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Analysis of Experimental Data IV
Christoph Engel
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non-linear dpg
I. binary dvII. dv with ordered discrete stepsIII. censored dvIV. dv with unordered discrete
choices
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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
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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?
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I. binary
dgp hdv = 5 + .5*level + error dv = 1 if hdv > 30
020
4060
hdv
0 20 40 60 80 100level
latent
0.2
.4.6
.81
dv
0 20 40 60 80 100level
observed
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linear probability model0
2040
60
0 20 40 60 80 100level
hdv Fitted values
latent
-.50
.51
1.5
0 20 40 60 80 100level
dv Fitted values
observed
interpretation of prediction as probabilitybut: some predicted values out of range(< 0 or > 1)
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additional problem0
2040
60
0 20 40 60 80 100level
hdv Fitted values
latent
0.5
11.
5
0 20 40 60 80 100level
dv Fitted values
observed
bias if a lot of mass on one end use non-linear model
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non-linear model-.5
0.5
11.
5
0 20 40 60 80 100level
data OLS Logit
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mass on one end0
.51
1.5
0 20 40 60 80 100level
data OLS Logit
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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
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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
ii
iii x
xdv
1
1
exp(1exp(
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standard output
_cons -10.32078 1.947648 -5.30 0.000 -14.1381 -6.503457 level 1.134157 .2107922 5.38 0.000 .7210122 1.547302 dv Coef. Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -29.001404 Pseudo R2 = 0.9011 Prob > chi2 = 0.0000 LR chi2(1) = 528.37Logistic regression Number of obs = 1000
Iteration 8: log likelihood = -29.001404 Iteration 7: log likelihood = -29.001404 Iteration 6: log likelihood = -29.001452 Iteration 5: log likelihood = -29.052485 Iteration 4: log likelihood = -30.739153 Iteration 3: log likelihood = -57.215141 Iteration 2: log likelihood = -111.90476 Iteration 1: log likelihood = -196.48485 Iteration 0: log likelihood = -293.18427
. logit dv level
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odds ratio
level 3.108553 .6552586 5.38 0.000 2.056514 4.698777 dv Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -29.001404 Pseudo R2 = 0.9011 Prob > chi2 = 0.0000 LR chi2(1) = 528.37Logistic regression Number of obs = 1000
. logit, or
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rewrite
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interpretation
marginal effect of 1 unit change in iv
on odds ratio
ii
i xpp
111ln
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log-linear
ln is good approximation of % change
020
4060
8010
0
0 10 20 30 40id
percent change ln(level)
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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!
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why so complicated?
change in probability not the same all over 10 11
.8961308-.7351278 = .1610030
19 20 .9999957-.999986
7 = .00000900
.51
1.5
0 20 40 60 80 100level
data OLS Logit
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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
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mathematics: OLS
model
marginal effect of 1 unit change in x1 = partial first derivative wrt x1
= beta1
iii xy 1
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logit
model
marginal effect of 1 unit change in x1
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illustration
dgp hdv = 40 - 3*treat + .5*level + error dv = (hdv > 50)
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graph0
.2.4
.6.8
1
0 20 40 60 80 100level
dv Pr(dv) prediction from logit
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marginal effect at means
level .0218054 .0072921 2.99 0.003 .0075132 .0360976 dy/dx Std. Err. z P>|z| [95% Conf. Interval] Delta-method
level = 50.5 (mean)at : treat = 4.509 (mean)dy/dx w.r.t. : levelExpression : Pr(dv), predict()
Model VCE : OIMConditional marginal effects Number of obs = 1000
. margins, dydx(level) atmeans
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average marginal effect
. margins, dydx(level) atmeansConditional marginal effects Number of obs = 1000Model VCE : OIMExpression : Pr(dv), predict()dy/dx w.r.t. : levelat : 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 = 1000Model VCE : OIMExpression : 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------------------------------------------------------------------------------
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ME level,conditional on treat
0.0
2.0
4.0
6.0
8.1
mar
gina
l effe
ct o
f 1 u
nit i
ncre
ase
in le
vel
0 2 4 6 8 10treatment
marginal effect confidence interval
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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
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ME with interactions
dgp hdv = 40 +2*treat + .25*level
- .05*treat*level + error dv = (hdv > 47)
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predicted from logit0
.2.4
.6.8
1P
r(dv
)
0 20 40 60 80 100level
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statistical model
_cons -13.40162 1.269447 -10.56 0.000 -15.88969 -10.91355 c.level -.0977411 .00795 -12.29 0.000 -.1133227 -.0821594 c.treat# level .4938895 .07228 6.83 0.000 .3522233 .6355556 treat 3.793513 .6003316 6.32 0.000 2.616884 4.970141 dv Coef. Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -136.00823 Pseudo R2 = 0.8038 Prob > chi2 = 0.0000 LR chi2(3) = 1114.08Logistic regression Number of obs = 1000
Iteration 6: log likelihood = -136.00823 Iteration 5: log likelihood = -136.00823 Iteration 4: log likelihood = -136.00827 Iteration 3: log likelihood = -136.04407 Iteration 2: log likelihood = -141.21212 Iteration 1: log likelihood = -188.96252 Iteration 0: log likelihood = -693.04918
. logit dv c.treat##c.level
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ME must take into account
mathematics using chain rule partial derivative wrt one main effect
Stata does if properly informed about interaction
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average MEs
level .0021692 .0023519 0.92 0.356 -.0024405 .0067788 treat -.0431288 .0223182 -1.93 0.053 -.0868717 .0006141 dy/dx Std. Err. z P>|z| [95% Conf. Interval] Delta-method
dy/dx w.r.t. : treat levelExpression : Pr(dv), predict()
Model VCE : OIMAverage marginal effects Number of obs = 1000
. margins, dydx(*)
on average marginal change in level immaterial
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hides more complex story
-.05
0.0
5
0 2 4 6 8 10treatment
marginal effect confidence interval
marginal effect of 1 unit increase in level
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opposite story for treat
-.001
0.0
01.0
02.0
03.0
04
0 2 4 6 8 10level
marginal effect confidence interval
marginal effect of 1 unit increase in treat
on average significantly different from 0but not conditional on specific levels
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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
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linear model0
2040
60
0 20 40 60 80 100level
hdv Fitted values
latent
0.5
11.
52
2.5
0 20 40 60 80 100level
dv Fitted values
observed
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ordered logit0
.2.4
.6.8
1
0 20 40 60 80 100level
Pr(dv==0) Pr(dv==1) Pr(dv==2)
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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
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intreg
sigma .0488678 .0052795 .0395423 .0603926 /lnsigma -3.018636 .1080365 -27.94 0.000 -3.230383 -2.806888 _cons -.2177289 .0149256 -14.59 0.000 -.2469825 -.1884752 level .0245559 .0002828 86.83 0.000 .0240016 .0251102 Coef. Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -72.312708 Prob > chi2 = 0.0000 LR chi2(1) = 2167.01Interval regression Number of obs = 1000
much more statistical power
. intreg ldv udv level
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III. censored
dgp hdv = -20 + .5*level + error dv = hdv if hdv > 0 else dv = 0
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censored-2
0-1
00
1020
30
0 20 40 60 80 100level
hdv Fitted values
latent
-20
-10
010
2030
0 20 40 60 80 100level
dv Fitted values
observed
linear model biased
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solution: Tobit
0 right-censored observations 611 uncensored observations Obs. summary: 389 left-censored observations at dv<=0 /sigma .9402592 .0267063 .8878523 .9926662 _cons -19.77266 .1487295 -132.94 0.000 -20.06452 -19.48081 level .4971101 .002079 239.11 0.000 .4930305 .5011897 dv Coef. Std. Err. t P>|t| [95% Conf. Interval]
Log likelihood = -843.49549 Pseudo R2 = 0.7035 Prob > chi2 = 0.0000 LR chi2(1) = 4002.16Tobit regression Number of obs = 1000
. tobit dv level, ll(0)
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prediction-2
0-1
00
1020
30
0 20 40 60 80 100level
data linear prediction Tobit prediction
Tobit
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procedure
maps zeros into negatives assumes
latent variable incompletely observed
all data points are observed but some are only observed to be
censored
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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?
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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
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Tobit biased0
2040
60
0 20 40 60 80 100level
data predicted
Tobit
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single hurdle model
/sigma .9963187 .0256565 38.83 0.000 .9460329 1.046605 _cons 5.06777 .0733597 69.08 0.000 4.923988 5.211552 level .4993736 .0012627 395.47 0.000 .4968987 .5018486 dv Coef. Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -1067.0988 Prob > chi2 = 0.0000 upper = +inf Wald chi2(1) = 1.6e+05Limit: lower = 0 Number of obs = 754Truncated regression
Iteration 2: log likelihood = -1067.0988 Iteration 1: log likelihood = -1067.0988 Iteration 0: log likelihood = -1067.1001
Fitting full model:
(note: 246 obs. truncated). truncreg dv level, ll(0)
_cons .0343676 .1172594 0.29 0.769 -.1954565 .2641918 pers -1.864497 .1599703 -11.66 0.000 -2.178033 -1.550961 zero Coef. Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -486.48539 Pseudo R2 = 0.1280 Prob > chi2 = 0.0000 LR chi2(1) = 142.82Logistic regression Number of obs = 1000
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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
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second process also generates zeros
0.1
.2.3
Den
sity
0 10 20 30 40hdv
first hurdle
0.1
.2.3
Den
sity
0 10 20 30 40dv
second hurdle
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estimation
. dhreg dv level, hd(pers)
_cons -.5181453 .0876164 -5.91 0.000 -.6898703 -.3464203 pers 2.134447 .1254733 17.01 0.000 1.888524 2.380371hurdle _cons 1.977432 .0565331 34.98 0.000 1.866629 2.088235sigma _cons -9.984145 .2095 -47.66 0.000 -10.39476 -9.573533 level .4999016 .0032727 152.75 0.000 .4934872 .506316above Coef. Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -1574.3771 Prob > chi2 = 0.0000 Wald chi2(1) = 23332.16 Number of obs = 1000
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prediction0
1020
3040
0 20 40 60 80 100level
data predicted
Double Hurdle Model
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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
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estimation
_cons 11.2145 1.398386 8.02 0.000 8.473716 13.95529 type 8.328941 1.025337 8.12 0.000 6.319317 10.33857 level -.4560097 .0551126 -8.27 0.000 -.5640284 -.34799112 _cons 16.56466 1.760686 9.41 0.000 13.11378 20.01554 type 14.60946 1.57338 9.29 0.000 11.52569 17.69322 level -1.517425 .1997494 -7.60 0.000 -1.908926 -1.1259231 0 (base outcome) dv Coef. Std. Err. z P>|z| [95% Conf. Interval]
Log likelihood = -100.83695 Pseudo R2 = 0.8958 Prob > chi2 = 0.0000 LR chi2(4) = 1734.50Multinomial logistic regression Number of obs = 1000
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prediction0
.2.4
.6.8
1
0 20 40 60 80 100level
0 1 2
type 1
0.2
.4.6
.81
0 20 40 60 80 100level
0 1 2
type 2
0.2
.4.6
.81
0 20 40 60 80 100level
0 1 2
type 3
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example0
.2.4
.6.8
1
0 20 40 60 80 100level
0 1 2
type 1
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The end