JOURNAL OF APPLIED ECONOMETRICS
J. Appl. Econ. (2010)
Published online in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/jae.1231
‘DUAL’ GRAVITY: USING SPATIAL ECONOMETRICS
TO CONTROL FOR MULTILATERAL RESISTANCE
KRISTIAN BEHRENS,a – c CEM ERTURd AND WILFRIED KOCHa,b *
a
Département des Sciences Economiques, Université du Québec à Montréal (UQAM), Canada
b CIRPÉE ESG-UQAM, Montréal, Canada
c CEPR, London, UK
d LEO, Faculté de Droit d’Economie et de Gestion, Université d’Orléans, France
SUMMARY
We derive a quantity-based structural gravity equation system in which both trade flows and error terms
are cross-sectionally correlated. This system can be estimated using techniques borrowed from the spatial
econometrics literature. To illustrate our methodology, we apply it to a well-known Canada–US trade dataset.
We find that border effects between the USA and Canada are smaller than suggested by previous studies:
about 7.5 for Canadian provinces and about 1.3 for US states. Hence controlling directly for cross-sectional
interdependence among both trade flows and error terms reduces measured border effects by capturing
‘multilateral resistance’. Copyright  2010 John Wiley & Sons, Ltd.
Supporting information may be found in the online version of this article.
1. INTRODUCTION
The gravity equation provides a remarkable empirical framework for analyzing bilateral trade flows.
Having been derived from various trade models under a wide range of modeling assumptions
(e.g., Anderson, 1979; Eaton and Kortum, 2002; Anderson and van Wincoop, 2003; Feenstra,
2004; Helpman et al., 2008; Chaney, 2008; Melitz and Ottaviano, 2008; Behrens et al., 2009), it
is nowadays firmly rooted in mainstream economic theory and has, as such, become an essential
part of every applied trade theorist’s toolbox. Despite its wide applicability and excellent fit, the
gravity equation suffers from several well-known and from several less well-known shortcomings.
The former category comprises mainly empirical issues, such as the treatment of zero trade flows,
the construction of own absorption, the measurement of internal distances, and concerns about
the plausibility of various parameter estimates. These problems have been extensively discussed
elsewhere in the literature (see Anderson and van Wincoop, 2004, pp. 729–733, for a recent
overview). This paper focuses instead on the less well-known problem of how to take into account
the interdependence between trade flows when estimating what is inherently a general equilibrium
system.
Anderson and van Wincoop (2003) have argued that dealing with the regional interaction
structure is important when estimating gravity equation systems. They show that the inclusion of
multilateral resistance terms, i.e., terms which capture the fact that bilateral trade flows do not only
depend on bilateral trade barriers but also on trade barriers across all trading partners, is crucial for
the results one obtains. In other words, bilateral predictions do not readily extend to a multilateral
world because of complex interactions linking all the trading partners. Although such a finding
is hardly surprising in a general equilibrium setting, it has been largely neglected until now in
applied work. Interdependence has, however, to be somehow controlled for in the gravity equation
Ł Correspondence to: Wilfried Koch, Département des Sciences Economiques, Université du Québec à Montréal, Case
postale 8888, Succursale Centre-Ville Montréal, Québec H3C 3P8, Canada. E-mail: [email protected]
Copyright  2010 John Wiley & Sons, Ltd.
K. BEHRENS, C. ERTUR AND W. KOCH
to obtain consistent estimates. Some previous studies aim at doing so by including ad hoc regional
remoteness indices, even though those indices have no theoretical foundation. Other studies try
to capture interdependence across trade flows by including origin- and destination-specific fixed
effects. The somewhat worrisome feature of both of these approaches is the underlying implicit
assumption that trade flows between two trading partners are independent of what happens to the
rest of the trading world. This is clearly a strong assumption that is unlikely to hold and therefore
may bias estimates.
Our paper is a methodological contribution to the rapidly expanding literature on the theorybased estimation of gravity equation systems. Building upon the observation that the structural
estimation of such systems usually hinges on the correct treatment of unobservable prices, our
modeling strategy consists in taking a ‘dual’ approach that relies on observable trade flows only.
More concretely, we derive a gravity equation from the quantity-based version of the constant
elasticity of substitution (CES) model by exploiting the property that the price indices in that
model are themselves implicit functions of trade flows. The resulting equilibrium system allows
us to recover, quite naturally, an econometric specification in which bilateral trade flows between
two regions depend on trade flows involving all the other trading partners. Put differently, our
econometric specification includes cross-sectional correlations among trade flows as structurally
implied by our theoretical model. This interdependence, which reflects complex interactions
between all trading partners, can be dealt with in a manner similar to that used in the spatial
econometrics literature to deal with simultaneous spatial autoregressive structures.1 Controlling
for cross-section correlations with the help of spatial econometric techniques then amounts to
controlling for ‘multilateral resistance’ and yields improved estimates of the gravity equation.2
Unlike in time series analysis, there is no natural ordering of cross-sectional observations. The
fundamental tool used in spatial econometrics to impose a ‘relevant ordering’ is the so-called spatial
weight matrix, which connects each observation to a set of ‘neighboring observations’ by means
of an exogenous pattern. Various spatial weight matrices based on some form of geographical
connectivity (contiguity, nearest neighbors, geographical distance) have been used to study a wide
range of topics including growth and convergence (Ertur and Koch, 2007), spatial patterns of
foreign direct investment (Blonigen et al., 2007), retail price competition (Pinkse et al., 2002),
and strategic interactions between local governments (Case et al., 1993; Brueckner, 1998). To
the best of our knowledge there have been, until now, no applications to trade and the gravity
equation, which may be due to the fact that origin–destination based models have not yet been
much developed in spatial econometrics with the exception of LeSage and Pace (2008), who model
spatial correlations among migration flows between nearby origin–destination dyads. Connectivity
across observations has been usually interpreted in a narrow sense as geographical proximity,
although Anselin (2006, p. 910) acknowledges the interest to consider more abstract spaces such
as those arising in sociology or in political science. What really matters, in the end, is to identify
the appropriate space and the most relevant associated interaction measure. Akerlof (1997), for
instance, points out that countries may be viewed as localized in some general socio-economic and
institutional or political space defined by a range of factors. Determining how countries relate to
each other then requires the choosing of a suitable metric. Conley and Ligon (2002), for example,
1 See Anselin and Bera (1998) for an overview and the recent book by LeSage and Pace (2009) for an up-to-date and
comprehensive presentation of the state of the art in spatial econometrics. The asymptotic properties of some estimators
have been derived by Kelejian and Prucha (1998) and Lee (2004).
2 More recently, Pesaran and Tosetti (2010) and Chudik et al. (2010) propose an alternative approach to modeling crosssection dependence in a panel setting. They show that under some specific assumptions like granularity, some spatial
econometric models can be interpreted as particular cases of weak cross-sectionally dependent (CWD) processes. They
can thus be represented by factor processes with an infinite number of weak factors and no idiosyncratic error.
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
DUAL GRAVITY
define an economic distance based on transport costs, whereas Conley and Topa (2002) define a
socio-economic distance based on social networks.
In this paper, we pursue this latter approach which aims at extending the methodology to deal
with any kind of interactions between observations in any network structure on the basis of a
similarity measure. To this end, we adopt a much broader definition of the spatial weight matrix
that captures those complex interaction structures, and propose to call it the interaction matrix.
We use techniques borrowed from the spatial econometrics literature to estimate our theory-based
gravity equation system characterized by cross-sectional correlation of trade flows. The interaction
coefficients we obtain, by analogy to the spatial autocorrelation coefficients, capture the strength
of cross-sectional correlation resulting from the interaction structure imbedded in the interaction
matrix. Furthermore, in contrast to the spatial econometrics literature, the interaction structure
arises from our theoretical model and is not chosen in an ad hoc way. We believe that this is an
important step forward since in the absence of a theoretical model the lack of a natural ordering
in cross-sections implies that many different interaction matrices may be used to study the same
issue. It is then difficult to identify the most ‘relevant’ one, thereby requiring extensive sensitivity
analysis and leaving the door open to arbitrariness.
On top of controlling directly for cross-sectional interdependence, our approach has several
additional advantages. First, it does not enforce a strict pattern of interdependence as, for example,
the model by Anderson and van Wincoop (2003) does. Our approach is therefore more robust to
potential misspecification concerning the form of the interdependence. Second, it reveals that all
coefficients, including the interaction coefficients for both trade flows and the error terms, are
generally region specific. Hence the proper estimation of the model requires the use of local
techniques that can deal with parameter heterogeneity across observations. Last, we model more
carefully the error structure, thereby also controlling for cross-sectional correlations that may arise
in the error terms.
To illustrate the usefulness of our approach, we apply it to the well-known Canada–US dataset
used by, among others, Anderson and van Wincoop (2003) and Feenstra (2004) to assess the
magnitude of Canada–US border effects. We first estimate a model where all coefficients are
constrained to be identical across regions (‘homogeneous coefficient case’). Doing so simplifies
the econometric implementation and yields results that are comparable to those in the literature.
We then provide estimates for a model with region-specific coefficients, except for the interaction
coefficients which are country specific (‘heterogeneous coefficient case’). Our key results may
be summarized as follows. First, we show that there remains cross-sectional correlation in the
ordinary least squares (OLS) residuals of the gravity equation, given our theoretical interaction
structure, even when including origin- and destination-specific fixed effects. Put differently, OLS
estimates are at best inefficient and at worst inefficient and biased, because the fixed effects fail
to capture the cross-sectional interdependence among trade flows. This finding vindicates the use
of techniques borrowed from spatial econometrics, as well as a more careful modeling of the
error structure. Second, we estimate the homogeneous coefficient specification of the model and
show that, as predicted by theory, there is significant negative cross-sectional correlation between
trade flows. Controlling for this correlation reduces measured border effects between the USA
and Canada, which drop to about 7.5 for Canadian provinces and 1.3 for US states. Last, we
provide results for the heterogeneous coefficient specification of the model under the restriction of
country-specific interaction parameters. Our estimates reveal significant variations in both distance
elasticities and border effects across provinces and states. While border effects for almost all US
states are small and statistically insignificant, those for Canadian provinces are generally larger
and almost all statistically significant.
The remainder of the paper is organized as follows. Section 2 presents the model and derives
our gravity equation system. Section 3 derives the estimating equation from the theoretical model.
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
K. BEHRENS, C. ERTUR AND W. KOCH
Section 4 presents the data, summarizes briefly previous estimation methods, and discusses our
empirical results for both the homogeneous and the heterogeneous coefficient cases. Section
5 concludes. All technical developments are relegated to a web appendix available online as
supporting information.
2. A ‘DUAL’ GRAVITY MODEL
We begin by presenting a novel way of deriving a system of gravity equations that does not
depend on unobservable price indices, yet encapsulates the general equilibrium interdependencies
of the full trading system. Contrary to Anderson and van Wincoop (2003), who derive gravity
equations from a CES expenditure system with goods that are differentiated by region of origin
and the supply of which is fixed, we rely on a CES monopolistic competition model with free
entry. Using the inverse demand functions and exploiting the fact that price indices depend on
trade flows, we derive an implicit equation system that depends on observables only and that can
be estimated using techniques borrowed from the spatial econometrics literature. In a nutshell,
whereas previous research either derives a gravity equation system with nonlinear constraints in
unobservable price indices (Anderson and van Wincoop, 2003) or uses a linearization with respect
to prices (Baier and Bergstrand, 2009), we derive an ‘unconstrained’ linearized gravity equation
with correlated trade flows.
2.1. Preferences
Consider an economy with n regions. Each region i is endowed with Li workers/consumers, who
each supply inelastically one unit of labor. Labor is the only production factor, so that Li stands
for both the size of, and the aggregate labor supply in, region i. All consumers have identical CES
preferences over a continuum of horizontally differentiated product varieties. A representative
consumer in region j solves the following problem:
1
max Uj qij v
dv subject to
qij vpij vdv D yj
i
i
i
i
where > 1 denotes the constant elasticity of substitution between any two varieties; yj stands
for individual income in region j; pij v and qij v denote the consumer (i.e., the delivered) price
and the per capita consumption of variety v produced in region i and sold in region j; and where
i denotes the set of varieties produced in region i. Varieties produced in the same region are
assumed to be symmetric, which allows us to alleviate notation by dropping the variety index v.
Let mk stand for the measure of k (i.e., the mass of varieties produced in region k). It is readily
verified that the aggregate inverse demand functions for each variety are then given by
1/
pij D Qij
11/
mk Qkj
Yj
1
k
where Qij Lj qij and Yj Lj yj denote the aggregate demand in region j for a variety produced
in region i and the aggregate income in region j, respectively.
2.2. Technology
Each firm produces a single product variety. Thus there is a one-to-one correspondence between
varieties and firms, i.e., mk also stands for the mass of firms operating in region k. To produce q
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
DUAL GRAVITY
units of output requires cq C F units of labor, where c is the marginal and F is the fixed input
requirement.3 Shipping varieties both within and across regions is costly. More precisely, shipping
one unit of any variety between regions j and k requires dispatching jk > 1 units from the region
of origin, while the rest ‘melts away’ in transportation (the so-called ‘iceberg’ assumption). It is
worth pointing out that we need not make a priori any assumption on either the value of trade
costs ii within regions, nor on the symmetry of trade costs ij across regions.
A firm located in region j maximizes its profit, given by
j D
k
pjk cwj jk Qjk Fwj
with respect to the quantities Qjk and subject to the inverse demand schedule (1). Because price
and quantity competition are equivalent when there is a continuum of firms, the profit-maximizing
prices display a constant markup over marginal cost: pjk D jk pj , where pj cwj / 1
stands for the producer (i.e., the mill) price in region j. Free entry and exit drive profits to zero,
which implies that each firm must produce the break-even quantity
k
jk Qjk D
F 1
Q
c
2
irrespective of the region j it is located in.4
2.3. Equilibrium
To derive the gravity equation system requires determination of the value of trade flows from i to
j. The latter is given by Xij mi pij Qij which, using (1), can be expressed as follows:
11/
Qij
Y
Xij D mi 11/ j
mk Qkj
3
k
Aggregate income
constraints, the equilibrium prices, and the zero profit condition (2) then
imply that Yi D k mi pik Qik D mi pi Q. Solving for mi D Yi /pi Q and substituting the result
into (3) allows us to eliminate the unobservable mass of firms to obtain
11/
Qij
Xij D Yi Yj pi
11/
Yk Qkj
pk
k
4
By definition of the trade flows Xij and the mass of firms mi , it must be that
Qij D
Xij
Xij Q
D
mi pij
Yi ij
5
3 All firms are identical in our model. For gravity equations with heterogeneous firms see Chaney (2008), Helpman et al.
(2008), Melitz and Ottaviano (2008) and Behrens et al. (2009).
4 Strictly speaking, this condition only holds when all regions have at least some firms (interior equilibrium). In what
follows, we focus exclusively on such interior equilibria as they are the empirically relevant ones for our subsequent
analysis.
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
K. BEHRENS, C. ERTUR AND W. KOCH
Plugging (5) into (4) and simplifying then yields
11/
11/
Xij Q
1/1 Xij
ij
Yi ij
Yi
Xij D Yi Yj
11/ D Yj Lk 1/1 Xkj 11/
pi
Xkj Q
Yk
Li kj
Yk
pk
Yk kj
k
k
6
where we have used the equilibrium relationship pi /pk D wi /wk and the aggregate income
constraint wi D Yi /Li . Expression (6) can be rewritten as follows:
Lk kj Yk 1/1 11/
Xij D Yj
Xkj
8i, j
7
k Li
ij Yi
which is a system of equations capturing the interdependence of all trade flows towards region j.
To close the general equilibrium system, we impose the aggregate income constraints
Yi Xik D 0, 8i
8
k
As can be seen from expressions (7) and (8), all trade flows Xij (including own absorption
Xii ) are linked in equilibrium, both directly (since varieties are substitutes) and indirectly (via the
aggregate income constraints). Formally, we can think about such a system as being a directed
graph, where the Xij are the flows between regions (the ‘nodes’) along trading routes (the ‘edges’),
and where the Yi play the role of flow conservation constraints.
3. ECONOMETRIC SPECIFICATION
We now propose an econometric method for estimating the gravity equation system (7). This
method builds on the foregoing observation that trade flows are interdependent and that this
interdependence needs to be somehow taken into account. Our approach draws quite naturally
on spatial econometric techniques, which are precisely designed to deal with cross-sectional
interdependence in both variables and error terms. When compared to other estimation methods,
we believe that ours offers four advantages:
1. It directly deals with cross-sectional interdependence among trade flows, as implied by our
theoretical model.
2. It uses a more careful modeling of the error structure, thereby controlling for possible crosssectional correlations in the error terms.
3. It reveals that all coefficients, including the distance elasticities and border effects, are generally
region-specific (see Anderson and Smith, 1999; Helpman et al., 2008) and allows for statistical
inference on estimated border effects and distance elasticities.
4. it allows us to sidestep the thorny issue of how to choose by working with a linearized version
of the equilibrium system. As is known in the literature, estimation results for depend both
on the level of aggregation and the estimation method and vary widely from about 1.3 to over
22 (Hummels, 2001; Broda and Weinstein, 2006).
We now linearize the model of Section 2, derive an econometric specification and discuss in
more detail the error structure.
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
DUAL GRAVITY
3.1. Linearization and Matrix Form
Taking the logarithm of (7), we readily obtain:


1
Lk kj Yk 1 1 1
Xkj  f
ln Xij D ln Yj ln 
k Li
ij Yi
9
Clearly, there is interdependence across trade flows as Xij depends negatively on the nominal
sales of the other regions in market j. To obtain a specification that is amenable to estimation with
spatial econometric techniques, we linearize f around D 1. As shown in supporting information
Appendix A, doing so yields the following equation:
Lk
Lk
ln Zij D ln L 1 ln ij ln kj ln wi 1
ln Zkj
k L
k L
10
where Zij Xij /Yi Yj is a GDP-standardized
trade
flow
(but
which
we
will
refer
to
as
trade
flow for short); and where L k Lk denotes the total population. Expression (10) reveals the
essence of interdependence in the gravity equation system: the trade flow Xij from region i to
region j also depends on all the trade flows from the other regions k to region j. Several comments
are in order. First, trade flows from i to j are affected by relative trade barriers, as measured by the
deviation of bilateral trade barriers ij from the population weighted average (second term). Put
differently, relative accessibility matters. Second, trade flows from i to j are negatively affected by
wages wi in the origin region (third term). Higher wages raise production costs and make region
i’s firms less competitive in market j, thereby reducing trade flows. Last, trade flows from i to j
decrease with trade flows Zkj from any third region k into the destination market, because varieties
are substitutes. This effect is stronger the closer substitutes the varieties are (i.e., the larger the
value of ).5 In our estimations, interdependence will be captured by an autoregressive interaction
coefficient, and this coefficient can be seen as a measure of ‘spatial competition’ encapsulating
both aspects related to market power and consumer preference for diversity (via the parameter ).
To make notation more compact, we recast (10) into matrix form as follows:
Z D ln L1I 1I W w 1WZ
11
In expression (11), we have defined Z lnXij /Yi Yj as the n2 ð 1 vector of the logarithms
of trade flows; 1I as the n2 ð 1 vector whose components are all equal to 1; I as the n2 ð n2 identity
matrix; W as the n2 ð n2 matrix whose contents will be made precise below; t ln ij as the
n2 ð 1 vector of the logarithms of trade costs, whose contents will also be made precise below;
and w ln wi as the n2 ð 1 vector of the logarithms of origin wages. Some simple algebraic
manipulations show that W D [S diagL] In , where S is the n ð n matrix whose elements are
all equal to 1; denotes the Kronecker (tensor) product; diag(L) is defined as the n ð n diagonal
matrix of the Lk /L terms; and In is the n ð n identity matrix.
Turning to the functional form of trade costs, we follow standard practice by assuming that ij
is a log-linear function of distance and border effects that can be expressed as follows:
ij dij ebij
12
5 When approaches 1, the linear approximation of the model improves but the autoregressive interaction term disappears,
whereas the approximation gets worse despite more interdependence when is large. When gets very large, trade flows
fall to zero, which corresponds to an extreme form of interdependence where trade frictions almost completely inhibit
interregional trade.
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
K. BEHRENS, C. ERTUR AND W. KOCH
where dij denotes the distance between regions i and j; and where bij is a dummy variable
taking the value 1 if the flow Xij crosses an international border (the Canada–US border in our
application below), and 0 otherwise. Taking logarithms of (12), we can rewrite the trade cost
specification in matrix form as follows:
D d C b
13
where d ln dij is the n2 ð 1 vector of the logarithms of distance; and where b is the n2 ð 1
vector of dummy variables indicating cross-border flows. We then have
C
D I W D I Wd C b D d
b
14
where variables superscripted with a tilde are measured as deviations from their population
weighted averages. We stick to this notation in the remainder of the paper to ease the exposition.
Substituting (14) into (11) then yields the following estimating equation:
C ˇ2 w C b
C WZ
Z D ˇ0 1I C ˇ1 d
15
where ˇ0 ln L < 0 is the constant term; ˇ1 1 < 0 is the distance coefficient of
deviations from population weighted average distances (which, because of the implicit structure
of the model, differs from the true distance elasticity); and where ˇ2 < 0 is the coefficient
for the wage in the origin region. Turning to the impacts of crossing an international border, i.e.,
the border effects, their coefficient is given by 1 < 0. How to precisely compute and
decompose the border effects into intra- and international components is explained in more detail
in Section 5. Finally, the autoregressive interaction coefficient 1 < 0, which captures
the interdependence across trade flows, is smaller the closer substitutes the varieties are. Hence provides an intuitive measure of ‘spatial competition’.
3.2. Econometric Specification and Error Structure
To obtain a specification that can be estimated by techniques borrowed from the spatial econometrics literature requires rewriting the implicit equation (15) in explicit form, i.e., to move all
the ln Xij terms to the left-hand side which then only depends on the other flows. Furthermore,
we have to introduce the error structure into the model.
Let Wdiag diagL In denote the matrix containing only the diagonal elements of W, each
repeated n times by block. Equation (15) can then be rewritten as follows:
C ˇ2 w C b
C W Wdiag Z
I Wdiag Z D ˇ0 1I C ˇ1 d
Because I Wdiag is an invertible diagonal matrix provided that 1 Li /L 6D 0 for all i, we
can pre-multiply it by its inverse to obtain the following expression:
C ˇ w C b
C Wd Z
Z D ˇ0 1I C ˇ1 d
2
16
where Wd D W Wdiag is our theory-based interaction matrix. It is worth pointing out that Wd
is not row-normalized (i.e., the rows do not sum to one). Note also that Wd does not contain any
form of geographic connectivity, in contrast to the traditional practice in the spatial econometrics
literature. The similarity measure that flows from our theoretical model is based on the relative
size of regions as reflected by population shares.
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
DUAL GRAVITY
The n elements between positions ni 1 C 1 and ni for i D 1, . . . , n of I Wdiag 1 ,
1
which are given by 1 C 1 LLi
, depend on the origin index i only, which is fixed and
identical for all destinations. Observe that in expression (16) the components of the transformed
(overlined) matrices of coefficients are given by
ˇ1i D [1 Li /L]1 < 0,
i D [1 Li /L]1 < 0,
ˇ2i D [1 Li /L]1 < 0,
i D [1 Li /L]1 < 0
17
We have a specification with a distinct set of parameters for each region. The full model therefore
has a ‘club’ structure since all parameters (including the autoregressive ones) should ideally be
estimated separately (‘locally’) for each region. Quite naturally, we refer to the model (16) as the
heterogeneous coefficients model. Since it is econometrically complex to handle, we first estimate
a simpler benchmark in which we constrain all coefficients to be identical across regions, which
we refer to as the homogeneous coefficients model. Formally, constraining the coefficients to be
identical amounts to assuming that the diagonal of W is equal to zero in equation (15), i.e., that
W D Wd . In that case, the model simplifies substantially and can be written as
C ˇ2 w C b
C Wd Z
Z D ˇ0 1I C ˇ1 d
18
where the different coefficients are now identical across regions.
To estimate equation (16) or (18), we need to spell out the error structure underlying the model.
Though fundamental to the analysis, this modeling aspect has received only little attention. This
is quite surprising because when the error terms are introduced into the econometric specification
via the trade costs ij or the trade flows Zij , as usually done in the literature, one must take
into account the fact that ‘the multilateral resistance variables also depend on these error terms’
(Anderson and van Wincoop, 2004, p. 713). The same holds true for the border effects, since
these effects in any region depend in a complex way on a spatially weighted average of the effects
in all the other regions. Consequently, the error terms will exhibit some form of cross-sectional
correlation that has to be dealt with. To the best of our knowledge, this point has largely gone
unnoticed until now in the gravity literature. Although ‘errors can enter the model in many [. . .]
ways of course, about which the theory has little to say’ (Anderson and van Wincoop, 2003,
p. 180), it is likely that the exact way the error terms are introduced into the model is crucial to
the estimates one obtains.
There are several ways to introduce the error terms into the model. First, we could assume that
they arise from the imperfect measurement of trade costs, i.e., ij D dij ebij eεij , where εij is an
i.i.d. normal error term. Second, the error terms could arise in the trade flows Zij themselves.
obs εij
stand for the unobserved ‘real’ trade flow, where Zobs
Let Zreal
ij Zij e
ij denotes the observed
trade flow and where εij is again an i.i.d. normal error term. The resulting error structure that
would, for example, appear in expression (18) for the homogeneous coefficient model under both
assumptions can then be expressed as follows:
u D ε C Wd ε
19
where the coefficient must be estimated from the model. Hence the error terms uij are
autocorrelated under the form of a first-order moving average across interacting regions. Our
econometric specification corresponds therefore to the so-called SARMA model in the spatial
econometric literature (Huang, 1984). Although moving averages are quite common in structural
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
K. BEHRENS, C. ERTUR AND W. KOCH
econometric models, especially in time series, they are less so in spatial econometrics. An
explanation for this fact could be the clear lack of structural models.
4. EMPIRICAL IMPLEMENTATION
We apply our methodology to the well-known Canada–US dataset used, among others, by
Anderson and van Wincoop (2003) and Feenstra (2004). We begin by briefly reviewing the
data, previous estimation methods and results. We then estimate the OLS benchmark, both
with and without importer–exporter fixed effects and show that the OLS residuals remain
autocorrelated. This finding vindicates the use of techniques similar to those used in spatial
econometrics for estimating such equations because OLS estimators are at best inefficient and at
worst inefficient and biased. Second, we estimate, using maximum likelihood, our preferred theorybased specification, namely the SARMA model, under the assumption of homogeneous coefficients
(given by equations (18) and (19)). We also run a series of robustness checks by estimating the
model using alternative error structures which are commonly used in spatial econometrics. As will
become clear, the empirical results clearly back the theoretical specification. Last, we estimate,
again using maximum likelihood, the SARMA model in its more general form with heterogeneous
coefficients (given by equations (16) and (19)). Since estimating the fully heterogeneous model
requires estimating as many autoregressive interaction coefficients for the endogenous variable
and the error terms as there are regions, we restrict ourselves to the case where we estimate only
country-specific autoregressive interaction coefficients, whereas all other coefficients are allowed
to vary across regions.6
4.1. Data and Controls
A description of the dataset and the data sources is provided in supporting information Appendix
B. Our dataset features aggregate manufacturing exports Xij between regions, regional GDPs Yi ,
internal absorption Xii (all measured in million US dollars for the year 1993), and great circle
distances dij (measured in kilometers) between regional and provincial capitals for 30 US states
and 10 Canadian provinces. The full list of regions is provided in Table IV.
Estimating the model raises three different sets of issues. First, unlike other gravity models,
which disregard own absorption Xii in the estimation, we require a measure of internal trade costs
because we have to take into account the full interdependence
structure. Following Redding and
p
Venables (2004), we measure internal trade costs as ii surfacei /, where surfacei denotes the
region’s surface (measured in square kilometers). As estimation results are known to be somewhat
sensitive to the measure of internal distance (see, for example, Head and Mayer, 2002) we use
both values of 1/3 and 2/3 for as robustness checks.
Second, we require data on regional wages wi , which we measure using average hourly
manufacturing wages, and regional populations Li . These two regressors may be potentially
endogenous. For example, some unobserved shock to production in a region i may lead to an
increase in trade flows from region i to region j, triggering at the same time an inflow of population
and wage increases as labor demand rises. To control for the potential endogeneity of wages and
populations, we use 5-year lagged values for these two variables.7
6 To the best of our knowledge, ours are the first estimates of SARMA models with heterogeneous autoregressive
coefficients. Technical details, the Matlab codes, and the data in both .xls and .dta format are available online as supporting
information.
7 Both 1988 wages and populations are highly correlated with 1993 wages and populations. We experimented with different
lags up to 10 years, and our qualitative results (including the magnitudes of the border effects) are unaffected. Results
are also robust when using the 1993 values for wages and populations.
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
DUAL GRAVITY
Last, since our estimation method requires all bilateral trade flows across all pairs of regions,
to account for their interdependence, we further have to deal with the well-known problem of
zero trade flows. Indeed, there are 49 zero observations out of 1600 in our dataset, which requires
an appropriate treatment. Since there is no generally agreed upon method for doing so (see, for
example, Anderson and van Wincoop, 2004; Disdier and Head, 2008), we proceed as follows.
First, we augment the trade flows by adding 1, such that that their log is equal to zero. Second,
we control for this adjustment by including a zero-flow dummy variable in the regression, which
takes value 1 if the log-flow is zero and which takes value 0 otherwise. Although this is a crude
way of controlling for zero trade flows, truncating the sample would not perform better or be
theoretically more sound. There are two justifications for our procedure. First, there are only few
zeros in the sample (about 3%, against more than 50% for studies using international trade data).
Hence the data are not dominated by the occurence of zero values and the need for a corrective
procedures is thus less stringent. Second, our zeros are unlikely to be ‘true zeros’ as this would
entail no aggregate manufacturing trade between several US states. While the occurrence of zero
flows is indeed more likely for regions that are further away from each other or for which the
partner GDP is smaller, it seems highly unlikely that there is no aggregate trade between, for
example, Ohio and Maine or Arizona and Texas, as reported in our data.8 We may thus consider
that some flows are just not reported, and the zero-flow dummy controls for these outliers.
4.2. Previous Estimation Methods
We now briefly review previous estimation methods. The first one is based upon the strong
assumption that trade flows are independent: estimating the determinants of Xij can then be done
without taking into account any information provided by Xkl . McCallum (1995), among others,
makes this assumption to estimate by OLS the following empirical gravity equation for Canada–US
interregional trade:
ln Xij D ˛1 C ˛2 ln Yi C ˛3 ln Yj C ˛4 ln dij C ˛5 bij C εij
20
Doing so yields paradoxically large values for the border coefficient ˛5 , ranging from 3.07 to
3.30 (McCallum, 1995). Consequently, Canadian provinces seem to trade 21.5 to 27 times more
with themselves than with US states of equal size and distance, an unrealistically large value for
two well-integrated and culturally similar countries.
In columns 1–3 of Table I, we replicate McCallum-type OLS regressions of the form (20) as
our benchmark. To stay as close as possible to the original analysis, we define the border effects as
in Anderson and van Wincoop (2003). Hence we introduce two sets of dummy variables, bordCAij
and bordUSij , for Canada–US and US–Canada flows, respectively. The implied border effects
can be retrieved as the exponential of minus the coefficient of bordCAij and bordUSij . As can be
seen from Table I, all coefficients have the correct sign, reasonable magnitudes, and are precisely
estimated. Results for the distance elasticity are somewhat sensitive to the definition of internal
distance. As can be further seen from Table I, the magnitude of the border effects for Canadian
provinces ranges from about 14.5 to 15.4, depending on the definition of internal distance. These
estimates are in line with the McCallum-type regressions of Anderson and van Wincoop (2003,
Table I, p. 173), which obtain border effects in the range of about 15.7.9
8 In the presence of a large share of true zeros, such as in international trade data, the use of a Tobit estimator would be
required (Felbermayr and Kohler, 2006; Helpman et al., 2008).
9 The slight difference between our OLS estimates and those of Anderson and van Wincoop (2003), despite using the same
dataset, is due to: (i) inclusion of own trade flows Xii ; (ii) accounting for intra-regional distances; and (iii) controlling for
zero trade flows.
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
K. BEHRENS, C. ERTUR AND W. KOCH
Table I. OLS regressions
Model
OLS–McCallum
OLS–Feenstra
ln Xij
ln Zij
Dependent variable
Constant
ln Yi
ln Yj
ln dij
bordCAij
bordUSij
bordij
Fixed effects
Internal distance
Adjusted R2
Border effects
Canada
USA
‘Average border’
Moran’s I stat.
8.830a
(0.526)
1.045a
(0.028)
0.920a
(0.024)
1.172a
(0.031)
2.674a
(0.122)
0.393a
(0.056)
—
8.387a
(0.561)
1.044a
(0.028)
0.919a
(0.024)
1.228a
(0.036)
2.734a
(0.129)
0.397a
(0.057)
—
8.888a
(0.548)
1.038a
(0.028)
0.913a
(0.024)
1.144a
(0.033)
2.655a
(0.125)
0.412a
(0.056)
—
No
1 surfi
3
0.937
No
2 surfi
3
0.935
No
1 min fd g
j ij
4
0.936
14.496a
(1.773)
1.482a
(0.083)
4.634a
(0.349)
0.039a
(0.007)
15.401a
(1.979)
1.487a
(0.084)
4.786a
(0.371)
0.044a
(0.007)
14.219a
(1.782)
1.510a
(0.085)
4.633a
(0.355)
0.036a
(0.007)
7.594a
(0.250)
—
6.906a
(0.292)
—
7.852a
(0.282)
—
—
—
—
1.289a
(0.032)
—
1.385a
(0.038)
—
1.248a
(0.037)
—
—
—
—
1.482a
(0.069)
Yes
1 surfi
3
0.921
1.504a
(0.071)
Yes
2 surfi
3
0.919
1.487a
(0.071)
Yes
1
4 minj fdij g
0.920
—
—
—
—
—
—
4.404a
(0.073)
0.015a
(0.001)
4.501a
(0.075)
0.015a
(0.001)
4.422a
(0.075)
0.015a
(0.001)
Note: The first three columns correspond to the estimation of equation (20), whereas the last three columns correspond
to the estimation of equation (23). Standard errors are given in parentheses (with a D 0.01, b D 0.05, c D 0.1); those
for border effect coefficients are computed using the Delta method. The number of observations is 1600. OLS–Feenstra
include importer–exporter fixed effects. Following Feenstra (2004), average border effects are computed as the geometric
mean of the individual border effects. Moran’s I statistic is computed using our theory-based interaction matrix Wd .
However, as can also be seen from the last line of Table I, not a single OLS specification à
la McCallum passes Moran’s I test for the absence of cross-sectional correlation of the residuals
given our theory-based interaction structure (Cliff and Ord, 1981). Stated differently, there remains
a significant amount of cross-sectional correlation in the OLS residuals, which leads at best to
inefficient and at worst to both inefficient and biased estimates (with omitted variable bias because
of the missing spatially lagged variable). The presence of cross-sectional correlation suggests that
the use of appropriate econometric techniques dealing with this problem is required.
To deal with interdependence and with McCallum’s ‘border effect puzzle’, Anderson and van
Wincoop (2003) build on the ‘price version’ of the CES model presented in Section 2. Assuming
equal wages and symmetric trade costs that are a log-linear function of bilateral distance and
the existence of an international border between i and j, they derive the following instance of a
gravity equation system:
1 ln 1 C εij
ln Zij D k C a1 ln dij C a2 1 bij ln i
j
21
1 and 1 are the
where Zij are the GDP-standardized trade flows; k is a constant; and i
j
multilateral resistance terms of regions i and j, which, apart from unitary income elasticities,
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
DUAL GRAVITY
represent the key difference with equation (20) as estimated by McCallum. The multilateral
resistance terms are linked by a system of nonlinear equations involving all regions’ expenditure
shares k and the whole trade cost distribution:
1 D
1 ea1 ln dik Ca2 1bik 8i
k 22
i
k
k
Equations (21) and (22) reveal that the determinants of Zij cannot be consistently estimated
without taking into account the conditions prevailing in the origin and destination markets i and
j, as captured here by a simple transformation of the CES price indices. These price indices
themselves depend on the inverse demands and therefore on the different trade flows. Hence the
independence assumption underlying the McCallum-type estimates is not valid, which is likely to
bias the estimation results.
Anderson and van Wincoop (2003) estimate equation (21) using nonlinear least squares, where
the multilateral resistance terms are solved for in a first step using (22). While this procedure
accounts for interdependence, it has at least three drawbacks. First, it makes strong structural
assumptions on the form of the cross-sectional interdependence between the multilateral resistance
terms that stem directly from the CES specification. Our approach does not impose a priori such a
structure and is therefore more robust to misspecification. Second, Anderson and van Wincoop’s
method does not allow for parameter heterogeneity across regions, whereas ours does. As will be
shown later, parameter estimates widely vary across regions, and imposing common coefficients
may thus be unwarranted. Last, as the multilateral resistance terms are solved for numerically,
their statistical significance is usually not tested (though this could be done). We will provide such
tests and show that US border effects are mostly insignificant, whereas Canadian border effects
are not.
A third estimation method has been suggested by Anderson and van Wincoop (2003) and
Feenstra (2004). This method replaces the multilateral resistance terms with region-specific
importer–exporter fixed effects. In this case, (21) can be written as
j j
ln Zij D k C a1 ln dij C a2 1 bij C ˇ1i υi1 C ˇ2 υ2 C εij
23
where υi1 denotes an indicator variable that equals one if region i is the exporter, and zero otherwise;
j
and where υ2 denotes an indicator variable that equals one if region j is the importer, and zero
i and ˇj D 1 ln j then provide estimates of
otherwise. The coefficients ˇ1i D 1 ln 2
the multilateral resistance terms.
Although the fixed-effects procedure yields theoretically consistent estimates of the average
border effect in a CES model à la Anderson and van Wincoop with uncorrelated error terms (see
Feenstra, 2004), it disregards some part of the interdependence across trade flows. Hence, while
the fixed effects method has the advantage of being simple to implement, as OLS can be used
under the traditional assumptions on the error terms, its main drawback is that it does not capture
all the interactions of the model. Columns 4–6 of Table I show results for OLS fixed effects
regression. As can be seen from the last line in Table I, even after controlling for multilateral
resistance by using region-specific importer–exporter fixed effects, there remains a significant
amount of cross-sectional correlation in the OLS residuals. This finding suffices to show that fixed
effects capture heterogeneity but do not capture interdependence. In other words, although fixed
effects allow one to partly control for interdependence, they are by no means sufficient from both
a theoretical and an econometric point of view.10
10 Anderson and van Wincoop (2003, p. 180) point out that the fixed-effects estimator could be less efficient than the
nonlinear least squares estimator, which uses information on the full structure of the model. We furthermore show in this
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
K. BEHRENS, C. ERTUR AND W. KOCH
Table II. Homogeneous coefficients, SARMA regressions
Model
Dependent variable
18.799a
(1.004)
0.944a
ln wi
(0.176)
ij
d
1.223a
(0.033)
bij
1.127a
(0.064)
0.208a
(0.042)
0.001a
(0.0001)
1 surfi
Internal distance
3
AIC
2.342
BIC
2.325
Border effects (total) (intra)/(inter)
CA
7.353a
(1.280)
CA–CA/CA–US
2.712a /0.369a
(0.056)/(0.056)
US
1.296a
(0.019)
US–US/US–CA
1.138a /0.879a
(0.007)/(0.007)
‘Average border’
3.086a
(0.246)
Constant
SARMA
ln Zij
ln Zij (US–US flows ð 1.3)
18.495a
(1.015)
1.026a
(0.179)
1.298a
(0.037)
1.146a
(0.065)
0.202a
(0.043)
0.001a
(0.0001)
2 surfi
3
2.364
2.347
18.892a
(1.012)
0.945a
(0.177)
1.188a
(0.033)
1.133a
(0.064)
0.214a
(0.043)
0.001a
(0.0001)
1
4 minj fdij g
2.350
2.333
19.322a
(1.000)
0.735a
(0.176)
1.212a
(0.033)
1.277a
(0.064)
0.218a
(0.042)
0.001a
(0.0001)
1 surfi
3
2.357
2.341
19.053a
(1.012)
0.815a
(0.179)
1.286a
(0.036)
1.296a
(0.065)
0.214a
(0.043)
0.001a
(0.0001)
2 surfi
3
2.280
2.363
19.450a
(1.007)
0.733a
(0.177)
1.177a
(0.033)
1.282a
(0.064)
0.226a
(0.043)
0.001a
(0.0001)
1
4 minj fdij g
2.366
2.349
7.607a
(1.360)
2.758a /0.363a
(0.057)/(0.057)
1.301a
(0.020)
1.141a /0.877a
(0.007)/(0.007)
3.146a
(0.258)
7.425a
(1.304)
2.725a /0.367a
(0.057)/(0.057)
1.297a
(0.019)
1.139a /0.878a
(0.007)/(0.007)
3.104a
(0.249)
9.590a
(1.848)
3.097a /0.323a
(0.056)/(0.056)
1.341a
(0.020)
1.158a /0.864a
(0.007)/(0.007)
3.586a
(0.319)
9.912a
(1.963)
3.148a /0.318a
(0.057)/(0.057)
1.347a
(0.020)
1.161a /0.862a
(0.007)/(0.007)
3.654a
(0.334)
9.677a
(1.881)
3.111a /0.322a
(0.057)/(0.057)
1.343a
(0.020)
1.159a /0.863a
(0.007)/(0.007)
3.604a
(0.324)
Note: All six columns correspond to the estimation of the model given by equations (18) and (19). Standard errors are
given in parentheses (with a D 0.01, b D 0.05, c D 0.1); those for border effect coefficients are computed using the Delta
method. See supporting information Appendix C.1 for an explanation of how to compute the border effects. Following
Feenstra (2004), average border effects are computed as the geometric mean of the individual border effects. The number
of observations is 1600. The interaction matrix is Wd in all specifications, it is normalized by its spectral radius (see, for
example, Kelejian and Prucha, 2009) so that the normalized autoregressive and moving average coefficients are both in
the interval (1,1). AIC and BIC stand for the Akaike and the Schwarz information criteria, respectively.
4.3. Econometric Estimation: Homogeneous Coefficients
We now estimate the gravity equation system using techniques similar to those used in spatial
econometrics. First, we impose homogeneous coefficients to estimate the theory-based SARMA
specification (18) and (19). Columns 1–3 in Table II summarize our results. As can be seen, all
coefficients, including the autoregressive ones, have the correct signs, plausible magnitudes, and are
ij , measured as a deviation from the population
precisely estimated. The distance coefficient for d
weighted average, increases slightly in absolute value when compared to the OLS specification
but remains overall fairly stable. Turning to the wage term, it is worth noting that it is negative
and highly significant. Put differently, higher-origin wages reduce trade flows because of increased
production costs. Although one might a priori suspect that interregional wage differentials should
not significantly affect interregional trade flows in an integrated economic environment like North
America, where interregional wages differentials are relatively small, our results show that this
is not the case: interregional wage differentials are significant enough across North American
paper that it could also be biased. Fixed effects allow to control for heterogeneity but generally not for interdependence,
even when more complex fixed-effects specifications are used (e.g., Baltagi et al., 2003).
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
DUAL GRAVITY
regions to affect trade flows. One of the key empirical results in the SARMA specification is that
< 0), as
there is a significant amount of negative cross-sectional correlation among trade flows (
predicted by our model. There is also significant positive cross-sectional correlation among error
> 0), which suggests that controlling for that correlation is important.
terms (
As can further be seen from Table II, capturing the interdependence of the equilibrium system
in the SARMA model reduces the border effects with respect to the OLS estimates, but also with
respect to Anderson and van Wincoop (2003). Supporting information Appendix C.1 provides a
detailed explanation of how to compute and to decompose the border effects. In our theory-based
specification, the border effects for Canadian provinces range from about 7.3 to 7.6, whereas
those for US states cluster around 1.3. These values must be compared with Anderson and van
Wincoop’s measured border effects of 10.5 for Canada and 2.6 for the USA.
Balistreri and Hillberry (2007) argue that the construction of the state–state trade flows by
Anderson and van Wincoop (2003), using shipping data from the Commodity Flow Survey
(CFS), relies on a scalar numerical scaling to make the data ‘comparable’ to the Canada–US
trade data. However, it is likely that Anderson and van Wincoop’s pre-treatment of the data
has overcorrected the CFS shipping data and therefore understates US state–state trade flows.
Consequently, measured Canadian border effects would appear artificially lower since the intraCanadian trade intensity seems significantly larger when compared to the US one. As a robustness
check with respect to the magnitude of US flows, columns 4–6 of Table II replicate our estimates
when all US state–state trade flows are inflated by a factor of 1.3 (i.e., instead of adjusting the
CFS data by a factor of 0.53, we adjust it by a factor of only 0.69). As one can see, measured
Canadian border effects indeed increase to reflect the larger US trade flows, which is consistent
with the findings by Balistreri and Hillberry (2007).
As stated in the foregoing, there are many ways of modeling the error structure about which
theory has little to say. To see how sensitive the results are to the precise type of error structure,
we now run two robustness checks. First, we approximate the moving average by a more general
autoregressive error structure, which leads to a model similar to the so-called general spatial model
(Anselin, 1988) or SARAR model (Kelejian and Prucha, 2009). Consider a vector of error terms
u that is cross-sectionally correlated according to the autoregressive structure:
u D Wd u C ε
24
where ε is i.i.d. and normally distributed with zero mean and variance 2 I. Provided that jj < 1/,
where is the spectral radius of Wd (see for example, Kelejian and Prucha, 2009), we can write:
u D [In2 Wd ]1 ε D
1
jD1
j
j Wd ε C ε
When the successive powers in the sum converge to 0 sufficiently quickly, the autoregressive
structure approximates appropriately the first-order moving average, i.e., u ³ ε C Wd ε as in (19).
Observe that this approximation is reasonably accurate provided that is small enough and that
the elements of the successive powers of Wd converge to zero sufficiently quickly.
We estimate the SARAR specification with homogeneous coefficients (given by equations (18)
and (24)). The results are summarized in columns 1–3 of Table III. Observe that, as in the
SARMA model, the autoregressive interaction coefficient is negative and highly significant in
all estimations, which is in accord with the underlying theory stipulating that trade flows towards
the same market are substitutes. The magnitude of is slightly larger but reasonably close to the
one obtained in the SARMA model. All remaining coefficients are precisely estimated and the
signs are identical to those obtained under the SARMA specification. The distance coefficients
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
K. BEHRENS, C. ERTUR AND W. KOCH
Table III. Homogeneous coefficients, SARAR and SAR regressions
Model
Dependent variable
17.220a
(1.028)
1.023a
ln wi
(0.177)
ij
d
1.246a
(0.034)
bij
1.102a
(0.064)
0.141a
(0.047)
0.504a
(0.054)
1 surfi
Internal distance
3
AIC
8.508
BIC
8.491
Border effects (total) (intra)/(inter)
CA
7.035a
(1.208)
CA–CA/CA–US
2.652a /0.377a
(0.057)/(0.057)
US
1.288a
(0.019)
US–US/US–CA
1.135a /0.881a
(0.007)/(0.007)
‘Average border’
3.010a
(0.236)
Constant
SARAR
SAR
ln Zij
ln Zij
17.049a
(1.040)
1.099a
(0.179)
1.321a
(0.037)
1.123a
(0.065)
0.142a
(0.047)
0.497a
(0.055)
2 surfi
3
8.537
8.520
17.260a
(1.033)
1.031a
(0.178)
1.213a
(0.033)
1.108a
(0.064)
0.145a
(0.047)
0.503a
(0.055)
1
4 minj fdij g
8.518
8.501
13.869a
(0.709)
1.163a
(0.176)
1.255a
(0.035)
1.046a
(0.066)
0.033
(0.029)
—
1 surfi
3
2.299
2.283
13.740a
(0.720)
1.238a
(0.179)
1.331a
(0.038)
1.067a
(0.067)
0.030
(0.030)
—
2 surfi
3
2.327
2.310
13.890a
(0.712)
1.173a
(0.177)
1.223a
(0.034)
1.052a
(0.066)
0.030
(0.030)
—
1
4 minj fdij g
2.309
2.292
7.296a
(1.288)
2.701a /0.370a
(0.058)/(0.058)
1.294a
(0.020)
1.138a /0.879a
(0.008)/(0.008)
3.073a
(0.248)
7.115a
(1.233)
2.667a /0.375a
(0.057)/(0.057)
1.290a
(0.019)
1.136a /0.880a
(0.007)/(0.007)
3.030a
(0.240)
6.366a
(1.080)
2.523a /0.396a
(0.058)/(0.058)
1.272a
(0.019)
1.128a /0.887a
(0.008)/(0.008)
2.845a
(0.220)
6.611a
(1.151)
2.571a /0.389a
(0.059)/(0.059)
1.278a
(0.020)
1.130a /0.885a
(0.008)/(0.008)
2.907a
(0.231)
6.441a
(1.102)
2.538a /0.394a
(0.058)/(0.058)
1.274a
(0.019)
1.129a /0.886a
(0.008)/(0.008)
2.864a
(0.223)
Note: The first three columns correspond to the estimation of the model given by equations (18) and (24), whereas
the last three columns correspond to the estimation of the model given by equation (25). Standard errors are given in
parentheses (with a D 0.01, b D 0.05, c D 0.1); those for border effect coefficients are computed using the Delta method.
See supporting information Appendix C.1 for an explanation of how to compute the border effects. Following Feenstra
(2004), average border effects are computed as the geometric mean of the individual border effects. The number of
observations is 1600. The interaction matrix is Wd in all specifications; it is normalized by its spectral radius (see, for
example, Kelejian and Prucha, 2009) so that the normalized autoregressive coefficients are in the interval (1,1). AIC
and BIC stand for the Akaike and the Schwarz information criteria, respectively.
and border effects remain fairly similar, and the wage coefficient is again negative and highly
significant.
As a second robustness check, we re-estimate the model by introducing the error terms in an
ad hoc way. The easiest way of doing so is to rewrite (18) as follows:
C ˇ2 w C b
C Wd Z C ε
Z D ˇ0 1I C ˇ1 d
25
which simply amounts to adding the i.i.d. normal error term ε to the estimating equation. The
resulting specification (25) is similar to a standard spatial autoregressive model (SAR model)
(Lee, 2004). Columns 4–6 in Table III summarize estimation results for the SAR model with
homogeneous coefficients. Observe that, although the other coefficients remain fairly stable, the
autoregressive interaction coefficient is not significantly different from zero (with even positive
point estimates). This result runs plainly against the underlying theory which predicts a negative
cross-sectional correlation across trade flows. Hence the ad hoc introduction of the error term is
not backed by the data in the sense that it is incompatible with the qualitative predictions of the
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
DUAL GRAVITY
theory presented in Section 2. As can be further seen, both the wage and border coefficients are
biased.
4.4. Econometric Estimation: Heterogeneous Coefficients
All previous estimates are based on the assumption that coefficients are identical across regions.
Yet, as one can see from equation (16), theory predicts that coefficients are region specific. This is
in accord with recent findings by Helpman et al. (2008, p. 474), who note in an international
context that ‘the elasticities vary widely across different country pairs’. We now re-estimate
the model by allowing for region-specific coefficients as implied by the underlying theoretical
specification. In so doing, we restrict ourselves to the SARMA model (given by (16) and (19))
because: (i) OLS have no theoretical foundation; (ii) the SARAR approximation yields very similar
results; and (iii) the SAR specification yields results that run against the theory.
As already mentioned in the foregoing, we estimate a simpler heterogeneous coefficients model
in which only the non-autoregressive parameters ˇi are allowed to vary across regions, whereas
the autoregressive interaction coefficients and vary by country only (
CA and CA for Canada,
and US and US for the USA). The estimation results for this ˇ-heterogeneous SARMA model are
summarized in Table IV, where we give the estimated coefficients for the distances and the borders,
the true regional distance elasticities εdij , as well as the border effects and their decomposition (see
supporting information Appendix C.2 for additional details). We also report the country-specific
autoregressive coefficients.
Several comments are in order. First, it is worth noting that there is a substantial amount of
heterogeneity in the estimated coefficients, both for distance elasticities, border effects, and autoregressive coefficients. As can be seen from Table IV, the autoregressive interaction coefficients
for both Canada and the USA are negative and significant. Furthermore, as O US < O CA , the US
market appears ‘more competitive’ than the Canadian market. The estimated distance coefficients
(which differ from the true elasticities εdij because of the cross-sectional interdependence and
the implicit form of the estimating equation) range from 0.639 for California to 2.615 for
Newfoundland and Labrador.11 The real elasticities εdij are very similar, with values in the same
range.
The regional structure of distance elasticities is depicted in Figure 1. The Canadian core regions
(Ontario and Québec), as well as the northeast of the USA (Maryland, New York, Pennsylvania)
form a cluster of regions with small distance elasticities (in absolute value) of trade flows. The
same holds true for the western states and provinces (British Columbia, Washington, California),
whereas the Great Plains and the remote Canadian provinces have relatively larger distance
elasticities (in absolute value). Table IV and Figure 2 reveal that, as expected, the Canadian
provinces face larger border effects than the US states. The reason for this is as in Anderson
and van Wincoop (2003) and explained in detail in supporting information Appendix C.2. As
further shown by Table IV, the intranational trade-boosting effect of the border is much larger
for Canadian provinces than for US states. Put differently, ‘trade barriers raise size adjusted trade
within small countries more than within large countries’ (Anderson and van Wincoop, 2003,
p. 176). Furthermore, the trade-reducing international effect of the border for Canadian exports
is larger than that for US exports, which illustrates again that the border has a stronger effect
on Canadian firms than on US firms. The reason is that the internal market of the USA is much
11 As the estimating equation is given in implicit form, the true distance elasticities differ from the estimated coefficients.
Starting from (16), they can be computed as d [I Wd ]1 ˇ1 , which is derived from the explicit solution to the
estimating equation. Note that d is generally an n2 ð n2 matrix of distance elasticities. Yet, since all distance elasticities
with the same origin index are identical, there are only n distinct elasticities.
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
K. BEHRENS, C. ERTUR AND W. KOCH
Table IV. ˇ-heterogeneous SARMA regressions
Model
SARMA
Code
Region name
AB
BC
MB
NB
NL
NS
ON
PE
QC
SK
AL
AZ
CA
FL
GA
ID
IL
IN
KY
LA
ME
MA
MI
MN
MO
MT
MD
NC
ND
NH
NJ
NY
OH
PA
TN
TX
VT
VA
WA
WI
Alberta
British Columbia
Manitoba
New Brunswick
Newfoundland and Labrador
Nova Scotia
Ontario
Prince Edward Island
Quebec
Saskatchewan
Alabama
Arizona
California
Florida
Georgia
Idaho
Illinois
Indiana
Kentucky
Louisiana
Maine
Massachusetts
Michigan
Minnesota
Missouri
Montana
Maryland
North Carolina
North Dakota
New Hampshire
New Jersey
New York
Ohio
Pennsylvania
Tennessee
Texas
Vermont
Virginia
Washington
Wisconsin
ij
d
1.446a
0.985a
1.883a
1.754a
2.615a
1.460a
1.155a
1.729a
1.152a
1.727a
1.354a
1.158a
0.639a
1.018a
1.359a
1.180a
1.001a
1.287a
1.340a
1.472a
1.333a
0.787a
1.269a
1.424a
1.488a
1.701a
0.933a
1.112a
2.026a
1.025a
0.758a
0.674a
1.095a
0.743a
1.331a
0.958a
1.161a
1.013a
1.167a
1.298a
bij
1.268a
0.976a
1.294a
1.615a
0.061
1.351a
1.635a
1.092a
1.389a
1.326a
2.842
3.700
2.305
0.001
2.787
2.563
0.400
3.141
1.723
2.429
7.115a
2.481
0.867
1.881
0.549
0.959
0.433
2.750
4.393c
1.151
3.697
0.562
1.036
1.282
1.292
1.527
4.264
5.369b
3.264
2.164
CA D 0.001a
US D 0.003a
CA D 0.003a
US D 0.016a
εdij
bij intra
bij inter
Total border
1.449
0.987
1.884
1.755
2.616
1.460
1.162
1.729
1.157
1.729
1.362
1.164
0.659
1.035
1.371
1.182
1.016
1.297
1.346
1.481
1.335
0.793
1.285
1.432
1.498
1.702
0.939
1.122
2.027
1.027
0.765
0.689
1.111
0.754
1.340
0.979
1.161
1.021
1.174
1.307
3.139
2.423
3.190
4.232
1.064
3.349
4.605
2.653
3.594
3.279
1.383
1.528
1.355
0.990
1.380
1.332
1.040
1.437
1.212
1.318
2.256
1.330
0.892
0.794
0.928
0.887
0.941
1.374
1.644
1.132
1.544
1.063
0.872
1.159
1.154
1.201
1.619
1.875
1.456
1.279
0.319
0.413
0.314
0.236
0.940
0.299
0.217
0.377
0.278
0.305
0.723
0.654
0.738
1.010
0.725
0.751
0.962
0.696
0.825
0.759
0.443
0.752
1.122
1.260
1.077
1.128
1.062
0.728
0.608
0.884
0.648
0.940
1.146
0.863
0.867
0.833
0.618
0.533
0.687
0.782
9.850a
5.871a
10.174a
17.906a
1.133
11.216a
21.207a
7.037a
12.918a
10.752a
1.914
2.336
1.835
0.980
1.905
1.775
1.081
2.065
1.470
1.738
5.089a
1.769
0.795
0.630
0.862
0.787
0.886
1.889
2.703c
1.281
2.382
1.131
0.761
1.344
1.331
1.442
2.622
3.516b
2.119
1.635
Note: Results correspond to the estimation of the model given by equations (16) and (19), where we impose countryspecific autoregressive parameters. Significance is displayed as a D 0.01, b D 0.05, and c D 0.1. The interaction matrix
is Wd in all specifications; it is normalized by its spectral radius (see, for example, Kelejian and Prucha, 2009) so that
the normalized autoregressive and moving average coefficients are both in the interval (1,1). See supporting information
Appendix C.2 for an explanation of how to compute the border effects. Distance elasticities εdij are computed from the
explicit form of the model. See supporting information Appendix D for details on the maximum likelihood estimation of
the heterogeneous SARMA model.
larger, so that borders affect only a much smaller part of sales from US firms than from Canadian
firms.
Last, as can be seen from Table IV, almost all border coefficients for US states are statistically
insignificant. There are some statistically significant and relatively large positive measured border
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
DUAL GRAVITY
-1.88
-1.73
-1.45
-0.98
-2.62
-1.15
-1.16
-1.17
-2.03
-1.70
-1.73
-1.75
-1.16-1.33
-1.46
-0.67
-1.02
-1.27
-0.79
-1.00-1.29-1.09 -0.74
-0.76
-1.49
-1.34
-1.01 -0.93
-1.33
-1.1
1
-1.42
-1.30
-1.18
-0.64
-1.16
-0.96
-1.35-1.36
-1.47
-1.02
Figure 1. Regional structure of distance elasticities
10.75
9.85
5.87
10.17
1.13
21.21
2.12
0.79
1.77
2.70
0.63
17.917.04
2.62 5.09 11.22
1.13 1.28
1.34
1.77
1.082.06 0.76
2.38
0.89
0.86
1.47
3.52
1.33
1.89
1.64
-0.46
1.83
2.34
1.44
12.92
0.79
1.91 1.91
1.74
0.98
Figure 2. Regional structure of total border effects
effects for states such as Maine, North Dakota and Virginia, but these are rather exceptions.12 On
the contrary, the border effects for Canadian provinces are almost all highly significant and there
is a lot of variation. Magnitudes for the border effects range from insignificant in Newfoundland
to about 21.2 in Ontario, yet most values are clustered around 10–12. On the contrary, border
12 Although we obtain some ‘negative border effects’ (i.e., border effects smaller than 1) for several US states, which
runs against theory, these are always statistically insignificant.
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
–.5
K. BEHRENS, C. ERTUR AND W. KOCH
CA
NY
–1
BC
VT
ID
–1.5
ME
NS AB
MD
AZ WA
IL FL
OH
ON
VA
NC
QC
WIIN
KY
ALTN GA
MN
LAMO
TX
MI
MT
SK
PENB
MB
–2
distance elasticity
NH
PA
NJ
MA
–2.5
ND
NL
0
.02
.04
.06
.08
.1
.12
population share
Figure 3. Distance elasticities and regional size
effects for the US states are uniformly smaller, ranging from insignificant to a high of about
5. To sum up, border effects are generally small for US exports to Canada, whereas they do
exist for Canadian exports to the USA. Their magnitude, however, seems smaller than previously
thought once ‘multilateral resistance’ is captured via the cross-sectional interdependence of trade
flows.
Last, as shown by expression (17), the model predicts the existence of a positive and concave
relationship between a region’s relative size Li /L and its distance coefficient ˇ1i . Figure 3, which
summarizes our estimation results from Table IV, reveals the existence of this pattern relating
regional sizes to distance elasticities: larger regions face lower distance elasticities than smaller
regions do. One possible explanation for this finding is that smaller markets are less competitive,
so that less productive firms are selected into those markets (Melitz, 2003; Melitz and Ottaviano,
2008). Provided there is a positive correlation between a firm’s productivity and its shipping costs,
these firms then have a greater handicap in serving foreign markets, thus facing higher distance
elasticities than more productive firms in larger and more competitive markets.
5. CONCLUSIONS
Building on a ‘dual’ version of the gravity equation, we have shown how techniques adapted
from the spatial econometrics literature can be used to control for cross-sectional interdependence
among trade flows. Handling such interdependence is a major issue for consistent estimation but
has been rather elusive until now. Our results suggest that, as in Anderson and van Wincoop
(2003), theory-based estimation of the gravity equation leads to smaller border effects than those
obtained with ad hoc specifications or fixed-effects methods. Put differently, there seems to be
less of a ‘border puzzle’ once the cross-sectional correlations in the data have been controlled for.
Besides reducing the magnitudes of the border effects, our methodology offers a number of
additional advantages when compared to previous approaches: (i) it accounts for cross-sectional
interdependence among trade flows, as implied by our model, and thus directly controls for
multilateral resistance; (ii) it uses a more careful modeling of the error structure, thereby controlling
for possible cross-sectional interdependence in the error terms; (iii) it does not impose a rigid
pattern on the interaction structure and is therefore more robust to potential misspecification;
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
DUAL GRAVITY
furthermore, our interaction structure is theory-based; and (iv) it allows for estimation of regionspecific coefficients.
While we have tried to stay as close as possible to the structural model, we have not yet
estimated the fully heterogeneous model with region-specific autoregressive coefficients. Doing so
is a daunting task that is left for future work.
ACKNOWLEDGEMENTS
We thank the editor M. Hashem Pesaran, two anonymous referees, as well as Peter Egger, Giordano
Mion, Yasusada Murata, Jan Mutl, Giovanni Peri, Michael Pfaffermayr, seminar participants in
Louvain-la-Neuve, Innsbruck, and conference participants in Kiel, Toronto, Louvain-la-Neuve,
Cambridge, Tokyo, Paris, and the 62nd European Meetings of the Econometric Society for very
helpful comments and suggestions. Behrens is holder of the Canada Research Chair in Regional
Impacts of Globalization. Financial support from the CRC Program of the Social Sciences and
Humanities Research Council (SSHRC) of Canada is gratefully acknowledged. Behrens also
acknowledges financial support from the European Commission (Grant MEIF-CT-2005-024266)
and from FQRSC Québec (Grant NP-127178). Behrens and Koch gratefully acknowledge financial
support from CNRS-GIP-ANR (Grant JC-05-439904). Part of this research was undertaken while
Wilfried Koch was visiting CORE, Université catholique de Louvain, and UQAM. The usual
disclaimer applies.
REFERENCES
Akerlof GA. 1997. Social distance and social decision. Econometrica 65: 1005–1027.
Anderson JE. 1979. A theoretical foundation for the gravity equation. American Economic Review 69:
106–116.
Anderson JE, van Wincoop E. 2003. Gravity with gravitas: a solution to the border puzzle. American
Economic Review 93: 170–192.
Anderson JE, van Wincoop E. 2004. Trade costs. Journal of Economic Literature 42: 691–751.
Anderson MA, Smith SLS. 1999. Canadian provinces in world trade: engagement and detachment. Canadian
Journal of Economics 32: 22–38.
Anselin L. 1988. Spatial Econometrics: Methods and Models. Kluwer: Dordrecht.
Anselin L. 2006. Spatial econometrics. In Palgrave Handbook of Econometrics, Vol. I: Econometric Theory,
Mills TC, Patterson K (eds). Palgrave Macmillan: New York; 901–969.
Anselin L, Bera A. 1998. Spatial dependence in linear regression models with an introduction to spatial
econometrics. In Handbook of Applied Economic Statistics, Ullah A, Giles D. (eds). Marcel Dekker: New
York; 237–289.
Baier SL, Bergstrand JH. 2009. Bonus vetus OLS: a simple method for approximating international trade-cost
effects using the gravity equation. Journal of International Economics 77: 77–85.
Balistreri EJ, Hillberry R. 2007. Structural estimation and the border puzzle. Journal of International
Economics 72: 451–463.
Baltagi BH, Egger P, Pfaffermayr M. 2003. A generalized design for bilateral trade flow models. Economics
Letters 80: 391–397.
Behrens K, Mion G, Murata Y, Südekum J. 2009. Trade, wages, and productivity. CEPR Discussion Paper
#7369, Center for Economic Policy Research, London.
Blonigen BA, Davies RB, Naughton HT, Waddell GR. 2007. FDI in space: spatial autoregressive relationships in foreign direct investment. European Economic Review 51: 1303–1325.
Broda C, Weinstein D. 2006. Globalization and the gains from variety. Quarterly Journal of Economics 121:
541–585.
Brueckner JK. 1998. Testing for strategic interaction among local governments: the case of growth controls.
Journal of Urban Economics 44: 438–467.
Case AC, Rosen HS, Hines JR. 1993. Budget spillovers and fiscal policy interdependence: Evidence from
the states. Journal of Public Economics 52: 285–307.
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae
K. BEHRENS, C. ERTUR AND W. KOCH
Chaney T. 2008. Distorted gravity: the intensive and extensive margins of international trade. American
Economic Review 98: 1707–1721.
Chudik A, Pesaran H, Tosetti E. 2010. Weak and strong cross section dependence and estimation of large
panels. Econometrics Journal (forthcoming)..
Cliff AD, Ord GK. 1981. Spatial Processes: Models and Applications. Pion: London.
Conley TG, Ligon E. 2002. Economic distance and cross-country spillovers. Journal of Economic Growth
7: 157–187.
Conley TG, Topa G. 2002. Socio-economic distance and spatial patterns in unemployment. Journal of Applied
Econometrics 17: 303–327.
Disdier A-C, Head K. 2008. The puzzling persistence of the distance effect on bilateral trade. Review of
Economics and Statistics 90: 37–48.
Eaton J, Kortum S. 2002. Technology, geography and trade, Econometrica 70: 1741–1779.
Ertur C, Koch W. 2007. Growth, technological interdependence and spatial externalities: theory and evidence.
Journal of Applied Econometrics 22: 1033–1062.
Feenstra RC. 2004. Advanced International Trade. Princeton University Press: Princeton, NJ.
Felbermayr GJ, Kohler W. 2006. Exploring the intensive and extensive margins of world trade. Review of
World Economics 142: 642–674.
Head K, Mayer T. 2002. Illusory border effects: distance mismeasurement inflates estimates of home bias in
trade. CEPII Discussion Paper #2002/1.
Helpman E, Melitz MJ, Rubinstein Y. 2008. Estimating trade flows: trading partners and trading volumes.
Quarterly Journal of Economics 123: 441–487.
Huang SJ. 1984. The autoregressive moving average model for spatial analysis. Australian Journal of
Statistics 26: 169–178.
Hummels D. 2001. Towards a geography of trade costs. Mimeo, Krannert School of Management, Purdue
University, West Lafayette, IN.
Kelejian HH, Prucha IR. 1998. A generalized spatial two-stage least square procedure for estimating a spatial
autoregressive model with autoregressive disturbances. Journal of Real Estate Finance and Economics 17:
99–121.
Kelejian HH, Prucha IR. 2009. Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances. Journal of Econometrics 157: 53–67.
Lee L-F. 2004. Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive
models. Econometrica 72: 1899–1925.
LeSage JP, Pace RK. 2008. Spatial econometric modeling of origin–destination flows. Journal of Regional
Science 48: 941–967.
LeSage JP, Pace RK. 2009. Introduction to Spatial Econometrics. Chapman & Hall/CRC: New York.
McCallum JM. 1995. National borders matter: Canada–U.S. regional trade patterns. American Economic
Review 85: 615–623.
Melitz MJ. 2003. The impacts of trade on intra-industry reallocations and aggregate industry productivity.
Econometrica 71: 1695–1725.
Melitz MJ, Ottaviano GIP. 2008. Market size, trade, and productivity. Review of Economic Studies 75:
295–316.
Pesaran H, Tosetti E. 2010. Large panels with spatial correlation and common factors. CESifo Working Paper
No. 2103, September 2007, revised January 2010.
Pinkse J, Slade M, Brett C. 2002. Spatial price competition: a semiparametric approach. Econometrica 70:
1111–1153.
Redding S, Venables AJ. 2004. Economic geography and international inequality. Journal of International
Economics 62: 53–82.
Copyright  2010 John Wiley & Sons, Ltd.
J. Appl. Econ. (2010)
DOI: 10.1002/jae