Download The distance puzzle: on the interpretation of the distance coefficient

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. Buch et al. / Economics Letters 83 (2004) 293–298
References
Anderson, J.E., Van Wincoop, E., 2003. Gravity with gravitas: a solution to the border puzzle. American Economic Review 93
(1), 170 – 192.
Buch, C.M., Kleinert, J., Toubal, F., 2003. The Distance Puzzle: On the Interpretation of the Distance Coefficient in Gravity
Equations. Kiel Institute for World Economics, Working Paper 1159, Kiel, Germany.
Cairncross, F., 1997. The Death of Distance: How the Communications Revolution Will Change Our Lives. Harvard Business
School Publications, Boston, MA.
Deardorff, A.V., 1998. Determinants of bilateral trade: does gravity work in a neoclassical world? In: Jeffrey, F. (Ed.), The
Regionalization of the World Economy. National Bureau of Economic Research, pp. 7 – 22.
Egger, P., 2000. A note on the proper econometric specification of the gravity equation. Economics Letters 66, 25 – 31.
Egger, P., 2002. An econometric view on the estimation of gravity models and the calculation of trade potentials. World
Economy 25 (2), 297 – 312.
Feenstra, R., Markusen, J., Rose, A., 2001. Using the gravity equation to differentiate among alternative theories of trade.
Canadian Journal of Economics 34 (2), 430 – 447.
Frankel, J., 1997. Regional Trading Blocks. Institute for International Economics, Washington, DC.
Freund, C., Weinhold, D., 2000. On the Effect of the Internet on International Trade. Board of Governors of the Federal Reserve
System. International Finance Discussion Papers 693, Washington, DC.
Leamer, E.E., Levinsohn, J., 1995. International trade theory: the evidence. In: Grossman, G., Rogoff, K. (Eds.), Handbook of
International Economics, vol. III. Elsevier, Amsterdam, pp. 1339 – 1394.
Portes, R., Rey, H., 2001. The Determinants of Cross-Border Equity Flows, Working Paper 00-111. Center for International and
Development Economics Research, Department of Economics, University of California, Berkeley, CA.
World Bank, 1995. World Development Report. Oxford University Press, New York, NY.