Controlling for Heterogeneity in Gravity Models of Trade

WORKING PAPER SERIESControlling for Heterogeneity in Gravity Models of TradeI-Hui ChengHoward J. WallWorking Paper 99-010Ahttp://www.stls.frb.org/research/wp/99-010.htmlFebruary 1999FEDERAL RESERVE BANK OF ST. LOUISResearch Division411 Locust StreetSt. Louis, MO 63102The views expressed are those of the individual authors and do not necessarily reflectofficial positions of the Federal Reserve Bank of St. Louis, the Federal Reserve System,or the Board of Governors.

Controlling for Heterogeneity in Gravity Models of TradeI-Hui ChengBirkbeck College, University of LondonHoward J. WallFederal Reserve Bank of St. LouisFebruary 1999This paper argues that it is necessary to allow for country-pair heterogeneity when using thegravity model to estimate international trade flows. We propose and estimate a fixed-effectsmodel that eliminates the heterogeneity bias inherent in standard methods. Further, we show thatthere is no statistical support for the restrictions necessary to obtain existing empirical models,which are special cases of our model. Because the gravity model has become the ‘workhorse’baseline model for estimating the effects of international integration, this has important empiricalimplications. In particular, our results suggest that standard gravity estimates of the effects ofintegration can differ a great deal from what is obtained when heterogeneity is accounted for.(JEL F15, F17)Corresponding author: Howard J. Wall, Research Division, Federal Reserve Bank of St.Louis, P.O. Box 442, St. Louis, MO 63166-0442, United StatesE-mail: wall@stls.frb.org; Phone: (314)444-8533; Fax: (314)444-8731We would like to thank Ron Smith for his insightful and helpful suggestions. We are alsograteful for comments from the participants at the Midwest International Economics Conferenceat Purdue University, May 1999. The views expressed are those of the authors and do notnecessarily represent official positions of the Federal Reserve Bank of St. Louis, nor of theFederal Reserve System.

Controlling for Heterogeneity in Gravity Models of TradeI-Hui Cheng and Howard J. Wall1. IntroductionStarting in the 1860s when H. Carey first applied Newtonian Physics to the study ofhuman behavior, the so-called “gravity equation” has been widely used in the social sciences.More recently, gravity model studies have achieved empirical success in explaining various typesof inter-regional and international flows, including labor migration, commuting, customers,hospital patients, and international trade. 1 The widespread use of gravity equations is despite thefact that they have tended to lack strong theoretical bases.The gravity model of international trade was developed independently by Tinbergen(1962) and Pöyhönen (1963). In its basic form, the amount of trade between two countries isassumed to be increasing in their sizes, as measured by their national incomes, and decreasing inthe cost of transport between them, as measured by the distance between their economic centers. 2Following this work, Linnemann (1966) included population as an additional measure of countrysize, employing what we will call the augmented gravity model. 3It is also common to insteadspecify the augmented model using per capita income, which captures the same effects. 4Whichever specification of the augmented model is used, the purpose is to allow for non-1 See Sen and Smith (1995) for a survey.2 For recent examples of the basic gravity model see McCallum (1995), Helliwell (1996), and Boisso andFerrantino (1997).3 For recent uses of the augmented gravity model with population see Oguledo and MacPhee (1994), Boisso andFerrantino (1997), and Bayoumi and Eichengreen (1997).4 Examples of the augmented model with per capita income include Sanso, Cuairan, and Sanz (1993), Frankeland Wei (1998), Frankel, Stein, and Wei (1995,1998), Eichengreen and Irwin (1998).1

homothetic preferences in the importing country, and to proxy for the capital/labor ratio in theexporting country (Bergstrand, 1989).The gravity model has been used widely as a baseline model for estimating the impact ofa variety of policy issues, such as regional trading groups, political blocs, patent rights, andvarious trade distortions. 5Typically, these events and policies are modeled as deviations fromthe volume of trade predicted by the baseline gravity model, and, in the case of regionalintegration, are captured by dummy variables. The recent popularity of the gravity model ishighlighted by Eichengreen and Irwin (1997, p.33) who call the it the “workhorse of empiricalstudies of (regional integration) to the virtual exclusion of other approaches.” This is despite thefact that, as Deardorff (1984) points out, most early papers were ad hoc rather than being basedon theoretical foundations. Exceptions to this include Anderson (1979), Bergstrand (1985),Hummels and Levinsohn (1995), Deardorff (1998), and Feenstra, Markusen, and Rose (1998),whose models are consistent with the gravity model. See also Evenett and Keller (1998) who,along with Deardorff (1998), evaluate the usefulness of gravity models in testing alternativetheoretical models of trade. The recent flurry of theoretical work has led Frankel (1998, p.2) tosay that the gravity equation has “gone from an embarrassment of poverty of theoreticalfoundations to an embarrassment of riches.”The perceived empirical success of the gravity model has come without a great deal ofanalysis regarding its econometric properties, as its empirical power has usually been stated5 See Aitken (1973), Brada and Mendez (1983), Bikker (1987), Sanso, Cuairan, and Sanz (1993), Oguledo andMacPhee (1994), McCallum (1995), Helliwell (1996), Wei and Frankel (1997), Bayoumi and Eichengreen(1997), Mátyás (1997), Frankel and Wei (1998), Frankel, Stein, and Wei (1998), and Smith (1999).2

simply on the basis of goodness of fit; i.e. a relatively high2R . 6The lack of attention paid to theempirical properties of the model is despite the fact that the strength of any baseline model lies inthe accuracy of its estimates. The aim of this paper is to begin to fill the gap regarding theempirical estimation of gravity models of trade. In particular, we demonstrate that standardmethods for estimating the gravity model produce biased estimates, tending to overestimate tradebetween low-trade countries, and to underestimate it between high-trade countries. We arguethat the primary source of this bias is the failure of standard methods to account for the pairwiseheterogeneity of bilateral trade relationships.The solution we propose uses simple panel data methods to allow for the intercepts of thegravity equation to be specific to each trading pair. We demonstrate how with this empiricalmodel the correlation between the residuals and the volume of trade disappears. To illustrate theempirical significance of our findings, we apply our model to the question of the effects ofregional integration on trade volumes. We find that standard methods find a strong negativerelationship between membership in a trading bloc and intra-bloc trade, whereas thiscounterintuitive finding is eliminated when heterogeneity is controlled for.Section 2 briefly sets out the various statistical models we examine. Section 3 presentsstandard empirical results for the basic and augmented gravity models, and illustrates the inherentestimation bias. In Section 4 we offer a solution to this, namely using a panel with fixed effectsto estimate the bilateral trade relationship. In Section 5 we compare our model to alternatives6 See Sanso, Cuairan, and Sanz (1993) for an examination of the predictive power of various specifications of theaugmented gravity model. Also see Oguledo and MacPhee (1994) for a survey of pre-1990 empirical results.3

which also control for heterogeneity. Section 6 illustrates the importance of controlling forheterogeneity when estimating the effects of trade blocs. Concluding remarks are provided inSection 7.2. A Statistical OverviewThis section briefly sets out the various forms of gravity models that have beenconstructed to estimate bilateral trade flows. These models can be considered as restrictedversions of a general gravity model, which has a log-linear specification, 7 but places norestrictions on the parameters. In the general model, the volume of trade between countries i andj in year t can be characterized by= α + αt+ α ′ij+ ijtZijt+ ε , t = 1,…,T; (1)ln Xijt 0 ijtwhereXijtis exports from country i to country j in year t, and Z′ijt= [zit z jt...] the 1 × k rowvector of gravity variables (GDP, population, and distance). The intercept has three parts, onewhich is common to all years and country pairs, α0, one which is specific to year t and commonto all pairs,αt, and one which is specific to the country pairs and common to all years,αij. Thedisturbance term εijtis assumed to be normally distributed with zero mean and constant variance2for all observations, i.e. εijt~ IN(0,σ t) , E ( εijt, εij′t) = 0 and E ( εijt, εijt −1 ) = 0 . It is also assumedthat the disturbances are pairwise uncorrelated.Obviously, because (1) has only one observation, it is not useful for estimation unless7 Sanso, Cuairan, and Sanz (1993) conclude that the log-linear specification, while not optimal, is a fair and readyapproximation of the optimal form.4

independent variables are assumed to be correlated withwhich can be estimated using OLS.α ij , and is a classical regression modelBayoumi and Eichengreen (1997) and Mátyás (1997) have also proposed models tohandle country-pair heterogeneity, each of which can be modeled as a restricted version of the FEmodel. In the Bayoumi and Eichengreen (BE) model the differences in the dependent andindependent variables are used to eliminate the fixed variables, including the country-pairdummies and distance. Specifically,∆lnijt 0 tijt ijtX = γ + γ + ∆Z + µ , t = 1, ..., T; (BE)ZKHUH LVWKHGLIIHUHQFHRSHUDWRUDQG γ0+ γt= αt− αt−1. In this model the intercept has twoparts: γ0is the change in the period-specific effect that is common across years, and γtis thechange that is specific to year t. As is well known, when there are no time dummies, such adifferencing model should yield results identical to a model with dummy variables to control forfixed effects. However, with time dummies it is necessary to impose restrictions on the timeeffects so as to avoid collinearity, which in turn makes the BE model a restricted form of the FEmodel. If the collinearity restriction is that the first time dummy in the BE model is equal tozero, this is equivalent to restricting the common component of the change in the period-specificeffects as equal to the difference in the first two period-specific effects, i.e. γ = α − . If0 2α1instead the collinearity restriction is that the sum of the time dummies in the BE model is zero,this is equivalent to restricting the common component as equal to the difference between thefirst and last time dummies, i.e. γ = α − .0 Tα 16

Mátyás (1997) proposesln Xijt= α0+ αt+ θi+ ωj+ijt ijt′ Z + ε , t = 1, ..., T; (M)as the correct specification of the gravity model, where the country-specific effect when a countryis an exporter is θi, and when it is an importer isωj. Note also that in this specification,distance, contiguity, and language are eliminated because they are fixed over time, even thoughthey are not collinear with the country-specific effects. This model is a special case of the FEmodel in that it imposes arbitrary restrictions on the country-pair effects; i.e. becauseα = θ + ω and α = θ + ω ; it must also be true that α = α − ω + θ . These cross-pairijijikikrestrictions do not change the coefficient estimates, but instead lead to odd residuals, and greatlyinaccurate predictions of trade flows.ijikkj3. An Overview of the Standard Empirical ResultsThis section presents regression results for the basic and the augmented versions of thestandard empirical models, CS and PCS. The data set is a balanced panel with 2110observations, and includes countries with many different levels of economic development andperformance for the period 1991-1995. It includes observations on exports from 22 countries to116 destination countries, although we do not have data on all possible pairs. Descriptions of thedata and their sources are provided in the Data Appendix.3.1. Single-year cross-section dataIn the augmented version of the gravity model, the gravity variables are the countries’7

GDPs, their populations, and the distance between them. Thus, the augmented CS model (CSa)assumes that in a given year trade flows from exporting country i to importing country j can beestimated using: 8ln Xij= α + β1 lnYi+ β2lnYj+ β3ln Ni+ β4ln Nj+ δ1ln Dij+ δ2Cij+ λLij+ εij; (2)where Y i andY j are the two countries’ GDPs;Nianddistance between their economic centers (their capital cities);N are their populations, D is thejC ij is a contiguity dummy; and Lijis a common-language dummy. As trade flows are expected to be positively related to nationalincomes, and negatively related to distance, it is expected that β 1, β 2 , and δ 2 are positive, andthat δ 1 is negative. Also, estimation typically yields a negative sign for β 3 , which wouldindicate that exported goods tend to be capital-intensive. It is also common to obtain a negativesign for β 4 , which would indicate that traded goods tend to have income-elastic demands.Finally, because Lijis meant to capture cultural and historical similarities between the tradingSDLUVZKLFKDUHWKRXJKWWRLQFUHDVHWKHYROXPHRIWUDGH LVH[SHFWHGWREHSRVLWLYHThe basic version of the gravity model does not include the populations of the twocountries, so it can be viewed as a special case of the augmented model in which the coefficientson population are restricted to zero. Thus, the basic CS model (CSb) assumes that bilateral tradecan be estimated with the following regression:ln Xij= α + β1 lnYi+ β2lnYj+ δ1ln Dij+ δ2Cij+ λLij+ εij.(3)The expected signs for the coefficients are as in the augmented model.ij8 Note that because ln ( per capita incomei ) = ln Yi − ln N , the regression could be suitably rearrangedito instead obtain the augmented model with per capita income.8

ln Xijt= α0+ αt+ β1lnYit+ β2lnYjt+ β3ln Nit+ β4ln Njt+ δ1ln Dij+ δ2Cij+ λLij+ εijt; (4)where α 0 is the portion of the intercept that is common to all years and trading pairs, anddenotes the year-specific effect common to all trading pairs. Note that we omit the dummy for1991 so as to avoid collinearity. We suppress the regression equation for the basic version of themodel (PCSb), as it is the same as (4) except for the restriction that β = β 0 . The expected3 4 =signs for the coefficients are the same as for the CS models, except that the PCS models havetime dummies to consider. We take the time dummies as an indicator of the extent of“globalization”, which we define as the common trend towards greater real trading volumes,independent of the sizes of the economies.The regression results for PCSa and PCSb are reported in the first two columns of Table2. Unsurprisingly, the results are similar to those from the single-year cross-sections. The CSand PCS models yield roughly the same elasticities on GDPs and distance, and have roughly thesame predictive power as measured by2R . Note, however, that because of the large number ofobservations relative to the CS models, the coefficients on the countries’ populations and oncontiguity are statistically significant. Comparing the two versions of the PCS model, althoughPCSa and PCSb yield very similar results, a likelihood ratio test rejects the null hypothesis thatαtthey are statistically the same. 10We therefore conclude that the augmented version of the gravitymodel is preferred to the basic model when using a pooled cross-section.According to the estimates of the preferred PCSa model: (i) a 10% rise in a country’s10 This is with a critical value of 5.99 at the 5% level, and χ 2 (2) = 14.94.10

GDP should be associated with a 6.2% rise in its exports and an 8.5% rise in its imports, all elseconstant; (ii) exports tend to be labor-intensive and income elastic (luxury goods), as indicated byWKHSRVLWLYHVLJQIRU 3DQGWKHQHJDWLYHVLJQIRU 4; and (iii) a country will export 82% less to amarket that is twice as distant as another otherwise-identical market, 20% more to a country thatis contiguous, and 70% more to a country with the same first language. Finally, we take the factthat our time dummies are not statistically different from zero to mean that globalization, asdefined above, was not an important factor in increasing trading volumes during the period.The general conclusion from our estimation of standard gravity models is that they yieldstable results that do not differ a great deal over the sample period, nor between the basic andaugmented versions. We also conclude that the augmented version is preferred statistically,although the basic version provides predictions of trade volumes that are nearly as accurate.Finally, the estimates we obtain are not greatly out of line with those obtained previously in theliterature.One should be cautious before concluding that there are no empirical problems with thesestandard methods. This is clear from the upper-left panel of Figure 1, which plots the residualsfor the PCSa model. The strong positive relationship between the residuals and the level ofexports indicates that the PCSa model tends to underestimate the level of trade when the actuallevel is high, and overestimates it when the actual level is low. To our knowledge, this bias hasnot been recognized in the literature. Because the gravity model is used to establish baselinelevels of trade, it is important to have unbiased estimates the coefficients on the gravity variables,11

even if one is not interested in these coefficients themselves. It is difficult to argue that a modelcan be useful for establishing a baseline when it yields such obviously biased predictions.4. The Model with Pairwise Heterogeneitya. The modelAs we describe in the previous section, standard cross-section estimates of the gravitymodel yield biased estimates of the volume of bilateral trade. One possible source of this bias isthat heterogeneity is not allowed for by the regression equations. With such heterogeneity acountry may export different amounts to two countries, even though the two export markets havethe same GDPs and are equidistant from the exporter. This can be because there can behistorical, cultural, ethnic, political, or geographic factors that affect the level of trade, and arecorrelated with the gravity variables (GDP, population, distance). If so, then estimates that donot account for these factors will suffer from heterogeneity bias.Various studies have to some extent tried to control for this by including things such aswhether trading partners share a common language, have had a colonial history, are in militaryalliance, etc. However, cultural, historical, and political factors are often difficult to observe, letalone quantify. This is why we will control for these factors using a simple fixed-effects modelthat assumes that there are fixed pair-specific factors that may be correlated with levels ofbilateral trade and with the right-hand-side variables.We assume that the gravity equation for a country pair may have a unique intercept, and12

that it may be different for each direction of trade (i.e.αij≠ αji). However, we retain theassumptions of the PCS model that the slope coefficients are constant over time and acrosstrading pairs. Our specification of the augmented gravity model with fixed effects (FEa) is:whereln Xijt= αij+ αt+ β1 lnYit+ β2lnYjt+ β3ln Nit+ β4ln Njt+ εijt;(5)α ij is the specific “country-pair” effect between the trading partners. The basic version(FEb) is the same as this, except for the constraint that β = β 0 . The country-pair intercepts3 4 =include the effects of all omitted variables that are cross-sectionally specific but remain constantover time, such as distance, contiguity, language, culture, etc. Using the pooled data describedabove, we have 422 country-pair intercepts.Because there is a long-standing problem with determining the appropriate measure ofeconomic distance so as to capture transportation and information costs, an added benefit of thefixed effects model is that it eliminates the need to include distance in the regression. The mostcommon method for handling distance is to do as we have above and simply measure it betweenthe economic centers (assumed to be the capital cities) of the two countries. There are obviousproblems with this, such as the implicit assumptions that overland transport costs are the same asthose over sea, and that all overland/oversea distances are equally costly. To provide just oneobvious example, Los Angeles is about 1300 kms further from Tokyo than is Moscow, but it isdifficult to believe that the economic distance between Tokyo and Los Angeles is not muchlower than that between Tokyo and Moscow. Our fixed-effects approach eliminates the need toinclude a distance variable, as it controls for all variables that do not change over time.13

Another difficulty with standard measures of economic distance is the simple assumptionthat the capital city is a useful proxy for the economic center. While this may be useful for smallcountries with one major city, it is wide of the mark for countries like Canada and the US, whichhave major cities thousands of miles apart on different oceans, and which serve as centers fortrade with completely different countries. By using Washington, DC or Ottawa to measuredistance between the US or Canada and its Pacific trading partners is to overstate distance by theentire breadth of the North American continent. As the US has the highest GDP and the highestvolume of trade, the mis-measure of economic distance can bias the estimation of the coefficientson the other variables in the gravity model. 11Another advantage of our approach is that it removes the problem of controlling forcontiguity. Although it is clearly important, as a great deal of the trade can occur from peoplecrossing the border to make everyday purchases, it is accounted for only sometimes. Even whenit is accounted for with a dummy variable as we do above, it still assumes that all contiguity isequivalent in terms of its effect on trade. Considering that Canada and the US, China and Russia,and Argentina and Chile are all equivalently contiguous pairs, this is difficult to abide by.b. The resultsThe middle columns of Table 2 report the estimation results for the augmented and basicversions of the fixed-effects model (FEa and FEb). Note that for comparison with the pooled11 As a practical matter this mis-measure of distance is magnified by the fact that data sets tend to haveproportionally more data on US trade. For example, in our data set 700 of the 2110 observations (33%) have theUS as either the importing or exporting country.14

cross-section results, the year dummies are measured relative to that of 1991. Also, the estimatesof the country-pair intercepts are omitted for space considerations. The difference between theaugmented and basic versions of the FE model is not glaring, but is nonetheless clear statistically.A likelihood ratio test rejects the null hypothesis that the two models are statistically the same. 12In other words, it rejects the restriction that the coefficients on population are zero (as in FEb).We therefore conclude that FEa is the preferred version of the FE model.According to the results for the preferred FEa model: (i) a 10% rise in a country’s GDPshould be associated with a 2.9% rise in exports and a 5.2% rise in imports; (ii) exports tend tobe labor-intensive, as indicated by a positive sign of origin population, and income-elastic(normal non-luxury goods), as indicated by the negative sign on destination population; and (iii)globalization increased the real volume of trade by nearly 20% between 1991 and 1995.Comparing the results of the FEa and PCSa models, allowing for trading-pairheterogeneity, as in the FEa model, lowers the estimated income elasticities of trade, greatlyincreases the absolute value of the coefficients on the countries’ populations, and greatly increasethe estimated role of globalization. It is obvious from the results that restricting the country-paireffects to zero, as does the PCSa model, has significant effects on the results, and this is easilyconfirmed by a likelihood ratio test. Further, as shown by the upper-right panel of Figure 1, thereis no obvious correlation between the residuals and the log of exports, indicating that the FEamodel does not suffer from the estimation bias exhibited by the PCSa model. It is also obvious12 This is with a critical value of 5.99 at the 5% level, and χ 2 (2) = 46.42.15

from Figure 1 that the residuals from the FEa model tend to be much smaller than those from thePCSa model.To summarize, because the PCSa model is a restricted form of the FEa model, and therestrictions are not supported statistically, we conclude that the FEa model is the preferredspecification of the gravity model. Also, the FEa model does not exhibit the obviousheterogeneity bias of the PCSa model, and is preferred on the basis of traditional measures ofgoodness of fit in that it provides a higher2R and a lower sum of squared residuals. Notethough that this improved statistical performance arises from the FEa model having 422 moreindependent variables than does the PCSa model. Nonetheless, on the basis of having smallervalues for the Akaike Information Criteria and the Amemiya Probability Criteria, which havemore severe penalties for the number of parameters, the FEa model is still easily preferred.In short, there is no statistical support for imposing the parameter restrictions required bythe standard procedures for estimating the gravity model of trade. In the absence of anyeconomic arguments for believing that the intercepts of the gravity equation are the same acrosstrading pairs, we conclude that the fixed effects model is the more appropriate specification.Oddly, Wei and Frankel (1997, p.125) reject the inclusion of country-pair dummies apriori on the basis that doing so would undermine their efforts at estimating the effects ofvariables that are constant over the sample period. Presumably their worry is that because thesevariables are subsumed into the country-pair effects they are hidden from analysis. This isunfounded because the effects of these variables are easily estimated by regressing them on the16

country-pair effects from the FE model. Specifically, where the estimates of the 422 country-paireffects are denoted asαˆij, and including the log of distance and the contiguity and languagedummies as independent variables, we obtainαˆij= 4.45 −1.04lnD(2.57) (0.296)ij+ 1.01C(1.19)ij− 0.51L(0.63)ij.The numbers in parentheses are standard errors, and the2R = 0.049. According to these results,only the distance variable is a statistically significant determinant of the country-pair effects.Contiguity and a common language do not therefore appear to be important determinants of thevolume of bilateral trade. Further, the low2R indicates that very little of the country-pair effectsare explained by the variables traditionally included in the standard cross-section models. Notethat these estimates are quite different from those obtained from the PCSa model, in whichestimates of the effects of time-invariant factors suffer from the same heterogeneity bias as thetime-variant factors. So, far from undermining estimation efforts, it is instead necessary tocontrol for country-pair heterogeneity to obtain unbiased estimates of the importance of timeinvariantfactors. Curiously, though, the coefficient on the distance variable is practically thesame as obtained from the PCSa model. It is not possible to tell if this is a coincidence, or if it isdue to the distance variable being uncorrelated with the other independent variables. However,because the distance variable is so suspect, we would not want to push this result in either case.5. Alternatives with HeterogeneityAs discussed earlier, two other papers have proposed empirical models for dealing with17

heterogeneity, and these models can be regarded as restricted forms of the FE model. Usingmany of the same arguments we use to argue for the FE model, Bayoumi and Eichengreen (1997)propose estimating the gravity equation in first differences. This method of handlingheterogeneity is nearly identical to ours, except in the restrictions that it imposes on the effects oftime. An additional difference between their model and our FEa model is that they use theirindependent variables are the products of the GDPs and populations, therefore imposing theadditional restrictions that β1= β2and β3= β4. Because we wish to focus on their treatment ofheterogeneity only, we will not estimate the model under those restrictions. However, as is clearfrom the results, these additional restrictions are easily rejected statistically.Taking the time difference of (5), the model that we will estimate (BEa) is∆ ln Xijt= γ0+ γt+ β1∆lnYit+ β2∆lnYjt+ β3∆lnNit+ β4∆lnNjt+ µijt; (6)where the intercept is as defined in Section 2, γ0+ γt= αt− αt−1. To prevent collinearity, we setthe time dummy for 1992 equal to zero, meaning that other time dummies are measured relativeto it. In terms of the more-general FEa model, this is equivalent to restricting the commoncomponent of the change in the period-specific effects as equal to the difference in the first twoperiod-specific effects, i.e. γ = α − .13 The empirical results are presented in Table 2.0 2α1The results for the FEa and BEa models are very similar in terms of the signs and order ofmagnitude of the coefficients. Also, as illustrated by the lower-left panel of Figure 1, the BEamodel is not subject to the obviously biased residuals of the PCSa model. Nonetheless, the FEa13 The alternative assumption that the sum of the year dummies is zero means that γ0= αT − α1, and yields thesame results except for the time dummies and the constant.18

and BEa results differ enough to reject the restrictions needed to obtain BEa model. This isconfirmed by a likelihood ratio test. Further, in terms of fitting the data, the FEa model ispreferred in terms of the sum of squared residuals, the Akaike Information Criteria, and theAmemiya Probability Criteria. As for the further restrictions in Bayoumi and Eichengreen (1997)that β1= β2and β3= β4, our results indicate that this would clearly have significant effects onthe estimates, and therefore should not be imposed.The other alternative to the FEa model is due to Mátyás (1997), who proposes using thespecificationln Xijt= α0+ αt+ θi+ ωj+ β1lnYit+ β2lnYjt+ β3ln Nit+ β4ln Njt+ εijt; (7)where the effect when a country is an exporter is θi, and when it is an importer isωj. Thismodel is a restricted form of the FEb model in that it imposes arbitrary cross-pair restrictions onthe country-pair effects; α = α − ω + θ .ijikkjThe empirical results, summarized by the last column of Table 2, show that the coefficientsare identical to those from the FEa model, although their standard errors are very large. In fact,they are large enough to reject the statistical significance of all but the coefficient on destinationGDP. The lower-right panel of Figure 1 plots the residuals of this model against the log ofexports, and shows that this model has very peculiar results. The wide dispersion of the residualsalso indicates a very poor fit relative to FEa. Further, a likelihood ratio test easily rejects the nullhypotheses that the cross-pair restrictions do not change the results in a statistically importantway. So, although this model eliminates the bias in the estimates of the coefficients on the19

independent variables, it has little predictive power.6. Implications for Estimating the Effects of IntegrationAs we discuss in the Introduction, the gravity model has become the primary tool forestimating the effects of regional integration on trade volumes. Up to this point, we have omittedintegration variables in order to focus on the importance of controlling for country-pairheterogeneity when estimating gravity models. However, now that we have established that theFEa model is statistically preferred to standard cross-sectional methods, we introduce integrationinto our model and demonstrate the striking effect that heterogeneity bias has on the results. Wewould also like to alleviate the legitimate concern that the heterogeneity bias we detected abovewas due to our implicit assumption that regional integration is uncorrelated with the independentvariables.The most common and straightforward method for estimating the effects of integration ina gravity model is to include dummy variables for each integration regime in place during thesample period. Each of these dummies takes the value of 1 for each observation for which thetwo countries are members of the regime, with the expectation that the coefficients on thesedummies are positive. We include three such dummy variables in our model, one each for theEuropean trading bloc, the North American trading bloc, and the South American trading bloc(MERCOSUR).Although there has been some deepening of trade integration in the European bloc, the20

primary change over the period was an expansion in the number of countries covered under thecustoms union. The twelve countries of the European Community (EC) renamed themselves theEuropean Union (EU) in 1992, but this had relatively little effect on internal trade policy, as itwas already nearly unfettered under the EC. Expansion of the bloc came in 1994 with theEuropean Economic Area (EEA), which extended the free trade zone to include Austria, Iceland,Finland, Norway, and Sweden. To capture the effect of this trading bloc, our European blocdummy variable takes the value of 1 when trade is between members of the EC or EU for 1991-93, and between members of the EEA for 1994-95.The North American trading bloc included only Canada and the United States for 1991-93, under the Canada-US Trade Agreement of 1988. The North American Free Trade Agreement(NAFTA) expanded the free trade zone in 1994 to include Mexico. For present purposes, wewill ignore NAFTA’s relatively mild deepening of US-Canada integration, and focus instead onit as an extension of the free trade bloc to Mexico. To capture the effects of North Americanintegration, our North American bloc dummy takes the value of 1 for trade between the US andCanada for 1991-95, and between Mexico, Canada, and the US for 1994-95.The third significant trade bloc during the period was MERCOSUR, which came intoforce in 1995, reducing trade barriers between Argentina, Brazil, Paraguay, and Uruguay. OurMERCOSUR dummy takes the value of 1 for trade between any two of these countries in 1995.We include these three trade bloc dummies in the PCSa and FEa models, and report theempirical results in Table 3. Note that inclusion of these dummies makes little difference for the21

PCSa model. Nonetheless, a likelihood ratio test rejects the null hypotheses that including thetrade bloc dummies in the PCSa model does not alter the results to a statistically significantextent. 14The results for the FEa model are also not dramatically different when the trade blocdummies are included. In fact, the null hypothesis that the inclusion of these variables has nostatistically significant effect on the results cannot be rejected. 15The dramatic change in the empirical results is in the comparison of the FEa and PCSamodels. The negative effects for all three trade blocs in the PCSa model are certainly contrary toexpectations. Not only are the coefficients on the European bloc and MERCOSUR dummiesstatistically different from zero, they are also very large. The results suggest that membership inthe European trade bloc decreases trade with another bloc member by roughly the same as wouldoccur if the distance between the countries doubled. The predicted decrease in trade due tomembership in MERCOSUR is three time this.Although these predictions have interesting implications regarding the future of the worldtrading system, they do not hold up when the estimation allows for heterogeneity. Using the FEamodel, all three trade blocs have positive effects on the volume of trade, although none arestatistically different from zero. However, this is likely due to the simplistic nature of thedummies, which do not account for trade diversion effects, rather than to the absence of realeffects (Cheng and Wall, 1999). For our present purposes though, the dummies we use are usefulto illustrate how the FEb model at least brings the results into the realm of believability.14 7KLVLVZLWKDFULWLFDOYDOXHRIDWWKHOHYHODQG 2 (3) = 79.38.15 7KLVLVZLWKDFULWLFDOYDOXHRIDWWKHOHYHODQG 2 (3) = 4.12.22

7. ConclusionsThe objective of this paper is to argue that heterogeneity needs to be allowed for whenusing the gravity model to estimate bilateral trade flows. Our empirical analysis indicates thatstandard methods for estimating gravity models of trade suffer from heterogeneity bias due toomitted or misspecified variables. To address the problem we adopt a two-way fixed-effectsmodel in which country-pair and period dummies are used to reflect the bilateral relationshipbetween trading partners. The fixed effects capture those factors such as physical distance, thelength of border (or contiguity), history, culture, language, etc., that are constant over the span ofthe data, and which are correlated with the volume of bilateral trade.We show that existing empirical models are special cases of our model, and that therestrictions necessary to obtain these special cases are not supported statistically. We concludethat the preferred specification of the gravity model takes into account the heterogeneity of thecountry pairs using country-pair dummies, and includes the populations of the two countries. Asthe gravity model has become the “workhorse” of empirical work on the effects of integration,we also check the importance of allowing for heterogeneity when doing this work. Our resultssuggest that the results when heterogeneity is allowed for can differ wildly from when it is not.23

Data Appendix1. Definitions of variablesVolume of Exports, measured in millions of US dollars, from the International Monetary Fund’sDirection of Trade Statistics. Bilateral Exchange Rates are from the IMF’s InternationalFinancial Statistics. Both are downloaded from Datastream’s IMF database, and deflatedusing the $-deflator from the World Tables 97 CD-ROM.Gross Domestic Product is in millions of 1987 US dollars, and Population is in thousands ofinhabitants. Both are from the World Bank’s World Tables 97 CD-ROM.Distance, expressed in kilometers, is the distance between capital cities, obtained from JohnHaveman’s web site at ftp://intrepid.mgmt.purdue.edu/pub/Trade.Data/dist.txt, and fromhttp://www.indo.com/distance/.Contiguity is equal to 1 if two trading partners share common border.Common Language is equal to 1 if two trading partners share a common first language.European Bloc is equal to 1 when both countries are members of the EC for 1991, the EU, for1992-93, or the EEA for 1994-95.North American Bloc is equal to 1 for Canada-US trade for all years, and Canada-Mexico andUS-Mexico trade for 1994-95.MERCOSUR is equal to 1 in 1995 for trade between Argentina, Brazil, Paraguay, and Uruguay.2. Countries included in data set22 Exporters: Argentina, Australia, Canada, China, Finland, France, Hong Kong, Italy, Japan,Kenya, South Korea, Malaysia, The Netherlands, Philippines, Portugal, Singapore, Spain,Sweden, Switzerland, Thailand, United Kingdom, United States116 Importers: Albania, Algeria, Angola, Argentina, Australia, Austria, Bahamas, Bahrain,Bangladesh, Barbados, Belgium, Bolivia, Botswana, Bulgaria, Burkina Faso, Burundi, Brazil,Brunei, Cameroon, Canada, Central African Republic, Chad, Chile, China, Colombia, Congo,Costa Rica, Czech Republic, Denmark, Djibouti, Dominican Republic, Ecuador, Egypt, ElSalvador, Ethiopia, Finland, France, Gabon, Gambia, Germany, Ghana, Greece, Guinea, Guinea-Bissau, Guyana, Haiti, Honduras, Hong Kong, Hungary, India, Indonesia, Ireland, Israel, Italy,Jamaica, Japan, Kenya, S. Korea, Kuwait, Lebanon, Luxembourg, Macao, Madagascar, Malawi,Malaysia, Mauritania, Mexico, Moldova, Mongolia, Morocco, Mozambique, Namibia,Netherlands, New Zealand, Nicaragua, Nigeria, Norway, Oman, Pakistan, Panama, Papua NewGuinea, Paraguay, Peru, Philippines, Poland, Portugal, Qatar, Romania, Russia, Rwanda, SaudiArabia, Senegal, Sierra Leone, Singapore, Solomon Islands, South Africa, Spain, Sri Lanka,Suriname, Swaziland, Sweden, Switzerland, Syria, Tanzania, Thailand, Togo, Tunisia, Turkey,Uganda, United Kingdom, United States, Uruguay, Venezuela, Yemen, Zambia, Zimbabwe24

ReferencesAitken, N. D., 1973, “The Effect of the EEC and EFTA on European Trade: A Temporal Cross-Section Analysis,” American Economic Review, 63, 5, 881-892.Anderson, J. E., 1979, “A Theoretical Foundation for the Gravity Equation,” American EconomicReview, 69, 1, 106-116.Bayoumi, T. and B. Eichengreen, 1997, “Is Regionalism Simply a Diversion? Evidence from theEvolution of the EC and EFTA,” in T. Ito and A. O. Krueger, Eds., Regionalism versusMultilateral Trade Arrangements, University of Chicago Press.Bergstrand, J. H., 1985, “The Gravity Equation in International Trade: Some MicroeconomicFoundations and Empirical Evidence,” Review of Economics and Statistics, 67, 474-481.Bergstrand, J. H., 1989, “The Generalized Gravity Equation, Monopolistic Competition, and theFactor-Proportions Theory of International Trade,” Review of Economics and Statistics, 71,143-153.Bikker, J. A., 1987, “An International Trade Flow Model with Substitution: An Extension of theGravity Model,” Kyklos, 40, 315-337.Boisso, D. and M. Ferrantino, 1997, “Economic Distance, Cultural Distance, and Openness inInternational Trade: Empirical Puzzles,” Journal of Economic Integration, 12, 456-484.Brada, J. C. and J. A. Mendez, 1983, “Regional Economic Integration and the Volume of Intra-Regional Trade: A Comparison of Developed and Developing Country Experience,” Kyklos,36, 589-603.Cheng, I.H. and H.J. Wall, 1999, “Estimating the Effects of Regional Integration on TradeVolumes,” working paper.Deardorff, A. V., 1984, “Testing Trade Theories and Predicting Trade flows,” in R. W. Jones andP. B. Kenen, Eds., Handbook of International Economics, Vol. I, Elsevier.Deardorff, A. V., 1998, “Determinants of Bilateral Trade: Does Gravity Work in a NeoclassicalWorld?” in J. A. Frankel, Ed., The Regionalization of the World Economy, University ofChicago Press.Eichengreen, B. and D. A. Irwin, 1998, “The Role of History in Bilateral Trade Flows,” in J. A.Frankel, Ed., The Regionalization of the World Economy, University of Chicago Press.Evenett, S. J. and W. Keller, 1998, “On Theories Explaining the Success of the GravityEquation,” NBER Working Paper 6529.25

Feenstra. R. C., J.A. Markusen, and A.K. Rose, 1998, “Understanding the Home Market Effectand the Gravity Equation: The Role of Differentiating Goods,” NBER Working Paper 6804.Frankel, F., Stein, E. and S. Wei, 1995, “Trading Blocs and Americas: the Natural, the Unnatural,and the Super-natural,” Journal of Development Economics, 47, 61-95.Frankel, F., Stein, E. and S. Wei, 1998, “Continental Trading Blocs: Are they Natural orSupernatural?,” in J.A. Frankel, Ed., The Regionalization of the World Economy, Universityof Chicago Press.Frankel, F. and S. Wei, 1998, “Regionalization of World Trade and Currencies,” in J. A. Frankel,Ed., The Regionalization of the World Economy, University of Chicago Press.Helliwell, J., 1996, “Do National Borders Matter for Quebec’s Trade?” Canadian Journal ofEconomics, 29, 507-522.Hummels, D. and J. Levinsohn, 1995, “Monopolistic Competition and International Trade:Reconsidering the Evidence,” Quarterly Journal of Economics, 110, 799-836.Linnemann, H., 1966, An Econometric Study of International Trade Flows, North-Holland.McCallum, J., “National Borders Matter: Canada-U.S. Regional Trade Patterns,” AmericanEconomic Review, 85, 615-623.Mátyás, L., 1997, “Proper Econometric Specification of the Gravity Model,” The WorldEconomy, 20, 363-368.Oguledo, V. I. and C. R. MacPhee, 1994, “Gravity Model: A Reformulation and an Applicationto Discriminatory Trade Arrangements,” Applied Economics, 40, 315-337.Pöyhönen, P., 1963, “A Tentative Model for the Volume of Trade Between Countries,”Weltwirtschaftliches Archive, 90, 93-100.Sanso, M., R. Cuairan, and F. Sanz, 1993, “Bilateral Trade Flows, the Gravity Equation, andFunctional Form,” Review of Economics and Statistics, 75, 266-275.Sen, A. and T. E. Smith, 1995, Gravity Models of Spatial Interaction Behavior, Springer-Verlag.Smith. P.J., 1999, “Are Weak Patent Rights a Barrier to US Exports?” Journal of InternationalEconomics, forthcoming.Tinbergen, J., 1962, Shaping the World Economy - Suggestions for an International EconomicPolicy, The Twentieth Century Fund.Wei, S.J. and J.A. Frankel, 1997, “Open versus Closed Trading Blocs,” in T. Ito and A. Krueger,Eds., Regionalism versus Multilateral Trade Arrangements, University of Chicago Press.26

Table 1: Regression results for single-year cross-section; augmented and basic versions; 1991-95dependent variable = log of exports1991 1992 1993 1994 1995CSa CSb CSa CSb CSa CSb CSa CSb CSa CSbconstant -5.154*(1.058)-4.954*(0.997)-5.380*(1.068)-5.184*(1.014)-4.913*(1.049)-4.943*(0.993)-4.798*(1.073)-4.797*(1.014)-4.686*(1.099)-4.694*(1.036)origin GDP 0.635*(0.050)0.670*(0.040)0.641*(0.051)0.671*(0.041)0.609*(0.050)0.658*(0.039)0.612*(0.053)0.663*(0.041)0.607*(0.055)0.662*(0.042)destination GDP 0.853*(0.046)0.805*(0.034)0.828*(0.046)0.784*(0.034)0.840*(0.046)0.807*(0.034)0.858*(0.046)0.820*(0.034)0.857*(0.046)0.817*(0.034)origin population 0.073(0.063)0.061(0.064)0.099(0.064)0.102(0.065)0.106(0.066)dest. population -0.075(0.055)-0.069(0.055)-0.044(0.055)-0.055(0.056)-0.060(0.056)distance -0.819*(0.081)-0.823*(0.079)-0.765*(0.078)-0.771*(0.080)-0.811*(0.081)-0.806*(0.080)-0.849*(0.082)-0.847*(0.081)-0.850*(0.083)-0.849*(0.081)contiguity 0.160(0.316)0.151(0.316)0.212(0.320)0.204(0.320)0.208(0.318)0.200(0.319)0.236(0.322)0.225(0.323)0.188(0.323)0.175(0.324)common language 0.704*(0.169)0.701*(0.169)0.729*(0.171)0.725*(0.171)0.740*(0.170)0.736*(0.171)0.661*(0.172)0.658*(0.172)0.662*(0.173)0.660*(0.173)log-likelihood -709.37 -710.92 -714.69 -715.90 -712.71 -714.19 -717.65 -719.33 -718.88 -720.712R 0.642 0.641 0.625 0.624 0.635 0.634 0.641 0.640 0.638 0.637All variables except for the contiguity and language dummies are in logs. Standard errors are in parentheses. * denotes significance at5% level. The basic version (CSa) is a form of the augmented version (CSa) with the coefficients on the populations restricted to zero.There are 422 observations for each year.27

Table 2: Regression results for models using pooled data; 1991-95;dependent variable = log of exportsconstant -5.007*(0.477)origin GDP 0.621*(0.023)destination GDP 0.847*(0.020)origin population 0.088*(0.029)dest. population -0.061*(0.024)distance -0.819*(0.036)contiguity 0.200(0.142)common language 0.700*(0.076)1992 -0.005(0.091)1993 0.027(0.091)1994 0.022(0.091)1995 0.056(0.091)pooled cross-section fixed effects alternatives w/heterog.PCSa PCSb FEa FEb BEa Ma-4.931*(0.453)0.664*(0.018)0.807*(0.015)-0.819*(0.036)0.190(0.143)0.696*(0.076)-0.004(0.091)0.028(0.091)0.022(0.091)0.056(0.091)- - 0.034†(0.020)0.293*(0.066)0.520*(0.054)1.762*(0.685)-2.014*(0.352)0.030(0.019)0.050*(0.023)0.099*(0.029)0.198*(0.035)0.323*(0.063)0.506*(0.054)0.021(0.018)0.035*(0.018)0.073*(0.019)0.164*(0.022)0.195*(0.083)0.427*(0.066)2.293†(1.195)-1.458*(0.587)-0.023(0.022)0.023(0.021)0.080*(0.021)-4.505(10.887)0.293(0.263)0.520*(0.214)1.769(2.732)-2.016(1.403)0.030(0.076)0.050(0.094)0.099(0.114)0.198(0.138)observations 2110 2110 2110 2110 1688 2110parameters 12 10 430 428 8 147log-likelihood -3575.10 -3582.57 131.01 107.80 -331.93 -2950.432R 0.641 0.638 0.987 0.986 0.073 0.788Akaike Info. Crt. 3.400 3.405 0.283 0.304 0.403 2.935Amemiya Prob. Crt. 1.755 1.764 0.078 0.080 0.088 1.102sum of sqd. resids. 3660.23 3686.24 109.11 111.54 146.45 2024.71All non-dummy variables are in logs. Standard errors are in parentheses. * and † denote significanceat 5% and 10% levels. For the BEa model all variables are in differences from the previous year.28

Table 3: Regression results with integration dummies; 1991-95;dependent variable = log of exportspooled cross-sectionPCSafixed effectsFEaconstant -3.572* (0.490) -origin GDP 0.649* (0.023) 0.312* (0.067)destination GDP 0.870* (0.020) 0.524* (0.054)origin population 0.054† (0.028) 1.864* (0.690)destination population -0.083* (0.024) -1.948* (0.356)distance -1.022* (0.041)contiguity 0.025 (0.144)common language 0.622* (0.075)European bloc -1.112* (0.115) 0.064 (0.044)North American bloc -0.062 (0.397) 0.157 (0.233)MERCOSUR -3.165* (0.924) 0.166 (0.202)1992 -0.006 (0.089) 0.027 (0.019)1993 0.027 (0.089) 0.047* (0.024)1994 0.104 (0.089) 0.087* (0.030)1995 0.149† (0.089) 0.181* (0.037)observations 2110 2110parameters 15 433log-likelihood -3524.73 132.972R 0.657 0.987Akaike Info. Crt. 3.355 0.284Amemiya Prob. Crt. 1.678 0.078sum of sqd. resids. 3489.61 108.91All non-dummy variables are in logs. Standard errors are in parentheses. * and †denote significance at 5% and 10% levels.29

Figure 1: Plots of Residuals; Various ModelsPooled Cross-Section (PCSa)6420-3 -2-42 7 12-6-8log of exportsFixed Effects (FEa)6420-3 -2 2 7 12-4-6-8log of exportsRestricted Time Effects (BEa)6420-3 -2 2 7 12-4-6-8log of exportsRestricted Fixed Effects (Ma)6420-3 -2 2 7 12-4-6-8log of exports30