1. Analysis of Experimental Data IV Christoph Engel

2. non-linear dpg I.binary dv II.dv with ordered discrete steps III.censored dv IV.dv with unordered discrete choices

3. illustrations  binary  responder in ultimatum game  ordered  vote for contribution level  0 / 10 / 20  censored  contributions to public good  unordered  public official in bribery experiment  reject  accept, but cheat  accept and grant favor

4. reason to go non-linear?  outside the lab  sample as proxy for true dgp  in the lab  dgp follows from design  e.g.: ? is action space constrained  public good  contributions in [0, 20]  is problem small enough to be ignored?

5. I. binary  dgp  hdv = 5 +.5*level + error  dv = 1 if hdv > 30

6. linear probability model interpretation of prediction as probability but: some predicted values out of range ( 1)

7. additional problem bias if a lot of mass on one end  use non-linear model

8. non-linear model

9. mass on one end

10. which model?  logit or probit  a matter of taste  different distributional assumption  probit  normality  logit  logistic distribution  logit  more robust  faster  coefficients can be directly interpreted

11. statistical model  nice mathematical properties  exp(.) is positive  exp(.)/(1+exp(.)) goes to  1 if (.) is positive and large  0 if (.) is negative and large

12. standard output

13. odds ratio

14. rewrite

15. interpretation  marginal effect of 1 unit change in iv  on odds ratio

16. log-linear  ln is good approximation of % change

17. example  predicted prob at level = 10 .735  odds 735/265 = 2.775  predicted prob at level = 11 .896  odds 896/104 = 8.627  odds ratio of 1 unit change  8.627/2.775 = 3.1085  holds for all comparisons!

18. why so complicated?  change in probability not the same all over  10  11 .8961308-.7351278 =.1610030  19  20 .9999957-.9999867 =.0000090

19. drawbacks  change in odds ratio not overly intuitive  only works for logit  not for any other non-linear model  different questions  marginal effect at average  of all ivs  average marginal effect  ~ conditional on one iv  OLS  coef = answer to all of them  not true for non-linear model

20. mathematics: OLS  model  marginal effect of 1 unit change in x 1  = partial first derivative wrt x 1  = beta 1

21. logit  model  marginal effect of 1 unit change in x 1

22. illustration  dgp  hdv = 40 - 3*treat +.5*level + error  dv = (hdv > 50)

23. graph prediction from logit

24. marginal effect at means

25. average marginal effect. margins, dydx(level) atmeans Conditional marginal effects Number of obs = 1000 Model VCE : OIM Expression : Pr(dv), predict() dy/dx w.r.t. : level at : treat = 4.509 (mean) level = 50.5 (mean) ------------------------------------------------------------------------------ | Delta-method | dy/dx Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- level |.0218054.0072921 2.99 0.003.0075132.0360976 ------------------------------------------------------------------------------. margins, dydx(level) Average marginal effects Number of obs = 1000 Model VCE : OIM Expression : Pr(dv), predict() dy/dx w.r.t. : level ------------------------------------------------------------------------------ | Delta-method | dy/dx Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- level |.0239235.001772 13.50 0.000.0204504.0273965 ------------------------------------------------------------------------------

26. ME level, conditional on treat

27. explanation  recall dgp  hdv = 40 - 3*treat +.5*level + error  dv = 1 if hdv > 50  if treat is large  hdv always < 50  if treat is very small  hdv always close to 50

28. ME with interactions  dgp  hdv = 40 +2*treat +.25*level -.05*treat*level + error  dv = (hdv > 47)

29. predicted from logit

30. statistical model

31. ME must take into account  mathematics  using chain rule  partial derivative wrt one main effect  Stata does  if properly informed about interaction

32. average MEs on average marginal change in level immaterial

33. hides more complex story

34. opposite story for treat on average significantly different from 0 but not conditional on specific levels

35. II. ordered  dgp  hdv = 5 +.5*level + error  dv = 0 if hdv < 20  dv = 1 if hdv in [20,40]  dv = 2 if hdv > 40

36. linear model

37. ordered logit

38. too conservative  if steps in dv have cardinal interpretation  1-0 = 2-1  two options  count model  interval regression  differences  distributional assumptions  count model: Poisson  intreg: normal  intreg is linear  coefficients can be directly interpreted

39. intreg much more statistical power

40. III. censored  dgp  hdv = -20 +.5*level + error  dv = hdv if hdv > 0  else dv = 0

41. censored linear model biased

42. solution: Tobit

43. prediction

44. procedure  maps zeros into negatives  assumes  latent variable  incompletely observed  all data points are observed  but some are only observed to be censored

45. assumption appropriate?  dictator game  dictators would even want to take  warranted  Bardsley ExpEc 2008  punishment in public good  non-punishers would even want to reward  warranted?

46. what if not?  dgp  hurdle  hdv = pers + error1  dv = 0 if hdv < 0  conditional on hurdle being passed  dv = 5 +.5*level + error2 if hdv > 0

47. Tobit biased

48. single hurdle model

49. double hurdle model  dgp  first hurdle  hhd = -.5 + 2*pers + error1  hd = (hhd > 0)  second hurdle and above  hhdv = -10 +.5*level + error2  hdv = hhdv*(hhdv > 0)  dv = hdv*hd

50. second process also generates zeros

51. estimation

52. prediction

53. IV. unordered discrete  dgp  latent  hdv0 = - 10 + 1.2*level - 8*type + error1  hdv1 = 5 -.2*level + 5*type + error2  hdv2 =.8*level -.7*type + error3  observed  dv = 0 if hdv0 > hdv1 & hdv2  dv = 1 if hdv1 > hdv0 & hdv2  dv = 2 if hdv2 > hdv0 & hdv1

54. estimation

55. prediction

56. example