1 Introduction
In this paper we investigate international trade flows of Major Conventional Weapons (MCW) using data provided by the Stockholm International Peace Research Institute (SIPRI). MCW include armoured vehicles, aircrafts, naval vessels etc. and SIPRI has collected all international arms transfers from 1950 to 2016 in a comprehensive database. The database includes information on the sending and the receiving country as well as the volume of the trades in a certain year. The volume is measured in so called TIV, shorthand for trendindicator value(s), and represents the value of military resources that are exported. The data can be regarded as a year wise sequence of weighted networks where the countries are the nodes and the arms trade flows among them are the valued edges.
Several scholars have started to investigate arms trade using a network framework, but the available studies were restricted to binary relations, i.e. trade or no trade, see for example Willardson (2013), Akerman and Seim (2014), Kinne (2016) and Thurner et al. (2018). The central workhorse in inferential binary network analysis is thereby the Exponential Random Graph Model (ERGM), as introduced by Holland and Leinhardt (1981) (see also Frank and Strauss, 1986, Kolaczyk, 2009 and Lusher et al., 2012). Recent proposals for modelling valued networks within the ERGM class are provided by Krivitsky (2012) or the Generalized Exponential Random Graph Model (GERGM) by Desmarais and Cranmer (2012). The first model allows for discrete valued counts while in the GERGM the weights on the edges are transformed into interval data between zero and one. Both approaches are not suitable for modelling of arms trade data because the TIV is continuous and zero inflated (i.e. TIV stands for no trade). Nonetheless, the binary analysis, i.e. trade or no trade, is central, since trading of arms implies governmental agreement and contracts and therefore represents direct and indirect trust relations, regardless of the amount of trading (see e.g. Jackson, 2010).
In this paper, however, we focus on the amount of trading (if there is trading). Of particular interest is thereby the dependence structure of trading amounts, that is how is the trading amount of a country (to a country) influenced by other trading amounts. In order to do so, we change the viewpoint of the network and go from a representation with the nodes as countries and the edges being arms trade flows to a network where the nodes are the trade flows between two countries and the edges represent the dependencies among them. This yields a model that at least conceptually resembles models from spatial econometrics (see e.g. LeSage and Pace, 2009 and Kauermann et al., 2012). This model class was developed for capturing spatial effects and spillovers from nearby sites and can be transformed to model the dependence of similar arms trade flows. Models from spatial econometrics even have their own name in the social network methodology and are commonly called Network Autocorrelation Models (see for example Leenders, 2002 and Hays et al., 2010).
The paper is organized as follows. In Section 2, we give a short description of the data, then in Section 3 we introduce our model and give different specifications and explanations for possible dependence structures. In Section 4 we present the results of our analysis. Section 5 concludes the paper.
2 Data description
The Stockholm International Peace Research Institute (SIPRI) is a unique data source on the international transfers of major conventional weapons. It covers more than 200 countries for the time period from 19502016 and includes information on the sending country, the receiving country as well as the type of MCW that is traded. Table 2 in Annex A.1 gives an overview of weapon systems included. SIPRI has developed a measure that represents arms trade flows as so called trend indicator values (TIV), based on production costs. The advantage of this measure is consistency over time and comparability of different arms systems. For detailed explanation of the data and methodology see SIPRI (2017b) or Holtom et al. (2012). The database can be accessed free of charge online at SIPRI (2017a).
In Figure 1 the development of the aggregated TIV for the time period from 1950 to 2016 is shown for the two most important exporters United States and the Soviet Union (Russia since 1992). In Figure 2 a loglog plot of the outdegree and the aggregated export volume on the left and the indegree and the aggregated import volume on the right is given. The solid line in the middle gives the mean of exports or imports that is associated with the respective in or outdegree, the whiskers indicate the maximum and the minimum value observed. In terms of the slope and range it can be seen that the connection between aggregated export volume and the outdegree is somewhat stronger than the connection between import volume and indegree. Nevertheless, both volumes increase on average in the degree.
3 Model
3.1 Definitions and Basic Formulation
In time periods, we observe flows where is the valued arms trade flow from country to country among the countries that exist in . In the following, we suppress the time index for ease of the notation. Let be the (directed) index set for all country pairs where valued trade flows exists so that . Let be of dimension and numerate the elements in so that .
As a tool to analyse the dependence structure amongst we are using a graph consisting of nodes and edges, where we take as node set and define as symmetric edge set: . We define the neighbourhood in the model through
The information given in the neighbourhood can be summarized in a matrix . Typically the rows of the matrix are normalized such that they sum up to one by defining if and zero otherwise. This is also the definition we are using. As we are interested in the dependencies among arms trade flows we will henceforth attach the name tradecorrelation to the concept of related (neighbouring) flows. The intention of the subsequent analysis is to find a reasonable weight matrix , which defines an appropriate dependence structure.
3.2 Regression Model
Arms trade flows are nonnegative, leading us to consider a multiplicative model of the form
(1) 
Here
represents a vector of some covariates,
gives the parameter vector and is the error term. The residuals are assumed to be correlated andprovides the variancecovariance matrix encoding the relation of different arms trade flows. Suitable candidates for estimation of (
1) are models from spatial econometrics or spatial statistics, respectively. We will employ the former. Spatial econometric models are common in network analysis, see for instance Leenders (2002), Leifeld et al. (2016), Hays et al. (2010), Metz and Ingold (2017), Silk et al. (2017) and Ba (2017). We follow a Spatial Error Model (SEM, see LeSage and Pace, 2009) which is given by defining the error term as(2) 
where
is the identity matrix of dimension
and provides a measure for the dependency among the related error terms . The error terms are assumed to be unstructured so that equation (2) can be reformulated to(3) 
This implies the covariance matrix . Hence, matrix expresses the dependence structure and the explicit choice of is the central model selection task that will be discussed in Section 3.3. The modelling approach is also known as Network Disturbance Model (NDM) in the network literature, delegating the dependencies among trade flows into the error term. Hence, the approach does not aim to model a direct dependence between the trade flows but implies that the exogenous covariates determine the ”right” value for an arms trade flow, but one can find clusters of trade flows where the residuals are unusually high or low (see Leenders, 2002 and Freeman, 2017 for further discussion). We sketch some details on estimation and software in the Annex A.2.
3.3 Specifications of the Weight Matrix
We now focus on the central modelling task which is the appropriate specification of matrix and hence the definition of the neighbourhood . We thereby focus on interpretable structures. Given that in the network context the weight matrices are rather abstract, we illustrate our approach with the arms trade network of the year 1952, being the simplest network under study. The network is shown in Figure 3. Note that all following illustrations of weight matrices have their origin in the network shown in Figure 3 and are therefore not representative for other years. Still, they allow to visually understand the dependence structure.
Tradecorrelation 1 (attached to the Sending Country)
The first dependence structure relates an arms trade flow from country to country to all other exports going out of country and a possible reciprocal trade flow . Notationally this means
The corresponding graph representation for the year 1952 is shown in Figure 4. The trade flows are grouped within clusters that are related to single exporting countries, most notably, the trade clusters of the big exporters: United States (USA), United Kingdom (GBR), Canada (CAN) and the Soviet Union (SUN). Via the dependence on reciprocal flows, the clusters are connected among each other if there is betweencluster trade. This explains the connection between the cluster of the United States (USA), Great Britain (GBR) and Canada (CAN). The trade activity of the Soviet Union (SUN), however, is disconnected.
Tradecorrelation 2 (attached to the Receiving Country)
The concept from above can be amended such that the tradecorrelation of an arms trade is related to the trades of the importing country and not the exporting one. This means
and implies that the arms trade amount from to is correlated with other imports of as well as the potential reciprocal trade flow . This is visualized in Figure 5. The resulting graph representation is very different from the previous one. There is no tendency of strong clustering as most importing countries do only have one or two suppliers and no exports at all which excludes connections via reciprocal flows. The greatest clusters consist of arms exports to Israel (ISR) and imports of Switzerland (CHE). Great Britain (GBR), United States (USA) and Canada (CAN) show their strong connections in arms trade by representing their own importrelated cluster. Note that this structure disregards an important characteristic of the arms trade network: Receiving countries are often restricted to be dependent on one single exporter. Most important, trade flows of the Soviet Union (SUN) are missing.
Tradecorrelation 3 (attached to the Trade Activity of Sender and Receiver)
The structure of tradecorrelation 1 and 2 can be combined by the inclusion of the whole trade activity of and , that is
In this case an arms trade from country to country is dependent on the whole trade activity of countries and , regardless whether they are reciprocal or not. This results in a very dense graphical representation, as shown in Figure 6. Here, the clusters are again very clearly visible but the interconnection within and between clusters is much stronger. Central are the trade flows United States (USA)  Great Britain (GBR) and Great Britain (GBR)  United States (USA) as they link the two countries with the most trade activity. The flows related to Switzerland (CHE) serve as a bridge between the clusters around the western producers and the Soviet Union (SUN). With this specification the graph is connected and in principle each trade flow is allowed to be correlated with an arbitrary other arms trade flow.
Tradecorrelation 4 (related to Defence Agreements)
It can be discussed whether arms exchange is related to military alliances as they might agree to build up comparable military strength which potentially leads to correlated imports. The Correlates of War Project (2017a) provides data on formal alliances. We define a structure that relates the imports of country to imports of its allies. Let be an indicator that is one if countries and have a formal alliance. Then we define
This assumption imposes a structure where importers regard the arms imports of their allies when buying arms. In the corresponding plot in Figure 7 the trade clusters are not any longer attached to the sending countries but are rather clustered among countries that are connected in the alliances network. We see that the United States (USA), United Kingdom (GBR) and Canada (CAN) represent a (bipartite) cluster with their closest allies while the arms trade activity of the Soviet Union (SUN) forms a disconnected cluster. With this structure we additionally see a cluster containing the alliances of Canada (CAN), United Kingdom (GBR) and the United States (USA) together with countries from the Middle East. Naturally it is also possible to relate the exports of sending countries that have an alliance, resulting in
Tradecorrelation 5 (related to Spatial Distance)
Using data provided by Gleditsch (2013a) we can incorporate spatial distances in the correlation structure. We assume that arms imports into spatially close areas are correlated and define as the distance between the two capital cities of and . With representing a threshold we define
Setting km leads to spatially related trade activity clusters as shown in Figure 8, where in middle a cluster of tradeflows starts with countries from western Europe and relates them over eastern Europe to the Middle East. The choice of is discussed in the Annex A.3. Note that it is also possible to relate flows of spatially close exporters, that is
3.4 Exogenous Covariates
We control for the influence from economics and politics by including additional covariates in model (1). Since there is an average time lag of roughly two years between ordering and delivering of weapons and some covariates are only available from 1950 on, the first network under study is the one of 1952 and covariates are lagged by two years.
Economic Quantities: The standard measure for economic size is the gross domestic product (GDP). We include this measure in logarithmic form for the sender as well as the receiver. The real GDP data is measured in thousands USD and are taken from Gleditsch (2013b). The data are merged from the year 2011 on with recent real GDP data from the World Bank real GDP dataset (see World Bank, 2017). It seems plausible that the military expenditures of the receiving countries are relevant. The data is available from Correlates of
War Project (2017b) with Singer
et al. (1972) as the basic reference for the data. As the time series for military expenditure ends in 2012 we have to assume that the values stay constant until 2014.
Political Quantities
: We use a dummy variable being one if the countries
and have a formal alliance and zero otherwise. Given the restriction that the data is available only until 2012 (Correlates of War Project (2017a) for the most recent version of the data and Gibler (2009) for the article of record for the data set) we assume that the alliances did not change between 2012 and 2015. In order to control for regime dissimilarity, we include the absolute difference of the so called polity score, ranging from the spectrum (hereditary monarchy) to (consolidated democracy). The data can be downloaded as annual timeseries. See Marshall (2017) for the data and basic reference.4 Results
4.1 Model Selection
Our ultimate goal is to select a suitable dependence model from above. As recommended by Leenders (2002) for this model class with several candidate weight matrices, we compare the proposed models using the Akaike Information Criteria (AIC) (see for example Claeskens and Hjort, 2008). For comparison we also include a simple linear model with unstructured error term (OLS, ). Estimation is done year wise and the AIC is calculated for all estimated models from 1952 to 2016, all in all covering observations.
•
In Table 1 the results for the aggregated AIC, rescaled by the lowest one (AICAIC) are presented. AIC corresponds to tradecorrelation structure 3.
In order to investigate the different models more in detail, we also look at year wise AIC values. To do so, we compute Akaike weights, which can be interpreted as the probability of model
being the best (i.e. minimizing the KullbackLeibler Divergence given the data) among the proposed ones. The weights are given by
(see Burnham and Anderson, 2004 for a further discussion). The weights are shown in Figure 9 for all eight specifications.We see that the specification where is restricted to zero (OLS) is among those models that have a low probability of being the best one. Only in the years 19902000 the probability for this model rises slightly as a consequence of the collapse of the Eastern Bloc. Overall, this provides evidence for our initial hypothesis that the trade flows are indeed correlated.
Specification (dependency of exports from spatially close countries), shows almost identical Akaike weights as the OLS specification. A plausible result, given that the capital cities of the big exporters United States and Soviet Union/Russia do not have nearby capital cities of other strong arms exporters and hence the greatest share of trade flows is restricted to be uncorrelated. This stands in contrast to (dependency of imports of spatially close countries) that shows up to be among the better models in the beginning of the observational period and after 1989. Hence, this indicates that arms imports into spatially near countries are indeed related to some extent after 1989, which may be interpreted as a spatial clustering of arms imports.
While in tradecorrelation structure 5, the importer related assumption () gives the higher probability to be the best one as compared to the exporter related assumption (), the opposite is the case for tradecorrelation structure 4, that is related to formal alliances. Here the model that assumes dependency of arms trade flows of allied exporting countries () performs better than the one that relates those of allied importing countries (). This leads to the presumption that formal alliances mostly consist of partners with rather similar possibilities to export arms, i.e. either high ones, which leads to exports with high residuals or low ones, leading to exports with low residuals. The importers that are connected via formal alliances, seem to be more heterogeneous with respect to the height of their imports.
For the tradecorrelation structure 2 (, relating the imports and a potential reciprocal arms trade flow), we find a similar behaviour as for OLS and . Leading to the conclusion that there is no strong evidence that the importing arms flows are strongly correlated.
The two specifications that compete about being the best model are tradecorrelations and . They alternate in being the best model in most time points. Mostly if the probability of the model with tradecorrelation decreases, the one with tradecorrelation increases and vice versa. But from a global perspective it is tradecorrelation that provides the best model, which leads to the conclusion that a single arms trade flow is related to the whole trade activity of the sender as well as the receiver.
We infer that the exchange of international arms leads to a rather complex dependence structure. This shows that, similar to the binary analysis, the assumption of independent observations, given the covariates, is a strong simplification of the real process. Given this result we further analyse the model with tradecorrelation structure .
4.2 Coefficients
Tradecorrelation: The top left panel of Figure 10 shows the trade correlation coefficient using the structure . The coefficient is consistently positive but exhibits some variation. The strongest downward movement can be found in the years 1990  1996, showing how massively the end of the cold war altered the structure of the international trade of arms. From 1996 on, the coefficient increases again. In the most recent years the tradecorrelation fluctuates around , close to the values before 1990.
Economic Quantities: The coefficient on the logarithmic GDP of the sender is, with the exception of the year 1952, consistently positive and significant, showing a slight tendency to decrease with time. This can be seen from Figure 10, left panel, middle plot and reflects the fact that mostly wealthy countries have a highly developed arms industry which are able to produce and export expensive military equipment. The slight decrease of the coefficient is probably related to the increase of exporting countries that have smaller economic size. When it comes to the GDP of the receiver (left panel, bottom plot), we see a more pronounced downward trend of the coefficients. It is interesting to observe that from 1994 on, the coefficient becomes permanently insignificant. All in all there is no strong evidence that the TIV of arms imports is strongly influenced by the GDP of the receiving country since the 90s. It is, however influenced by the logarithm of military expenditure of the respective country as can be seen in the bottom plot of the right panel. There the coefficient on the logarithmic military expenditures is almost always positive since 1967 and mostly significant from 1990 on. This indicates a change in the arms trading mechanism. Before the end of the cold war, high arms trade flows went to countries with big economic size and after that, high flows are directed more strongly towards countries with high spending for military equipment.
Political Quantities: Before 1990, having an alliance (right panel, top plot) in tendency increased the amount of arms trading but at least since 1990 this effect is almost zero. The coefficient on the absolute differences of the polity score (right panel, middle plot) shows a time related trend. In the beginning (19531973) the coefficient is rather less than zero, giving the intuitive result that differences in political regimes reduce the amount of arms traded. From 1976 until 1991 the coefficient can be regarded as zero, which mirrors the fact that at the height of the cold war the belonging to a political bloc was more important then distances in the polity scores. After the end of the Soviet Union, the coefficient remained close to zero until 2011 where the coefficients becomes significantly positive for six consecutive years, providing the surprising result that differences in the polity score do not lead to a reduction of the traded amount of arms.
4.3 Residual Analysis
We give a graphical summary in Figure 11 for the pooled unstructured estimated residuals
. The plot on the left gives a QQplot with the quantiles of the standardized residuals on the vertical axes against the quantiles from a standard normal distribution on the horizontal axis. The right panel shows a histogram of the residuals together with a standard normal. Besides some irregularities in the tails, the results look quite acceptable.
As shown in the previous section, the coefficient on the tradecorrelation is always positive and it is therefore natural to ask whether it is possible to identify trade clusters with higher or lower than expected trade flows. In order to do so, we attribute to each country those elements of the vector of estimated ”tradecorrelationresiduals” () where the respective country is either the sender or the receiver. This is illustrated in Figure 12. For the periods 19521991 (left) and 19922016 (right) kernel density estimates of the the estimated ”tradecorrelationresiduals” for the most important countries are given. A strong difference in the distributions of the residuals can be found. Most notably the residuals are mostly negative for the United States (USA), Great Britain (GBR), Germany (DEU), France (FRA), and Canada (CAN) while the distribution for the Soviet Union (SUN) is shifted to the right and the one of China (CHN) is somewhat inbetween In the period 19922016, the distributions are shifted closer together but the remarkable divide between the big western exporters and Russia (RUS) is still visible. This leads to the suggestion that the positive coefficient on tradecorrelation is partly driven by the differences of the tradeclusters of the western world and the one around the Soviet Union/Russia
5 Conclusion
In this paper we analysed the volume of international arms trade flows by employing a Network Disturbance Model. Seven different dependence structures are proposed and the best one is selected by the minimization of the AIC. Using this approach we find that a specification that relates the error term of an arms trade flow to the whole trade activity of the sending and receiving state works well in most years. This reveals that the network of international arms trade imposes a very complex dependence structure among the arms trade flows.
The analysis of the tradecorrelation coefficient shows that the correlation pattern among the flows rather stable and breaks only for the transition period after the end of the cold war. The development of the coefficients on the covariates shows how the influence of economic and political factors has changed with time.
Acknowledgements
The authors gratefully acknowledge funding provided by the German Research Foundation (DFG) for the project International Trade of Arms: A Network Approach.
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Appendix A Annex
a.1 Descriptives
Table 2 gives the categories of arms that are included in the analysis. All types with explanations are taken from SIPRI (2017b). The 171 countries that are included in our analysis can be found in Table 3, together with the threedigit country codes that are used to abbreviate countries in the paper. In addition to that, the time periods, for which we coded the countries as existent are included. Note that the SIPRI data set contains more than 171 arms trading entities but we excluded nonstates and countries with no (reliable) covariates available. In the covariates some missings are present in the data. No time series of covariates for the selected countries is completely missing (those countries are excluded from the analysis) and the major share of them is complete but there are series with some missing values. This is sometimes the case in the year and/or
where the former socialist countries split up or had some transition time. In other cases values at the beginning or at the end of the series are missing. We have decided to impute the missing values via linear interpolation, using the
R package imputeTS by Moritz (2016).Type  Explanation 

Aircraft  All fixedwing aircraft and helicopters, including unmanned aircraft with a minimum loaded weight of 20 kg. Exceptions are microlight aircraft, powered and unpowered gliders and target drones. 
Airdefence systems  (a) All landbased surfacetoair missile systems, and (b) all antiaircraft guns with a calibre of more than 40 mm or with multiple barrels with a combined caliber of at least 70 mm. This includes selfpropelled systems on armoured or unarmoured chassis. 
Antisubmarine warfare weapons  Rocket launchers, multiple rocket launchers and mortars for use against submarines, with a calibre equal to or above 100 mm. 
Armoured vehicles  All vehicles with integral armour protection, including all types of tank, tank destroyer, armoured car, armoured personnel carrier, armoured support vehicle and infantry fighting vehicle. Vehicles with very light armour protection (such as trucks with an integral but lightly armoured cabin) are excluded. 
Artillery  Naval, fixed, selfpropelled and towed guns, howitzers, multiple rocket launchers and mortars, with a calibre equal to or above 100 mm. 
Engines  (a) Engines for military aircraft, for example, combatcapable aircraft, larger military transport and support aircraft, including large helicopters; (b) Engines for combat ships ,fast attack craft, corvettes, frigates, destroyers, cruisers, aircraft carriers and submarines; (c) Engines for most armoured vehicles  generally engines of more than 200 horsepower output. 
Missiles  (a) All powered, guided missiles and torpedoes, and (b) all unpowered but guided bombs and shells. This includes manportable air defence systems and portable guided antitank missiles. Unguided rockets, freefall aerial munitions, antisubmarine rockets and target drones are excluded. 
Sensors  (a) All land, aircraft and shipbased active (radar) and passive (e.g. electrooptical) surveillance systems with a range of at least 25 kilometres, with the exception of navigation and weather radars, (b) all firecontrol radars, with the exception of rangeonly radars, and (c) antisubmarine warfare and antiship sonar systems for ships and helicopters. 
Satellites  All reconnaissance and communications satellites. 
Ships  (a) All ships with a standard tonnage of 100 tonnes or more, and (b) all ships armed with artillery of 100mm calibre or more, torpedoes or guided missiles, and (c) all ships below 100 tonnes where the maximum speed (in kmh) multiplied with the full tonnage equals 3500 or more. Exceptions are most survey ships, tugs and some transport ships 
Other  (a) All turrets for armoured vehicles fitted with a gun of at least 12.7 mm calibre or with guided antitank missiles, (b) all turrets for ships fitted with a gun of at least 57mm calibre, and (c) all turrets for ships fitted with multiple guns with a combined calibre of at least 57 mm, and (d) air refueling systems as used on tanker aircraft. 
In cases where the system is fitted on a platform (vehicle, aircraft or ship), the database only includes those systems that come from a different supplier from the supplier of the platform.  
The Arms Transfers Database does not cover other military equipment such as small arms and light weapons (SALW) other than portable guided missiles such as manportable air defence systems and guided antitank missiles. Trucks, artillery under 100mm calibre, ammunition, support equipment and components (other than those mentioned above), repair and support services or technology transfers are also not included in the database.  
Source: SIPRI (2017b) 
Country  Code  Included  Country  Code  Included  Country  Code  Included 

Afghanistan  AFG  1950  2016  German Dem. Rep.  GDR  1950  1991  Pakistan  PAK  1950  2016 
Albania  ALB  1950  2016  Germany  DEU  1950  2016  Panama  PAN  1950  2016 
Algeria  DZA  1962  2016  Ghana  GHA  1957  2016  Papua New Guin.  PNG  1975  2016 
Angola  AGO  1975  2016  Greece  GRC  1950  2016  Paraguay  PRY  1950  2016 
Argentina  ARG  1950  2016  Guatemala  GTM  1950  2016  Peru  PER  1950  2016 
Armenia  ARM  1991  2016  Guinea  GIN  1958  2016  Philippines  PHL  1950  2016 
Australia  AUS  1950  2016  GuineaBissau  GNB  1973  2016  Poland  POL  1950  2016 
Austria  AUT  1950  2016  Guyana  GUY  1966  2016  Portugal  PRT  1950  2016 
Azerbaijan  AZE  1991  2016  Haiti  HTI  1950  2016  Qatar  QAT  1971  2016 
Bahrain  BHR  1971  2016  Honduras  HND  1950  2016  Romania  ROM  1950  2016 
Bangladesh  BGD  1971  2016  Hungary  HUN  1950  2016  Russia  RUS  1992  2016 
Belarus  BLR  1991  2016  India  IND  1950  2016  Rwanda  RWA  1962  2016 
Belgium  BEL  1950  2016  Indonesia  IDN  1950  2016  Saudi Arabia  SAU  1950  2016 
Benin  BEN  1961  2016  Iran  IRN  1950  2016  Senegal  SEN  1960  2016 
Bhutan  BTN  1950  2016  Iraq  IRQ  1950  2016  Serbia  SRB  1992  2016 
Bolivia  BOL  1950  2016  Ireland  IRL  1950  2016  Sierra Leone  SLE  1961  2016 
Bosnia Herzegov.  BIH  1992  2016  Israel  ISR  1950  2016  Singapore  SGP  1965  2016 
Botswana  BWA  1966  2016  Italy  ITA  1950  2016  Slovakia  SVK  1993  2016 
Brazil  BRA  1950  2016  Jamaica  JAM  1962  2016  Slovenia  SVN  1991  2016 
Bulgaria  BGR  1950  2016  Japan  JPN  1950  2016  Solomon Islands  SLB  1978  2016 
Burkina Faso  BFA  1960  2016  Jordan  JOR  1950  2016  Somalia  SOM  1960  2016 
Burundi  BDI  1962  2016  Kazakhstan  KAZ  1991  2016  South Africa  ZAF  1950  2016 
Cambodia  KHM  1953  2016  Kenya  KEN  1963  2016  Soviet Union  SUN  1950  1991 
Cameroon  CMR  1960  2016  North Korea  PRK  1950  2016  Spain  ESP  1950  2016 
Canada  CAN  1950  2016  South Korea  KOR  1950  2016  Sri Lanka  LKA  1950  2016 
Cape Verde  CPV  1975  2016  Kuwait  KWT  1961  2016  Sudan  SDN  1956  2016 
Central Afr. Rep.  CAF  1960  2016  Kyrgyzstan  KGZ  1991  2016  Suriname  SUR  1975  2016 
Chad  TCD  1960  2016  Laos  LAO  1950  2016  Swaziland  SWZ  1968  2016 
Chile  CHL  1950  2016  Latvia  LVA  1991  2016  Sweden  SWE  1950  2016 
China  CHN  1950  2016  Lebanon  LBN  1950  2016  Switzerland  CHE  1950  2016 
Colombia  COL  1950  2016  Lesotho  LSO  1966  2016  Syria  SYR  1950  2016 
Comoros  COM  1975  2016  Liberia  LBR  1950  2016  Taiwan  TWN  1950  2016 
DR Congo  ZAR  1960  2016  Libya  LBY  1951  2016  Tajikistan  TJK  1991  2016 
Congo  COG  1960  2016  Lithuania  LTU  1990  2016  Tanzania  TZA  1961  2016 
Costa Rica  CRI  1950  2016  Luxembourg  LUX  1950  2016  Thailand  THA  1950  2016 
Cote dIvoire  CIV  1960  2016  Macedonia  MKD  1991  2016  TimorLeste  TMP  2002  2016 
Croatia  HRV  1991  2016  Madagascar  MDG  1960  2016  Togo  TGO  1960  2016 
Cuba  CUB  1950  2016  Malawi  MWI  1964  2016  Trinidad Tobago  TTO  1962  2016 
Cyprus  CYP  1960  2016  Malaysia  MYS  1957  2016  Tunisia  TUN  1956  2016 
Czech Republic  CZR  1993  2016  Mali  MLI  1960  2016  Turkey  TUR  1950  2016 
Czechoslovakia  CZE  1950  1991  Mauritania  MRT  1960  2016  Turkmenistan  TKM  1991  2016 
Denmark  DNK  1950  2016  Mauritius  MUS  1968  2016  Uganda  UGA  1962  2016 
Djibouti  DJI  1977  2016  Mexico  MEX  1950  2016  Ukraine  UKR  1991  2016 
Dominican Rep.  DOM  1950  2016  Moldova  MDA  1991  2016  Un. Arab Emirates  ARE  1971  2016 
Ecuador  ECU  1950  2016  Mongolia  MNG  1950  2016  United Kingdom  GBR  1950  2016 
Egypt  EGY  1950  2016  Morocco  MAR  1956  2016  United States  USA  1950  2016 
El Salvador  SLV  1950  2016  Mozambique  MOZ  1975  2016  Uruguay  URY  1950  2016 
Equatorial Guin.  GNQ  1968  2016  Myanmar  MMR  1950  2016  Uzbekistan  UZB  1991  2016 
Eritrea  ERI  1993  2016  Namibia  NAM  1990  2016  Venezuela  VEN  1950  2016 
Estonia  EST  1991  2016  Nepal  NPL  1950  2016  Vietnam  VNM  1976  2016 
Ethiopia  ETH  1950  2016  Netherlands  NLD  1950  2016  South Vietnam  SVM  1950  1975 
Fiji  FJI  1970  2016  New Zealand  NZL  1950  2016  Yemen  YEM  1991  2016 
Finland  FIN  1950  2016  Nicaragua  NIC  1950  2016  North Yemen  NYE  1950  1991 
France  FRA  1950  2016  Niger  NER  1960  2016  South Yemen  SYE  1950  1991 
Gabon  GAB  1960  2016  Nigeria  NGA  1960  2016  Yugoslavia  YUG  1950  1992 
Gambia  GMB  1965  2016  Norway  NOR  1950  2016  Zambia  ZMB  1964  2016 
Georgia  GEO  1991  2016  Oman  OMN  1950  2016  Zimbabwe  ZWE  1950  2016 
a.2 Technical Details on the Estimation
The estimation of the models is done in R (see R Development Core Team, 2014). For models with one weight matrix, there is an implementation by Bivand et al. (2013) (see also Bivand and Piras, 2015), based on a twostep method. First, the profiled likelihood is maximized with respect to , by using the simplex method of Nelder and Mead (1965). In the second step the other parameters are found by weighted least squares. Given those estimates, the concentrated likelihood is updated for . The major computational burden comes with the calculation of the logdeterminant of
. The problem is solved by using the eigenvalues of
, i.e. , with being the eigenvalues of .a.3 Optimal Cutoff for the Spatial Distance
The optimal cutoff distances are found by maximization of Moran’s (see Moran, 1950) for the whole time period. In order to do so we construct a blockdiagonal weight matrix for all observations and time periods and calculate Moran’s for varying threshold levels. The distance is discretized on a grid, ranging from km to km, incrementing in steps of km. The highest value for Moran’s is found for km for the weight matrix that relates the distances of the receiver () and km for the weight matrix that relates the distances of the sender (). Note that we have chosen the maximum value in the senderrelated model, although there is also a minimum value that is comparable in absolute numbers and could be interpreted as negative spatial correlation. This value is however realized for a cutoff value around km which seems to be unrealistic (e.g. the distance between Moscow and London is roughly km). The result is visualized in Figure 13.
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