The gravity model of trade in international economics, similar to other gravity models in social science, predicts bilateral trade flows based on the economic sizes of (often using GDP measurements) and distance between two units. The model was first used by Tinbergen in 1962.^{[1]} The basic theoretical model for trade between two countries (i and j) takes the form of:
 F_{ij} = G \frac{M_i M_j}{D_{ij}}
Where F is the trade flow, M is the economic mass of each country, D is the distance and G is a constant. The model has also been used in international relations to evaluate the impact of treaties and alliances on trade, and it has been used to test the effectiveness of trade agreements and organizations such as the North American Free Trade Agreement (NAFTA) and the World Trade Organization (WTO).
Theoretical justifications and research
The model has been an empirical success, but the theoretical justifications for the model are the subject of some dispute. The model clearly has a relationship with a geographic view of trade, but other theoretical justifications for the model have also been proposed.
The gravity model estimates the pattern of international trade. While the model s basic form consists of factors that have more to do with geography and spatiality, the gravity model has been used to test hypotheses rooted in purer economic theories of trade as well. One such theory predicts that trade will be based on relative factor abundances. One of the common relative factor abundance models is the HeckscherOhlin model. This theory would predict that trade patterns would be based on relative factor abundance. Those countries with a relative abundance of one factor would be expected to produce goods that require a relatively large amount of that factor in their production. While a generally accepted theory of trade, comparative advantage has suffered empirical problems. Investigations into real world trading patterns have produced a number of results that do not match the expectations of comparative advantage theories. Notably, a study by Wassily Leontief found that the United States, the most capital endowed country in the world, actually exports more in labor intensive industries. Comparative advantage in factor endowments would suggest the opposite would occur. Other theories of trade and explanations for this relationship were proposed in order to explain the discrepancy between Leontief s empirical findings and economic theory. The problem has become known as the Leontief paradox.
An alternative theory, first proposed by Staffan Linder, predicts that patterns of trade will be determined by the aggregated preferences for goods within countries. Those countries with similar preferences would be expected to develop similar industries. With continued similar demand, these countries would continue to trade back and forth in differentiated but similar goods since both demand and produce similar products. For instance, both Germany and the United States are industrialized countries with a high preference for automobiles. Both countries have automobile industries, and both trade cars. The empirical validity of the Linder hypothesis is somewhat unclear. Several studies have found a significant impact of the Linder effect, but others have had weaker results. Studies that do not support Linder have only counted countries that actually trade; they do not input zero values for the dyads where trade could happen but does not. This has been cited as a possible explanation for their findings. Also, Linder never presented a formal model for his theory, so different studies have tested his hypothesis in different ways.
Elhanan Helpman and Paul Krugman asserted that the theory behind comparative advantage does not predict the relationships in the gravity model. Using the gravity model, countries with similar levels of income have been shown to trade more. Helpman and Krugman see this as evidence that these countries are trading in differentiated goods because of their similarities. This casts some doubt about the impact HeckscherOhlin has on the real world. Jeffrey Frankel sees the HelpmanKrugman setup here as distinct from Linder s proposal. However, he does say HelpmanKrugman is different from the usual interpretation of Linder, but, since Linder made no clear model, the association between the two should not be completely discounted. Alan Deardorff adds the possibility, that, while not immediately apparent, the basic gravity model can be derived from HeckscherOhlin as well as the Linder and HelpmanKrugman hypotheses. Deardorff concludes that, considering how many models can be tied to the gravity model equation, it is not useful for evaluating the empirical validity of theories.
Bridging economic theory with empirical tests, James Anderson and Jeffrey Bergstrand develop econometric models, grounded in the theories of differentiated goods, which measure the gains from trade liberalizations and the magnitude of the border barriers on trade (see McCallum Border puzzle).
Adding to the problem of bridging economic theory with empirical results, some economists have pointed to the possibility of intraindustry trade not as the result of differentiated goods, but because of reciprocal dumping. In these models, the countries involved are said to have imperfect competition and segmented markets in homogeneous goods, which leads to intraindustry trade as firms in imperfect competition seek to expand their markets to other countries and trade goods that are not differentiated yet for which they do not have a comparative advantage, since there is no specialization. This model of trade is consistent with the gravity model as it would predict that trade depends on country size.
The reciprocal dumping model has held up to some empirical testing, suggesting that the specialization and differentiated goods models for the gravity equation might not fully explain the gravity equation. Feenstra, Markusen, and Rose (2001) provided evidence for reciprocal dumping by assessing the home market effect in separate gravity equations for differentiated and homogeneous goods. The home market effect showed a relationship in the gravity estimation for differentiated goods, but showed the inverse relationship for homogeneous goods. The authors show that this result matches the theoretical predictions of reciprocal dumping playing a role in homogeneous markets.
Past research using the gravity model has also sought to evaluate the impact of various variables in addition to the basic gravity equation. Among these, price level and exchange rate variables have been shown to have a relationship in the gravity model that accounts for a significant amount of the variance not explained by the basic gravity equation. According to empirical results on price level, the effect of price level varies according the relationship being examined. For instance, if exports are being examined, a relatively high price level on the part of the importer would be expected to increase trade with that country. A nonlinear system of equations are used by Anderson and van Wincoop (2003) to account for the endogenous change in these price terms from trade liberalization. A more simple method is to use a first order loglinearization of this system of equations (Baier and Bergstrand (2009)), or exportercountryyear and importercountryyear dummy variables. For counterfactual analysis, however, one would still need to account for the change in world prices.
United States trade
Rebecca M. Summary specifically investigated American trade in an attempt to examine the specific role of political factors on United States trade. Her model consisted of the basic gravity equation, but did not multiply GDP or population (she used these separately and excluded income level) to find an interaction since she only looked at United States data and all data would be weighted by the same, crosssectional factor. Summary s study concluded that American exports and imports are affected by a number of political and social factors related to business ties, alliances, and foreign policy as well as the basic factors usually included in the gravity model.
Econometric Estimation of Gravity Equations
Since the gravity model for trade does not hold exactly, in econometric applications it is customary to specify

F_{ij} = G (M_i^{\beta_1} M_j^{\beta_2} / D_{ij}^{\beta_3})\eta_{ij}^{\ },
where F_{ij} represents volume of trade from country i to country j, M_i and M_j typically represent the GDPs for countries i and j, D_{ij} denotes the distance between the two countries, and \eta represents an error term with expectation equal to 1.
The traditional approach to estimating this equation consists in taking logs of both sides, leading to a loglog model of the form (note: constant G becomes part of \beta_0):

\ln (F_{ij}) = \beta_0 + \beta_1 \ln (M_i) + \beta_2 \ln (M_{j})  \beta_3 \ln (D_{ij}) + \epsilon^{\ }_{ij}.
However, this approach has two major problems. First, it obviously cannot be used when there are observations for which F_{ij} is equal to zero. Second, it has been argued by Santos Silva and Tenreyro (2006) that estimating the loglinearized equation by least squares (OLS) can lead to significative biases. As an alternative, these authors have suggested that the model should be estimated in its multiplicative form, i.e.,

F_{ij} = \exp [ \beta_0 + \beta_1 \ln (M_i) + \beta_2 \ln (M_{j})  \beta_3 \ln (D_{ij})] \eta_{ij}^{\ },
using a Poisson pseudomaximum likelihood (PPML) estimator usually used for count data (see the original paper for details). One of the authors' more surprising findings was that, when controlling for sharing a common language, having past colonial ties does not increase trade a finding which contrasts with what more basic methods, such as OLS or even scatter plots of trade data, would indicate. Martin and Pham (2008) argued that using PPML on gravity severely biases estimates when zero trade flows are frequent. However, their results were challenged by Santos Silva and Tenreyro (2011), who pointed out that the simulation results of Martin and Pham (2008) are based on misspecified models and confirmed that the PPML estimator performs well even when the proportions of zeros is very large.
In applied work, the model is often extended by including variables to account for language relationships, tariffs, contiguity, access to sea, colonial history, exchange rate regimes, and other variables of interest.
See also
Notes
References
 Anderson, J., van Wincoop, E. "Gravity with Gravitas: A Solution to the Border Puzzle." American Economic Review (2003). http://www.ingentaconnect.com/content/aea/aer/2003/00000093/00000001/art00009>
 SL Baier, JH Bergstrand. "Bonus Vetus OLS:"A Simple Method for Approximating International TradeCost Effects Using the Gravity Equation." Journal of Internation Economics (2009). http://linkinghub.elsevier.com/retrieve/pii/S0022199608001062>
 Bergstrand, Jeffrey H. The Gravity Equation in International Trade: Some Microeconomic Foundations and Empirical Evidence. The Review of Economics and Statistics, Vol. 67, No. 3. (Aug., 1985), pp. 474 481. http://links.jstor.org/sici?sici=00346535%28198508%2967%3A3%3C474%3ATGEIIT%3E2.0.CO%3B22>
 Caruso, R. "The impact of International Economic Sanctions on Trade. An Empirical Analysis.", Peace Economics, Peace Science and Public Policy, vol.9, no.2 (2003). http://www.bepress.com/peps/vol9/iss2/1/
 Deardorff, Alan V. Determinants of Bilateral Trade: Does Gravity Work in a Neoclassical World? In The Regionalization of the World Economy, edited by J.A. Frankel. Chicago: University of Chicago Press. 1998, 21.
 Frankel, Jeffery A. Regional Trading Blocs: In the World Economic System. Washington, DC: Institute of International Economics. October 1997.
 Feenstra, Robert C., James R. Markusen, and Andrew K. Rose. Using the Gravity Equation to Differentiate among Alternative Theories of Trade. The Canadian Journal of Economics, Vol. 34, No. 2. (May, 2001), pp. 431. http://links.jstor.org/sici?sici=00084085%28200105%2934%3A2%3C430%3AUTGETD%3E2.0.CO%3B26>
 Hacker, R Scott, Einarsson, Henrik. The Pattern, Pull, and Potential of Baltic Sea Trade. The Annals of Regional Science, Vol. 37, No. 1, pp. 15 29. http://www.springerlink.com/content/b9wq4batxqpmbdya/>
 Isard, W., "Location Theory and Trade Theory: ShortRun Analysis". Quarterly Journal of Economics, vol. 68, 1954, p. 305 322.
 Martin, William and Cong S. Pham, "Estimating the Gravity Model When Zero Trade Flows are Frequent." Working Paper. http://mpra.ub.unimuenchen.de/9453/1/MPRA_paper_9453.pdf>.
 McPherson,M. A., M. R. Redfearn and M. A. Tieslau. A Reexamination of the Linder Hypothesis: a RandomEffects Tobit Approach. Working Paper from the website of the Department of Economics; University of North Texas. http://www.econ.unt.edu/research/pdf/0009MATlinder1.PDF>
 Santos Silva, J.M.C. and Tenreyro, Silvana (2006), The Log of Gravity, The Review of Economics and Statistics, 88(4), pp. 641 658. http://www.mitpressjournals.org/doi/abs/10.1162/rest.88.4.641>
 Santos Silva, J.M.C. and Tenreyro, Silvana (2011), Further simulation evidence on the performance of the Poisson pseudomaximum likelihood estimator, Economics Letters, 112(2), pp. 220222. http://www.sciencedirect.com/science/article/pii/S0165176511001741>
 Summary, Rebecca M. A PoliticalEconomic Model of U.S. Bilateral Trade. The Review of Economics and Statistics, Vol. 71, No. 1. (Feb., 1989), pp. 179 182. http://links.jstor.org/sici?sici=00346535%28198902%2971%3A1%3C179%3AAPMOUB%3E2.0.CO%3B2J>
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