Gravity Redux Structural Estimation Of Gravity Equations .
Gravity Redux :Structural Estimation of Gravity Equations withAsymmetric Bilateral Trade Costs Jeffrey H. Bergstrand†, Peter Egger‡, and Mario Larch§December 20, 2007AbstractTheoretical foundations for estimating gravity equations were enhanced recentlyin Anderson and van Wincoop (2003). Though elegant, the model assumes symmetric bilateral trade costs to generate an estimable set of structural equations. Inreality, however, trade costs (and trade flows) are not bilaterally symmetric. To allow for asymmetric bilateral trade costs, we provide an alternative framework usingthe simple workhorse Krugman-type e model of trade assuming only multilateral trade balance. A Monte Carloanalysis of our general equilibrium model demonstrates – in the presence of asymmetric bilateral trade costs – that the bias of the Anderson-van Wincoop approach(assuming either symmetric or asymmetric bilateral trade costs) is at least an orderof-magnitude larger than that using our approach for computing general equilibriumcomparative statics. We then confirm empirically the difference of our approach andthat of Anderson and van Wincoop in the Canadian-U.S. ”border puzzle” case allowing asymmetric effects of national borders. Furthermore, we apply our approachempirically to the more general case of trade flows among 67 countries in the presence of asymmetric bilateral tariff rates.Key words: International trade; Gravity equation; Trade costs; StructuralestimationJEL classification: F10; F12; F13 Acknowledgements: To be added.Affiliation: Department of Finance, Mendoza College of Business, and Kellogg Institute for International Studies, University of Notre Dame, and CESifo Munich. Address: Department of Finance, Mendoza College of Business, University of Notre Dame, Notre Dame, IN 46556 USA. E-mail:bergstrand.1@nd.edu.‡Affiliation: Ifo Institute for Economic Research, Ludwig-Maximilian University of Munich, CESifo,and Centre for Globalization and Economic Policy, University of Nottingham. Address: Ifo Institute forEconomic Research, Poschingerstr. 5, 81679 Munich, Germany. E-mail: egger@ifo.de.§Affiliation: Ifo Institute for Economic Research and CESifo. Address: Ifo Institute for EconomicResearch, Poschingerstr. 5, 81679 Munich, Germany. E-mail: larch@ifo.de.†
1Introduction”Our analysis suggests that inferential identification of the asymmetry [in bilateral trade costs] is problematic.”(Anderson and van Wincoop, 2003, p. 175)For nearly a half century, the ”gravity equation” has been used to explain economet-rically the ex post effects of economic integration agreements, national borders, currencyunions, language, and other measures of trade costs on bilateral trade flows, cf., Rose(2004). While two early formal theoretical foundations for the gravity equation withtrade costs – first Anderson (1979) and later Bergstrand (1985) – addressed the role of”multilateral prices,” Anderson and van Wincoop (2003) refined the theoretical foundations for the gravity equation to emphasize the importance of accounting properly for theendogeneity of prices. Two major conclusions surfaced from the now seminal Andersonand van Wincoop (henceforth, A-vW) study, ”Gravity with Gravitas.” First, a completederivation of a standard Armington (conditional) general equilibrium model of bilateraltrade in a multi-region (N 2) setting with iceberg trade costs suggests that traditionalcross-section empirical gravity equations have been misspecified owing to the omission oftheoretically-motivated multilateral (price) resistance terms for exporting and importingregions. Second, to estimate properly the full general equilibrium comparative-static effects of a national border or an economic integration agreement, one needs to estimatethese multilateral resistance (MR) terms for any two regions with and without a border oragreement, respectively, in a manner consistent with theory. Due to the underlying nonlinearity of the structural relationships, A-vW suggest a custom nonlinear least squares(NLS) program to account properly for the endogeneity of prices and to estimate thecomparative-static effects of a trade cost.However, though A-vW (2003) is elegant and motivated by only four assumptions,one assumption is that every pair of regions has perfectly symmetric bilateral trade costs.11The other three assumptions are that all goods are produced in an endowment economy and aredifferentiated by origin, preferences are CES, and market clearance holds. This approach is summarized2
Hence, in a world-trade setting with N countries, the tariff rate (and ad valorem-equivalentnon-tariff rate) on products from Japan to the United States equals exactly that onproducts from the United States to Japan, and so forth. Clearly, this assumption isgrossly at odds with reality; data supporting this is provided in Figure 1, using bilateraltariff data from the Global Trade Analysis Project (GTAP) on 67 economies in 2001.There is large heterogeneity bilaterally in tariff rates. In Figure 1, only 42 percent of thebilateral tariff rates are symmetric; 58 percent are not. Also, the figure illustrates thatthe asymmetry can be as large as 150 percent. Moreover, an important implication ofthis assumption is that every pair of countries’ bilateral trade will be balanced. This isalso grossly at odds with reality. However, the symmetric bilateral trade costs (SBTC)assumption was useful to derive an elegant system of structural equations that provided alogically-consistent formal theoretical foundation for proper estimation of a gravity model.Since the SBTC assumption is often violated in the real world, we address three questions in this paper. First, is there a set of plausible alternative assumptions that cangenerate a theoretical foundation for the gravity equation without SBTC? Second, in aworld where we know the true data-generating process, can this alternative theoreticalfoundation provide unbiased coefficient estimates and precisely-estimated general equilibrium comparative statics? Third, in a world with asymmetric bilateral trade costs(ABTC), does the A-vW approach yield biased coefficient estimates and comparativestatics, and are such biases avoided under the alternative approach?In this paper, we suggest two fairly standard assumptions as alternatives to SBTC tomotivate a theoretical foundation for the gravity equation. First, we assume the simpleKrugman (1980) model of increasing returns to scale with monopolistic competition (IRMC), as summarized in Baier and Bergstrand (2001) and Feenstra (2004), that has becomethe workhorse for studying bilateral intra-industry trade. This workhorse IR-MC modelpins down the relationship between the exporting country’s economic size and the numberof varieties consumed by the representative consumer in the importing country. Thein equations (12) and (13) of A-vW (2003). We also discuss later the A-vW approach allowing asymmetricborder barriers, equations (9)-(11).3
second assumption is multilateral trade balance. Of course, this assumption has a longhistory in the pure theory of international trade, unlike the assumption of bilateral tradebalance implied by symmetric bilateral trade costs. Even open-economy macroeconomicsmodels assume multilateral trade balance in the long run. By assuming multilateral tradebalance, we can address the endogeneity of prices raised by A-vW without assumingsymmetric bilateral trade costs.We show in this paper that replacing A-vW’s endowment economy with a Krugman IRMC economy assuming only multilateral trade balance generates a theoretical foundationfor the gravity equation where structural estimation of the model yields both unbiasedcoefficient estimates and even more precisely estimated general equilibrium comparativestatics than (either version of) A-vW’s model – when bilateral trade costs are asymmetric.We demonstrate this in the context of a Monte Carlo analysis allowing either symmetricor asymmetric bilateral trade costs. Finally, we apply the approach in the context of twowidely-recognized empirical examples with symmetric and asymmetric trade costs.The remainder of this paper is as follows. Section 2 establishes the theoretical framework. Section 3 presents the Monte Carlo analysis. Section 4 provides empirical analyses.Section 5 concludes.2Gravity ReduxThe purpose of this section is to show that the theoretical model of Krugman (1980),summarized in Baier and Bergstrand (2001) and Feenstra (2004, Ch. 5), generates astraightforward gravity equation for bilateral trade flows allowing for endogeneity of pricesand GDPs without assuming symmetric bilateral trade costs.2.1UtilityFollowing Krugman (1980), Baier and Bergstrand (2001), and Feenstra (2004), thereexists a single industry where preferences are constant-elasticity-of-substitution (CES).As typical to the Dixit-Stiglitz (1977) class of models, we assume that preferences are4
determined by a ”love of variety.” We assume that utility of consumers in country j isgiven by:"Uj niN XXσ 1σσ# σ 1,cijk(1)i 1 k 1where cijk is the consumption of consumers in country j of variety k from country i, ni isthe number of varieties of the single good produced in country i, which is endogenous inthe model, and N is the number of countries (or regions).2As typical, we assume iceberg transport costs and symmetric firms within each country,and hence all products in country i sell at the same price, pi . Consequently, the utilityfunction simplifies to:"Uj NXσ 1σσ# σ 1ni cij.(2)i 1Maximizing equation (2) subject to the budget constraint:Yj NXni pi tij cij ,(3)i 1where tij is one plus the iceberg trade costs (the latter a fraction) and Yj is nationalincome, yields the demand functions:µcij pi tijPj¶ σYj,Pj(4)where Pj is the CES price index:"Pj NX1# 1 σni (pi tij )1 σ.(5)i 1As in Krugman (1980), Baier and Bergstrand (2001), and Feenstra (2004), the value2We begin with utility function (5.21) from Feenstra (2004, p. 152). We could easily introduce acountry-specific preference parameter βi to the function as in A-vW. However, A-vW effectively circumvent estimating βi by treating prices for each good i as ”scaled prices (βi pi )” in their solution, withoutloss of generality, cf., A-vW (2003, p. 175). Following Krugman (1980), Baier and Bergstrand (2001),and Feenstra (2004), we assume for simplicity that the βi are unity for all i.5
of aggregate exports from country i to country j, Xij , equals ni pi tij cij . Substitutingequation (4) into this expression for Xij yields:µXij ni Yjpi tijPj¶1 σ,(6)which is identical to equation (5.26) in Feenstra (2004, p. 153).2.2Production: Alternative Assumption 1The assumption of a monopolistically competitive market with increasing returns to scalein production (internal to the firm) and a single factor (labor) is sufficient to identifythe exporting country’s number of varieties, cf., Krugman (1980), Baier and Bergstrand(2001), and Feenstra (2004). The representative firm in country i is assumed to maximizeprofits subject to the workhorse linear cost function:li α φyi ,(7)where li denotes labor used by the representative firm in country i and yi denotes theoutput of the firm.Two conditions characterize equilibrium in this class of models. First, profit maximization ensures that prices are a markup over marginal costs:pi σφwi ,σ 1(8)where wi is the wage rate in country i, determining the marginal cost of production.3Second, under monopolistic competition, zero economic profits in equilibrium ensures:yi α(σ 1) ȳ,φso that the output of each firm is a constant, ȳ.3The wage rate in country 1 serves as the numeraire.6(9)
An assumption of full employment of labor in each country ensures that the size ofthe exogenous factor endowment, Li , determines the number of varieties:ni Li.α φȳ(10)We can now derive a gravity equation. First, we can show that the trade flow from ito j is a function of GDPs, labor endowments, and trade costs. With labor the only factorof production, Yi wi Li or wi Yi /Li . Using equations (8) and (10), we can substituteσφwi /(σ 1) for pi in equation (6) and substitute Yi /Li for wi in the resulting equationto yield:Xij Yi Yj hPN(Yi /Li ) σ t1 σij σ t1 σkjk 1 Yk (Yk /Lk )1 .i 1 σ(11)However, we can easily show that equation (11) is identical to the gravity equation in Feenstra (2004) with GDPs and prices. Using equation (8), we can substitute pi /[(σφ)/(σ 1)]for wi in Li Yi /wi and then substitute the resulting equation, Yi /[(σ 1)pi /(σφ)] , forLi in equation (10) to yield:ni γYi,pi(12)where γ φσ/[(σ 1)(α φȳ)]. Substituting equation (12) into equation (6) yields:1 σYi Yj p σi tij.Xij PN σ 1 σk 1 Yk pk tkj(13)which is identical to equation (5.26’) in Feenstra (2004, p. 154).42.3Multilateral Trade Balance: Alternative Assumption 2Equation (13) is a standard representation of the gravity equation. Feenstra (2004) summarized the three methods that have been used up to this point in the literature toaddress the role of prices. The first approach, used in Bergstrand (1985, 1989) and Baier4To see this, note that – using our notation – the denominator of (13) is identical toPNȳ k 1 nk (pk tkj )1 σ .7
and Bergstrand (2001), was to assume that prices are exogenous and use available priceindex data to account for the role of prices. This method is now acknowledged to workpoorly for two reasons, the first is that conceptually such prices are endogenous and thesecond is that available price indexes are crude approximations. The second approach hasbeen to account for the price terms using region-specific fixed effects. While such fixedeffects can account for the influence of the price terms in estimation, the shortcoming ofthis method is that – without estimates of the prices before and after the counterfactualexperiment – one cannot calculate the appropriate general equilibrium comparative statics using fixed effects (or method 1 above). The third method is to estimate a structuralset of nonlinear price equations – under the assumption of symmetric bilateral trade costs(SBTC) – which then generate multilateral price terms before and after the counterfactual experiment to conduct finally the general equilibrium comparative statics, cf., A-vW(2003, eqs. 12 and 13). While this approach provides unbiased estimates and generalequilibrium comparative statics, it does so under the SBTC assumption, which also implies bilateral trade balance, cf., A-vW (2003, eq. 13) for xij and xji . Both considerationsare typically violated in the real world.An alternative assumption, which has a long history in the pure theory of internationaltrade, is to assume multilateral trade balance. While also violated in the real world, itis less restrictive than bilateral trade balance. Multilateral trade balance is ensured byassuming N equations:NXj 1Xij NXXjii 1, ., N.(14)j 1Hence, our gravity model is equations (11) subject to (14), analogous to A-vW’sequations (12) and (13) for SBTC. Our N (N 1) equations (11) along with N equations(14) comprise a system of N 2 equations in N (N 1) endogenous bilateral trade flows, Xij(excluding as in A-vW a country’s internal trade), and N GDPs, Yi . However, unlike AvW, we do not assume symmetric bilateral trade costs.5 Rather, we arrive at our system5A-vW’s (2003) equations (9)-(11) also comprise a structural system, but allowing ABTC. However,8
of equations using the Krugman IR-MC market structure to identify ni combined withthe (less restrictive) multilateral trade balance assumption.62.4Estimating Elasticities of Substitution and Comparative StaticsAn important aspect of the recent gravity-equation literature is going beyond just estimation of unbiased coefficient estimates (or the partial effects of trade costs); country fixedeffects can be used to obtain unbiased bilateral trade cost parameter estimates. Rather,the unique feature of this literature is calculating general equilibrium comparative statics– including potentially welfare effects. A-vW (2003) went beyond estimation to compute comparative statics using actual and counterfactual MR terms. However, estimatesof comparative-static effects require an assumption regarding elasticities of substitution,because the elasticities could not be estimated, cf., A-vW (2001).In our approach, the elasticities of substitution can be estimated. Given data onGDPs, populations and cif-fob factors and given estimates of trade-cost parameters, thenin our model minimizing the absolute values of the differences of exports and imports forall N countries yields an estimate of the elasticity of substitution. These will be provided.7Consequently, the comparative-static effects of trade-cost changes can be estimated usingthe estimated elasticities that surface from our approach. Using the estimated elasticitiesof substitution, we provide estimates of two comparative statics. One is the change intrade relative to the products of GDPs, Xij /(Yi Yj /YW ). The other is the welfare effectas their footnote 11 explains, if bilateral trade costs are asymmetric across countries, the interpretation oftheir border barrier’s effect is restricted to be only an ”average” of the barrier’s effects in both directions.We will contrast the implictions of our model with those of A-vW’s equations (12) and (13) assumingSBTC and A-vW’s equations (9)-(11) allowing ABTC using Monte Carlo analyses in section 3.6While the assumption of bilateral trade balance is very restrictive, some recent evidence that theassumption of multilateral trade balance is not very restrictive is found in Dekle, Eaton and Kortum(2007). In that paper, the authors use a calibrated general equilibrium model of world trade to considerhow much wage rates and prices would have to change from current levels if all multilateral trade balanceswere eliminated (the counterfactual). The authors find that wage rates and prices do not change verymuch. For instance, elimination of China’s and the United States’ large multilateral trade imbalancesrequires wage rate adjustments of less than 10 percent.7Appendix A describes an alternative method; both approaches yield consistent estimates.9
due to a change in trade costs, based on the equivalent variation for country i (EVi ),defined as: EVi 100 ·c YiYiÃPN σ(tki )1 σk 1 Yk (Yk /Lk )PNcc σ (tc )1 σkik 1 Yk (Yk /Lk )1! 1 σ 1 ,(15)where superscript c indicates counterfactual values of trade costs and GDP.The remainder of our paper demonstrates our approach under both symmetric andasymmetric bilateral trade costs. In the following section, we provide a Monte Carloanalysis to demonstrate our approach relative to A-vW’s (to avoid data measurementissues). Section 4 applies our approach to two widely-recognized empirical contexts.3Monte Carlo AnalysisTo avoid data measurement issues, we conduct a large-scale Monte Carlo study to evaluateour approach relative to several alternatives: A-vW, a traditional OLS gravity specification without multilateral resistance terms (labeled, for brevity, OLS), and a recent linearapproximation approach suggested by Baier and Bergstrand (2006) (described below andreferred to henceforth, for brevity, as BV-OLS).The Monte Carlo analysis proceeds in two steps. First, we use alternative sets ofparameter values described in detail below to generate sets of all endogenous variables ofthe theoretical model (Yi , pi , wi , ni , Xij ) as functions of the model’s exogenous variables,Li , and tij , in a baseline general equilibrium. Then, we change exogenous bilateral tradecosts tij , holding the model parameters and all Li constant to obtain counterfactual valuesfor all the endogenous variables.In a second step, we use these generated general equilibrium data and add a stochasticerror term as in traditional Monte Carlo studies.8 The major advantage of this procedure8An additive log-linear error term is conventional to the general-equilibrium-based literature ong
Gravity Redux: Structural Estimation of Gravity Equations with . University of Notre Dame, Notre Dame, IN 46556 USA. E-mail: bergstrand.1@nd.edu. zA–liation: Ifo Institute for Economic Research, Ludwig-Maximilian University of Munich, CESifo, and Centre for Globalization and Ec
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Bergstrand, J.H., P Egger and M. Larch (2007), “Gravity Redux: Structural Estimation of Gravity Equations with Asymmetric Bilateral Trade Costs”, University of Notre Dame and Ifo Institute for Economic Res
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