Johan Koskinen & Alessandro Lomi

Transcription

1 The Local Structure of Globalization Johan Koskinen & Alessandro Lomi Journal of Statistical Physics 1 ISSN J Stat Phys DOI /s x 1 23

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3 JStatPhys DOI /s x The Local Structure of Globalization The Network Dynamics of Foreign Direct Investments in the International Electricity Industry Johan Koskinen Alessandro Lomi Received: 8 August 2012 / Accepted: 25 February 2013 Springer Science+Business Media New York 2013 Abstract We study the evolution of the network of foreign direct investment (FDI) in the international electricity industry during the period We assume that the ties in the network of investment relations between countries are created and deleted in continuous time, according to a conditional Gibbs distribution. This assumption allows us to take simultaneously into account the aggregate predictions of the well-established gravity model of international trade as well as local dependencies between network ties connecting the countries in our sample. According to the modified version of the gravity model that we specify, the probability of observing an investment tie between two countries depends on the mass of the economies involved, their physical distance, and the tendency of the network to self-organize into local configurations of network ties. While the limiting distribution of the data generating process is an exponential random graph model, we do not assume the system to be in equilibrium. We find evidence of the effects of the standard gravity model of international trade on evolution of the global FDI network. However, we also provide evidence of significant dyadic and extra-dyadic dependencies between investment ties that are typically ignored in available research. We show that local dependencies between national electricity industries are sufficient for explaining global properties of the network of foreign direct investments. We also show, however, that network dependencies vary significantly over time giving rise to a time-heterogeneous localized process of network evolution. Keywords Dynamic stochastic models for networks Electricity industry Foreign direct investments Globalization Gravity model Longitudinal exponential random graph models Ensemble J. Koskinen ( ) Social Statistics Discipline Area, University of Manchester, Humanities Bridgeford Street, Manchester M13 9PL, UK johan.koskinen@manchester.ac.uk A. Lomi Faculty of Economics, University of Lugano, Lugano, Switzerland

4 J. Koskinen, A. Lomi 1 Introduction and Motivation The aggregate dynamics of socio-economic systems is often modeled as emerging from the behavior of connected agents capable only of local initiative [62, 80, 81]. This view has stimulated a new generation of analytical approaches to social interaction and economic exchange explicitly inspired by models developed in statistical mechanics [15, 27]. Research on the International Trade Network (ITN), also known as the World Trade Web (WTW), represents a prime example of these recent developments [13, 30, 82, 97]. The cornerstone of empirical economic research on the WTW is the so called gravity model of international trade [5, 11]. First suggested by physicist turned economist Jan Tinbergen [96], the model posits that the volume of trade between two countries (bilateral trade) is proportional to their economic masses as measured, for example, by their Gross Domestic Product, GDP and inversely proportional to the distance between them as measured, for example, by the distance between the main cities [25]. The basic idea is that a mass of resources supplied at the origin ( sender ) country (V i ) is attracted by demand in the destination ( receiver ) country (V j ), but the potential flow of resources between the origin and destination countries (Y ij ) is reduced by their distance (D ij ). In its simplest form, the gravity equation specified and estimated in empirical models is therefore: Y ij = V i V j /D ij. The gravity model of trade is one of the most successful empirical models in economics [4, 57]. For this reason, it is frequently adopted as a benchmark for more complex models that try to reproduce the observed network structure of the WTW [26]. The empirical success of the gravity model also stimulated recent attempts to extend its domain of application to binary networks such as, for example, the network of trade agreements [11]. The main objective of this paper is to advance this contemporary line of research by examining the extent to which the dyadic (or local ) network connections between countries implied by the gravity model of international trade are consistent with the global network structures that are actually observed. More specifically, in this paper we estimate a modified gravity equation that we specify to model change in the binary architecture of the global network of Foreign Direct Investment (FDI) in the international electricity industry. FDI involves the acquisition of direct ownership (represented by voting securities) by a single company located in one country (the parent) of a company located in a different country (the foreign affiliate or target). We show that the modified gravity equation reproduces the insight of the original economic model, while adding important detail about the local structure of dependence relations among countries. Our work extends previous research in three ways. First, we adopt a new class of exponential random graph models (ERGMs) [36, 46, 66 68, 88, 101] to represent explicitly the network dependencies between countries linked by a FDI relationship. While some research is available that has considered the role of spatial dependencies in the form of neighbours effects [28], to the best of our knowledge this is the first study that attempts to incorporate endogenous degree-based network effects explicitly in the gravity model. Second, we examine how third-country effects shape the global structure of the FDI network. A limited number of contemporary studies have examined the role of third country effects in the formation of dyadic FDI relations between countries [9]. Yet, an underlying assumption of both the theoretical as well as the empirical literature is that FDI decisions are dyadic in nature with no interdependence beyond bilateral trade relations [14]. We are not aware of studies that have linked specific forms of extra-dyadic dependence among countries to the global structure of ITN. Third, the longitudinal ERGM that we specify assumes that tie-variables between each pair of countries evolve in continuous time and an update to the system is governed by the conditional probability of the limiting ERGM distribution. Estimation of the model

5 The Local Structure of Globalization is performed by means of a Bayesian Markov chain Monte Carlo (MCMC) scheme that iterates between drawing unobserved sample paths connecting the observations and drawing parameters. There is only little experience of using of the statistical model we present [48, 60, 86] and we are not aware of prior empirical studies adopting this model specification and estimation strategy to analyze the dynamics of connected socio-economic systems. The rest of the paper is structured as follows. We begin by introducing and describing the data set. We then emphasize features of the temporal sequence of networks such as skewed degree distributions and clustering that represent evident departures from existing model formulations and estimations. We then suggest new model specifications that incorporate both the basic gravity elements as well as extra-dyadic network dependencies. We produce empirical results supportive of our model. We discuss the results with reference to predictive properties and conclude with a discussion that sets a possible agenda for further research. 2 Empirical Setting and Data 2.1 Empirical Setting The dependence relations that are at the core of our modelling efforts are generated by Foreign Direct Investments (FDI) defined as investments: Involving a long-term relationship and reflecting a lasting interest and control by a resident entity in one economy ([the] foreign direct investor or parent enterprise) in an enterprise resident in an economy other than that of the foreign direct investor... (UNCTAD 2003: 231 [98]). Current research recognizes FDI as an important financial mechanism for the embedding of local economies into a global network of international relations [94]. We restrict the focus of our analysis of FDI which involve the transfer of control of local companies to foreign owners [8]. Defined in this way FDIs imply an inflow of capital from a country into another and a corresponding outflow of property rights [56]. During the period that we analyze, FDI has progressively become one of the main factors underlying the increased interdependence of national economies [50]. We focus on the electricity industry for two main reasons. The first is more empirical. FDI has been a major force in the globalization of the electricity industry. For example, in 2004 (the last year of observation in our sample) the total value of mergers and acquisitions (the main forms of FDI) in the global electricity industry (including gas) was USD bn, 46 % of which were cross borders (USD 57.2 bn), i.e., involved companies in different countries [71]. The second reason relates more directly to our modelling objectives. For a mix of technological, economic, political, and historical factors the international electricity industry has long been considered as an aggregate of national (or local ) industries with almost no connection across national borders. Before the early 1990s it was almost unconceivable to have national ( local ) electricity companies owned and operated by rival companies located in foreign countries. After two decades of progressive internationalization the electricity industry is recurrently portrayed as a representative example of a truly global industry [7]. Unlike the majority of studies on International Trade Networks (ITN) we are not interested in predicting the aggregate (non-zero) flows of capital between countries [13, 31, 41]. Rather we focus on the change in the binary architecture underlying such flows [26]. Our exclusive focus on network structure is motivated, in part, by recent results suggesting that the structure of international trade networks may be fully characterized in terms of their local topological properties alone [92]. According to Squartini, Fagiolo, and Garlaschelli, for example [93] (2011: ): in order to properly understand the structure of the international trade system it is essential to reproduce its binary topology, even if one is interested in a weighted description.

6 J. Koskinen, A. Lomi Table 1 Density: (i,j) V (2) x ij (t m )/(n(n 1)); and Hamming distance: (i,j) V (2) x ij (t m ) x ij (t m 1 ) ; for Foreign direct investment ties between 97 countries over 10 years, t m, m = 0,...,9. Starting year t 0 = 1994 m Density Distance Two main questions guide our empirical analysis: how did the current structure of the FDI network come about? What are the micro-mechanisms driving its change over time? The answers we provide emphasize the endogenous network mechanisms underlying change in each individual tie as a function of the presence or absence of other ties in its neighborhood [69]. 2.2 Data Description Here we provide some basic descriptive characterization of data to motivate the statistical model. The change in density of the FDI networks over time provides a first characterization of the process (Table 1). Network density increases with time until 1999, after which the density decreases. There is thus a considerable number of ties being created and ties being deleted across time. This is mirrored by an increase in the Hamming distances between consecutive years until 1999, after which it decreases. There is thus considerable change over time as well as between consecutive time-points. The networks after 1994 are all characterized by having one large component and many isolates (on average 58 isolates and six countries that remain isolates for all ten years). The in- and out-degree distributions across the years are given in Fig. 1. The distribution of the number of investment ties received is relatively homogeneous and does not change noticeably over the years. The maximum number of investment ties received from other countries is 7 (the USA in 1999). The number of FDI ties sent varies a great deal more and the distribution of out-degrees is much more skewed than the in-degree (one country, the USA, sent ties to 23 other countries in 1999). The network aggregated over the years with nodes positioned geographically is reported in Fig. 2. The USA and Europe are the two main geographical hubs. Ties are generally geographically clustered (especially within Europe but also in the Americas and for example in Austral-Asia) but there are also some longer-range (inter-continental) ties. The geographical hubs and clustering are in accordance with the gravity model ( mass and distance ) but some triangles, indicating extra-dyadic clustering are also evident. To examine evidence of dependence beyond the dyads we compare the triad-census (or triadic motifs [65]) for the observed networks against the corresponding triad-census predicted by a uniform distribution. In particular, we generate reference networks from the U MAN distribution [44], which is a conditionally uniform distribution over all graphs that have a prescribed dyad census. The dyad census counts the number of mutual (M), asymmetric (A), and null (N) dyads. The triad census for normalized for each year by the mean and standard deviation according to the corresponding U MAN distribution is given in Fig. 3 (the triads are labeled according to [44]). This gives a structural profile of the network over time. Across the board it is apparent that there is considerable residual structure beyond the dyadic level. As an example, there are many more null triads ( 003 ) in data cross-sectionally than we would expect had the dyads been distributed at random.

7 The Local Structure of Globalization Fig. 1 The in- and out-degree distributions for the FDI networks with log-log plot (insert) Fig. 2 The symmetrized network (dotted lines are reciprocated ties) of FDI ties aggregated across time with nodes positioned in the geographical centers of the countries This suggests that dyads tend to cluster together. An example of how they cluster together is the out-star ( 021D ) where one country invests in two countries that are themselves neither directly tied, nor linked to the origin country by reciprocated ties. This may suggest outdegree centralization. We also note that reciprocated dyads do not tend to exist in isolation ( 102 ) but tend to involve one node that invests in a third party ( 111U ). Of particular interest is the large number of transitive triads ( 030T ). The fact that these are over represented in combination with cycles ( 030C ) being few might suggest that the process of foreign direct investment is locally hierarchal.

8 J. Koskinen, A. Lomi Fig. 3 Observed triad census standardized by the predicted census under the conditional uniform U MAN distribution (i.e., a uniform distribution over graphs with a prescribed number of mutual, asymmetric and null dyads [44]). The horizontal dashed line is the origin for reference and counts are given as violin-plots sequentially for years 1994 through Vertical axis is given as standard deviation units and MCMC p-values (multiplied by 100) are provided below and atop violins (alternatingly). Violin-plots display data as both a box plot and a kernel density estimate [42] Thus, according to the patterns in Fig. 3 there is residual structure over and above the dyadic level. A question central to the analysis of FDI concerns the possible sources of this residual structure. The corresponding dependencies may be due to: (i) over-propensities prescribed by the gravity model; (ii) endogenous processes such as preferential attachment or triadic closure; and, (iii) dependence on initial conditions, i.e. any structure is due to the first observation. It has been argued that triadic features may be captured by spatial embedding [18], a key feature of the gravity model, and it is well known that extra-dyadic dependencies may stem from unobserved nodal and dyadic variates [43] (substantively, these variates represent for example homophily or propinquity [64]). As Daraganova et al. [24] demonstrate, spatial embedding and endogenous dependencies are complimentary in explaining triadic and other extra-dyadic features. The initial condition is not modelled, hence subsequent observations may exhibit extra-dyadic structure even if the process that generated them is to all intents and purposes random whenever the first observation has features that depart from random. In order to address these questions we employ a statistical procedure that allows us to analyze all of these aspects simultaneously. We discuss the modeling framework next (Sect. 3). To summarize: the descriptive analysis shows that there is change over time in the FDI network, that the ties are spatially embedded and that there is evidence of extra-dyadic dependencies liking the countries.

9 The Local Structure of Globalization 3 Longitudinal Exponential Random Graph Models 3.1 Notational Preliminaries We consider di-graphs G(V, E) on a fixed set of nodes V ={1, 2,...,n}, with a stochastic arc-set E V (2) (V (2) ={(i, j) V V : i j}). For the purposes of modelling, a graph G(V, E) is represented by its n n binary adjacency matrix X = (X ij : (i, j) V 2 ), where the tie-indicator X ij is equal to one or zero according to whether (i, j) E or not, respectively. The space of all adjacency matrices X ={0, 1} V (2) has dimensions 2 n(n 1). 3.2 Exponential Random Graph Models We define and ERGM process as the process on X for which the limiting distribution is an ERGM. ERGMs [36, 46, 88, 101] for cross-sectional networks are a well-researched family of models that evolved out of statistical mechanics and are becoming increasingly popular in the social sciences [66 68]. The ERGM is an ensemble model on X, where the probability mass function is of an exponential family distribution form, Pr θ (X = x) = exp { g(x; θ) ψ(θ) }, where g( ; θ) is the potential function and ψ(θ) = log x X eg(x;θ) is the (log) partition function, ensuring that the distribution sums to unity on X. The potential function may be a fairly general real-valued function of the network (and fixed graph attributes) that reflect substantively interesting properties of the network. However, principled considerations suggests that g( ; θ) should be a linear combination of a collection of graph statistics that reflect dependencies between tie-variables. While the Bernoulli-graph [29, 91], with g(x; θ) = x ij, has been used extensively in social network analysis as a null-model, the first model to fully draw on properties of the exponential family and go beyond independent tie-variables was proposed by [46] and [34]. This model was developed from the point of view of standard statistical techniques for modelling binary data and it was not until the seminal paper by Frank and Strauss [36] that the methods for statistical mechanics developed by [12] and[95] were brought to bear on network data. Frank and Strauss [36] proposed that tie-variables may be interdependent but that two tie-variables X ij and X kl (for undirected graphs) are conditionally independent given {X uv : {u, v} ( V 2) /{i, j, k, l}} if {i, j} {k,l}=. Imposing some homogeneity restriction, this ( xi+ ) k dependence assumption implies an ERGM with sufficient statistics k-stars s k (x) = i (k = 1, 2,...,n 1), and triangles t(x) = i<j<k x ij x jk x ik (they provide the analogous extension to directed graph; in the following the subscript + indicates summation over the corresponding index). This class of ERGMs is called Markov random graphs. Pattison and Robins [69] proposed to extend the Markov assumption by allowing tievariables to be dependent given the presence of other network tie variables. Through a partial conditional independence assumption tie-variables may be made dependent if they share edges rather than nodes. This provides a hierarchy of dependence assumptions [70, 99] and suggests a number of statistics in addition to the Markov statistics. The most frequently used dependence assumption is the so-called social circuit dependence assumption (SCDA): two tie-variables X ij and X kl may be conditionally dependent if x ik = x jl = 1 or x il = x jk = 1. In combination with the Markov assumption, SCDA admits a number of higher-order statistics such as for example independent 2-paths P k = ( L2ij ) i<j k and triangles T k = i<j x ( L2ij ) ij k of order k = 1, 2,...,n 2, where L2ij = h i,j x ihx hj (their

10 J. Koskinen, A. Lomi directed graph equivalents are given by [79]). Following [88], stars and higher order parameters typically are assumed to have alternating form, so that the parameter for the k th order, isassumedtobe( 1)/λ times the parameter for order k 1, for a smoothing constant λ>0. The alternating form allows us to parameterize all the statistics while having an estimable number of free parameters. The homogeneity assumptions of the model may be relaxed by introducing node-level covariates [77]. As we illustrate in the empirical part of our paper, interest may lie specifically in inferring the effect of exogenous variables while controlling for the aforementioned dependencies. Degeneracy, near degeneracy and related issues have been extensively treated in the recent literature [16, 38, 47, 51, 66, 78, 85, 88]. Most of the results focus on Markov random graph models which are prone to near degeneracy, making them generally problematic fits for real social network data. Social circuit models, particularly with the alternating parameterizations, are much more robust, but still may be liable to degeneracy. The relation to traditional models in statistical mechanics, such as the Ising model, were already evident in the seminal paper by [36]. Although some large-sample properties are known [20, 95] it is not obvious how these relate to the type of models typically employed in the social and behavioral sciences which tend to be complex and based on relatively small samples. Applied to small social networks ERGMs have proved effective in answering substantive questions [61, 75]. Relatively parsimonious models have also been shown to reproduce local as well as global features of empirical networks. 3.3 The Model When we have explicit information on tie-creation and tie-deletion in the form of observed changes to a network, we may model these transitions without having to resort to assumptions about limiting properties. This serves as practical motivation for the ERGM process but also has substantive importance from the point of view of social science (further discussed in Sect ). For the purpose of defining the ERGM process, assume a process X(t), for a timeparameter t T R +, that is a continuous-time Markov chain with intensity matrix Q. We shall define this process in terms of toggles of tie-variables. Continuous-time models that evolve through incremental changes to the network have also been proposed by Holland and Leinhardt [45] and for the frequently used stochastic actor-oriented model (SAOM) [84]. The key differences between the tie-based model proposed here and the SAOM are detailed in [86, 87] and Sect For any given network x, wedenotebyx ij the incomplete network from which the information about the value of the tie variable x ij is deleted. Define the operator sign ij,that is such that for x, y = sign ij x has y ij = x ij and y ij = 1, y ij = 0, or y ij = 1 x ij according to whether sign is equal to +,,or, respectively. In particular ij x toggles the entry (i, j) of x. The neighborhood of a graph x,isdefinedasn(x)={y X : y = ij x, for some (i, j) V (2) } and we define the Markov chain in terms of a process that stays in the current state x for some time and then jumps to a state in the neighborhood of x. This is a random walk on a binary n(n 1)-cube [3]. More specifically we assume the rate: q ij (x), if y = ij x q(x,y) = y N(x) q(x,y), if y = x (1) 0, y / N(x) {x}

11 The Local Structure of Globalization which defines the intensity matrix Q, and the corresponding transition probability matrix P(t) = e tq, with elements P x,y (t) = Pr(X(t) = y X(0) = x), forx,y X.TheERGM process has rates defined by q ij (x) = ρpr θ (X ij = 1 x ij X ij = x ij ),where Pr θ (X = + ij Pr θ (X ij = 1 X ij = x ij ) = x) Pr θ (X = + ij x) + Pr θ(x = ij x) (2) is the conditional tie-probability of the ERGM. We call this an ERGM process as the Markov chain is equivalent to the Gibbs sampler for the ERGM and as a consequence the limiting distribution is Pr θ (X = x) = exp{g(x; θ) ψ(θ)}. 3.4 Estimation The cross-sectional ERGM, the model for t large, enjoys the standard properties of an exponential family distribution and in particular the method of moments is equal to the likelihood equation. A number of likelihood-based Markov chain Monte Carlo estimation procedures have been proposed and are in common use [19, 21 23, 38, 54, 85]. The ERGM process is no longer an exponential family model and requires an alternative method. (The method of moments may be used but is less efficient and the properties of the estimator are not fully known [86, 87]. 1 ) For M 1 observation points t 0,t 1,...,t M in time (t m 1 <t m ), inference is performed conditionally on x(t 0 ), with a likelihood defined by L(θ; x(t 0 ),..., x(t M )) = M m=1 P x(t m 1 ),x(t m) (t m t m 1 ). As the transition matrix is analytically intractable, we perform inference by augmenting data x(t 0 ),...,x(t M ) with sample paths v M m=1 X (x(t m 1), x(t m )), where X (x, y) ={(v 0,v 1,...,v R ) X R : v 0 = x,v R = y,v r N(v r 1 ) {v r 1 }}, and alternate between drawing v π(v θ,x(t 0 ),..., x(t M )),andθ π(θ v,x(t 0 ),..., x(t M )). Thus a variate v X (x, y) is an unobserved sample path that is constrained to start in x and end in y, the observed states we know the path must connect. The full conditional posteriors are proportional to the augmented data likelihood L ( θ; v,x(t 0 ),..., x(t M ) ) e ρ/[n(n 1)](t M t 0 ) [ ρ(tm t 0 ) ] R Pr θ (X = v r X = v r 1 ) R! r As v r N(v r 1 ) {v r 1 }, the transition probability Pr θ (X = sign ij v r 1 X ij = v r 1, ij ) is given by Eq. (2), for some (i, j) and sign {+, }. The augmented data likelihood can be seen as consisting of one factor stemming from the Poisson process, and one that relates to the Gibbs updating step of the embedded chain. Updates to v are performed by proposing a move from the current state v to a new state v,wherev is generated from a proposal distribution D(v v). The update is accepted and v := v with probability min{1,h},where H = π(v θ,x(t 0 ),..., x(t M )) D(v v ) π(v θ,x(t 0 ),..., x(t M )) D(v v) The proposals are local in the sense that v is constructed by lengthening v by two selfcancelling moves (v 0,v 1,...,v s, ij v s, ij v s+1,..., ij v r, ij ij v r = v r,...,v R ),orremoving two cancelling moves. Further details are given in [53]. As the process is tiebased, some updates result in loops in the state space graph, i.e. there is a transition from a 1 The programs PNet [100] andrsiena [73] may be used for tie-based processes. Currently both programs have limited functionality and the former only uses the method of moments. Estimation in the current paper is carried out in Matlab.

12 J. Koskinen, A. Lomi graph to itself. To allow for loops we follow [89] and include diagonal proposals v to v, (v 0,v 1,...,v s 1,v s,v s,v s+1,...,v R ). 3.5 Model Properties Time Dimension We can make the following observations regarding assumptions for the time dimension. A. Stationarity: The process may already be in equilibrium at t 0, in which case the rest of the observations are also from a stationary process. B. Time-homogeneous: The process is time-homogeneous, in the sense that Q is not a function of t. C. Time-heterogeneous: The process changes over time in the sense that Q is a function of t. For case A the model is fitted as set out above with an additional inference step needed to account for x(t 0 ) being stochastic. This entails accounting for the initial state contribution to the likelihood Pr θ (X = x) = exp{g(x; θ) ψ(θ)}. It is an empirical issue whether data are compatible with the stationarity assumption but it is a theoretical issue whether the assumption is plausible for any given socio-economic process. If the process is assumed to be stationary we also need to take into account the model degeneracy issues that are frequently associated with ERGM [22, 38, 47, 85]. For case B no assumption of stationarity is made. The process may be drifting in some direction but the probabilistic law governing the evolution is fixed. Hence, we do not require our model to be stable in the sense that the limiting distribution is non-degenerate. For SAOM the argument is often made that limiting properties of the process are of little relevance for describing the evolution of observed data. For case C we assume that the probabilistic behavior may change over time. It could be that the process is affected by exogenous factors, such as cycles, or endogenous processes. In the most general case this is a state space process. In the simplest case we may assume that the process is piece-wise constant, with Q being constant in the intervals [t m 1,t m ). Attention has only recently been given to this issue in the context of SAOMs, where focus has been on diagnosing differenced between parameters in different intervals [59]. An inhomogeneous process severely restricts our ability to do forecasts and predictions based on limiting properties Global Patterns and Local Configurations The model assumed is local in character [69], in the sense that the propensity to change a tie-variable is determined by the local configurations that make up the potential function. Nevertheless, the local rules for updating ties make giver rise to non-trivial global patters [78]. Features such as short path-lengths and clustering that may characterize globalization, can be emergent features of local processes Tie-Based Versus Stochastic Actor-Oriented Models Stochastic actor-oriented models (SAOM) [84] are widely used for modelling longitudinal network data. These assume that the actors do not only re-evaluate their neighborhood dyadwise, but also that they chose an optimal change from all the out-going ties. Here we are dealing with composite agents that are internally heterogeneous and ties that are aggregated

13 The Local Structure of Globalization and binarized. With this in mind we are reluctant to take an actor-oriented approach and prefer to make weaker assumptions. In general SAOMs are not appropriate when there is too much change in-between successive observations [90]. Here the Jaccard index, defined as N 11 /[N 11 +N 01 +N 10 ],forn uv = {(i, j) : x ij (t m ) = u, x ij (t m+1 ) = v}, ranges from to 0.24 indicating a very large turn-over of ties. The ERGM process is not sensitive to this as the limiting distribution is known to be an ERGM. As the number of changes in-between observations increases the model tends to a sequence of independent ERGMs Discrete-Time Versus Continuous-Time A natural alternative to a continuous-time model would be a discrete-time model as proposed in [76], [55], and [40]. While these models offer some simplifications, it is not straightforward to define a generative model with tie-interdependencies in discrete-time. A more detailed argument is given in [89] (Sect. 2.1). The interpretation of the ERGM process in terms of an agent-based model is that a directed dyad is allowed to make an update to the FDI at exponential holding-times, with rate ρ. The time until any dyad makes an update has rate ρn(n 1). Given that i considers (i, j), the tie is created or retained with probability given by Eq. (2). Note that even if we had time-stamped data, the model would not be directly applicable as a relational event model [17, 87]. As the model does not force changes to be made given that an opportunity presents itself, a time-stamped version would require time-stamped observations on opportunities to change. 4 AnalysisofFDI We fit the model as set out in the previous section to the FDI data set using the estimation algorithm in Sect For pragmatic reasons we use constant priors. We present, in order, a dyad-independent model that includes the gravity-model and reciprocity, an SCDA model, and finally an SCDA model where we relax the time-homogeneity constraint. The first two models thus relies on time as in B in Sect , and the latter model as in C. We use the gravity equation to guide our empirical model specification as we explain the next section. 4.1 Model Specification According to the original gravity model the volume of trade between two countries (bilateral trade) is proportional to their economic masses and inversely proportional to the distance between them. Somewhat simplifying, the basic model assumes that the trade flow Y ij from country i to country j may be described by the equation Y ij = e α V β 1 i V β 2 j N γ 1 i N γ 2 j D δ ij, where V i is the gross domestic product (GDP) of country i, N i is the population of country i,andd ij is the geographic distance of country i to country j. As the outcome variable of interest in our study is binary, it seems natural to adopt a log-linear model form for X ij, with linear predictor g(x ij ; θ)= x ij (α + β 1 log V i + β 2 log V j γ 1 log N i γ 2 log N j δ log D ij ), in the first instance. This is also the linear predictor typically assumed when (the logarithm of) Y ij is regressed on the predictors with an identity link function. Note that including the logarithm of distance in g( ; θ) in an ERGM means that the implied distance interaction function assumes an attenuated power-law form [24]. This is also the case for SAOMs [72]. Different functional forms for the distance interaction function are defined and investigated in [18] and the attenuated power-law has some desirable properties. In the empirical part of

14 J. Koskinen, A. Lomi the study, we rely on the so called CEPII (Center d études prospectives et d informations internationales) distance which is routinely used in studies of economic geography and international trade [63]. The presence of potential dependencies linking panel units whose behaviour is believed to respond to gravity forces has been acknowledged before (e.g. [33]). Adding ERGM terms, such as those mentioned in Sect. 3.2, to the linear predictor, allows us to explicate these dependencies. The basic gravity-model does not account for these forms of dependencies and new generations of gravity models typically account for third party dependencies through spatial auto-correlation [9]. For the gravity-model the mass of the countries are approximated by their Gross Domestic Product (GDP) the market value of all the goods and services produced by each country every year. Distances between countries are calculated following the great circle formula, which uses latitudes and longitudes of the most important cities/agglomerations (in terms of population) [63]. As explained in Sect. 3.4, the logarithm of these variables is included in the linear predictor g( ; θ). We also include the interaction term log V i log V j as a covariate for the tie-variable X ij in g( ; θ). This may be seen as a homophily effect [64], whether countries that are similar with respect to GDP are more or less likely to be involved in FDI with each other. We include this effect as a control and prefer to remain agnostic as to theoretical motivations. The structural parameters used are the following. Density, defined as i x i+, where x i+ = j x ij, which represents the baseline tie-probability. Reciprocity, defined as i<j x ij x ji, which is the most basic of the directed Markov statistics. This reflects the simplest form of dyadic dependence. In addition to these two effects, we chose to include two effects that model the degree distributions and one that models transitive closure (with a control). This is a typical model specification for a directed network [74] which here is motivated by the descriptive analysis. There are additional triadic and 2-path statistic available [79] and the most important one from the perspective of our data would be 3-cycles (or alternating forms of which). The number of cycles is however too small for the corresponding effect to be estimated (see the count of 030C in Fig. 3). The alternating out-stars salt out(x) = n 2 k=2 ( 1)k sk out (x)/λ k 2,wheresk out (x) = ( xi+ ) i k (λ = 2) parameterizes the entire out-degree distribution with geometrically decreasing weights through combining out-k-stars (Fig. 4(c)) [88]. In general a positive sign indicates presence of out-degree centralization, with some sending a larger number of FDIs than others. In terms of FDI the associated dependence assumption has relevance as we expect that investments are costly and the knowledge that one country already have invested in k countries would limit the capacity of forming an additional tie to yet another country. The reverse may also be of relevance, namely knowledge that one country has made unusually many investments already, might lead us to conclude that an additional tie is more likely than for a country known only to have made few if any investments. The alternating in-stars (Fig. 4(d)), defined analogously to the alternating out-stars as s in Alt (x) = n 2 k=2 ( 1)k s in k (x)λk 2, models the dispersion in the number of investments received. A directed form of the alternating triangles (Fig. 4(a)) presented in Sect. 3.2 is λ i,j x ij [1 (1 1/λ) L 2ij ],wherel 2ij = h i,j x ihx hj [88]. The alternating triangle accounts for the Markov dependence and SCDA and is interpreted in general as a tendency to (transitive) clustering (parameter positive), and the tendency to form multiply clustered regions. The effect of the alternating form is to dampen the contribution to the statistic of additional two-paths. Substantively the effect is consistent with local hierarchy, where investments tent to flow from some countries towards countries lower in the hierarchy. Independent 2-paths (Fig. 4(a)), λ i,j [1 (1 1/λ)L 2ij ], represents multiple connectivity. The 2-path in itself measures the association between in and out-degree. Here a positive effect

15 The Local Structure of Globalization Fig. 4 Configurations in the ERGM process for FDI: k-triangles (a), independent k 2-paths (b), out-k-stars (c), in-k-stars (d) would mean that countries would tend to be connected indirectly through many intermediaries. We include this effect primarily as a control for alternating triangles (seeing as we need to control for the number of open triads when assessing the transitivity effect). For modelling the rate at which countries reassess their ties to other countries we follow the standard practice in stochastic actor-oriented models and allow for ρ to be dependent on m and to set all intervals to unity t m t m 1 = 1[84]. Note that the rate of the process is given by Eq. (1). 4.2 Model Estimates The posteriors for the dyad-independent model that only includes the gravity variables and reciprocity (dyad independence model) are given in Fig. 6 and numerical summaries in Table 2. Both models assume time-homogeneity in the sense of B in Sect (with the standard exception of different rates). Most notable is the strong effect of distance (posterior mean: 0.68; with standard deviation: 0.03). The effects are aligned with the predictions of the gravity model. In ERGM terms, the effects of GDP capture heterogeneity in the sending and receiving of ties [77]. Countries with high GDP both send and receive more ties but there is no homophily on GDP. In addition to these standard gravity effects we may also note that there is a tendency to reciprocation: FDI relations are more likely between countries linked by reciprocated network ties.

16 J. Koskinen, A. Lomi Table 2 Posterior means and standard deviations for the dyad-independent model and the extra-dyadic (SCDA) model. Models are time-homogeneous in the sense of B in Sect Mean Std Mean Std Rate Rate Rate Rate Rate Rate Rate Rate Rate Density Mutuality Alt. in-stars Alt. out-stars Alt. transitive triangles Alt. indep. two-paths Interaction GDP Sender GDP Receiver GDP Log distance Taking extra-dyadic dependencies into account, the posteriors of the full ERGM process in Fig. 6 indicate that the main effects of the first model remain. The endogenous processes do not explain away the strong dependence on distance. The parameters for the alternating stars reveal a tendency towards centralization in the sending and receiving of ties, a result that is aligned with the triad census. Even though GDP explains a lot of the heterogeneity in the sending of ties, there is thus evidence for an independent endogenous rich get richer effect. The effects for GDP are somewhat smaller than for the dyad-independent model as a result of some of the heterogeneity being explained by the alternating starts. The point estimate for the homophily effect of GDP is positive but interpretation of the effect is inconclusive as a result of the large uncertainty. That both the GDP effects and the alternating star parameter accounts for heterogeneity in the degree distribution is reflected in the fact that the parameters are correlated in the posterior. There is a positive correlation between the alternating out-stars and interaction GDP, and a negative between the alternating out-star and sender GDP. To illustrate the combined effects of GDP on tie-probability and how this is extended by the endogenous dependencies of the ERGM process we provide an extension of the ego-alter selection tables typically used in stochastic actor-oriented models [90]. Based on Eq. (2) we may construct conditional log-odds of the Gibbs distribution. For the dyadindependent model (with x ji = 0), the conditional log-odds of a tie (i, j) is given by log Pr θ(x = + ij x) Pr θ (X = ij x) = θ dens + θ send log V i + θ rec log V j + θ int log V i log V i + θ dist log D ij (3)

17 The Local Structure of Globalization Fig. 5 Ego-alter selection table: Posterior conditional log odds of tie probability from sender i to receiver j. Dyad independent model (dotted) gives conditional log odds for combinations of log V i and log V j for low (1.18), mean (3.44), andhigh(5.71) GDP. The other posterior conditional log odds include the additional effects + ij sout Alt (x) and + ij sin Alt (x), for values of dout, j i and d in, i j as specified in legend For the model with extra-dyadic SCDA dependencies, effects for Alternating in-stars, outstars, triangles and two-paths enter into the conditional log odds. Conditional on adding (i, j) not changing the alternating triangle or two-path statistics, the additional terms to Eq. (3) areθ out + ij sout Alt (x) + θ in + ij sin Alt (x),where + ij sout Alt (x) = λ[ ] 1 (1/1/λ) dout, j i and + ij sin Alt (x) = λ[ ] 1 (1/1/λ) din, i j and where d out, j i = k j x ik and d in, i j = k i x kj. A comparison of the selection tables for the dyad independent model and the full ERGM is given in Fig. 5. The magnitude of the GDP effects are similar for the receiver and sender effect in the dyad-independent model (as evidenced in Fig. 6) which leads to the log odds for high sender/low receiver GDP being close to low sender/high receiver GDP (Fig. 5). Since the interaction effect is centred on zero, the high sender/high receiver GDP is more or less the sum of the main effects. For the ERGM process with extra-dyadic dependencies Fig. 5 gives the conditional logodds, conditional on (d out, j i,d in, i j ) being equal to (0, 0), (10, 0), (0, 5), and(10, 5), respectively, representing the conditional probability of a tie from a state with d out, j i outgoing ties to a state with d in, i j incoming ties. The values correspond to low and high inand out-degree relative to the degree distributions (Fig. 1). When i sends few ties and j receives few ties, the probability of creating the tie (i, j) is lower across all combinations of GDP (thick solid curve in Fig. 5) compared to the dyad-independent model (dash-dot curve in Fig. 5). When j receives a large number of ties, the probability that the tie (i, j) will be created (dashed curve) is considerably increased. This may reflect a form of preferential attachment [1, 39, 83], where popular states receive more ties. Note that in the

18 J. Koskinen, A. Lomi Fig. 6 Posterior densities for model parameters for time-homogeneous ERGM process (light grey) and dyad-independent model (dark grey) with 95 % highest posterior density (HPD) regions longitudinal framework this also means that already popular states retain their popularity. If i already sends many ties and j receive many ties, tie probability is increased even further. The conditional log-odds of Fig. 5 reflect the effects of GDP and degree ceteris paribus and there are too many conditioning states to allow for a complete investigation. The multitude of possible conditionals makes interpretation of parameter magnitude more straightforward in terms of predictive distributions instead. The interpretation of the magnitude of estimates is the focus of much current research (e.g. for SAOM [49]) and typically relies on MCMC-estimates based on GOF distributions. While the point estimate for the transitivity parameter is positive, there is considerable uncertainty about the parameter (the 95 % HPD interval covers zero). We can therefore not say for sure whether the discrepancies we saw in Fig. 3 are confirmed. On the basis of this model we cannot rule out the possibility that the observed profile in Fig. 3 is an artefact of dependence on initial observation, or the consequence of spatial embeddedness. The high degree of turn-over in ties between years (as reflected in the low Jaccard indices mentioned in Sect ) means that the rates are high. An estimate of the log rate of 0, does for example mean that a tie is updated on average once in any unit time interval. The consequence for estimation is that the paths v connecting observations are long. To improve mixing v were updated times for every update to θ. To estimate the rate parameters with a high degree of precision is hard but the rate is a nuisance parameter and the other parameters vary little across different values of the rates. For reasons that will be discussed in Sect. 4.3, we relax the assumption of the parameters being identical over time and Q being constant (apart from the rates). The posteriors for a time-heterogeneous model (type C as defined in 3.5.1) are given in Fig.7. The HPD intervals for the homogeneous model are provided in the plot for reference. The magnitude of the parameters and the associated uncertainty differ between years but for some parameter more than for others. The alternating out-star effect appears to be stable as the posteriors are similar across years. Similarly, the geographical distance effect is stable over

19 The Local Structure of Globalization Fig. 7 Boxplots and 95 %-HPD intervals (light grey bands) of posterior densities for model parameters for time-heterogeneous (type C) ERGM process by interval m. Thedark grey band is the highest posterior density region for homogeneous model for the corresponding parameter. Bottom right panel gives the posterior probability that θ k is non-negative by year for the mutuality, alternating in-star, and triangle parameters time. The receiver and sender effects for GDP fluctuate considerably, in the sense that the associated parameters change signs, especially so for the latter. The most important aspect is the unpacking of the structural effects of mutuality and transitivity. For the mutuality of FDI ties, there is great uncertainty about the parameter but we also see, tentatively, a preference against mutual ties at m = 4. Aggregating the evidence for mutuality across years, the HPD for the homogeneous model does not cover zero. For transitivity, the parameter is positive (with more or less uncertainty) for the first 5 periods, then this tendency go away (in ) to finally become negative. The posterior probabilities of non-negative parameters for mutuality, alternating stars and triangles are provided in the bottom right panel of Fig. 7. When the evidence for transitivity is aggregated over the years as is done in the homogeneous model the conflicting evidence of different periods combines to render the effect non detectable. In the literature there are several competing explanations for why we might observe clustering in a network that is not due to transitive closure (as discussed with respect to Fig. 3 in Sect. 2.2). While the substantive mechanism goes under many different names, such as homophily, shared affiliations, spatial propinquity [64], foci [32] or settings[69], from an empirical point of view these are all to do with unobserved variables. The effects in the models are included because they are of particular interest and while we cannot discount the possibility of unobserved confounders we may perform a check of the robustness of the results. For the particular case of FDI in the electricity market the mass of a country could alternatively be specified in terms of production capacity, production and/or consumption. Population size is also commonly included as a measure of the size of the country. These variables are not likely to explain away closure in terms of homophily but have the potential of inducing a hierarchy in terms of demand, supply and resources. It has been established that cultural distance is an important factor for homophily in international relations. For the interval , we tested alternative explanations for the strong effect of transitive closure by adding effects for population size, electricity generation and cultural distance (as