Estimation of Treatment Effects in Repeated Public Goods Experiments
Abstract
:1. Introduction
2. Canonical Examples
3. Statistical Model
3.1. Statistical Modelling Nonstationary Initial Condition
3.2. Features of New Statistical Model
3.3. Estimation of Overall Treatment Effects
4. Asymptotic Theory
- (A.1)
- is independent and identically distributed with meanand variance, but it is bounded between 0 and 1:
- (A.2)
- forwhereis independently distributed with mean zero and variance, but it is bounded betweenandThat is,
5. Return to Empirical Examples
6. Monte Carlo Study
7. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Proofs of Theorem
Appendix B. Proof of Remark 5
Appendix C. Proof of Remark 6
References
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1 | When the cross-sectional average is increasing over rounds, (but is weakly -converging), the trend regression needs to be modified as = + + error. Furthermore, the long run overall outcome becomes |
2 | Note that dynamic panel regressions or dynamic truncated regressions are invalid since usually the decay rate—AR(1) coefficient—is assumed to be homogeneous across different games. In addition, see Chao et al. (2014) for additional issues regarding the estimation of the dynamic panel regression under non-stationary initial conditions. |
3 | |
4 | Assume that the sample cross-sectional average estimates the unknown common stochastic function consistently for every t as Then, by using a conventional spline method, we can approximate the unknown The overall effects can be estimated by the sum of the approximated function and statistically evaluate it by using an HAC estimator defined by Newey and West (1987). However, this technical approach does not provide any statistical advantage over the AR(1) fitting, which we will discuss in the next section. |
5 | The decay function for the controlled and treated experiments are set as and respectively. |
6 | However, it is not always the case. More recent experimental studies show widely heterogeneous divergent behaviors. See Kong and Sul (2013) for detailed discussion. |
7 | Note that if the condition in (9) does not hold, then it implies that the variance of is increasing over time if the variance of is time invariant. To be specific, let for all t. Then which is an increasing function over |
8 | If , can be identified as long as is known, but cannot be identified. If both and cannot be identified jointly. |
9 | Since = where under (11), as jointly. |
10 | If subjects do not know the number of repeated games, the dominant strategy could change to Pareto-optimum in an infinitely repeated game. See Dal Bó and Fréchette (2011) for a more detailed discussion. |
11 | More precisely speaking, we assume that the fraction of free riders in the long run becomes unity as the number of subjects goes to infinity. This assumption allows a few outliers such as altruists. As long as the number of altruists does not increase as the number of subjects increases, the asymptotics studied in the next section are valid. |
12 | Taking logarithm in both sides of Equation (16) yields |
13 | We will show later that the point estimates of for all three empirical examples are around 0.9. However, the choice of does not matter much when comparing the two variances. |
Authors | Croson3 | Keser and Winden | ||
---|---|---|---|---|
Strangers | Partners | Strangers | Partners | |
subjects no. | 24 | 24 | 120 | 40 |
G | 4 | 4 | 4 | 4 |
T | 10 | 10 | 25 | 25 |
e | 25 | 25 | 10 | 10 |
Croson | Keser & Winden | |||
---|---|---|---|---|
Partner | Strangers | Partner | Strangers | |
Total Number of Subjects | 24 | 24 | 120 | 40 |
Number of Pure Altruistic | 1 | 0 | 0 | 1 |
−0.098 | −0.043 | −0.049 | −0.016 | |
(s.e × 10) | 0.007 | 0.020 | 0.007 | 0.009 |
0.614 | 0.459 | 0.618 | 0.381 | |
(s.e) | 0.080 | 0.081 | 0.033 | 0.036 |
0.912 | 0.884 | 0.972 | 0.934 | |
(s.e) | 0.021 | 0.018 | 0.014 | 0.002 |
Group/Treatment | Croson | Keser & Winden |
---|---|---|
(s.e) | 7.000 (2.309) | 20.74 (12.18) |
(s.e) | 3.951 (1.051) | 5.798 (0.617) |
(s.e) | 3.050 (2.537) | 14.94 (12.20) |
( ) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
V | V | |||||||||
0.15 | 25 | 10 | 0.499 | 0.499 | 0.900 | 0.900 | 4.205 | 4.188 | 4.927 | 3.891 |
0.15 | 50 | 10 | 0.500 | 0.500 | 0.900 | 0.900 | 2.262 | 2.252 | 2.466 | 1.910 |
0.15 | 100 | 10 | 0.500 | 0.500 | 0.900 | 0.900 | 1.169 | 1.159 | 1.150 | 0.900 |
0.15 | 200 | 10 | 0.500 | 0.500 | 0.900 | 0.900 | 0.567 | 0.557 | 0.600 | 0.455 |
0.15 | 25 | 20 | 0.501 | 0.500 | 0.900 | 0.900 | 4.518 | 4.442 | 1.259 | 0.502 |
0.15 | 50 | 20 | 0.501 | 0.500 | 0.900 | 0.900 | 2.328 | 2.261 | 0.588 | 0.224 |
0.15 | 100 | 20 | 0.500 | 0.500 | 0.900 | 0.900 | 1.132 | 1.109 | 0.296 | 0.116 |
0.15 | 200 | 20 | 0.501 | 0.500 | 0.900 | 0.900 | 0.549 | 0.535 | 0.149 | 0.056 |
0.12 | 25 | 10 | 0.500 | 0.499 | 0.900 | 0.900 | 3.752 | 3.736 | 5.030 | 3.977 |
0.12 | 50 | 10 | 0.500 | 0.500 | 0.900 | 0.900 | 2.025 | 2.014 | 2.535 | 1.958 |
0.12 | 100 | 10 | 0.500 | 0.500 | 0.900 | 0.900 | 1.050 | 1.038 | 1.186 | 0.931 |
0.12 | 200 | 10 | 0.500 | 0.500 | 0.900 | 0.900 | 0.510 | 0.499 | 0.620 | 0.470 |
0.12 | 25 | 20 | 0.501 | 0.500 | 0.900 | 0.900 | 4.041 | 3.964 | 1.286 | 0.516 |
0.12 | 50 | 20 | 0.501 | 0.500 | 0.900 | 0.900 | 2.076 | 2.008 | 0.607 | 0.231 |
0.12 | 100 | 20 | 0.500 | 0.500 | 0.900 | 0.900 | 1.012 | 0.987 | 0.307 | 0.120 |
0.12 | 200 | 20 | 0.501 | 0.500 | 0.900 | 0.900 | 0.491 | 0.476 | 0.155 | 0.058 |
0.1 | 25 | 10 | 0.500 | 0.499 | 0.900 | 0.900 | 3.392 | 3.376 | 5.118 | 4.056 |
0.1 | 50 | 10 | 0.500 | 0.500 | 0.900 | 0.900 | 1.834 | 1.822 | 2.591 | 1.996 |
0.1 | 100 | 10 | 0.500 | 0.500 | 0.900 | 0.900 | 0.951 | 0.939 | 1.218 | 0.958 |
0.1 | 200 | 10 | 0.500 | 0.500 | 0.900 | 0.900 | 0.463 | 0.453 | 0.636 | 0.482 |
0.1 | 25 | 20 | 0.501 | 0.500 | 0.900 | 0.900 | 3.655 | 3.576 | 1.311 | 0.526 |
0.1 | 50 | 20 | 0.501 | 0.500 | 0.900 | 0.900 | 1.872 | 1.802 | 0.620 | 0.237 |
0.1 | 100 | 20 | 0.500 | 0.500 | 0.900 | 0.900 | 0.915 | 0.889 | 0.313 | 0.122 |
0.1 | 200 | 20 | 0.501 | 0.500 | 0.900 | 0.900 | 0.444 | 0.430 | 0.158 | 0.060 |
Size (5%): | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
0.05 | 0.03 | 0.01 | ||||||||
10 | 15 | 20 | 10 | 15 | 20 | 10 | 15 | 20 | ||
0.15 | 25 | 0.018 | 0.040 | 0.042 | 0.028 | 0.051 | 0.050 | 0.041 | 0.062 | 0.058 |
0.15 | 50 | 0.018 | 0.023 | 0.035 | 0.025 | 0.031 | 0.044 | 0.043 | 0.042 | 0.052 |
0.15 | 100 | 0.018 | 0.027 | 0.040 | 0.024 | 0.034 | 0.045 | 0.038 | 0.047 | 0.051 |
0.15 | 200 | 0.016 | 0.028 | 0.036 | 0.023 | 0.032 | 0.043 | 0.040 | 0.039 | 0.051 |
0.12 | 25 | 0.015 | 0.035 | 0.041 | 0.026 | 0.047 | 0.048 | 0.038 | 0.062 | 0.056 |
0.12 | 50 | 0.016 | 0.020 | 0.034 | 0.022 | 0.027 | 0.041 | 0.040 | 0.042 | 0.052 |
0.12 | 100 | 0.014 | 0.025 | 0.038 | 0.021 | 0.032 | 0.045 | 0.036 | 0.048 | 0.051 |
0.12 | 200 | 0.012 | 0.024 | 0.033 | 0.022 | 0.030 | 0.038 | 0.038 | 0.041 | 0.052 |
0.10 | 25 | 0.013 | 0.030 | 0.039 | 0.022 | 0.041 | 0.046 | 0.038 | 0.061 | 0.054 |
0.10 | 50 | 0.014 | 0.020 | 0.031 | 0.021 | 0.024 | 0.037 | 0.038 | 0.039 | 0.049 |
0.10 | 100 | 0.012 | 0.023 | 0.033 | 0.019 | 0.031 | 0.042 | 0.036 | 0.047 | 0.050 |
0.10 | 200 | 0.010 | 0.020 | 0.031 | 0.018 | 0.028 | 0.036 | 0.037 | 0.040 | 0.049 |
Power (5%): | ||||||||||
0.15 | 25 | 0.392 | 0.476 | 0.534 | 0.472 | 0.517 | 0.562 | 0.566 | 0.558 | 0.597 |
0.15 | 50 | 0.715 | 0.801 | 0.812 | 0.769 | 0.828 | 0.829 | 0.841 | 0.853 | 0.845 |
0.15 | 100 | 0.960 | 0.980 | 0.983 | 0.971 | 0.986 | 0.986 | 0.985 | 0.989 | 0.990 |
0.15 | 200 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
0.12 | 25 | 0.422 | 0.516 | 0.571 | 0.509 | 0.556 | 0.604 | 0.612 | 0.606 | 0.632 |
0.12 | 50 | 0.743 | 0.835 | 0.845 | 0.806 | 0.860 | 0.859 | 0.872 | 0.893 | 0.877 |
0.12 | 100 | 0.967 | 0.990 | 0.990 | 0.982 | 0.993 | 0.993 | 0.993 | 0.994 | 0.996 |
0.12 | 200 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
0.10 | 25 | 0.446 | 0.546 | 0.603 | 0.537 | 0.599 | 0.642 | 0.647 | 0.656 | 0.677 |
0.10 | 50 | 0.776 | 0.865 | 0.873 | 0.839 | 0.895 | 0.890 | 0.904 | 0.929 | 0.909 |
0.10 | 100 | 0.979 | 0.994 | 0.996 | 0.988 | 0.995 | 0.998 | 0.998 | 0.999 | 0.998 |
0.10 | 200 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
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Kong, J.; Sul, D. Estimation of Treatment Effects in Repeated Public Goods Experiments. Econometrics 2018, 6, 43. https://doi.org/10.3390/econometrics6040043
Kong J, Sul D. Estimation of Treatment Effects in Repeated Public Goods Experiments. Econometrics. 2018; 6(4):43. https://doi.org/10.3390/econometrics6040043
Chicago/Turabian StyleKong, Jianning, and Donggyu Sul. 2018. "Estimation of Treatment Effects in Repeated Public Goods Experiments" Econometrics 6, no. 4: 43. https://doi.org/10.3390/econometrics6040043
APA StyleKong, J., & Sul, D. (2018). Estimation of Treatment Effects in Repeated Public Goods Experiments. Econometrics, 6(4), 43. https://doi.org/10.3390/econometrics6040043