1769:
1538:
1024:
894:
751:
60:. Unlike approaches that deliver point-identification of the model parameters, methods from the literature on partial identification are used to obtain set estimates that are valid under weaker modelling assumptions.
1186:
209:
157:
261:
1598:
1822:
1393:
320:
573:
1250:
1590:
357:
1342:
939:
809:
1277:
1213:
1055:
778:
424:
664:
599:
500:
462:
1378:
1138:
1118:
1098:
1078:
639:
619:
524:
377:
944:
814:
1887:
673:
2334:
2263:
1143:
40:
to environments where the model and the distribution of observable variables are not sufficient to determine a unique value for the model
1980:
Bonhomme, Stephane; Shaikh, Azeem (2017). "Keeping the econ in econometrics:(micro-) econometrics in the journal of political economy".
162:
110:
214:
1764:{\displaystyle \Theta _{I}=\{p\in :p=\mathrm {P} (Y=1\mid Z=1)\mathrm {P} (Z=1)+q\mathrm {P} (Z=0),{\text{ for some }}q\in \}.}
2144:
2219:
2353:
2118:
Powell, James (2017). "Identification and
Asymptotic Approximations: Three Examples of Progress in Econometric Theory".
1533:{\displaystyle \mathrm {P} (Y=1)=\mathrm {P} (Y=1\mid Z=1)\mathrm {P} (Z=1)+\mathrm {P} (Y=1\mid Z=0)\mathrm {P} (Z=0).}
1777:
270:
48:
of the parameter space. Statistical models that are set (or partially) identified arise in a variety of settings in
2358:
2187:
532:
1222:
2004:; Hong, Han; Tamer, Elie (2007). "Estimation and Confidence Regions for Parameter Sets in Econometric Models".
1546:
1384:
325:
1923:
1292:
2097:
Marschak, Jacob; Andrews, Williams (1944). "Random
Simultaneous Equations and the Theory of Production".
780:
denotes the true (i.e. data-generating) structure. The model is said to be point-identified if for every
1840:
41:
1310:
95:
named partial identification as an example of theoretical progress in the econometrics literature, and
911:
2001:
1844:
2310:
2169:
2085:
1255:
1191:
57:
1033:
1030:
of structures is the collection of admissible structures that are observationally equivalent to
756:
382:
2330:
2302:
2259:
2161:
2031:
1943:
1848:
783:
37:
578:
467:
429:
2294:
2251:
2200:
2196:
2153:
2127:
2106:
2064:
2021:
2013:
1989:
1939:
1935:
1895:
1836:
2238:
2234:
99:
list partial identification as “one of the most prominent recent themes in econometrics.”
33:
1357:
644:
2322:
2273:
2230:
1832:
1123:
1103:
1083:
1063:
624:
604:
509:
362:
77:
2182:
1855:
constructs confidence regions that cover the identified set with a given probability.
2347:
2173:
2139:
2043:
2017:
45:
2142:(2019-12-01). "The Identification Zoo: Meanings of Identification in Econometrics".
1019:{\displaystyle {\mathcal {P}}_{Y\mid Z}(s)\neq {\mathcal {P}}_{Y\mid Z}(s^{\star })}
889:{\displaystyle {\mathcal {P}}_{Y\mid Z}(s)\neq {\mathcal {P}}_{Y\mid Z}(s^{\star })}
2285:
1345:
21:
2278:"Monotone Instrumental Variables: With an Application to the Returns to Schooling"
2226:
1060:
In most cases the definition can be substantially simplified. In particular, when
91:
Partial identification continues to be a major theme in research in econometrics.
1100:
and has a known (up to some finite-dimensional parameter) distribution, and when
211:
denote a vector of observed (possibly endogenous) explanatory variables, and let
1592:, which is constrained to lie between 0 and 1. Therefore, the identified set is
53:
2255:
17:
2306:
2165:
2035:
1947:
2298:
1120:
is known up to some finite-dimensional vector of parameters, each structure
49:
2157:
2131:
2242:
2026:
1774:
Given the missing data constraint, the econometrician can only say that
1252:, is the set of parameter values that are observationally equivalent to
746:{\displaystyle {\mathcal {P}}_{Y\mid Z}(s)={\mathcal {P}}_{Y\mid Z}(s')}
2314:
2277:
2215:
2089:
2048:
Statistical
Confluence Analysis by means of Complete Regression Systems
1835:
cannot rely on the usual tools for statistical inference developed for
1215:
denotes the true (i.e. data-generating) vector of parameters, then the
1899:
2220:"Partial Identification in Applied Research: Benefits and Challenges"
2076:
Manski, Charles (1990). "Nonparametric Bounds on
Treatment Effects".
68:
Early works containing the main ideas of set identification included
2110:
1888:"Generalized Instrumental Variable Models - The Econometric Society"
1181:{\displaystyle \theta \in \Theta \subset \mathbb {R} ^{d_{\theta }}}
76:. However, the methods were significantly developed and promoted by
2068:
1993:
1843:
in the context of set-identified models, focusing on constructing
1839:. A literature in statistics and econometrics studies methods for
204:{\displaystyle Z\in {\mathcal {Z}}\subseteq \mathbb {R} ^{d_{z}}}
152:{\displaystyle U\in {\mathcal {U}}\subseteq \mathbb {R} ^{d_{u}}}
1851:
with appropriate properties. For example, a method developed by
1140:
can be characterized by a finite-dimensional parameter vector
256:{\textstyle Y\in {\mathcal {Y}}\subseteq \mathbb {R} ^{d_{y}}}
2055:
Manski, Charles (1989). "Anatomy of the
Selection Problem".
1924:"Nonparametric Identification in Structural Economic Models"
983:
951:
853:
821:
712:
680:
539:
332:
292:
263:
denote a vector of observed endogenous outcome variables. A
226:
174:
122:
2250:. Cambridge: Cambridge University Press. pp. 307–359.
359:
represents a collection of conditional distributions, and
506:
is a collection of admissible (i.e. possible) structures
575:
denote the collection of conditional distributions of
217:
1852:
1780:
1601:
1549:
1396:
1360:
1313:
1258:
1225:
1194:
1146:
1126:
1106:
1086:
1066:
1036:
947:
914:
817:
786:
759:
676:
647:
627:
607:
581:
535:
512:
470:
432:
385:
365:
328:
273:
165:
113:
1959:
1957:
2327:
44:, but instead constrain the parameters to lie in a
1816:
1763:
1584:
1532:
1372:
1336:
1271:
1244:
1207:
1180:
1132:
1112:
1092:
1072:
1049:
1018:
933:
888:
803:
772:
745:
658:
633:
613:
593:
567:
518:
494:
456:
418:
371:
351:
314:
255:
203:
151:
1817:{\displaystyle \mathrm {P} (Y=1)\in \Theta _{I}}
1824:. This makes use of all available information.
73:
315:{\displaystyle s=(h,{\mathcal {P}}_{U\mid Z})}
96:
2152:(4). American Economic Association: 835–903.
8:
1755:
1615:
568:{\displaystyle {\mathcal {P}}_{Y\mid Z}(s)}
2050:. University Institute of Economics, Oslo.
1245:{\displaystyle \Theta _{I}\subset \Theta }
896:. More generally, the model is said to be
2105:(3/4). The Econometric Society: 143–205.
2025:
2012:(5). The Econometric Society: 1243–1284.
1808:
1781:
1779:
1729:
1706:
1680:
1648:
1606:
1600:
1585:{\displaystyle \mathrm {P} (Y=1\mid Z=0)}
1550:
1548:
1507:
1475:
1452:
1420:
1397:
1395:
1359:
1314:
1312:
1263:
1257:
1230:
1224:
1199:
1193:
1170:
1165:
1161:
1160:
1145:
1125:
1105:
1085:
1065:
1041:
1035:
1007:
988:
982:
981:
956:
950:
949:
946:
925:
913:
877:
858:
852:
851:
826:
820:
819:
816:
785:
764:
758:
717:
711:
710:
685:
679:
678:
675:
646:
626:
606:
580:
544:
538:
537:
534:
511:
469:
431:
384:
364:
337:
331:
330:
327:
297:
291:
290:
272:
245:
240:
236:
235:
225:
224:
216:
193:
188:
184:
183:
173:
172:
164:
159:denote a vector of latent variables, let
141:
136:
132:
131:
121:
120:
112:
2183:"Partial Identification in Econometrics"
908:if there exists at least one admissible
352:{\displaystyle {\mathcal {P}}_{U\mid Z}}
2201:10.1146/annurev.economics.050708.143401
1940:10.1146/annurev-economics-082912-110231
1864:
2244:Advances in Economics and Econometrics
1963:
1307:. The econometrician is interested in
92:
85:
81:
69:
1871:
1853:Chernozhukov, Hong & Tamer (2007)
1288:
7:
1917:
1915:
1882:
1880:
379:is a structural function such that
1805:
1782:
1707:
1681:
1649:
1603:
1551:
1508:
1476:
1453:
1421:
1398:
1315:
1239:
1227:
1153:
14:
1337:{\displaystyle \mathrm {P} (Y=1)}
2120:Journal of Economic Perspectives
2018:10.1111/j.1468-0262.2007.00794.x
1982:The Journal of Political Economy
934:{\displaystyle s\neq s^{\star }}
2276:; Pepper, John V. (July 2000).
1922:Matzkin, Rosa L. (2013-08-02).
36:(or "point identification") in
2145:Journal of Economic Literature
2057:The Journal of Human Resources
1798:
1786:
1752:
1740:
1723:
1711:
1697:
1685:
1677:
1653:
1636:
1624:
1579:
1555:
1524:
1512:
1504:
1480:
1469:
1457:
1449:
1425:
1414:
1402:
1331:
1319:
1013:
1000:
974:
968:
883:
870:
844:
838:
740:
729:
703:
697:
601:consistent with the structure
562:
556:
489:
471:
451:
433:
407:
389:
309:
280:
1:
2329:. New York: Springer-Verlag.
74:Marschak & Andrews (1944)
2078:The American Economic Review
621:. The admissible structures
97:Bonhomme & Shaikh (2017)
1543:The only unknown object is
1272:{\displaystyle \theta _{0}}
1208:{\displaystyle \theta _{0}}
2375:
2188:Annual Review of Economics
1928:Annual Review of Economics
1892:www.econometricsociety.org
1050:{\displaystyle s^{\star }}
773:{\displaystyle s^{\star }}
668:observationally equivalent
419:{\displaystyle h(y,z,u)=0}
2256:10.1017/9781108227223.010
2218:; Rosen, Adam M. (2017).
32:) extends the concept of
1385:law of total probability
1354:can only be observed if
1291:. Suppose there are two
804:{\displaystyle s\neq s'}
2299:10.1111/1468-0262.00144
1293:binary random variables
1287:This example is due to
594:{\displaystyle Y\mid Z}
495:{\displaystyle (Y,Z,U)}
457:{\displaystyle (y,z,u)}
1818:
1765:
1586:
1534:
1374:
1338:
1273:
1246:
1209:
1182:
1134:
1114:
1094:
1074:
1051:
1020:
935:
890:
805:
774:
747:
660:
635:
615:
595:
569:
520:
496:
464:of the random vectors
458:
420:
373:
353:
316:
257:
205:
153:
30:partial identification
1841:statistical inference
1828:Statistical inference
1819:
1766:
1587:
1535:
1375:
1339:
1283:Example: missing data
1274:
1247:
1210:
1183:
1135:
1115:
1095:
1075:
1052:
1021:
936:
891:
806:
775:
748:
661:
636:
616:
596:
570:
521:
497:
459:
426:for all realizations
421:
374:
354:
317:
258:
206:
154:
2354:Econometric modeling
2181:Tamer, Elie (2010).
2158:10.1257/jel.20181361
2132:10.1257/jep.31.2.107
2002:Chernozhukov, Victor
1845:confidence intervals
1778:
1731: for some
1599:
1547:
1394:
1358:
1311:
1256:
1223:
1192:
1144:
1124:
1104:
1084:
1064:
1034:
945:
912:
815:
784:
757:
674:
645:
625:
605:
579:
533:
510:
468:
430:
383:
363:
326:
271:
215:
163:
111:
1373:{\displaystyle Z=1}
1219:, often denoted as
2323:Manski, Charles F.
2274:Manski, Charles F.
1849:confidence regions
1814:
1761:
1582:
1530:
1370:
1348:problem, however:
1334:
1269:
1242:
1205:
1178:
1130:
1110:
1090:
1080:is independent of
1070:
1047:
1016:
931:
886:
801:
770:
743:
659:{\displaystyle s'}
656:
631:
611:
591:
565:
516:
492:
454:
416:
369:
349:
312:
253:
201:
149:
58:Rubin causal model
38:statistical models
26:set identification
2359:Estimation theory
2336:978-0-387-00454-9
2265:978-1-108-22722-3
1900:10.3982/ecta12223
1732:
1133:{\displaystyle s}
1113:{\displaystyle h}
1093:{\displaystyle Z}
1073:{\displaystyle U}
634:{\displaystyle s}
614:{\displaystyle s}
519:{\displaystyle s}
372:{\displaystyle h}
80:, beginning with
2366:
2340:
2318:
2282:
2269:
2249:
2239:Samuelson, Larry
2235:Piazzesi, Monika
2224:
2204:
2177:
2135:
2114:
2093:
2072:
2051:
2039:
2029:
1997:
1988:(6): 1846–1853.
1967:
1961:
1952:
1951:
1919:
1910:
1909:
1907:
1906:
1884:
1875:
1869:
1837:point estimation
1823:
1821:
1820:
1815:
1813:
1812:
1785:
1770:
1768:
1767:
1762:
1733:
1730:
1710:
1684:
1652:
1611:
1610:
1591:
1589:
1588:
1583:
1554:
1539:
1537:
1536:
1531:
1511:
1479:
1456:
1424:
1401:
1379:
1377:
1376:
1371:
1353:
1343:
1341:
1340:
1335:
1318:
1306:
1300:
1278:
1276:
1275:
1270:
1268:
1267:
1251:
1249:
1248:
1243:
1235:
1234:
1214:
1212:
1211:
1206:
1204:
1203:
1187:
1185:
1184:
1179:
1177:
1176:
1175:
1174:
1164:
1139:
1137:
1136:
1131:
1119:
1117:
1116:
1111:
1099:
1097:
1096:
1091:
1079:
1077:
1076:
1071:
1056:
1054:
1053:
1048:
1046:
1045:
1025:
1023:
1022:
1017:
1012:
1011:
999:
998:
987:
986:
967:
966:
955:
954:
940:
938:
937:
932:
930:
929:
895:
893:
892:
887:
882:
881:
869:
868:
857:
856:
837:
836:
825:
824:
810:
808:
807:
802:
800:
779:
777:
776:
771:
769:
768:
752:
750:
749:
744:
739:
728:
727:
716:
715:
696:
695:
684:
683:
665:
663:
662:
657:
655:
640:
638:
637:
632:
620:
618:
617:
612:
600:
598:
597:
592:
574:
572:
571:
566:
555:
554:
543:
542:
525:
523:
522:
517:
501:
499:
498:
493:
463:
461:
460:
455:
425:
423:
422:
417:
378:
376:
375:
370:
358:
356:
355:
350:
348:
347:
336:
335:
321:
319:
318:
313:
308:
307:
296:
295:
262:
260:
259:
254:
252:
251:
250:
249:
239:
230:
229:
210:
208:
207:
202:
200:
199:
198:
197:
187:
178:
177:
158:
156:
155:
150:
148:
147:
146:
145:
135:
126:
125:
2374:
2373:
2369:
2368:
2367:
2365:
2364:
2363:
2344:
2343:
2337:
2321:
2293:(4): 997–1010.
2280:
2272:
2266:
2247:
2222:
2214:
2211:
2209:Further reading
2180:
2138:
2117:
2111:10.2307/1905432
2096:
2075:
2054:
2042:
2000:
1979:
1976:
1971:
1970:
1962:
1955:
1921:
1920:
1913:
1904:
1902:
1886:
1885:
1878:
1870:
1866:
1861:
1830:
1804:
1776:
1775:
1602:
1597:
1596:
1545:
1544:
1392:
1391:
1356:
1355:
1349:
1309:
1308:
1302:
1296:
1285:
1259:
1254:
1253:
1226:
1221:
1220:
1195:
1190:
1189:
1166:
1159:
1142:
1141:
1122:
1121:
1102:
1101:
1082:
1081:
1062:
1061:
1037:
1032:
1031:
1003:
980:
948:
943:
942:
921:
910:
909:
873:
850:
818:
813:
812:
793:
782:
781:
760:
755:
754:
732:
709:
677:
672:
671:
666:are said to be
648:
643:
642:
623:
622:
603:
602:
577:
576:
536:
531:
530:
508:
507:
466:
465:
428:
427:
381:
380:
361:
360:
329:
324:
323:
289:
269:
268:
241:
234:
213:
212:
189:
182:
161:
160:
137:
130:
109:
108:
105:
66:
34:identifiability
12:
11:
5:
2372:
2370:
2362:
2361:
2356:
2346:
2345:
2342:
2341:
2335:
2319:
2270:
2264:
2210:
2207:
2206:
2205:
2195:(1): 167–195.
2178:
2140:Lewbel, Arthur
2136:
2126:(2): 107–124.
2115:
2094:
2084:(2): 319–323.
2073:
2069:10.2307/145818
2063:(3): 343–360.
2052:
2044:Frisch, Ragnar
2040:
1998:
1994:10.1086/694620
1975:
1972:
1969:
1968:
1953:
1934:(1): 457–486.
1911:
1876:
1863:
1862:
1860:
1857:
1833:Set estimation
1829:
1826:
1811:
1807:
1803:
1800:
1797:
1794:
1791:
1788:
1784:
1772:
1771:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1728:
1725:
1722:
1719:
1716:
1713:
1709:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1683:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1651:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1609:
1605:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1553:
1541:
1540:
1529:
1526:
1523:
1520:
1517:
1514:
1510:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1485:
1482:
1478:
1474:
1471:
1468:
1465:
1462:
1459:
1455:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1423:
1419:
1416:
1413:
1410:
1407:
1404:
1400:
1369:
1366:
1363:
1333:
1330:
1327:
1324:
1321:
1317:
1284:
1281:
1266:
1262:
1241:
1238:
1233:
1229:
1217:identified set
1202:
1198:
1173:
1169:
1163:
1158:
1155:
1152:
1149:
1129:
1109:
1089:
1069:
1044:
1040:
1028:identified set
1015:
1010:
1006:
1002:
997:
994:
991:
985:
979:
976:
973:
970:
965:
962:
959:
953:
928:
924:
920:
917:
885:
880:
876:
872:
867:
864:
861:
855:
849:
846:
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279:
276:
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78:Charles Manski
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93:Powell (2017)
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83:
82:Manski (1989)
79:
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70:Frisch (1934)
63:
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46:strict subset
43:
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2286:Econometrica
2284:
2243:
2231:Pakes, Ariel
2192:
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2099:Econometrica
2098:
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2027:1721.1/63545
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2006:Econometrica
2005:
1985:
1981:
1931:
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1903:. Retrieved
1891:
1867:
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1542:
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1346:missing data
1303:
1297:
1289:Tamer (2010)
1286:
1216:
1059:
1027:
905:
901:
897:
667:
528:
503:
264:
106:
90:
67:
52:, including
29:
25:
22:econometrics
15:
1964:Lewbel 2019
54:game theory
2348:Categories
2227:Honore, Bo
1974:References
1905:2024-01-05
1872:Tamer 2010
941:such that
906:identified
267:is a pair
103:Definition
42:parameters
18:statistics
2307:0012-9682
2174:125792293
2166:0022-0515
2036:0012-9682
1948:1941-1383
1806:Θ
1802:∈
1738:∈
1666:∣
1622:∈
1604:Θ
1568:∣
1493:∣
1438:∣
1261:θ
1240:Θ
1237:⊂
1228:Θ
1197:θ
1172:θ
1157:⊂
1154:Θ
1151:∈
1148:θ
1043:⋆
1009:⋆
993:∣
978:≠
961:∣
927:⋆
919:≠
902:partially
879:⋆
863:∣
848:≠
831:∣
791:≠
766:⋆
722:∣
690:∣
586:∣
549:∣
342:∣
302:∣
265:structure
232:⊆
222:∈
180:⊆
170:∈
128:⊆
118:∈
50:economics
2325:(2003).
2241:(eds.).
2216:Ho, Kate
2046:(1934).
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64:History
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1026:. The
753:. Let
2311:JSTOR
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2248:(PDF)
2225:. In
2223:(PDF)
2170:S2CID
2086:JSTOR
1859:Notes
1188:. If
504:model
2331:ISBN
2303:ISSN
2260:ISBN
2162:ISSN
2032:ISSN
1944:ISSN
1301:and
900:(or
641:and
529:Let
502:. A
107:Let
84:and
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