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No. 9322
GRAVITY EQUATIONS: WORKHORSE,
TOOLKIT, AND COOKBOOK
Keith Head and Thierry Mayer
INTERNATIONAL TRADE AND
REGIONAL ECONOMICS
ISSN 0265-8003
GRAVITY EQUATIONS: WORKHORSE, TOOLKIT,
AND COOKBOOK
Keith Head, Sauder School of Business and CEPR
Thierry Mayer, Sciences Po, CEPII and CEPR
Discussion Paper No. 9322
January 2013
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Copyright: Keith Head and Thierry Mayer
CEPR Discussion Paper No. 9322
January 2013
ABSTRACT
Gravity Equations: Workhorse, Toolkit, and Cookbook*
This chapter focuses on the estimation and interpretation of gravity equations
for bilateral trade. This necessarily involves a careful consideration of the
theoretical underpinnings since it has become clear that naive approaches to
estimation lead to biased and frequently misinterpreted results. There are
now several theory-consistent estimation methods and we argue against sole
reliance on any one method and instead advocate a toolkit approach. One
estimator may be preferred for certain types of data or research questions but
more often the methods should be used in concert to establish robustness. In
recent years, estimation has become just a first step before a deeper analysis
of the implications of the results, notably in terms of welfare. We try to
facilitate diffusion of best-practice methods by illustrating their application in a
step-by-step cookbook mode of exposition.
JEL Classification: F1
Keywords: bilateral, gravity and trade
Keith Head
University of British Columbia
Sauder School of Business
2053 Main Hall
Vancouver, B.C. V6T1Z2
CANADA
Email: keith.head@sauder.ubc.ca
For further Discussion Papers by this author see:
www.cepr.org/pubs/new-dps/dplist.asp?authorid=154019
Thierry Mayer
Sciences Po
28 rue des Saints-Pères
75007 Paris
FRANCE
Email: thierry.mayer@sciences-po.fr
For further Discussion Papers by this author see:
www.cepr.org/pubs/new-dps/dplist.asp?authorid=152707
Submitted 22 January 2013
Gravity Equations:
Workhorse,Toolkit, and Cookbook
∗
Keith Head
†
Thierry Mayer
‡
January 26, 2013
Contents
1 Introduction 2
1.1 Gravity features of trade data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 A brief history of gravity in trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Microfoundations for Grav i ty Equations 7
2.1 Three Definitions of t he Gravity Equation . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Assumptions underlying struc t ur al gravity . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Main variants of gravity for trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Gravity models beyond trade in goods . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Theory-consistent estimation 19
3.1 Proxies for multilateral resistance terms . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Iterative structural estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Fixed effects estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 Ratio-type estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
∗
This is a draft of a chapter to appear the Handbook of International Economics Vol. 4, eds. Gopin a th, Help-
man, and Rogoff. The chapter has a companion website, https://sites.google.com/site/hiegravity/,withan
appendix, Stata code, and related links. We thank Leo Fankh¨anel and Camilo Uman a for outst a n d in g assistance with
the programming and meta-analysis in this chapter, Soledad Zignago for great help with providing and understand i n g
subtleties of some of the data used, and Julia Jauer for her update of the gravity data. Scott Baier, Sebastian Sotelo,
Jo˜ao Santos Silva generously provided computer code. A n d res Rodr´ıguez-Clare answered many questions we had
about welfare calculations but is not responsible of co u rse, for any mistakes we may have made. Arnaud Costinot,
Gilles Duranton, Thibault Fally, Mario Larch, Marc Melitz, Gianmarco Ottaviano, and Daniel Trefler made very
useful comments on previous drafts. We are especially grateful to Jose de Sousa: his careful reading identified many
necessary corrections in an early dra ft. Participants at presentations Hito t su b a sh i GCOE Confe ren c e on International
Trade and FDI 2012, National Bank of Belgium, Clemson University also contributed to improving the draft. Finally,
we thank our discussants at the handbook conference, Rob Feenstra and Jim Anderson, for many helpful suggestions.
We regret that because of limitations of time and space, we have not been able to fully respond to all of the valuable
suggestions we received. This research has received funding from the European Research Council under the European
Community’s Seventh Framework Programme (FP7/2007-2013 ) Grant Agreement no. 313522.
†
Sauder School of Business, Un iversity of British Columbia and CEPR, keith.head @ sa u d er. u bc.ca
‡
Sciences Po, CEPII, and CEPR, thierry.mayer@sciences-po.fr
1
3.5 Other methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.6 Monte Carlo study of alt er nat ive estimators . . . . . . . . . . . . . . . . . . . . . . . 23
3.7 Identification and estimation of country-spec ifi c effects . . . . . . . . . . . . . . . . . 26
4 Gravity estimates of policy impacts 28
4.1 Meta-analysis of poli cy dummies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 The elasticity of trade with respect to trade costs . . . . . . . . . . . . . . . . . . . . 31
4.3 Partial vs general equilibriu m impacts on trade . . . . . . . . . . . . . . . . . . . . . 34
4.4 Testing structural gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5 Frontiers of gravity research 40
5.1 Gravity’s errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2 Causes and consequences of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3 Firm-level gravity, extensi ve and intensive margins . . . . . . . . . . . . . . . . . . . 50
6 Directions for futu re research 56
7 Conclusions 57
References 57
1 Introduction
As the name suggests, gravity equations are a model of bilateral interactions in which size and
distance effects enter multiplicatively. They have been used as a workhorse for analyzing the
determinants of bilateral trade flows for 50 years since being introduced by Ti nb er gen (1962).
Krugman (1997) referred to grav i ty equati on s as examples of “social physics,” the relatively few law-
like empirical regularities that characterize social interactions.
1
Over the last decade, concentrated
efforts of trad e theorists have established that gravity equations emerge from mainstream modeling
frameworks in economics and should no longer be thou ght of as deriving from some murky analogy
with New ton i an physics. Meanwhile empirical work—guided in varying degrees by the new theory—
has proceed ed to lay down a raft of stylized facts about the determinants of bilateral trade. As a
result of recent mode l li n g, we now know that gravity estimates can be combined with trade policy
experiments to calculate implied welfare chan ges.
This chapter focuses on the estimati on and interpretation of gravity equations for bilat er al
trade. This necessarily involves a careful c onsideration of the theoretical underpinnings since i t
has bec ome clear that naive approaches to estimation lead to biased and f re qu ently misinterpreted
results. There are now several theory-consistent estimation methods and we argue against sole
reliance on any one method and in st ead advocate a toolkit approach. One estimator may be
1
Other examples of social physics include power function distributions thought to characterize incomes, firm and
city sizes, and network linka g es .
2
