VOL 87 NO.5 PALFREY AND PRISBREY:PUBLIC GOODS EXPERIMENTS R7 50 percent from 2.7 to 1.6.This suggests that subject confus account for a rienced subiec better?There are several ways to conduct such The bottom lines from the aggregate probit est and in all analysis are:(1)there is strong evidence for a warm-glow t leading to voluntary contr for an alt test between the probit model including only that much of the decline in contribution fron experience and repetition is due to decline in del the benefit of t han a c inge in the unde sign the likelihood of contribution at the cut ror rates is a planation forth point(diff= effects observed in some past experiments.a 93.36.T ratio is equal to se to models 2g.198 of reputation】 adiustment (rather than a standard chi. E.The Heterogeneous Subjects Model tion assumes We next run a probit including the addi in tional control variables exper,endow,and pe ilar indications have also been noted in man ab/so vey an ey,19 2a2emmen2 the effect of the variables on the coefficient of riments (Isaac et al.1984) diff.with a negative coefficient indicating that Here,the aggregate analysis of the previ- the the rand yterm is get Ou5eCtionisbrokendowmatthendividul ummy variab into m for eac actual distribution of individu effects. expenence van the pe e predic over time.Also of interest is the fact that none ients are onabe this ible explanati fo of reduced variance and the fact that th ign. ta d di-glow estimat matching des which i enced subjects in round one and expe andom ol in rec subjects in round ten is quite large, with the and estimated warm-glow term dropping by nearly 10)fo r this e写 are other VOL 87 NO. 5 PALFREY AND PRISBREY: PUBUC GOODS EXPERIMENTS 837 Here we see that the estimated warm-glow term is more than twice as large in magnitude in the probit model compared with the con￾stant error {g, q) model. Which estimate is better? There are several ways to conduct such a test, and in all those that we tried, a likeli￾hood ratio test shows the probit model to be the clear winner, at highly significant levels. For example, we conducted a likelihood ratio test between the probit model including only the constant and diff \ahables and the (g, q) model with g = 1 and q = 0.105. To give the (g, q) model the benefit of the doubt, we as￾sign the likelihood of contribution at the cut￾point (diff = 1) to simply equal the empirical frequency. The likelihood ratio is equal to 93.36. Since the two models are strictly non￾nested, we use the Quang H. Vuong (1989) adjustment (rather than a standard chi-square test) to conduct a formal statistical test. This produces a z-statistic of 7.30 {significant at p< 10-''), We next run a probit including the addi￾tional control variables exper, endow, and pe￾riod, and also including the interaction of these variables with diff.^^ (See column 2 of Table 3.) The interaction coefficients measure the effect ofthe variables on the coefficient of diff, with a negative coefficient indicating that the variance of the random utility term is get￾ting smaller, Behaviorally, this lower variance translates into more predictable behavior by subjects, or steeper probit response curves. Not surprisingly, the interaction coefficients for both the experience variable and the period variable show such an effect, indicating that subject behavior is becoming more predictable over time. Also of interest is the fact that none of the noninteraction coefficients are signifi￾cant. Jointly, this implies that the overall effect of experience and repetition is to reduce ag￾gregate contributions, but that this reduction effect is indirect and due to the combination of reduced variance and the fact that the warm-glow level is positive. The estimated difference between cutpoints for inexperi￾enced subjects in round one and experienced subjects in round ten is quite large, with the estimated warm-glow term dropping by nearly '^ Interactions with V are not included because the ef￾fect of V is insignificant. 50 percent from 2.7 to 1,6, This suggests that subject confusion may indeed account for a large portion of the contributions by inexpe￾rienced subjects."" The bottom lines from the aggregate probit analysis are: (I) there is strong evidence for a warm-glow effect leading to voluntary contri￾bution, and (2) there is no significant evidence for an altruism effect. The results also show that much of the decline in contribution from experience and repetition is due to decline in error rates rather than a change in the under￾lying decision rule. As such, the decline in er￾ror rates is a possible explanation for the decay effects observed in some past experiments, an explanation that avoids any recourse to models of reputation building or repeated games," E. The Heterogeneous Subjects Model The analysis in the previous section assumes that individuals are identical. In fact, there are indications of heterogeneity in our data. Sim￾ilar indications have also been noted in many other economics and decision experiments (McKelvey and Palfrey, 1992; El-Gamal and David M, Grether. 1995) and in public goods experiments (Isaac et al,, 1984), Here, the aggregate analysis of the previ￾ous section is broken down at the individual level by including a dummy variable for each individual, from which we can estimate the actual distribution of individual warm-glow effects.'" The last column of Table 3 reports the coefficients for the included variables. '" The coefficient on the endow treatment variable is insignificant and the coefficient on the interaction between endow and diff'\^ very smali ( <,01) and barely significant at the 5-percent level. In the later analysis with individual effects, this small effect vanishes, " This also provides a possible explanation for Andreoni's (1988) finding that in a random matching de￾sign, there is less decay than in the standard repeated￾group design. This could happen if subject learning occurs more slowly in the random matching design, which is plausible since the random matching protocol introduces another source of noise in the feedback received by sub￾jects after each period of play. See Palfrey and PHsbrey (1996) for additional evidence for this explanation. '^ There are other conceivable sources of heterogeneity in these experiments, including cohort effects, nonlinear warm-glow terms, different varieties of altruistic prefer￾ences, or differential error rates across subjects, but an exploration of multidimensional heterogeneity is well