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Economics Letters 83 (2004) 293 – 298 www.elsevier.com/locate/econbase The distance puzzle: on the interpretation of the distance coefficient in gravity equations Claudia M. Buch a, Jörn Kleinert a,*, Farid Toubal b b a Kiel Institute for World Economics, Düsternbrooker Weg 120, D-24105 Kiel, Germany Christian-Albrechts-University Kiel, Department of Economics, Wilhelm-Seelig-Platz 1, D-24118 Kiel, Germany Received 20 May 2003; accepted 21 October 2003 Abstract Although globalization has diminished the importance of distance, empirical gravity models find little change in distance coefficients. We argue that changing distance costs are largely reflected in the constant term. A proportional fall in distance costs is consistent with constant distance coefficients. D 2004 Elsevier B.V. All rights reserved. Keywords: Distance coefficients; Gravity equations; Globalization JEL classification: F0; F21 1. Motivation The globalization of the world economy seems to have diminished the economic importance of geographical distance. Technological improvements have potentially facilitated the decline in distance costs. Lower distance costs are likely to be behind the sharp increase in gross trade, capital, and migration flows that characterize the globalization process (World Bank, 1995; Cairncross, 1997). To assess the effect of distance, empirical work often uses gravity equations. These mainly specify the log of bilateral activities as a function of log distance, log GDP, and a set of control variables. Many of these studies find that the coefficient on distance remains unchanged when comparing cross-section estimates for different time periods. We argue that little can actually be learned with regard to changes in distance costs from comparing distance coefficients for different time periods. Changes in distance costs are to a large extent subsumed in the constant term of gravity models specified in logs and are thus confounded with a number of other * Corresponding author. Tel.: +49-431-8814-325; fax: +49-431-8814-500. E-mail address: [email protected] (J. Kleinert). 0165-1765/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2003.10.022 294 C.M. Buch et al. / Economics Letters 83 (2004) 293–298 time-varying factors. The distance coefficient measures the relative importance of economic relationships between the source countries and those host countries that are located far away, as opposed to those located nearby (Frankel, 1997). If distance costs shrink for all countries proportionally, no change in the distance coefficient would be observable. The paper is structured as follows. In Part Two, we sketch earlier evidence on distance coefficients. In Part Three, we lay down the conceptual argument that demonstrates why the interpretation of coefficients on geographical distance as evidence in favor or against changes in ‘distance costs’ is problematic. Part Four concludes. 2. Empirical evidence According to the gravity model of foreign trade, trade between two countries is proportional to the size of their markets, and it is inversely related to geographical distance between the markets.1 Whereas the foreign trade literature has interpreted the distance coefficient as evidence for the presence of physical transportation costs (see, e.g. Freund and Weinhold, 2000), Portes and Rey (2001) argue that distance also captures information costs. A few papers look at changes in distance coefficients over time. Results by Frankel (1997) regarding international trade suggest that the distance coefficient increased in absolute terms from 0.4 in 1965 to 0.7 in 1992. He notes that the interpretation of the decrease in distance coefficients as a decrease in distance costs is problematic. We will pick up this argument below in a more formal framework. Freund and Weinhold (2000) use cross-section trade data for the years 1995 through 1999 and find a relatively constant coefficient on distance of 0.8. More recently, the gravity equation has also been used to study international investment decisions. Studies on international equity flows (Portes and Rey, 2001) and on foreign direct investments (Buch et al., 2003) find empirical evidence for a constant distance coefficient in gravity models. 3. Distance coefficients versus distance costs 3.1. Distance costs and increased integration This section shows that bilateral economic activities such as trade, capital flows, or migration might increase due to decreasing distance costs while estimated distance coefficients remain unchanged. We assume that bilateral economic activities, Yi, j, between country i and a number of partner countries j are, as in standard gravity models, a function of only two variables: GDP of country j and geographic distance, Distij, between the two countries. Some unobservable variables are collected in the constant (b0): lnðYi;j Þ ¼ b0 þ b1 lnðGDPj Þ þ b2 lnðDisti;j Þ þ ui;j ; ð1Þ where ui,j is the residual. This logarithmic specification is the econometric model generally used. Eq. (1) assumes that distance costs, DC, are not readily observable. The underlying assumption is that distance 1 See Anderson and Van Wincoop (2003); Deardorff (1998); or Feenstra et al. (2001) for theoretical underpinnings. For surveys of the empirical literature see Egger (2000, 2002); Frankel (1997); or Leamer and Levinsohn (1995). C.M. Buch et al. / Economics Letters 83 (2004) 293–298 295 costs are a linear function of geographical distance, DC = f(Disti,j) = kDisti,j, and bilateral economic activities are inversely proportional to distance costs, i.e. Yi,j = f(1/DC). Hence, lower distance costs (reflected in a decline in k) would increase bilateral economic activities. However, proportional changes in distance costs, which lead to a proportional change in Yi,j, do not affect the point elasticity of distance in cross-section equations. Hence, changes in the distance coefficient cannot be interpreted as evidence for or against changes in distance costs. To see this, note that in standard OLS model, b1 and b2 are calculated as: X 0 1 2X 31 2 X 3 gdpj disti;j gdpj yi;j ðgdpj Þ2 b̂1 @ A ¼ 4X 5 4X 5; ð2Þ X 2 gdpj disti;j ðdisti;j Þ disti;j yi;j b̂2 where lower cases denote deviations from the logged mean. The constant b0 is calculated as b0 ¼ ȳi;j b1 gdpj b2 disti;j , where upper bar variables denote the logged mean. Now, let the parameter k decline over time such that DC low in t = 1 is only a fraction of the DC high in t = 0. Thus, geographic distance does not change, but distance costs decrease to 1/k of their level in t = 0. The change is assumed to be proportional across all partner countries: k1 = k0/k. For simplicity, suppose the GDP of each partner country j to be constant.2 Due to the decline in distance costs, the level of bilateral economic cross-border activity, Yi,j changes but, if Eq. (1) is specified in logs, the deviation of y from its mean, ȳ, does not: yDC low = yDC high. If the value of all bilateral relationships increases by the same rate, the deviation from the mean remains the same. Hence, b1 and b2 do not change, although distance costs have decreased drastically. However, the mean of ȳ increases, ȳDC low>ȳDC high, which is reflected in an increase in the constant term b0. 3.2. A numerical example This section gives a numerical example which shows that the distance coefficient might remain unchanged even if distance costs decline significantly. We consider bilateral exports of a hypothetical country. Table 1 shows data for two time periods. In the first period, distance costs are eight times higher than in the second period, i.e. k = 8. GDP remains constant. For each country pair, exports are eight times larger in the second period than in the first. Thus, the change in distance costs leads to a proportional increase in all export values, in spite of all exogenous variables remaining unchanged. Using the data of Table 1, estimation of Eq. (1) yields (t-values in parenthesis): Ex0i;j ¼ 1:12ð10:7Þ þ 1:38ð24:4ÞGDPj 0:36ð8:7ÞDisti;j Ex1i;j ¼ 0:96ð9:2Þ þ 1:38ð24:4ÞGDPj 0:36ð8:7ÞDisti;j for t ¼ 0; ð3Þ for t ¼ 1: Although distance costs decrease by a factor of eight, this does not show up in the distance coefficient, b2, which gives the partial effect of distance on exports, but rather in the constant term, b0. The correct interpretation of b2 is that a 10% increase in distance between the two trading partners lowers exports by 3.6%. This effect remains unchanged over time. However, decreasing distance costs have a tremendous effect on export levels, which increase by a factor of eight. This strong increase can 2 Relaxing this assumption would not change the results of the following argument. 296 C.M. Buch et al. / Economics Letters 83 (2004) 293–298 Table 1 A numerical example YDC mean high/YDC low = 1/k = 8, n YDC YDC 1 2 3 4 5 6 7 8 9 10 11 Mean 6.75 5.70 5.31 6.36 4.20 3.12 4.02 3.00 3.42 2.61 1.71 4.2 high low 54.00 45.60 42.48 50.88 33.60 24.96 32.16 24.00 27.36 20.88 13.68 33.6 yDC high, yDC low logarithmic endogenous variable given in deviation from the GDP Distance ln( YDC 15 14 13 12 11 10 9 8 7 6 5 10 7.5 9.0 8.3 2.8 8.0 9.6 4.6 6.0 2.6 3.3 4.3 6 1.91 1.74 1.67 1.85 1.44 1.14 1.39 1.10 1.23 0.96 0.54 1.36 high) ln( YDC low) 3.99 3.82 3.75 3.93 3.51 3.22 3.47 3.18 3.31 3.04 2.62 3.44 YDC high 0.55 0.38 0.31 0.49 0.08 0.22 0.03 0.26 0.13 0.40 0.82 0 YDC low 0.55 0.38 0.31 0.49 0.08 0.22 0.03 0.26 0.13 0.40 0.82 0 be seen in the constant, b0. The change in b0 would be the right measure for the effect of changes in distance cost on exports but, when applied to real data, this effect is mixed with omitted variables which might also be included in b0. 3.3. Robustness of results Qualitatively, the above argument holds regardless of whether distance costs change in a nonproportional way and whether other exogenous variables change as well. Fig. 1 depicts the effect of distance costs on bilateral economic activities such as exports. The graph illustrates four cases which differ in total export levels and in the distribution of these exports over different distances. In all cases, Fig. 1. Proportional vs. non-proportional distance cost changes. C.M. Buch et al. / Economics Letters 83 (2004) 293–298 297 changing average levels of exports are reflected in a changing intercept, while a changing slope reflects a change in b2. Using case I as a benchmark, three scenarios are distinguished: In scenario 1, distance costs decrease proportionally from I to IV. The distance coefficient b2 remains unchanged. All information about the positive effect of decreasing distance costs on Exports is contained in the constant b0IV, which is larger than the constant b0I. In scenario 2, distance costs decrease non-proportionally from I to II, and the decrease is greater for smaller distances. Now, the distance coefficient b2II is larger than b2I. Export levels are higher for small distances in case II than in the benchmark, but they are lower for large distances. In scenario 3, distance costs also decrease non-proportionally from I to III, but now the decrease is smaller for smaller distances. The distance coefficient b2III is now smaller than b2I in absolute terms. Note that for both scenarios involving non-proportional changes, from I to II and from I to III, the total effect of distance on exports always depends on b0 and on b2. Thus, changes in b2 alone do not completely reflect the effect of distance on exports. 3.4. What does the distance coefficient measure? Changes in the distance coefficient over time cannot be interpreted in terms of rising or falling distance costs. Rather, b2 measures how important bilateral economic activities with partners that are far away are relative to those with partners that are close to the home country. A decrease of b2 indicates that trade with countries far away from the home country increases relative to trade with countries closer to the home country. An increase in b2, in contrast, indicates that trade with countries far away decreases relative to trade with countries closer to the home country. 4. Conclusion Increasing volumes of global trade and capital flows witness the globalization of the world economy. Deregulation and technological progress are likely to have led to a decrease in distance costs. Beyond this conventional wisdom, economists are interested in empirically assessing the magnitude of these changes. Since good direct measures of distance costs are unavailable, geographic distance between countries is often used as a proxy in gravity equations. Many empirical studies suggest that the coefficient on distance has not changed significantly over time. In this paper, we have argued that the interpretation of distance coefficients as indicators of a change in distance costs is misleading. Essentially, we cannot infer changes in distance costs from changes in distance coefficients obtained from cross-section equations for different years. In the extreme case of a proportional decline in distance costs and a proportional increase in bilateral economic linkages, the effects of changes in distance costs would show up solely in the constant term of gravity equations. Acknowledgements This paper has partly been written while the authors were visiting the Research Center of the Deutsche Bundesbank. The hospitality of the Bundesbank is gratefully acknowledged. 298 C.M. 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