THE LOG OF GRAVITY J. M. C. Santos Silva and Silvana Tenreyro* Abstract—Although economists have long been aware of Jensen’s in- models are severely biased, distorting the interpretation of equality, many econometric applications have neglected an important implication of it: under heteroskedasticity, the parameters of log- the model. These biases might be critical for the compara- linearized models estimated by OLS lead to biased estimates of the true tive assessment of competing economic theories, as well as elasticities. We explain why this problem arises and propose an appropri- for the evaluation of the effects of different policies. In ate estimator. Our criticism of conventional practices and the proposed solution extend to a broad range of applications where log-linearized contrast, our method is robust to the different patterns of equations are estimated. We develop the argument using one particular heteroskedasticity considered in the simulations. illustration, the gravity equation for trade. We find significant differences We next use the proposed method to provide new esti- between estimates obtained with the proposed estimator and those ob- tained with the traditional method. mates of the gravity equation in cross-sectional data. Using standard tests, we show that heteroskedasticity is indeed a severe problem, both in the traditional gravity equation I. Introduction introduced by Tinbergen (1962), and in a gravity equation CONOMISTS have long been aware that Jensen’s in- that takes into account multilateral resistance terms or fixed Eequality implies that E(ln y) ln E(y), that is, the effects, as suggested by Anderson and van Wincoop (2003). expected value of the logarithm of a random variable is We then compare the estimates obtained with the proposed different from the logarithm of its expected value. This PML estimator with those generated by OLS in the log basic fact, however, has been neglected in many economet- linear specification, using both the traditional and the fixed- ric applications. Indeed, one important implication of Jen- effects gravity equations. sen’s inequality is that the standard practice of interpreting Our estimation method paints a very different picture of the parameters of log-linearized models estimated by ordi- the determinants of international trade. In the traditional nary least squares (OLS) as elasticities can be highly mis- gravity equation, the coefficients on GDP are not, as gen- leading in the presence of heteroskedasticity. erally estimated, close to 1. Instead, they are significantly Although many authors have addressed the problem of smaller, which might help reconcile the gravity equation obtaining consistent estimates of the conditional mean of with the observation that the trade-to-GDP ratio decreases the dependent variable when the model is estimated in the with increasing total GDP (or, in other words, that smaller log linear form (see, for example, Goldberger, 1968; Man- countries tend to be more open to international trade). In ning & Mullahy, 2001), we were unable to find any refer- addition, OLS greatly exaggerates the roles of colonial ties ence in the literature to the potential bias of the elasticities and geographical proximity. estimated using the log linear model. Using the Anderson–van Wincoop (2003) gravity equa- In this paper we use the gravity equation for trade as a tion, we find that OLS yields significantly larger effects for particular illustration of how the bias arises and propose an geographical distance. The estimated elasticity obtained appropriate estimator. We argue that the gravity equation, from the log-linearized equation is almost twice as large as and, more generally, constant-elasticity models, should be that predicted by PML. OLS also predicts a large role for estimated in their multiplicative form and propose a simple common colonial ties, implying that sharing a common pseudo-maximum-likelihood (PML) estimation technique. colonial history practically doubles bilateral trade. In con- Besides being consistent in the presence of heteroskedas- trast, the proposed PML estimator leads to a statistically and ticity, this method also provides a natural way to deal with economically insignificant effect. zero values of the dependent variable. The general message is that, even controlling for fixed Using Monte Carlo simulations, we compare the perfor- effects, the presence of heteroskedasticity can generate mance of our estimator with that of OLS (in the log linear strikingly different estimates when the gravity equation is specification). The results are striking. In the presence of log-linearized, rather than estimated in levels. In other heteroskedasticity, estimates obtained using log-linearized words, Jensen’s inequality is quantitatively and qualitatively important in the estimation of gravity equations. This sug- Received for publication March 29, 2004. Revision accepted for publi- gests that inferences drawn on log-linearized regressions cation September 13, 2005. can produce misleading conclusions. * ISEG/Universidade Te´cnica de Lisboa and CEMAPRE; and London School of Economics, CEP, and CEPR, respectively. Despite the focus on the gravity equation, our criticism of We are grateful to two anonymous referees for their constructive the conventional practice and the solution we propose ex- comments and suggestions. We also thank Francesco Caselli, Kevin tend to a broad range of economic applications where the Denny, Juan Carlos Hallak, Daniel Mota, John Mullahy, Paulo Parente, Manuela Simarro, and Kim Underhill for helpful advice on previous equations under study are log-linearized, or, more generally, versions of this paper. The usual disclaimer applies. Jiaying Huang transformed by a nonlinear function. A short list of exam- provided excellent research assistance. Santos Silva gratefully acknowl- ples includes the estimation of Mincerian equations for edges the partial financial support from Fundac¸a˜o para a Cieˆncia e Tecnologia, program POCTI, partially funded by FEDER. A previous wages, production functions, and Euler equations, which are version of this paper circulated as “Gravity-Defying Trade.” typically estimated in logarithms. The Review of Economics and Statistics, November 2006, 88(4): 641–658 © 2006 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/rest.88.4.641 by guest on 30 September 2021 642 THE REVIEW OF ECONOMICS AND STATISTICS The remainder of the paper is organized as follows. where ␩ij is an error factor with E(␩ij͉Yi,Yj,Dij) ϭ 1, Section II studies the econometric problems raised by the assumed to be statistically independent of the regressors, estimation of gravity equations. Section III considers leading to constant-elasticity models in general; it introduces the PML ␣ ␣ ␣ ͑ ͉ ͒ ϭ ␣ 1 2 3 estimator and specification tests to check the adequacy of E Tij Yi,Yj,Dij 0Y i Y j Dij . the proposed estimator. Section IV presents the Monte Carlo simulations. Section V provides new estimates of both the There is a long tradition in the trade literature of log- traditional and the Anderson–van Wincoop gravity equa- linearizing equation (2) and estimating the parameters of tion. The results are compared with those generated by interest by least squares, using the equation OLS, nonlinear least squares, and tobit estimations. Section ln Tij ϭ ln ␣0 ϩ ␣1 ln Yi ϩ ␣2 ln Yj VI contains concluding remarks. (3) ϩ␣3 ln Dij ϩ ln ␩ij. II. The Econometrics of the Gravity Equation The validity of this procedure depends critically on the assumption that ␩ij, and therefore ln ␩ij, are statistically A. The Traditional Gravity Equation independent of the regressors. To see why this is so, notice that the expected value of the logarithm of a random The pioneering work of Jan Tinbergen (1962) initiated a variable depends both on its mean and on the higher-order vast theoretical and empirical literature on the gravity equa- moments of the distribution. Hence, for example, if the tion for trade. Theories based on different foundations for variance of the error factor ␩ij in equation (2) depends on Yi, trade, including endowment and technological differences, Yj, or Dij, the expected value of ln ␩ij will also depend on the increasing returns to scale, and Armington demands, all regressors, violating the condition for consistency of OLS.2 predict a gravity relationship for trade flows analogous to In the cases studied in section V we find overwhelming Newton’s law of universal gravitation.1 In its simplest form, evidence that the error terms in the usual log linear speci- the gravity equation for trade states that the trade flow from fication of the gravity equation are heteroskedastic, which country i to country j, denoted by Tij, is proportional to the violates the assumption that ln ␩ij is statistically indepen- product of the two countries’ GDPs, denoted by Yi and Yj, dent of the regressors and suggests that this estimation and inversely proportional to their distance, Dij, broadly method leads to inconsistent estimates of the elasticities of construed to include all factors that might create trade interest. resistance. More generally, A related problem with the analogy between Newtonian gravity and trade is that gravitational force can be very ␣ ␣ ␣ ϭ ␣ 1 2 3 Tij 0Y i Y j Dij , (1) small, but never zero, whereas trade between several pairs of countries is literally zero. In many cases, these zeros occur simply because some pairs of countries did not trade where ␣0, ␣1, ␣2, and ␣3 are unknown parameters. The analogy between trade and the physical force of in a given period. For example, it would not be surprising to 3 gravity, however, clashes with the observation that there is find that Tajikistan and Togo did not trade in a certain year. no set of parameters for which equation (1) will hold exactly These zero observations pose no problem at all for the for an arbitrary set of observations. To account for devia- estimation of gravity equations in their multiplicative form.