45
B Supplementary Tables and Figures
Table B.1: Summary of the
parameters for the modified
CTB
1
2
3
4
t
0
0
35
35
k
35
63
35
63
P
1
1.05 1.00 1.05 1.00
P
2
1.11 1.05 1.11 1.05
P
3
1.18 1.11 1.18 1.11
P
4
1.25 1.33 1.25 1.33
P
5
1.43 1.67 1.43 1.67
P
6
1.82 2.22 1.82 2.22
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46
Figure B.1: Histogram of fraction of monotonic choices by individuals for full sample and
analysis sample.
C Additional Exercises and Robustness Tests
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47
Table C.1: Plan Choice and Experimental Behavior 2 (for restricted sample)
Dependent Variable:
Chose Smooth Plan
(1)
(2)
(3)
(4)
Interiorness
0.523***
(0.135)
Patience
0.488***
(0.136)
Present Biasedness
0.111
(0.121)
Curvature: quintile rank of
α
-0.035*
(0.019)
Discount Factor: quintile rank of
δ
30
0.041**
(0.019)
Present Bias: quintile rank of
β
-0.004
(0.018)
Quintile rank of Plan Value Ratio (PVR)
0.051***
(0.016)
PVR
>
1
0.160***
(0.047)
Constant
0.322*** 0.779*** 0.659*** 0.676***
(0.122)
(0.087)
(0.043)
(0.039)
R
2
0.067
0.034
0.033
0.035
Observations
343
343
343
343
Joint significance test for interiorness and patience /
α
and
δ
30
F
statistic
8.641
5.073
Two sided
p
-value
<
0.001
0.007
Notes:
The dependent variable takes the value 1 if a participant selected one of the smooth payment plans, 0
otherwise. Interiorness takes values between 0 and 1, and it denotes the average share of the chosen option
that is allocated equally between soon and later payments. Patience takes values between 0 and 1, and it
denotes the average share of budget allocated to later payment. Present biasedness takes values between -1
and 1, and it denotes the average difference in patience between choices among options that have the same
delay between soon and later payments and the same interest rate (the corresponding choices among options
differ only because one involves the present in the soon amount, and the other does not). Ordinary least
squares (OLS) estimates are presented, for which heteroscedasticity-consistent standard errors are reported in
parentheses. *, ** and *** indicate significance at the 10%, 5% and 1% levels, respectively.
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48
C.1 Individual Measurement Error
Our aggregate analysis provides statistical comparisons for estimated preference parameters
and plan values between individuals who chose smooth plans and those who chose the single
plan. As such, this aggregate analysis accounts for estimation errors in the preferences and
plan values. The individual analysis presented in Table 4 shows differences in preferences and
plan values at the
point
estimates of individual preferences. Individual estimation error could
potentially alter the conclusions reached.
To evaluate the robustness to individual measurement error in preferences, we draw 1,000
simulants for each of the 348 individuals in our individual analysis. Each simulation has pa-
rameter values drawn from a multivariate normal distribution centered at the point estimates
for
α
and
δ
of the individual with covariance matrix determined by the estimated covariance
matrix for the individual parameters. At these simulated parameter values, we construct a
simulated Plan Value Ratio (PVR). Figure C.1 graphs the CDF of the 348
×
1000 = 348,000
simulated PVR values separately by individuals who chose the single plan or one of the smooth
payment plans. Allowing for the variation in PVRs induced by individual estimation errors, we
continue to reject the null hypothesis of equal distributions of PVRs across these two groups
(Mann–Whitney test,
z
=
−
103
.
48
,
(
p <
0
.
001)).
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49
Figure C.1: CDF of simulated Plan Value Ratios (PVRs) for chose single and chose smooth
groups.
Notes
: The figure graphs the CDF of the 348
×
1000 = 348,000 simulated PVR values
separately by individuals who chose the single plan or one of the smooth payment plans.
C.2 Transaction Costs and Plan Choice
Our implemented CTB design for eliciting preferences focuses on equalizing transaction costs
at each payment date by providing minimum payments. For payment plan choice, we do
not require subjects to receive some minimum payment at every potential date. As such,
some payment plans may incur different transaction costs than others. For example, the single
payment plan only requires one trip to the bank, while the twelve payment plan requires twelve.
These additional transaction costs are not modeled in our main analysis estimating plan values.
In Figure C.2, we incorporate transaction costs into our analysis determining plan values.
The left hand side of Figure C.2 presents the distribution of predicted plan choices for each
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50
individual, based on the estimated
PV R
i
for the pooled sample (top row), and separately for
the individuals who chose the single (second row) and smooth (bottom row) payment plans.
This graph highlights the stark prediction that all participants are either predicted to have
the single or the twelve payment plan as the highest value plan. Indeed, within the range of
our parameters, a stark cutoff exists in (
α,δ
) space below which the single plan is predicted
to have the highest value, otherwise the twelve payment plan is valued highest. In Figure C.3,
Panel A, we make this clear by simulating 100,000 preferences from two independent uniform
distributions:
α
∈
[0
.
1
,
1] and
δ
∈
[0
.
8
,
1
.
1]. Either the single plan or the twelve payment plan
should be chosen with no intermediate choices generated. Figure C.3, Panel B also plots the
next best alternative, with intermediate options of the two and six payment plans being the
second highest value plan for large swathes of the parameter space.
Figure C.2: Predicted plan choices from simulations based on group estimates.
51
Figure C.3: Simulated Preferences and Plan Choice.
Next, we incorporate transaction costs into the analysis. We use each individual’s estimated
parameters 24 times and draw random transaction cost payment parameter from a Poisson
distribution with a mean of GTQ25. We substract the transaction cost parameter from the
money amount in each date presribed by a plan, and are thus included in each simulant’s plan
values.
26
Using the same simulants that generated the left side of Figure C.2 and transaction costs,
we then calculate payment plan values including these transaction costs as:
V
j,i
=
[
l
∑
k
=
s
δ
k
−
s
i
(
x
t
+
k
;
j
−
1
x
t
+
k,j
>
0
∗
c
i
)
α
i
]
1
/α
i
,
(5)
where
i
refers to the simulant, and
c
i
is the simulated transaction cost incorporated into plan
j
if a payment is prescribed. The right side of Figure C.2 shows the influence of these modest
transaction costs. Generally, transaction costs increase the attractiveness of intermediate plans
with fewer payments. Moreover, for those individuals who chose the single payment plan, it
26
Naturally, these additional simulated costs are arbitrary and non-exhaustive of all possible costs and benefits
that can affect payment plan choice in addition to preference parameters and are intended for illustrative
purposes.
52
makes that plan even more attractive.
The analysis to here shows that moderate transaction costs may make intermediate and
single plans more attractive, but incorporating such transaction costs into the analysis does
not alter our core conclusion. Plan values for individuals who chose single and smooth plans
are predicted to deviate substantially.
Interestingly, these transactions costs may actually be an important driver of behavior
within the smooth plans. Only a small fraction of individuals actually choose the twelve pay-
ment plan. Moreover, following the results in Tables 2 and 4, correlations exist between CTB
behavior and plan choice within intermediate plans. For smooth payment plan subjects, the
number of payments correlates significantly with estimates of
α
and
δ
30
.
27
Such correlations
would be expected if there was an overarching avoidance of the most smooth plan due to fre-
quent incurrence of transaction costs. As a further investigation of the possibility that plan
choice is related to transaction costs empirically, we asked subjects the amount of time it would
take to get to a bank. For the subsample who chose smooth payment plans, regressing the
number of payments on an indicator for whether the individual would need more than 30 min-
utes to get to the bank (controlling for estimated preference parameters) yields a coefficient of
−
0
.
495 (robust s.e. = 0.19), which is statistically significant at the one percent level.
C.3 Robustness to Alternate Presentation Treatments for CTB
Our CTB design features several, cross-randomized presentation treatments. As noted in Sec-
tion 2.2, subjects were randomized into seeing budgets with the sooner amount either increasing
or decreasing as they moved down the task; seeing budgets in order of increasing or decreas-
ing marginal rate of transformation,
P
; and seeing the budget options with or without their
participation payment included.
Table C.2 presented below replicates the structure of Table 5, adding covariates to the
27
A regression of number of payments on these two preference parameters and a constant for the smooth
payment subsample yields a coefficient for
δ
30
of
−
0
.
11 (robust s.e. = 0.02) and a coefficient for
α
of
−
3
.
88 (1
.
64).
Both coefficients differ significantly from zero at the one percent level.
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53
regression (in column (4) of the results of Table 3) of observed plan choice on predicted plan
choice. Specifically it adds a dummy of the presentation treatments to address concerns of
alternate presentation effects affecting our results. Each column adds a single presentation
treatment covariate and its interaction with the PVR based prediction of a smooth payment
plan choice (
PV R
i
>
1).
Table C.2: Robustness and Alternative Presentation Treatments for CTB
Dep. Var.:
Chose Smooth Plan
Alternate CTB Presentation Treatments
(1)
(2)
(3)
PV R
i
>
1
0.274***
0.200***
0.132**
(0.062)
(0.061)
(0.062)
Decreasing soon amount
0.215***
(0.071)
P
decreasing order
0.116
(0.073)
Participation Payment included
-0.031
(0.074)
Interactions
-0.261***
-0.114
0.038
(0.084)
(0.087)
(0.088)
Constant
0.591***
0.637***
0.702***
(0.053)
(0.051)
(0.050)
# of Observations
408
408
408
Adjusted R-squared
0.053
0.032
0.025
Notes:
The
dependent variable takes the value 1 if a participant selected one of the smooth payment plans, 0 otherwise.
PVR
>
1 is a predicted smooth choice dummy; it takes the value of 1 if the average Plan Value Ratio for
smooth is greater than for single, otherwise, it takes the value of 0. Interactions present the interaction of the
relevant independent variable for each column, interacted with the dummy for predicted smooth choice (based
on PVR
>
1). Ordinary least squares (OLS) estimates are presented, for which heteroscedasticity-consistent
standard errors are reported in parentheses. *, ** and *** indicate significance at the 10%, 5% and 1% levels,
respectively.
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54
One of the presentational treatments does alter plan value choices and estimated values of
smooth and single payment plans. The first column presents the treatment varying the order of
the sooner amount. The results show that the decreasing soon amount presentation does have a
differential effect on the predictive power of our PVR measure. Columns 3 and 4 illustrate that
the other two presentation treatments do not have a relevant influence on payment plan choices
nor on estimated plan values (PVR). Importantly, however, in every specification, the coefficient
of
PV R
i
>
1 remains economically and statistically significant across all specifications. These
results help to ensure the robustness of our results, as the differences between payment groups
are not localized to a single type of CTB elicitation.
This may be instructive for knowing which type of presentation is most well-calibrated for
subsequent prediction. Our findings suggest that presentations with sooner amounts increas-
ing within budgets may be best suited for delivering well-calibrated estimates and discernible
differences across groups.
C.4 Robustness to Alternate Estimation Strategies
Andreoni and Sprenger (2012) and Andreoni, Kuhn and Sprenger (2015) discuss a number of
estimation strategies for CTB data. Although we follow Andreoni, Kuhn and Sprenger (2015)
in using simple OLS analysis, our exercise could be conducted with alternative methods. In
Table C.3, we provide an aggregate analysis using non-linear least squares estimation of the
solution function for
x
∗
t
,
x
∗
t
=
ω
+ (
Pβ
t
0
δ
k
)
1
/
(
α
−
1)
(
M
−
ω
)
1 + (
Pβ
t
0
δ
k
)
1
/
(
α
−
1)
,
where
ω
is a Stone–Geary background parameter term (Geary, 1950; Stone, 1954) that is fre-
quently found in the literature and imposed to be either minus some daily level of consumption
or a minimum subsistence level (Andersen, Harrison, Lau and Rutstrom, 2008; Andreoni, Kuhn
and Sprenger, 2015) or estimated from the data (Andreoni and Sprenger, 2012). With
ω
= 0,
we reproduce the directional differences in curvature and discounting previously observed, al-
though with less precision. And, although differences in plan values are observed across groups,
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55
the single payment choice group is estimated to value smooth payment plans more than the
single payment plan. Varying the assumption of the background parameter
ω
, as in Andreoni
and Sprenger (2012), alters the conclusions with respect to curvature, discounting and plan
values. For example, with
ω
= 10, curvature estimates are quite similar across the payment
groups, while differences in discounting and plan values are broadly in line with our prior aggre-
gate results. Estimating
ω
from the data
28
we find
ω
around GTQ13-14 for each group. Based
on monthly income data, this is around one-third to one-half of a day’s household income, a
plausible minimum subsistence level. When estimating
ω
, we find curvature estimates close to
those from our initial specifications similar across groups, clear differences in discounting across
groups, and plan values broadly in line with our prior estimates.
Harrison, Lau and Rutstrom (2012) present an alternative estimation strategy rather than
the strategies implemented by Andreoni and Sprenger (2012), which yields increasing, rather
than modestly diminishing, marginal utility. The estimation strategy is similar to the random
choice models of Holt and Laury (2002) and Andersen, Harrison, Lau and Rutstrom (2008),
applied to the Andreoni and Sprenger (2012) data.
29
Recent theoretical work has called into
question the use of such random choice models for estimating preferences given a demonstrated
non-monotonicity in choice probabilities with respect to key parameters of interest (Apesteguia
and Ballester, 2018). Nonetheless, Table C.4 compares the results of the Harrison, Lau and
Rutstrom (2012) estimator with the OLS estimator presented in Table 4, and the NLS estimator
presented in Table C.3. As in Harrison, Lau and Rutstrom (2012), using such ML methods on
CTB data yields convex utility estimates, as the estimator attempts to match the slight majority
(54%) of observations at budget corners. These convex utility estimates are virtually unaffected
by the differential price sensitivity across groups, with both single and smooth payment subjects
having substantial estimated convexity that cannot be differentiated statistically. Although
28
Andreoni and Sprenger (2012) note that unlike the other parameters of interest, there is no experimental
variation that identifies
ω
. It is identified from the functional form.
29
From this exercise Harrison, Lau and Rutstrom (2012) conclude the following: “And we believe that rejecting
concave utility in favor of convex utility, in settings such as these, will strike most economists as a priori
implausible, and raise further questions about the comprehension of subjects of this new experimental task.”
(p. 21)
56
Table C.3: Alternative Estimation Strategies
Estimation Strategy
NLS
NLS
NLS
NLS
NLS
Background Parameter (
ω
)
ω
= 0
ω
= 10
ω
= 5
ω
=
−
5
Estimated
(1)
(2)
(3)
(4)
(5)
Curvature:
α
Chose Single Payment
0.762
0.856
0.812
0.711
0.881
(0.020)
(0.016)
(0.018)
(0.022)
(0.013)
Chose Smooth Payments
0.737
0.867
0.805
0.665
0.904
(0.013)
(0.010)
(0.012)
(0.015)
(0.006)
Monthly Discount Factor:
δ
30
Chose Single Payment
0.924
0.931
0.928
0.922
0.933
(0.017)
(0.015)
(0.016)
(0.018)
(0.014)
Chose Smooth Payments
0.992
0.983
0.976
0.997
0.982
(0.009)
(0.006)
(0.008)
(0.011)
(0.005)
Present Bias:
β
Chose Single Payment
1.095
1.075
1.084
1.103
1.070
(0.029)
(0.024)
(0.027)
(0.031)
(0.023)
Chose Smooth Payments
1.068
1.040
1.054
1.080
1.031
(0.014)
(0.010)
(0.013)
(0.016)
(0.009)
Background Parameter:
ω
Chose Single Payment
13.352
(1.336)
Chose Smooth Payments
14.363
(0.696)
Plan Value Ratio (PVR)
Chose Single Payment
1.315
1.087
1.179
1.500
1.044
(0.095)
(0.048)
(0.066)
(0.139)
(0.044)
Chose Smooth Payments
1.729
1.215
1.370
2.252
1.126
(0.090)
(0.028)
(0.048)
(0.171)
(0.018)
# Observations
9,789
9,789
9,789
9,789
9,789
# Clusters
408
408
408
408
408
H
0
:
α
ChoseSingle
=
α
ChoseSmooth
;
χ
2
(1)
1.11
0.30
0.11
2.99
2.57
(
p
= 0
.
29) (
p
= 0
.
58) (
p
= 0
.
74) (
p
= 0
.
08) (
p
= 0
.
11)
H
0
:
δ
30
ChoseSingle
=
δ
30
ChoseSmooth
;
χ
2
(1)
11.90
10.46
7.39
12.52
10.48
(
p <
0
.
01) (
p <
0
.
01) (
p <
0
.
01) (
p <
0
.
01) (
p <
0
.
01)
H
0
:
β
ChoseSingle
=
β
ChoseSmooth
;
χ
2
(1)
0.69
1.79
1.05
0.45
2.49
(
p
= 0
.
41) (
p
= 0
.
18) (
p
= 0
.
30) (
p
= 0
.
50) (
p
= 0
.
12)
H
0
:
PV R
ChoseSingle
=
PV R
ChoseSmooth
;
χ
2
(1)
9.95
5.31
5.48
11.65
2.99
(
p <
0
.
01) (
p <
0
.
05) (
p <
0
.
05) (
p <
0
.
01) (
p <
0
.
10)
Notes
: Estimates are based on non-linear least squares (NLS) regression of solution function with alternative values for a Stone–
Geary background parameter (
ω
), with standard errors that are clustered on individual level. Plan values are calculated from
non-linear combinations of estimated parameters.
57
there are sizable estimated differences in discounting across groups, the ML method overall
leads to sharp mispredictions for the level of payment plan values. Both groups are predicted to
value the single payment contract substantially more than smooth contracts, as the
PV R <
1
for both groups. However, the
Choose Single
group still has a lower PVR than the
Choose
Smooth
group (
χ
2
= 9
.
59
,p
−
value
<
0
.
01).
C.5 Exploring Alternative Information on Curvature
Our exercise identifies curvature from a desire to smooth intertemporal payments in the CTB.
Experimental designs that leave curvature unmeasured require additional assumptions to make
predictions for decisions such as payment plan choice. Table C.5 examines the effects of impos-
ing alternative assumptions for curvature. Specifically, we fix
α
at different values and show
that these assumptions deeply influence the results.
In the first column we impose
α
= 0
.
5 for all subjects.
30
In line with the discussed con-
founding effects of curvature, if marginal utility diminishes at the implied rate, individuals
are estimated to be extremely patient. Indeed, for the 78% of subjects who choose smooth
payments we estimate an aggregate monthly discount factor of around 1.7 and a remarkably
high degree of future bias. Furthermore, the implied valuations for smooth payments strain
plausibility with both smooth and single payment groups estimated to value smooth payment
plans much more than the single payment.
31
In columns (2) to (5) of Table C.5, we fix
α
at additional values of 0
.
75
,
0
.
9
,
0
.
95
,
and
0
.
99. This assumption tunes estimates of patience, payment plan values, and the differences
between groups. Based upon the assumption of curvature, researchers can predict that both
groups will prefer smooth contracts or both groups will prefer single contracts with potentially
indiscernible differences in the strength of preference. Only in special case of
α
at around 0.9
30
This is in line with estimates from risky choice tasks such as those estimated by Andreoni and Sprenger
(2012); Andreoni, Kuhn and Sprenger (2015).
31
Given the prior findings on levels of risk aversion and a lack of correlation between risky choice tasks and
curvature in CTBs, this suggests that using risky choice information could lead to substantial misprediction for
such large-stakes intertemporal choices.
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Table C.4: Comparison with ML Methods
Estimation Strategy
HLR (2012) MLE (
ω
= 0) OLS (
ω
= 0) NLS (estimated
ω
)
(1)
(2)
(3)
Curvature:
α
Chose Single Payment
1.752
0.911
0.881
(0.169)
(0.007)
(0.013)
Chose Smooth Payments
1.503
0.878
0.904
(0.089)
(0.007)
(0.006)
Monthly Discount Factor:
δ
30
Chose Single Payment
0.835
0.893
0.933
(0.043)
(0.029)
(0.014)
Chose Smooth Payments
1.007
1.020
0.982
(0.023)
(0.023)
(0.005)
Present Bias:
β
Chose Single Payment
1.218
1.177
1.070
(0.088)
(0.056)
(0.023)
Chose Smooth Payments
1.190
1.159
1.031
(0.045)
(0.032)
(0.009)
Noise Parameter:
μ
Chose Single Payment
0.426
(0.066)
Chose Smooth Payments
0.427
(0.046)
Background Parameter:
ω
Chose Single Payment
13.352
(1.336)
Chose Smooth Payments
14.363
(0.696)
Plan Value Ratio (PVR)
Chose Single Payment
0.451
0.900
1.044
(0.038)
(0.064)
(0.044)
Chose Smooth Payments
0.623
1.306
1.126
(0.040)
(0.085)
(0.018)
# Observations
9,789
9,789
9,789
# Clusters
408
408
408
H
0
:
α
ChoseSingle
=
α
ChoseSmooth
;
χ
2
(1)
1.69
12.22
2.57
(
p
= 0
.
19)
(
p <
0
.
01)
(
p
= 0
.
11)
H
0
:
δ
30
ChoseSingle
=
δ
30
ChoseSmooth
;
χ
2
(1)
12.20
12.12
10.48
(
p <
0
.
01)
(
p <
0
.
01)
(
p <
0
.
01)
H
0
:
β
ChoseSingle
=
β
ChoseSmooth
;
χ
2
(1)
0.09
0.08
2.49
(
p
= 0
.
77)
(
p
= 0
.
78)
(
p
= 0
.
12)
H
0
:
PV R
ChoseSingle
=
PV R
ChoseSmooth
;
χ
2
(1)
9.59
14.48
2.29
(
p <
0
.
01)
(
p <
0
.
01)
(
p <
0
.
10)
Notes
: Estimates are based on Maximum Likelihood (ML) methods of Harrison, Lau and Rutstrom (2012) (HLR), ordinary least
squares (OLS) regression, or non-linear least squares (NLS) regression of solution function with standard errors that are clustered on
individual level. Plan values are calculated from non-linear combinations of estimated parameters.
59
Table C.5: Alternative Curvature Information
Fixing
α
Estimated
α
(1)
(2)
(3)
(4)
(5)
(6)
Curvature:
α
Chose Single Payment
0.500
0.750
0.900
0.950
0.990
0.887
(NA)
(NA)
(NA)
(NA)
(NA)
(0.005)
Chose Smooth Payments
0.500
0.750
0.900
0.950
0.990
0.887
(NA)
(NA)
(NA)
(NA)
(NA)
(0.005)
Monthly Discount Factor:
δ
30
Chose Single Payment
1.051
0.952
0.897
0.879
0.865
0.901
(0.185)
(0.084)
(0.031)
(0.015)
(0.003)
(0.036)
Chose Smooth Payments
1.718
1.216
0.989
0.923
0.873
1.007
(0.140)
(0.050)
(0.016)
(0.008)
(0.001)
(0.020)
Present Bias:
β
Chose Single Payment
2.822
1.657
1.204
1.083
0.994
1.238
(0.785)
(0.231)
(0.067)
(0.030)
(0.006)
(0.077)
Chose Smooth Payments
1.988
1.391
1.122
1.045
0.987
1.144
(0.226)
(0.079)
(0.025)
(0.012)
(0.002)
(0.029)
Plan Value Ratio (PVR)
Chose Single Payment
7.434
1.471
0.924
0.824
0.760
0.955
(6.882)
(0.369)
(0.069)
(0.028)
(0.005)
(0.085)
Chose Smooth Payments
198.778
3.350
1.155
0.909
0.773
1.239
(128.823)
(0.527)
(0.046)
(0.015)
(0.002)
(0.069)
# Observations
9,789
9,789
9,789
9,789
9,789
9,789
# Clusters
408
408
408
408
408
408
H
0
:
δ
30
ChoseSingle
=
δ
30
ChoseSmooth
;
χ
2
(1)
6.49
6.46
6.44
6.44
6.47
6.51
(
p <
0
.
05) (
p <
0
.
05) (
p <
0
.
05) (
p <
0
.
05) (
p <
0
.
05)
(
p <
0
.
05)
H
0
:
β
ChoseSingle
=
β
ChoseSmooth
;
χ
2
(1)
1.04
1.19
1.30
1.35
1.48
1.29
(
p
= 0
.
31) (
p
= 0
.
27) (
p
= 0
.
25) (
p
= 0
.
25) (
p
= 0
.
22)
(
p
= 0
.
26)
H
0
:
PV R
ChoseSingle
=
PV R
ChoseSmooth
;
χ
2
(1)
2.20
8.54
7.83
7.13
6.61
7.91
(
p
= 0
.
14) (
p <
0
.
01) (
p <
0
.
01) (
p <
0
.
01) (
p <
0
.
05)
(
p <
0
.
01)
Notes
: Not Available (NA). Estimates are based on ordinary least squares (OLS) regression of equation (3) with standard errors that are clustered
on individual level. Utility estimates and plan values are calculated from non-linear combinations of regression coefficients. The standard errors
reported in parentheses are calculated using the delta method. Null hypotheses tested after regression of equation (3) with interactions for plan
choice with
k
,
t
0
and ln(
P
), with standard errors clustered at individual level.
60
will the assumed value of curvature correctly tune patience to successfully differentiate between
payment plan groups. Incidentally, as shown in column (6) of Table C.5, this is close to what is
estimated on aggregate using curvature information inherent to our CTB elicitation strategy.