Modeling European regional FDI flows using a Bayesian spatial Poisson interaction model
aa r X i v : . [ ec on . E M ] O c t Modeling European regional FDI flows using a Bayesianspatial Poisson interaction model
Tamás Krisztin ∗ and Philipp Piribauer International Institute for Applied Systems Analysis (IIASA) Austrian Institute of Economic Research (WIFO)
Abstract
This paper presents an empirical study of spatial origin and destination effects of Europeanregional FDI dyads. Recent regional studies primarily focus on locational determinants, butignore bilateral origin- and intervening factors, as well as associated spatial dependence.This paper fills this gap by using observations on interregional FDI flows within a spatiallyaugmented Poisson interaction model. We explicitly distinguish FDI activities between threedifferent stages of the value chain. Our results provide important insights on drivers of regionalFDI activities, both from origin and destination perspectives. We moreover show that spatialdependence plays a key role in both dimensions.
Keywords:
Spatial interaction model, Bayesian Poisson model, regional FDI flows, Euro-pean regions, spatial random effects.
JEL Codes:
C11, C21, F23, R11 ∗ Corresponding author : Tamás Krisztin, International Institute for Applied Systems Analysis (IIASA), Schlossplatz1, 2361 Laxenburg, Austria.
E-mail : [email protected]. The research carried out in this paper was supported byfunds of the Oesterreichische Nationalbank (Jubilaeumsfond project number: 18116), and of the Austrian Science Fund(FWF): ZK 35 Introduction
Recent decades have shown a rapid growth of worldwide foreign direct investment (FDI), whichled to increased efforts in research to understand the economic determinants of FDI activities.Classical explanations focus on the factors driving firms to become multinational. The Ownership-Localization-Internalization theory (see Dunning 2001) explains firms’ motivation as an effort tointernalize transaction costs and reap the benefits of externalities stemming from strategic assets.A large alternative strand of empirical literature builds on trade theory. In this context the driversof FDI activity are the need for larger sales markets, cheaper source markets, and the willingnessto reach a technological frontier (Markusen 1995). Following empirical international economicsliterature, FDI flows are usually captured within the context of a bilateral spatial interaction modelframework. The main advantage of this approach is that it specifically accounts for the role oforigin- and destination-specific factors, as well as intervening opportunities. For an overview onthe determinants of FDI activities and the location choice of multinationals, see Basile and Kayam(2015), Blonigen and Piger (2014), or Blonigen (2005).Due to the scarcity of data on FDI activities on a subnational scale, the vast majority of theempirical literature focuses on country-specific FDI patterns. A subnational perspective, how-ever, would allow for in-depth decomposition of the spatial patterns of FDI flows, since FDIsources and destinations are not uniformly distributed within a country, but tend to be spatiallyclustered. Multiple studies focusing on regional investment decisions of multinational companies(Crescenzi et al., 2013; Ascani et al., 2016a) emphasize within-country heterogeneity of FDI de-cisions, which can exceed cross-country differences. However, a major gap in the literature is thatregional level studies only focus on the destination of FDI decisions, and largely neglect to accountfor origin-specific factors, as well as intervening opportunities in a subnational context. However,a simultaneous treatment appears particularly important for providing a complete picture on third-regional spatial interrelationships in both source- as well as destination-specific characteristics(Leibrecht and Riedl, 2014). Moreover, neglecting to take into account both origin, destination,and third region effects, can lead to biased parameter estimates (Baltagi et al., 2007).The present paper aims to fill these gaps by focusing on subnational FDI flows in a Europeanmulti-regional framework and explicitly accounting for origin-, destination-, as well as third region-specific factors. In this paper we make use of subnational data from the fDi Markets database, whichreports on bilateral FDI flows, with detailed information on the source and destination city. Thiscan be compiled to multiple dyadic format, that is each region pair appears twice, corresponding toFDI flowing from one region to the other and vice versa. A specific virtue of the database is that itdistinguishes FDI flows by their respective business activity. This allows us to contrast the impactof origin, destination, and third region effects across multiple stages of the global values chain.When adopting a subnational perspective, it is crucial to control for spatial dependence, as itspresence in regional data is well documented (LeSage and Pace, 2009). Even national-level empiri-cal applications clearly document the presence of spatial spillovers on FDI activities. An influentialexample is the work by Blonigen et al. (2007), who analyse the determinants of US outbound FDIactivities in a cross-country framework, while explicitly accounting for spatial dependence among2estinations. Further studies which document the presence of spatial issues amongst bilateral (na-tional) FDI activities include Pintar et al. (2016), Regelink and Elhorst (2015), Chou et al. (2011),Garretsen and Peeters (2009), Poelhekke and van der Ploeg (2009), or Baltagi et al. (2007).We therefore employ an econometric framework in the spirit of Koch and LeSage (2015) andLeSage et al. (2007) which captures not only third-regional effects but also spatial dependence usingspatially-augmented random effects. Estimation is achieved using work by Frühwirth-Schnatter et al.(2009), allowing us to deal with the high-dimensional specifications in a flexible and computation-ally efficient way.The remainder of the paper is organized as follows. Section 2 presents the proposed spatialinteraction model, which is augmented by spatial autoregressive origin- and destination-specificrandom effects, intended to capture spatially dependencies, as well as so-called third region effects.Section 3 details the FDI data, the considered determinants, as well as our selection of regions. InSection 4 we assess the determinants of European interregional FDI flows across different stagesof the global value chain. The analysis is performed using information on FDI dyads covering 266NUTS-2 regions in the period 2003 to 2011. Section 5 concludes.
This section presents the model specification used for the empirical analysis. It is worth notingthat the spatial econometric model is similar to work by LeSage et al. (2007), who aimed atmodelling regional knowledge spillovers in Europe. An efficient Bayesian estimation approach forthe employed multiplicative form of the Poisson model with spatial random effects is provided inthe Appendix. Let 𝒚 denote an 𝑁 × 𝑛 regions. In the classic spatial interaction model framework the flows are regressed on corre-spondingly stacked origin-, destination-, and distance-specific explanatory variables, as well astheir spatially lagged counterparts. 𝑿 𝑜 and 𝑿 𝑑 denote 𝑁 × 𝑝 𝑋 origin- and destination-specificmatrices of explanatory variables, respectively. Distances and further intervening factors betweenthe 𝑛 regions are captured by the 𝑁 × 𝑝 𝐷 matrix 𝑫 . Extending the standard model specificationwith local spillover effects as well as spatial random effects, we consider a Poisson specification ofthe form: 𝒚 ∼ P ( 𝝀 ) 𝝀 = exp (cid:0) 𝛼 + 𝑿 𝑜 𝜷 𝑜 + 𝑿 𝑑 𝜷 𝑑 + 𝑫𝜸 𝐷 + 𝑾 𝑜 𝑿 𝑜 𝜹 𝑜 + 𝑾 𝑑 𝑿 𝑑 𝜹 𝑑 + 𝑽 𝑜 𝜽 𝑜 + 𝑽 𝑑 𝜽 𝑑 (cid:1) , (2.1) Detailed R codes for running the proposed model are available upon request. It is worth noting that in the present study 𝑁 is of lower dimension than 𝑛 , since FDI dyads by construction exhibitno own-regional and no own-country flows. Detailed information on the straightforward construction of the origin- and destination-specific matrices of explana-tory variables 𝑿 𝑜 and 𝑿 𝑑 from an 𝑛 × 𝑝 𝑋 dimensional matrix of explanatory variables is provided in LeSage and Pace(2009). LeSage and Pace (2009) also provide detailed guidelines on the convenient construction of origin- anddestination-specific spatial weight matrices. P (·) denotes the Poisson distribution and 𝛼 is an intercept parameter. 𝜷 𝑜 , 𝜷 𝑑 , and 𝜸 𝐷 areparameter vectors corresponding to 𝑿 𝑜 , 𝑿 𝑑 , and 𝑫 , respectively. The spatial lags of the covariatesare captured by 𝑾 𝑜 𝑿 𝑜 and 𝑾 𝑑 𝑿 𝑑 , with 𝜹 𝑜 and 𝜹 𝑑 denoting the respective 𝑝 𝑋 × 𝑾 𝑜 and 𝑾 𝑑 to account fororigin- and destination-specific third regional effects, respectively.Origin-based random effects are captured by the term 𝑽 𝑜 𝜽 𝑜 , where 𝑽 𝑜 denotes an 𝑁 × 𝑛 matrixof origin-specific dummy variables with a corresponding 𝑛 × 𝜽 𝑜 . Similarly, the 𝑛 × 𝜽 𝑑 captures regional effects associated with the destination regions’ matrix of dummy variables 𝑽 𝑑 . We follow work by LeSage et al. (2007) and introduce a further source of spatial dependencevia the 𝑛 × 𝜽 𝑜 and 𝜽 𝑑 , which are assumed to follow a first-order spatialautoregressive process: 𝜽 𝑜 = 𝜌 𝑜 𝑾𝜽 𝑜 + 𝝂 𝑜 𝝂 𝑜 = N (cid:16) , 𝜙 𝑜 𝑰 𝑛 (cid:17) (2.2) 𝜽 𝑑 = 𝜌 𝑑 𝑾𝜽 𝑑 + 𝝂 𝑑 𝝂 𝑑 = N (cid:16) , 𝜙 𝑑 𝑰 𝑛 (cid:17) , (2.3)where 𝜌 𝑜 and 𝜌 𝑑 denote origin- and destination-specific spatial autoregressive (scalar) parameters,respectively. 𝑾 denotes an 𝑛 × 𝑛 row-stochastic spatial weight matrix with known constants andzeros on the main diagonal.The disturbance error vectors 𝝂 𝑜 and 𝝂 𝑑 are both assumed to be independently and identicallynormally distributed, with zero mean and 𝜙 𝑜 and 𝜙 𝑑 variance, respectively. Note that this assump-tion implies a one-to-one mapping to origin- and destination-specific normally distributed randomeffects in the case of 𝜌 𝑜 = 𝜌 𝑑 =
0. For a row-stochastic 𝑾 , a sufficient stability conditionmay be employed by assuming the spatial autoregressive parameters 𝜌 𝑜 and 𝜌 𝑑 to lie in the interval − < 𝜌 𝑜 , 𝜌 𝑑 < Our data set comprises observations on regional FDI dyads for 266 European NUTS-2 regions inthe period 2003 to 2011. A complete list of the regions in our sample is provided in Table A2 inthe Appendix.Observations on regional cross-border greenfield FDI investments stem from the fDi Markets database. This database is maintained by fDi Intelligence, which is a specialist division of theFinancial Times Ltd. The provided data draws on media and corporate sources to report on thesources and hosts of FDI flows (detailed by country, region, and city), industry classifications, aswell as the level of capital investment. Crescenzi et al. (2013) report several robustness tests anddetailed comparisons with official data sources. They confirm the reliability of the fDi Markets data set, especially with regard to the reported spatial distribution of FDI investments.4ur dependent variables are based on the total amount of inflows from European regions inthe period 2003 to 2011. Since the fDi Markets data base also contains information on severaldistinct business activities for both origin and host companies, we follow previous studies byAscani et al. (2016a) and study the determinants of regional FDI dyads at different stages ofthe value chain. This information is valuable as investor companies maximize their utility withrespect to their position along the value chain. Since specifics of the investor company, as well asdetails on the FDI investment are largely unobserved, it is crucial to account for the heterogeneityin investor decisions by subdividing industry activities relative to their position along the valuechain (see, for example, Ascani et al. 2016a). We therefore define three different classifications:
Upstream , Downstream , and
Production . The classification adopted in this paper builds on generalclassifications of the value chain by Sturgeon (2008) and closely tracks the ones employed byCrescenzi et al. (2013) and Ascani et al. (2016a).Specifically, the upstream category comprises conceptual product development including de-sign and testing, as well as management and business administration activities. The downstreamcategory summarizes consumer-related activities such as sales, product delivery, or support. Fi-nally, the production category includes activities related to physical product creation, includingextraction, manufacturing, as well as recycling activities. A complete list of the employed globalvalue chain classification is provided in Table A1 in the Appendix.Our choices for explanatory variables are motivated by recent literature on (regional) FDI flowsas well as regional growth empirics (see, for example, Crespo Cuaresma et al. 2018, Blonigen and Piger2014, Leibrecht and Riedl 2014, or Blonigen 2005). In most gravity-type models a region’s abilityto emit and attract FDI flows is chiefly captured by its economic characteristics. Our main indicatorfor economic characteristics is the regions’ market size, proxied by regional gross value added.To control for the degree of urbanization both in origin and host regions we also include regionalpopulation densities as an additional covariate. Empirical evidence suggests (Coughlin et al., 1991;Huber et al., 2017) that higher wages have a deterrent effect on investment. We proxy this in ourmodel by including the average compensation of employees per hour worked as an explanatoryvariable.We account for the regional industry mix by including the share of employment in manufac-turing and construction (NACE classifications B to F), as well as services (NACE G to U). Wemoreover include typical supply-side quantities such as regional endowments of human and knowl-edge capital. To proxy regional human capital endowments we include two different variables.The first variable measures regional tertiary education attainment shares labelled higher educationworkers. A second variable labelled lower education workers is proxied by the share of the workingage population with lower secondary education levels or less.We use data on patent numbers to proxy regional knowledge capital endowments. Patent dataexhibit particularly desirable characteristics for this purpose, since they can be viewed as a directresult of research and development activities (LeSage and Fischer 2012). In order to constructregional knowledge stocks we use the perpetual inventory method. We follow Fischer and LeSage(2015) and LeSage and Fischer (2012) to construct knowledge capital stocks 𝐾 𝑖𝑡 for region 𝑖 in5eriod 𝑡 . Specifically, we define 𝐾 𝑖𝑡 = ( − 𝑟 𝐾 ) 𝐾 𝑖𝑡 − + 𝑃 𝑖𝑡 , where 𝑟 𝐾 = .
10 denotes a constantdepreciation rate and 𝑃 𝑖𝑡 denotes the number of patent applications in region 𝑖 at time 𝑡 .The matrix 𝑫 includes several different distance metrics. First and foremost, we include thegeodesic distance between parent and host regions. Recent empirical literature also consider com-mon language as a potential quantity in 𝑫 (see Krisztin and Fischer 2015, or Blonigen and Piger2014). We measure whether the same official language is present in the source and host regionsthrough a dummy variable. Information on official national and minority languages is obtainedfrom the European Commission .Several studies on FDI flows also highlight the importance of corporate tax rates as a potentialkey quantity to attract FDI inflows (see Blonigen and Piger 2014, Leibrecht and Riedl 2014, andBellak and Leibrecht 2009). Lower corporate income tax rates in the host region as compared tothe origin region are thus expected to increase the potential attractiveness of FDI inflows. Matrix 𝑫 therefore also contains the (country-specific) difference in corporate income tax rates betweenorigin and destination regions. Larger differences are expected to be associated with increasingFDI inflows.In order to alleviate potential endogeneity problems, we moreover measure all explanatoryvariables at the beginning of our sample (that is in 2003). For specification of the spatial weightmatrix we rely on a row-stochastic seven nearest neighbour specification. Data on the variablesused stem from the fDi Markets , Cambridge Econometrics , as well as the
Eurostat regionaldatabases. Detailed information on the construction of the dependent and explanatory variablesused are presented in Table 1. To assess the robustness of the results we also estimated a model where the explanatory variables were averagesfrom 2003 to 2011. Overall the estimated quantities and their statistical significance remained unchanged. A series of tests using different number of nearest neighbours for the neighbourhood structure appeared to affectthe results in a negligible way. able 1: Variables used in the empirical illustration
Variable Description 𝒚 Upstream FDI inflows associated with upstream activities.
Source: fDi Markets
Downstream FDI inflows associated with downstream activities.
Source: fDi Mar-kets
Production FDI inflows associated with production activities.
Source: fDi Markets 𝑿 Market size Proxied by means of regional gross value added, in log terms.
Source:Cambridge Econometrics
Population density Population per square km, in log terms.
Source: Cambridge Econo-metrics
Compensation per hour Compensation of employees per hours worked, in log terms.
Source:Cambridge Econometrics
Employment in industry Share of NACE B to F (industry and construction) in total employment.
Source: Cambridge Econometrics
Employment in services Share of NACE G to U (services) in total employment.
Source:Cambridge Econometrics
Lower education workers Share of population (aged 25 and over) with lower education (ISCEDlevels 0-2).
Source: Eurostat
Higher education workers Share of population (aged 25 and over) with higher education (ISCEDlevels 6+).
Source: Eurostat
Regional knowledgecapital Knowledge stock formation measured in terms of patent accumulation,in log terms.
Source: Eurostat 𝑫 Geographic distance Geodesic distance between source and host region.
Source: Eurostat
Difference in tax rates Country-specific top statutory corporate income tax rates (includingsurcharges). Measured by means of difference between source andhost region.
Source: Eurostat
Common language Dummy variable, 1 denotes that the regions share a common officiallanguage, 0 otherwise.
Source: European Commission
Notes : ISCED and NACE refer to the international standard classification of education and the second revision of thestatistical classification of economic activities in the European community, respectively. Empirical results
This subsection presents the empirical results obtained from 15,000 posterior draws after discardingthe first 10,000 as burn-ins. Running multiple chains with alternating starting values did not affectthe empirical results, which also provides evidence for sampler convergence.Posterior quantities for upstream-, downstream-, and production-related investment flows arepresented in Tables 2, 3, and 4, respectively. Each table reports posterior means and posteriorstandard deviations for the quantities of interest. Statistical significance of the respective posteriormean estimates is based on a 90% credible interval and highlighted in bold. The first block ineach table presents origin- and destination-specific slope parameter estimates, respectively. Theseestimates are reported for both own region characteristics as well as their spatial lags or thirdregion characteristics (Baltagi et al. 2007). In the spatial econometrics literature, the former areoften referred to as average direct impacts. The spatially lagged counterparts denote averageindirect (or spillover) impacts (LeSage and Pace 2009). The second block in each table reportsposterior summary metrics for the spatial autoregressive origin and destination random effects. Thethird and last block in each table shows posterior inference for the variables used in the distancematrix 𝑫 . Origin- and destination-specific core variables
Table 2 reports posterior parameter estimates for upstream FDI (most notably consisting of businessservices and headquarters). Starting with the key drivers for regions producing FDI outflows inupstream-related activities, Table 2 shows particularly strong evidence for the importance of theown-regional market size and population density . In addition, the corresponding third-regionaleffects are significant and negative. For example, an increase in the market size restricted only toneighbouring regions thus decreases the amount of FDI outflows from a given region. The table alsosuggests a particularly accentuated importance of a well educated working age population ( highereducation workers ) in the origin region. The estimated impact appears much more pronouncedas compared to downstream and production FDI. Moreover, for upstream FDI the third regioneffect associated with the higher education workers variable also appears to be positive and highlysignificant. Own-regional knowledge capital endowments appear to be positively associated withthe generation of upstream FDI outflows. However, the impacts of regional knowledge capital endowments for upstream FDI outflows appear rather muted as compared to the other types of FDIconsidered. Interestingly, Table 2 shows negative third-regional impacts for knowledge capital .Unlike other types of FDI under scrutiny, the compensation per hour variable only appears to havea significant impact for own-regional upstream FDI outflows.Inspection of the regional determinants to attract upstream FDI inflows shows some interestingsimilarities to the origin-specific characteristics. This holds particularly true for the market size and population density variables. Both destination-specific variables show a positive and highlysignificant own-regional impact, with negative (and significant) spatial lags. Similar to the ori-gin specific determinants of upstream FDI, the corresponding host-specific impacts appear morepronounced as in other activity types. This finding is in line with Henderson and Ono (2008),8efever (2006), or Duranton and Puga (2005), who highlight that the location choice of businessservices and headquarters related activities are particularly driven by functional aspects (ratherthan by sectoral aspects) and typically tend to be located in urban agglomerations. Regional FDIinflows associated with upstream investment activities moreover appear to be particularly attractedby regions with a higher specialization in the services sector ( employment in services ), relative tothe agriculture sector (which serves as the benchmark in the specifications).
Table 2:
Posterior parameter estimates for FDI associated with upstream value chains.
Variable Origin DestinationMean Std. Dev. Mean Std. Dev.Market size -0.59 -0.94 𝑾 Market size -2.04 -0.53 𝑾 Population density -0.47 -0.35 𝑾 Compensation per hour -0.14 0.26 -0.63 𝑾 Employment in industry -2.04 1.83 0.20 1.31 𝑾 Employment in services 1.47 1.57 -2.79 𝑾 Lower education workers 0.42 0.93 -0.01 1.01 𝑾 Higher education workers 𝑾 Regional knowledge capital -0.94 𝜌 𝑜 , 𝜌 𝑑 𝜙 𝑜 , 𝜙 𝑑 -1.01 Notes : The model includes a constant. Results based on 15,000 Markov-chain Monte Carloiterations, where the first 10,000 were discarded as burn-in. Estimates in bold are statisticallysignificant under a 90% confidence interval.
From a theoretical point of view, we would also expect labour costs, measured in termsof compensation per hours , to be an important determinant for attracting FDI inflows. Thishypothesis is confirmed by inspecting the destination-specific results across all tables. Significantnegative direct impacts of this variable can be observed throughout all stages of the value chain,both concerning the own region, as well as third regions. This corroborates the findings ofAscani et al. (2016b), who study the location determinants of Italian multinational enterprises.
Regional knowledge capital as a pull-factor for upstream FDI inflows appears less relevant. Onlythe respective third-regional impact is significant, however, it appears comparatively muted.Overall, the results for downstream FDI reported in Table 3 show a strong similarity to those ofupstream FDI (Table 2). This resemblance can be observed for both origin- and destination-specificspatial determinants. For regions as a source of downstream FDI, Table 3 also highlights the key9mportance of agglomeration forces, proxied by the variables market size and population density .Both variables show a positive and significant direct impact for the generation of downstream FDIoutflows, along with negative third-regional effects. These impacts, however, appear somewhatless pronounced as compared to upstream FDI. Similarly, the impact of regional tertiary educa-tion attainment ( higher education workers ) for downstream FDI outflows appears less accentuatedas compared to upstream FDI outflows. As opposed to the results for origin-specific upstreamFDIs, the third-regional effects of tertiary education attainment are insignificant.
Regional knowl-edge capital endowments, on the other hand, appear somewhat more important for generatingdownstream FDI as compared to upstream FDI, with positive direct, and negative third-regionaleffects.
Table 3:
Posterior parameter estimates for FDI associated with downstream value chains.
Variable Origin DestinationMean Std. Dev. Mean Std. Dev.Market size -0.92 -1.10 -0.06 𝑾 Market size -1.34 -0.55 𝑾 Population density -0.63 𝑾 Compensation per hour -0.58 -0.48 𝑾 Employment in industry -2.14 𝑾 Employment in services -1.15 1.68 -1.31 0.98 𝑾 Lower education workers 1.01 1.01 𝑾 Higher education workers 1.78 1.04 1.24 0.94 𝑾 Regional knowledge capital -0.92 𝜌 𝑜 , 𝜌 𝑑 𝜙 𝑜 , 𝜙 𝑑 -0.85 Notes : The model includes a constant. Results based on 15,000 Markov-chain Monte Carloiterations, where the first 10,000 were discarded as burn-in. Estimates in bold are statisticallysignificant under a 90% confidence interval.
In line with the prevalent literature (see, among others, Leibrecht and Riedl 2014, Casi and Resmini2010, or Baltagi et al. 2007), the destination-specific regional determinants for downstream FDIalso show a strong importance of the market size and population density variables as a means toattracting downstream-related FDI inflows. Similar to destination-specific upstream FDI, educa-tional attainment ( lower and higher education workers ) and the compensation per hour variableappear as important pull-factors. Concerning the regional industry mix, Table 3 suggests that10igher shares in the industry and service sectors ( employment in industry and services ) appearto be significantly and positively associated with attracting downstream-related FDI inflows. Aninteresting result is given by a negative and statistically significant own-regional impact of the regional knowledge capital variable. The estimated impacts, however, appear rather offset by thepositive third-regional impacts. Similar results can also be found in work by Dimitropoulou et al.(2013), a study on the location determinants of FDI for UK regions.Empirical results for production-related FDI are summarized in Table 4. Starting with theorigin-specific determinants of generating production FDI outflows, Table 4 shows not surprisinglya pronounced importance of regional market size and population density . Similar to the other typesof FDI, both variables also exhibit significant negative third-regional effects. Interestingly, thesource regional industry mix also appears to play a key role. Specifically, the employment inindustry variable shows a positive and highly significant direct impact of the origin region. Theremaining origin-specific drivers are basically in line with those of the other types of FDI, mostnotably positive impacts of tertiary education attainment ( higher education workers ) levels and regional knowledge capital endowments.
Table 4:
Posterior parameter estimates for FDI associated with production value chains.
Variable Origin DestinationMean Std. Dev. Mean Std. Dev.Market size -0.13 -1.21 𝑾 Market size -1.11 -0.90 𝑾 Population density -0.41 𝑾 Compensation per hour -0.38 0.29 -0.61 0.50 𝑾 Employment in industry 0.55 1.28 -1.27 1.08 𝑾 Employment in services 0.13 1.40 -0.85 0.81 𝑾 Lower education workers 𝑾 Higher education workers 𝑾 Regional knowledge capital -1.21 𝜌 𝑜 , 𝜌 𝑑 𝜙 𝑜 , 𝜙 𝑑 -0.96 Notes : The model includes a constant. Results based on 15,000 Markov-chain Monte Carloiterations, where the first 10,000 were discarded as burn-in. Estimates in bold are statisticallysignificant under a 90% confidence interval. market size shows a similar importance, along with negative third-regional effects, the direct impact of the population density variable shows a negative and significant sign. Our estimation results thus showthat production-oriented FDI activities are predominantly attracted by smaller regions in proximityto urban agglomerations. For upstream and downstream activities, however, urban agglomerationsseem to play a more central role. Moreover, our results imply that regional human capital en-dowments are particularly important for explaining upstream and downstream-oriented investmentdecisions. For production activities, the importance of regional human capital endowments ap-pears slightly less pronounced. These results corroborate the findings of Strauss-Kahn and Vives(2009), and Defever (2006) by highlighting that industry-related location decisions typically fo-cus on sectoral, rather than on functional aspects. The significant and positive own-regional,destination-specific industry mix ( employment in industry and services ) further underpins thesefindings.For attracting production-related FDI, Table 4 shows a particularly pronounced negative impactof the compensation per hour variable of the host region. The negative direct impact on inflows isthe strongest with a posterior mean of − .
21 for production-related activities. However, it is worthnoting that the associated third-regional impacts on inflows are insignificant for production, whereasboth downstream and upstream related FDI flows exhibit significant negative third-regional impacts.Our findings are moreover in line with Fallon and Cook (2014) and Crescenzi et al. (2013), whoboth find that locational drivers for production-related FDI flows differ from those associated withbusiness service activities.
Spatial-dependence and distance metrics
This subsection discusses the results for the spatial autoregressive origin and destination randomeffects, as well as the estimates of intervening opportunities from the distance matrix 𝑫 . In-spection of posterior estimates for the spatial latent random effects provides significant evidencefor pronounced spatial dependence patterns in the random effects across all stages of the valuechain. This finding holds true for both source- and host-regional heterogeneity in the sample. Acomparison of their corresponding posterior means and standard deviations shows that all spatialautoregressive parameters are estimated with a high precision. The intensity of spatial dependencein the upstream- and downstream-specific latent unobservable effects appear similarly pronounced,with values ranging from 0 .
42 to 0 .
58. For production-related investment activities, the differencebetween 𝜌 𝑜 and 𝜌 𝑑 appears more pronounced, with the former being particularly sizeable (0 . 𝑫 . As expected, the posterior mean estimates for geographicaldistance are negative and significantly differ from zero for all types of investment activities.Moreover, the posterior standard deviations are comparatively small, indicating that the impactof geographic distance is estimated with a high precision. Higher geographic separation of two12egions is thus associated with lower FDI activities, as increased distance often raises transportation,monitoring and thus investment costs. The negative impacts reported in Tables 2, 3, and 4are in line with recent empirical results in FDI (Leibrecht and Riedl 2014) and trade literature(Krisztin and Fischer 2015).Our dummy variable measuring whether a pair of regions shares an official common language proxies the cultural distance between regions in the sample. As expected, the reported posteriormeans show a positive sign and are significantly different from zero. The third distance variablein the matrix 𝑫 measures the (country-specific) difference in corporate tax rates between sourceand target regions. In line with theoretical and empirical literature on the location choice ofmultinationals, the tables report significant and positive impacts to regional FDI flows whencorporate tax rates in the target region are lower than in the source region (see Bellak and Leibrecht2009 and Strauss-Kahn and Vives 2009). The estimated posterior means for the difference in taxrates suggest that a 1% decrease in the tax rate difference between source and destination regionsresults in a 1 .
3% and 3 .
5% increase in the number of FDI flows for downstream and upstreamrelated activities, respectively.
This paper presents an empirical study on the spatial determinants of bilateral FDI flows amongEuropean regions. Due to data scarcity on the subnational level, previous papers typically adopta national perspective when analysing FDI dyads (see, for example, Leibrecht and Riedl 2014).This paper thus provides a first spatial econometric analysis on the European regional level byexplicitly accounting for origin-, destination-, and third region-specific factors in the analysis. Thesubnational perspective of our analysis allows us to study the spatial spillover mechanisms ofregional FDI flows in more detail. Unlike recent studies on the locational determinants of FDIinflows (see, for example, Ascani et al. 2016b, or Crescenzi et al. 2013), we model FDI decisiondeterminants not only across destination regions but also across the origin regional dimension.Moreover, due to the well-known need to control for spatial dependence when modelling regionaldata (LeSage and Pace, 2009), we also capture spatial dependence through spatially structuredrandom effects associated with origin and destination regions.Our data comes from the fDi Markets database, which contains detailed information on regionalFDI activities using media sources and company data. The data from the fDi Markets database alsocontains detailed sectoral information on the functional form of the FDI activity, which allows usto explicitly focus on FDI flows across different stages of the value chain. Specifically, the paperstudies the origin- and destination-specific determinants of upstream, downstream, and productionactivities.Our empirical results clearly indicate that both source and destination spatial dependenceplays a key role for all investment activities under scrutiny. In line with recent literature, wefind that regional market size, corporate tax rates, as well as third region effects appear to be ofparticular importance for all stages in the value chain. We moreover find that production-orientedFDI activities are predominantly attracted by smaller regions in proximity to urban agglomerations.13or upstream and downstream activities, however, being in the same region as urban agglomerationsseem to play a key role. Moreover, our results imply that regional human capital endowments areparticularly important for explaining upstream and downstream-oriented investment decisions. Forproduction activities, the importance of regional human capital endowments are less accentuated.These results corroborate the findings of Strauss-Kahn and Vives (2009), or Defever (2006) byhighlighting that industry-related location decisions typically focus on sectoral, rather than onfunctional aspects. From an origin-specific perspective of FDI activities, our empirical resultsmoreover clearly show that regional knowledge capital endowments appear crucial for host regionsto produce FDI outflows. Similar to the results on the destination-specific factors for FDI inflows,we also find high education and agglomeration forces as particularly important aspects for hostregional FDI outflows.
DeclarationsFunding:
The research carried out in this paper was supported by funds of the OesterreichischeNationalbank (project number: 18116), and of the Austrian Science Fund (FWF): ZK35.
Conflict of interest:
The authors declare that they have no conflict of interest.
Ethical approval:
This article does not contain any studies with human participants or animalsperformed by any of the authors.
References
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International Journal of Technological Learning, Innovation and Development ppendix Table A1:
Classification of fDi Markets business functions
Classification Business activities % of classification
Upstream Business Services 64 . . . . . . . . . . . . . . . . . Notes : The last column indicates the percent of industry activities per FDI classification. The values are based on thetotal observed FDI flows in the fDi Markets database targeting the selected NUTS-2 regions in the period 2003-2011. able A2: List of regions in the study.
Austria France [continued] Hungary Poland [continued] UK
Burgenland (AT) Languedoc-Roussillon Dél-Alföld Lódzkie Bedfordshire and HertfordshireKärnten Limousin Dél-Dunántúl Lubelskie Berkshire, Buckinghamshire andNiederösterreich Lorraine Észak-Alföld Lubuskie OxfordshireOberösterreich Midi-Pyrénées Észak-Magyarország Malopolskie CheshireSalzburg Nord - Pas-de-Calais Közép-Dunántúl Mazowieckie Cornwall and Isles of ScillySteiermark Pays de la Loire Közép-Magyarország Opolskie CumbriaTirol Picardie Nyugat-Dunántúl Podkarpackie Derbyshire and NottinghamshireVorarlberg Poitou-Charentes
Ireland
Podlaskie DevonWien Provence-Alpes-Côte d’Azur Border, Midland and Western Pomorskie Dorset and Somerset
Belgium
Rhône-Alpes Southern and Eastern Slaskie East AngliaProv. Antwerpen
Germany Italy
Swietokrzyskie East WalesProv. Brabant Wallon Arnsberg Abruzzo Warminsko-Mazurskie East Yorkshire andProv. Hainaut Berlin Basilicata Wielkopolskie Northern LincolnshireProv. Liège Brandenburg Calabria Zachodniopomorskie Eastern ScotlandProv. Limburg (BE) Braunschweig Campania
Portugal
EssexProv. Luxembourg (BE) Bremen Emilia-Romagna Alentejo Gloucestershire, Wiltshire andProv. Namur Chemnitz Friuli-Venezia Giulia Algarve BristolProv. Oost-Vlaanderen Darmstadt Lazio Área Metropolitana de Lisboa Greater ManchesterProv. Vlaams-Brabant Detmold Liguria Centro (PT) Hampshire and Isle of WightProv. West-Vlaanderen Dresden Lombardia Norte Herefordshire, WorcestershireRégion de Bruxelles-Capitale Düsseldorf Marche
Romania and Warwickshire
Bulgaria
Freiburg Molise Bucuresti - Ilfov Highlands and IslandsSeveren tsentralen Gießen Piemonte Centru Inner LondonSeveroiztochen Hamburg Provincia Autonoma di Bolzano/ Nord-Est KentSeverozapaden Hannover Bozen Nord-Vest LancashireYugoiztochen Karlsruhe Provincia Autonoma di Trento Sud - Muntenia Leicestershire, Rutland andYugozapaden Kassel Puglia Sud-Est NorthamptonshireYuzhen tsentralen Koblenz Sardegna Sud-Vest Oltenia Lincolnshire
Czech Republic
Köln Sicilia Vest MerseysideJihovýchod Leipzig Toscana
Slovakia
North Eastern ScotlandJihozápad Lüneburg Umbria Bratislavský kraj North YorkshireMoravskoslezsko Mecklenburg-Vorpommern Valle d’Aosta/Vallée d’Aoste Stredné Slovensko Northern Ireland (UK)Praha Mittelfranken Veneto Východné Slovensko Northumberland and Tyne andSeverovýchod Münster
Latvia
Západné Slovensko WearSeverozápad Niederbayern Latvija
Slovenia
Outer LondonStrední Cechy Oberbayern
Lithuania
Vzhodna Slovenija Shropshire and StaffordshireStrední Morava Oberfranken Lietuva Zahodna Slovenija South Western Scotland
Denmark
Oberpfalz
Luxemburg Sweden
South YorkshireHovedstaden Rheinhessen-Pfalz Luxemburg Mellersta Norrland Surrey, East and West SussexMidtjylland Saarland
Netherlands
Norra Mellansverige Tees Valley and DurhamNordjylland Sachsen-Anhalt Drenthe Östra Mellansverige West MidlandsSjælland Schleswig-Holstein Flevoland Övre Norrland West Wales and The ValleysSyddanmark Schwaben Friesland (NL) Småland med öarna West Yorkshire
Estonia
Stuttgart Gelderland StockholmEesti Thüringen Groningen Sydsverige
Finland
Trier Limburg (NL) VästsverigeÅland Tübingen Noord-Brabant
Spain
Etelä-Suomi Unterfranken Noord-Holland AndalucíaHelsinki-Uusimaa Weser-Ems Overijssel AragónLänsi-Suomi
Greece
Utrecht CantabriaPohjois-ja Itä-Suomi Anatoliki Makedonia, Thraki Zeeland Castilla y León
France
Attiki Zuid-Holland Castilla-la ManchaAlsace Dytiki Ellada
Norway
CataluñaAquitaine Dytiki Makedonia Agder og Rogaland Comunidad de MadridAuvergne Ionia Nisia Hedmark og Oppland Comunidad Foral de NavarraBasse-Normandie Ipeiros Nord-Norge Comunidad ValencianaBourgogne Kentriki Makedonia Oslo og Akershus ExtremaduraBretagne Kriti Sør-Østlandet GaliciaCentre (FR) Notio Aigaio Trøndelag Illes BalearsChampagne-Ardenne Peloponnisos Vestlandet La RiojaCorsica Sterea Ellada
Poland
País VascoFranche-Comté Thessalia Dolnoslaskie Principado de AsturiasHaute-Normandie Voreio Aigaio Kujawsko-Pomorskie Región de MurciaÎle de France .1 Detailed description of the Bayesian Markov-chain Monte Carlo algorithm This section provides a detailed description of the employed Bayesian Markov-chain MonteCarlo (MCMC) algorithm. A similar version is employed by LeSage et al. (2007), who usesuch a modelling strategy for estimation of knowledge spillovers (measured in terms of patent-ing dyads) in European regions. Specifically, their estimation approach relies on work byFrühwirth-Schnatter and Wagner (2006), who introduce a Bayesian auxiliary mixture samplingapproach for non-Gaussian distributed data. This approach builds on a hierarchical data augmen-tation procedure by introducing 𝑦 𝑖 + 𝑦 𝑖 , where 𝑦 𝑖 denotesthe 𝑖 -th element of 𝒚 (with 𝑖 = , ..., 𝑁 ).In order to alleviate the implied computational burden, we rely on an improved version of thisauxiliary mixture sampling algorithm (Frühwirth-Schnatter et al., 2013). The algorithm tremen-dously reduces the number of latent parameters per observation. Specifically, the required numberof latent parameters is reduced from 𝑦 𝑖 + 𝜆 𝑖 from Eq. (2.1) can be interpreted as a parameter in a Poissonprocess describing occurring events in a given time interval, where 𝜆 𝑖 denotes the 𝑖 -th elementof the Poisson mean 𝝀 . For illustration, imagine sorting all unique values of the observed FDIflows from lowest to highest. The Poisson process can be viewed as modeling – given a specificnumber of FDI flows – the probability of jumping from one unique value to the next. Thesetwo quantities can be characterized as so-called arrival and inter-arrival times. Motivated by thisformulation, the distribution itself can be described using merely arrival and inter-arrival times,derived from the rate of the process 𝜆 𝑖 . The expected value of arrival time of 𝑦 𝑖 is / 𝜆 𝑖 and itfollows a Gamma distribution with shape one and rate equal to 𝑦 𝑖 . The inter-arrival times are bydefinition independent and arise from an exponential distribution with rate equal to 𝜆 𝑖 . Based onthis definition, we can model 𝜆 𝑖 if we sample from the inter-arrival time 𝜏 𝑖 between 𝑦 𝑖 and 𝑦 𝑖 + 𝑦 𝑖 >
0, the arrival 𝜏 𝑖 time for 𝑦 𝑖 . The main contribution of Frühwirth-Schnatter et al.(2009) is that they introduce auxiliary variables for 𝜏 𝑖 and 𝜏 𝑖 , conditional on 𝑦 𝑖 .For this purpose let us define the latent variables 𝜏 𝑖 and 𝜏 𝑖 , based on the properties of arrivaland inter-arrival times: 𝜏 𝑖 = 𝜉 𝑖 𝜆 𝑖 , 𝜉 𝑖 ∼ E ( ) (A.1) 𝜏 𝑖 = 𝜉 𝑖 𝜆 𝑖 , 𝜉 𝑖 ∼ G ( 𝑦 𝑖 , ) ∀ 𝑦 𝑖 > , (A.2)where E (·) denotes the exponential and
G (· , ·) the Gamma distribution. The arrival times 𝜏 𝑖 onlyapply for 𝑦 𝑖 >
0, since zero values have by definition no arrival time. Eqs. (A.1) and (A.2) can belog-linearized in the following fashion: − ln 𝜏 𝑖 = ln 𝜆 𝑖 + 𝜀 𝑖 , 𝜀 𝑖 = − ln 𝜉 𝑖 (A.3) − ln 𝜏 𝑖 = ln 𝜆 𝑖 + 𝜀 𝑖 , 𝜀 𝑖 = − ln 𝜉 𝑖 ∀ 𝑦 𝑖 > , (A.4)19here for 𝑦 𝑖 = 𝜀 𝑖 and 𝜀 𝑖 would be Gaussian this wouldimply a linear model, which could be easily sampled from. While 𝜀 𝑖 and 𝜀 𝑖 are not Gaussianper se, the distributions can be approximated by a mixture of Gaussians, from which sampling caneasily be achieved (Frühwirth-Schnatter et al., 2009).In order to obtain a model which is conditionally Gaussian, the non-normal density can beapproximated by a mixture of 𝑄 ( 𝜈 ) normal components, where 𝜈 denotes the shape parameter ofa Gamma distribution. For sampling 𝜀 𝑖 we can set 𝜈 =
1, and in the case of sampling 𝜀 𝑖 the rate 𝜈 would be equal to 𝑦 𝑖 . Therefore, the mixture of normal components can be generalised for bothdistributions. Thus, the mixture distribution is given by: 𝑝 𝜀 ( 𝜀 | 𝜈 ) ∼ 𝑄 ( 𝜈 ) Õ 𝑞 = 𝑤 𝑞 ( 𝜈 )N (cid:2) 𝜀 | 𝑚 𝑞 ( 𝜈 ) , 𝑠 𝑞 ( 𝜈 ) (cid:3) , (A.5)where 𝑤 𝑞 ( 𝜈 ) denotes the weight, 𝑚 𝑞 ( 𝜈 ) the mean, and 𝑠 𝑞 ( 𝜈 ) the variance. These components, aswell as 𝑄 ( 𝜈 ) directly depend on the choice of 𝜈 . Values for all these parameters conditional on 𝜈 are provided in Frühwirth-Schnatter et al. (2009). To approximate the Poisson process through theGaussian mixture in Eq. (A.5), the additional latent discrete variable 𝜈 𝑖 , and additionally in casesof 𝑦 𝑖 > 𝜈 𝑖 are introduced.Given 𝜏 𝑖 and 𝜈 𝑖 and additionally for the case of 𝑦 𝑖 > 𝜏 𝑖 and 𝜈 𝑖 , the conditional posteriorof the Poisson model’s slope parameters are Gaussian: − ln 𝜏 𝑖 = ln 𝜆 𝑖 + 𝑚 ( ) + 𝜀 𝑖 , 𝜀 𝑖 | 𝜈 𝑖 ∼ N [ , 𝑠 ( )] (A.6) − ln 𝜏 𝑖 = ln 𝜆 𝑖 + 𝑚 ( 𝜈 𝑖 ) + 𝜀 𝑖 , 𝜀 𝑖 | 𝜈 𝑖 ∼ N [ , 𝑠 ( 𝜈 𝑖 )] ∀ 𝑦 𝑖 > . (A.7)We can easily sample from the distributions given in Eqs. (A.6) and (A.7) and therefore constructan efficient Gibbs sampling algorithm (for a detailed description, see Section A.1 in the Appendix).For Bayesian estimation, we have to define prior distributions for all parameters in the model.We follow the canonical approach and use a Gaussian prior setup for the parameters 𝛼 , 𝜷 𝑜 , 𝜷 𝑑 , 𝜸 𝐷 , 𝜹 𝑜 , and 𝜹 𝑑 with zero mean and a relatively large prior variance of 10 . We follow LeSage et al.(2007) in our choice of priors for the spatially structured random effect vectors and set a normalprior structure 𝜽 𝑜 and 𝜽 𝑑 , with with zero mean and 𝜙 𝑥 ( 𝑨 𝑥 𝑨 𝑥 ) − variance, where 𝑥 ∈ [ 𝑜, 𝑑 ] and 𝑨 𝑥 = 𝑰 𝑛 − 𝜌 𝑥 𝑾 . For the variance of the random effects 𝜙 𝑥 we employ an inverse Gamma priorwith rate equal to 5 and the shape parameter to 0 .
05. Following LeSage et al. (2007), we elicit anon-informative uniform prior specification 𝜌 𝑥 ∼ U (− , ) . A.2 The Gibbs sampling scheme
Let us collect the explanatory variables from Eq. (2.1) in an 𝑁 × 𝑃 (with 𝑃 = + 𝑝 𝑋 + 𝑝 𝐷 ) matrix 𝒁 = [ 𝜾 𝑁 , 𝑿 𝑜 , 𝑿 𝑑 , 𝑫 , 𝑾 𝑜 𝑿 𝑜 , 𝑾 𝑑 𝑿 𝑑 ] and 𝜸 = [ 𝛼 , 𝜷 ′ 𝑜 , 𝜷 ′ 𝑑 , 𝜸 ′ 𝐷 , 𝜹 ′ 𝑜 , 𝜹 ′ 𝑑 ] ′ . Thus, 𝝀 = exp ( 𝒁𝜸 + 𝑽 𝑜 𝜽 𝑜 + 𝑽 𝑑 𝜽 𝑑 ) .Moreover, let us denote the number of non-zero observations in 𝒚 as 𝑁 𝑦> . Then, let 𝑁 + = 𝑁 + 𝑁 𝑦> and let the 𝑁 + × 𝒚 + be 𝒚 + = [ 𝒚 ′ , 𝒚 ′ 𝑦> ] ′ , where 𝒚 𝑦> contains all elements of 𝒚 𝑁 + × 𝑃 matrix 𝒁 + be 𝒁 + = [ 𝒁 ′ , 𝒁 ′ 𝑦> ] ′ , where thematrix 𝒁 𝑦> contains all rows of 𝒁 corresponding to 𝑦 𝑘 >
0. In a similar fashion, we augment thedummy observation matrices 𝑽 𝑜 and 𝑽 𝑑 and denote the resulting 𝑁 + × 𝑛 matrices as 𝑽 + 𝑜 and 𝑽 + 𝑑 .Accordingly we order the auxiliary variables corresponding to 𝜀 𝑖 and 𝜀 𝑖 and collect theminto the following 𝑁 + × 𝝉 = [ 𝜏 , ..., 𝜏 𝑁 , 𝜏 , ..., 𝜏 𝑁 𝑦> ] and 𝝂 = [ 𝜈 , ..., 𝜈 𝑁 , 𝜈 , ..., 𝜈 𝑁 𝑦> ] . Based on this, we define the 𝑁 + × 𝑁 + variance matrix 𝛀 .Additionally – based on the definition of the Gaussian mixtures in Eqs. (A.6 - A.7) – an 𝑁 + × 𝒚 can be obtained conditional on 𝝉 and 𝝂 , so that ˜ 𝒚 = 𝑚 ( 𝝂 ) − ln 𝝉 .Given appropriate starting values the following Gibbs sampling algorithm can be devised:I. Sample 𝜸 from its conditional Gaussian distribution 𝑝 ( 𝜸 |·) ∼ N ( 𝚺 𝜸 𝝁 𝜸 , 𝚺 𝜸 ) , where 𝚺 𝜸 = (cid:16) 𝒁 ′+ 𝛀 − 𝒁 + + 𝚺 − 𝜸 (cid:17) − 𝝁 𝜸 = 𝒁 ′+ 𝛀 − ( ˜ 𝒚 − 𝑽 + 𝑜 𝜽 𝑜 − 𝑽 + 𝑑 𝜽 𝑑 ) + 𝚺 − 𝜸 𝝁 𝜸 . 𝚺 𝜸 denotes the 𝑃 × 𝑃 prior variance matrix and 𝝁 𝜸 the 𝑃 × 𝜽 𝑥 from their conditional distributions 𝑝 ( 𝜽 𝑥 |·) ∼ N ( 𝚺 𝜽 𝑥 𝝁 𝜽 𝑥 , 𝚺 𝜽 𝑥 ) , where 𝚺 𝜽 𝑥 = (cid:16) 𝜙 − 𝑥 𝑨 𝑥 ′ 𝑨 𝑥 + 𝑽 + ′ 𝑥 𝛀 − 𝑽 + 𝑥 (cid:17) − 𝝁 𝜽 𝑥 = 𝑽 + ′ 𝑥 𝛀 − (cid:0) ˜ 𝒚 − 𝒁 + 𝜸 − 𝑽 + 𝑥 𝜽 𝑥 (cid:1) . III. We sample 𝜙 𝑥 from the conditional posterior, which is inverse Gamma distributed and givenas 𝑝 ( 𝜙 𝑥 |·) ∼ IG ( 𝑠 𝑥 , / 𝑣 𝑥 ) where: 𝑠 𝑥 = ( 𝑛 + 𝑠 𝑥 )/ 𝑣 𝑥 = (cid:0) 𝜽 ′ 𝑥 𝑨 ′ 𝑥 𝑨 𝑥 𝜽 𝑥 + 𝑠 𝑥 𝑣 𝑥 (cid:1) / .𝑠 𝑥 and 𝑣 𝑥 denote the prior rate and shape parameters of the inverse gamma distribution IG (· , ·) .IV. For 𝜌 𝑥 the conditional posterior distribution is: 𝑝 ( 𝜌 𝑥 |·) ∝ | 𝑨 𝑥 | exp (cid:18) − 𝜙 𝑥 𝜽 ′ 𝑥 𝑨 ′ 𝑥 𝑨 𝑥 𝜽 𝑥 (cid:19) . Unfortunately, this is not a well-known distribution, thus – as is standard in the spatialeconometric literature – we resort to a griddy Gibbs step (Ritter and Tanner 1992) in orderto sample from the conditional posterior for 𝜌 𝑥 . For this purpose candidate values 𝜌 ∗ 𝑥 are sampled from 𝜌 ∗ 𝑥 = N ( 𝜌 𝑥 , 𝜋 𝜌 𝑥 ) , where 𝜋 𝜌 𝑥 is the proposal density variance, which isadaptively adjusted using the procedure from LeSage and Pace (2009) and thus is constrainedto a desired interval by the means of rejection sampling. The candidate values are evaluatedusing their full posterior distributions . An alternative, however, computationally more intensive approach also frequently used in the spatial econometricliterature involves a Metropolis-Hastings step for the spatial autoregressive parameter (see, for example, LeSage and Pace2009). In practice it is costly to evaluate the log-determinant directly. Instead we use an adapted version of the log-determinant approximation by Pace and Barry (1997) for pre-calculation.
21. For 𝑖 = , ..., 𝑁 we sample 𝜏 𝑖 from 𝜉 𝑖 ∼ Ex ( 𝝀 𝑖 ) and set 𝜏 𝑖 = + 𝜉 𝑖 . If 𝑦 𝑘 > 𝜏 𝑖 from B ( 𝑦 𝑖 , ) (where B (·) denotes the Beta distribution) and set 𝜏 𝑖 = − 𝜏 𝑖 + 𝜉 𝑖 .VI. For 𝑖 = , ..., 𝑁 we sample 𝜈 𝑖 from the discrete distribution involving the mixture of normaldistributions with 𝑟 = , ..., 𝑄 ( ) : 𝑝 ( 𝜈 𝑖 = 𝑟 |·) ∝ 𝑤 𝑟 ( )N [− ln 𝜏 𝑖 − ln 𝜆 𝑖 | 𝑚 𝑟 ( ) , 𝑠 𝑟 ( )] and for 𝑦 𝑖 > 𝜈 𝑖 from the discrete distribution (with 𝑟 = , ..., 𝑄 ( 𝑦 𝑖 ) ): 𝑝 ( 𝜈 𝑖 = 𝑟 |·) ∝ 𝑤 𝑟 ( 𝑦 𝑖 )N [− ln 𝜏 𝑖 − ln 𝜆 𝑖 | 𝑚 𝑟 ( 𝑦 𝑖 ) , 𝑠 𝑟 ( 𝑦 𝑖 )] . With the sampled values for 𝝉 and 𝝂 , we update ˜ 𝒚 = ln 𝝉 − 𝑚 ( 𝝂 ) and 𝛀 .This concludes the Gibbs sampling algorithm. The Markov-chain Monte Carlo algorithm cyclesthrough steps I. to VI. 𝐵 times and excludes the initial 𝐵 draws as burn-ins. Inference regardingthe parameters is subsequently conducted using the remaining 𝐵 − 𝐵 draws. Whether the MCMC algorithm converged can be easily verified using convergence diagnostics by Geweke (1991)or Raftery and Lewis (1992). For the present application we utilised an implementation of these convergence diagnosticsfrom the R coda package.package.