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# Gravity model of trade

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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 (often using GDP measurements) and distance between two units. The model was first used by Jan Tinbergen in 1962.^{[1]} The basic model for trade between two countries (i and j) takes the form of:

<math>F_{ij} = G (M_i^{\beta_1} M_j^{\beta_2} / D_{ij}^{\beta_3}){\ }</math>

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).

## Contents

## Theoretical justifications and research

The model has been an empirical success in that it accurately predicts trade flows between countries for many goods and services, but for a long time some scholars believed that there was no theoretical justification for the gravity equation.^{[2]} However, a gravity relationship can arise in almost any trade model that includes trade costs that increase with distance.

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 Heckscher-Ohlin 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, many economists in the Chicago School believed that the Heckscher-Ohlin model alone was sufficient to describe all trade, while Bertil Ohlin himself argued that in fact the world is more complicated. 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 Heckscher-Ohlin has on the real world. Jeffrey Frankel sees the Helpman-Krugman setup here as distinct from Linder’s proposal. However, he does say Helpman-Krugman 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 Heckscher-Ohlin as well as the Linder and Helpman-Krugman 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 intra-industry 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 intra-industry 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 non-linear 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 log-linearization of this system of equations (Baier and Bergstrand (2009)), or exporter-country-year and importer-country-year dummy variables. For counterfactual analysis, however, one would still need to account for the change in world prices.

## Econometric Estimation of Gravity Equations

Since the gravity model for trade does not hold exactly, in econometric applications it is customary to specify

- <math>F_{ij} = G (M_i^{\beta_1} M_j^{\beta_2} / D_{ij}^{\beta_3})\eta_{ij}^{\ }</math>,

where <math>F_{ij}</math> represents volume of trade from country <math>i</math> to country <math>j</math>, <math>M_i</math> and <math>M_j</math> typically represent the GDPs for countries <math>i</math> and <math>j</math>, <math>D_{ij}</math> denotes the distance between the two countries, and <math>\eta</math> 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 log-log model of the form (note: constant G becomes part of <math>\beta_0</math>):

- <math>\ln (F_{ij}) = \beta_0 + \beta_1 \ln (M_i) + \beta_2 \ln (M_{j}) - \beta_3 \ln (D_{ij}) + \epsilon^{\ }_{ij}</math>.

However, this approach has two major problems. First, it obviously cannot be used when there are observations for which <math>F_{ij}</math> is equal to zero. Second, it has been argued by Santos Silva and Tenreyro (2006) that estimating the log-linearized equation by least squares (OLS) can lead to significant biases. As an alternative, these authors have suggested that the model should be estimated in its multiplicative form, i.e.,

- <math> F_{ij} = \exp [ \beta_0 + \beta_1 \ln (M_i) + \beta_2 \ln (M_{j}) - \beta_3 \ln (D_{ij})] \eta_{ij}^{\ }</math>,

using a Poisson pseudo-maximum 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. This is despite the fact that simpler methods, such as taking simple averages of trade shares of countries with and without former colonial ties suggest that countries with former colonial ties continue to trade more. Santos Silva and Tenreyro (2006) did not explain where their result came from and even failed to realize their results were highly anomalous. 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 argued that the simulation results of Martin and Pham (2008) are based on misspecified models and showed 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.

## Dynamic Gravity Equation

The gravity equation of international trade is often motivated using New Trade Theory models, which are models of increasing returns.^{[3]} Many increasing returns models feature costs that either fixed or sunk, while it has long been known that trade is a dynamic process. Recently, several authors have proposed a dynamic gravity equation in place of the traditional static gravity equation, including Yotov and Olivero (2012),^{[4]} Campbell (2010),^{[5]} and Campbell (2013).^{[6]} The dynamic gravity equation, in its most general form, posits that bilateral trade between country i and j is a function of the size of each country, the current trade costs, and also the past history of trade costs.

- <math>ln(F_{ij,t}) = ln(Y_i Y_j) - a ln(\tau_{ij,t}) - b ln(\tau_{ij,t-1})</math>

Empirically, the idea that trade flows are determined by historical forces is confirmed by the evidence offered in Eichengreen and Irwin (1996),^{[7]} Campbell (2010), and Campbell (2013). Campbell (2013)^{[8]} shows that empirical estimations using a dynamic gravity equation can have an enormous impact on the measured impact of policy variables, such as the impact of currency unions on trade.

## See also

## Notes

- ↑ Nello, Susan S.,
*The European Union: Economics, Policies and History,*Maidenhead: McGraw Hill Education (2009) ISBN 0-07-711813-8 - ↑ Deardorff, Alan (1998). "Determinants of Bilateral Trade: Does Gravity Work in a Neoclassical World?" (PDF).
*The Regionalization of the World Economy*. - ↑ Feenstra, Robert (2003).
*Advanced International Trade*. Princeton University Press. ISBN 0691114102. - ↑ Olivero, Maria Pia; Yoto Yotov (2012). "Dynamic gravity: endogenous country size and asset accumulation".
*Canadian Journal of Economics***45**(1): 64–92. doi:10.1111/j.1540-5982.2011.01687.x. - ↑ Campbell, Douglas (2010). "History, Culture, and Trade: A Dynamic Gravity Approach".
*Working Paper*. - ↑ Campbell, Douglas (2013). "Relative Prices, Hysteresis, and the Decline of American Manufacturing".
*Working Paper*. - ↑ Eichengreen, Barry; Douglas A. Irwin (1998). "The Role of History in Bilateral Trade Flows".
*The Regionalization of the World Economy*. - ↑ Campbell, Douglas (2013). "Estimating the Impact of Currency Unions on Trade: Solving the Glick and Rose Puzzle".
*The World Economy***36**(10): 1278–1293.

## 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 Trade-Cost Effects Using the Gravity Equation." Journal of International 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=0034-6535%28198508%2967%3A3%3C474%3ATGEIIT%3E2.0.CO%3B2-2>
- 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=0008-4085%28200105%2934%3A2%3C430%3AUTGETD%3E2.0.CO%3B2-6>
- 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: Short-Run 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.uni-muenchen.de/9453/1/MPRA_paper_9453.pdf>.
- McPherson,M. A., M. R. Redfearn and M. A. Tieslau. “A Re-examination of the Linder Hypothesis: a Random-Effects Tobit Approach.” Working Paper from the website of the Department of Economics; University of North Texas. <http://www.econ.unt.edu/research/pdf/00-09MATlinder1.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/pdfplus/10.1162/rest.88.4.641>
- Santos Silva, J.M.C. and Tenreyro, Silvana (2011), Further simulation evidence on the performance of the Poisson pseudo-maximum likelihood estimator, Economics Letters, 112(2), pp. 220–222. <http://www.sciencedirect.com/science/article/pii/S0165176511001741>
- Summary, Rebecca M. “A Political-Economic 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=0034-6535%28198902%2971%3A1%3C179%3AAPMOUB%3E2.0.CO%3B2-J>
- Balding, C., & Dauchy, E. P. (2013). Asymmetric Trade Estimator in Modified Gravity: Corporate Tax Rates and Trade in OECD Countries.
*Available at SSRN 2249536*. <http://www.cefir.ru/papers/WP200.pdf>

## External links

### Information

- World Bank presentation on the gravity model
- Global multi-market simulation using World Bank's World Integrated Trade Solution *Global Tariff Cuts and Trade Simulator
- A page on the implementation of the Poisson pseudo-maximum likelihood estimator