A New Technique to Estimate Treatment Effects in Repeated Public Goods Experiments

Transcription

1 A New Technique to Estimate Treatment Effects in Repeated Public Goods Experiments Jianning Kong (with Donggyu Sul) Department of Economics University of Texas at Dallas Nov, 2012

2 Outline The New Technique Tested on Isaac, Walker and Williams (1994) data Conclusion

3 Average Contribution IWW (1994): MPCR= Average Contributions for the Two Games G=10 G= Round

4 Average Contribution Standard t-test of the Difference Average Contributions for the Two Games G=10 G= Round difference Round z-score None of the differences are significantly different from 0.

5 Econometric Modeling Test and Estimations Individual s Pattern of Contribution We consider three contribution patterns in public good donation games: Nash Group (G 1 ), Pareto Group (G 3 ), and Confused Group (G 2 ). Model these contribution patterns utilizing Sul(2012)'s exponential decay model y it i t 1 e it if i G 1 i e it if i G i t 1 e it if i G 3 The percent contribution for subject i at time t Econometric theory for unknown common factor: Want to know?

6 Average Contribution Econometric Modeling Test and Estimations How to measure the Treatment Effect? 1 Game A, Free Frider: Slow Decay 0.5 Game A, Altruist Slow Decay Game B, Free Rider Fast Decay Game B, Altruist Fast Decay Round

7 Average Contribution Econometric Modeling Test and Estimations How to measure the Treatment Effect? 0.5 Game A: Less Pareto but Slow Decay 0.4 Game B: More Pareto but Fast Decay Round

8 Average Contribution Econometric Modeling Test and Estimations How to measure the Treatment Effect? Game A: Less Pareto but Slow Decay 0.3 Game B: More Pareto but Fast Decay Round

9 Econometric Modeling Test and Estimations Asymptotic Average Treatment Effect To get a robust measure, we define the asymptotic average treatment effect (AATE), which is the overall average of the contributions, letting N, T Asym ATE plim N,T 1 TN N i 1 T t 1 y it n 2 n 3. AATE captures the long run treatment effect. AATE depends only on three parameters: the unconditional mean of the first outcome, the fractions of two groups. To evaluate AATE, we have to estimate the three key parameters, n 2,. 2

10 Econometric Modeling Test and Estimations Test and Estimation Flow (Game A and B) Convergence Test Both Convergent A converges but B diverges A &B diverge both Run Trend Regression B dominates A always Cluster subjects into Nash, Pareto and Confused groups Estimate the key parameter

11 Econometric Modeling Test and Estimations Convergence Test Confused Free Riders

12 Cross Sectional Variance Econometric Modeling Test and Estimations Empirical Example: IWW(1994) Cross Sectional Variances for the Two Games G=10 G= Round

13 Econometric Modeling Test and Estimations Convergence Test (Formally) The test can be done by running the following trend regression where H N,t H N,t t u t N 1 N i 1 y it N 1 N 2 i 1 y it Testing the existence of multiple equilibrium becomes the following test t can be calculated in an ordinary way. If t >1.65, the null hypothesis is rejected at the 5% level, which provide evidence of existence of multiple equilibriums. The t-statistic H 0 : 0, v.s. H A : 0 Asymptotic Justification: Want to know? Divergence

14 Econometric Modeling Test and Estimations Convergence Test using STATA The following is the Stata code for convergence test. Note: convt.ado should be put in the working directory. insheet using mydata.csv forvalues i=1/240{ replace v`i'=v`i'/100 } The data set in file mydata.csv is a T*N matrix with no headings Convert the data to ratios by dividing 100 convt v1-v240 Convergence test Result:, t

15 Econometric Modeling Test and Estimations Empirical Example Results: Result of Convergence Test GroupSize10 GroupSize100 β t-ratio Both of the t-ratios are greater than Strong evidence that there are multiple equilibriums in the two games. Back to flow

16 Econometric Modeling Test and Estimations Clustering Method For each subject s outcome, run trend regression y it a bt it Construct the conventional t-statistic for b t b t b t b Nash Group Confused Group Pareto Group Calculate n 2 Econometric Theory: Want to know?

17 Econometric Modeling Test and Estimations Estimation of Asymptotic Average Treatment Effect Step 1: Estimate by using the first round observations. N 1 y i,1 Step 2: Use the core groups subjects. Run the following pooled regression. log y cn n,t log t 1 v 1t, log 1 y cp n,t log 1 t 1 v 3t, where y cn cp n,t, y n,t are from the core Nash and Pareto subjects, respectively. And can be estimated by exp

18 Econometric Modeling Test and Estimations Estimation of Asymptotic Average Treatment Effect Step 3: Estimation of AATE. Run the following time series regression with all the subjects outcome. y n,t t 1 e t 1. The estimate of becomes AATE T 2. Construct the ordinary sample variance of, 2. Note 2 is inconsistent Theorem 1 & 2

19 Econometric Modeling Test and Estimations Estimation of Asymptotic Average Treatment Effect Step 4: Estimate the variance of AATE. n 2 2 /N 2 /T n 2 and 2 from clustering analysis 2 : First take time series mean of y it for each i, next, take its cross sectional variance.

20 Econometric Modeling Test and Estimations Estimation of AATE using STATA The following is the Stata code for estimation of Asymptotic Average Treatment Effect: Note: Put the treatment.ado file in the working directory, insheet using mydata.csv forvalues i=1/240{ replace v`i'=v`i'/100 } treatment v1-v240 Estimation Results: n 2 V

21 Econometric Modeling Test and Estimations Empirical Example Results: Estimation From Clustering GroupSize 10 GroupSize 100 μ The initial means ρ The decay rates n Fraction of the confused groups Estimation of Asymptotic Average Treatment Effect (AATE) τ % tokens allocated to the public account with Group Size=10. 32% tokens allocated to the public account with Group Size=100. V(τ)x Will be used for a formal test

22 Econometric Modeling Test and Estimations Comparison of AATE between Two Games If both game A and game B diverge, the difference between the two AATE can be denoted as A B The t-ratio of the difference can be constructed as t / V A V B If t then the difference between the two experiments becomes significantly different from 0 at 5% level.

23 Econometric Modeling Test and Estimations IWW (1994) Data Estimation From Clustering GroupSize 10 GroupSize 100 μ Initial mean contribution ρ Decay rates n Fraction individuals flat contribution patterns Estimation of Asymptotic Average Treatment Effect (AATE) τ % tokens allocated to the public account with Group 0. Size= % tokens allocated to the public account V with Group Size=100. V(τ)x Will be used for a formal test t 0. 11/

24 Average Contribution Econometric Modeling Test and Estimations Empirical Example Result Average Contributions for the Two Games 0.55 G= G= Our 0.3 conclusion: The average outcome of blue is significantly higher than that of the red Round

25 : We develop a new statistical test to identify differences between treatments in public goods games We propose a new pre-test for divergence (for testing multiple equilibriums), and provide estimation methods to measure treatment effects robustly and efficiently under multiple equilibriums. shows that the finite sample properties of the proposed methods are performed reasonably well. The newly suggested methods work well in practice.

26

27 Finite Sample Performance of Clustering (n , n , n ) Estimates False Inclusion Rate Variance N T n 1 n 2 n 3 n 1 n 3 V n c

28 Finite Sample Performance of Clustering (n , n , n ) Estimates False Inclusion Rate Variance N T n 1 n 2 n 3 n 1 n 3 V n c

29 Effectiveness of Core Membership on the Estimation of Decay Rates (T 10, n , n , n , 0. 85) Estimates False Inclusion Rate N c n 1 n 2 n 3 n 1 n

30 Size of the Tests (5%) N/T Size: n 2 p , n 3 p , Size: n 2 p , n 3 p ,

31 Power of the Tests (5%) N/T Power: n , n , p , p , Power: n , n , p , p ,

32 Assumptions (A) i an i.i.d bounded and non-negative random variable which has a finite mean with a finite variance 2, and is independent from e k,it. (B) e it has mean 0 and a finite variance 2 ; e it and e js are independent for all i,j,t,s and i j, t s. As N, we assume that N 1/2 N i 1 e 2 it 2 d N 0, 4 where 4 is a finite positive constant. When is known, the treatment regression in (ref: treatreg) can be run directly. Here we rewrite (ref: treatreg) as y N,t N N t 1 e Nt where N N N. The limit theory in this case can be formally stated as Theorem 1 (Limit Theory under Infeasible Estimation) Under Assumption A and B, as N,T jointly, the limiting distribution of N is given by N N N N d N 0 0, n n

33 y N,t N N t 1 e Nt, where e Nt e Nt N t 1 t 1. As long as O p N 1/2, the limiting distributions of N and N in (ref: feas) follow (ref: The1). That is, Theorem 2 (Feasible Estimation) Suppose that O p N 1/2. Then (i) as N,T jointly, the limiting distributions of N and N in (ref: feas) becomes N N N N d N 0 0, n n (ii) If N with a fixed T, the limiting distribution of N becomes N N d N 0, n /T. Back

34 Assumption C: (Weak to Unity) T 1 ct k for 1 2 k 1 and c 0. Note that when 0. 5 k 1, the process is called weak to unity. See Phillips and Magdalinos (2007) for more detailed discussion on this. This device allows T becomes more persistent as T increases. Then the asymptotic properties of b 1 and it s t-statistic are given by Theorem 3 (Limit Theory for Clustering Method) Under Assumptions A through C, as T, (i) the probability limit of b 1 becomes plim T T 2 k b 1 6b 0 c T 2 k b T, let say (ii) its limiting distribution is given by T 3/2 b 1 b T d N 0,12 2 (iii) the limiting distribution of its t statistic becomes t b b 1 V b 1 N 0, 1 if i G 2 if i G 2. Back

35 Remark 2 (Fixed Alternative) Even though the asymptotic justification of the clustering mechanism in (ref: clus) holds only under weak to unity, it is useful to know how t b 1 behaves with a fixed. Let assume T 0 and define t b as t b T3/2 b 1 b T Hence the expectation of t ratio and b 1 become N 0,1. Eb 1 6b 0 1 T 2 1, Et b 1 3 b 0 1 T. To evaluate this, let T 10, 0. 8, and b which come from the estimation results of our empirical examples. Then the mean of b 1 becomes 0.15, meanwhile the t ratio becomes centered around Then the 5% critical value for the t ratio can be approximated as However if T 20, then the 5% value drops to Of course, this calculation is totally relying on the value of b 0, and and. Back

36 Remark 3 (Estimation of and We can estimate consistently by using the first observation of y it. That is, N 1 y i1. Next use higher c in (ref: club) to identify core Nash and Pareto groups. Note that a subject i must be in Nash or Pareto group if y it 0 or 1 for all t, respectively. And then run the following pooled OLS to estimate log. ln 1 N 1 c c N 1 c i G1 ln 1 1 N 3 c y it ln t 1 v 1t, c N 3 c i G3 y it ln 1 t 1 v 3t, where log, G 1 c and G 3 c are core Nash and Pareto groups, respectively. N 1 c and N 3 c are the total numbers of the core Nash and Pareto groups, respectively. As N with a fixed T, the limiting distribution of is approximated by N d N 0, 2, where 2 3 n n T 2 T 4 1. The estimate of becomes exp.

37 Theorem 4: (Asymptotic Properties of Convergence Test under the Null) Under the Assumption A and B, as N, T, (i) the probability limit of T is given by plim N T 1 T T 3 T 1 when 0 1 or i G 1,G 3 0 when 1 or i G 2, where T 2, T (ii) the probability limit of t statistic is converging to a negative constant if 0 1, plim N,T t plim N,T T/ V T when 0 1, but when 1, t converges in the following limiting distribution. t d N 0, 1 when 1. Back

38 Remark 4 (Limiting Distribution of T with Warm-Glow Givers) Suppose that all subjects are warm-glow givers. Then as N, the limiting distribution of T is given by N T 0 d N 0, T 3 O T 4. In fact, as N,T jointly, the limiting distribution is also well defined as N T 3/2 T 0 d N 0, Back

39 Theorem 5: (Asymptotic Properties of Convergence Test under the Alternative) Under the Assumption A, B and D, as N,T jointly (i) the probability limit of T is given by plim N T 6 where is defined in (ref: App4) in Appendix A. (ii) the probability limit of the t statistic is given by 2a 1 b 1 2 T as T, plim N,T t plim N,T T/ V T when 0 1, Back

40 Remark 6 (Heterogeneous and Unknown Decay Rates) Suppose that there are two equilibria but the decay rates are unknown and different. To be specific, consider the following data generating process. y it a 1 b i t e it for i G 1 a 2 c i t e it for i G 2, where we assume that b i iid b, b 2 and c i iid c, c 2. Further assume that t / t 0 but t / t 0 for all t. Then from the direct calculation, the probability limit of the cross sectional variance becomes plim N H N,t b 2 n 1 t 2 c 2 n 2 t 2 n 1 n 2 a 1 b t a 2 c t 2 2. Hence if 1 1 but t t for t 1, then it is easy to see that the probability limit of H N,t is an increasing function of t. In this case, we don t need Assumption C Back