The Space of Gravity: Spatial Filtering Estimation of a Gravity Model for Bilateral Trade

The Space of Gravity: Spatial Filtering Estimation of a Gravity Model for Bilateral Trade A First Note on Estimation and Empirical Results Gert-Jan Linders, a1 Roberto Patuelli b,c and Daniel A. Griffith d a Department of Spatial Economics, VU University Amsterdam, The Netherlands b Institute for Economic Research (IRE), University of Lugano (USI), Switzerland c The Rimini Centre for Economic Analysis, Italy d School of Economic, Political and Policy Sciences, University of Texas at Dallas, USA ABSTRACT Bilateral trade flows traditionally have been analysed by means of the spatial interaction gravity model. Such analyses mostly used to be based upon an atheoretical version of this model often referred to as the empirical gravity equation. This equation essentially, an unconstrained gravity specification has been criticized largely for leading to biased parameter estimates. Over the last decade, the gravity model has experienced renewed attention in the empirical literature about international trade. While some studies have focused on issues such as border effects or intangible trade barriers, recent studies also have returned attention to the theoretical gravity model of bilateral trade and the use of multilateral resistance terms (that is, balancing factors) to correct for accessibility effects. In a cross-section context, such models are often estimated by OLS, with the inclusion of country-specific fixed effects. A disadvantage of this approach is that country-specific variables such as (per capita) GDP or institutional quality cannot be identified. The use of an alternative estimator, using a firstorder Taylor-series expansion approach, has been suggested, in order to allow identification of country-specific variables. This paper offers an alternative but complementary approach, based upon spatial filtering techniques, which reinterprets the balancing factors as indicators of origin- and destinationbased spatial dependence in trade flows. Spatial filtering estimation enables us to control for spatial dependence, and to estimate an unconstrained gravity model that includes countryspecific variables. JEL codes: C14, C21, F10 Keywords: bilateral trade, unconstrained gravity model, spatial filtering 1 Corresponding author. E-mail address: glinders@feweb.vu.nl. Tel.: +31-(0)20-598-6088. 1

1. Introduction The gravity model of bilateral trade is the workhorse model for the empirical analysis of bilateral trade patterns (Eichengreen and Irwin, 1998). This model describes the volume of bilateral trade as a function of economic size of origin and destination, and transactional distance between them. Over the past decade, renewed interest has emerged for the theoretical foundations and the correct estimation of the gravity model (e.g., Matyas, 1998; Anderson and van Wincoop, 2003; Egger, 2004). There are several motivations for this. First, the interest in modelling trade flows has increased with questions on the effectiveness of trade liberalization and integration, the modelling of determinants of exporting choices, and the persistent effects of largely unobserved trade costs on trade patterns. The result has been a set of complex partial equilibrium models, underpinning the gravity equation for bilateral trade, and extending its empirical specification. Although this renewed interest has been of great benefit to the consistent modelling of bilateral trade flows, it has some drawbacks. Estimation increases in complexity, or the specification looses a lot of its explanatory power in order to retain relative simplicity of estimation. The main research question we intend to address in this paper is: Can we find a linear empirical estimator that provides consistent parameter estimates, but not at the cost of loosing explanatory power? This paper proposes an alternative approach to the estimation of the gravity equation of bilateral trade, which acknowledges the importance of theoretical consistency, while retaining a simple estimation framework, and without losing valuable information about the determinants of trade patterns. The suggested estimation approach is based upon recent advances in other fields, such as commuting and patent citations (Fischer and Griffith, 2009b; Griffith, 2009a), with regard to the spatial modelling of origindestination flows and the spatial filtering estimation of gravity equations. We proceed as follows: first, the theoretically consistent gravity model is introduced in Section 2. We illustrate the currently most widely advocated approach to the consistent estimation of the gravity model: fixed-effects estimation. Section 3 discusses recent developments in the literature concerning the estimation of the theoretical gravity model. In Section 4, we present our argument for considering a spatial filtering estimator of the gravity model. We discuss the main problem associated with the fixed-effects approach, that is, the loss of variables that can be assessed in estimation. The spatial filtering estimation of the gravity equation yields simple, theoretically consistent estimation of the gravity equation. In Section 5, we turn to the estimation results. Section 6 concludes the paper. 2

2. A Theoretical Trade Gravity Model Gravity equations for analysing bilateral trade flows have been estimated since the 1960s (Tinbergen, 1962; Poyhonen, 1963). For instance, the effects of (changes in) trade costs can be analysed using the gravity model of bilateral trade. Recently, several papers have contributed to providing the gravity model with a coherent theoretical foundation, which allows an internally consistent analysis of such trade cost effects on trade patterns. The most notable contribution is this context is the seminal paper by Anderson and van Wincoop (henceforth AvW) (2003). 2 The reduced-form gravity equation following from the AvW model explicitly takes into account the role played by country-specific price indices (called multilateral resistance (MR) terms). Assuming an N-country endowment economy, CES 3 preferences, and symmetric bilateral trade costs (t ij = t ji ) formulated as a tariff equivalent using an iceberg trade-cost specification, they derive a theoretical gravity equation of the form: x ij yy i j ij = y t PP w i j 1 σ. (1) Here, x ij is the value of the flow of goods from country i to country j, y is GDP (w stands for world), and P is a country-specific multilateral resistance term (where P i measures outward accessibility of country i, and P j measures inward accessibility of country j). The term σ is the elasticity of substitution (σ > 1). The log-linear equivalent of Equation (1) is: 1 σ 1 σ ( ) ln x = ln y + ln y + ln y + 1 σ t ln P ln P. (2) ij w i j ij i j The first two determinants of bilateral trade are the size of the origin and destination markets. These forces of flow generation and attraction are represented by GDP. The third factor of interest is bilateral trade costs (t ij ). Trade costs can depend on a host of variables ranging from bilateral characteristics, such as transport costs and tariffs, to cultural or 2 See also, for example, Bergstrand (1985). 3 The acronym CES denotes constant elasticities of substitution. 3

institutional trade barriers, to geographical characteristics, like landlockedness and overall area. A representative specification of trade costs, for our purposes, would be (see Deardorff, 1998; Anderson and Van Wincoop, 2004): β1 ij β2 CBij β3 CLij β4 CHij β5 FTAij t = D e e e e, (3) ij where D stands for geographical distance, determining transport costs and possibly cultural trade barriers; CB stands for an indicator variable equal to 1 if two countries share a (land) border (and zero otherwise); CL, CH and FTA are a set of similar indicator variables indicating whether or not two countries share a common language, common colonial history, and/or common free-trade agreement. Anderson and van Wincoop (2003) show that for the MR terms P i and P j, P P 1 σ i 1 σ j N 1 σ tij = θ j, P (4) j= 1 j N 1 σ tij = θ i, P, (5) i= 1 i where θ = y y, i. i i w Note that the equations for the MR terms include the sum of GDP-share-weighted bilateral trade costs relative to the price index. That is why these variables can be seen as multilateral resistance terms: they reflect outward resistance for country i and inward resistance for country j, respectively. The higher the MR terms (that is, the higher P i and P j ), the higher bilateral trade is, all else being equal. This reflects that higher resistance means that, given bilateral trade costs t ij, other countries k are less attractive trading partners. 4 A first important and empirically relevant deduction from this model is that the MR terms and the other regressor variables may be correlated, because trade costs and GDP occur both as separate regressors and embedded in the MR terms. In empirical estimation, this would imply that failure to control for multilateral resistance might result in omitted variable bias. Second, as noted by AvW (2003) and Feenstra (2004), estimation of the system of Equations 4 It is no coincidence that the MR terms obtained by solving the reduced-form model for bilateral flows impose the constraints: x = y and i ij j x = y ij i. In similar applications of the model in the regional science field, j this type of gravity model is known as a doubly-constrained gravity model. 4

(2) (5) involves non-linearities in the parameters, from the set of Equations (4) and (5). AvW (2003) implemented a non-linear estimation procedure for the estimation of their specific application of the gravity model. 3. Recent Developments in Estimating the Theoretical Trade Gravity Model The theoretical gravity model shows that consistent estimation of the parameters requires us to take into account the price indices. As discussed in Feenstra (2004), the computational complexity of the non-linear estimation procedure has prevented its widespread use in the applied international trade literature. Still, Avw (2003) show that estimation of the more traditional empirical gravity equation (omitting the MR terms) yields inconsistent parameter estimates on the key regressor variables. A simple solution that results in consistent parameter estimates is to use a set of country-specific indicator variables for the exporting and importing countries (Broecker and Rohweder, 1990; Feenstra, 2004). The indicator variables capture the country-specific MR terms, and control for omitted variables bias related to the countryspecific intercepts. The main advantage is that the resulting specification can be estimated by familiar methods such as OLS or Poisson regression. However, the disadvantage of this solution is that the parameters of country-specific determinants of trade cannot be estimated. Variables such as GDP, per capita income, landlockedness, land area, and so on are captured by the country-specific indicator variables. Still, empirical estimation of the effect of these variables may be relevant depending on the topic under investigation. Hence, a solution that would share the basic simplicity of estimation with the indicator variable specification, and allow retention of the countryspecific regressors is needed. Several recent developments in the trade gravity model literature have focused on combining consistent estimation and flexibility in the specification of the gravity equation. Egger (2005) argues that a Hausman-Taylor approach, which allows for country-specific covariates, is consistent even if unobserved country-specific heterogeneity exists. This provides an alternative to the indicator variables specification that controls for omitted variables bias due to omitted MR terms, and allows for the estimation of the parameters related to the country-specific variables. The method is based upon an approach similar to instrumental variables, which relies on instruments from inside the model. 5

Baier and Bergstrand (BB) (2009), instead, log-linearize the multilateral resistance terms using a first-order Taylor series approximation. This yields exogenous bilateral multilateralworld-resistance (MWR) variables that proxy the endogenous country-specific multilateral resistance (MR) variables in AvW (2003). The resulting reduced-form gravity equation can be estimated with OLS. This method is termed bonus vetus ( good-old ) OLS (BV-OLS). The approach yields log-linear approximations of the multilateral resistance terms, using Taylor series expansion around a centre of identical and symmetric trade costs, t ij = t, but differing economic sizes (θ i = y i /y w ). Starting from a reformulated Equation (2): ( ) ( ) ( ) ln x = ln y + ln y + ln y σ 1lnt + σ 1lnP + σ 1ln P, ij w i j ij i j the equation that BB derive is: N 1 N N ln xij = ln yw + ln yi + ln yj ( σ 1ln ) tij + ( σ 1) θ j ln tij θθ i j ln tij j= 1 2 i= 1 j= 1 N N N 1 + ( σ 1) θiln tji θθ i jln tij. i= 1 2 i= 1 j= 1 (6) The terms in square brackets are the MR terms. They contain a first component that captures multilateral trade frictions for each exporting or importing country, relative to a second part that reflects world trade costs. A third approach to the consistent cross-sectional estimation of the gravity model is proposed in Behrens et al. (henceforth, BEK) (2007). Their approach is closely related to our approach of spatial filtering. Starting from the AvW formulation of the theoretical gravity equation, they show that the MR terms can be shown to reflect a correlation structure between trade flows that can be modelled as spatial autocorrelation (SAC). 5 They suggest a spatialautoregressive moving-average specification for the gravity model, which results in consistent estimates of the standard gravity equation parameters. BEK argue that the baseline fixedeffects specification discussed earlier does not fully succeed in capturing the MR 5 Spatial autocorrelation is the correlation that occurs among the values of a georeferenced variable, and that can be attributed to the proximity of the units. The concept of SAC can be related to the first law of geography, stating that everything is related to everything else, but near things are more related than distant things (Tobler, 1970, p. 236). 6

dependencies in the error structure introduced by the general equilibrium nature of trade patterns modelling. The residuals still show a significant amount of SAC (BEK, 2007, p. 10). We now proceed to discuss the methodology followed in this paper. The alternative we propose, spatial filtering, combines two attractive features. First, it is fairly simple to apply, much like OLS with indicator variables; second, it takes into account the general equilibrium interdependence of trade flows that can be modelled as SAC, like spatial econometric origindestination specifications. 4. Proposed Methodology: Spatial Filtering Estimation The theoretical gravity model includes origin- and destination-specific MR variables that reflect the export and import accessibility of countries. Omitting these endogenous MR variables from the specification results in potential omitted variables bias, both for the trade costs variables and for the size variables in the gravity equation. Consistent estimation requires some way to capture the endogeneity between MR terms and standard regressors. We propose to make use of the fact that this dependency structure is likely to manifest as SAC in the residuals of the traditional specification of the gravity model. The reasoning is that many trade costs variables, such as geographical distance, adjacency, trade agreements, and common language, are spatially correlated: countries close in space are more likely to share the same (or similar) characteristics. This likewise implies that both inward and outward accessibility are spatially correlated: close countries are likely to have more similar accessibility. We deal with SAC by using an origin- and a destination-specific spatial filter, which serve to capture the spatially autocorrelated parts of the residuals. When including these spatial filters as additional origin- and destination-specific regressors (much like the origin and destination specific MR variables), the model can be estimated by standard regression techniques, such as OLS or Poisson regression, which are common in the literature about spatial interaction patterns. The parameters of the standard regressor variables are unrelated to the remaining residual term, and standard estimation yields consistent parameter estimates as a result. We refer to this estimation method as spatial filtering estimation of origin-destination models (see Griffith, 2007; Fischer and Griffith, 2008). Basically, spatial filtering estimation of georeferenced data regressions (such as international trade) can reduce to defining a geographically varying mean and a variance on the basis of an exogenous spatial weights matrix. In other words, the spatially correlated 7

residuals from an otherwise non-spatial regression model are partitioned into two synthetic variables: (i) a spatial filter which captures latent spatial dependence; and, (ii) a non-spatial variable (free of SAC), which will be the newly obtained residuals. The workhorse for this spatial filtering decomposition is a transformation procedure based upon eigenvector extraction from the matrix (I 11 T /n) C (I 11 T /n), where C is a generic n x n spatial weights matrix; I is an n x n identity matrix; and, 1 is an n x 1 vector containing 1s. The spatial weights matrix C defines the relationships of proximity between the n georeferenced units (e.g., points, regions, and countries). The transformed matrix appears in the numerator of Moran s coefficient (MC), which is a commonly used measure of SAC. Because of the above transformation, the resulting eigenvectors represent distinct map pattern descriptions of SAC underlying georeferenced variables (Griffith, 2003). Moreover, the first extracted eigenvector, say e 1, is the one showing the highest positive MC that can be achieved by any spatial recombination induced by C. The subsequently extracted eigenvectors will maximize MC while being orthogonal to the previously extracted eigenvectors. Finally, the last extracted eigenvector will maximize negative MC, while still being orthogonal to and uncorrelated with the previous (n 1) eigenvectors. Having extracted the above eigenvectors, a spatial filter is constructed by judiciously selecting a subset of the n eigenvectors. In detail, for our empirical application, we select a first subset of eigenvectors (which we will call candidate eigenvectors ), by means of the following threshold: MC(e i )/MC(e 1 ) > 0.25. This threshold yields a spatial filter that approximately replicates the amount of variance explained by a spatial autoregressive model (SAR) (Anselin, 1988). Subsequently, a stepwise regression model may be employed to further reduce the first subset (till here completely unrelated to the data) to just the (smaller) subset of eigenvectors that are statistically significant as additional regressors in the model to be evaluated. The resulting group of eigenvectors is what we call our spatial filter. This estimation technique has been applied, both in autoregression and in traditional modelling terms, to various fields, including regional labour markets (Patuelli, 2007), regional systems of innovation (Grimpe and Patuelli, 2009), and ecology (Monestiez et al., 2006. The added challenge, with regard to the case at hand, is that trade data are not point data, but flow data. Therefore, the eigenvectors are linked to the flow data by means of Kronecker 8

products: the product e K 1, where e K is the n x k matrix of the candidate eigenvectors, may be linked to the origin-specific information (for example, GDP per exporting countries), while the product 1 e K may be linked to destination-specific information (again, for example, the GDP of importing countries) (Fischer and Griffith, 2008). As a result, we have two sets of origin- and destination-specific variables, which aim to capture the SAC patterns commonly captured by the indicator variables of a doubly-constrained gravity model (Griffith, 2009a), therefore avoiding omitted variable bias. The main advantages of the proposed estimation method are: (a) this approach can be applied to any type of regression, including simple OLS and generalized linear models such as Poisson or negative binomial regressions, for which usually dedicated spatial econometric applications do not exist; (b) by avoiding the use of indicator variables, we are able to save degrees of freedom, and, (c) the approach can be used to estimate regression parameters for origin- and destination-specific variables, such as GDP or trade agreements indicators. For our case study, because of the nature of trade data, as suggested by Santos Silva and Tenreyro (2006), we estimate a count data model. While the natural choice would be Poisson regression, in order to take into account overdispersion in the data (the sample mean differs greatly from the sample variance), we choose to estimate a quasi-poisson model (Venables and Ripley, 2002); that is, a quasi-likelihood, non-parametric estimation solution to the overdispersion problem. Alternatively, a negative binomial model, which, however, is significantly heavier computation-wise, could be estimated. 5. Empirical application We apply the spatial filtering estimation to a cross-section of bilateral trade flows between 137 countries for the year 2000. In this section, we discuss the empirical specification, data and the estimation results. 5.1 Data and Model Specification For estimation, we follow a standard specification of the gravity equation of bilateral trade. Starting from the trade costs variables identified in equation (3), we further extend the specification with some additional variables commonly mentioned in the literature (see, for example, Frankel, 1997; Raballand, 2003). We use the following standard specification of the gravity equation: 9

ln X ln( GDP GDP ) = α + α ln( GDPCAP GDPCAP ) + β ln( D ) ij i j 0 1 i j 1 ij + β CB + β CL + β CH + β FTA + β ISL + β ISL + β ln( Area) 2 ij 3 ij 4 ij 5 ij 6 i 7 j 8 i + β ln( Area) + β LL + β LL + δ MWRCB + δ MWRCL 9 j 10 i 11 j 2 ij 3 ij + δ MWRCH + δ MWRFTA + ε, 4 ij 5 ij ij (7) where GDPCAP represents per capita GDP, ISL is an indicator variable that equals 1 if the country is an island, Area is the land area of a country, and LL equals 1 for landlocked countries. The other variables are as defined earlier. The data for trade are from the World Trade Database compiled on the basis of COMTRADE data by Feenstra et al. (2005). GDP and per capita GDP data are from the World Bank s WDI database. Distance, language, colonial history, landlocked countries, and land area data are from the CEPII institute. 6 Whether pairs of countries take part in a common regional integration agreement (FTA) has been determined on the basis of OECD data about major regional integration agreements. 7 A dummy variable indicates whether a pair of countries has membership in at least one common FTA. Data on island status have been kindly provided by Hildegunn Kyvik-Nordas (from Jansen and Kyvik Nordas, 2004). We first estimate Equation (7) using Poisson regression including country-specific indicator variables. GDP is used as an offset, which implies we move the log-sum of GDP to the left of Equation (7), assuming it has a proportional effect on trade (AvW, 2003). This is our first benchmark model, which, according to Feenstra (2004), yields consistent parameter estimates, but is criticized by BEK (2007). Second, we estimate Equation (7), extending it with approximations of MR terms obtained using the Taylor series approximation proposed by BB (2009). This is our second benchmark model. These results, as well as the ones for the spatial filtering approach, are discussed in Section 5.2. 5.2 Estimation Results: Spatial Filtering and Benchmark Models The first benchmark model includes origin and destination specific indicator variables. As shown in AvW (2003) and Feenstra (2004), this specification accounts for multilateral resistance terms, and yields consistent parameter estimates. The disadvantage is that countryspecific variables cannot be included in the specification, as their effect cannot be identified separately. This implies that explanatory variables that are potentially relevant for explaining 6 See http://www.cepii.fr. 7 See http://www.oecd.org/dataoecd/39/37/1923431.pdf. 10

variation in bilateral trade patterns, such as GDP per capita, land area and landlockedness, cannot be investigated empirically. A second disadvantage is the loss of degrees of freedom for estimation, because a substantial number of indicator variables (2n 1) is needed. Usually, however, the degrees of freedom remain large enough, since observations are bilateral (i.e., n 2 n). The second benchmark model is the specification developed in BB (2009), which includes first-order Taylor series approximations of the MR variables. This specification, follows from Equation (6). Further manipulation of it (substituting Equation (3) for bilateral trade costs) allows us to combine both terms between square brackets into a set of bilateral variables, one for each bilateral trade costs variable determining trade costs (such as geographical distance). The reduced-form double-log gravity equation is as follows: ln X ln( GDP GDP ) = α + α ln( GDPCAP GDPCAP ) + β ln( D ) + β CB + β CL ij i j 0 1 i j 1 ij 2 ij 3 ij + β CH + β FTA + β ISL + β ISL + β ln( Area) + β ln( Area) + β LL + β LL 4 ij 5 ij 6 i 7 j 8 i 9 j 10 i 11 j + δ MWRCB + δ MWRCL + δ MWRCH + δ MWRFTA + ε, 2 ij 3 ij 4 ij 5 ij ij (8) in which: N N N N MWRDij = θ j ln( Dij ) + θi ln( Dij ) θθ i j ln( Dij ), j= 1 i= 1 i= 1 j= 1 (9) and likewise for the remaining MWR variables. BB (2009) show that theory imposes the restrictions δ k = β k for each k. The equations specify the model in double-logarithmic transformation. We estimated the benchmark models multiplicatively, using Poisson regression. This method allows a direct treatment of zero-valued trade flows, and enables us to correct for overdispersion of trade flows (see Santos Silva and Tenreyro, 2006). The empirical estimation results are presented in Table 1. Model (1) in Table 1 presents the regression results for the first benchmark model, including country-specific indicator variables. Following AvW (2003), we estimated the model using GDP as an offset variable (restricting the coefficient on GDP variables to equal 1). The parameter estimates are in line with the findings elsewhere in the literature (see, for example, Anderson and van Wincoop, 2004; Disdier and Head, 2008). Geographical distance has a negative effect on trade, with an 11

estimated elasticity of 0.63. This result is comparable to other Poisson estimates, such as the ones presented in Santos Silva and Tenreyro (2006). The effect of proximity on trade is reinforced by a positive and significant effect of contiguity on trade. Proximity in terms of language, preferential trade policy, and colonial links also positively affects bilateral trade, though the effect is not always statistically (highly) significant. These results confirm previous findings about the importance of these dimensions of transactional distance on trade (for example, Obstfeld and Rogoff, 2000; Loungani, 2002). Table 1: Estimation results (1) (2) (3) (4) Fixed effects (GDP offset) Spatial filter BB-estimation (GDP offset) BB-estimation Distance 0.63 *** 0.61 *** 0.54 *** 0.52 *** Common border 0.59 *** 0.58 *** 0.90 *** 0.76 *** Common language 0.10 0.16 *** 0.32 *** 0.23 ** Common history 0.15 * 0.18 *** 0.03 0.08 Free trade 0.43 *** 0.41 *** 0.41 *** 0.55 *** GDP exporter - 0.81 *** - 0.78 *** GDP importer - 0.82 *** - 0.82 *** GDP per cap. exporter - 0.01 0.22 *** 0.03 GDP per cap. importer - 0.00 0.10 *** 0.05 * Island exporter - 0.20 *** 0.12 0.22 *** Island importer - 0.12 *** 0.03 0.06 Area exporter - 0.05 *** 0.18 *** 0.07 *** Area importer - 0.14 *** 0.16 *** 0.08 *** Landlocked exporter - 0.23 *** 0.01 0.15 Landlocked importer - 0.34 *** 0.09 0.01 Constant 35.23 *** 29.31 *** 32.40 *** 27.06 *** Adj. pseudo R 2 0.792 0.930 0.604 0.916 AIC 1.97e+09 n.a. 3.80e+09 3.40e+09 12

(1) (2) (3) (4) Fixed effects Spatial filter BB-estimation BB-estimation (GDP offset) (GDP offset) Observations 18632 18632 18632 18632 ***, **, * denote parameter estimates statistically significantly at 1, 5 and 10 per cent, respectively. Model (3) confronts these findings with the regression outcomes for the second benchmark model, the BB estimation. This method proxies for the endogenous and unobserved MR terms by including exogenous linear approximations based upon bilateral trade costs variables. Provided that the approximation is sufficiently adequate, this specification results in consistent estimates as well (BB, 2009). Once again, GDP has been used as an offset variable, and the model is estimated by Poisson regression. Although some parameter estimates are comparable to the estimates for the first benchmark model (notably for free-trade blocs), the BB estimation results differ quantitatively and sometimes qualitatively from the specification with country-specific effects (Model (1)). On the one hand, the BB specification has an advantage, because it enables us to include country-specific regressors explicitly; on the other hand, the results do not always appear to be satisfactory. Closer inspection of BB estimation, dropping the offset assumption on the product of exporter and importer GDP in Model (4), yields more plausible results, and a better fit. For example, although a negative effect of GDP per capita variables on trade is not uncommon in some specifications (see, for example, Anderson and Marcouiller, 2002), the effect in Model (3) seems to be driven mainly by the offsetting that GDP imposes a GDP elasticity of trade, which empirically is too high. Summarizing, the two benchmark models yield different results. Although some effects may be more plausible in the BB estimation results (for example, the effect of common language), the more traditional specification using country-specific indicator variables results in more plausible parameter estimates for most variables. The disadvantages of this model, though, are the loss of country-specific variables, and a loss of (adjusted) goodness of fit resulting from the loss of degrees of freedom in the model estimation. Results emerging from the spatial filtering estimation of the gravity model, which combines the consistent estimation of the first benchmark model with the flexibility of specification of the second benchmark model, are shown for Model (2) in Table 1. The results presented here 13

are obtained for a rook contiguity 8 spatial weights matrix C, and for a non-parametric, quasi- Poisson estimation, employed in order to cope with overdispersion in the trade flows. With regard to the coefficients of bilateral resistance variables, we note that all are highly significant, and their values are consistent with the ones found for Model (1). Noteworthy is that the variable referring to colonial links has the strongest (and most significant) coefficients out of all models estimated. With regard to the importer- and exporter-specific variables, we are able to identify highly significant and positive coefficients for GDP, while GDP per capita is insignificant in both cases. This result is consistent with the ones for the benchmark models, in which the same variable is statistically significant only when GDP is employed as an offset. The spatial filtering estimation also allows us to estimate significant parameters for the variables identifying the geographical characteristics of importer and exporter countries. The signs obtained are consistent with the ones found for our benchmarks, and show that larger and landlocked countries tend to trade less, while islands tend to import more and export less. Also noteworthy is that only the spatial filtering model is able to find significant effects for landlocked countries (both importing and exporting). Finally, the pseudo-r 2 of the spatial filtering model appears to be very high, because of the high amount of variance explained by the origin- and destination-specific spatial filters, which are also highly significant from a statistical viewpoint (not shown in Table 1). In summary, the proposed spatial filtering approach to the estimation of a gravity model of trade allows identification of the regression parameters related to the bilateral variables, as well as those related to the origin- and destination-specific variables. Moreover, the model has a higher pseudo-r 2 than competing models, and uses a limited number of degrees of freedom. 6. Conclusions Recent contributions to the modelling of bilateral trade have shown the importance of sound theoretical underpinnings for obtaining consistent parameter estimates for the determinants of trade in the gravity model of bilateral trade. This paper addresses the issue of how to achieve empirical consistency without the need to estimate a full general equilibrium system of equations, and without the loss of specification flexibility that results from the use of origin- 8 For rook contiguity, two regions/countries are defined as neighbours if they share a border. For islands, the nearest country was selected as a neighbour. Alternative definitions of proximity based upon, for example, k- nearest neighbours or distance decay, could be tested in order to assess the sensitivity of the model to the spatial specification. 14

and destination-specific indicator variables. We argue that endogeneity of regressors and residuals due to omitted multilateral resistance variables in the traditional gravity model is likely to manifest in the form of spatial autocorrelation (SAC) in both regressors and residuals. By including an origin-specific and a destination-specific spatial filter as additional regressors, spatial filtering estimation of the gravity equation enables us to filter SAC out of the residuals. As a result, the residuals and the regressors are no longer correlated, and standard estimation methods can be applied to obtain consistent parameter estimates for the determinants of bilateral trade. We demonstrate the use of spatial filtering estimation in a Poisson regression of the gravity equation of bilateral trade. The comparison with two benchmark models, which are theoretically consistent in estimation, reveals that spatial filtering yields results that are very comparable to the estimation using country-specific indicator variables. Moreover, spatial filtering estimation does not suffer from the drawbacks of using indicator variables. It allows explicit estimation of the effect of country-specific variables that are potentially important determinants of bilateral trade, such as GDP, per capita GDP and landlockedness. Future research should focus, on the methodological side, on expanding the analyses above to the spatial-filtering network-autocorrelation approach suggested by Chun (2008), and on the empirical side, on exploiting the methodology proposed toward investigating specific research questions in the bilateral trade field. References Anderson, J.E. and E. Van Wincoop (2003): Gravity with Gravitas: A Solution to the Border Puzzle, American Economic Review, 93, pp. 170 192. Anderson, J.E. and E. Van Wincoop (2004): Trade Costs, Journal of Economic Literature, 42, pp. 691 751. Anselin, L. (1988): Spatial Econometrics: Methods and Models. Dordrecht Boston: Kluwer Academic Publishers. Baier, S. and Bergstrand, J.H. (2009): Bonus Vetus OLS: A Simple Method for Approximating International Trade-Cost Effects Using the Gravity Equation, Journal of International Economics, 77, pp. 77 85. Behrens, K., C. Ertur and W. Koch (2007): Dual gravity: Using spatial econometrics to control for multilateral resistance, Ecore Discussion Paper 2007/79. 15

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