasiyana-siyana kunouya apo kukurudzirwa kusina kukonzera kushandiswa kwemishonga (apo \(\text{ITT}_W\) iduku). Izvi zvinonzi chiridzwa chisina simba , uye chinotungamirira kune zvinetso zvakasiyana-siyana (Imbens and Rosenbaum 2005; Murray 2006) . Imwe nzira yekufunga nayo dambudziko nemidziyo isina simba ndeyokuti \(\widehat{\text{CACE}}\) inogona kuve nehanya nemadiki mashoma mu \(\widehat{\text{ITT}_Y}\) -kuda kukanganisa kwekuregererwa kwekuregererwa-nokuti izvi zvinokanganisa zvinokudzwa nediki \(\widehat{\text{ITT}_W}\) (ona chi 2.13). Zvichida, kana kurapwa kwezvinhu zvakasikwa kusina kukanganisa kurapwa kwaunotarisira, iwe uchave wakaoma nguva yekudzidza pamusoro pehutano hwako hwaunofarira.

Ona chitsauko 23 ne24 Imbens and Rubin (2015) kuti ive nehurukuro yakawanda yehurukuro iyi. Nzira yekugadzirisa mari inoshandiswa pakushandura zvishandiso inowanzobudiswa maererano nekufungidzira kuenzanisa, kwete zvingagoneka. Kuti uwane sumo kubva kune imwe imwe pfungwa, ona Angrist and Pischke (2009) , uye kuti muenzanisire pakati pemitambo miviri, ona chikamu 24.6 Imbens and Rubin (2015) . Imwe nzira, zvishomanana zvisingasviki mharidzo yeshanduko dzakasiyana-siyana zvinoshandiswa zvinopiwa muchitsauko 6 Gerber and Green (2012) . Nokuda kwekuwedzera kwekuregererwa kwekuregererwa, ona D. Jones (2015) . Aronow and Carnegie (2013) vanotsanangura chimwe chimwe chezvifungidziro zvinogona kushandiswa kuverenga ATE pane CACE. Nokuda kwekuti sei kuedza kwepanyama kungave kwakanyanyisa kududzira, ona Sekhon and Titiunik (2012) . Nokuda kwemashoko anowanzowedzera kuongororwa kwepanyama-imwe inopfuura kungosiyana-siyana kwezvinhu zvinoshandiswa pakuisawo zvakare zvirongwa zvakadai sokuregererwa kwekuregerera-ona Dunning (2012) .