David M. Drukker Italian Stata Users Group meeting 15 November Executive Director of Econometrics Stata

Transcription

1 Estimating the ATE of an endogenously assigned treatment from a sample with endogenous selection by regression adjustment using an extended regression models David M. Drukker Executive Director of Econometrics Stata 2018 Italian Stata Users Group meeting 15 November 2018

2 Fictional data on wellness ram from large company. use wram2. describe Contains data from wram2.dta obs: 3,000 vars: 8 28 Jul :13 size: 96,000 storage display value variable name type format label variable label wchange float %9.0g changel Weight change level age float %9.0g Years over 50 over float %9.0g Overweight (tens of pounds) phealth float %9.0g Prior health score float %9.0g yesno Participate in wellness ram wt float %9.0g yesno Offered work time to participate in ram wtsamp float %9.0g Offered work time to participate in sample insamp float %9.0g In sample: attended initial and final weigh in Sorted by: 1 / 41

3 Three levels of wchange. tabulate wchange Weight Participate in change wellness ram level No Yes Total Loss ,114 No change Gain Total 589 1,295 1,884 2 / 41

4 Three levels of wchange. tabulate wchange Weight Participate in change wellness ram level No Yes Total Loss ,114 No change Gain Total 589 1,295 1,884 Data are observational 2 / 41

5 Dealing with observational data. tabulate wchange Weight Participate in change wellness ram level No Yes Total Loss ,114 No change Gain / 41 Total 589 1,295 1,884 Table does not account for how observed covariates that affect ram participation also affect the potential outcome variables Assume the treatment is as good as random after conditioning on covariates Conditional mean independence Exogenous treatment assignment teffects

6 Dealing with observational data. tabulate wchange Weight Participate in change wellness ram level No Yes Total Loss ,114 No change Gain Total 589 1,295 1,884 Table does not account for how observed unobserved error that affect ram participation also affect the potential outcome variables Endogenous treatment assignment etefffects and etregress for continuous outcomes etpoisson for count outcomes Need Stata command for ordinal outcome 4 / 41

7 Dealing with observational data. tabulate wchange Weight Participate in change wellness ram level No Yes Total Loss ,114 No change Gain Total 589 1,295 1,884 Table does not account for the possibility that unobserved errors in the process that caused some of 3,000 individuals not to show for the final weigh in may also affect the potential outcome variables Endogenous loss to follow up Endogenous sample selection 5 / 41

8 Ordinal Potential outcomes Because the outcome wchange is ordinal, there are really three binary outcomes wchange== Loss, wchange== No Change, and wchange== Gain 6 / 41

9 Ordinal Potential outcomes In the potential outcome framework, there is an outcome for each person when they participate and when the do not participate 7 / 41

10 Ordinal Potential outcomes In the potential outcome framework, there is an outcome for each person when they participate and when the do not participate Thus, there are really three binary outcomes for each potential outcome Participate Not participate wchange p == Loss wchange np == Loss wchange p == No change wchange np == No change wchange p == Gain wchange np == Gain 7 / 41

11 Potential outcome framework For each outcome (Loss, No change, and Gain), we only observe one of these two potential outcomes for each individual 8 / 41

12 Potential outcome framework For each outcome (Loss, No change, and Gain), we only observe one of these two potential outcomes for each individual We estimate the parameters of a model and use the estimated parameters to predict what each person does in the unobserved potential outcome Regression adjustment 8 / 41

13 Average treatment effects In the case of one outcome, the average treatment effect (ATE) is E[y p y np ] 9 / 41

14 Average treatment effects In the case of one outcome, the average treatment effect (ATE) is E[y p y np ] As there are three outcomes, there are three ATEs one for Loss, one for No Change, and one for Gain 9 / 41

15 Average treatment effects In the case of one outcome, the average treatment effect (ATE) is E[y p y np ] As there are three outcomes, there are three ATEs one for Loss, one for No Change, and one for Gain ATE Loss = E[(wchange p == Loss ) (wchange np == Loss )] ATE Nochange = E[(wchange p == No change ) (wchange np == No change )] ATE Gain = E[(wchange p == Gain ) (wchange np == Gain )] 9 / 41

16 Average treatment effects I will provide some details about the average treatment effect for Loss The details for the outcomes of No change and Gain are analogous 10 / 41

17 the average treatment effect (ATE) of the ram on the Loss outcome ATE Loss ATE Loss = E[(wchange p == Loss ) (wchange np == Loss )] The first line says that ATE Loss is the mean diffence in the outcomes when everyone participates instead of no one participates 11 / 41

18 the average treatment effect (ATE) of the ram on the Loss outcome ATE Loss ATE Loss = E[(wchange p == Loss ) (wchange np == Loss )] = E[wchange p == Loss ] E[wchange np == Loss ] The second line says that the mean of the differences is the difference in the means 12 / 41

19 the average treatment effect (ATE) of the ram on the Loss outcome ATE Loss ATE Loss = E[(wchange p == Loss ) (wchange np == Loss )] = E[wchange p == Loss ] E[wchange np == Loss ] = Pr[wchange p == Loss ] Pr[wchange np == Loss ] The third line says that because the mean of binary outcome is the probability that the event is true, the ATE Loss is the difference in the probability an individual is in the state of Loss when everyone participates instead of no one participates 13 / 41

20 14 / 41 I am going to use the ERM comand eoprobit to estimate the parameters of Pr[wchange p == Loss x] and Pr[wchange np == Loss x] and

21 I am going to use the ERM comand eoprobit to estimate the parameters of Pr[wchange p == Loss x] and Pr[wchange np == Loss x] and Then I use margins or estat teffects to estimate E[Pr[wchange p == Loss x]] E[Pr[wchange np == Loss x]] = Pr[wchange p == Loss ] Pr[wchange np == Loss ] = ATE Loss 14 / 41

22 I am going to use the ERM comand eoprobit to estimate the parameters of Pr[wchange p == Loss x] and Pr[wchange np == Loss x] and Then I use margins or estat teffects to estimate E[Pr[wchange p == Loss x]] E[Pr[wchange np == Loss x]] = Pr[wchange p == Loss ] Pr[wchange np == Loss ] = ATE Loss The ATE Loss is the mean difference in the probability an individual is in the state of Loss when everyone participates instead of no one participates 14 / 41

23 Models for the ordinal outcome For exogenous treatment, we do a one-step equivalent to fitting two separate ordinal probit models One fit to partipants Another fit to non partipants 15 / 41

24 Model for partipants Loss if xβ 0 + ɛ 0 cut1 0 wchange = No change if cut1 0 < xβ 0 + ɛ 0 cut2 0 Gain if cut2 0 < xβ 0 + ɛ 0 xβ 0 = β 1,0 age + β 2,0 over + β 3,0 phealth for the observations at which =0, and ɛ 0, is standard normal 16 / 41

25 Model for nonparticipants Loss if xβ 1 + ɛ 1 cut1 1 wchange = No change if cut1 1 < xβ 1 + ɛ 1 cut2 1 Gain if cut2 1 < xβ 1 + ɛ 1 xβ 1 = β 1,1 age + β 2,1 over + β 3,1 phealth for the observations at which =1 ɛ 1 is standard normal 17 / 41

26 Loss if xβ 0 + ɛ 0 cut1 0 wchange = No change if cut1 0 < xβ 0 + ɛ 0 cut2 0 Gain if cut2 0 < xβ 0 + ɛ 0 xβ 0 = β 1,0 age + β 2,0 over + β 3,0 phealth for the observations at which =0, and Loss if xβ 1 + ɛ 1 cut1 1 wchange = No change if cut1 1 < xβ 1 + ɛ 1 cut2 1 Gain if cut2 1 < xβ 1 + ɛ 1 xβ 1 = β 1,1 age + β 2,1 over + β 3,1 phealth for the observations at which =1 ɛ 0, and ɛ 1 are normal corr(ɛ 0, ɛ 1 ) is not identified or estimated 18 / 41

27 . eoprobit wchange age over phealth, extreat() vsquish nolog Extended ordered probit regression Number of obs = 1,884 Wald chi2(6) = Log likelihood = Prob > chi2 = wchange Coef. Std. Err. z P> z [95% Conf. Interval] #c.age No Yes #c.over No Yes # c.phealth No Yes /wchange #c.cut1 No Yes #c.cut2 No Yes estimates store oprobit 19 / 41

28 . estat teffects Predictive margins Number of obs = 3,000 ATE_Pr0 : Pr(wchange=0=Loss) ATE_Pr1 : Pr(wchange=1=No change) ATE_Pr2 : Pr(wchange=2=Gain) Unconditional Margin Std. Err. z P> z [95% Conf. Interval] ATE_Pr0 (Yes vs No) ATE_Pr1 (Yes vs No) ATE_Pr2 (Yes vs No) When everyone joins the ram instead of when no one participants in the ram, 20 / 41

29 . estat teffects Predictive margins Number of obs = 3,000 ATE_Pr0 : Pr(wchange=0=Loss) ATE_Pr1 : Pr(wchange=1=No change) ATE_Pr2 : Pr(wchange=2=Gain) Unconditional Margin Std. Err. z P> z [95% Conf. Interval] ATE_Pr0 (Yes vs No) ATE_Pr1 (Yes vs No) ATE_Pr2 (Yes vs No) When everyone joins the ram instead of when no one participants in the ram, On average, the probablity of Loss goes up by / 41

30 . estat teffects Predictive margins Number of obs = 3,000 ATE_Pr0 : Pr(wchange=0=Loss) ATE_Pr1 : Pr(wchange=1=No change) ATE_Pr2 : Pr(wchange=2=Gain) Unconditional Margin Std. Err. z P> z [95% Conf. Interval] ATE_Pr0 (Yes vs No) ATE_Pr1 (Yes vs No) ATE_Pr2 (Yes vs No) / 41 When everyone joins the ram instead of when no one participants in the ram, On average, the probablity of Loss goes up by.44 On average, the probablity of No change goes down by.17

31 When everyone joins the ram instead of when no one participants in the ram, On average, the probablity of Loss goes up by.44 On average, the probablity of No change goes down by.17 On average, the probablity of Gain goes down / 41. estat teffects Predictive margins Number of obs = 3,000 ATE_Pr0 : Pr(wchange=0=Loss) ATE_Pr1 : Pr(wchange=1=No change) ATE_Pr2 : Pr(wchange=2=Gain) Unconditional Margin Std. Err. z P> z [95% Conf. Interval] ATE_Pr0 (Yes vs No) ATE_Pr1 (Yes vs No) ATE_Pr2 (Yes vs No)

32 None lost to follow up Some observations on wchange are missing No observations on covariates are missing Can do predictions for all cases 21 / 41

33 ATE: How (1). generate _original =. replace = 0 (1,700 real changes made). predict double pr_loss_0, outlevel("loss") (option pr assumed; predicted probabilities). replace = 1 (3,000 real changes made). predict double pr_loss_1, outlevel("loss") (option pr assumed; predicted probabilities). replace = _original (1,300 real changes made). drop _original. mean pr_loss_0 pr_loss_1 Mean estimation Number of obs = 3,000 Mean Std. Err. [95% Conf. Interval] pr_loss_ pr_loss_ / 41

34 ATE: How (2). estimates restore oprobit (results oprobit are active now). margins, /// > predict(outlevel("loss")) /// > predict(outlevel("no change")) /// > predict(outlevel("gain")) noesample Predictive margins Number of obs = 3,000 Model VCE : OIM 1._predict : Pr(wchange==Loss), predict(outlevel("loss")) 2._predict : Pr(wchange==No change), predict(outlevel("no change")) 3._predict : Pr(wchange==Gain), predict(outlevel("gain")) Delta-method Margin Std. Err. z P> z [95% Conf. Interval] _predict# 1#No #Yes #No #Yes #No #Yes / 41

35 ATE: How (3). margins r., /// > predict(outlevel("loss")) /// > predict(outlevel("no change")) /// > predict(outlevel("gain")) /// > contrast(nowald) /// > noesample Contrasts of predictive margins Model VCE : OIM 1._predict : Pr(wchange==Loss), predict(outlevel("loss")) 2._predict : Pr(wchange==No change), predict(outlevel("no change")) 3._predict : Pr(wchange==Gain), predict(outlevel("gain")) Delta-method Contrast Std. Err. [95% Conf. (Yes vs No) (Yes vs No) (Yes vs No) / 41

36 Endogenous Treatment model The potential-outcome model for an endogenous treatment Allows the coefficients to differ for the treated and not-treated state Allows the cut offs to differ for the treated and not-treated state Allows for distinct (nonzero) correlations between the errors driving treatment assignment and the errors driving the ordinal outcomes for the treated and not-treated states 25 / 41

37 26 / 41 = (xγ + γ 1 wt + η > 0)

38 = (xγ + γ 1 wt + η > 0) Loss if xβ 0 + ɛ 0 cut1 0 wchange = No change if cut1 0 < xβ 0 + ɛ 0 cut2 0 Gain if cut2 0 < xβ 0 + ɛ 0 xβ 0 = β 1,0 age + β 2,0 over + β 3,0 phealth for the observations at which =0, and 26 / 41

39 = (xγ + γ 1 wt + η > 0) Loss if xβ 0 + ɛ 0 cut1 0 wchange = No change if cut1 0 < xβ 0 + ɛ 0 cut2 0 Gain if cut2 0 < xβ 0 + ɛ 0 xβ 0 = β 1,0 age + β 2,0 over + β 3,0 phealth for the observations at which =0, and Loss if xβ 1 + ɛ 1 cut1 1 wchange = No change if cut1 1 < xβ 1 + ɛ 1 cut2 1 Gain if cut2 1 < xβ 1 + ɛ 1 xβ 1 = β 1,1 age + β 2,1 over + β 3,1 phealth for the observations at which =1 26 / 41

40 26 / 41 = (xγ + γ 1 wt + η > 0) Loss if xβ 0 + ɛ 0 cut1 0 wchange = No change if cut1 0 < xβ 0 + ɛ 0 cut2 0 Gain if cut2 0 < xβ 0 + ɛ 0 xβ 0 = β 1,0 age + β 2,0 over + β 3,0 phealth for the observations at which =0, and Loss if xβ 1 + ɛ 1 cut1 1 wchange = No change if cut1 1 < xβ 1 + ɛ 1 cut2 1 Gain if cut2 1 < xβ 1 + ɛ 1 xβ 1 = β 1,1 age + β 2,1 over + β 3,1 phealth for the observations at which =1 ɛ 0, ɛ 1, and η are correlated and joint normal ρ 0 correlation between ɛ 0 and η ρ 1 correlation between ɛ 1 and η

41 Endogenous treatment model. eoprobit wchange age over phealth, /// > entreat( = age over phealth wt, pocorr ) /// > vce(robust) vsquish nolog Extended ordered probit regression Number of obs = 1,884 Wald chi2(6) = Log pseudolikelihood = Prob > chi2 = Robust Coef. Std. Err. z P> z [95% Conf. Interval] wchange #c.age No Yes #c.over No Yes # c.phealth No Yes age over phealth / 41 wt

42 Robust Coef. Std. Err. z P> z [95% Conf. Interval] wchange #c.age No Yes #c.over No Yes # c.phealth No Yes age over phealth wt _cons /wchange #c.cut1 No Yes #c.cut2 No Yes corr(e., e.wchange) 28 / 41

43 No Yes #c.over No Yes # c.phealth No Yes age over phealth wt _cons /wchange #c.cut1 No Yes #c.cut2 No Yes corr(e., e.wchange) No Yes / 41

44 . estat teffects Predictive margins Number of obs = 3,000 ATE_Pr0 : Pr(wchange=0=Loss) ATE_Pr1 : Pr(wchange=1=No change) ATE_Pr2 : Pr(wchange=2=Gain) Unconditional Margin Std. Err. z P> z [95% Conf. Interval] ATE_Pr0 (Yes vs No) ATE_Pr1 (Yes vs No) ATE_Pr2 (Yes vs No) When everyone joins the ram instead of when no one participants in the ram, 30 / 41

45 . estat teffects Predictive margins Number of obs = 3,000 ATE_Pr0 : Pr(wchange=0=Loss) ATE_Pr1 : Pr(wchange=1=No change) ATE_Pr2 : Pr(wchange=2=Gain) Unconditional Margin Std. Err. z P> z [95% Conf. Interval] ATE_Pr0 (Yes vs No) ATE_Pr1 (Yes vs No) ATE_Pr2 (Yes vs No) When everyone joins the ram instead of when no one participants in the ram, On average, the probablity of Loss goes up by.1 30 / 41

46 . estat teffects Predictive margins Number of obs = 3,000 ATE_Pr0 : Pr(wchange=0=Loss) ATE_Pr1 : Pr(wchange=1=No change) ATE_Pr2 : Pr(wchange=2=Gain) Unconditional Margin Std. Err. z P> z [95% Conf. Interval] ATE_Pr0 (Yes vs No) ATE_Pr1 (Yes vs No) ATE_Pr2 (Yes vs No) / 41 When everyone joins the ram instead of when no one participants in the ram, On average, the probablity of Loss goes up by.1 On average, the probablity of No change does not change by much

47 When everyone joins the ram instead of when no one participants in the ram, On average, the probablity of Loss goes up by.1 On average, the probablity of No change does not change by much 30 / 41 On average, the probablity of Gain goes down.09. estat teffects Predictive margins Number of obs = 3,000 ATE_Pr0 : Pr(wchange=0=Loss) ATE_Pr1 : Pr(wchange=1=No change) ATE_Pr2 : Pr(wchange=2=Gain) Unconditional Margin Std. Err. z P> z [95% Conf. Interval] ATE_Pr0 (Yes vs No) ATE_Pr1 (Yes vs No) ATE_Pr2 (Yes vs No)

48 . margins r., /// > predict(fix() outlevel("loss")) /// > predict(fix() outlevel("no change")) /// > predict(fix() outlevel("gain")) /// > contrast(nowald) vce(unconditional) noesample Contrasts of predictive margins 1._predict : Pr(wchange==Loss), predict(fix() outlevel("loss")) 2._predict : Pr(wchange==No change), predict(fix() outlevel("no change")) 3._predict : Pr(wchange==Gain), predict(fix() outlevel("gain")) Unconditional Contrast Std. Err. [95% Conf. (Yes vs No) (Yes vs No) (Yes vs No) / 41

49 32 / 41 fix() gets us the effect of the ram that is not contaminated by the selection effect/correlation between ɛ and η that increases the participation among people more likely to lose weight

50 fix() gets us the effect of the ram that is not contaminated by the selection effect/correlation between ɛ and η that increases the participation among people more likely to lose weight predict(fix()) tells margins to specify fix() to predict when computing each predicted probability 32 / 41

51 fix() gets us the effect of the ram that is not contaminated by the selection effect/correlation between ɛ and η that increases the participation among people more likely to lose weight predict(fix()) tells margins to specify fix() to predict when computing each predicted probability fix() causes the value of not to affect ɛ, even though they are correlated 32 / 41

52 fix() gets us the effect of the ram that is not contaminated by the selection effect/correlation between ɛ and η that increases the participation among people more likely to lose weight predict(fix()) tells margins to specify fix() to predict when computing each predicted probability fix() causes the value of not to affect ɛ, even though they are correlated fix() specifies that the part of ɛ that is correlated with be integrated out 32 / 41

53 This type of prediction is sometimes called the structural prediction or an average structural function; see Blundell and Powell (2003), Blundell and Powell (2004), Wooldridge (2005), Wooldridge (2010), and Wooldridge (2014), The difference between the mean of the average of the structural predictions when =1 and the mean of the average of the structural predictions when =0 is an average treatment effect (Blundell and Powell (2003) and Wooldridge (2014)) 33 / 41

54 Endogenous sample selection Reconsider our fictional weight-loss ram 34 / 41

55 Endogenous sample selection Reconsider our fictional weight-loss ram Some ram participants and some nonparticipants will not show up for the final weigh in This is commonly known as lost to follow up 34 / 41

56 Endogenous sample selection Reconsider our fictional weight-loss ram Some ram participants and some nonparticipants will not show up for the final weigh in This is commonly known as lost to follow up If unobservables that affect whether someone is lost to follow up 34 / 41

57 Endogenous sample selection Reconsider our fictional weight-loss ram Some ram participants and some nonparticipants will not show up for the final weigh in This is commonly known as lost to follow up If unobservables that affect whether someone is lost to follow up are independent of the unobservables that affect ram participantion 34 / 41

58 Endogenous sample selection Reconsider our fictional weight-loss ram Some ram participants and some nonparticipants will not show up for the final weigh in This is commonly known as lost to follow up If unobservables that affect whether someone is lost to follow up are independent of the unobservables that affect ram participantion and they are independent of the unobservables that affect the outcomes with and without the ram, 34 / 41

59 Endogenous sample selection Reconsider our fictional weight-loss ram Some ram participants and some nonparticipants will not show up for the final weigh in This is commonly known as lost to follow up If unobservables that affect whether someone is lost to follow up are independent of the unobservables that affect ram participantion and they are independent of the unobservables that affect the outcomes with and without the ram, the previously discussed estimator consistently estimates the effects 34 / 41

60 Endogenous sample selection Reconsider our fictional weight-loss ram Some ram participants and some nonparticipants will not show up for the final weigh in This is commonly known as lost to follow up If unobservables that affect whether someone is lost to follow up are independent of the unobservables that affect ram participantion and they are independent of the unobservables that affect the outcomes with and without the ram, the previously discussed estimator consistently estimates the effects Any dependence among the unobservables must be modeled 34 / 41

61 insamp = (xα + α 1 wtsamp + ξ > 0) 35 / 41

62 insamp = (xα + α 1 wtsamp + ξ > 0) = (xγ + γ 1 wt + η > 0) 35 / 41

63 insamp = (xα + α 1 wtsamp + ξ > 0) = (xγ + γ 1 wt + η > 0) Loss if xβ 0 + ɛ 0 cut1 0 wchange = No change if cut1 0 < xβ 0 + ɛ 0 cut2 0 Gain if cut2 0 < xβ 0 + ɛ 0 xβ 0 = β 1,0 age + β 2,0 over + β 3,0 phealth for the observations at which =0, and 35 / 41

64 insamp = (xα + α 1 wtsamp + ξ > 0) = (xγ + γ 1 wt + η > 0) Loss if xβ 0 + ɛ 0 cut1 0 wchange = No change if cut1 0 < xβ 0 + ɛ 0 cut2 0 Gain if cut2 0 < xβ 0 + ɛ 0 xβ 0 = β 1,0 age + β 2,0 over + β 3,0 phealth for the observations at which =0, and Loss if xβ 1 + ɛ 1 cut1 1 wchange = No change if cut1 1 < xβ 1 + ɛ 1 cut2 1 Gain if cut2 1 < xβ 1 + ɛ 1 xβ 1 = β 1,1 age + β 2,1 over + β 3,1 phealth for the observations at which =1 35 / 41

65 insamp = (xα + α 1 wtsamp + ξ > 0) = (xγ + γ 1 wt + η > 0) Loss if xβ 0 + ɛ 0 cut1 0 wchange = No change if cut1 0 < xβ 0 + ɛ 0 cut2 0 Gain if cut2 0 < xβ 0 + ɛ 0 xβ 0 = β 1,0 age + β 2,0 over + β 3,0 phealth for the observations at which =0, and Loss if xβ 1 + ɛ 1 cut1 1 wchange = No change if cut1 1 < xβ 1 + ɛ 1 cut2 1 Gain if cut2 1 < xβ 1 + ɛ 1 xβ 1 = β 1,1 age + β 2,1 over + β 3,1 phealth for the observations at which =1 ξ, ɛ 0, ɛ 1, and η are correlated and joint normal distinct correlations between each treatment error and others 35 / 41

66 . eoprobit wchange age over phealth, /// > entreat( = age over phealth wt, pocorr ) /// > select(insamp = age over phealth wtsamp ) /// > vce(robust) vsquish nolog Extended ordered probit regression Number of obs = 3,000 Selected = 1,884 Nonselected = 1,116 Wald chi2(6) = Log pseudolikelihood = Prob > chi2 = Robust Coef. Std. Err. z P> z [95% Conf. Interval] wchange #c.age No Yes #c.over No Yes # c.phealth No Yes insamp age over phealth wtsamp 36 /

67 c.phealth No Yes insamp age over phealth wtsamp _cons age over phealth wt _cons /wchange #c.cut1 No Yes #c.cut2 No Yes corr(e.ins ~ p, e.wchange) No Yes corr(e., e.wchange) 37 / 41

68 #c.cut1 No Yes #c.cut2 No Yes corr(e.ins ~ p, e.wchange) No Yes corr(e., e.wchange) No Yes corr(e., e.insamp) Nonzero correlations between e.insamp and e.wchange imply endogenous sample selection for outcomes Nonzero correlations between e. and e.wchange imply endogenous treatment assignment 38 / 41

69 . estat teffects Predictive margins Number of obs = 3,000 ATE_Pr0 : Pr(wchange=0=Loss) ATE_Pr1 : Pr(wchange=1=No change) ATE_Pr2 : Pr(wchange=2=Gain) Unconditional Margin Std. Err. z P> z [95% Conf. Interval] ATE_Pr0 (Yes vs No) ATE_Pr1 (Yes vs No) ATE_Pr2 (Yes vs No) When everyone joins the ram instead of when no one participants in the ram, 39 / 41

70 . estat teffects Predictive margins Number of obs = 3,000 ATE_Pr0 : Pr(wchange=0=Loss) ATE_Pr1 : Pr(wchange=1=No change) ATE_Pr2 : Pr(wchange=2=Gain) Unconditional Margin Std. Err. z P> z [95% Conf. Interval] ATE_Pr0 (Yes vs No) ATE_Pr1 (Yes vs No) ATE_Pr2 (Yes vs No) When everyone joins the ram instead of when no one participants in the ram, On average, the probablity of Loss goes up by / 41

71 . estat teffects Predictive margins Number of obs = 3,000 ATE_Pr0 : Pr(wchange=0=Loss) ATE_Pr1 : Pr(wchange=1=No change) ATE_Pr2 : Pr(wchange=2=Gain) Unconditional Margin Std. Err. z P> z [95% Conf. Interval] ATE_Pr0 (Yes vs No) ATE_Pr1 (Yes vs No) ATE_Pr2 (Yes vs No) When everyone joins the ram instead of when no one participants in the ram, On average, the probablity of Loss goes up by.14 On average, the probablity of No change does not change 39 / 41

72 . estat teffects Predictive margins Number of obs = 3,000 ATE_Pr0 : Pr(wchange=0=Loss) ATE_Pr1 : Pr(wchange=1=No change) ATE_Pr2 : Pr(wchange=2=Gain) Unconditional Margin Std. Err. z P> z [95% Conf. Interval] ATE_Pr0 (Yes vs No) ATE_Pr1 (Yes vs No) ATE_Pr2 (Yes vs No) / 41 When everyone joins the ram instead of when no one participants in the ram, On average, the probablity of Loss goes up by.14 On average, the probablity of No change does not change On average, the probablity of Gain goes down.16

73 . margins r., /// > predict(fix() outlevel("loss")) /// > predict(fix() outlevel("no change")) /// > predict(fix() outlevel("gain")) /// > contrast(nowald) vce(unconditional) noesample Contrasts of predictive margins 1._predict : Pr(wchange==Loss), predict(fix() outlevel("loss")) 2._predict : Pr(wchange==No change), predict(fix() outlevel("no change")) 3._predict : Pr(wchange==Gain), predict(fix() outlevel("gain")) Unconditional Contrast Std. Err. [95% Conf. (Yes vs No) (Yes vs No) (Yes vs No) When everyone joins the ram instead of when no one participants in the ram, 40 / 41

74 . margins r., /// > predict(fix() outlevel("loss")) /// > predict(fix() outlevel("no change")) /// > predict(fix() outlevel("gain")) /// > contrast(nowald) vce(unconditional) noesample Contrasts of predictive margins 1._predict : Pr(wchange==Loss), predict(fix() outlevel("loss")) 2._predict : Pr(wchange==No change), predict(fix() outlevel("no change")) 3._predict : Pr(wchange==Gain), predict(fix() outlevel("gain")) Unconditional Contrast Std. Err. [95% Conf. (Yes vs No) (Yes vs No) (Yes vs No) When everyone joins the ram instead of when no one participants in the ram, On average, the probablity of Loss goes up by / 41

75 . margins r., /// > predict(fix() outlevel("loss")) /// > predict(fix() outlevel("no change")) /// > predict(fix() outlevel("gain")) /// > contrast(nowald) vce(unconditional) noesample Contrasts of predictive margins 1._predict : Pr(wchange==Loss), predict(fix() outlevel("loss")) 2._predict : Pr(wchange==No change), predict(fix() outlevel("no change")) 3._predict : Pr(wchange==Gain), predict(fix() outlevel("gain")) Unconditional Contrast Std. Err. [95% Conf. (Yes vs No) (Yes vs No) (Yes vs No) When everyone joins the ram instead of when no one participants in the ram, On average, the probablity of Loss goes up by.14 On average, the probablity of No change does not change 40 / 41

76 . margins r., /// > predict(fix() outlevel("loss")) /// > predict(fix() outlevel("no change")) /// > predict(fix() outlevel("gain")) /// > contrast(nowald) vce(unconditional) noesample Contrasts of predictive margins 1._predict : Pr(wchange==Loss), predict(fix() outlevel("loss")) 2._predict : Pr(wchange==No change), predict(fix() outlevel("no change")) 3._predict : Pr(wchange==Gain), predict(fix() outlevel("gain")) Unconditional Contrast Std. Err. [95% Conf. (Yes vs No) (Yes vs No) (Yes vs No) / 41 When everyone joins the ram instead of when no one participants in the ram, On average, the probablity of Loss goes up by.14 On average, the probablity of No change does not change On average, the probablity of Gain goes down.16

77 More about ERM commands The commands eregress, eprobit, and eintreg fit ERMs handle continuous-and-unbounded, binary, and censored/corner outcomes Look at for more examples and a wealth of details 41 / 41

78 Bibliography Blundell, R. W., and J. L. Powell Endogeity in nonparametric and semiparametric regression models. In Advances in Economics and Econometrics: Theory and Applications, Eighth World Congress, ed. M. Dewatripont, L. P. Hansen, and S. J. Turnovsky, vol. 2, Cambridge: Cambridge University Press Endogeneity in semiparametric binary response models. Review of Economic Studies 71: Wooldridge, J. M Unobserved heterogeneity and estimation of average partial effects. In Identification and Inference for Econometric Models: Essays in Honor of Thomas Rothenberg, ed. D. K. Andrews and J. H. Stock, Cambridge, UK: Cambridge: Cambridge University Press Econometric Analysis of Cross Section and Panel Data. 2nd ed. Cambridge, Massachusetts: MIT Press Quasi-maximum likelihood estimation and testing for nonlinear models with endogenous explanatory variables. Journal of Econometrics 182: / 41