Multiple equilibria in social interaction models: estimating a model of adolescent smoking behavior.

Transcription

1 Multiple equilibria in social interaction models: estimating a model of adolescent smoking behavior. Andrea Moro, Vanderbilt University (with Alberto Bisin and Giorgio Topa) Michigan State University, April 3, 213

2 Question Social interaction problems typically display multiple equilibria Problem often ignored in empirical applications Want to quantitatively assess the importance of multiplicity in a model of smoking with social interactions Introduction

3 Identification problems in social interaction models Manski s reflection problem Distinguish endogenous group effects from contextual effects individual effects Correlated unobservables Selection into group membership Multiplicity Introduction

4 Multiplicity is policy-relevant What equilibrium are we playing? 1% 45% Avg. smoking Introduction

5 Multiplicity is policy-relevant What equilibrium are we playing? The equilibrium is policy-relevant 1% 45% Avg. smoking Introduction

6 Multiplicity is policy-relevant What equilibrium are we playing? The equilibrium is policy-relevant Extrapolating results from a linear specification might be misleading 1% 45% Avg. smoking Introduction

7 Outline (of the paper) Present a general framework for models with multiple equilibria in economies with social interactions Introduce a consistent and computationally feasible two-step estimator Show Monte Carlo analysis of the estimator s properties Present an application to teenage smoking using Add Health data Introduction

8 Estimation of models with multiple equilibria Various approaches: Impose conditions for equilibrium uniqueness Characterize eq m structure and make assumptions about equilibrium selection Exploit information from data to learn about equilibrium Nakajima (27), Soetevent-Kooreman (27), Jovanovic (1989) - Dagsvik-Jovanovic (1994) - Moro (22), Fang (25) I/O: Bresnahan and Reiss (1991) - Tamer (23) - Bajari-Hong-Ryan (27) - Aguirregabiria-Mira (27), Bajari-Hong-Krainer-Nekipelov (26); Introduction

9 Brock and Durlauf s binary choice model Smoking is a binary choice y i { 1, 1} Model

10 Brock and Durlauf s binary choice model Smoking is a binary choice y i { 1, 1} School-wide interactions π = average smoking in school Model

11 Brock and Durlauf s binary choice model Smoking is a binary choice y i { 1, 1} School-wide interactions π = average smoking in school Agents maximize max V (y i, X i, Z, π, ε i ) = y i (cx i + Jπ) + ε i (y i ) y i { 1,1} Model

12 Brock and Durlauf s binary choice model Smoking is a binary choice y i { 1, 1} School-wide interactions π = average smoking in school Agents maximize max V (y i, X i, Z, π, ε i ) = y i (cx i + Jπ) + ε i (y i ) y i { 1,1} ε i depends on choice y i and follows an extreme-value distribution: Pr (ε i ( 1) ε i (1) z) = exp ( z) Model

13 Model solution From the First Order Conditions, Pr (y i = 1) = exp ( 2 (cx i + Jπ)) Assuming that #I is large enough that a LLN applies, we obtain the following characterization of equilibrium: π = tanh (cx i + Jπ). i I Model

14 Equilibrium mapping π = tanh (cx i + Jπ) i I The map moves down as c decreases and flattens as J decreases 1% 45% Avg. smoking Model

15 Estimation First step: obtain π n from fraction of smokers in each school n Estimation

16 Estimation First step: obtain π n from fraction of smokers in each school n Second step: use π in the FOC s. Log likelihood: log L n = i 1+yi 2 log 1 + exp 2 c n X in + J n π n + 1 y i 2 log 1 + exp 2 c n X in + J n π n Estimation

17 Estimation First step: obtain π n from fraction of smokers in each school n Second step: use π in the FOC s. Log likelihood: log L n = i 1+yi 2 log 1 + exp 2 c n X in + Jn Gπ n + Jn Lπ g(i) + 1 y i 2 log 1 + exp 2 c n X in + Jn Gπ n + Jn Lπ g(i) Can account for local interactions π g(i) Estimation

18 Selection into group membership prob((i, k) are friends) = W ik γ + u ik prob(i smokes k is a friend) = (model) Exclusion restrictions: 1. k s characteristics and difference in i and k s characteristics 2. k s friends average characteristics (Bramouille ) Selection model can be estimated with standard parametric or non-parametric methods Estimation

19 Identification Agents choose y i { 1, 1} to maximize y i (c + J π n )+shock Single school: intercept in c not identified Estimation

20 Identification Agents choose y i { 1, 1} to maximize y i (cx i + J π n )+shock Single school: intercept in c not identified Slopes in c identified from interactions J from variation in smoking behavior between people with different X (J affects all) Estimation

21 Identification Agents choose y i { 1, 1} to maximize y i (cx i + J π n )+shock Single school: intercept in c not identified Slopes in c identified from interactions J from variation in smoking behavior between people with different X (J affects all) With multiple schools: intercept in c is identified (constant affects all schools equally, J proportionally to the average smoking π n ) Estimation

22 Teen smoking and friendship data Data from the National Longitudinal Study of Adolescent Health ( Add Health ) Nationally representative sample of adolescents in grades 7-12 in the US during the school year Combines longitudinal survey data on respondents attributes with contextual data on the family, neighborhood, school, individual networks, and peer groups. Our current data consist of about 46,6 observations in 45 schools We define smoking as smoking at least once or twice a week schools ( 18% of students) Results: teen smoking behavior

23 Specifications Preference parameters c n J n 1. School-by-school estimation Results: teen smoking behavior

24 Specifications Preference parameters c n = α + α k Zn k J n = γ + γ k Zn k k k 1. School-by-school estimation 2. Multiple schools, parameters depend on school or neighborhood attributes Z Results: teen smoking behavior

25 Specifications Preference parameters c n = α + α n, with α n N(,σ α ) J n = γ + γ n, with γ n N(,σ γ ) 1. School-by-school estimation 2. Multiple schools, parameters depend on school or neighborhood attributes Z 3. Multiple schools with random coefficients in parameters Results: teen smoking behavior

26 Specifications Preference parameters c n = α + α k Zn k + α n, with α n N(,σ α ) k J n = γ + γ k Zn k + γ n, with γ n N(,σ γ ) k 1. School-by-school estimation 2. Multiple schools, parameters depend on school or neighborhood attributes Z 3. Multiple schools with random coefficients in parameters 4. Multiple schools with random parameters that depend on school+neighborhood attributes Results: teen smoking behavior

27 Specifications Preference parameters c n = α + α k Zn k + α n, with α n N(,σ α ) k J n = γ + γ k Zn k + γ n, with γ n N(,σ γ ) k 1. School-by-school estimation 2. Multiple schools, parameters depend on school or neighborhood attributes Z 3. Multiple schools with random coefficients in parameters 4. Multiple schools with random parameters that depend on school+neighborhood attributes We estimated versions with school-wide interactions only, as well as both school-wide and local interactions Results: teen smoking behavior

28 Results, separate schools Variable Mean Median Black Asian Hispanic Female Age.11.8 Not belong to club GPA Mom college Dad at home Local Inter Constant One additional smoking friend increases probability of smoking by 12% Results: teen smoking behavior

29 Black Hispanic Age GPA Dad at home Asian Female No Club Mom college 6 Distribution of parameter estimates Results: teen smoking behavior J n (local) Constant

30 Multiple schools, parameters fct of Z Female Age No club Intercept (.13) -.37 (.67) (.2113) % Urban.34 (.61).333 (.8) (.726) % Poverty (.3644).1857 (.4).4575 (.4327) % College over (.3485).37 (.238).2228 (.4275) Female Labor Force P.R (.2978) -.9 (.133).9968 (.5351) Male Labor Force P.R (.1841).16 (.76).2586 (.31) Tobacco training (.379).223 (.58).122 (.482) Tobacco student policy.922 (.88) (.68).256 (.14) Tobacco staff policy (.882).388 (.61) (.956) Positive relationship between age and Results: teen smoking behavior

31 Multiple schools, parameters fct of Z Female Age No club Intercept (.13) -.37 (.67) (.2113) % Urban.34 (.61).333 (.8) (.726) % Poverty (.3644).1857 (.4).4575 (.4327) % College over (.3485).37 (.238).2228 (.4275) Female Labor Force P.R (.2978) -.9 (.133).9968 (.5351) Male Labor Force P.R (.1841).16 (.76).2586 (.31) Tobacco training (.379).223 (.58).122 (.482) Tobacco student policy.922 (.88) (.68).256 (.14) Tobacco staff policy (.882).388 (.61) (.956) Female smoking related to socio-economic status Results: teen smoking behavior

32 Multiple schools, parameters fct of Z Female Age No club Intercept (.13) -.37 (.67) (.2113) % Urban.34 (.61).333 (.8) (.726) % Poverty (.3644).1857 (.4).4575 (.4327) % College over (.3485).37 (.238).2228 (.4275) Female Labor Force P.R (.2978) -.9 (.133).9968 (.5351) Male Labor Force P.R (.1841).16 (.76).2586 (.31) Tobacco training (.379).223 (.58).122 (.482) Tobacco student policy.922 (.88) (.68).256 (.14) Tobacco staff policy (.882).388 (.61) (.956) Positive relationship between age and smoking stronger in poor areas Results: teen smoking behavior

33 Multiple schools, parameters fct of Z (cont.) GPA Mom college Dad Home Intercept.1427 (.325) (.172) (.966) % Urban.1 (.37).1338 (.73) (.77) % Poverty.711 (.174) (.399).3613 (.3915) % College over (.1322) (.3739).2692 (.3856) Female Labor Force P.R (.734) -.49 (.4452) (.2632) Male Labor Force P.R (.449).9316 (.2464).412 (.1495) Tobacco training (.245) (.457).121 (.485) Tobacco student policy (.386) -.75 (.114) (.996) Tobacco staff policy.16 (.42) -.25 (.165).1388 (.923) Female smoking related to socio-economic status Results: teen smoking behavior

34 Multiple schools, parameters fct of Z (cont.) GPA Mom college Dad Home Intercept.1427 (.325) (.172) (.966) % Urban.1 (.37).1338 (.73) (.77) % Poverty.711 (.174) (.399).3613 (.3915) % College over (.1322) (.3739).2692 (.3856) Female Labor Force P.R (.734) -.49 (.4452) (.2632) Male Labor Force P.R (.449).9316 (.2464).412 (.1495) Tobacco training (.245) (.457).121 (.485) Tobacco student policy (.386) -.75 (.114) (.996) Tobacco staff policy.16 (.42) -.25 (.165).1388 (.923) Dad at home -> Less smoking; more so when Fem LFPR high Results: teen smoking behavior

35 Multiple schools, parameters fct of Z (cont./2) Local Inter. Global Inter. Constant Intercept.3394 (.1373) (.1562).1882 (.199) % Urban (.517).959 (.1878) % Poverty (.3193) (.7985) % College over (.2953) (.69) Female Labor Force P.R (.3383) (.3243) Male Labor Force P.R..58 (.216) (.1961) Tobacco training -.43 (.397).6166 (.1487) Tobacco student policy.38 (.821) (.1554) Tobacco staff policy (.816) (.156) School policies can have large impact on school-wide interaction Results: teen smoking behavior

36 The prevalence of multiplicity Specification: only global interaction, no school covariates, only random effects Counter-factuals

37 The prevalence of multiplicity Specification: only global interaction, no school covariates, only random effects Multiplicity in 4 out of 41 cases In 35 out or 4, intermediate equilibrium Counter-factuals

38 The prevalence of multiplicity Specification: only global interaction, no school covariates, only random effects Multiplicity in 4 out of 41 cases In 35 out or 4, intermediate equilibrium Example: School N J Percentage of smokers Data Equil. 1 Equil. 2 Equil If all school were in the lowest equilibrium smoking would be 14.3 percentage points lower (from 6 to 32 percentage points) Counter-factuals

39 The effect of social interactions The mapping flattens as J increases Counter-factuals

40 Counterfactual: reduction in global interaction School ID 41 Interactions Percentage of smokers Data Equil 1 Equil 2 Equil 3 Baseline J reduced 3% J reduced 5% 99.8 J reduced 1% 99.8 Counter-factuals

41 Counterfactual: reduction in global interaction School ID Interactions Percentage of smokers Data Equil 1 Equil 2 Equil 3 Baseline J reduced 3% J reduced 5% 99.8 J reduced 1% 99.8 Baseline J reduced 3% J reduced 5% J reduced 1% Counter-factuals

42 What explains the variation of smoking? Most school in the intermediate equilibrium No correlation between interaction par. J n and smoking π n Counter-factuals

43 What explains the variation of smoking? Most school in the intermediate equilibrium No correlation between interaction par. J n and smoking π n Variation in c x (school heterogeneity), or student heterogeneity Standard deviation of smoking across schools Simulation Eq. 1 Eq. 2 Eq. 3 Original estimates All schools same c, J All students like school # (school heterogeneity more important) Counter-factuals

44 Counterfactual with both local and global interactions Here we need to simulate the Markov process letting students update their smoking state based on configuration of smoking in the previous period Let the process run many periods until convergence Results do not change if alllow random agent to change state in each period Multiplicity: different starting percentage of smokers Various exercises using school #56 as baseline: changes in J G, J L, number of friends, utility cost of smoking. arrangement of students within school: perfect integration vs. segregation. Counter-factuals

45 Multiplicity and nonlinearities: local interactions Local interaction parameter 3J 2J 1.5J 1.2J 1.1J 1.5J 1.1J.99J J.95J.9J.8J.5J.2J.1J.5J Percentage of smokers in low-level equil. Length of line represents the basin of attraction of low-level equilibrium Initial percentage of smokers Counter-factuals

46 Global interactions Global interaction parameter 1.5G 2G 3G 1.2G 1.1G 1.5G 1.1G.99G G.95G.9G.8G.5G.2G.1G.5G Only high-level equilibrium (1% smokers) Only high-level equilibrium (1% smokers) Only high-level equilibrium (99.99% smokers) Only high-level equilibrium (99.98% smokers) Only high-level equilibrium (99.96% smokers) Initial percentage of smokers Counter-factuals

47 Number of friends Number of friends Baseline One fewer 2 fewer 3 fewer.7 4 fewer Initial percentage of smokers Counter-factuals

48 What we learn about interactions Multiple equilibria are pervasive, nonlinearities are important Reducing strength of global or local interaction, or the number of friends makes low-level equilibria less likely Interactions lead to less smoking Counter-factuals

49 Direct utility from smoking Utility from smoking C i + 2med(C) C i + med(c) C i +.5med(C) C i +.2med(C) C i +.1med(C) C i +.5med(C) C i.5med(c) C i.1med(c) C i.2med(c) C i.5med(c) C i med(c) C i max(c) C i = Only high-level equilibrium (1% smokers) Initial percentage of smokers Counter-factuals

50 Artificial school School similar to #56. 8 students on a circle, with friends on her right Baseline: 4 friends (median in that school) Initially: homogenous students (median student: white 16yo female, 3.GPA...) Change demographics Change number of friends and patterns of friendships Counter-factuals

51 Integrated school: effect of friendship patterns L: least likely to smoke (based on individual characteristics M: most likely LLLLLLLM 1 friend LLLLLLM 4 friends LLLLLLM 8 friends LLLM 1 friend LLLM 4 friends LLLM 8 friends LMLM 1 friend LMLM 4 friends LMLM 8 friends More friends: less smoking More integration of smokers: more smoking Initial percentage of smokers Counter-factuals

52 Segregated school Fig 17 in the paper Two subgroups (least and most likely to smoke), local interactions only within groups A new equilibrium appears with the fraction of smokers equal to the proportion of high smokers The arrangement of students within the school and the composition of the students personal network matters Counter-factuals

53 Conclusions Nonlinearities matter, multiplicities matter Dangerous to extrapolate policy conclusions from a linear specification Conclusions