A Semiparametric Network Formation Model with Multiple Linear Fixed Effects

Transcription

1 A Semiparametric Network Formation Model with Multiple Linear Fixed Effects Luis E. Candelaria University of Warwick January 7, 2018

2 Overview This paper is the first to analyze a static network formation model with two main features: 1 / 42

3 Overview This paper is the first to analyze a static network formation model with two main features: 1. Multiple and unobserved agent-specific components. 1 / 42

4 Overview This paper is the first to analyze a static network formation model with two main features: 1. Multiple and unobserved agent-specific components. 2. Semiparametric approach. 1 / 42

5 Overview This paper is the first to analyze a static network formation model with two main features: 1. Multiple and unobserved agent-specific components. 2. Semiparametric approach. Motivation: Friendships network. Homophily and unobserved agent-specific heterogeneity. 1 / 42

6 Overview This paper is the first to analyze a static network formation model with two main features: 1. Multiple and unobserved agent-specific components. 2. Semiparametric approach. Motivation: Friendships network. Homophily and unobserved agent-specific heterogeneity. This paper: One large network is observed. 1 / 42

7 Overview This paper is the first to analyze a static network formation model with two main features: 1. Multiple and unobserved agent-specific components. 2. Semiparametric approach. Motivation: Friendships network. Homophily and unobserved agent-specific heterogeneity. This paper: One large network is observed. Unrestricted dependence. 1 / 42

8 Overview This paper is the first to analyze a static network formation model with two main features: 1. Multiple and unobserved agent-specific components. 2. Semiparametric approach. Motivation: Friendships network. Homophily and unobserved agent-specific heterogeneity. This paper: One large network is observed. Unrestricted dependence. No distributional assumptions on the unobserved components. 1 / 42

9 Introduction Network formation models study the creation of relationships. e.g. friendships, partnerships, scientific collaborations. 2 / 42

10 Introduction Network formation models study the creation of relationships. e.g. friendships, partnerships, scientific collaborations. Why are they important? 1. Peer effects: network endogeneity. Goldsmith-Pinkham and Imbens Policy: social programs. Banerjee et al Social meaning: homophily. McPherson et al / 42

11 Introduction Network formation with unobserved agent-specific attributes. 3 / 42

12 Introduction Network formation with unobserved agent-specific attributes. Challenges: Arbitrarily correlation with the observed attributes. Semiparametric framework: identification? 3 / 42

13 Introduction Network formation with unobserved agent-specific attributes. Challenges: Arbitrarily correlation with the observed attributes. Semiparametric framework: identification? Implications: Biased and inconsistent results if these attributes are omitted. 3 / 42

14 Friendship network Definition (Network) A network is an ordered pair, (N n, D n ), where N n = {1,, n} is a set of nodes and D n = (D n ij ) is a n n adjacency matrix. 4 / 42

15 Friendship network Definition (Network) A network is an ordered pair, (N n, D n ), where N n = {1,, n} is a set of nodes and D n = (D n ij ) is a n n adjacency matrix. Assume the network is Undirected: D n ij = Dn ji for any i, j N n. Unweighted: D n ij {0, 1} for any i, j N n. Normalize D n ii = 0 for any i N n. 4 / 42

16 Example: Friendship network Figure: Undirected Network 5 / 42

17 Example: Friendship network Figure: Homophily on Observed Characteristics 5 / 42

18 Example: Friendship network Figure: Unobserved Agent-Specific Heterogeneity 5 / 42

19 Model of Interest Agents i, j N n form an undirected link according to the equation: for i j. D n ij =1 [ X n ij β 0 + µ i + µ j ε n ij 0 ], (NF) 6 / 42

20 Model of Interest Agents i, j N n form an undirected link according to the equation: for i j. D n ij =1 [ X n ij β 0 + µ i + µ j ε n ij 0 ], (NF) X n ij β 0: systematic part of the net benefit. 6 / 42

21 Model of Interest Agents i, j N n form an undirected link according to the equation: for i j. D n ij =1 [ X n ij β 0 + µ i + µ j ε n ij 0 ], (NF) X n ij β 0: systematic part of the net benefit. µ i, µ j : unobserved agent-specific factors. 6 / 42

22 Model of Interest Agents i, j N n form an undirected link according to the equation: for i j. D n ij =1 [ X n ij β 0 + µ i + µ j ε n ij 0 ], (NF) X n ij β 0: systematic part of the net benefit. µ i, µ j : unobserved agent-specific factors. ε n ij : pair-specific exogenous factor. 6 / 42

23 Model of Interest Agents i, j N n form an undirected link according to the equation: for i j. D n ij =1 [ X n ij β 0 + µ i + µ j ε n ij 0 ], (NF) X n ij β 0: systematic part of the net benefit. µ i, µ j : unobserved agent-specific factors. ε n ij : pair-specific exogenous factor. β 0 R K : unknown parameter. 6 / 42

24 Overview This paper is the first to analyze a static network formation model [ ] D n ij =1 Xij n β 0 + µ i + µ j ε n ij 0. with two main features: 1. Multiple and unobserved agent-specific fixed effects: µ i + µ j. 2. Semiparametric approach: F ε n ij x,µ and F µ x are unrestricted. Objective: Identification and estimation of β 0. 7 / 42

25 Main Results 1. New identification strategy. Point identification of β0. Identified set and bounds on each element of β0. Semiparametric pairwise estimator. Computationally tractable. Asymptotics: Growing number of agents. Empirical application. Friendships network: Add Health dataset. Homophily: age, GPA, and the level of father s education. 8 / 42

26 Main Results 1. New identification strategy. Point identification of β0. Identified set and bounds on each element of β0. 2. Semiparametric pairwise estimator. Computationally tractable. Asymptotics: growing number of agents. Empirical application. Add Health dataset: Friendships network. Homophily: age, GPA, and the level of father s education. 8 / 42

27 Main Results 1. New identification strategy. Point identification of β0. Identified set and bounds on each element of β0. 2. Semiparametric pairwise estimator. Computationally tractable. Asymptotics: growing number of agents. 3. Empirical application. Friendship network: Add Health dataset. Evidence for homophily on age, Hispanic, and father s education. 8 / 42

28 Literature Review I. Network Formation. Observed Heterogeneity: Brock and Durlauf (2005), Christakis, Fowler, Imbens, and Kalyanaraman (2010), Sheng (2012), Boucher and Mourifié (2013), Chandrasekhar and Jackson (2014), Souza (2014), Leung (2015a,b, 2016), Menzel (2015), Ridder and Sheng (2015), Hsieh and Lee (2016), de Paula, Richards-Shubik, and Tamer (2017), and Mele (2017). Unobserved Heterogeneity: Goldsmith-Pinkham and Imbens (2013), Charbonneau (2014), Auerbach (2016), Dzemski (2017), Graham (2017) and Jochmans (2017). II. Semiparametric Methods. Identification: Andersen (1973), Manski (1985, 1987) and Vytlacil and Yildiz (2007). Maximum Rank: Han (1987) and Abrevaya (1999). Inference: Andrews and Schafgans (1998), Newey (1990), Chamberlain (2010), Khan and Tamer (2010) and Khan and Nekipelov (2015). 9 / 42

29 Outline 1. Network Formation Model 2. Identification 3. Inference 4. Simulations 5. Application 6. Conclusions and Extensions

30 Outline 1. Network Formation Model 2. Identification 3. Inference 4. Simulations 5. Application 6. Conclusions and Extensions

31 Model - Framework 1. Triangular array of random networks {(N n, D n ) : n N}. 10 / 42

32 Model - Framework 1. Triangular array of random networks {(N n, D n ) : n N}. 2. A dyad is a pair (i, j) of agents with i, j N n and i j. 10 / 42

33 Model - Framework 1. Triangular array of random networks {(N n, D n ) : n N}. 2. A dyad is a pair (i, j) of agents with i, j N n and i j. Unique dyads: N (2) n {(1, 2), (1, 3),, (n 1, n)}. 10 / 42

34 Model - Framework 1. Triangular array of random networks {(N n, D n ) : n N}. 2. A dyad is a pair (i, j) of agents with i, j N n and i j. Unique dyads: N (2) n {(1, 2), (1, 3),, (n 1, n)}. Cardinality: N #N (2) n = O(n 2 ). 10 / 42

35 Model - Framework 1. Triangular array of random networks {(N n, D n ) : n N}. 2. A dyad is a pair (i, j) of agents with i, j N n and i j. Unique dyads: N (2) n {(1, 2), (1, 3),, (n 1, n)}. Cardinality: N #N (2) n = O(n 2 ). Each (i, j) N (2) n is endowed with X n ij, and let X n ( X n 12,, X n n 1,n). 10 / 42

36 Model - Preferences Agent i s latent marginal benefit of adding the link {ij} to D n is V ij (X n, η ij ; β 0 ) = u ij (X n ; β 0 ) + η ij. 11 / 42

37 Model - Preferences Agent i s latent marginal benefit of adding the link {ij} to D n is V ij (X n, η ij ; β 0 ) = u ij (X n ; β 0 ) + η ij. u ij (X n ; β 0 ) denotes the systematic part: u ij (X n ; β 0 ) 1 2 X ij β / 42

38 Model - Preferences Agent i s latent marginal benefit of adding the link {ij} to D n is V ij (X n, η ij ; β 0 ) = u ij (X n ; β 0 ) + η ij. u ij (X n ; β 0 ) denotes the systematic part: u ij (X n ; β 0 ) 1 2 X ij β 0. η ij denotes the unobserved valuation component: η ij µ j 1 2 ε ij. 11 / 42

39 Model - Preferences Agent i s latent marginal benefit of adding the link {ij} to D n is V ij (X n, η ij ; β 0 ) = u ij (X n ; β 0 ) + η ij. u ij (X n ; β 0 ) denotes the systematic part: u ij (X n ; β 0 ) 1 2 X ij β 0. η ij denotes the unobserved valuation component: η ij µ j 1 2 ε ij. Remarks: Rules out: Network externalities: uij (X n, D n ; β 0 ). Unobserved complementarity: g(µi, µ j ) as in Candelaria (2016). 11 / 42

40 Model - Stability Condition A network D n is stable with transfers if for each (i, j) N (2) n : 1. for all D ij = 1, V ij (X n, η ij ; β 0 ) + V ji (X n, η ji ; β 0 ) 0; 2. for all D ij = 0, V ij (X n, η ij ; β 0 ) + V ji (X n, η ji ; β 0 ) < / 42

41 Model - Stability Condition A network D n is stable with transfers if for each (i, j) N (2) n : 1. for all D ij = 1, V ij (X n, η ij ; β 0 ) + V ji (X n, η ji ; β 0 ) 0; 2. for all D ij = 0, V ij (X n, η ij ; β 0 ) + V ji (X n, η ji ; β 0 ) < 0. Equivalently, the network D n is stable with transfers if: [ ] D ij =1 X ij β 0 + µ i + µ j ε ij 0, (i, j) N n (2). (NF) 12 / 42

42 Model-Assumptions Assumption (A1) The following hold for any n. 1 For any distinct (i, k), (j, l) N (2) n : ε ik ε jl X n = x, µ n = µ, and F εik x,µ = F εjl x,µ. 2 The pdf f εi1 x,µ is positive everywhere for all (x, µ). A1 used in Graham (2017) and Menzel (2015). Agnostic about F εi1 x,µ and F µ x. 13 / 42

43 Identification Strategy Consider the subnetwork given by J, R, E N n. D JR = 1 J D JE = 0 R E D 14 / 42

44 Identification Strategy Consider the subnetwork given by J, R, E N n. D JR = 1 J D JE = 0 R E D 14 / 42

45 Identification Strategy Conditional on {X n = x, µ n = µ}: µ J + µ R J µj + µ E R E D 14 / 42

46 Identification Strategy Conditional on {X n = x, µ n = µ}: E [D JR D JE D JR D JE, X n = x, µ n = µ] µ R µ E µ J + µ R J µj + µ E R E D µ 3 µ 4 E [D 23 D 24 D 23 D 24, X n = x, µ n = µ] 14 / 42

47 Identification Strategy Consider the tetrad given by {J, R, E, D}. J R E D 15 / 42

48 Identification Strategy Conditional on {X n = x, µ n = µ}: J R E µd + µ R D µ D + µ E 15 / 42

49 Identification Strategy Conditional on {X n = x, µ n = µ}: E [D JR D JE D JR D JE, X n = x, µ n = µ] µ R µ E J R E D µ R µ E E [D DR D DE D DR D DE, X n = x, µ n = µ] 15 / 42

50 Assumptions Let kl X i X ik X il for any distinct (i, k), (i, l) N (2) n ; 16 / 42

51 Assumptions Let kl X i X ik X il for any distinct (i, k), (i, l) N (2) n ; kl X i = ( kl X (1) i, kl X ( 1) i ). 16 / 42

52 Assumptions Let kl X i X ik X il for any distinct (i, k), (i, l) N (2) n ; kl X i = ( kl X (1) i, kl X ( 1) i ). Assumption (A2) The following hold for any n, and any distinct (i, k), (i, l) N (2) n. 1 kl X i is not contained in a proper subspace of R K. 16 / 42

53 Assumptions Let kl X i X ik X il for any distinct (i, k), (i, l) N (2) n ; kl X i = ( kl X (1) i, kl X ( 1) i ). Assumption (A2) The following hold for any n, and any distinct (i, k), (i, l) N (2) n. 1 kl X i is not contained in a proper subspace of R K. 2 Exists kl X (1) i with β (1) 0 0 s.t. the cond. density of kl X (1) i is positive everywhere for any kl x ( 1) i. 16 / 42

54 Assumptions Let kl X i X ik X il for any distinct (i, k), (i, l) N (2) n ; kl X i = ( kl X (1) i, kl X ( 1) i ). Assumption (A2) The following hold for any n, and any distinct (i, k), (i, l) N (2) n. 1 kl X i is not contained in a proper subspace of R K. 2 Exists kl X (1) i with β (1) 0 0 s.t. the cond. density of kl X (1) i is positive everywhere for any kl x ( 1) i. Sign of β (1) 0 is identified, and scale is normalized: β (1) 0 = 1. A2 used in Manski(1985,1987), Han (1987) and Abrevaya (1999). 16 / 42

55 Assumption (A3) For any i N n, supp(µ i X n = x) [ B, B], for any x supp(x n ), and some B <. 17 / 42

56 Assumption (A3) For any i N n, supp(µ i X n = x) [ B, B], for any x supp(x n ), and some B <. Allows for continuous or a discrete fixed effects. Intuitively: supp(µ k µ l X n = x) supp( kl X i β 0) 17 / 42

57 Assumption (A3) For any i N n, supp(µ i X n = x) [ B, B], for any x supp(x n ), and some B <. Allows for continuous or a discrete fixed effects. Intuitively: supp(µ k µ l X n = x) supp( kl X i β 0) Let: X B = {x X n : for any i, j, k, l N n, kl x i β 0 2B, and sign { kl x i β 0 } sign { kl x j β 0 }} 17 / 42

58 Point Identification For any distinct i, j, l, k N n, let: Y (s) kl (D sk D sl ) for s = i, j, 18 / 42

59 Point Identification For any distinct i, j, l, k N n, let: Y (s) kl (D sk D sl ) for s = i, j, Ω(ijlk) { D ik D il, D jl D jk D ik D jk } 18 / 42

60 Point Identification For any distinct i, j, l, k N n, let: Y (s) kl (D sk D sl ) for s = i, j, Ω(ijlk) { } D ik D il, D jl D jk D ik D jk }{{} Within-ind i k l j 18 / 42

61 Point Identification For any distinct i, j, l, k N n, let: Y (s) kl (D sk D sl ) for s = i, j, Ω(ijlk) { } D ik D il, D jl D jk D ik D jk }{{} Across-inds i k l j 18 / 42

62 Point Identification Theorem 1 Let assumptions 1-3 hold. Then, for any n, and any i, j, k, l N n : [ Med Y (i) kl Y (j) where x X B. ] kl Xn = x, Ω(ijlk) = 2 sign { [ kl x i kl x j ] β0 }, (MC) 2 Let assumptions 1-3 hold. Then β 0 is point identified. 19 / 42

63 Point Identification Ω(ijlk) contains the subnetworks with enough variation to identify β / 42

64 Point Identification Ω(ijlk) contains the subnetworks with enough variation to identify β 0. i i i k l k l k l j j j Structure 1. Structure 2. Structure 3. i i i k l k l k l j j j Structure 4. Structure 5. Structure 6. Figure: Subnetwork by the tetrad (i, j, k, l) in Ω(ijlk). 20 / 42

65 Identification Failure I. Thin Set Let Ω n {Ω(ijlk) : for any distinct i, j, k, l N n }. 21 / 42

66 Identification Failure I. Thin Set Let Ω n {Ω(ijlk) : for any distinct i, j, k, l N n }. Theorem (Thin Set) Under Ass If the class Ω n has probability zero, then: 21 / 42

67 Identification Failure I. Thin Set Let Ω n {Ω(ijlk) : for any distinct i, j, k, l N n }. Theorem (Thin Set) Under Ass If the class Ω n has probability zero, then: { } Med Y (i) kl Y (j) kl X n = x does not have identification power. 21 / 42

68 Identification Failure I. Thin Set Let Ω n {Ω(ijlk) : for any distinct i, j, k, l N n }. Theorem (Thin Set) Under Ass If the class Ω n has probability zero, then: { } Med Y (i) kl Y (j) kl X n = x does not have identification power. In the empirical application: P(Ω n ) = 2.24%. Thin set identification as in Khan and Tamer (2010). 21 / 42

69 Thin Set Lemma (Sufficient Conditions) For any n, the class Ω n has probability zero if for any (i, j) N (2) n : 1 D n is empty, i.e., ( ) supp X ij β 0 µ n = µ, ε ij = e = (, µ i + µ j e ] 22 / 42

70 Thin Set Lemma (Sufficient Conditions) For any n, the class Ω n has probability zero if for any (i, j) N (2) n : 1 D n is empty, i.e., ( ) supp X ij β 0 µ n = µ, ε ij = e = (, µ i + µ j e ] 2 D n is dense, i.e., ( ) supp X ij β 0 µ n = µ, ε ij = e = [ µ i + µ j e, ) 22 / 42

71 Thin Set Lemma (Sufficient Conditions) For any n, the class Ω n has probability zero if for any (i, j) N (2) n : 1 D n is empty, i.e., ( ) supp X ij β 0 µ n = µ, ε ij = e = (, µ i + µ j e ] 2 D n is dense, i.e., ( ) supp X ij β 0 µ n = µ, ε ij = e = [ µ i + µ j e, ) 3 D n is homogeneous, i.e., supp ( µ i + µ j X ij = x, ε ij = e ) = [ e x β 0, ) 22 / 42

72 Additional Identification Results Is the large support assumption necessary for identification? 23 / 42

73 Additional Identification Results Is the large support assumption necessary for identification? Suppose all the covariates have bounded support, and 23 / 42

74 Additional Identification Results Is the large support assumption necessary for identification? Suppose all the covariates have bounded support, and 1. A2 : there exists at least one continuous variable with ( ) supp kl X i β 0 kl X ( 1) i = kl x ( 1) i = [ δ, δ] β 0 is still identified 23 / 42

75 Additional Identification Results Is the large support assumption necessary for identification? Suppose all the covariates have bounded support, and 1. A2 : there exists at least one continuous variable with ( ) supp kl X i β 0 kl X ( 1) i = kl x ( 1) i = [ δ, δ] β 0 is still identified 2. A2 : they are all discrete variables. Bounds for each element of β 0 are obtained. 23 / 42

76 Outline 1. Network Formation Model 2. Identification 3. Inference 4. Simulations 5. Application 6. Conclusions and Extensions

77 Inference The identification condition in (MC) suggests an estimator for β 0. Limiting objective function: [ { [ kl ] } ( Q(b) 2E S(X B ) sign X i kl X j b Y (i) kl where, S(X B ) = 1 if x X B, and 0 otherwise. ) ] Y (j) kl Ω(ijlk), 24 / 42

78 Inference The identification condition in (MC) suggests an estimator for β 0. Limiting objective function: [ { [ kl ] } ( Q(b) 2E S(X B ) sign X i kl X j b Y (i) kl where, S(X B ) = 1 if x X B, and 0 otherwise. ) ] Y (j) kl Ω(ijlk), Q(b) is uniquely maximized at b = β / 42

79 Inference The identification condition in (MC) suggests an estimator for β 0. Limiting objective function: [ { [ kl ] } ( Q(b) 2E S(X B ) sign X i kl X j b Y (i) kl where, S(X B ) = 1 if x X B, and 0 otherwise. ) ] Y (j) kl Ω(ijlk), Q(b) is uniquely maximized at b = β 0. The semiparametric pairwise difference estimator is ˆβ n = arg max Q n (b) b B 24 / 42

80 Inference Given a random sample of n agents, let { } { } z n ij = D n (i,j) N (2) ij, x ij n (i,j) N (2) n. 25 / 42

81 Inference Given a random sample of n agents, let The sample analog of Q(b): { } { } z n ij = D n (i,j) N (2) ij, x ij n (i,j) N (2) n. Q n (b) ( ) 1 n h(z i1,3, z i1,4, z i2,3, z i2,4, b), 4 C n,4 (Qn) where C n,4 indexes all the unique tetrads in {1, 2,, n}. 25 / 42

82 Inference Given a random sample of n agents, let The sample analog of Q(b): { } { } z n ij = D n (i,j) N (2) ij, x ij n (i,j) N (2) n. Q n (b) ( ) 1 n h(z i1,3, z i1,4, z i2,3, z i2,4, b), 4 C n,4 (Qn) where C n,4 indexes all the unique tetrads in {1, 2,, n}. Kernel function: h(z i1,3, z i1,4, z i2,3, z i2,4, b) 2 { { sign [ 3,4 x 1 3,4 x 2 ] b } 4! P 4 { } } (y (1) 3,4 y(2) 3,4 ) 1 y (1) 3,4 y(2) 3,4 = 2 S(x i1,3, x i1,4, x i2,3, x i2,4, B), where P 4 denotes the 4! permutations of {i 1,3, i 1,4, i 2,3, i 2,4 }. 25 / 42

83 Choice of B 1. If B is known X B = {x X n : for any i, j, k, l N n, kl x i b 2B, and sign { kl x i b} sign { kl x j b }} 26 / 42

84 Choice of B 1. If B is known X B = {x X n : for any i, j, k, l N n, kl x i b 2B, and sign { kl x i b} sign { kl x j b }} 2. Trimming X B (γ n ) = {x X n : for any i, j, k, l N n, kl x i b γ n, and sign { kl x i b} sign { kl x j b }}, with γ n. 26 / 42

85 Choice of B 1. If B is known X B = {x X n : for any i, j, k, l N n, kl x i b 2B, and sign { kl x i b} sign { kl x j b }} 2. Trimming X B (γ n ) = {x X n : for any i, j, k, l N n, kl x i b γ n, and with γ n. sup γ n Γ sign { kl x i b} sign { kl x j b }}, sup Q n (b; γ n ) E [Q n (b; γ n )] b B p / 42

86 Assumptions Assumption (B1) The researcher observes a random sample of n agents. For each dyad in N n (2), the researcher observes the link status and dyad-level attributes. { } Dij, x ij. (i,j) N (2) n Used in Graham (2017), Leung (2015b) and Menzel (2015). 27 / 42

87 Assumptions Assumption (B1) The researcher observes a random sample of n agents. For each dyad in N n (2), the researcher observes the link status and dyad-level attributes. { } Dij, x ij. (i,j) N (2) n Used in Graham (2017), Leung (2015b) and Menzel (2015). Assumption (B2) The parameter space B is compact and β 0 is an interior point of B. Used in Han (1987), Sherman (1993, 1994) and Abrevaya (1999). 27 / 42

88 Assumptions Assumption (B3) Let p n P (Ω n ), where 1 p n p 0 0, as n. 2 Npn, as n. The probability p n is allowed to decay as n. 28 / 42

89 Consistency Theorem (Consistency) Let assumptions A1, A2, B1-B3 hold. Then, p ˆβ n β 0 0 as n. 29 / 42

90 Theorem (Asymptotic Normality) If assumptions A1, A2, B1-B4. hold, then: p n N( ˆβn β 0 ) d N (0, V 1 V 1 ) (AN) with 4V = E [ 2 τ 2 (, β 0 ) Ω n ], = E [ 1 τ(, β 0 )] [ 1 τ(, β 0 )]. Recall that N = O(n 2 ). 30 / 42

91 Convergence Rate The convergence rate depends on the limit of p n P (Ω n ). 1. Regular Estimator: p n p > 0, as n. 31 / 42

92 Convergence Rate The convergence rate depends on the limit of p n P (Ω n ). 1. Regular Estimator: p n p > 0, as n. 2. Irregular Estimator: p n 0, as as n. (Newey 1990, Andrews and Schafgans 1998 and Khan and Tamer 2010). Theorem (Information bound) In model given by equation (NF), under assumptions A1, A2, B1-B4. If p n 0, then the information bound for β 0 is zero. 31 / 42

93 Adaptive Rate Inference Consider the studentized estimator, as in Andrews and Schafgans (1998) and Khan and Tamer (2010), ˆΣ 1/2 n N( ˆβn β 0 ) d N (0, I), as n where ˆΣ n, and Ŝn is the Bootstrap estimate of ˆΣ n = Ŝn/ˆp 2 n, S = V 1 V 1. Subbotin (2007): Bootstrap validity for Maximum Rank estimators. 32 / 42

94 Outline 1. Network Formation Model 2. Identification 3. Inference 4. Simulations 5. Application 6. Conclusions and Extensions

95 Computation The objective function Q n (b) is a 4th order U-statistic. O(n 4 ) operations. Proposition The estimator ˆβ n can be equivalently computed from: Q n (b) 1 n(n 1)(n 2) i j k S(B)Rank j,k [ (xik x il ) b ] y (i) k,l Q n (b) can be computed in O(n 3 log(n)) operations. 33 / 42

96 Simulations Consider the following model: [ ] D ij = 1 X ij β 0 + µ i + µ j ε ij 0, for (i, j) N n (2). 1. Dyad-specific attributes, X ij for (i, j) N (2) n : X ij = [ z i1 z j1, z i2 z j2, z i3 z j3 ]. where the individual-specific attributes are drawn as: z i1 Normal(0, 3), z i2 Uniform { 1, 0, 1} with p k = 1/3, z i3 Uniform( 2, 2). 34 / 42

97 Simulations 2. Fixed effects: α i = λ (z i1 + z i2 + z i3 ) /3 + (1 λ)normal(0, 1), where λ {1/4, 1/2, 3/4} measures the degree of dependence. B if α i < B µ i = α i if B α i B, B if B < α i with B = Link-specific disturbance term: ε (2) ij Normal(0, 2). True DGP: β 0 = [1, 1.5, 1.5] 35 / 42

98 MC Simulations: Normal(0,2) Pairwise Difference Graham (2015) P(Ω n ) Median Mean Bias(%) RMSE Median Mean Bias(%) RMSE N = % β 2 /β 1 = β 3 /β 1 = N = % β 2 /β 1 = β 3 /β 1 = M=500, λ = / 42

99 MC Simulations: Normal(0,2) Pairwise Difference Graham (2015) P(Ω n ) Median Mean Bias(%) RMSE Median Mean Bias(%) RMSE N = % β 2 /β 1 = β 3 /β 1 = N = % β 2 /β 1 = β 3 /β 1 = N = % β 2 /β 1 = β 3 /β 1 = M=500, λ = / 42

100 Simulation: Discrete and Bounded Support: Consider the next specification for the observed covariates: X (1) ij takes the values {0, 1, 2, 3, 4}. X (2) ij takes the values { 1, 0, 1}. X (3) ij takes the values { 1, 0, 1, 2}. Thus, the support of X ij contains 60 points. 37 / 42

101 Discrete and Bounded Support: Sharp Bounds Figure: Bounds and Rectangular Superset 38 / 42

102 Outline 1. Network Formation Model 2. Identification 3. Inference 4. Simulations 5. Application 6. Conclusions and Extensions

103 Empirical Application This application estimates a model of friendships formation using the Add Health dataset. 39 / 42

104 Empirical Application This application estimates a model of friendships formation using the Add Health dataset. Objective: Estimate the preferences for homophily. 39 / 42

105 Empirical Application This application estimates a model of friendships formation using the Add Health dataset. Objective: Estimate the preferences for homophily. Dataset: Add Health is a longitudinal national survey. 39 / 42

106 Empirical Application This application estimates a model of friendships formation using the Add Health dataset. Objective: Estimate the preferences for homophily. Dataset: Add Health is a longitudinal national survey. High school students: Grades 7-12 during the school years. 39 / 42

107 Empirical Application This application estimates a model of friendships formation using the Add Health dataset. Objective: Estimate the preferences for homophily. Dataset: Add Health is a longitudinal national survey. High school students: Grades 7-12 during the school years. Observed network: availability of respondents friendship network. 39 / 42

108 Empirical Application This application estimates a model of friendships formation using the Add Health dataset. Objective: Estimate the preferences for homophily. Dataset: Add Health is a longitudinal national survey. High school students: Grades 7-12 during the school years. Observed network: availability of respondents friendship network. Saturated high schools: each student nominates at most 5 male and 5 female friends. 39 / 42

109 Empirical Application This application estimates a model of friendships formation using the Add Health dataset. Objective: Estimate the preferences for homophily. Dataset: Add Health is a longitudinal national survey. High school students: Grades 7-12 during the school years. Observed network: availability of respondents friendship network. Saturated high schools: each student nominates at most 5 male and 5 female friends. Wave I In-home interview: One high school with 319 students. 39 / 42

110 Exogenous Covariates Table: Descriptive Statistics Variable Mean Std. Dev. Min Max Household Income Age Female Grade Hispanic White Black Asian Indian Other races Overall GPA Mother s Education Father s Education Sample size = / 42

111 Estimation Results Logistic Pairwise Difference Graham (2015) Age Female Grade Hispanic White Black Asian Indian Other races Overall GPA Mother Education Father Education P(Ω) = 2.24% Average Degree = Number of Students = 319. Number of dyads = 50,721. *,**,*** represents the significant at 10%, 5%, and 1% level. 41 / 42

112 Conclusions 1. Semiparametric network model with unobserved heterogeneity. 2. Point identification and sharp bounds for each component of β Semiparametric pairwise difference estimator. 4. Empirical application considers a friendship network. 42 / 42

113 Thanks! 1 / 11

114 Appendix 2 / 11

115 Covariates with Bounded Support I. At Least One Continuous Covariate Assumption (A2 ) The following hold for any n, and any i, l, k N n, with l k. 1 The random vector kl X i has a bounded support on R K. 2 For some δ > 0, there exists an interval I δ = [ δ, δ] and a set N δ R K 1 such that Nδ is not contained in any proper linear subspace of R K 1. P ( kl Xi N δ ) > 0. For almost every kl x N δ, the distribution of kl X i β 0 conditional on kl Xi = kl x i has a probability density that is everywhere positive on I δ. Proposition Let Assumptions A1, A2, and A3 hold; then β 0 is point identified. 3 / 11

116 Covariates with Bounded Support II. Discrete Support I obtain sharp bounds for each component in β 0 using Komarova (2013). Assumption (A2 ) For any n, and any i, k, l N n, with k l. 1 The support of F Xik is not contained in any proper linear space of R K. 2 The profile vector of observed attributes X n (X 12,, X n 1,n ) has a discrete support given by for a finite D. supp(x n ) = { x 1,, x D}, 4 / 11

117 Thin Set Table: Stochastic Dominance and Sparsity Empty Sparse Dense E [Degree] P [Ω n ] E [Degree] P [Ω n ] E [Degree] P [Ω n ] (%) (%) (%) λ = 0.25 Log LnN N Gam T λ = 0.5 Log LnN N Gam T λ = 0.75 Log LnN N Gam T Notes: N=100, M= / 11

118 Thin Set Table: Thin Set Simulations: Homogeneous Network µ = 10 Bernoulli(p) + ( 5) (1 Bernoulli(p)) N=100 E [Degree] P [Ω(ijkl)] (%) Jaccard SI (Mean) Cosine SI (Mean) (%) (Mean) (Mean) p = 0.2 Log LnN N Gam T p = 0.8 Log LnN N Gam T Notes: M= / 11

119 Identification Failure II. Nonlinear Panel Data Identification Strategy Proposition 1. Let assumption 1 hold; then, for any n, and any i, l, k N n. Med(D ik D il X n = x, D il + D ik = 1) = sign [(x ik x il ) β 0 + (µ k µ l )] (MS) 2. Let Assumptions 1 and 2 hold; then, the equation (MS) does not have identification power. 7 / 11

120 References I Abrevaya, J. (1999). Leapfrog estimation of a fixed-effects model with unknown transformation of the dependent variable. Journal of Econometrics 93(2), Andersen, E. B. (1973). Conditional inference for multiple-choice questionnaires. British Journal of Mathematical and Statistical Psychology 26(1), Andrews, D. W. and M. M. Schafgans (1998). Semiparametric estimation of the intercept of a sample selection model. The Review of Economic Studies 65(3), Auerbach, E. (2016). Identification and estimation of models with endogenous network formation. Technical report, Working Paper. Banerjee, A., A. G. Chandrasekhar, E. Duflo, and M. O. Jackson (2013). The diffusion of microfinance. Science 341(6144), Boucher, V. and I. Mourifié (2013). My friend far far away: Asymptotic properties of pairwise stable networks. Working Paper. Brock, W. A. and S. N. Durlauf (2005). Multinomial choice with social interactions. The Economy As an Evolving Complex System, III: Current Perspectives and Future Directions, 175.

121 References II Candelaria, L. E. (2016). Network formation models with interactive fixed effects. Working Paper. Chamberlain, G. (2010). Binary response models for panel data: Identification and information. Econometrica 78(1), Chandrasekhar, A. G. and M. O. Jackson (2014). random graph models. Working Paper. Tractable and consistent Charbonneau, K. (2014). Multiple fixed effects in nonlinear panel data models. Working Paper. Christakis, N. A., J. H. Fowler, G. W. Imbens, and K. Kalyanaraman (2010). An empirical model for strategic network formation. Working Paper. de Paula, A., S. Richards-Shubik, and E. Tamer (2017). Identifying preferences in networks with bounded degree. forthcoming in. Econometrica. Dzemski, A. (2017). An empirical model of dyadic link formation in a network with unobserved heterogeneity. Goldsmith-Pinkham, P. and G. W. Imbens (2013). Social networks and the identification of peer effects. Journal of Business & Economic Statistics 31(3), Graham, B. S. (2017). An econometric model of link formation with degree heterogeneity. Working Paper.

122 References III Han, A. K. (1987). Non-parametric analysis of a generalized regression model: the maximum rank correlation estimator. Journal of Econometrics 35(2), Hsieh, C.-S. and L. F. Lee (2016). A social interactions model with endogenous friendship formation and selectivity. Journal of Applied Econometrics 31(2), Jochmans, K. (2017). Semiparametric analysis of network formation. Journal of Business & Economic Statistics (just-accepted). Khan, S. and E. Tamer (2010). Irregular identification, support conditions, and inverse weight estimation. Econometrica 78(6), Komarova, T. (2013). Binary choice models with discrete regressors: Identification and misspecification. Journal of Econometrics 177(1), Leung, M. (2015a). A random-field approach to inference in large models of network formation. Available at SSRN. Leung, M. (2015b). Two-step estimation of network-formation models with incomplete information. Journal of Econometrics 188(1), Leung, M. (2016). Working Paper. A weak law for moments of pairwise-stable networks.

123 References IV Manski, C. F. (1985). Semiparametric analysis of discrete response: Asymptotic properties of the maximum score estimator. Journal of Econometrics 27(3), Manski, C. F. (1987). Semiparametric analysis of random effects linear models from binary panel data. Econometrica, McPherson, M., L. Smith-Lovin, and J. M. Cook (2001). Birds of a feather: Homophily in social networks. Annual Review of Sociology, Mele, A. (2017). A structural model of dense network formation. Econometrica 85(3), Menzel, K. (2015). Stategic network formation with many agents. Working Paper. Newey, W. K. (1990). Semiparametric efficiency bounds. Journal of Applied Econometrics 5(2), Ridder, G. and S. Sheng (2015). Estimation of large network formation games. Technical report, Working papers, UCLA. Sheng, S. (2012). Identification and estimation of network formation games. Working Paper. Sherman, R. P. (1993). The limiting distribution of the maximum rank correlation estimator. Econometrica,

124 References V Sherman, R. P. (1994). Maximal inequalities for degenerate U-processes with applications to optimization estimators. The Annals of Statistics, Souza, P. (2014). Estimating network effects without network data. LSE Working. Subbotin, V. (2007). Asymptotic and bootstrap properties of rank regressions. Working Paper.