Understanding Interactions in Social Networks and Committees

Abstract

While much of the literature on cross-section dependence has focused on estimation of the regression coefficients in the underlying model, estimation and inferences on the magnitude and strength of spillovers and interactions has been largely ignored. At the same time, such inferences are important in many applications, not least because they have structural interpretations and provide useful inferences and structural explanation for the strength of any interactions. In this paper we propose GMM methods designed to uncover underlying (hidden) interactions in social networks and committees. Special attention is paid to the interval censored regression model. Small sample performance is examined through a Monte Carlo study. Our methods are applied to a study of committee decision making within the Bank of England's Monetary Policy Committee.

RÉSUMÉ Bien qu'une grande partie de la littérature sur la dépendance transversale se soit concentrée sur l'estimation des coefficients de régression dans le modèle sous-jacent, l'estimation et les inférences sur la magnitude et la force des retombées et des interactions ont été, en grande partie, ignorées. Parallèlement à ceci, ces inférences jouent un rôle important dans un grand nombre d'applications, ne serait-ce que parce qu'elles présentent des interprétations structurelles et fournissent des inférences utiles ainsi qu'une explication structurelle pour l'intensité des interactions. Dans la présente communication, nous proposons des méthodes GMM conues pour mettre à nu les interactions sous-jacentes (masquées) dans les réseaux et comités sociaux. On se penche tout particulièrement sur le modèle de régression à intervalle censuré, et on examine les performances de petits échantillons par le biais d'une étude Monte Carlo. Nos méthodes sont appliquées à une étude des prises de décision de comités au sein du comité de politique monétaire (Monetary Policy Committee) de la Banque d'Angleterre.

EXTRACTO Aunque gran parte de la bibliografía sobre la dependencia transversal se ha centrado en estimar los coeficientes de regresión en el modelo subyacente, la estimación e inferencias de la magnitud y fortaleza de los spillovers e interacciones han sido ampliamente ignoradas. Al mismo tiempo, tales inferencias son importantes en muchas aplicaciones, entre otros motivos, porque tienen interpretaciones estructurales y proporcionan inferencias útiles y explicación estructural de la fortaleza de cualquier interacción. En este estudio, proponemos métodos GMM diseñados para descubrir interacciones subyacentes (ocultas) en redes sociales y comités. Se presta particular atención al modelo de regresión censurada a intervalos. Se examina el rendimiento de muestras pequeñas a través de un estudio de Monte Carlo. Nuestros métodos se aplican a un estudio de toma de decisiones de comité dentro del Comité de Política Monetaria del Banco de Inglaterra

摘要:

目前大量对跨部门依赖的研究文献都集中在估计模型的回归系数 , 估计和推导影响的幅度和强度 , 而相互影响几乎被忽略。但在同时 , 这种对相互影响的推导在许多场合十分重要 , 因为它不仅揭示内在结构 , 还可推测其他任何相互作用的强度并揭示内在结构。本文中 , 我们提出的GMM方法可以揭示社会网络和XX的隐藏相互作用。我们特别考虑了区间截尾回归模型。通过蒙特卡罗研究方法我们检验了小样本性能。应用本方法研究了英格兰银行货币政策委员会的决策制定。

Keywords:

• Committee decision making
• social networks
• cross-section and spatial interaction
• generalised method of moments
• censored regression model
• expectation-maximisation algorithm
• monetary policy committee

JEL CLASSIFICATION:

• D71
• D85
• E43
• E52
• C31
• C34

Acknowledgments

The detailed review, comments and constructive criticism by two anonymous referees helped us extend, revise and improve the paper substantially. Their contribution is gratefully acknowledged. The paper has also benefited from comments by Jushan Bai, George Evans and Bernie Fingleton, as well as participants at the Econometric Society World Congress (Shanghai, 2010), International Panel Data Conference (Bonn, 2009), NTTS Conference (European Commission, Brussels, 2009) and seminars at Durham University and University of St Andrews. The usual disclaimer applies.

Notes

1. We prefer to use the term interaction rather than spatial since the latter term denotes some notion of physical proximity when there are many circumstances in which an interaction takes place in a much broader sense.

2. In our application later in the paper we consider interactions between members of the Monetary Policy Committee of the Bank of England. The membership changes over time, so during the period that we study there are members joining and leaving who provide instruments for the subset who remain on the Committee for the whole of the period.

3. Following Ahn & Schmidt (1995), one can add further moment conditions under the assumption that u _it  are homoscedastic over time. Likewise, lags of exogenous regressors can constitute additional moment conditions/instruments.

4. We are grateful to an anonymous referee for suggesting the Monte Carlo study, which helped us understand the finite sample performance of our estimation strategy.

5. Intuitively, a ‘very exogenous regressor’ has three properties. First, it is a regressor in the latent variable model and has coefficient unity. Second, conditional on all other endogenous, exogenous and instrumental variables in the model for the regression errors (8), it is independent of the error term u. Third, it has a support large enough to counterbalance the effect of other regressors and the errror. The second condition is similar to exogeneity but stronger—exogeneity requires conditioning only on the exogenous variables in the model. The regressor must also take both positive and negative values, which is easily achieved by location shifts.

6. Austen-Smith & Banks (1996) point out that we need each committee member to be open in revealing his estimate of the output gap and sincere in casting a vote for an interest rate decision that corresponds to the infomation available. Although we consider only the one period problem here, in a multi-period context we assume that reputational considerations are sufficiently powerful to ensure fair play.

7. This is the standard formula for the optimal combination of linear signals and was first introduced into economics by Bates & Granger (1969) as a way of combining forecasts.

8. The weights that members attach to the estimate of the output gap can in principle be negative if there is a sufficiently large negative covariance, or larger than one if there is a large positive covariance.

9. The derivation of the above inflation ‘feed forward’ rule (22) is based on Svensson (1997), where the policymaker only targets inflation, and the central bank can (in expectation) use the current interest rate to hit the target for inflation two periods hence.

10. Two assumptions are made in the above derivation. First, the rows of V are linearly independent, so that V ^{− 1} exists. Intuitively, this implies that each of the selected MPC members (j = 1,2,…,m) hold beliefs on accuracy of the initial forecasts that are different from each other. Second, none of the MPC members put zero weight on their own forecast, or technically speaking, all diagonal cofactors of V are non-zero. This ensures that diagonal elements of V ^{− 1} are non-zero. Given that the selection of members is made from a group of internal and external experts it would be unlikely that any Committee member attached no credence to his own opinions, or held private beliefs that were exactly the same as another member.

11. The pooling of information avoids any of the complications that arise in Townsend's (1983) model of ‘forecasting the forecasts of others’.

12. As before we assume openness and sincerity in providing information about what the output gap is.

13. The MPC met twice in September 2001. The special meeting was called after the events of 09/11.

14. We thank an anonymous referee for valuable suggestions that encouraged us to extend our analysis of the MPC network structure to the Governor King period. This enabled us to examine issues of changes in the network over time, the role of the Governor, and the effect of entry and exit of members. Further, this helped us clarify the important issue of strong versus weak spatial dependence in the context of the MPC.

15. On the issue of using real-time data rather than ex post revised data for analysing monetary policy decisions, see, for example, Orphanides (2003) and Bhattacharjee & Holly (2011).

16. See the working paper version for more extensive discussion of this approach in the current context.

17. We thank an anonymous referee for pointing this issue out to us.

18. For example, suppose the observed response for the jth member in a given month t is 0.25. By our assumed censoring mechanism (27), this response is assigned to the interval (0.20,0.375]. Suppose also that the linear prediction of the policy response, based on estimates of the interval regression model is . Then the residual cannot be assigned a single numerical value, but can be assigned to the interval (0.20−0.22,0.375−0.22]. In other words, the residual is interval censored: .

19. Note that the idiosyncratic errors in our spatial model are heteroscedastic across the members. Further, our inference focuses on cross-section dynamics while temporal dynamics is not fully modelled here. Hence, it is important to use HAC standard errors.

20. Strictly speaking, we test for only one part of the spatial granularity condition, that the row norm of W is less than unity, leaving the similar condition for the column norm aside. Consider, for example, an estimated row of interaction weights (−0.11,0.40,0.62,0,−0.49), aggregating to a sum of absolute values of 1.62, which is much larger than 1. Based on the estimated values and covariance matrix of interaction weights, we simulate values for the above row. Based on these simulations, we report as G(p)=0.102 the proportion of cases where the signed linear combination lies within the permissible range (−1,1) under the null hypothesis of spatial granularity. In other words, since the null hypothesis has a coverage of 10.2%, we cannot reject the null at 5% and 1% confidence levels.

21. Specifically, Bhattacharjee & Holly (2009) assumed that: (a) the sum of squares for each row was unity (row standardisation), (b) idiosyncratic error variances for the equations corresponding to the three internal members were equal (partial homoscedasticity), and (c) interactions between the internal members were symmetric (partial symmetry). Bhattacharjee & Holly (2011) assumed a symmetric interaction weights matrix.