Internal and External Distance:
Gravity Depends on It!
W ERNER A NTWEILER∗
University of British Columbia
October 30, 2007
Abstract
Gravity models of international trade rely crucially on measures of distance, both internal within a country and external between pairs of countries.
Yet, empirical work on the gravity model makes use of rather imperfect approximations such as the distance between countries’ capital cities and ad-hoc
assumptions about the shape of countries. This results in distance measures
that use different methodologies to derive internal and external measures and
do not allow for shifting patterns of economic activity within a country. Using
the Gridded Population of The World (GPWv3) database, this paper introduces
a distance measure that is based on a weighted harmonic mean of distances
between small latitude-longitude squares within each country that overcomes
these limitations. This paper shows that internal and external distance vary
over time as populations move within countries, and that these distance measures affect results of different types of gravity equation estimations in a significant manner. It also becomes evident that country-internal migration affects
bilateral trade friction. A further purpose of this paper is to document the
new time-varying distance measures and make them available freely to other
researchers.
VERY PRELIMINARY. PLEASE DO NOT CITE.
∗
Sauder School of Business, University of British Columbia, 2053 Main Mall, Vancouver, BC,
V6T 1Z2, Canada. Phone: 604-822-8484. E-mail: werner.antweiler@ubc.ca. This paper owes a debt
of gratitude to Keith Head, Thierry Mayer, and John Ries. Some of the critical ideas in this paper
were inspired by their recent work involving gravity equations and models of economic geography.
This paper was written while I was on sabbatical visiting the University of Kiel in Germany. I would
like to express sincere thanks to my host Horst Raff and his staff for their hospitality.
1
1
Introduction
The gravity equation of international trade remains one of the workhorses of empirical international trade research. In recent years it has found a solid theoretical
underpinning as the gravity equation can be derived from a model of differentiated goods with increasing returns to scale in production and iceberg transportation costs (Anderson and van Wincoop, 2003, 2004). Gravity equations can also be
derived from modern versions of Ricardian (technology- or productivity-driven)
trade; see Eaton and Kortum (2002), Melitz and Ottaviano (2005), and Dekle, Eaton,
and Kortum (2007). All of the empirical work related to the gravity model depends
crucially on a measure of distance. Moreover, modifications of the gravity model
that estimate the border effect also distinguish between internal distance Dii of a
country i and external distance Dij between countries i and j.
Most work simply uses the great-circle distance between capital cities of countries as a ready approximation for the external distance between countries. As the
location of capital cities in large countries may not be central, country distances
may be mismeasured substantially. Some capital cities are not the main agglomerations for historical or political reasons. Even taking the largest urban agglomeration instead of the capital city does not overcome the spatial mismeasurement
problem. Many large countries have multiple large agglomerations.
The internal distance of a country is approximated with a variety of measures,
in particular the area of the country or its square root, multiplied by a suitably
defined proportionality factor. As is the case for external distance, these approximations of internal distance may be quite inaccurate. Furthermore, internal and
external distances are measured using different methods, therefore introducing
further data problems.
The questions thus are: How large are these inaccuracies? Do these inaccuracies matter? If yes, how much do they matter? Can we do better and develop
more accurate measures of internal and external distance? This paper aims to answer these questions by introducing a new set of internal and external distance
measures that are computed consistently. Using the Gridded Population of the World
version 3 (GPWv3) database, it is feasible to calculate average internal and external distances that weight distances by population in origin and destination regions. The relatively fine grid of the GPWv3 database makes these calculations
computationally expensive, but these calculations generate a much more reliable
set of distances measures which conventional methods of distance calculation can
be judged against.
This paper is not the first to address the problem of internal and external distances. Head and Mayer (2000, 2002) provide a survey of some of the problems
and attempts at addressing them. Referring to the mismeasurement of internal distances, the Nitsch (2001) paper was entitled “It’s Not Right but It’s Okay.” Using a
much superior method of calculating internal and external distances consistently,
this paper finds that mismeasurement problems are serious, and that internal and
external distances that take account of shifts of population within countries vary
significantly over time. This result is consistent with Head and Disdier (2007), who
2
report time variation in estimates of the distance effect in gravity equations.
A further intriguing question lurks behind measuring internal and external distance. Does economic distance—as opposed to geographic distance—change over
time? Economic distance is shaped by where economic activity takes place and
thus where in a country people choose to settle. Internal migration within countries has never been linked to international trade. The effect of immigration on
trade has already been studied elsewhere; see for example Wagner, Head, and Ries
(2002) and Mundra (2005). However, this is the first paper to demonstrate that not
just international migration but also country-internal population migration affects
trade friction between countries and thus trade flows.
The new measures of internal and internal distance are made available freely to
other researchers on the author’s web site [URL to be announced]. This web page
also contains ancillary information and further documentation. Estimation of the
gravity equation remains a workhorse of empirical international trade research, including policy analysis. Improving the quality of this work is therefore important.
Whereas Silva and Tenreyro (2006) take a critical look at estimation strategies, this
paper takes a critical look at part of the underlying data.
2
Strategies for Estimating Gravity Equations
In its simplest form, the gravity model predicts that exports Xij from region i and
j is proportional to the output Yi of the exporter region and the expenditures Yj of
the importer region and inverse proportional to the distance Dij between the two
regions. The classic gravity equation can be written as
Xij = A
Yi Y j
Dij
(1)
Estimating a log-linear fully-parameterized version of this equation
ln Xijt = β0 + β1 Yit + β2 + Yjt + β3 Dij[t] + µi + µj + µt + ijt
(2)
has a number of pitfalls, however. As modern derivations of the gravity equation
show, the proportionality factor A in equation (1) is not constant at all, and the
GDP-trade elasticities ought to be constrained to unity. Furthermore, in a multiyear panel of equation (1) many authors choose inconsistent deflators for the volume of trade Xij . As Baldwin and Taglioni (2006) show, this is potentially troublesome. We will return to some of these estimation problems later and address
how the inclusion of various types of dummy variables (µi , µj , and µt ) can mitigate
some of these problems.
There are several approaches to derive modern versions of the gravity equation. Fortunately, they all generate expressions—or estimating equations—that are
quite similar. An elegant way of deriving the gravity equation has been popularized by Anderson and van Wincoop (2003), following in the footsteps of Krugman
(1980), Deardorf (1998), and others. Their derivation of the gravity model assumes
3
that consumers have CES love-of-variety preferences for differentiated goods and
that trade exhibits iceberg transportation costs φij = 1+τij with τij > 0. With product differentiation captured through the substitution elasticity σ, and introducing
world output Yw , the modern gravity equation can be written as
"
Yi Yj φij
Xij =
Yw R j P i
#1−σ
(3)
Here, Rj and Pi are the inward and outward multilateral resistance terms that
capture the notion that trade barriers are relative to those of alternative export or
import destinations. The presence of these two multilateral resistance terms makes
it difficult to estimate (3) directly, and thus Anderson and van Wincoop (2003) relied on an intricate iterative estimation procedure. The estimation problems can
be overcome, however. One empirical shortcut is to estimate the gravity equation
with country fixed effects, country-pair fixed effects, or country-pair-time fixed
effects. The fixed effects are meant to capture the (time-varying) multilateral resistance terms. The fixed effects specification provides unbiased albeit somewhat
less efficient estimates of the parameters of interest than the original Anderson and
van Wincoop (2003) specification.
Alternative procedures transform the gravity equation (3) in a variety of useful ways in order to eliminate the multilateral resistance terms. There are several
advantages of such a transformation in addition to eliminating the multilateral resistance terms. Consider the trade ratio approach developed in Head and Mayer
(2002) and Head and Ries (2001):
"
#
"
φij φji
Xii Xjj
= (σ − 1) ln
ln
Xij Xji
φii φjj
#
(4)
This approach focuses on the product of two ratios, namely the odds Xii /Xji of
buying domestic goods relative to foreign goods in country i with the odds Xjj /Xij
of buying domestic relative to foreign goods in country j. Head and Mayer (2002)
refer to the square root (the geometric mean) of the two odds ratios as bilateral
“trade friction”:
s
Xii Xjj
Ξij ≡
>0
(5)
Xij Xji
If one assumes symmetry in transportation costs (φij = φji ) and zero countryinternal transportation costs (φii = φjj = 1), the expression on the right-hand side
of equation 4 simplifies to 2(σ − 1) ln(φij ). This makes it possible to estimate the
border effect by regressing the left-hand side of equation 4 on a log-linear function of trade impediments such as distance, language, common borders, currency
unions, FTA memberships, and so on. In addition to being able to estimate the
border effect with ordinary least squares, a further advantage of this specification
is the elimination of price effects on the left-hand side of the equation, and thus the
need to find suitable price deflators for the value of exports and imports.
Assuming zero country-internal trade costs is certainly a rather crude approximation, the more so for large countries such as the United States. If one assumes
4
that φij = κDij is merely a function of distance, then (4) can be estimated as
s
ln


Xii Xjj
Dij 
= ln Ξij = β ln  q
Xij Xji
Dii Djj
(6)
where β ≡ (σ − 1)κ > 0. Note the appearance of internal distances Dii and Dij in
the equation. Whereas the left-hand side of the equation is the “measured trade
friction” (ln Ξij ), the right-hand side of the equation is the “predicted trade friction,” which in turn is the product of the substitution elasticity factor (σ − 1) and
the relative distance ratio
Dij
>0
(7)
Ψij ≡ q
Dii Djj
If distance varies over time, so does the relative distance ratio. As in Wei (1996),
internal trade Xii is calculated as output minus all exports, i.e.,
Xii = Yi −
X
(8)
Xij
j
A simple approximation of this is to subtract total exports from GDP. At the industry level, this is more suitably replaced by industry output minus industry exports.
There is yet another useful transformation which has been suggested by Head,
Mayer, and Ries (2007). Instead of using country dyads (pairs), one can also use
country tetrads involving two exporters and two importers. Then
"
#
"
Xij Xlk
φik φlj
ln
= (σ − 1) ln
Xik Xlj
φij φlk
#
(9)
This expression involves only external distances, but for a country pair of interest
this involves choosing two reference countries. Unless the reference country is
held fixed, the above expression involves estimating the border effect from a data
set of dimension O(n4 ) instead of O(n2 ) with respect to the number of countries n.
A variation of this approach chooses the same reference country l = k, in which
case one needs one internal distance and three external distances rather than four
external distances.
Lastly, it is also possible to estimate the gravity equation in a time-differenced
form. If the trade resistance term φij is modeled in log-linear form such that
(10)
ln φijt = α + βUij + γ ln Vijt + δ ln Dijt + it
then the time-differenced version of (6) can be written as
∆ ln Ξijt
Vijt Vjit
(σ − 1)γ
∆ ln
= (σ − 1)δ∆ ln Ψijt +
2
Viit Vjjt
!
+ ijt
(11)
If distance was time invariant, ∆ ln Φijt would be zero. Time differencing equation
(10) cancels out the intercept and time-invariant terms, leaving merely the timevariant explanatory variables. For example, if Vijt captures the tariff mark-up 1 +
τijt that country j imposes on imports from country i, then (11) would appear as
q
∆ ln Ξijt = (σ − 1)δ∆ ln Ψijt + (σ − 1)γ∆ ln (1 + τijt )(1 + τjit ) + µijt
5
(12)
The square root expression is thus the geometric average of the bilateral tariffs.
The time-differenced gravity equation (12) is able to answer one further important question. If internal and external distances change over time, then the relative
distance ratio Ψ should also positively and significantly change the trade friction
Ξ. This amounts to a test of the hypothesis that country-internal migration affects
trade flows by increasing or decreasing trade friction. Testing this hypothesis is a
central theme of this paper.
The discussion above has focused primarily on the Anderson and van Wincoop
(2003) approach for deriving the modern gravity equation. As already observed
in Feenstra, Markusen, and Rose (2001), the gravity equation is consistent with a
number of microeconomic foundations. Given the specific assumptions underlying the model, the Anderson and van Wincoop approach is by no means entirely
satisfactory. Their model is not a full-fledged model of international trade because
it provides no source of comparative advantage (beyond proximity or remoteness), thus ignoring technology and productivity differences (Ricardo) and factor
endowments (Heckscher-Ohlin). Unrealistically, the Anderson and van Wincoop
model predicts that all countries trade with each other, even if only tiny amounts.
This ignores the fact that there is a very large number of zero-trade country pairs.
From this perspective, the Anderson and van Wincoop model might be viewed
as more fitting to differentiated goods trade among OECD countries than interindustry trade among Northern and Southern countries. Significant progress has
been made to remedy some of the problems. Eaton and Kortum (2002) derive a
gravity equation from a Ricardian trade model with a continuum of goods and
probabilistic technology. Melitz and Ottaviano (2005) introduce quadratic utility
with variety preferences for consumers, thus explicitly modeling when bilateral
trade may become zero. Firms compete monopolistically and exhibit dispersion in
technology and productivity. Helpman, Melitz, and Rubinstein (2007) also account
for zero trade, but in a CES utility model. They also allow for firm heterogeneity so
that only the highest productivity (lowest cost) firms export. This brief selection of
recent research papers demonstrates the substantial progress that has been made
finding a sound theoretical underpinning of the gravity equation.
3
Measuring Distance
How can one construct a measure of distance that appropriately reflects the spatial
structure of a country? A useful measure is obviously a weighted average of the
distance between cities: either all the cities in a particular country to get a measure
of internal distance, or all the cities in a pair of countries to get a measure of external
distance. But which weights are appropriate? It may appear that trade flows provide good weights. However, actual trade between cities is usually unobservable.
Furthermore, using a measure of predicted rather than actual trade prevents contaminating the distance measure with trade barriers that are unrelated to distance.
The gravity model provides a useful starting point to obtain a measure of predicted
trade opportunities in the sense that it can be viewed as a search-and-match model.
6
Let Ωi denote the set of locations in country i, and let Ωj denote the set of locations in country j. Then define the distance weight
Wij ≡
Pi Pj
Dij
(13)
as the gravity-style approximation of interaction opportunities between regions i
and j. Hence,
P
Dij =
P
P
P
Dkl Wkl
k∈C
l∈C Pk Pl
= P i P j
P
P
k∈Ci
l∈Cj Wkl
k∈Ci
l∈Cj Dkl
k∈Ci
l∈Cj
(14)
This means that distance Dij is the weighted harmonic mean of the pairwise distances of regions within countries i and j.
Other authors have constructed distance measures using arithmetic means, that
is, using weights Wija ≡ Pi Pj instead of Wij . The main difference between the
two distance weights is that the harmonic mean gives greater weight to small distances, whereas the arithmetic mean gives greater weight to large distances. Which
one should one favour? Helliwell and Verdier (2001) make the case for arithmetic
means. However, since distance is meant to capture “economic distance” in the
context of international trade, the harmonic mean is consistent with the ‘gravity’
potential of trade links.1
4
Empirical Implementation
The CIESIN (2005) GPWv3 database provides a novel opportunity to calculate internal and external distances consistently. The database projects the entire human
population on to squares of 2.5 arc-minutes of longitude and latitude, that is, 24
times 24 squares for each degree of longitude and latitude. Currently, these data
are available for three years: 1990, 1995, and 2000.2 This allows not only for the calculation of distances, but also for the determination of how much these distances
vary over the course of the last decade. Adding the time dimension to distance
improves on earlier work by Mayer and Zignago (2005) and explicitly allows for
the effects of country-internal population migration.
The distance between each populated square is calculated using great circle
distances between midpoints. Let φi and λi denote the latitude and longitude of
location i. Then the great circle distance between the two locations i and j is given
by3
6372.795km arccos (sin φi sin φj + cos φi cos φj cos(λi − λj ))
(15)
1
Nevertheless, the new distance database made available in conjunction with this paper makes
both types of distance measures (harmonic and arithmetic means) available.
2
Population projections are used to generate data sets for quinquennial periods past 2000.
3
Equation (15) shows the approximation for a perfect sphere. For large distances a precise ellipsoidal calculation would be superior, although computationally a lot more expensive. As the
evidence shows, external distance calculations are the least sensitive for large distances, and therefore the spherical approximation suffices.
7
Table 1: Gridded Population Data by Population Thresholds
Population
Latitude-Longitude Squares
# of Countries
Threshold
number
share
cumul.
share included excluded
100,000
1,942
0.02%
1,942
0.02%
71
160
50,000
4,420
0.05%
6,362
0.08%
111
120
20,000
26,859
0.32%
33,221
0.39%
149
82
10,000
76,632
0.91%
109,853
1.30%
168
63
5,000
148,284
1.76%
258,137
3.06%
196
35
2,000
354,688
4.20%
612,825
7.26%
216
15
1,000
361,061
4.28%
973,886
11.54%
219
12
500
494,329
5.86% 1,468,215
17.39%
226
5
200
822,993
9.75% 2,291,208
27.14%
227
4
100
719,466
8.52% 3,010,674
35.66%
228
3
50
638,969
7.57% 3,649,643
43.23%
228
3
20
793,233
9.40% 4,442,876
52.63%
229
2
10
632,681
7.49% 5,075,557
60.12%
229
2
5
558,982
6.62% 5,634,539
66.74%
229
2
2
612,694
7.26% 6,247,233
74.00%
230
1
1
401,205
4.75% 6,648,438
78.75%
230
1
> 0 1,793,670 21.25% 8,442,108 100.00%
231
0
Of course, great circle distances remain a less than ideal approximation of shipping distances for manufactured goods. For now, the computational complexity of
calculating the distances between all the squares in the GPWv3 database limits the
use of superior methods based on geo-computation (those provided by GPS navigation systems, for example, or the popular Google-Maps interface on the web).
Table 1 illustrates the computational complexity of calculating distances between pairs of latitude-longitude squares. If only squares are included that have
a population of 10,000 people or more, we require (109,853)x(109,852)/2, or about
6 billion, calculations. However, at this resolution 63 countries would not be covered because they have low population densities. When including all 8,442,108
latitude-longitude squares, we require (8,442,108)x(8,442,107)/2, or 35.6 trillion,
calculations. Ultimately, I calculate all external distances at a resolution that includes all squares with 50 people or more, at a computational expense of 6.7 trillion
calculations. Even on a very fast Apple G5 (64-bit) computer, acquired in 2006, this
task took three weeks to complete. When considering country internal distances
alone, I include all longitude-latitude squares as the computational complexity of
this problem is considerably smaller.
Figure 1 shows the frequency distribution of external distances for all country
pairs. The median distance is 7,808 km. The distribution of distances trails off
for long inter-continental distances. The smallest distance is 46 km (between the
U.S. and British Virgin Islands) and the largest is 19,744 km (between Ghana and
Tuvalu).
8
Relative Frequency
Figure 1: Distribution of External Distances
1800
1700
1600
1500
1400
1300
1200
1100
1000
900
800
700
600
500
400
300
200
100
0
0
<1
1
<2
2
<3
3
<4
4
<5
5
<6
6
<7
7
<8
8
9 10 11 12 13 14 15 16 17 18 19
<9 <10 <11 <12 <13 <14 <15 <16 <17 <18 <19 <20
Country−Pair Distances (1000km)
Figure 2: Population Thresholds and Internal Distance Calculation for the United
States
700
600
Distance [km]
500
400
300
200
100
0
1
10
100
1k
Population Threshold
9
10k
100k
Choosing the population threshold for excluding latitude-longitude squares is
not unproblematic. Figure 2 illustrates how the choice of threshold affects the internal distances measure for the United States. The distance measure decreases
monotonically as the threshold increases (shown on a logarithmic scale). The slope
of the curve is very flat up to a population threshold of about 50 and then becomes
increasingly steep. This result is a good indication that the population threshold of
50 for the external distance calculation does not compromise accuracy significantly.
5
5.1
How Much Mismeasurement?
Internal Distance
Versions of the gravity equation that make use of the dyadic transformation (6) require measures that relate country-internal distances to external distances of countries. As was shown in Wei (1996), the magnitude of the border effect depends
crucially on the way internal distances are calculated. As Head and Mayer (2000)
point out:
If this internal distance is overestimated, then holding international distance
constant, the negative effect of distance will be underestimated as the cost of
shipping a good internally becomes closer to the cost of shipping it to another country. As a result, the border effect—which accounts for any excessive
amount of trade within a country—will be given more weight in the regression, leading ceteris paribus to an overestimated border effect.
Several approximations have been suggested to address this problem.
(a) Wei (1996) suggest using one-quarter of the distance to the nearest foreign economic center.
(b) Wolf (1997, 2000) calculates internal distance as the distance between the two
largest cities in each country.
(c) Nitsch (2000) and Leamer (1997) assume that internal distance is proportional
to the square root of the area of the country, using a proportionality factor of
0.56 = π −1/2 . Nitsch has previously worked with ratios of 0.2 and 0.6 as well,
derived from Canadian provincial data. Using a more rigorous derivation,
Head and Mayer (2000) calculate a proportionality factor of 0.376 = (2/3)π −1/2 .
(d) Head and Mayer (2000) use employment-weighted means to calculate distances between European Union regions.
(e) Helliwell and Verdier (2001) derive and calculate arithmetic mean distances using population weights for Canadian intra-city, city-rural, and inter-provincial
distances.
(f) Head and Mayer (2002) develop the gravity-style harmonic mean distance and
use it for European data. However, they calculate averages based on a large
set of cities at a more aggregate scale.
(g) Chen (2004) also calculates region-weighted internal distances for European
Union countries, yielding somewhat different results than Head and Mayer
(2000).
10
Table 2: Regression of Circular Area Internal Distance Approximation on Gravity Distances
Intercept
Log Distance
Log Population
Observations
R2
6= 0
6= 1
6= 0
(A)
−0.828c (7.17)
1.291c (10.4)
223
0.906
(B)
−0.823c (7.22)
1.121a (2.19)
0.086c (3.49)
223
0.909
Note: Dependent variable is the circular
area approximation from
p
Head and Mayer (2000), Dic = (2/3) Ai /π, where A is the country
area in square kilometers. Ordinary least squares regressions are performed on all countries except those that are profound outliers. The
outlier truncation rule is abs(ln(Dic /Dig )) > 2, where Dic and Dgc are
internal distances calculated through the circular area approximation
and through population-weighted averages, respectively. Absolute tratios for the tests shown in the table (either 6= 0 or 6= 1) are given in
parentheses. Statistical significance at the 95%, 99%, and 99.9% confidence levels are indicated by the superscripts a, b, and c, respectively.
How well do these approximations capture internal distance? Internal distance
that explicitly allows for the spatial distribution of population (and thus economic
activity) within countries provides a useful benchmark against which one can compare these approximations.
Figure 3 illustrates the degree of mismeasurement4 when comparing the circular area approximation (using the 2/3 proportionality factor) with the populationweighted country-internal distances for the year 2000. Obvious outliers in this
diagram are several island nations where populations are scattered over several
(sometimes distant) islands, for example Tuvalu, the Maldives, or the Marshall
Islands. The other extreme is Suriname. In this mid-sized South-American country (and former Dutch colony) the population is highly concentrated around the
capital Paramaribo, whereas the vast interior is largely unpopulated.
What is immediately apparent in figure 3 is that the data points scatter near
the 45-degree line but with a downward bias for small countries and an upward
bias for large countries. This visually compelling results is confirmed by econometric analysis. Table 2 shows the results from estimating a log-linear model
with the circular area approximation of internal distance as the dependent variable
(2/3)(Ai /π)1/2 ) and the population-weighted internal distance as the independent
variable. Results are show in column (A). In column (B) population has been added
as a second regressor. Outliers were removed from the regression as indicated in
the table notes. The t-test for a 45-degree line (unity of the log distance coefficient)
is soundly rejected. The circular area approximation biases internal distance up4
Referring to these discrepancies as “mismeasurement” implies that the new distance measures
are superior; it is not meant to imply that they are perfect, though. Their limitations have been
acknowledged above.
11
Figure 3: Relative Accuracy of Circular Area Approximation of Internal Distance
Circular Distance (2/3)sqrt(Area[sq.km.]/Pi)
1k
SUR
100
10
MHL
MDV
GTM LBR
TUV
TKL
1
10
100
Population−Weighted Distance [km]
12
ward for large countries and downward for small countries. Notwithstanding the
significant heterogeneity in population densities across countries, the aforementioned bias is also linked to population size. The log distance coefficient in column
(A) can be disaggregated almost completely into a size effect and and a population
effect in column (B). Figure 3 and the results in table 2 show that internal distance
approximations suffer from a potentially troubling bias.
5.2
External Distance
By far the most popular way to approximate external distances for gravity equations has been the great circle distance between the capital cities of countries. Alternatively, some authors have used the distance between the main agglomeration,
which often is not the capital city.5 This method is marginally better, but still ignores that many large countries have multiple important agglomerations, often
hundreds or even thousands of kilometers apart.
Prior to the work presented here, Mayer and Zignago (2005) provide the most
serious attempt to put distance measures on a better footing. In addition to providing distance measures for 225 countries, Mayer and Zignago (2005) also provide an
additional 13 distance measures for countries where the capital city is not the main
city. Furthermore, they provide two weighted distance measures (harmonic and
arithmetic means) that are based on the same idea as the measures proposed in this
paper. The key difference is that the measures proposed in this paper take into account all of the world’s population rather than just a list of urban agglomerations.
The two weighted distance measures used by Mayer and Zignago (2005) calculate distance between two countries based on bilateral distances between the
largest cities of those two countries. The authors stress that this procedure can
be used in a consistent way for both internal and international distances. This
point is echoed loudly in this paper: to use distance measures appropriately
and effectively, internal and external distances must be calculated in a consistent manner. Mayer and Zignago (2005) obtain geographic locations and population data of main agglomerations from the database available on the www.worldgazetteer.com web site. In contrast, the new data set provided here improves upon
these measures and introduces one new distinct and important feature, namely,
that internal and external distances vary over time due to the effects of countryinternal migration.
Capital city distances have been used widely as approximations of external
distance. Compared to the new distance measure, how far off the mark are these
conventional approximations? Table 3 shows the mismeasurement of external distance by comparing capital-city distances with population-weighted distances. To
present results compactly, distances are averaged for each country, using country population as weights. Only those countries are shown where the average
distance difference exceeds 200 km, and countries with less than 5 million inhabi5
For example: Toronto vs. Ottawa in Canada, New York vs. Washington in the United States,
Rio de Janeiro vs. Brasilia in Brazil.
13
Table 3: Mismeasurement of Country Distances by Capital City Distances
Capital GPWv3 Diff.
Min.
Max.
Rel.
Country
[km]
[km]
[km]
[km]
[km]
[%]
HKG Hong Kong
3,051
2,238
813
-379
1,258
36.3
AUS Australia
10,465
9,754
711
-545
1,513
7.3
MOZ Mozambique
7,031
6,336
695 -1,643
1,375
11.0
LKA Sri Lanka
4,232
3,545
687
-445
1,141
19.4
MMR Myanmar
3,389
2,802
586
-447
1,144
20.9
THA Thailand
3,722
3,216
506
-454
1,135
15.7
KHM Cambodia
3,669
3,172
496
-418
1,037
15.6
IDN Indonesia
6,066
5,591
475
-508
1,098
8.5
BGD Bangladesh
2,722
2,247
474
-311
855
21.1
LAO Laos
3,239
2,786
453
-425
1,101
16.3
COD Congo/Zaire
6,189
5,764
425 -1,677
1,494
7.4
MYS Malaysia
4,396
3,979
417
-535
1,083
10.5
TWN Taiwan
3,329
2,945
384
-364
1,372
13.1
CHN China
4,835
4,496
339
-939
1,846
7.6
VNM Viet Nam
3,421
3,136
285
-753
1,451
9.1
PHL Philippines
4,622
4,384
238
-559
1,486
5.4
ZMB Zambia
6,295
6,088
207 -1,376
717
3.4
ARG Argentina
10,578 10,377
201 -1,664
744
1.9
DNK Denmark
3,787
3,998
-211
-762
1,465
-5.3
SWE Sweden
4,168
4,385
-217
-819
1,403
-5.0
GBR Great Britain
4,371
4,607
-236
-741
1,686
-5.1
VEN Venezuela
8,307
8,564
-258
-948
489
-3.0
ECU Ecuador
8,951
9,215
-263
-933
511
-2.9
FIN Finland
4,358
4,644
-286
-744
1,597
-6.2
CAN Canada
6,720
7,063
-344 -1,109
1,760
-4.9
USA United States
8,782
9,257
-475 -1,677
2,121
-5.1
Note: The table is sorted in descending order of average distance mismeasurement. Differences are weighted by population of partner country divided by gridded (GPWv3) distance
in order to capture the ‘trade potential’ of partner countries. Only differences in excess of
200 km are shown for countries that have at least 5 million inhabitants (thus excluding
most island states).
14
tants (mostly island states) are suppressed. The percentage differences can be quite
large. Capital city distances err in particular on the side of putting countries too
far at a distance. The most noticeable case is Hong Kong, whose proximity to a
densely populated area in mainland China, the Pearl River Delta, puts the capital
city of China, Beijing, too far at a distance. On the other end of the scale, Canada
and the United States are more remote to the rest of the world as the location of the
capital cities suggest.
A closer look at the United States illustrates the problem, decomposing the distance measurement by partner country. Table 4 shows the distance difference in descending order of percentage difference where this difference exceeds 10 percent.
There are two large outliers. The capital city measure puts Mexico too far away
from the United States (by 40%), and Canada too close (by 49%)! Intra-NAFTA
trade flows are quite large, and gravity equation estimates that get the distances
between these countries ‘wrong’ by that order of magnitude will lose some of their
credibility. Table 4 shows that the location of Washington, DC on the east coast
of the United States makes European countries appear roughly 10-14% closer than
the population-weighted average distances suggest. Likewise, this effect is also
noticeable for a number of African countries.
To treat these findings more rigorously, table 5 estimates a log-linear model
with the capital-city distance as the dependent variable and the populationweighted distance (linear and squared) as the independent variables, progressively limiting the sample from column (A) through column (E) by lowering the
threshold for excluding country pairs from 50,000 km to 500 km. Columns (A)
through (C) suggest that the relationship between the two measures is both nonlinear (due to the significance of the quadratic term) and subject to a noticeable
positive intercept. The interpretation of this positive intercept captures the story
between Hong-Kong and China’s Pearl River Delta. Major agglomerations are often closer to the border between two countries than the distance between the two
countries’ capital cities suggests. Lowering the distance threshold for including
country pairs has the effect of lowering the correlation between the two series. For
long distances, the mismeasurement problem is negligible, and the R2 is near perfect. However, when looking only at near countries, the mismeasurement problem
is quite pronounced and the R2 drops noticeably.
The measurement problem for external distances is much less a problem than
for internal distances, simply because large distances dominate the picture. The
question then becomes: how much does it matter? The section after next will return to this problem. Before doing so, it is necessary and useful to explore the
overlooked dimension of economic distance: time.
6
Does Distance Vary Over Time?
One of the key contributions of this paper is to add the time dimension to measures
of (economic) distance. With the distance measures introduced in this paper, distance variation over time is caused by internal population migration within coun15
Table 4: Mismeasurement of Country Distances by Capital City Distances: A
Closer Look at the United States of America and her Trading Partners
Country (Capital)
Mexico (Mexico)
Guatemala (Guatemala)
El Salvador (San Salvador)
Honduras (Tegucigalpa)
Venezuela (Caracas)
Austria (Vienna)
Italy (Rome)
Dominican Republic (Santo Domingo)
Hungary (Budapest)
Brazil (Brasilia)
Nigeria (Abuja)
Great Britain (London)
Ukraine (Kiev)
Bulgaria (Sofia)
Burkina Faso (Ouagadougou)
Serbia and Montenegro (Belgrade)
Ivory Coast (Yamoussoukro)
Slovakia (Bratislava)
Angola (Luanda)
Tunisia (Tunis)
Czech Republic (Prague)
Switzerland (Bern)
Belgium (Brussels)
Mali (Bamako)
Chad (N’Djamena)
Russia (Moskva)
Netherlands (Amsterdam)
Libya (Tripoli)
Congo/Zaire (Kinshasa)
Algeria (Algiers)
Portugal (Lisbon)
Guinea (Conakry)
France (Paris)
Niger (Niamey)
Spain (Madrid)
Morocco (Rabat)
Senegal (Dakar)
Canada (Ottawa)
Capital GPWv3
[km]
[km]
3,123
2,230
3,114
2,750
3,176
2,874
3,040
2,757
3,451
3,844
7,043
7,846
7,159
7,998
2,507
2,808
7,271
8,162
6,872
7,722
8,883
9,982
5,835
6,560
7,744
8,708
7,847
8,829
7,932
8,939
7,529
8,495
8,000
9,032
7,098
8,054
10,627 12,061
7,299
8,322
6,809
7,770
6,536
7,474
6,147
7,036
7,386
8,473
9,283 10,656
7,737
8,886
6,096
7,023
7,773
8,978
10,492 12,169
6,767
7,850
5,710
6,627
7,098
8,243
6,100
7,112
8,142
9,500
6,065
7,077
6,138
7,200
6,422
7,537
610
1,205
Diff.
[km]
893
364
302
283
-393
-804
-840
-301
-890
-849
-1,099
-725
-965
-982
-1,007
-966
-1,032
-956
-1,434
-1,023
-960
-938
-888
-1,086
-1,373
-1,149
-927
-1,205
-1,677
-1,084
-918
-1,145
-1,012
-1,358
-1,012
-1,063
-1,115
-595
Rel.
[%]
40.0
13.2
10.5
10.3
-10.2
-10.2
-10.5
-10.7
-10.9
-11.0
-11.0
-11.1
-11.1
-11.1
-11.3
-11.4
-11.4
-11.9
-11.9
-12.3
-12.4
-12.5
-12.6
-12.8
-12.9
-12.9
-13.2
-13.4
-13.8
-13.8
-13.8
-13.9
-14.2
-14.3
-14.3
-14.8
-14.8
-49.4
Note: The table is sorted in descending order of distance mismeasurement. Only differences in excess of 200 km are shown for countries that have at least 5 million inhabitants
(thus excluding most island states).
16
Table 5: Regression of Capital-City Distances on Gravity
Thresholds
(A)
(B)
(C)
Threshold [km]
50,000
10,000
5,000
c
c
Intercept
0.456
0.454
0.569c
(13.3)
(8.14)
(4.56)
c
c
0.867c
0.900
Log Distance
0.899
(25.2)
(63.0)
(108)
c
c
Log Distance Squared
0.006
0.006
0.008c
(11.0)
(6.07)
(3.31)
Observations
22,155
14,668
6,315
2
R
0.991
0.985
0.964
Distances by Distance
(D)
1,000
−0.803
(.788)
1.317c
(3.84)
−0.029
(.996)
716
0.742
(E)
500
−1.711
(.595)
1.672
(1.56)
−0.063
(.633)
231
0.535
Note: Ordinary least squares regressions are performed on all observations where the gravity distance is smaller than or equal to the threshold indicated in the first data row of the
table. Dependent variable is the capital-city distance. Absolute t-ratios are given in parentheses. Statistical significance at the 95%, 99%, and 99.9% confidence levels are indicated by
the superscripts a, b, and c, respectively.
tries. There are several well-known economic processes that can shape internal
migration:
1. Urbanization (primarily related to the transition from agricultural production to manufacturing)
2. Regional economic ascent and decline, often due to natural resources availability (minerals, oil & gas, fisheries, forestry, etc.)
3. Agglomeration (external economies of scale in production)
4. Demographics (e.g., retirees prefer warmer climates)
5. Policy-induced migration (subsidies, taxes)
6. Trade liberalization with neighbouring countries
As people move within countries, they can move closer together or further apart.
This affects average internal distance. Similarly, they can move closer to one border
or another. This affects external distances between countries. The GPWv3 database
provides the tool to investigate the magnitude of internal migration and how it
affects distances.
6.1
Internal Distance
Table 6 shows internal country distances, sorted in descending order of the percentage increase in the population-weighted harmonic (‘gravity’) mean between
1990 and 2000. Countries with less than 3% change are suppressed, as are a number of very small island and city-state countries.
The percentage changes can be remarkably large. Several countries have
changes in excess or ten percent such as Guinea, Argentina, Venezuela, Russia,
17
Table 6: Internal Country Distance
Country
Guinea
Argentina
Venezuela
Armenia
Cuba
Côte d’Ivoire
Mexico
Uruguay
Latvia
United States
Israel
Peru
Ireland
Lesotho
Mauritania
Canada
Georgia
Norway
Kyrgyzstan
Equatorial Guinea
El Salvador
Sudan
South Africa
Malaysia
Turkey
United Arab Emirates
Nicaragua
Panama
Mongolia
Dominican Republic
Sweden
Greenland
Botswana
Zimbabwe
Finland
Paraguay
New Zealand
Iceland
Haiti
Senegal
Bahamas
Somalia
Russian Federation
Gambia
Albania
Greece
Harmonic Average
in 2000 ∆ 1990-2000
[km] [km]
[%]
147.2
28.9
24.4
107.6
11.1
11.5
130.6
12.1
10.2
30.3
2.0
6.9
98.4
5.0
5.4
118.0
4.9
4.4
196.7
8.2
4.4
41.3
1.6
3.9
38.7
1.4
3.8
607.9
21.5
3.7
34.5
1.2
3.5
106.4
3.3
3.2
49.6
1.6
3.2
54.5
-1.7
-3.0
111.7
-3.5
-3.0
179.4
-5.8
-3.1
55.1
-1.8
-3.2
90.9
-3.1
-3.3
82.7
-3.0
-3.5
62.4
-2.3
-3.5
33.0
-1.3
-3.9
307.9 -12.7
-4.0
194.7
-8.2
-4.0
128.1
-5.5
-4.1
262.5 -12.4
-4.5
66.7
-3.3
-4.7
61.1
-3.1
-4.8
55.8
-3.0
-5.0
31.8
-1.8
-5.5
53.7
-3.2
-5.7
110.2
-6.9
-5.9
31.3
-2.1
-6.2
124.7
-8.4
-6.3
117.9
-8.2
-6.5
84.9
-6.6
-7.2
55.4
-4.5
-7.5
71.2
-6.2
-8.0
17.2
-1.5
-8.1
47.3
-4.2
-8.2
65.3
-6.0
-8.4
15.1
-1.4
-8.7
177.9 -19.1
-9.7
444.4 -54.2 -10.9
30.0
-3.9 -11.4
50.9
-7.5 -12.8
52.7
-8.9 -14.5
Arithmetic Average
in 2000 ∆ 1990-2000
[km] [km]
[%]
306.0
11.7
4.0
632.9
20.3
3.3
384.2
7.3
1.9
75.1
2.9
4.0
364.6
-1.2
-0.3
268.8
4.6
1.7
705.6
22.9
3.4
174.2
0.3
0.2
126.5
0.6
0.4
1778.7
24.5
1.4
81.3
1.0
1.2
573.9
8.0
1.4
135.6
-1.0
-0.7
85.0
-2.5
-2.9
390.7
1.2
0.3
1552.2
18.5
1.2
148.9
-7.4
-4.8
330.4
-4.4
-1.3
232.4
-2.3
-1.0
170.4
6.7
4.1
70.9
-1.6
-2.2
666.5
-7.0
-1.0
562.0 -17.6
-3.0
636.6
27.0
4.4
543.3
0.9
0.2
163.6
-4.3
-2.6
134.4
-0.9
-0.7
160.8
-4.7
-2.9
89.6
-7.4
-7.7
103.2
-1.5
-1.4
315.1
-2.4
-0.8
515.2
2.0
0.4
262.8
-7.1
-2.6
284.3
0.7
0.3
230.1
-2.2
-0.9
175.8
-3.0
-1.7
401.0 -10.0
-2.4
74.2 -10.6 -12.5
105.7
-4.4
-4.0
191.5
-2.9
-1.5
94.3
-9.7
-9.3
538.2 -13.0
-2.4
1705.5 -92.9
-5.2
113.6
-1.3
-1.1
86.5 -11.1 -11.3
243.2
-7.5
-3.0
Note: Only changes in excess of plus or minus three percent in harmonic
distance are shown. Islands, city states, as well as small territories and countries (including Bahrain, Brunei, Djibouti, Hong-Kong, Macao, Qatar, and
the Vatican) were excluded from this analysis.
18
Gambia, Albania, and Greece. Some of the distance changes can also be sizeable
in absolute terms. For example, Russia’s internal distance shrank by 54 km (or
about 11%) in the last decade. Russia’s economic turmoil following the end of
communism contributed to significant population movements and accelerated agglomeration.
Table 6 also shows the difference between harmonic and arithmetic means of
internal distance. Arithmetic averages tend to be significantly larger because they
give greater prominence to long distances. By comparison, harmonic means give
greater weight to short distances. Most changes agree in direction, although not
always. For example, Canada has shrunk by 3.1% in terms of its harmonic average distance, but has expanded by 1.2% in terms of its arithmetic average distance.
However, the United States has expanded consistently by 3.7% and 1.4%, depending on the type of average.
Changes in internal distance over a single decade are sufficiently large in magnitude to suggest that the time dimension matters. Without having data for a
longer time period, one can only speculate how much larger the effect of countryinternal migration may have been. Nevertheless, the changes between 1990 and
2000 show how sensitive empirical work will be that makes use of internal country
distances, and how this work will lead to questionable results if the time dimension (and thus internal population migration) is ignored.
6.2
External Distance
As the results in the previous section show, population migration within countries
is sufficient to change internal distances measurably even over a single decade. If
the population ‘centre of gravity’ of countries changes, so does the distance relative to other countries. Table 7 reveals the changes for the United States between
1990 and 2000, sorted in descending order of the distance change. Only changes
in excess of 25km are shown. Also indicated are the percentage changes. Most noticeably, on average the populations of Canada and the United States have moved
28km further away from each other. With 2.4% this is also the largest percentage
change for any country pair. By comparison, the populations of Mexico and the
United States have moved 64km closer to each other; a change of 2% and the second largest change for any country pair.
From table 6 we know that the internal distances of the United States, Mexico,
and Canada have all changed over the last decade, indicating considerable population movements. Apparently, the ‘centre of gravity’ of the United States has
moved south, closer towards Mexico and further away from Canada. The magnitude of these changes is not negligible. A 2.4% change in economic distance is not
that far off in magnitude from average tariff reductions during the first decade of
NAFTA. Consequently, gravity equations would tend to underestimate the effect
of trade liberalization between Canada and the United States and overestimate the
effect of trade liberalization between Mexico and the United States.
Table 8 shows the largest distance changes for any country pair, sorted in descending order of percentage change, and suppressing changes smaller than one
19
Table 7: External Country Distances vis-à-vis United States
Harmonic Average
Arithmetic Average
in 2000 ∆ 1990-2000 in 2000 ∆ 1990-2000
Partner Country
[km] [km]
[%]
[km] [km]
[%]
Somalia
13,328 82.6
0.6 13,419 83.4
0.6
Guinea
8,243 66.5
0.8
8,453 69.3
0.8
Russian Federation
8,886 60.5
0.7
8,987 56.4
0.6
Niger
9,500 56.0
0.6
9,676 56.9
0.6
Spain
7,077 51.2
0.7
7,275 51.9
0.7
Mauritania
7,628 49.7
0.7
7,856 52.3
0.7
Greece
9,026 49.5
0.6
9,146 49.3
0.5
Mali
8,473 49.4
0.6
8,675 51.3
0.6
Algeria
7,850 48.9
0.6
8,027 49.5
0.6
Iran
10,946 47.8
0.4 11,019 48.0
0.4
Botswana
13,801 46.5
0.3 13,927 48.2
0.3
Serbia and Montenegro
8,495 46.4
0.5
8,615 46.4
0.5
Tunisia
8,322 46.3
0.6
8,477 46.7
0.6
France
7,112 46.0
0.7
7,280 46.4
0.6
United Kingdom
6,560 45.9
0.7
6,724 46.0
0.7
Zambia
13,387 45.2
0.3 13,515 46.6
0.3
Greenland
3,971 39.8
1.0
4,157 39.1
1.0
Canada
1,205 28.0
2.4
2,049 30.0
1.5
Timor-Leste
14,844 -26.7
-0.2 14,965 -24.5
-0.2
Nauru
10,818 -40.0
-0.4 11,037 -39.0
-0.4
Tonga
10,686 -42.1
-0.4 10,875 -42.6
-0.4
Samoa
9,944 -44.6
-0.4 10,156 -44.9
-0.4
Fiji
11,033 -44.8
-0.4 11,230 -44.9
-0.4
Mexico
2,230 -46.1
-2.0
2,564 -25.7
-1.0
New Zealand
12,744 -57.0
-0.4 12,882 -57.3
-0.4
Australia
14,703 -58.4
-0.4 14,898 -55.7
-0.4
Note: Only changes in excess of plus 45 percent or minus 25 percent in harmonic distance are shown. Islands, city states, as well as small territories and countries (including Bahrain, Brunei, Djibouti, Hong-Kong, Macao, Qatar, and the Vatican) were excluded
from this analysis.
20
Table 8: Largest External Country Distance Changes
Harmonic Average
Arithmetic Average
in 2000 ∆ 1990-2000 in 2000 ∆ 1990-2000
Country Pair
[km] [km]
[%]
[km] [km]
[%]
Canada, United States
1,205 28.0
2.4
2,049 30.0
1.5
Greenland, United States
3,971 39.8
1.0
4,157 39.1
1.0
United States, Guinea
8,243 66.5
0.8
8,453 69.3
0.8
United States, Spain
7,077 51.2
0.7
7,275 51.9
0.7
United States, Ireland
6,182 44.3
0.7
6,359 44.5
0.7
United States, United Kingdom
6,560 45.9
0.7
6,724 46.0
0.7
United States, Russian Federation
8,886 60.5
0.7
8,987 56.4
0.6
United States, Iceland
5,191 35.1
0.7
5,350 35.2
0.7
Honduras, United States
2,757 -19.5
-0.7
2,978 -11.2
-0.4
Cuba, United States
1,990 -14.7
-0.7
2,471
8.0
0.3
Guatemala, United States
2,750 -20.8
-0.8
2,949 -14.6
-0.5
Mexico, United States
2,230 -46.1
-2.0
2,564 -25.7
-1.0
Note: Only changes in excess of plus/minus two-thirds of a percent in harmonic distance are shown.
Islands, city states, as well as small territories and countries (including Bahrain, Brunei, Djibouti,
Hong-Kong, Macao, Qatar, and the Vatican) were excluded from this analysis.
two-thirds of a percent. All the large changes involve the United States. This appears to be a reflection of the high level of internal mobility in the United States.
The U.S. Census Bureau reports continuing population losses for the Northeast
and Midwest, with significant population gains for the South and West. This general south-west movement of the ‘centre of gravity’ for the United States is reflected in changes in external distances. Distances to European countries, Canada,
and Russia have increased, and distances to numerous Latin-American countries
have decreased.
7
Re-Estimating Gravity: Does it really matter?
The previous sections have shown that internal and external distances vary over
time, and that approximations of these distances are subject to a degree of mismeasurement. These problems are particularly pronounced for internal distances and
short external distances. But how much do these mismeasurements really matter?
Do they affect estimation results in a significant manner? To explore this question it is useful to re-estimate both the modern and conventional types of gravity
equations.
The empirical work in this section makes use of additional data sources. Trade
and production data were obtained from Nicita and Olarrega (2007).
21
7.1
Modern Gravity Equation
Modern forms of the gravity equation can be explored usefully by regressing bilateral (geometric mean) trade friction (Ξ) on relative distance (Ψ) and other determinants of the border effect. As discussed earlier, the key benefit of this transformation is that this method avoids estimating nuisance parameters (inward and outward multilateral resistance) and finding a suitable price deflator for trade flows.
Even the GDP measures, which are constrained to unity elasticities in modern versions of the gravity equation, cancel out conveniently.
It is worthwhile exploring the raw data of measured trade friction and relative
distance ratios. Figure 4 plots the data points for the year 2000 in four panels.
Panels A through D present charts for all country pairs, pairs of OECD countries,
pairs of near countries (whose external distance is less than 3000 km), and for country pairs involving the United States, respectively. Measured trade friction (Ξ) is
shown on the vertical axis and the relative distance ratio (Ψ) is shown on the horizontal axis. The correlation between the data points is most striking when considering only OECD country pairs. In the other panels a positive correlation is visible,
but there are clearly other determinants at work that scatter the data points more
widely. For the trading partners of the United States (panel D), NAFTA partners
Mexico and Canada unsurprisingly exhibit the lowest relative distance and lowest
trade friction.
Estimating gravity equation (6) amounts to identifying the effect of distance
and other explanatory variables on bilateral (geometric mean) trade friction Ξ. The
simplest form of this model is
ln Ξijt = δ ln Ψij[t] + µt + ijt
(16)
where Ψij[t] is the relative distance ratio from (7), either captured by the harmonic
mean distance or by approximation through circular area distance (for internal distance) and capital city distance (for external distance). More sophisticated versions
of this model allow for further determinants of the border effect.
Table 9 shows the results of the regressions for a panel consisting of the years
1990-2000, with columns (A), (B) and (C) selecting all country pairs, country pairs
where both countries are OECD members, and country pairs whose external distance is less than 3000 km, respectively. Year fixed effects (µt ) were included in all
regressions, but their estimates are not shown.
How do the different relative distance measures perform in comparison? Quite
clearly, the population-weighted distance measures outperform the conventional
approximation distance measures. The new distance measures are estimated more
precisely (the t-ratios are higher) and the regression R2 are all significantly higher.
Unsurprisingly, the dyadic gravity equation performs best for OECD countries.
This may be attributed to the fact that the modern gravity equation is based on
differentiated goods trade rather than inter-industry commodity trade.
The magnitude of the estimates for Ψ are also quite different across types of
distance measures. When looking at the results in columns (A1) and (B1) the effect
of distance is quite consistent and around unity for the regressions involving all
22
Figure 4: Measured Trade Friction and Relative Distance Ratios
Panel A: All Country Pairs
Panel B: OECD Country Pairs
24
20
19
22
18
20
17
16
Trade Friction
Trade Friction
18
16
15
14
13
14
12
12
11
10
10
9
8
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8
8.0
0.0
1.0
2.0
Population−Weighted Distance Ratio
3.0
4.0
5.0
6.0
7.0
Population−Weighted Distance Ratio
Panel C: Near Country Pairs
Panel D: United States of America
22
22
BDI
SDN
VUT
20
CAF
20
ZWE
ERI
18
Trade Friction
Trade Friction
18
IRN
16
14
ETH
TON
TGO ALB
BFA
NER
CPV
GIN BEN
TJK
KGZ MWI
MLI
TZA
SEN
BWA
MOZVCT
CMR
MDALSO
GAB
PNG
NGA KEN NAM
TKM
FJI MDV NCL
GEO
AZE
ARM
BLR
SYR
OMN PYF
CYP
MDG
DMA
TUN
LBN
HRV
ATG
LTU
MNG
CIV
NPL
LCA
KWT
KAZ BGR LVA
MUS
JOR
SVK
PRYISL
MAR
VNM
EST
GHA
MKD
SVN
KNA
BGD
BRB
POL
NIC
URY
UKR
KHM SWZ
GRD
GRC
EGY
PAK
PAN DZA
MAC
ARE
PRT
BLZ
ECU
BOL NOR
CZE
DNK SUR
SLV SAUGUY
GTM
HUN
BHR
RUS
FIN ARG
TUR
AUT
ZAF
PER
COL BHS
ESP
NZL
HND
IDN CHL
IND
JAM
AUS
MLT
ITA SWE TTO
VEN
BRA CRI
FRA
CHE
NLD
CHN
ISR
DEU
GBR
THA
JPN
PHL
KOR
IRL
TWN
16
14
12
12
10
10
CAN
8
0.0
1.0
2.0
3.0
4.0
5.0
8
6.0
1.0
Population−Weighted Distance Ratio
GMB
UGA
1.5
MEX
2.0
MYS
2.5
3.0
3.5
4.0
4.5
5.0
5.5
Population−Weighted Distance Ratio
Note: Measured trade frictions Ξ is shown on the vertical axis, and relative distance ratio Ψ is
shown on the horizontal axis, for the country pairs indicated. All data points correspond to the
year 2000. In Panel B only country pairs where both countries were OECD members in 2000 are
included. In Panel C only country pairs whose external distance is less than 3000 km are included.
In Panel D the partner countries of the United States are labeled.
23
Table 9: Dyadic Gravity Equation Regression, 1990-2000
(A1)
Country Pairs
Intercept (1995)
Ψ Harmonic Mean
(B1)
All
(B2)
Both OECD
11.693c
12.816c
9.493c
10.102c
(208)
1.014c
(94.8)
(249)
(107)
1.080c
(62.8)
(105)
Ψ Approximated
Observations
R2
(A2)
37,518
0.196
0.860c
(83.0)
37,518
0.158
3,212
0.553
1.134c
(50.4)
3,212
0.444
(C1)
(C2)
≤ 3000km
10.750c
(115)
1.190c
(51.0)
9,872
0.212
12.056c
(140)
0.896c
(39.7)
9,872
0.141
Note: Dependent variable is the Head and Ries (2001) measured trade friction (ln Ξijt ). Key
regressors are the distance friction ratios (ln Φijt ) based on the population-weighted harmonic
mean distances (columns A1, B1, C1) or the capital-city/circular-area approximate distances
(columns A2, B2, C2). Estimation is by ordinary least squares. Year fixed effects are included
in all regressions. Annual data for 1990-200 are used, with distances interpolated exponentially between the 1990, 1995, and 2000 GPWv3-derived data. Absolute t-ratios are given in
parentheses. Statistical significance at the 95%, 99%, and 99.9% confidence levels are indicated by the superscripts a, b, and c, respectively.
countries or only OECD pairs. By comparison, in columns (A2) and (B2) there is
significant instability in the results, and the effect of distance varies between 0.86
and 1.13.
Lastly, the results in columns (C1) and (C2) show that when including only near
country pairs in the analysis, the gap in estimates of the distance effect widens (1.19
versus 0.90), while the R2 favors the new distance measures strongly.
7.2
Time-Differenced Modern Gravity Equation
The main theme of this paper has been that economic distance varies over time due
to the effect of population migration within countries. The time-differenced version of the gravity equation as expressed in equation (12) provides a suitable platform for testing the hypothesis that country-internal population migration changes
trade frictions, and thus the volume of trade.
Using a panel of only two time-differenced periods (1990-1995 and 1995-2000)
due to the fact that the GPWv3 population data are only available for 1990, 1995,
and 2000, table 10 shows estimation results for a simplified version of equation (12):
∆5 ln Ξijt = µt + ∆5 ln Ψijt + ijt
(17)
with two fixed effects (µt ) capturing overall changes in trade friction. In table 10
these fixed effects are negative and significant, demonstrating that trade frictions
have decreased in general. Much of this will be due to reductions in tariffs fol24
Table 10: Dyadic Time-Differenced Gravity Equation Regressions
Country Group
1990/1995 Dummy
1995/2000 Dummy
Distance Ratio
Observations
R2
(A)
(B)
(C)
All
both OECD
≤ 3000km
−0.339c
−0.203c
(12.7)
−0.272c
(13.2)
0.870c
(12.1)
4,920
0.092
(6.63)
−0.414c
(16.4)
5.700c
(4.97)
630
0.322
−0.365c
(7.23)
−0.311c
(8.56)
1.011c
(12.0)
1,290
0.196
Note: Dependent variable is the time-differenced Head and
Ries (2001) measured trade friction (∆ ln Ξijt ). Key regressor is
the time-differenced distance friction ratio (∆ ln Φijt ) based on
population-weighted harmonic mean distances. The data set is
constructed by calculating the differences between the years 1990,
1995, and 2000, resulting in two time-differenced data points for
each country. Estimation is by ordinary least squares. Absolute tratios are given in parentheses. Statistical significance at the 95%,
99%, and 99.9% confidence levels are indicated by the superscripts
a, b, and c, respectively.
lowing the conclusion of the Uruguay round of the GATT, and some will also be
due to the information technology revolution, which has facilitated outsourcing
and the integration of production across borders. Columns (A) through (C) show
results for all country pairs, OECD country pairs, and near country pairs (less than
3000 km apart), respectively. Results for OECD country pairs are again the most
compelling based on the regression R2 .
The effect of changes in the relative distance on trade frictions is positive and
significant in all cases, with the magnitude ranging from 0.87 for all countries over
1.01 for near countries to 5.70 for OECD countries. The magnitude for the OECD
countries seems to suggest that economic activity and trade are particularly sensitive to the effects of internal migration for these countries. As internal migration is
of course endogenous and depends on a variety of driving forces, discussed earlier
in this paper, the question arises what to make of this high level of sensitivity. Without any “natural experiments” in population dynamics in OECD countries during
the last decade, it is difficult to instrument the relative distance ratio changes with
any exogenous variables.
While the above results suggest that internal migration affects trade, there is
also the potential for reverse causality. Trade liberalization with neighbouring
countries may induce country-internal population migration if industries locate
close to the border. Examples of such cross-border agglomerations include the Ontario/Michigan auto industry, the Mexican maquiladora, and the Pearl River Delta
25
next to Hong Kong. Arguably, trade liberalization (e.g., the US-Canada auto pact,
NAFTA) may play an important role here. How can one account for this potential endogeneity? One way is to explicitly account for tariff reductions in (17), as
suggested in equation (12). Another way is to control for industry location. While
these important questions are subject matter for further research, this paper stops
at presenting prima facies evidence for internal migration’s effect on trade.
7.3
Conventional Gravity Equation
Estimation of the conventional gravity equation in log-linear form remains frought
with problems. Baldwin and Taglioni (2006) point out a number of common mistakes in estimating the gravity equation in this form. Typical fixes for estimating
a log-linear form of the gravity equation involve time dummies, country dummies (one each for each exporter and importer country, i.e. 2n), and country-pair
dummies (i.e. n(n − 1)/2). However, using country-pair fixed effects eliminates
time-invariant regressors such as capital-city distance.
Table 11 presents results that show how introducing a refined measure of external distance affects gravity equation estimates (columns A through D), and how
these results stack up against the conventional distance approximation via capital cities (columns E through H). The panel consists of all countries for the period 1990-2000. Columns (A) and (E) show results for a simple pooled estimation.
Columns (B) and (F) show results with time dummies included (except for 1995,
the base year). Columns (C) and (G) show results with both time dummies and
country dummies included. Country dummies are provided for each exporter and
importer country except the United States, which serves as the reference country. Columns (D) and (H) show results with both set of dummies, but now using
weighted least squares instead of ordinary least squares. Weights are the product
of exporter and importer GDP.
Using the new distance measures improves the R2 of each regression marginally,
and the distance measures tend to be slightly more significant. The magnitude of
the estimates varies only slightly. This will come as good news to the authors of the
huge number of papers that have used conventional gravity equations in the past.
They will not have to re-estimate their models. However, the results in table 11
show how much specification matters. Including dummy variables changes the
magnitude of the distance effect; it also hugely boosts the R2 . Nevertheless, looking at the R2 alone provides a rather incomplete picture of performance. If one
is interested in predicting trade flows rather than testing trade models, a natural
question to ask is how large the prediction error is. A simple but useful measure is
the absolute error
P ijt Xijt − X̂ijt (18)
A = 100% ·
P
ijt Xijt
that shows how much trade is mispredicted as a percentage relative to total world
trade. The corresponding numbers are shown in row “Absolute Error” in table 11.
Surprisingly, using time and country dummies increases the absolute error even
26
Table 11: Conventional Gravity Equation Estimation
Time Fixed Effects
Country Fixed Effects
Estimation Method
no
no
OLS
(A)
yes
yes
no
yes
OLS
OLS
Panel A: Time-Varying Distance Measure
(B)
(C)
Intercept (1995)
Log Exporter GDP
Log Importer GDP
Log Distance
R2
Absolute Error
−27.70c
−27.67c
10.761c
Intercept (1995)
Log Exporter GDP
Log Importer GDP
Log Distance
R2
Absolute Error
−27.62c (240)
1.202c (386)
0.770c (298)
−1.456c (198)
0.631
82.1%
−27.59c (237)
1.201c (386)
0.772c (299)
−1.463c (199)
0.632
83.9%
10.742c (7.20)
0.477c (11.9)
0.362c (11.7)
−1.880c (260)
0.744
348.8%
yes
yes
WLS
(D)
(242)
(238)
(7.23) −8.873c (12.4)
1.205c (387)
1.203c (387)
0.476c (11.9)
0.535c (32.0)
0.772c (299)
0.774c (300)
0.368c (11.9)
0.723c (44.9)
−1.458c (200) −1.465c (201) −1.867c (262) −1.053c (374)
0.632
0.633
0.745
0.926
84.0%
85.9%
331.8%
49.9%
Panel B: Capital City Distance Measure
(E)
(F)
(G)
(H)
−10.11c (13.7)
0.553c (32.2)
0.741c (44.7)
−1.056c (355)
0.922
51.6%
Note: Dependent variable is the log of exports. Estimation is by ordinary least squares (OLS) or
weighted least squares (WLS). When weighted least squares is used, the weights are the product of
the exporter and importer GDP. Time fixed effects mean that a dummy has been included for each
year except 1995. Country fixed effects mean that a separate dummy has been included for each exporter country and each importer country except for the United States. Absolute t-ratios are given in
parentheses. Statistical significance at the 95%, 99%, and 99.9% confidence levels are indicated by the
superscripts a, b, and c, respectively.
though the R2 improves significantly. This is because estimation of the log-linear
gravity equation minimizes relative deviations, not absolute deviations. A central
flaw in using the gravity equation to predict trade flows is that it treats relative errors alike, no matter how large the actual trade volume. Thus a relative error in the
large trade volume between Canada and the United States counts the same as the
same relative error in the small trade volume between Ghana and Burkina Faso.
This problem can be addressed by using weighted least squares when estimating
the gravity equation, giving greater weight to pairs of large countries. Columns
(D) and (H) show estimates when GDP product weights are used. The R2 goes up
remarkably while the absolute error shrinks enormously.
The bottom line of the results in table 11 is that using better distance measures
improves the estimates only marginally, whereas the real challenge lies in finding
the right model specification. This result is consistent with Feenstra, Markusen,
and Rose (2001), who also document that the estimates of the effect of distance de27
pend on the specification, the country sample, and the type of goods (differentiated
or homogeneous) that are being considered. Moreover, the results also show that
weighting large and small trading partners equally may not be the best estimation
method if what one really cares about is predicting total trade flows.
8
Conclusions
This paper has set out to introduce a new time-varying measure of internal (intracountry) distance and external (inter-country) distance. Using the Gridded Population of the World database that provides population figures for all 2.5-by-2.5 arcminute latitude-longitude squares it is possible to calculate (harmonic) mean distances within and between countries consistently. These new distance measures
constitute a significant improvement over previous ad-hoc approximations of internal and external distance. In addition to providing new time-varying internal
and external distance measures, this paper is also able to answer a number of related research questions.
First, do distance measures vary over time significantly? The answer is yes. In
particular, internal distances show significant time variation. Over the last decade,
numerous countries exhibited “expansion” or “shrinkage” of more than ten percent. For example, internal migration in Russia led to an eleven percent decrease
in average internal distance. External distance also varies over time, although at
a relatively smaller scale. The United States has experienced one of the largest
relative change in external distance. Populations in Canada and the United States
have moved 28km (or 2.4%) away from each other over the last decade. During the
same period populations in Mexico and the United States have moved 64km (or
2.0%) closer. These changes are primarily the result of large internal migration in
the United States, with the Northeast losing population and the Southwest gaining
population.
Second, how do conventional ad-hoc approximate measures of distance stack
up against the new measures of distance? Internal distance approximations using circular areas exhibit a noticeable bias compared to the new measures. These
approximations produce internal distances that are too high for large (and populous) countries and too low for small countries. External distance approximations
through capital city distances are likewise very problematic. Not only is the location of the capital city not always the main urban agglomeration in a country, but
where countries have numerous urban agglomerations, it matter if these agglomerations are closer to the border than the capital city. For example, Stockholm,
the capital of Sweden, lies almost on the periphery of the densely populated area
of this country. Populous cities such as Malmö and Göteborg are much closer to
neighbouring countries than Stockholm. Mismeasurements of external distance
can be quite large. Averaged across trading partners, differences range from –5%
(United States) to +36% (Hong Kong). The capital of the United States (Washington, DC) puts the country too close to Europe (roughly by 10%), too far away from
Mexico (by 40%), and too close to Canada (by 50%).
28
Third, does the use of conventional ad-hoc distance measures bias estimates
of the effect of distance when estimating both modern and conventional gravity
equations? The evidence in this paper shows that conventional gravity equation
estimates are quite robust to mismeasurements of external distance, in part because
of the relative prominence of long distance country pairs. However, estimating
modern versions of the gravity equation requires both internal and external distances. This paper shows that the method of distance calculation matters strongly
for country-internal distances and shorter distances between countries. In comparison to conventional ad-hoc measures, the new distance measures perform vastly
better.
Fourth, does time-varying distance affect trade? The answer is yes. Using a
time-differenced version of the modern gravity equation, there is solid evidence
that reductions in the relative distance ratio over time reduce trade frictions between countries. This result is tantamount to proving that country-internal population migration has the ability to affect trade. This effect appears particularly
pronounced for OECD countries. However, internal migration may also be a result
of trade liberalization with neighbouring countries. Controling for this potential
endogeneity is a task for further research.
This paper has shown how to use better distance measures, and that using them
is relevant. Calculating these improved distance measures is computationally expensive. However, improvements in computer technology hold the prospect of
even better measures in the future. New geo-computational techniques, currently
popular in car navigation systems and web sites such as Google Maps, can eventually be utilized to compute actual trucking, shipping, and air-cargo distances. The
problem will be to apply these techniques to trillions of potential location pairs on
the planet.
With respect to estimating gravity equations, this paper makes a strong case
for using time-varying measures of distance that allow for country-internal migration of populations. Even though this study focused only on the last decade, the
magnitude of the distance changes are large enough to make them economically
meaningful and relevant. Ignoring the time-varying nature of economic distance
may bias the results of empirical work that attempts to identify the effect of policies
such as tariff reductions. Putting the new distance data set into the public domain
may help improve the quality and reliability of empirical work with the gravity
equation in the future.
29
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