ashley, ball and eckel

8/7/2019 Ashley, Ball and Eckel

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Analysis of Public Goods Experiments

Using Dynamic Panel Regression Models*

Richard Ashley

Sheryl Ball

Catherine Eckel

Department of Economics, Virginia Tech

Abstract

Laboratory experiments on the provision of public goods follow each subject over time as he interacts

with a small group of others, deciding in each period how much of an initial allocation to contribute to a

group good that yields a return to all group members. These experiments produce data sets that are rich indynamics, as subjects respond not only to the parameters of the experiment, but also to previous

allocation decisions made by themselves and the other individuals in their group. In most early studies,

the data analysis consisted of an informal inspection of the time plots of data for each treatment, averaged

over all participants. Subsequent studies conducted simple statistical tests of average behavior. There hasbeen very little analysis of the individual data. Data sets from these experiments thus provide an

unexploited opportunity to understand how an individuals behavior evolves over time in response to

feedback about the behavior of others. A better understanding of the dynamics of behavior in these

games has the potential to lead to the design of institutions that improve the efficiency of the private

provision of public goods.

Proper analysis of these data presents several challenges. In addition to modeling the (possibly complex)

interactions among agents, the data are almost always substantially censored from both above and below.We analyze the results of two classic public goods experiments (Isaac and Walker (1988) and Andreoni

(1995)) explicitly as double-censored panel data in which both lagged dependent variables and laggedallocations from the remainder of the group play important roles as explanatory variables. We find that

modeling the dynamic interactions and double censoring extract a richer set of results than previously

seen. In particular, failure to take into account the censoring of the data may substantially underestimate

the magnitude of treatment effects. More importantly, there are economically and statistically significant,

and asymmetric, cross-subject dynamics in these data. Subjects respond much more dramatically whentheir contributions are above average than when they are below average. Thus heterogeneity in initial

contribution levels inherently leads to the deterioration of average contributions over time.

JEL Codes: C920, C230, C240, C700

Key Words: Public Good, Voluntary Contributions Mechanism, Panel Data, CensoringCorrespondence: Catherine C. Eckel, Department of Economics (0316), Virginia Tech, 3016 Pamplin

Hall, Blacksburg, VA 24061. ([email protected])

We thank Jim Andreoni and Jimmy Walker for giving us access to their data. John Pepper provided very helpful

comments on an earlier draft. Any remaining shortcomings are, of course, our own. Eckel was supported by a grant

from the National Science Foundation, SES 0094800. This manuscript is VPI Economics Department Working

Paper E2003-8 and is available for download at http://ashleymac.econ.vt.edu/working_papers/E2003_8.pdf.


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1. Introduction

One of the primary social problems studied by economists is the issue of how to fund the

provision of public goods. What makes the public goods problem so interesting to the dismal

science is the especially dismal theoretical prediction that the level of voluntary contributions

should be zero, resulting in the public good never being produced. The theorists approach to

solving this problem has been to design a mechanism that gives agents the incentive to contribute

enough to produce the socially optimal level of the public good (e.g., Laffont, 2000; Chander,

1993; Hurwicz and Walker, 1990; Groves and Ledyard, 1980).

Experimental economists have instead focused on the Voluntary Contribution

Mechanism (VCM), an institution that mimics the incentive structure of a public goods problem

(see Ledyard, 1995 for a survey). Experimental results typically show that contributions within

the VCM framework are not as low as theorists expect, but nevertheless fall much below socially

optimal levels. While a great deal has been discovered about average behavior in this simple

game across a broad range of variations, there is still much to be understood. For example,

overall contributions nearly always deteriorate over time, but as researchers we do not know why

or by what mechanism this occurs. Behavior is highly heterogeneous across subjects, but we

have been largely unable to explain the nature and sources of the heterogeneity. It is clear that

subjects respond to each others actions, but details of the interactions are not well understood.

In this paper we conduct a more in-depth analysis of VCM data from two classic studies in an

attempt to better understand the underlying behavior.

An understanding of what information agents take into account and how they respond to

information can help economists to design mechanisms that encourage the provision of public

goods. Public goods experiments have already shown the critical importance of structural


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factors such as provision points (Marks and Croson, 1999; Rondeau, Schultze and Poe, 1999),

and social factors such as communication (Ostrom, 1990; Ostrom, et al, 1994; Wilson and Sell,

1997) and knowledge of the identity of ones group members (Andreoni and Petri, 2003). Our

work highlights the importance of information about the behavior of other group members, and

the heterogeneity among agents in their initial contributions, as well as responses to changes in

the environment. Fundraisers seem to be well aware of the importance of information about

others contributions, and actively manipulate information in their fundraising activities.

Manipulating information can significantly affect the level of contributions through the selective

announcement of others contributions (Harbaugh, 1998) or the announcement of an initial large

contribution (Vesterlund, 2003; Andreoni, 2002).

In this paper we suggest that data sets from VCM experiments can best be understood by

approaching them explicitly as doubly-censored panels, in which both lagged dependent

variables and lagged allocations from the subjects group play important roles as explanatory

variables. We estimate these models using data from two classic VCM experiments conducted

by Isaac and Walker (1988) and Andreoni (1995). Our models allow us to make a rich set of

inferences about the intensity and asymmetry of individual responses to the recent behavior of

the remainder of the individuals group. Subjects respond much more dramatically when their

contributions are above average than when they are below average. Thus heterogeneity in initial

contribution levels inherently leads to the deterioration of average contributions over time.

In addition, our models reproduce the qualitative results of both studies as to the impacts of the

experimental treatments they considered, but in a more powerful and more statistically credible

framework. Finally, we show that failure to take into account the substantial censoring of the

data may lead to underestimates of the magnitude of treatment effects.


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2. VCM Games and Data Analysis

Each subject in a VCM experiment is followed over time as he interacts with a small

group of other subjects, deciding on how much of an initial allocation to contribute each period

to a group good that yields a return to all members of the group. The individual allocation

decisions in such settings typically exhibit very strong serial correlation. In addition because

subjects respond to the behavior of the rest of the group individual allocations are significantly

related to previous allocation decisions made by others. Moreover, data collected in these

settings are almost always substantially censored: from below by the fact that an individuals

allocation to the group good is constrained to be non-negative, and from above by the fact that an

individuals allocation to the group good cannot exceed his endowment for that period.

The analysis of these VCM data often consists of little more than an informal inspection

of a few time plots or testing whether the mean or median time-averaged allocations of

individuals differ by experimental treatment. The former approach is simple, but forgoes

statistical analysis altogether, thereby disregarding much of the information in the sample data.

The latter approach discards all of the dynamic information in the sample data and also violates

the assumptions underlying the means/medians tests by ignoring the across-individual

correlations in the data induced by their within-group interactions. Both of the studies whose

data we reanalyze present only limited analysis of their data, using techniques such as these.

(Andreoni, 1995; Issac and Walker, 1988).

Several recent papers take a more sophisticated approach. 1 Clark (2002) uses OLS

regression tocompare contribution rates in a VCM experiment when each groups highest

contributor in a given period is announced and when subjects can reward the highest contributor

1Anderson, et al, (1998) present an alternative approach to the analysis of VCM contributions that focuses on

decision errors.


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in their group. He fails to account for dynamics or for the apparent censoring in the data,

however. Solow and Kirkwood (2002) look for relationships between gender, group identity,

and economic behavior in VCM experiments. They account for the fact that their sample data is

censored but, like Clark (2002), do not consider the dynamics. Neither paper accounts for the

possible importance of individual effects due to the panel structure of the data.

Gunnthorsdottir, Houser, McCabe and Ameden (2001) estimate dynamic decision-rule

regressions using VCM data. In a fashion that they note is somewhat arbitrary, they first bisect

their sample into cooperator subjects (who contribute more than 30% of their initial allocation

in period one) and free rider subjects (who contribute less than this in the first period). They

then estimate two separate dynamic regression models to explain the sample variation in the

individual contributions in each period one regression for the cooperator data and one for the

free rider data. They use Tobit regression to account for the substantial double censoring in their

data and attempt to model both the serial and across-subject dynamics in the data using functions

of lagged dependent variables. They then informally compare the sizes and t-ratios of the

estimated coefficients in the two regressions. Gunnthorsdottir, et al, (2001) do not, however,

effectively consider the panel nature of their data in that they ignore the fact that regardless of

whether the ith subject is a cooperator or a free rider this subjects group-good allocation in

period t is substantially determined by a time-invariant person-specific (fixed) effect.

Consequently, the error term in their regression equations is correlated across time for each

person, which violates the assumptions of the Tobit regression framework. Our approach

substantially alleviates this problem.

We chose data sets from two well-known studies for our analysis (Isaac and Walker,

1988; and Andreoni, 1995) because of their importance in the area of VCM studies, and because


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the data sets are readily available. Both experiments were computerized. Subjects were given an

endowment (E) each period, which could be allocated between a private account and a group

account. Contributions to the group account paid to each member of the group the marginal per

capita return (MPCR) times the total of the groups contributions.

Isaac and Walker (IW) test the effect on contributions of changing group size from 4 to

10, and of changing the MPCR from .3 to .75, using a 2x2 experimental design. Within a

session, stable groups of a given size experience ten periods of one value for MPCR, followed by

ten periods of the other value; the order of the MPCR values is blocked. Thus their data consist

of 20 decision periods for each of 84 subjects. Andreoni (1995) tests the effect of positive and

negative framing on contributions, holding group size and MPCR constant. In the positive

frame, the level of the public good is determined by total contributions to the group account. In

the negative frame, subjects withdraw amounts from the group account and transfer it to their

private account. The amount remaining after withdrawals determines the level of the public

good. In Andreonis experiment, subjects are rematched each period into new groups.2

His is a

between-subjects design, so his data consist of ten periods for 80 subjects, 40 in each treatment.

3. A Panel Regression Model for VCM Contributions

We model tiC, , subject is contribution to the group good in period t, expressed as a

fraction of the subjects total period t endowment. The observed contribution is equal to Ci,t*

, a

latent variable that could be characterized as desired contribution, if and only if 10 *, tiC .

Thus,

tiC, = 1 ifCi,t*

> 1

2Some inconsistencies in Andreonis rematching protocol were found in our analysis, but none significantly

affected the statistical results.


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tiC, = Ci,t*

if 10 *, tiC

Ci,t = 0 ifCi,t*

< 0

and Ci,t

*is determined by:

ti

k

i

m

j

i

tiititititi DCCC ,1 1

,2,21,1

*

, +++++= = =

],0[~2

, NIIDti (1)

where the Xt1, Xt2,Xtn include both time invariant variables such as treatment effects, and

dynamic variables such as the deviation of subject is contribution from that of the remainder of

the group in round t-1. itD is a dummy variable specific to subject i, and subsumes all time-

invariant characteristics of the subject. This model allows for both the individual-specific fixed

effects (1, 2,...m) and the strong serial correlation that we expect (and find) in the IW and

Andreoni Ci,t data. This is a standard fixed-effects dynamic panel data regression model, but (so

far as we know) such models have never been considered before with double censoring. Still, in

principle, this model is estimable using the usual maximum likelihood methods for Tobit Type I

models.3

Because of the censoring, some care must be taking in interpreting the estimated

coefficients in this model. To see this consider a simpler model which retains the double

censoring but suppresses all but one explanatory variable.

ci*

= + xi

+ i i~ NIID(0,2) (2)

c i = c i*

for c i*

(0,E)

= 0 for c i* 0

3See Maddala (1983, pp. 160-1 and 186) for the likelihood function, and Amemiya (1973) for a theoretical analysis.

Further complications ensue if lagged values of the latent variable (Ci,t1*

and Ci,t2*

) are used instead of lagged

values of the observed variable. Arellano, et al. (1999) consider a random-effects formulation for this case with one-


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= E for ci* E

This model implies that the ci* are all independently distributed N[+ x

i,2]; thus

(ci* xi) / is a unit normal and ]|*[ ixicE is just + xi .

Clearly, E[ci* | x ] /x i is simply and quantifies the sensitivity of subject is desired

contribution to changes in xi . It is this sensitivity, of course, which is ordinarily the focus of

economic analysis.

Note, however, that the sensitivity of ci, subject is observed contribution to change in

xi, is not equal to the parameter , because observed contributions equal desired contributions

only when ci* is in the interval (0,E). Outside of this interval the observed contribution is a

constant either 0 or E, depending on whether ci* is less than zero or greater than E.

Consequently, the magnitude of the apparent dependence of subject is contribution on the

explanatory variable is (on average) diminished whenever + xi either falls close to (or lies

below) zero or approaches (or exceeds) E. In particular, since (ci* x

i) /~ N(0,1) , this

apparent sensitivity is

]|),0([*/]|[ *i

xEcyprobabiliti

xi

xi

cE i =

=

/)(

/)(

)(

i

i

xE

x

dzzf (3)

where f(z) is the unit normal density function. This probability peaks for xi such that

sided censoring and including only a single lag in the dependent variable; Honore (1993) considers a non-parametric

approach to this special case.


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+ xi = E/ 2 and diminishes to zero as |+ xi E/ 2 | increases. Intuitively, subject is observed

contribution becomes increasingly insensitive to xi as xi becomes sufficiently extreme that

subject i is likely to be either not contributing at all or contributing his entire endowment. This

apparent sensitivity is precisely what is needed so as to analyze the sensitivity of actual average

contributions to changes in explanatory variables - e.g. this sensitivity might be relevant in

predicting the cost of running a particular experiment - but it is itself which quantifies how the

subjects desired behavior depends on x.

Note also that using OLS to estimate the parameters and in the model

iibxa

ic ++= (4)

yields an estimate (bols) which is necessarily an inconsistent estimator ofE[ci

| x i]/x i. The

inconsistency follows directly from the fact that, as noted above, E[ci

| x i]/x i is a function of

xi , whereas plim )( olsb is a constant. In general bols is also an inconsistent estimator of , since it

ignores the distinction between ci

and ci*. The seriousness of these inconsistencies is most

sensibly gauged by comparing bols directly to the maximum likelihood estimator of in the

model given by equation (2); this is the comparison we make in our analysis below.

4. Descriptions of the data sets and variables:

The two data sets we examine are from experiments that were conducted to test different

hypotheses. IW examines the effect of group size (4 or 10) and MPCR (.3 and .75) on an

individuals contributions to the public good. As explained above, each session includes ten

periods with one value of MPCR followed by ten periods using the other value of MPCR. Group

size is varied across sessions. Explanatory variables for this data set include contributions, the


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treatment variables, and measures of how a subjects contribution in the previous round differed

from the average contributions for their group. MPCRhi is a dummy variable equal to 1 for the

high-mpcr sessions. Group size takes on the values 4 or 10 corresponding to the two group sizes.

The dummy variable First 10 periods was set to one in the first set of 10 periods and to zero in

the second set. For subject i, the lagged value of the deviation of his own contribution ( 1, tiC )

from the value of the average contribution of all other members of the group in that round is

denoted Deviation from Group(+) when this value is positive and is otherwise zero. Deviation

from Group(-) is similarly defined for the observations for which this value is negative and is

otherwise zero. These variables thus allow for a possible asymmetry in the responses. The

variable Ci,1 is the subjects initial contribution; this variable is used in Section 7.

Recall that the purpose of the Andreoni study is to test for the effect of positive or

negative framing on contributions to a pubic good. Positive framing is the standard VCM,

whereas subjects withdraw contributions from a common pool in the negative frame treatment.

Variables for this data set include the treatment variable Positive Frame, as well as the variables

defined above. The frame is varied across sessions, which last for ten periods.

Table 1 contains descriptive statistics on these variables for both data sets. Considering

how different these two experiments are, the descriptive statistics are remarkably similar.

Average contributions are 36.0 percent of the endowment in the IW data, and 24.9 percent in the

Andreoni data. The lower average contributions in the Andreoni data are probably are due to the

negative frame treatment, which reduces overall contributions. Some of the difference may also

be due to MCPR, which IW show affects contribution levels: IW use an MCPR of .3 and .75

while Andreoni uses a value of .5 for the MCPR. If we look at the average value the change in

contributions in each period, ci,t , we observe that it is identical in both data sets at -0.03,


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indicating that contributions typically fall by three percentage points per round. The deviations

from group variables also are very similar in the two data sets - 0.145 compared with 0.124 -

indicating a similar range of contributions for both studies. Finally, the initial contribution levels,

Ci,1, are higher, overall, for the IW data, again probably due to the lower contributions in

Andreonis negative frame treatment.

5. Estimation Results: Introducing Censoring and Panel Structure to the Basic Model

Table 2 contains OLS and censored panel estimates for what we term the basic model.

This model, estimated using OLS, is employed in most studies that go beyond graphical

comparisons or comparisons of central tendency (mean or median outcome comparisons) to

analyze individual data. The dependent variable for both data sets is tiC, the level of

contributions in round t expressed as a fraction of endowment. Independent variables include the

round number (a crude attempt capture any time trend in the data) and relevant treatment

variables: MPCRhi and Group size and their interaction in IW, and Positive Frame in Andreoni.

Model 2a is estimated using OLS and adjusts neither for two-sided censoring (at 0 or 100

percent donations) nor for the panel nature of the data. According to these estimates, in both

data sets contributions decrease over time, falling about 2.7 - 2.8 percentage points per round.

Both studies show statistically significant main treatment effects. In IW, a higher MPCR is

estimated to increase contributions by 51 percentage points. Larger group size has a positive

effect when MPCR is low moving from a group size of 4 to a group size of 10 increases

contributions by about 29 percentage points but this effect is offset by the negative coefficient

on the interaction term for the high MPCR case. This result explains why IW can only conclude

there appears to be no support for a pure numbers argument relating increases in group size to

increases in free riding behavior (p. 196). In Andreoni, the positive frame variable carries a


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positive, significant coefficient: the positive frame increases contributions by 17.4 percentage

points.

Model 2b for both data sets uses Tobit estimation to account for the two-sided censoring

in the data, and allows for the panel structure of the data sets by explicitly estimating individual-

specific fixed effects.4

A comparison of 2a and 2b allows us to examine the impact on the

estimates of these two modeling changes. First, note that censoring and fixed effects are clearly

present in both of these data sets: 49 percent of the IW observations are censored, as are 56

percent of the Andreoni observations. In addition, 52 and 67 percent of the fixed-effect dummy

variable coefficients in each of these data sets are significantly different from zero at the 5

percent level, and the joint null hypothesis that all of the fixed effects coefficients are zero is

rejected for each data set with p-value less than .001.5

Individual-specific effects and censoring are clearly present in both data sets. But is

failure to account for these features of the data consequential? We note that the estimates of the

coefficient on the round number variable are nearly twice as large (5-6 percentage points per

round versus 3 percentage points per round in the OLS estimates) once the fixed effects and

censoring are treated appropriately this is a difference of about 4 to 6 estimated standard errors

in each case. In the IW data the coefficient on the MPCR variable is nearly twice as large as in

the OLS estimates; again this is a difference of about 5 estimated standard errors. Furthermore,

the coefficient estimate on the group size variable is no longer significant. Turning to the

Andreoni data, the direction and significance of the treatment effect are unchanged, but as in the

4It should also be noted that fixed effect dummy variable coefficients are not identified for subjects whose

contribution is censored in every round, and are not, in practice, estimable for subjects whose contribution is

censored in every round but one. This was the case for a notable fraction of the subjects in the Andreoni study

(28/80) but only 2/84 for the IW study.5

The individual fixed effect coefficient estimates are suppressed in the table for brevity. The fixed effect dummy

for the last individual in each sample is omitted so that these estimates can be interpreted as the difference in the

intercept relative to this individual.


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IW data, failure to account for the censoring and panel structure substantially reduces the

estimated magnitude of the treatment effect: the coefficient on the positive frame variable is

.174.022 in the OLS model compared with .435.087 in the Tobit model. These results are

consistent with what one might expect from the analysis at the end of Section 3 and, in fact,

primarily stem from this source rather than from the inclusion of the fixed effect dummy

variables in the model.

6. Estimation Results: Introducing Lags and Group Feedback

Adjusting for panel structure and censoring is clearly appropriate for these data sets, but

the estimated models in Table 2 fail to allow for the possibility of correlation in a subjects

decisions over time. In theory the random rematching of the subjects should render the outcomes

independent in each round, since subjects should ignore information about one group in making

a decision with another. A cursory examination of the sample data indicates otherwise. To

allow for these serial correlations, we model the dynamics by including lagged values of the

observed contribution. Results incorporating these dynamics are reported in Tables 3 and 4.

Note that the number of observations is decreased to account for the calculation of the lagged

values. In both tables, results are again reported for both OLS and censored panel regressions.6

In the OLS models, the coefficients on the lagged values are positive and significant,

though their sum is less than one. (If the values sum to one, the model has a unit root.) Clearly

there is substantial correlation in contributions from one round to the next. By comparison with

Model 2a for both data sets in Table 2, we see that adjusting for autocorrelation improves the fit

of the model; the treatment effects now appear smaller in magnitude, however. Turning to the

censored panel estimation results, the biases in the estimates of the treatment effects from


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neglecting the censoring and panel effects is evident: the coefficients on all treatment variables

are larger in the censored panel models. The fixed effects are jointly significant in both data sets

as well.7

Recall that the IW subjects are in stable groups, while the Andreoni subjects are

randomly rematched. The dependence between rounds should be lower in the rematched data

(Table 4). However, that is not the case. The coefficients on the lagged contribution variables

are not substantially different across the two studies. Evidently, random rematching is not as

effective as one might have expected in inducing independence of observations between rounds.

Tables 5 and 6 report models that examine the dynamics in more detail. An additional

factor that is ignored by the basic model is the effect of recent history on subjects decisions.

Each subject experiences a different history, since the identities of the subjects constituting the

remainder of his group in the previous round are unique to him. We test the effect of this intra-

group interaction by incorporating the two additional variables, Deviation from Group (+) and

Deviation from Group (-) defined in Section 4. These estimates are reported for the censored

panel regression model in Tables 5 and 6. (OLS estimates are not reported; the biases are similar

to those reported above.)

Models 5a and 6a allow the subjects response to differ, depending whether his or her

contribution in the previous round was above or below the group average. A positive coefficient

on these variables indicates adjustment away from the group average, while a negative

coefficient means that the subject is adjusting his contribution toward the group average. In both

data sets, if a subjects contribution is above the group average, the estimated coefficients

6 Note that the variable Round Number is not included in the reported estimates. While retaining the Round Number variable inthe models improves the fit slightly, it leaves the coefficients unchanged in magnitude and significance, except for the firstvariable in the IW data, which changes sign.7 Adjusting for fixed effects alone increases the treatment effect slightly (to .078). Adding censoring alone raises the coefficientto .170. These alternative specifications are available on request from the authors.


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indicate that he will reduce his contribution significantly, by about 60 percent of the excess.

However, if his contribution is below the group average, there is no corresponding increase; the

insignificance of this coefficient for both studies indicates no response. The null hypothesis that

these two coefficients are equal is easily rejected: for the IW data, 2

(1) = 17.0, p < .001; and for

the Andreoni data, 2

(1) = 6.8, p = .009.8

As in the previous models, we again observe positive

and significant coefficients on the lagged values, and significant treatment effects.

Models 5b and 6b extend the analysis by allowing the response to group feedback to vary

by treatment. Here we interact the Deviation from Group variables with the treatment variables.

In the IW data, the estimated treatment effects appear relatively stable across the two

specifications. None of the interaction effects is significant on its own, but a LR test rejects the

null hypothesis that the effect of the interactions is zero (2

(4) = 11.2, p = .024).9

In the Andreoni data, for both frames the coefficient on Deviation from Group (+) is

negative, though the magnitude of the response is substantially greater in the negative frame.

Thus subjects in the negative frame are quicker to adjust their contributions downward, toward

the group average. If the subjects contribution last period was below average, the negative

coefficient on Negative Frame X Deviation from Group(-) indicates that the negative frame

subjects are still more responsive. They adjust their contributions toward the group average more

than the positive frame subjects, who do not respond significantly at all in this case. A

likelihood ratio test for the null hypotheses that coefficients are equal across treatments can be

rejected (2

(2) = 12.0, p = .002). Thus we can conclude that the response to feedback differs by

treatment; Andreonis framing treatments affect contributions in part by affecting the subjects

8In both cases the restricted model (not shown) yields a negative coefficient estimate that is smaller in magnitude

but still significantly different from zero at p


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response to what others are doing.

7. Does the initial contribution level signal type?

Each of the fixed effect models above shows that there is heterogeneity among subjects.

The evidence adduced above that subjects react to feedback about their groups contribution

level suggests that a stable group consisting of relatively high initial contributors might lead to

sustainable high contribution levels. GHMA test this hypothesis by sorting subjects into

groups of others with similar contribution levels based on their initial first-period contributions.

The validity of this procedure depends on whether a subjects initial contribution level contains a

unique signal of their type that is, their propensity to contribute in subsequent periods

which is distinct from the effects of the various treatments, such as MPCR, group size, or

positive/negative framing. Tables 7 and 8 contain estimates for models that include Ci,1, the first

period contribution level, as an explanatory variable. Note that the coefficient on Ci,1 is positive

and significant for both the IW and the Andreoni data.10

Using Ci,1 as an indicator of type is problematic, however, since the level of initial

contributions is no doubt affected by the treatments. If so, then Ci,1 andCi,t for t>1 are jointly

determined by the treatment effects, in which case the coefficient estimates on C i,1 become

difficult to interpret as quantifying the effect of subject type on contributions.

Ideally, one would like to have a measure of type that is independent of treatment, as in

Park (2000). Failing that, however, the hypothesis that Ci,1 reflects type (and that type matters)

can be tested by interacting Ci,1 with the treatments and observing whether or not the impact of

Ci,1 on contributions is significant (and in the same direction) regardless of the treatment.

10Ci,t is in part a substitute for individual effects and, as a consequence, we found that that the MLE routine did not

always converge for models including both the fixed effects and explanatory variables including Ci,1; where

necessary, results for models omitting the fixed effect dummy variables are quoted in Tables 7 and 8.


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Models 7b and 7c for the IW data and Model 8b for the Andreoni data address this issue.

The results in Model 7b indicate that the coefficient on Ci,1 is significantly different from zero

and essentially identical regardless of whether the MPCR is .75 (MCPRhi=1) or .30

(MCPRlow=1). Similarly, as shown in Model 7c, since the estimated coefficient on Ci,1 X

GroupSize is not significantly different from zero, the coefficient on Ci,1is not significantly

different for the subjects in groups of size 4 than for subjects in groups of size 10.

Turning to the results on the Andreoni data in Table 8, the estimated coefficient on Ci,1is

clearly positive and significant in Model 8a, but when the coefficient is separately estimated over

the data for the positive and negative framing treatments using Ci,1 X Positive Frame and Ci,1 X

Negative Frame, both coefficients become individually insignificant. They are also not

significantly different from each other using a likelihood ratio test.11

Thus, it appears that the estimated coefficient on Ci,1 is not significantly sensitive to the

treatment for either of these two data sets. Consequently, it seems reasonable to interpret the

positive estimated coefficient on Ci,1 in Model 5 as quantifying the effect of subject type on

contributions.

9. Conclusions

For the most part, our analysis confirms the direction of the treatment effects found by

the original investigators in both data sets. We find (as did IW) that the MPCR is an important

determinant of behavior in the VCM. Group size is a significant determinant of the allocation

level when MPCR is low, but not when it is high. IW could only conclude that the effect is

weak and ambiguous. We also are able to determine that the impact of MPCR on allocations

depends inversely on group size. Similarly, our results support Andreonis conclusion that

11The value of the log likelihood function for Model 8a without the fixed effect dummy variables is 375.53, so the

likelihood ratio test statistic is .32~2(1), p=.572.


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subjects contribute more when the public good is positively framed, but we also are able to show

that the frame affects the speed with which subjects adjust their contributions to match those of

the group.

Treating the data as a fixed-effects panel had limited effect on the qualitative results.

Although the fixed-effect dummy variables in the allocation regression are highly significant as a

group, the key economic results are not sensitive to them. Indeed, even omitting these variables

altogether does not substantially alter the conclusions, at least in these two data sets. Of course,

one cannot know that is the case without (appropriately) including them. However, this

robustness result reduces our concern that, because their number increases as one expand the

number of subjects, such coefficients cannot be consistently estimated.

In contrast, we find that failure to appropriately deal with the censored nature of VCM

data is quite consequential. While the direction of the treatment effect is rarely affected, its

magnitude is sometimes substantially different when censoring is properly specified. It seems

clear from our results that censoring is not an aspect of the data that can be safely ignored.

More importantly, we find economically and statistically significant, asymmetric cross-

subject dynamics in the data. When a subjects allocation is above the average of the other

members of the group, he reduces his contribution toward the group average; when he discovers

his contribution is below average, he does not adjust toward the others. In other words, when a

subject finds that he/she has contributed more than the rest of the group, that subject responds!

This pattern of behavior is consistent with recent work on inequality aversion (Fehr and Schmidt,

1999), which asserts that people care about inequality, but that they care more when their income

is below than when it is above others income. Statistically, is not appropriate to adopt the past

practice of analyzing data using methods that assume that the allocations are independently


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18

distributed across subjects. Allocations are correlated over time for a given individual, and

across individuals.

Comparing the two data sets, we also conclude that random rematching has virtually no

effect on the extent to which subjects respond to information about others contributions.

Random rematching is commonly used in economics experiments, and the practice is used to

justify the assumption of independence of decisions over time. Our results show that this is

clearly not the case. In our data, random rematching does nothing to alter the dependence of

decisions in a given period on feedback from the decisions in previous rounds. We also find that

random rematching fails to eliminate serial correlation in individual contribution decisions. Thus

dynamic regression analysis is imperative for valid statistical analysis of data generated by VCM

experiments.

Finally, we examine the validity of using initial contributions as an indicator of subject

type. We find that initial contributions do enter our models with a significant positive

coefficient, which does not seem to be sensitive to the treatment regime. We conclude that, for

these data sets at least, initial contribution may be a useful measure of subject type.

Our results suggest that the institutional design for the private provision of public goods

could be improved by careful attention to the types of the participants, and the information that

they receive. Selection of types into groups may result in at least some groups that attain Pareto

efficient outcomes. In addition, information feedback that reassures contributors can keep

contributions high.

This work builds on the points made by Roth (2002) regarding the use of experimental

data in engineering market and nonmarket institutions. Game theory is extremely helpful in

understanding the incentives and examining the equilibria inherent in any given situation.


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19

However, experiments play a unique role in understanding how an individuals preferences and

attitudes interact with the incentives in a game to produce actual behavior. Our results show that

much more is going on in these games than attention limited soley to the incentive structure of

the monetary payoffs would indicate. Subjects care about their own payoffs, but not so much

that they will allow free riding to keep them from exploiting the gains to cooperation. When

such gains are possible, subjects are willing to cooperate, but only if they observe others

cooperating at least as much as they do. Institutional design that takes these preferences into

consideration are likely to be much more effective at attaining appropriate levels of public goods

production.


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Table 1

Descriptive Statistics: Isaac-Walker Data

Variable N Mean Std.

Dev.

Min Max

Ci,,t Fraction of endowment contributed

to public good

1680 0.360 0.370 0 1

ci,t Change in contribution to public

good: Ci, t-1 - Ci,,t

1512 -0.030 0.319 -1 1

Group size Size of group: (4 or 10) 1680 8.286 2.711 4 10

MPCRhi Dummy variable = 1 for

High MPCR treatment

1680 0.494 0.500 0 1

Deviation

from Group(+)i,,t-1

Ci,,t-1 less (average contribution by

others in group)t-1, if >0

1512 0.145 0.216 0 0.907

Deviation

from Group

(-)i,,t-1

Ci,,t-1 less (average contribution by

others in group)t-1, if 0

800 0.124 0.235 0 100

Deviation from Group (-)i,t-1 Ci,t-1 less (average contribution by

others in group)t-1, if


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Table 2: Basic Models:

Dependent Variable, Ci,,t, Contribution Levels: Fraction of Endowment Contributed to Public Good

(Standard Errors in Parentheses)

Isaac-Walker Data: OLS and Censored Panel Regressions

Model 2a: OLS Model 2b: Censored Panel Regression

Variable Coefficient(standard

error)

p-value2-tailed test Coefficient(standard error) p-value2-tailed test

Round number -0.028

(0.003)

p < .001 -0.054

(0.004)

p < .001

First 10 rounds -0.296

(0.033)

p < .001 -0.536

(0.050)

p < .001

MPCR = HI 0.510

(0.053)

p < .001 0.957

(0.084)

p < .001

Group Size 0.029

(0.004)

p < .001 -0.017

(0.023)

p = .459

MPCRhi X Group Size -0.035

(0.006)

p < .001 -0.067

(0.009)

p < .001

Constant 0.460

(0.059)

p < .001 1.026

(0.203)

p < .001

Fixed Effects:

smallest p-value p < .001

% p-values < .05 52

p-value for Ho: all 0 2(51) = 741.6, p < .001Fraction censored at 0% .344

Fraction censored at 100% .142

N 1680 1680

Adjusted R2

0.158

Pseudo R2

0.296

LLF -1109.6

Andreoni Data: OLS and Censored Panel Regression

Model 2a: OLS Model 2b: Censored Panel Regression

Variable Coefficient

(standard error)

p-value

2-tailed test

Coefficient

(standard error)

p-value

2-tailed test

Round number -0.027

(0.004)

p < .001 -0.056

(0.008)

p < .001

Positive Frame

(=1 for this treatment)

0.174

(0.022)

p < .001 0.435

(0.087)

p < .001

Constant 0.313

(0.026)

p < .001 -0.217

(0.071)

p = .002

Fixed effects:


% p-values < .05 67p-value for Ho: all 0 2(79) = 183, p < .001



N 800 800

Adjusted R2

0.125

Pseudo R2

0.213

LLF -547.6

*This test is compared to a Tobit model omitting the fixed effect dummy variable.


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Table 3: Isaac-Walker Data: Introducing Lags: OLS and Censored Panel RegressionDependent Variable, Ci,t, Contribution Levels: Fraction of Endowment Contributed to Public Good


Model 3a: OLS Model 3b: Censored

Panel Regression


(standard

error)

p-value

2-tailed test

Coefficient

(standard error)

p-value

2-tailed test

Ci,t-1 0.441

(0.027)

p < .001 0.511

(0.052)

p < .001

Ci,t-2 0.207

(0.026)

p < .001 0.161

(0.052)

p = .002

First 10 rounds 0.013

(0.015)

p = .372 0.045

(0.028)

p = .100

MPCR = HI 0.190

(0.050)

p < .001 0.657

(0.100)

p < .001

Group Size 0.010

(0.004)

p = .009 -0.048

(0.027)

p = .077


(0.006)

p = .031 -0.047

(0.011)

p < .001

Constant -0.048

(0.035)

p = .171 0.233

(0.231)

p = .314

Fixed effects:


% p-values < .05 39.4

p-value for Ho: all 0

2(78) = 197.2

p < .001



N 1344 1312

Adjusted R2

0.273 -

Pseudo R2

- 0.329

LLF - -842.8



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Table 4: Andreoni Data: Introducing Lags: OLS and Censored Panel RegressionDependent Variable, Ci,t, Contribution Levels: Fraction of Endowment Contributed to Public Good


Model 4a: OLS Model 4b: Censored

Panel Regression


(standard error)

p-value

2-tailed test

Coefficient

(standard error)

p-value

2-tailed test

Ci,t-1 0.361

(0.035)

p < .001 0.529

(0.075)

p < .001

Ci,t-2 0.251

(0.004)

p < .001 0.449

(0.074)

p < .001

Positive Frame


0.057

(0.021)

p = .005 0.204

(0.087)

p = .028

Constant 0.031

(0.026)

p = .043 -0.588

(0.066)

p < .001

Fixed effects:smallest p-value p < .001

% p-values < .05 39.2


2(51) = 75.2

p = .015



N 640 640

Adjusted R2

0.371 -

Pseudo R2

0.34

LLF -348.7



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Table 5:Isaac-Waker Data: Introducing Group Feedback

Censored Panel RegressionDependent Variable, Ci,t, Contribution Levels: Fraction of Endowment Contributed to Public Good


Model 5a: Group Feedback Model 5b: Group Feedback

by TreatmentVariable Coefficient

(standard error)

p-value

2-tailed test

Coefficient

(standard error)

p-value

2-tailed test

Ci,t-1 0.848

(0.110)

p < .001 0.890

(0.112)

p < .001

Ci,t-2 0.150

(0.051)

p = .003 0.137

(0.050)

p = .007


(0.027)

p = .091 0.047

(0.027)

p = .083

MPCR = HI 0.569

(0.112)

p < .001 0.436

(0.122)

p < .001

Group Size -0.011

(0.004)p = .009 -0.030

(0.027)p = .276

MPCRhi X Group Size -0.041(0.011)

p < .001 -0.036(0.012)

p = .004

Deviation from Group (+) -0.649

(0.132)

p < .001 -

Deviation from Group (-) 0.016

(.132)

p = .902

MPCRhi X

Deviation from Group (+)

-0.059

(0.284)

p = .835

MPCRlow X


-0.447

(0.317)

p = .158

MPCRhi X

Deviation from Group (-)

-0.208

(0.330)

p = .528

MPCRlow X


-0.079

(0.373)

p = .883

Group Size X


-0.052

(0.032)

p = .098

Group Size X


0.014

(0.038)

p = .693

Constant -0.103

(0.203)

p = .614 -0.119

(0.233)

p = .608

Fixed effects:

smallest p-valuep < .001 p < .001

% p-values < .0534.2 27.4


2

(79) = 199.0p < .001

2

(51) = 79.4p = .007

Fraction censored at 0% .362 .362


N 1312 1312

Pseudo R2

0.339 0.344

LLF -829.8 -824.2



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Table 6: Andreoni Data: Introducing Group Feedback

Censored Panel RegressionDependent Variable, Ci,t, Contribution Levels: Fraction of Endowment Contributed to Public Good


Model 6a: Group Feedback Model 6b: Group Feedback

by Treatment


(standard error)

p-value

2-tailed test

Coefficient

(standard error)

p-value

2-tailed test

Ci,t-1 0.939

(0.164)

p < .001 0.906

(0.162)

p < .001

Ci,t-2 0.428

(0.074)

p < .001 0.372

(0.074)

p < .001

Positive Frame


0.217

(0.094)

p = .022 0.300

(0.126)

p = .017

Deviation from Group (+) -0.607

(0.189)

p = .001

Deviation from Group (-) 0.045

(.178)

p = .798

Positive Frame X


-.379

(0.214)

p = .078

Negative Frame X


-.721

(0.213)

p = .001

Positive Frame X


.203

(0.212)

p = .340

Negative Frame X


-.508

(0.298)

p = .089

Constant -0.570(0.072) p < .001 -0.638(0.085) p < .001

Fixed effects:

smallest p-value p < .001 p < .001

% p-values < .05 29.4 27.4


2(51) = 70.0

p = .040

2

(51) = 79.4

p = .007



N 640 640Pseudo R

20.347 0.358

LLF -343.5 -337.5



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Table 7 Isaac-Walker Data: Do initial contributions indicate type?

Dependent Variable, Ci,t, Contribution Levels: Fraction of Endowment Contributed to Public Good


Model 7a: Ci,1=type

Censored Regressiona

Model 7b: Ci,1=type


Interactions: Type X

MPCRhi

Model 7c: Ci,1=type


Interactions: Type X

Group Size


(standard

error)

p-value

2-tailed

test

Coefficient

(standard

error)

p-value

2-tailed

test

Coefficient

(standard

error)

p-value

2-tailed

test

Ci,t-1 0.924

(0.102)

p < .001 0.924

(0.102)

p < .001 0.930

(.103)

p < .001

Ci,t-2 0.318

(0.050)

p < .001 0.317

(0.050)

p < .001 .317

(.050)

p < .001


(0.028)

p = .090 0.046

(0.028)

p = .105 .048

(.028)

p = .088

MPCR = HI 0.343

(0.119)

p = .004 0.323

(0.124)

p = .009 .337

(.119)

p = .005

Group Size 0.029

(0.010)

p = .003 0.029

(0.010)

p = .003 .025

(.012)

p = .035


(0.012)

p = .014 -0.031

(0.012)

p = .014 -.030

(.012)

p = .016

MPCRhi X


-0.157

(0.280)

p = .573 -0.147

(0.280)

p = .573 -.152

(.280)

p = .585

MPCRlow X


-0.580

(0.317)

p = .067 -0.557

(0.319)

p = .067 -.572

(.317)

p = .072

MPCRhi X


0.162

(0.287)

p = .571 0.158

(0.287)

p = .571 .163

(.287)

p = .569

MPCRlow X


0.272

(0.342)

p = .428 0.266

(0.342)

p = .428 .273

(.342)

p = .425

Group Size X


-0.020

(0.031)

p = .513 -0.021

(0.031)

p = .513 -.022

(.031)

p = .483

Group Size X


-0.002

(0.033)

p = .961 -0.002

(0.033)

p = .961 -.002

(.033)

p = .944

Ci,1 0.161

(0.042)

p < .001 .095

(.136)

p = .483

Ci,1 X MPCRhi 0.185

(0.058)

p = .001

Ci,1 X MPCRlow 0.139

(0.057)

p = .016

Ci,1 X Group Size .008

(.015)

p = .610

Constant -0.523

(0.081)

p < .001 -0.513

(0.083)

p < .001 -.494

(.099)

p < .001

Fraction censored at 0% .378 .378 .378Fraction censored at 100% .131 .131 .131

N 1344 1344 1344

Pseudo R2

0.289 0.289 0.289

LLF -916.1 -915.9 -915.9

aNote that the censored panel regression failed to converge when C1 was included.


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Table 8 Andreoni Data: Do initial contributions indicate type?

Dependent Variable, Ci,t, Contribution Levels: Fraction of Endowment Contributed to Public Good


Model 8a: C1=type

Censored Panel Regression

Model 8b: C1=type

Censored Panel Regressiona

Interactions: Type X Frame X

Group Feedback


(standard

error)

p-value

2-tailed test

Coefficient

(standard

error)

p-value

2-tailed test

Ci,t-1 0.892

(0.160)

p < .001 .912

(.154)

p < .001

Ci,t-2 0.291

(0.074)

p < .001 .460

(.074)

p < .001

Positive Frame


0.264

(0.129)

p = .042 .160

(.088)

p = .068

Positive Frame X


-.503

(0.215)

p = .019 -.604

(.196)

p = .002

Negative Frame X


-.683

(0.211)

p = .001 -.454

(.195)

p = .020

Positive Frame X


0.188

(0.214)

p = .382 .397

(.194)

p = .041

Negative Frame X


-.601

(0.297)

p = .044 -.048

(.277)

p = .863

C1 0.474

(0.109)

p < .001

Ci,1 X Positive Frame .140

(.078)

p = .072

Ci,1 X Negative Frame .074

(.095)

p = .437

Constant -0.760

(0.093)

p < .001 -.379

(.066)

p < .001

Fixed effects:

smallest p-valuep < .001

% p-values < .0527.4

p-value for Ho: all 0 2(51) = 94.6

p < .001



N 640 640

Pseudo R2

0.376 0.422

LLF -328.2 -303.5


a

Note that the censored panel regression failed to converge when Ci,1 was replace by Ci,1 X Positive Frame and Ci,1 X Negative Frame.


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