1393:
639:
1388:{\displaystyle {\begin{aligned}n_{s}&={\text{ number of individuals in group }}s\\{\overline {y}}_{st}&={\frac {1}{n_{s}}}\sum _{i=1}^{n}y_{it}\ I(s(i)~=~s),\\{\overline {\gamma }}_{s}&={\frac {1}{n_{s}}}\sum _{i=1}^{n}\gamma _{s(i)}\ I(s(i)~=~s)~=~\gamma _{s},\\{\overline {\lambda }}_{st}&={\frac {1}{n_{s}}}\sum _{i=1}^{n}\lambda _{t}\ I(s(i)~=~s)~=~\lambda _{t},\\D_{st}&={\frac {1}{n_{s}}}\sum _{i=1}^{n}I(s(i)~=~{\text{ treatment, }}t{\text{ in after period}})\ I(s(i)~=~s)~=~I(s~=~{\text{ treatment, }}t{\text{ in after period}}),\\{\overline {\varepsilon }}_{st}&={\frac {1}{n_{s}}}\sum _{i=1}^{n}\varepsilon _{it}\ I(s(i)~=~s),\end{aligned}}}
5263:
106:
2091:
4942:
4359:
5472:
and
Pennsylvania have parallel trends over time, Pennsylvania's change in employment can be interpreted as the change New Jersey would have experienced, had they not increased the minimum wage, and vice versa. The evidence suggested that the increased minimum wage did not induce a decrease in employment in New Jersey, contrary to what some economic theory would suggest. The table below shows Card & Krueger's estimates of the treatment effect on employment, measured as
5258:{\displaystyle {\begin{aligned}{\widehat {E}}(y\mid T=1,~S=0)&={\widehat {E}}(y\mid {\text{ after period, control}})\\\\&={\frac {{\widehat {E}}(y\ I({\text{ after period, control}}))}{{\widehat {P}}({\text{ after period, control}})}}\\\\&={\frac {\sum _{i=1}^{n}y_{i,{\text{after}}}I(i{\text{ in control}})}{n_{\text{control}}}}={\overline {y}}_{\text{control, after}}\\\\&={\overline {y}}_{\text{12}}\end{aligned}}}
1498:
4936:. This nuance is important to understand when the user believes (weak) violations of parallel pre-trend exist or in the case of violations of the appropriate counterfactual approximation assumptions given the existence of non-common shocks or confounding events. To see the relation between this notation and the previous section, consider as above only one observation per time period for each group, then
3735:
2865:. This involves 'Matching' known 'treatment' units with simulated counterfactual 'control' units: characteristically equivalent units which did not receive treatment. By defining the Outcome Variable as a temporal difference (change in observed outcome between pre- and posttreatment periods), and Matching multiple units in a large sample on the basis of similar pre-treatment histories, the resulting
2662:
2869:(i.e. the ATT: Average Treatment Effect for the Treated) provides a robust difference-in-differences estimate of treatment effects. This serves two statistical purposes: firstly, conditional on pre-treatment covariates, the parallel trends assumption is likely to hold; and secondly, this approach reduces dependence on associated ignorability assumptions necessary for valid inference.
2086:{\displaystyle {\begin{aligned}&({\overline {y}}_{11}-{\overline {y}}_{12})-({\overline {y}}_{21}-{\overline {y}}_{22})\\={}&{\big }\\&\qquad {}-{\big }\\={}&\delta (D_{11}-D_{12})+\delta (D_{22}-D_{21})+{\overline {\varepsilon }}_{11}-{\overline {\varepsilon }}_{12}+{\overline {\varepsilon }}_{22}-{\overline {\varepsilon }}_{21}.\end{aligned}}}
141:) can be explained as being an effect of the treatment, because the treatment group and control group did not start out at the same point at time 1. DID, therefore, calculates the "normal" difference in the outcome variable between the two groups (the difference that would still exist if neither group experienced the treatment), represented by the dotted line
4354:{\displaystyle {\begin{aligned}{\hat {\beta }}_{0}&={\widehat {E}}(y\mid T=0,~S=0)\\{\hat {\beta }}_{1}&={\widehat {E}}(y\mid T=1,~S=0)-{\widehat {E}}(y\mid T=0,~S=0)\\{\hat {\beta }}_{2}&={\widehat {E}}(y\mid T=0,~S=1)-{\widehat {E}}(y\mid T=0,~S=0)\\{\hat {\beta }}_{3}&={\big }\\&\qquad {}-{\big },\end{aligned}}}
5471:
such as weather and macroeconomic conditions of the region. By including
Pennsylvania as a control in a difference-in-differences model, any bias caused by variables common to New Jersey and Pennsylvania is implicitly controlled for, even when these variables are unobserved. Assuming that New Jersey
112:
Difference in differences requires data measured from a treatment group and a control group at two or more different time periods, specifically at least one time period before "treatment" and at least one time period after "treatment." In the example pictured, the outcome in the treatment group is
113:
represented by the line P and the outcome in the control group is represented by the line S. The outcome (dependent) variable in both groups is measured at time 1, before either group has received the treatment (i.e., the independent or explanatory variable), represented by the points
2620:. Below it is shown how this estimator can be read as a coefficient in an ordinary least squares regression. The model described in this section is over-parametrized; to remedy that, one of the coefficients for the dummy variables can be set to 0, for example, we may set
2585:
2880:
would have been with parallel trends, had there been no treatment. The
Achilles' heel of DID is when something other than the treatment changes in one group but not the other at the same time as the treatment, implying a violation of the parallel trend assumption.
2859:
4947:
3524:
2302:
292:
5467:, in February 1992 and in November 1992, after New Jersey's minimum wage rose from $ 4.25 to $ 5.05 in April 1992. Observing a change in employment in New Jersey only, before and after the treatment, would fail to control for
127:. The treatment group then receives or experiences the treatment and both groups are again measured at time 2. Not all of the difference between the treatment and control groups at time 2 (that is, the difference between
66:) by comparing the average change over time in the outcome variable for the treatment group to the average change over time for the control group. Although it is intended to mitigate the effects of extraneous factors and
3740:
2884:
To guarantee the accuracy of the DID estimate, the composition of individuals of the two groups is assumed to remain unchanged over time. When using a DID model, various issues that may compromise the results, such as
1503:
5418:
644:
2460:
4406:
3411:
4864:
is not actually conditional on the treatment or control group. Consistently, a difference among the treatment and control groups would eliminate the need for treatment differentials (i.e.,
4754:
2715:
2452:
4934:
4898:
4862:
4826:
4790:
4672:
4636:
4574:
4538:
4494:
624:
3332:
3291:
3243:
3117:
2651:
570:
523:
3691:
446:
2393:
476:
2861:
may well be more realistic. In order to increase the likelihood of the parallel trend assumption holding, a difference-in-differences approach is often combined with
594:
3202:
3174:
3076:
3048:
2934:
2618:
1490:
325:
3723:
1460:
1428:
397:
4600:
4458:
4432:
3659:
3593:
3146:
3020:
2988:
2961:
2767:
2741:
2360:
2334:
2776:
5306:
5286:
3633:
3613:
3567:
3547:
543:
496:
417:
368:
348:
3424:
2106:
192:
5452:
5719:"Constructing a More Closely Matched Control Group in a Difference-in-Differences Analysis: Its Effect on History Interacting with Group Bias"
5890:
5805:
Card, David; Krueger, Alan B. (1994). "Minimum Wages and
Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania".
5697:
5314:
627:
5964:"How Does Charitable Giving Respond to Incentives and Income? Dynamic Panel Estimates Accounting for Predictable Changes in Taxation"
4496:
is an estimate of the counterfactual rather than the impact of the control group. The control group is often used as a proxy for the
5917:
3729:, furthermore, it turns out that the group and period averages in that section relate to the model parameter estimates as follows
97:
to measure the differences, between the treatment and control group, of the changes in the outcome variable that occur over time.
6040:
5941:
5767:
5651:
93:
of the treatment effect (which measures the difference between treatment and control groups), difference in differences uses
4540:
can be interpreted as the impact of both the control group and the intervention's (treatment's) counterfactual. Similarly,
2580:{\displaystyle {\hat {\delta }}~=~({\overline {y}}_{11}-{\overline {y}}_{12})-({\overline {y}}_{21}-{\overline {y}}_{22}),}
6030:
5476:. Card and Krueger estimate that the $ 0.80 minimum wage increase in New Jersey led to a 2.75 FTE increase in employment.
6045:
6035:
4367:
449:
3338:
4685:
5593:
2307:
6014:
6050:
4576:, due to the parallel trend assumption, is also the same differential between the treatment and control group in
2901:
The DID method can be implemented according to the table below, where the lower right cell is the DID estimator.
105:
5807:
5571:
5566:
4602:. The above descriptions should not be construed to imply the (average) effect of only the control group, for
4501:
2866:
2680:
2398:
70:, depending on how the treatment group is chosen, this method may still be subject to certain biases (e.g.,
5723:
5602:
2670:
2097:
75:
71:
4903:
4867:
4831:
4795:
4759:
4641:
4605:
4543:
4507:
4463:
5561:
5468:
2862:
328:
79:
39:
5451:, published in 1994, is considered one of the most famous DID studies; Card was later awarded the 2021
599:
6055:
5473:
2890:
59:
5607:
3297:
3256:
3208:
3082:
2623:
90:
47:
2872:
As illustrated to the right, the treatment effect is the difference between the observed value of
548:
5923:
5857:
5816:
5784:
5740:
5668:
5620:
501:
63:
55:
3664:
2773:
above accurately represents reality, this assumption automatically holds. However, a model with
422:
5913:
5886:
5880:
5693:
5687:
2365:
455:
579:
5994:
5971:
5950:
5905:
5847:
5776:
5732:
5660:
5612:
3180:
3152:
3054:
3026:
2909:
2593:
1465:
300:
5549:
3696:
2854:{\displaystyle \lambda _{st}~:~\lambda _{22}-\lambda _{21}\neq \lambda _{12}-\lambda _{11}}
2661:
1433:
1401:
3725:. Although it is not shown rigorously here, this is a proper parametrization of the model
2886:
373:
4579:
4437:
4411:
3638:
3572:
3125:
2999:
2967:
2940:
2746:
2720:
2339:
2313:
3519:{\displaystyle y~=~\beta _{0}+\beta _{1}T+\beta _{2}S+\beta _{3}(T\cdot S)+\varepsilon }
2297:{\displaystyle \operatorname {E} \left~=~\delta (D_{11}-D_{12})+\delta (D_{22}-D_{21}).}
572:
is the time trend shared by both groups according to the parallel trend assumption (see
5714:
5291:
5271:
4497:
3618:
3598:
3552:
3532:
633:
Consider the average of the dependent variable and dummy indicators by group and time:
528:
481:
402:
353:
333:
287:{\displaystyle y_{it}~=~\gamma _{s(i)}+\lambda _{t}+\delta I(\dots )+\varepsilon _{it}}
67:
50:, by studying the differential effect of a treatment on a 'treatment group' versus a '
6024:
5936:
5927:
5861:
5744:
51:
4638:, or only the difference of the treatment and control groups in the pre-period, for
5639:
5624:
5464:
5456:
5444:
5440:
4679:
35:
5987:"Inference with 'Difference in Differences' with a Small Number of Policy Changes"
5788:
5672:
5836:"diff: Simplifying the estimation of difference-in-differences treatment effects"
1492:
is not random; it just encodes how the groups and the periods are labeled. Then
89:
of the treatment effect on subjects (which analyzes differences over time) or a
86:
5852:
5835:
5780:
5664:
2590:
which can be interpreted as the treatment effect of the treatment indicated by
58:. It calculates the effect of a treatment (i.e., an explanatory variable or an
5448:
5436:
4675:
94:
43:
31:
5909:
5616:
5460:
3418:
Running a regression analysis gives the same result. Consider the OLS model
5736:
5718:
5591:
Abadie, A. (2005). "Semiparametric difference-in-differences estimators".
5423:
But this is the expression for the treatment effect that was given in the
170:.) The treatment effect is the difference between the observed outcome (P
5954:
5820:
5759:
5643:
5413:{\displaystyle {\hat {\beta }}_{3}~=~(y_{11}-y_{21})-(y_{12}-y_{22}).}
4408:
stands for conditional averages computed on the sample, for example,
5976:
5963:
16:
Statistical technique to use observational data for causal analysis
5999:
5986:
5545:
2660:
104:
5758:
Bertrand, Marianne; Duflo, Esther; Mullainathan, Sendhil (2004).
5760:"How Much Should We Trust Differences-In-Differences Estimates?"
5644:"How Much Should We Trust Differences-in-Differences Estimates?"
5544:
A software example application of this research is found on the
5937:"Recent Developments in the Econometrics of Program Evaluation"
5455:
in part for this and related work. Card and
Krueger compared
4682:, below, a first (time) difference of the outcome variable
5882:
Mostly
Harmless Econometrics: An Empiricist's Companion
5689:
Mostly
Harmless Econometrics: An Empiricist's Companion
419:
belongs (i.e. the treatment or the control group), and
5317:
5294:
5274:
4945:
4906:
4870:
4834:
4798:
4762:
4688:
4644:
4608:
4582:
4546:
4510:
4466:
4440:
4414:
4370:
3738:
3699:
3667:
3641:
3621:
3601:
3575:
3555:
3535:
3427:
3341:
3300:
3259:
3211:
3183:
3155:
3128:
3085:
3057:
3029:
3002:
2970:
2943:
2912:
2779:
2749:
2723:
2683:
2626:
2596:
2463:
2401:
2368:
2342:
2316:
2109:
1501:
1468:
1436:
1404:
642:
602:
582:
551:
531:
504:
484:
478:
is true, and 0 otherwise. In the plot of time versus
458:
425:
405:
376:
356:
336:
303:
195:
4504:
for a deeper understanding of this point). Thereby,
174:) and the "normal" outcome (the difference between P
3615:is a dummy variable for group membership, equal to
5412:
5300:
5280:
5257:
4928:
4892:
4856:
4820:
4784:
4748:
4666:
4630:
4594:
4568:
4532:
4488:
4452:
4426:
4400:
4353:
3717:
3685:
3653:
3627:
3607:
3587:
3561:
3541:
3518:
3405:
3326:
3285:
3237:
3196:
3168:
3140:
3111:
3070:
3042:
3014:
2982:
2955:
2928:
2853:
2761:
2735:
2709:
2673:apply equally to DID. In addition, DID requires a
2645:
2612:
2579:
2446:
2387:
2354:
2328:
2296:
2085:
1484:
1454:
1422:
1387:
618:
588:
564:
537:
517:
490:
470:
440:
411:
391:
362:
342:
319:
286:
5935:Imbens, Guido W.; Wooldridge, Jeffrey M. (2009).
4460:is an indicator for the control group. Note that
4401:{\displaystyle {\widehat {E}}(\dots \mid \dots )}
42:in the social sciences that attempts to mimic an
5904:. Cambridge university press. pp. 768–772.
5885:. Princeton University Press. pp. 227–243.
5692:. Princeton University Press. pp. 227–243.
5900:Cameron, Arthur C.; Trivedi, Pravin K. (2005).
3406:{\displaystyle (y_{11}-y_{21})-(y_{12}-y_{22})}
4749:{\displaystyle (\Delta Y_{i}=Y_{i,1}-Y_{i,0})}
62:) on an outcome (i.e., a response variable or
4339:
4233:
4213:
4107:
3549:is a dummy variable for the period, equal to
2665:Illustration of the parallel trend assumption
1913:
1773:
1753:
1613:
8:
525:is the vertical intercept for the graph for
5902:Microeconometrics: Methods and Applications
5962:Bakija, Jon; Heim, Bradley (August 2008).
4756:eliminates the need for time-trend (i.e.,
2677:. The parallel trend assumption says that
666: number of individuals in group
5998:
5975:
5851:
5606:
5453:Nobel Memorial Prize in Economic Sciences
5398:
5385:
5366:
5353:
5331:
5320:
5319:
5316:
5293:
5273:
5245:
5235:
5215:
5205:
5193:
5180:
5164:
5157:
5147:
5136:
5129:
5105:
5091:
5090:
5077:
5051:
5050:
5047:
5026:
5006:
5005:
4951:
4950:
4946:
4944:
4920:
4909:
4908:
4905:
4884:
4873:
4872:
4869:
4848:
4837:
4836:
4833:
4812:
4801:
4800:
4797:
4776:
4765:
4764:
4761:
4731:
4712:
4699:
4687:
4658:
4647:
4646:
4643:
4622:
4611:
4610:
4607:
4581:
4560:
4549:
4548:
4545:
4524:
4513:
4512:
4509:
4480:
4469:
4468:
4465:
4439:
4413:
4372:
4371:
4369:
4338:
4337:
4290:
4289:
4239:
4238:
4232:
4231:
4226:
4212:
4211:
4164:
4163:
4113:
4112:
4106:
4105:
4092:
4081:
4080:
4028:
4027:
3977:
3976:
3963:
3952:
3951:
3899:
3898:
3848:
3847:
3834:
3823:
3822:
3770:
3769:
3756:
3745:
3744:
3739:
3737:
3698:
3666:
3640:
3620:
3600:
3574:
3554:
3534:
3489:
3473:
3457:
3444:
3426:
3394:
3381:
3362:
3349:
3340:
3318:
3305:
3299:
3277:
3264:
3258:
3229:
3216:
3210:
3188:
3182:
3160:
3154:
3127:
3103:
3090:
3084:
3062:
3056:
3034:
3028:
3001:
2969:
2942:
2917:
2911:
2845:
2832:
2819:
2806:
2784:
2778:
2748:
2722:
2710:{\displaystyle \lambda _{2}-\lambda _{1}}
2701:
2688:
2682:
2631:
2625:
2601:
2595:
2565:
2555:
2545:
2535:
2519:
2509:
2499:
2489:
2465:
2464:
2462:
2432:
2419:
2406:
2400:
2373:
2367:
2341:
2315:
2282:
2269:
2247:
2234:
2201:
2191:
2181:
2171:
2155:
2145:
2135:
2125:
2108:
2070:
2060:
2050:
2040:
2030:
2020:
2010:
2000:
1987:
1974:
1952:
1939:
1925:
1912:
1911:
1902:
1892:
1882:
1866:
1853:
1834:
1824:
1814:
1798:
1785:
1772:
1771:
1766:
1752:
1751:
1742:
1732:
1722:
1706:
1693:
1674:
1664:
1654:
1638:
1625:
1612:
1611:
1605:
1589:
1579:
1569:
1559:
1543:
1533:
1523:
1513:
1502:
1500:
1473:
1467:
1435:
1403:
1333:
1323:
1312:
1300:
1291:
1275:
1265:
1249:
1241:
1170:
1162:
1129:
1118:
1106:
1097:
1081:
1064:
1009:
999:
988:
976:
967:
951:
941:
927:
863:
853:
842:
830:
821:
808:
798:
745:
735:
724:
712:
703:
687:
677:
664:
651:
643:
641:
607:
601:
581:
556:
550:
530:
509:
503:
483:
457:
424:
404:
375:
355:
335:
308:
302:
275:
244:
222:
200:
194:
5478:
2903:
5879:Angrist, J. D.; Pischke, J. S. (2008).
5800:
5798:
5686:Angrist, J. D.; Pischke, J. S. (2008).
5583:
4434:is the indicator for the after period,
452:equal to 1 when the event described in
5424:
3726:
2770:
2447:{\displaystyle D_{11}=D_{12}=D_{21}=0}
2893:, must be considered and dealt with.
7:
5991:NBER Technical Working Paper No. 312
3693:is a dummy variable indicating when
6015:Difference in Difference Estimation
5985:Conley, T.; Taber, C. (July 2005).
4929:{\displaystyle {\hat {\beta }}_{3}}
4893:{\displaystyle {\hat {\beta }}_{2}}
4857:{\displaystyle {\hat {\beta }}_{1}}
4821:{\displaystyle {\hat {\beta }}_{3}}
4785:{\displaystyle {\hat {\beta }}_{1}}
4667:{\displaystyle {\hat {\beta }}_{2}}
4631:{\displaystyle {\hat {\beta }}_{1}}
4569:{\displaystyle {\hat {\beta }}_{2}}
4533:{\displaystyle {\hat {\beta }}_{1}}
4489:{\displaystyle {\hat {\beta }}_{1}}
4900:) to form an unbiased estimate of
4792:) to form an unbiased estimate of
4692:
2110:
14:
619:{\displaystyle \varepsilon _{it}}
1398:and suppose for simplicity that
573:
5548:'s command -diff- authored by
5474:FTEs (or full-time equivalents)
4225:
1765:
6017:, Healthcare Economist website
5942:Journal of Economic Literature
5768:Quarterly Journal of Economics
5652:Quarterly Journal of Economics
5404:
5378:
5372:
5346:
5325:
5268:and so on for other values of
5185:
5174:
5110:
5102:
5085:
5082:
5074:
5062:
5031:
5017:
4995:
4962:
4914:
4878:
4842:
4806:
4770:
4743:
4689:
4652:
4616:
4554:
4518:
4474:
4395:
4383:
4334:
4301:
4283:
4250:
4208:
4175:
4157:
4124:
4086:
4072:
4039:
4021:
3988:
3957:
3943:
3910:
3892:
3859:
3828:
3814:
3781:
3750:
3680:
3668:
3507:
3495:
3400:
3374:
3368:
3342:
2571:
2531:
2525:
2485:
2470:
2288:
2262:
2253:
2227:
2207:
2167:
2161:
2121:
1993:
1967:
1958:
1932:
1908:
1846:
1840:
1778:
1748:
1686:
1680:
1618:
1595:
1555:
1549:
1509:
1375:
1360:
1354:
1348:
1254:
1226:
1211:
1196:
1190:
1184:
1175:
1150:
1144:
1138:
1048:
1033:
1027:
1021:
911:
896:
890:
884:
873:
867:
787:
772:
766:
760:
465:
459:
435:
429:
386:
380:
327:is the dependent variable for
265:
259:
232:
226:
156:is the same as the slope from
145:. (Notice that the slope from
1:
3327:{\displaystyle y_{11}-y_{12}}
3286:{\displaystyle y_{21}-y_{22}}
3238:{\displaystyle y_{11}-y_{21}}
3112:{\displaystyle y_{12}-y_{22}}
2646:{\displaystyle \gamma _{1}=0}
596:is the treatment effect, and
5968:NBER Working Paper No. 14237
5463:sector in New Jersey and in
5240:
5210:
2560:
2540:
2514:
2494:
2336:is the treatment group, and
2196:
2176:
2150:
2130:
2098:strict exogeneity assumption
2065:
2045:
2025:
2005:
1897:
1829:
1737:
1669:
1584:
1564:
1538:
1518:
1270:
946:
803:
682:
565:{\displaystyle \lambda _{t}}
44:experimental research design
5642:; Mullainathan, S. (2004).
5107: after period, control
5079: after period, control
5028: after period, control
2669:All the assumptions of the
2454:, giving the DID estimator
518:{\displaystyle \gamma _{s}}
6072:
5853:10.1177/1536867X1601600108
5781:10.1162/003355304772839588
5665:10.1162/003355304772839588
5594:Review of Economic Studies
3686:{\displaystyle (T\cdot S)}
2362:is the after period, then
2308:Without loss of generality
5308:, which is equivalent to
3661:. The composite variable
2675:parallel trend assumption
441:{\displaystyle I(\dots )}
20:Difference in differences
5910:10.1017/CBO9780511811241
5808:American Economic Review
5572:Synthetic control method
5567:Average treatment effect
5427:and in the above table.
4502:Synthetic control method
2388:{\displaystyle D_{22}=1}
471:{\displaystyle (\dots )}
48:observational study data
5834:Villa, Juan M. (2016).
5617:10.1111/0034-6527.00321
589:{\displaystyle \delta }
5414:
5302:
5282:
5259:
5152:
4930:
4894:
4858:
4822:
4786:
4750:
4668:
4632:
4596:
4570:
4534:
4490:
4454:
4428:
4402:
4355:
3719:
3687:
3655:
3629:
3609:
3589:
3563:
3543:
3520:
3407:
3328:
3287:
3239:
3198:
3197:{\displaystyle y_{11}}
3170:
3169:{\displaystyle y_{21}}
3142:
3113:
3072:
3071:{\displaystyle y_{12}}
3044:
3043:{\displaystyle y_{22}}
3016:
2984:
2957:
2930:
2929:{\displaystyle y_{st}}
2876:and what the value of
2855:
2763:
2737:
2711:
2666:
2647:
2614:
2613:{\displaystyle D_{st}}
2581:
2448:
2389:
2356:
2330:
2298:
2087:
1486:
1485:{\displaystyle D_{st}}
1456:
1424:
1389:
1328:
1243: treatment,
1164: treatment,
1134:
1004:
858:
740:
620:
590:
566:
539:
519:
492:
472:
448:is short-hand for the
442:
413:
399:is the group to which
393:
364:
344:
321:
320:{\displaystyle y_{it}}
288:
109:
91:cross-section estimate
6041:Design of experiments
5737:10.1353/obs.2020.0011
5724:Observational Studies
5562:Design of experiments
5415:
5303:
5283:
5260:
5132:
4931:
4895:
4859:
4823:
4787:
4751:
4669:
4633:
4597:
4571:
4535:
4491:
4455:
4429:
4403:
4356:
3720:
3718:{\displaystyle S=T=1}
3688:
3656:
3630:
3610:
3590:
3564:
3544:
3521:
3408:
3329:
3288:
3240:
3199:
3171:
3143:
3114:
3073:
3045:
3017:
2985:
2958:
2931:
2856:
2764:
2738:
2717:are the same in both
2712:
2664:
2648:
2615:
2582:
2449:
2390:
2357:
2331:
2299:
2088:
1487:
1457:
1455:{\displaystyle t=1,2}
1425:
1423:{\displaystyle s=1,2}
1390:
1308:
1251: in after period
1172: in after period
1114:
984:
838:
720:
621:
591:
567:
540:
520:
493:
473:
443:
414:
394:
365:
345:
322:
289:
108:
80:omitted variable bias
40:quantitative research
32:statistical technique
6031:Econometric modeling
5315:
5292:
5272:
4943:
4904:
4868:
4832:
4796:
4760:
4686:
4642:
4606:
4580:
4544:
4508:
4464:
4438:
4412:
4368:
3736:
3697:
3665:
3639:
3619:
3599:
3573:
3553:
3533:
3425:
3339:
3298:
3257:
3209:
3181:
3153:
3126:
3083:
3055:
3027:
3000:
2968:
2941:
2910:
2777:
2747:
2721:
2681:
2624:
2594:
2461:
2399:
2366:
2340:
2314:
2107:
1499:
1466:
1434:
1402:
640:
600:
580:
549:
529:
502:
482:
456:
423:
403:
392:{\displaystyle s(i)}
374:
354:
334:
301:
193:
87:time-series estimate
60:independent variable
6046:Observational study
6036:Regression analysis
4595:{\displaystyle T=1}
4453:{\displaystyle S=0}
4427:{\displaystyle T=1}
3654:{\displaystyle s=2}
3588:{\displaystyle t=2}
3141:{\displaystyle t=1}
3015:{\displaystyle t=2}
2983:{\displaystyle s=1}
2956:{\displaystyle s=2}
2762:{\displaystyle s=2}
2736:{\displaystyle s=1}
2355:{\displaystyle t=2}
2329:{\displaystyle s=2}
186:Consider the model
5955:10.1257/jel.47.1.5
5410:
5298:
5278:
5255:
5253:
4926:
4890:
4854:
4818:
4782:
4746:
4664:
4628:
4592:
4566:
4530:
4486:
4450:
4424:
4398:
4351:
4349:
3715:
3683:
3651:
3625:
3605:
3585:
3559:
3539:
3516:
3403:
3324:
3283:
3235:
3194:
3166:
3138:
3109:
3068:
3040:
3012:
2980:
2953:
2926:
2851:
2759:
2733:
2707:
2667:
2643:
2610:
2577:
2444:
2385:
2352:
2326:
2294:
2100:then implies that
2083:
2081:
1482:
1452:
1420:
1385:
1383:
616:
586:
562:
535:
515:
488:
468:
438:
409:
389:
360:
340:
317:
284:
110:
101:General definition
64:dependent variable
56:natural experiment
5892:978-0-691-12034-8
5840:The Stata Journal
5699:978-0-691-12034-8
5550:Juan Miguel Villa
5542:
5541:
5469:omitted variables
5425:formal definition
5345:
5339:
5328:
5301:{\displaystyle S}
5281:{\displaystyle T}
5248:
5243:
5218:
5213:
5199:
5196:
5183:
5167:
5114:
5108:
5099:
5080:
5070:
5059:
5029:
5014:
4985:
4959:
4917:
4881:
4845:
4809:
4773:
4655:
4619:
4557:
4521:
4477:
4380:
4324:
4298:
4273:
4247:
4198:
4172:
4147:
4121:
4089:
4062:
4036:
4011:
3985:
3960:
3933:
3907:
3882:
3856:
3831:
3804:
3778:
3753:
3727:formal definition
3628:{\displaystyle 1}
3608:{\displaystyle S}
3562:{\displaystyle 1}
3542:{\displaystyle T}
3439:
3433:
3416:
3415:
2801:
2795:
2771:formal definition
2769:. Given that the
2563:
2543:
2517:
2497:
2484:
2478:
2473:
2223:
2217:
2199:
2179:
2153:
2133:
2068:
2048:
2028:
2008:
1900:
1832:
1740:
1672:
1587:
1567:
1541:
1521:
1371:
1365:
1344:
1306:
1273:
1252:
1244:
1240:
1234:
1222:
1216:
1207:
1201:
1180:
1173:
1165:
1161:
1155:
1112:
1059:
1053:
1044:
1038:
1017:
982:
949:
922:
916:
907:
901:
880:
836:
806:
783:
777:
756:
718:
685:
667:
538:{\displaystyle s}
491:{\displaystyle Y}
412:{\displaystyle i}
363:{\displaystyle t}
343:{\displaystyle i}
217:
211:
182:Formal definition
85:In contrast to a
76:reverse causality
6063:
6051:Causal inference
6004:
6002:
5981:
5979:
5958:
5931:
5896:
5866:
5865:
5855:
5831:
5825:
5824:
5802:
5793:
5792:
5764:
5755:
5749:
5748:
5710:
5704:
5703:
5683:
5677:
5676:
5648:
5635:
5629:
5628:
5610:
5588:
5479:
5419:
5417:
5416:
5411:
5403:
5402:
5390:
5389:
5371:
5370:
5358:
5357:
5343:
5337:
5336:
5335:
5330:
5329:
5321:
5307:
5305:
5304:
5299:
5287:
5285:
5284:
5279:
5264:
5262:
5261:
5256:
5254:
5250:
5249:
5246:
5244:
5236:
5227:
5224:
5220:
5219:
5216:
5214:
5206:
5200:
5198:
5197:
5194:
5188:
5184:
5182: in control
5181:
5170:
5169:
5168:
5165:
5151:
5146:
5130:
5122:
5119:
5115:
5113:
5109:
5106:
5101:
5100:
5092:
5088:
5081:
5078:
5068:
5061:
5060:
5052:
5048:
5040:
5037:
5030:
5027:
5016:
5015:
5007:
4983:
4961:
4960:
4952:
4935:
4933:
4932:
4927:
4925:
4924:
4919:
4918:
4910:
4899:
4897:
4896:
4891:
4889:
4888:
4883:
4882:
4874:
4863:
4861:
4860:
4855:
4853:
4852:
4847:
4846:
4838:
4828:, implying that
4827:
4825:
4824:
4819:
4817:
4816:
4811:
4810:
4802:
4791:
4789:
4788:
4783:
4781:
4780:
4775:
4774:
4766:
4755:
4753:
4752:
4747:
4742:
4741:
4723:
4722:
4704:
4703:
4673:
4671:
4670:
4665:
4663:
4662:
4657:
4656:
4648:
4637:
4635:
4634:
4629:
4627:
4626:
4621:
4620:
4612:
4601:
4599:
4598:
4593:
4575:
4573:
4572:
4567:
4565:
4564:
4559:
4558:
4550:
4539:
4537:
4536:
4531:
4529:
4528:
4523:
4522:
4514:
4495:
4493:
4492:
4487:
4485:
4484:
4479:
4478:
4470:
4459:
4457:
4456:
4451:
4433:
4431:
4430:
4425:
4407:
4405:
4404:
4399:
4382:
4381:
4373:
4360:
4358:
4357:
4352:
4350:
4343:
4342:
4322:
4300:
4299:
4291:
4271:
4249:
4248:
4240:
4237:
4236:
4227:
4221:
4217:
4216:
4196:
4174:
4173:
4165:
4145:
4123:
4122:
4114:
4111:
4110:
4097:
4096:
4091:
4090:
4082:
4060:
4038:
4037:
4029:
4009:
3987:
3986:
3978:
3968:
3967:
3962:
3961:
3953:
3931:
3909:
3908:
3900:
3880:
3858:
3857:
3849:
3839:
3838:
3833:
3832:
3824:
3802:
3780:
3779:
3771:
3761:
3760:
3755:
3754:
3746:
3724:
3722:
3721:
3716:
3692:
3690:
3689:
3684:
3660:
3658:
3657:
3652:
3634:
3632:
3631:
3626:
3614:
3612:
3611:
3606:
3594:
3592:
3591:
3586:
3568:
3566:
3565:
3560:
3548:
3546:
3545:
3540:
3525:
3523:
3522:
3517:
3494:
3493:
3478:
3477:
3462:
3461:
3449:
3448:
3437:
3431:
3412:
3410:
3409:
3404:
3399:
3398:
3386:
3385:
3367:
3366:
3354:
3353:
3333:
3331:
3330:
3325:
3323:
3322:
3310:
3309:
3292:
3290:
3289:
3284:
3282:
3281:
3269:
3268:
3244:
3242:
3241:
3236:
3234:
3233:
3221:
3220:
3203:
3201:
3200:
3195:
3193:
3192:
3175:
3173:
3172:
3167:
3165:
3164:
3147:
3145:
3144:
3139:
3118:
3116:
3115:
3110:
3108:
3107:
3095:
3094:
3077:
3075:
3074:
3069:
3067:
3066:
3049:
3047:
3046:
3041:
3039:
3038:
3021:
3019:
3018:
3013:
2989:
2987:
2986:
2981:
2962:
2960:
2959:
2954:
2935:
2933:
2932:
2927:
2925:
2924:
2904:
2891:Ashenfelter dips
2860:
2858:
2857:
2852:
2850:
2849:
2837:
2836:
2824:
2823:
2811:
2810:
2799:
2793:
2792:
2791:
2768:
2766:
2765:
2760:
2742:
2740:
2739:
2734:
2716:
2714:
2713:
2708:
2706:
2705:
2693:
2692:
2652:
2650:
2649:
2644:
2636:
2635:
2619:
2617:
2616:
2611:
2609:
2608:
2586:
2584:
2583:
2578:
2570:
2569:
2564:
2556:
2550:
2549:
2544:
2536:
2524:
2523:
2518:
2510:
2504:
2503:
2498:
2490:
2482:
2476:
2475:
2474:
2466:
2453:
2451:
2450:
2445:
2437:
2436:
2424:
2423:
2411:
2410:
2394:
2392:
2391:
2386:
2378:
2377:
2361:
2359:
2358:
2353:
2335:
2333:
2332:
2327:
2303:
2301:
2300:
2295:
2287:
2286:
2274:
2273:
2252:
2251:
2239:
2238:
2221:
2215:
2214:
2210:
2206:
2205:
2200:
2192:
2186:
2185:
2180:
2172:
2160:
2159:
2154:
2146:
2140:
2139:
2134:
2126:
2092:
2090:
2089:
2084:
2082:
2075:
2074:
2069:
2061:
2055:
2054:
2049:
2041:
2035:
2034:
2029:
2021:
2015:
2014:
2009:
2001:
1992:
1991:
1979:
1978:
1957:
1956:
1944:
1943:
1926:
1917:
1916:
1907:
1906:
1901:
1893:
1887:
1886:
1871:
1870:
1858:
1857:
1839:
1838:
1833:
1825:
1819:
1818:
1803:
1802:
1790:
1789:
1777:
1776:
1767:
1761:
1757:
1756:
1747:
1746:
1741:
1733:
1727:
1726:
1711:
1710:
1698:
1697:
1679:
1678:
1673:
1665:
1659:
1658:
1643:
1642:
1630:
1629:
1617:
1616:
1606:
1594:
1593:
1588:
1580:
1574:
1573:
1568:
1560:
1548:
1547:
1542:
1534:
1528:
1527:
1522:
1514:
1505:
1491:
1489:
1488:
1483:
1481:
1480:
1461:
1459:
1458:
1453:
1429:
1427:
1426:
1421:
1394:
1392:
1391:
1386:
1384:
1369:
1363:
1342:
1341:
1340:
1327:
1322:
1307:
1305:
1304:
1292:
1283:
1282:
1274:
1266:
1253:
1250:
1245:
1242:
1238:
1232:
1220:
1214:
1205:
1199:
1178:
1174:
1171:
1166:
1163:
1159:
1153:
1133:
1128:
1113:
1111:
1110:
1098:
1089:
1088:
1069:
1068:
1057:
1051:
1042:
1036:
1015:
1014:
1013:
1003:
998:
983:
981:
980:
968:
959:
958:
950:
942:
932:
931:
920:
914:
905:
899:
878:
877:
876:
857:
852:
837:
835:
834:
822:
813:
812:
807:
799:
781:
775:
754:
753:
752:
739:
734:
719:
717:
716:
704:
695:
694:
686:
678:
668:
665:
656:
655:
625:
623:
622:
617:
615:
614:
595:
593:
592:
587:
571:
569:
568:
563:
561:
560:
544:
542:
541:
536:
524:
522:
521:
516:
514:
513:
497:
495:
494:
489:
477:
475:
474:
469:
447:
445:
444:
439:
418:
416:
415:
410:
398:
396:
395:
390:
369:
367:
366:
361:
349:
347:
346:
341:
326:
324:
323:
318:
316:
315:
293:
291:
290:
285:
283:
282:
249:
248:
236:
235:
215:
209:
208:
207:
6071:
6070:
6066:
6065:
6064:
6062:
6061:
6060:
6021:
6020:
6011:
5984:
5961:
5934:
5920:
5899:
5893:
5878:
5875:
5873:Further reading
5870:
5869:
5833:
5832:
5828:
5804:
5803:
5796:
5762:
5757:
5756:
5752:
5713:Basu, Pallavi;
5712:
5711:
5707:
5700:
5685:
5684:
5680:
5646:
5637:
5636:
5632:
5608:10.1.1.470.1475
5590:
5589:
5585:
5580:
5558:
5433:
5394:
5381:
5362:
5349:
5318:
5313:
5312:
5290:
5289:
5270:
5269:
5252:
5251:
5234:
5225:
5222:
5221:
5204:
5189:
5153:
5131:
5120:
5117:
5116:
5089:
5049:
5038:
5035:
5034:
4998:
4941:
4940:
4907:
4902:
4901:
4871:
4866:
4865:
4835:
4830:
4829:
4799:
4794:
4793:
4763:
4758:
4757:
4727:
4708:
4695:
4684:
4683:
4645:
4640:
4639:
4609:
4604:
4603:
4578:
4577:
4547:
4542:
4541:
4511:
4506:
4505:
4467:
4462:
4461:
4436:
4435:
4410:
4409:
4366:
4365:
4348:
4347:
4219:
4218:
4098:
4079:
4076:
4075:
3969:
3950:
3947:
3946:
3840:
3821:
3818:
3817:
3762:
3743:
3734:
3733:
3695:
3694:
3663:
3662:
3637:
3636:
3617:
3616:
3597:
3596:
3571:
3570:
3551:
3550:
3531:
3530:
3485:
3469:
3453:
3440:
3423:
3422:
3390:
3377:
3358:
3345:
3337:
3336:
3314:
3301:
3296:
3295:
3273:
3260:
3255:
3254:
3225:
3212:
3207:
3206:
3184:
3179:
3178:
3156:
3151:
3150:
3124:
3123:
3099:
3086:
3081:
3080:
3058:
3053:
3052:
3030:
3025:
3024:
2998:
2997:
2966:
2965:
2939:
2938:
2913:
2908:
2907:
2899:
2887:autocorrelation
2841:
2828:
2815:
2802:
2780:
2775:
2774:
2745:
2744:
2719:
2718:
2697:
2684:
2679:
2678:
2659:
2627:
2622:
2621:
2597:
2592:
2591:
2554:
2534:
2508:
2488:
2459:
2458:
2428:
2415:
2402:
2397:
2396:
2369:
2364:
2363:
2338:
2337:
2312:
2311:
2278:
2265:
2243:
2230:
2190:
2170:
2144:
2124:
2120:
2116:
2105:
2104:
2080:
2079:
2059:
2039:
2019:
1999:
1983:
1970:
1948:
1935:
1927:
1919:
1918:
1891:
1878:
1862:
1849:
1823:
1810:
1794:
1781:
1759:
1758:
1731:
1718:
1702:
1689:
1663:
1650:
1634:
1621:
1607:
1599:
1598:
1578:
1558:
1532:
1512:
1497:
1496:
1469:
1464:
1463:
1432:
1431:
1400:
1399:
1382:
1381:
1329:
1296:
1284:
1264:
1261:
1260:
1102:
1090:
1077:
1074:
1073:
1060:
1005:
972:
960:
940:
937:
936:
923:
859:
826:
814:
797:
794:
793:
741:
708:
696:
676:
673:
672:
657:
647:
638:
637:
603:
598:
597:
578:
577:
552:
547:
546:
527:
526:
505:
500:
499:
480:
479:
454:
453:
421:
420:
401:
400:
372:
371:
352:
351:
332:
331:
304:
299:
298:
271:
240:
218:
196:
191:
190:
184:
177:
173:
169:
162:
151:
140:
133:
126:
119:
103:
72:mean regression
17:
12:
11:
5:
6069:
6067:
6059:
6058:
6053:
6048:
6043:
6038:
6033:
6023:
6022:
6019:
6018:
6010:
6009:External links
6007:
6006:
6005:
5982:
5977:10.3386/w14237
5959:
5932:
5918:
5897:
5891:
5874:
5871:
5868:
5867:
5826:
5815:(4): 772–793.
5794:
5775:(1): 249–275.
5750:
5705:
5698:
5678:
5659:(1): 249–275.
5638:Bertrand, M.;
5630:
5582:
5581:
5579:
5576:
5575:
5574:
5569:
5564:
5557:
5554:
5540:
5539:
5536:
5533:
5530:
5524:
5523:
5520:
5517:
5514:
5508:
5507:
5504:
5501:
5498:
5492:
5491:
5488:
5485:
5482:
5432:
5429:
5421:
5420:
5409:
5406:
5401:
5397:
5393:
5388:
5384:
5380:
5377:
5374:
5369:
5365:
5361:
5356:
5352:
5348:
5342:
5334:
5327:
5324:
5297:
5277:
5266:
5265:
5242:
5239:
5233:
5230:
5228:
5226:
5223:
5217:control, after
5212:
5209:
5203:
5192:
5187:
5179:
5176:
5173:
5163:
5160:
5156:
5150:
5145:
5142:
5139:
5135:
5128:
5125:
5123:
5121:
5118:
5112:
5104:
5098:
5095:
5087:
5084:
5076:
5073:
5067:
5064:
5058:
5055:
5046:
5043:
5041:
5039:
5036:
5033:
5025:
5022:
5019:
5013:
5010:
5004:
5001:
4999:
4997:
4994:
4991:
4988:
4982:
4979:
4976:
4973:
4970:
4967:
4964:
4958:
4955:
4949:
4948:
4923:
4916:
4913:
4887:
4880:
4877:
4851:
4844:
4841:
4815:
4808:
4805:
4779:
4772:
4769:
4745:
4740:
4737:
4734:
4730:
4726:
4721:
4718:
4715:
4711:
4707:
4702:
4698:
4694:
4691:
4661:
4654:
4651:
4625:
4618:
4615:
4591:
4588:
4585:
4563:
4556:
4553:
4527:
4520:
4517:
4498:counterfactual
4483:
4476:
4473:
4449:
4446:
4443:
4423:
4420:
4417:
4397:
4394:
4391:
4388:
4385:
4379:
4376:
4362:
4361:
4346:
4341:
4336:
4333:
4330:
4327:
4321:
4318:
4315:
4312:
4309:
4306:
4303:
4297:
4294:
4288:
4285:
4282:
4279:
4276:
4270:
4267:
4264:
4261:
4258:
4255:
4252:
4246:
4243:
4235:
4230:
4224:
4222:
4220:
4215:
4210:
4207:
4204:
4201:
4195:
4192:
4189:
4186:
4183:
4180:
4177:
4171:
4168:
4162:
4159:
4156:
4153:
4150:
4144:
4141:
4138:
4135:
4132:
4129:
4126:
4120:
4117:
4109:
4104:
4101:
4099:
4095:
4088:
4085:
4078:
4077:
4074:
4071:
4068:
4065:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4035:
4032:
4026:
4023:
4020:
4017:
4014:
4008:
4005:
4002:
3999:
3996:
3993:
3990:
3984:
3981:
3975:
3972:
3970:
3966:
3959:
3956:
3949:
3948:
3945:
3942:
3939:
3936:
3930:
3927:
3924:
3921:
3918:
3915:
3912:
3906:
3903:
3897:
3894:
3891:
3888:
3885:
3879:
3876:
3873:
3870:
3867:
3864:
3861:
3855:
3852:
3846:
3843:
3841:
3837:
3830:
3827:
3820:
3819:
3816:
3813:
3810:
3807:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3777:
3774:
3768:
3765:
3763:
3759:
3752:
3749:
3742:
3741:
3714:
3711:
3708:
3705:
3702:
3682:
3679:
3676:
3673:
3670:
3650:
3647:
3644:
3624:
3604:
3584:
3581:
3578:
3558:
3538:
3527:
3526:
3515:
3512:
3509:
3506:
3503:
3500:
3497:
3492:
3488:
3484:
3481:
3476:
3472:
3468:
3465:
3460:
3456:
3452:
3447:
3443:
3436:
3430:
3414:
3413:
3402:
3397:
3393:
3389:
3384:
3380:
3376:
3373:
3370:
3365:
3361:
3357:
3352:
3348:
3344:
3334:
3321:
3317:
3313:
3308:
3304:
3293:
3280:
3276:
3272:
3267:
3263:
3252:
3246:
3245:
3232:
3228:
3224:
3219:
3215:
3204:
3191:
3187:
3176:
3163:
3159:
3148:
3137:
3134:
3131:
3120:
3119:
3106:
3102:
3098:
3093:
3089:
3078:
3065:
3061:
3050:
3037:
3033:
3022:
3011:
3008:
3005:
2994:
2993:
2990:
2979:
2976:
2973:
2963:
2952:
2949:
2946:
2936:
2923:
2920:
2916:
2898:
2897:Implementation
2895:
2848:
2844:
2840:
2835:
2831:
2827:
2822:
2818:
2814:
2809:
2805:
2798:
2790:
2787:
2783:
2758:
2755:
2752:
2732:
2729:
2726:
2704:
2700:
2696:
2691:
2687:
2658:
2655:
2642:
2639:
2634:
2630:
2607:
2604:
2600:
2588:
2587:
2576:
2573:
2568:
2562:
2559:
2553:
2548:
2542:
2539:
2533:
2530:
2527:
2522:
2516:
2513:
2507:
2502:
2496:
2493:
2487:
2481:
2472:
2469:
2443:
2440:
2435:
2431:
2427:
2422:
2418:
2414:
2409:
2405:
2384:
2381:
2376:
2372:
2351:
2348:
2345:
2325:
2322:
2319:
2310:, assume that
2305:
2304:
2293:
2290:
2285:
2281:
2277:
2272:
2268:
2264:
2261:
2258:
2255:
2250:
2246:
2242:
2237:
2233:
2229:
2226:
2220:
2213:
2209:
2204:
2198:
2195:
2189:
2184:
2178:
2175:
2169:
2166:
2163:
2158:
2152:
2149:
2143:
2138:
2132:
2129:
2123:
2119:
2115:
2112:
2094:
2093:
2078:
2073:
2067:
2064:
2058:
2053:
2047:
2044:
2038:
2033:
2027:
2024:
2018:
2013:
2007:
2004:
1998:
1995:
1990:
1986:
1982:
1977:
1973:
1969:
1966:
1963:
1960:
1955:
1951:
1947:
1942:
1938:
1934:
1931:
1928:
1924:
1921:
1920:
1915:
1910:
1905:
1899:
1896:
1890:
1885:
1881:
1877:
1874:
1869:
1865:
1861:
1856:
1852:
1848:
1845:
1842:
1837:
1831:
1828:
1822:
1817:
1813:
1809:
1806:
1801:
1797:
1793:
1788:
1784:
1780:
1775:
1770:
1764:
1762:
1760:
1755:
1750:
1745:
1739:
1736:
1730:
1725:
1721:
1717:
1714:
1709:
1705:
1701:
1696:
1692:
1688:
1685:
1682:
1677:
1671:
1668:
1662:
1657:
1653:
1649:
1646:
1641:
1637:
1633:
1628:
1624:
1620:
1615:
1610:
1608:
1604:
1601:
1600:
1597:
1592:
1586:
1583:
1577:
1572:
1566:
1563:
1557:
1554:
1551:
1546:
1540:
1537:
1531:
1526:
1520:
1517:
1511:
1508:
1506:
1504:
1479:
1476:
1472:
1451:
1448:
1445:
1442:
1439:
1419:
1416:
1413:
1410:
1407:
1396:
1395:
1380:
1377:
1374:
1368:
1362:
1359:
1356:
1353:
1350:
1347:
1339:
1336:
1332:
1326:
1321:
1318:
1315:
1311:
1303:
1299:
1295:
1290:
1287:
1285:
1281:
1278:
1272:
1269:
1263:
1262:
1259:
1256:
1248:
1237:
1231:
1228:
1225:
1219:
1213:
1210:
1204:
1198:
1195:
1192:
1189:
1186:
1183:
1177:
1169:
1158:
1152:
1149:
1146:
1143:
1140:
1137:
1132:
1127:
1124:
1121:
1117:
1109:
1105:
1101:
1096:
1093:
1091:
1087:
1084:
1080:
1076:
1075:
1072:
1067:
1063:
1056:
1050:
1047:
1041:
1035:
1032:
1029:
1026:
1023:
1020:
1012:
1008:
1002:
997:
994:
991:
987:
979:
975:
971:
966:
963:
961:
957:
954:
948:
945:
939:
938:
935:
930:
926:
919:
913:
910:
904:
898:
895:
892:
889:
886:
883:
875:
872:
869:
866:
862:
856:
851:
848:
845:
841:
833:
829:
825:
820:
817:
815:
811:
805:
802:
796:
795:
792:
789:
786:
780:
774:
771:
768:
765:
762:
759:
751:
748:
744:
738:
733:
730:
727:
723:
715:
711:
707:
702:
699:
697:
693:
690:
684:
681:
675:
674:
671:
663:
660:
658:
654:
650:
646:
645:
613:
610:
606:
585:
559:
555:
534:
512:
508:
487:
467:
464:
461:
450:dummy variable
437:
434:
431:
428:
408:
388:
385:
382:
379:
359:
339:
314:
311:
307:
295:
294:
281:
278:
274:
270:
267:
264:
261:
258:
255:
252:
247:
243:
239:
234:
231:
228:
225:
221:
214:
206:
203:
199:
183:
180:
175:
171:
167:
160:
149:
138:
131:
124:
117:
102:
99:
68:selection bias
15:
13:
10:
9:
6:
4:
3:
2:
6068:
6057:
6054:
6052:
6049:
6047:
6044:
6042:
6039:
6037:
6034:
6032:
6029:
6028:
6026:
6016:
6013:
6012:
6008:
6001:
6000:10.3386/t0312
5996:
5992:
5988:
5983:
5978:
5973:
5969:
5965:
5960:
5956:
5952:
5948:
5944:
5943:
5938:
5933:
5929:
5925:
5921:
5919:9780521848053
5915:
5911:
5907:
5903:
5898:
5894:
5888:
5884:
5883:
5877:
5876:
5872:
5863:
5859:
5854:
5849:
5845:
5841:
5837:
5830:
5827:
5822:
5818:
5814:
5810:
5809:
5801:
5799:
5795:
5790:
5786:
5782:
5778:
5774:
5770:
5769:
5761:
5754:
5751:
5746:
5742:
5738:
5734:
5730:
5726:
5725:
5720:
5716:
5709:
5706:
5701:
5695:
5691:
5690:
5682:
5679:
5674:
5670:
5666:
5662:
5658:
5654:
5653:
5645:
5641:
5634:
5631:
5626:
5622:
5618:
5614:
5609:
5604:
5600:
5596:
5595:
5587:
5584:
5577:
5573:
5570:
5568:
5565:
5563:
5560:
5559:
5555:
5553:
5551:
5547:
5537:
5534:
5531:
5529:
5526:
5525:
5521:
5518:
5515:
5513:
5510:
5509:
5505:
5502:
5499:
5497:
5494:
5493:
5489:
5486:
5483:
5481:
5480:
5477:
5475:
5470:
5466:
5462:
5458:
5454:
5450:
5446:
5442:
5438:
5430:
5428:
5426:
5407:
5399:
5395:
5391:
5386:
5382:
5375:
5367:
5363:
5359:
5354:
5350:
5340:
5332:
5322:
5311:
5310:
5309:
5295:
5275:
5237:
5231:
5229:
5207:
5201:
5190:
5177:
5171:
5161:
5158:
5154:
5148:
5143:
5140:
5137:
5133:
5126:
5124:
5096:
5093:
5071:
5065:
5056:
5053:
5044:
5042:
5023:
5020:
5011:
5008:
5002:
5000:
4992:
4989:
4986:
4980:
4977:
4974:
4971:
4968:
4965:
4956:
4953:
4939:
4938:
4937:
4921:
4911:
4885:
4875:
4849:
4839:
4813:
4803:
4777:
4767:
4738:
4735:
4732:
4728:
4724:
4719:
4716:
4713:
4709:
4705:
4700:
4696:
4681:
4677:
4659:
4649:
4623:
4613:
4589:
4586:
4583:
4561:
4551:
4525:
4515:
4503:
4499:
4481:
4471:
4447:
4444:
4441:
4421:
4418:
4415:
4392:
4389:
4386:
4377:
4374:
4344:
4331:
4328:
4325:
4319:
4316:
4313:
4310:
4307:
4304:
4295:
4292:
4286:
4280:
4277:
4274:
4268:
4265:
4262:
4259:
4256:
4253:
4244:
4241:
4228:
4223:
4205:
4202:
4199:
4193:
4190:
4187:
4184:
4181:
4178:
4169:
4166:
4160:
4154:
4151:
4148:
4142:
4139:
4136:
4133:
4130:
4127:
4118:
4115:
4102:
4100:
4093:
4083:
4069:
4066:
4063:
4057:
4054:
4051:
4048:
4045:
4042:
4033:
4030:
4024:
4018:
4015:
4012:
4006:
4003:
4000:
3997:
3994:
3991:
3982:
3979:
3973:
3971:
3964:
3954:
3940:
3937:
3934:
3928:
3925:
3922:
3919:
3916:
3913:
3904:
3901:
3895:
3889:
3886:
3883:
3877:
3874:
3871:
3868:
3865:
3862:
3853:
3850:
3844:
3842:
3835:
3825:
3811:
3808:
3805:
3799:
3796:
3793:
3790:
3787:
3784:
3775:
3772:
3766:
3764:
3757:
3747:
3732:
3731:
3730:
3728:
3712:
3709:
3706:
3703:
3700:
3677:
3674:
3671:
3648:
3645:
3642:
3622:
3602:
3582:
3579:
3576:
3556:
3536:
3513:
3510:
3504:
3501:
3498:
3490:
3486:
3482:
3479:
3474:
3470:
3466:
3463:
3458:
3454:
3450:
3445:
3441:
3434:
3428:
3421:
3420:
3419:
3395:
3391:
3387:
3382:
3378:
3371:
3363:
3359:
3355:
3350:
3346:
3335:
3319:
3315:
3311:
3306:
3302:
3294:
3278:
3274:
3270:
3265:
3261:
3253:
3251:
3248:
3247:
3230:
3226:
3222:
3217:
3213:
3205:
3189:
3185:
3177:
3161:
3157:
3149:
3135:
3132:
3129:
3122:
3121:
3104:
3100:
3096:
3091:
3087:
3079:
3063:
3059:
3051:
3035:
3031:
3023:
3009:
3006:
3003:
2996:
2995:
2991:
2977:
2974:
2971:
2964:
2950:
2947:
2944:
2937:
2921:
2918:
2914:
2906:
2905:
2902:
2896:
2894:
2892:
2888:
2882:
2879:
2875:
2870:
2868:
2864:
2846:
2842:
2838:
2833:
2829:
2825:
2820:
2816:
2812:
2807:
2803:
2796:
2788:
2785:
2781:
2772:
2756:
2753:
2750:
2730:
2727:
2724:
2702:
2698:
2694:
2689:
2685:
2676:
2672:
2663:
2656:
2654:
2640:
2637:
2632:
2628:
2605:
2602:
2598:
2574:
2566:
2557:
2551:
2546:
2537:
2528:
2520:
2511:
2505:
2500:
2491:
2479:
2467:
2457:
2456:
2455:
2441:
2438:
2433:
2429:
2425:
2420:
2416:
2412:
2407:
2403:
2382:
2379:
2374:
2370:
2349:
2346:
2343:
2323:
2320:
2317:
2309:
2291:
2283:
2279:
2275:
2270:
2266:
2259:
2256:
2248:
2244:
2240:
2235:
2231:
2224:
2218:
2211:
2202:
2193:
2187:
2182:
2173:
2164:
2156:
2147:
2141:
2136:
2127:
2117:
2113:
2103:
2102:
2101:
2099:
2076:
2071:
2062:
2056:
2051:
2042:
2036:
2031:
2022:
2016:
2011:
2002:
1996:
1988:
1984:
1980:
1975:
1971:
1964:
1961:
1953:
1949:
1945:
1940:
1936:
1929:
1922:
1903:
1894:
1888:
1883:
1879:
1875:
1872:
1867:
1863:
1859:
1854:
1850:
1843:
1835:
1826:
1820:
1815:
1811:
1807:
1804:
1799:
1795:
1791:
1786:
1782:
1768:
1763:
1743:
1734:
1728:
1723:
1719:
1715:
1712:
1707:
1703:
1699:
1694:
1690:
1683:
1675:
1666:
1660:
1655:
1651:
1647:
1644:
1639:
1635:
1631:
1626:
1622:
1609:
1602:
1590:
1581:
1575:
1570:
1561:
1552:
1544:
1535:
1529:
1524:
1515:
1507:
1495:
1494:
1493:
1477:
1474:
1470:
1449:
1446:
1443:
1440:
1437:
1417:
1414:
1411:
1408:
1405:
1378:
1372:
1366:
1357:
1351:
1345:
1337:
1334:
1330:
1324:
1319:
1316:
1313:
1309:
1301:
1297:
1293:
1288:
1286:
1279:
1276:
1267:
1257:
1246:
1235:
1229:
1223:
1217:
1208:
1202:
1193:
1187:
1181:
1167:
1156:
1147:
1141:
1135:
1130:
1125:
1122:
1119:
1115:
1107:
1103:
1099:
1094:
1092:
1085:
1082:
1078:
1070:
1065:
1061:
1054:
1045:
1039:
1030:
1024:
1018:
1010:
1006:
1000:
995:
992:
989:
985:
977:
973:
969:
964:
962:
955:
952:
943:
933:
928:
924:
917:
908:
902:
893:
887:
881:
870:
864:
860:
854:
849:
846:
843:
839:
831:
827:
823:
818:
816:
809:
800:
790:
784:
778:
769:
763:
757:
749:
746:
742:
736:
731:
728:
725:
721:
713:
709:
705:
700:
698:
691:
688:
679:
669:
661:
659:
652:
648:
636:
635:
634:
631:
629:
628:residual term
611:
608:
604:
583:
575:
557:
553:
532:
510:
506:
485:
462:
451:
432:
426:
406:
383:
377:
357:
337:
330:
312:
309:
305:
279:
276:
272:
268:
262:
256:
253:
250:
245:
241:
237:
229:
223:
219:
212:
204:
201:
197:
189:
188:
187:
181:
179:
166:
159:
155:
148:
144:
137:
130:
123:
116:
107:
100:
98:
96:
92:
88:
83:
81:
77:
73:
69:
65:
61:
57:
53:
52:control group
49:
45:
41:
37:
33:
29:
25:
21:
5990:
5967:
5946:
5940:
5901:
5881:
5846:(1): 52–71.
5843:
5839:
5829:
5812:
5806:
5772:
5766:
5753:
5728:
5722:
5715:Small, Dylan
5708:
5688:
5681:
5656:
5650:
5633:
5598:
5592:
5586:
5543:
5527:
5511:
5495:
5487:Pennsylvania
5465:Pennsylvania
5445:minimum wage
5434:
5422:
5267:
4363:
3528:
3417:
3249:
2900:
2883:
2877:
2873:
2871:
2674:
2668:
2589:
2306:
2095:
1462:. Note that
1397:
632:
296:
185:
164:
157:
153:
146:
142:
135:
128:
121:
114:
111:
84:
36:econometrics
27:
23:
19:
18:
6056:Subtraction
5949:(1): 5–86.
5731:: 103–130.
5601:(1): 1–19.
5490:Difference
5443:article on
2992:Difference
2657:Assumptions
574:Assumptions
6025:Categories
5578:References
5484:New Jersey
5457:employment
5449:New Jersey
498:by group,
329:individual
95:panel data
5928:120313863
5862:124464636
5745:221702893
5640:Duflo, E.
5603:CiteSeerX
5461:fast food
5392:−
5376:−
5360:−
5326:^
5323:β
5241:¯
5211:¯
5134:∑
5097:^
5057:^
5024:∣
5012:^
4969:∣
4957:^
4915:^
4912:β
4879:^
4876:β
4843:^
4840:β
4807:^
4804:β
4771:^
4768:β
4725:−
4693:Δ
4653:^
4650:β
4617:^
4614:β
4555:^
4552:β
4519:^
4516:β
4475:^
4472:β
4393:…
4390:∣
4387:⋯
4378:^
4308:∣
4296:^
4287:−
4257:∣
4245:^
4229:−
4182:∣
4170:^
4161:−
4131:∣
4119:^
4087:^
4084:β
4046:∣
4034:^
4025:−
3995:∣
3983:^
3958:^
3955:β
3917:∣
3905:^
3896:−
3866:∣
3854:^
3829:^
3826:β
3788:∣
3776:^
3751:^
3748:β
3675:⋅
3514:ε
3502:⋅
3487:β
3471:β
3455:β
3442:β
3388:−
3372:−
3356:−
3312:−
3271:−
3223:−
3097:−
2843:λ
2839:−
2830:λ
2826:≠
2817:λ
2813:−
2804:λ
2782:λ
2699:λ
2695:−
2686:λ
2671:OLS model
2629:γ
2561:¯
2552:−
2541:¯
2529:−
2515:¯
2506:−
2495:¯
2471:^
2468:δ
2276:−
2260:δ
2241:−
2225:δ
2197:¯
2188:−
2177:¯
2165:−
2151:¯
2142:−
2131:¯
2114:
2066:¯
2063:ε
2057:−
2046:¯
2043:ε
2026:¯
2023:ε
2017:−
2006:¯
2003:ε
1981:−
1965:δ
1946:−
1930:δ
1898:¯
1895:ε
1876:δ
1864:λ
1851:γ
1844:−
1830:¯
1827:ε
1808:δ
1796:λ
1783:γ
1769:−
1738:¯
1735:ε
1716:δ
1704:λ
1691:γ
1684:−
1670:¯
1667:ε
1648:δ
1636:λ
1623:γ
1585:¯
1576:−
1565:¯
1553:−
1539:¯
1530:−
1519:¯
1331:ε
1310:∑
1271:¯
1268:ε
1116:∑
1062:λ
1007:λ
986:∑
947:¯
944:λ
925:γ
861:γ
840:∑
804:¯
801:γ
722:∑
683:¯
605:ε
584:δ
554:λ
507:γ
463:…
433:…
350:and time
273:ε
263:…
254:δ
242:λ
220:γ
5717:(2020).
5556:See also
5512:November
5496:February
4674:. As in
2863:matching
576:below).
178:and Q).
34:used in
5821:2118030
5625:8801460
5459:in the
5441:Krueger
5431:Example
5195:control
4680:Krueger
626:is the
54:' in a
30:) is a
5926:
5916:
5889:
5860:
5819:
5789:470667
5787:
5743:
5696:
5673:470667
5671:
5623:
5605:
5528:Change
5522:−0.14
5506:−2.89
5344:
5338:
5069:
4984:
4500:(see,
4364:where
4323:
4272:
4197:
4146:
4061:
4010:
3932:
3881:
3803:
3595:, and
3529:where
3438:
3432:
3250:Change
2800:
2794:
2483:
2477:
2222:
2216:
1370:
1364:
1343:
1239:
1233:
1221:
1215:
1206:
1200:
1179:
1160:
1154:
1058:
1052:
1043:
1037:
1016:
921:
915:
906:
900:
879:
782:
776:
755:
545:, and
297:where
216:
210:
46:using
5924:S2CID
5858:S2CID
5817:JSTOR
5785:S2CID
5763:(PDF)
5741:S2CID
5669:S2CID
5647:(PDF)
5621:S2CID
5546:Stata
5538:2.75
5535:−2.16
5519:21.17
5516:21.03
5503:23.33
5500:20.44
5166:after
3635:when
3569:when
5914:ISBN
5887:ISBN
5694:ISBN
5532:0.59
5439:and
5437:Card
5435:The
5288:and
4678:and
4676:Card
2889:and
2743:and
2395:and
2096:The
1430:and
134:and
120:and
78:and
38:and
5995:doi
5972:doi
5951:doi
5906:doi
5848:doi
5777:doi
5773:119
5733:doi
5661:doi
5657:119
5613:doi
5447:in
2867:ATE
163:to
152:to
82:).
26:or
24:DID
6027::
5993:.
5989:.
5970:.
5966:.
5947:47
5945:.
5939:.
5922:.
5912:.
5856:.
5844:16
5842:.
5838:.
5813:84
5811:.
5797:^
5783:.
5771:.
5765:.
5739:.
5727:.
5721:.
5667:.
5655:.
5649:.
5619:.
5611:.
5599:72
5597:.
5552:.
5400:22
5387:12
5368:21
5355:11
5247:12
3396:22
3383:12
3364:21
3351:11
3320:12
3307:11
3279:22
3266:21
3231:21
3218:11
3190:11
3162:21
3105:22
3092:12
3064:12
3036:22
2847:11
2834:12
2821:21
2808:22
2653:.
2567:22
2547:21
2521:12
2501:11
2434:21
2421:12
2408:11
2375:22
2284:21
2271:22
2249:12
2236:11
2203:22
2183:21
2157:12
2137:11
2072:21
2052:22
2032:12
2012:11
1989:21
1976:22
1954:12
1941:11
1904:22
1884:22
1836:21
1816:21
1744:12
1724:12
1676:11
1656:11
1591:22
1571:21
1545:12
1525:11
630:.
370:,
74:,
28:DD
6003:.
5997::
5980:.
5974::
5957:.
5953::
5930:.
5908::
5895:.
5864:.
5850::
5823:.
5791:.
5779::
5747:.
5735::
5729:6
5702:.
5675:.
5663::
5627:.
5615::
5408:.
5405:)
5396:y
5383:y
5379:(
5373:)
5364:y
5351:y
5347:(
5341:=
5333:3
5296:S
5276:T
5238:y
5232:=
5208:y
5202:=
5191:n
5186:)
5178:i
5175:(
5172:I
5162:,
5159:i
5155:y
5149:n
5144:1
5141:=
5138:i
5127:=
5111:)
5103:(
5094:P
5086:)
5083:)
5075:(
5072:I
5066:y
5063:(
5054:E
5045:=
5032:)
5021:y
5018:(
5009:E
5003:=
4996:)
4993:0
4990:=
4987:S
4981:,
4978:1
4975:=
4972:T
4966:y
4963:(
4954:E
4922:3
4886:2
4850:1
4814:3
4778:1
4744:)
4739:0
4736:,
4733:i
4729:Y
4720:1
4717:,
4714:i
4710:Y
4706:=
4701:i
4697:Y
4690:(
4660:2
4624:1
4590:1
4587:=
4584:T
4562:2
4526:1
4482:1
4448:0
4445:=
4442:S
4422:1
4419:=
4416:T
4396:)
4384:(
4375:E
4345:,
4340:]
4335:)
4332:0
4329:=
4326:S
4320:,
4317:0
4314:=
4311:T
4305:y
4302:(
4293:E
4284:)
4281:0
4278:=
4275:S
4269:,
4266:1
4263:=
4260:T
4254:y
4251:(
4242:E
4234:[
4214:]
4209:)
4206:1
4203:=
4200:S
4194:,
4191:0
4188:=
4185:T
4179:y
4176:(
4167:E
4158:)
4155:1
4152:=
4149:S
4143:,
4140:1
4137:=
4134:T
4128:y
4125:(
4116:E
4108:[
4103:=
4094:3
4073:)
4070:0
4067:=
4064:S
4058:,
4055:0
4052:=
4049:T
4043:y
4040:(
4031:E
4022:)
4019:1
4016:=
4013:S
4007:,
4004:0
4001:=
3998:T
3992:y
3989:(
3980:E
3974:=
3965:2
3944:)
3941:0
3938:=
3935:S
3929:,
3926:0
3923:=
3920:T
3914:y
3911:(
3902:E
3893:)
3890:0
3887:=
3884:S
3878:,
3875:1
3872:=
3869:T
3863:y
3860:(
3851:E
3845:=
3836:1
3815:)
3812:0
3809:=
3806:S
3800:,
3797:0
3794:=
3791:T
3785:y
3782:(
3773:E
3767:=
3758:0
3713:1
3710:=
3707:T
3704:=
3701:S
3681:)
3678:S
3672:T
3669:(
3649:2
3646:=
3643:s
3623:1
3603:S
3583:2
3580:=
3577:t
3557:1
3537:T
3511:+
3508:)
3505:S
3499:T
3496:(
3491:3
3483:+
3480:S
3475:2
3467:+
3464:T
3459:1
3451:+
3446:0
3435:=
3429:y
3401:)
3392:y
3379:y
3375:(
3369:)
3360:y
3347:y
3343:(
3316:y
3303:y
3275:y
3262:y
3227:y
3214:y
3186:y
3158:y
3136:1
3133:=
3130:t
3101:y
3088:y
3060:y
3032:y
3010:2
3007:=
3004:t
2978:1
2975:=
2972:s
2951:2
2948:=
2945:s
2922:t
2919:s
2915:y
2878:y
2874:y
2797::
2789:t
2786:s
2757:2
2754:=
2751:s
2731:1
2728:=
2725:s
2703:1
2690:2
2641:0
2638:=
2633:1
2606:t
2603:s
2599:D
2575:,
2572:)
2558:y
2538:y
2532:(
2526:)
2512:y
2492:y
2486:(
2480:=
2442:0
2439:=
2430:D
2426:=
2417:D
2413:=
2404:D
2383:1
2380:=
2371:D
2350:2
2347:=
2344:t
2324:2
2321:=
2318:s
2292:.
2289:)
2280:D
2267:D
2263:(
2257:+
2254:)
2245:D
2232:D
2228:(
2219:=
2212:]
2208:)
2194:y
2174:y
2168:(
2162:)
2148:y
2128:y
2122:(
2118:[
2111:E
2077:.
2037:+
1997:+
1994:)
1985:D
1972:D
1968:(
1962:+
1959:)
1950:D
1937:D
1933:(
1923:=
1914:]
1909:)
1889:+
1880:D
1873:+
1868:2
1860:+
1855:2
1847:(
1841:)
1821:+
1812:D
1805:+
1800:1
1792:+
1787:2
1779:(
1774:[
1754:]
1749:)
1729:+
1720:D
1713:+
1708:2
1700:+
1695:1
1687:(
1681:)
1661:+
1652:D
1645:+
1640:1
1632:+
1627:1
1619:(
1614:[
1603:=
1596:)
1582:y
1562:y
1556:(
1550:)
1536:y
1516:y
1510:(
1478:t
1475:s
1471:D
1450:2
1447:,
1444:1
1441:=
1438:t
1418:2
1415:,
1412:1
1409:=
1406:s
1379:,
1376:)
1373:s
1367:=
1361:)
1358:i
1355:(
1352:s
1349:(
1346:I
1338:t
1335:i
1325:n
1320:1
1317:=
1314:i
1302:s
1298:n
1294:1
1289:=
1280:t
1277:s
1258:,
1255:)
1247:t
1236:=
1230:s
1227:(
1224:I
1218:=
1212:)
1209:s
1203:=
1197:)
1194:i
1191:(
1188:s
1185:(
1182:I
1176:)
1168:t
1157:=
1151:)
1148:i
1145:(
1142:s
1139:(
1136:I
1131:n
1126:1
1123:=
1120:i
1108:s
1104:n
1100:1
1095:=
1086:t
1083:s
1079:D
1071:,
1066:t
1055:=
1049:)
1046:s
1040:=
1034:)
1031:i
1028:(
1025:s
1022:(
1019:I
1011:t
1001:n
996:1
993:=
990:i
978:s
974:n
970:1
965:=
956:t
953:s
934:,
929:s
918:=
912:)
909:s
903:=
897:)
894:i
891:(
888:s
885:(
882:I
874:)
871:i
868:(
865:s
855:n
850:1
847:=
844:i
832:s
828:n
824:1
819:=
810:s
791:,
788:)
785:s
779:=
773:)
770:i
767:(
764:s
761:(
758:I
750:t
747:i
743:y
737:n
732:1
729:=
726:i
714:s
710:n
706:1
701:=
692:t
689:s
680:y
670:s
662:=
653:s
649:n
612:t
609:i
558:t
533:s
511:s
486:Y
466:)
460:(
436:)
430:(
427:I
407:i
387:)
384:i
381:(
378:s
358:t
338:i
313:t
310:i
306:y
280:t
277:i
269:+
266:)
260:(
257:I
251:+
246:t
238:+
233:)
230:i
227:(
224:s
213:=
205:t
202:i
198:y
176:2
172:2
168:2
165:S
161:1
158:S
154:Q
150:1
147:P
143:Q
139:2
136:S
132:2
129:P
125:1
122:S
118:1
115:P
22:(
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