WORKING PAPER SERIES
Controlling for Heterogeneity in Gravity Models of Trade
I-Hui Cheng
Howard J. Wall
Working Paper 99-010A
http://www.stls.frb.org/research/wp/99-010.html
February 1999
FEDERAL RESERVE BANK OF ST. LOUIS
Research Division
411 Locust Street
St. Louis, MO 63102
The views expressed are those of the individual authors and do not necessarily reflect
official positions of the Federal Reserve Bank of St. Louis, the Federal Reserve System,
or the Board of Governors.
Controlling for Heterogeneity in Gravity Models of Trade
I-Hui Cheng
Birkbeck College, University of London
Howard J. Wall
Federal Reserve Bank of St. Louis
February 1999
This paper argues that it is necessary to allow for country-pair heterogeneity when using the
gravity model to estimate international trade flows. We propose and estimate a fixed-effects
model that eliminates the heterogeneity bias inherent in standard methods. Further, we show that
there is no statistical support for the restrictions necessary to obtain existing empirical models,
which are special cases of our model. Because the gravity model has become the ‘workhorse’
baseline model for estimating the effects of international integration, this has important empirical
implications. In particular, our results suggest that standard gravity estimates of the effects of
integration can differ a great deal from what is obtained when heterogeneity is accounted for.
(JEL F15, F17)
Corresponding author: Howard J. Wall, Research Division, Federal Reserve Bank of St.
Louis, P.O. Box 442, St. Louis, MO 63166-0442, United States
E-mail: wall@stls.frb.org; Phone: (314)444-8533; Fax: (314)444-8731
We would like to thank Ron Smith for his insightful and helpful suggestions. We are also
grateful for comments from the participants at the Midwest International Economics Conference
at Purdue University, May 1999. The views expressed are those of the authors and do not
necessarily represent official positions of the Federal Reserve Bank of St. Louis, nor of the
Federal Reserve System.
Controlling for Heterogeneity in Gravity Models of Trade
I-Hui Cheng and Howard J. Wall
1. Introduction
Starting in the 1860s when H. Carey first applied Newtonian Physics to the study of
human behavior, the so-called “gravity equation” has been widely used in the social sciences.
More recently, gravity model studies have achieved empirical success in explaining various types
of inter-regional and international flows, including labor migration, commuting, customers,
hospital patients, and international trade.1 The widespread use of gravity equations is despite the
fact that they have tended to lack strong theoretical bases.
The gravity model of international trade was developed independently by Tinbergen
(1962) and Pöyhönen (1963). In its basic form, the amount of trade between two countries is
assumed to be increasing in their sizes, as measured by their national incomes, and decreasing in
the cost of transport between them, as measured by the distance between their economic centers.2
Following this work, Linnemann (1966) included population as an additional measure of country
size, employing what we will call the augmented gravity model.3 It is also common to instead
specify the augmented model using per capita income, which captures the same effects.4
Whichever specification of the augmented model is used, the purpose is to allow for non1
See Sen and Smith (1995) for a survey.
For recent examples of the basic gravity model see McCallum (1995), Helliwell (1996), and Boisso and
Ferrantino (1997).
3
For recent uses of the augmented gravity model with population see Oguledo and MacPhee (1994), Boisso and
Ferrantino (1997), and Bayoumi and Eichengreen (1997).
4
Examples of the augmented model with per capita income include Sanso, Cuairan, and Sanz (1993), Frankel
and Wei (1998), Frankel, Stein, and Wei (1995,1998), Eichengreen and Irwin (1998).
2
1
homothetic preferences in the importing country, and to proxy for the capital/labor ratio in the
exporting country (Bergstrand, 1989).
The gravity model has been used widely as a baseline model for estimating the impact of
a variety of policy issues, such as regional trading groups, political blocs, patent rights, and
various trade distortions.5 Typically, these events and policies are modeled as deviations from
the volume of trade predicted by the baseline gravity model, and, in the case of regional
integration, are captured by dummy variables. The recent popularity of the gravity model is
highlighted by Eichengreen and Irwin (1997, p.33) who call the it the “workhorse of empirical
studies of (regional integration) to the virtual exclusion of other approaches.” This is despite the
fact that, as Deardorff (1984) points out, most early papers were ad hoc rather than being based
on theoretical foundations. Exceptions to this include Anderson (1979), Bergstrand (1985),
Hummels and Levinsohn (1995), Deardorff (1998), and Feenstra, Markusen, and Rose (1998),
whose models are consistent with the gravity model. See also Evenett and Keller (1998) who,
along with Deardorff (1998), evaluate the usefulness of gravity models in testing alternative
theoretical models of trade. The recent flurry of theoretical work has led Frankel (1998, p.2) to
say that the gravity equation has “gone from an embarrassment of poverty of theoretical
foundations to an embarrassment of riches.”
The perceived empirical success of the gravity model has come without a great deal of
analysis regarding its econometric properties, as its empirical power has usually been stated
5
See Aitken (1973), Brada and Mendez (1983), Bikker (1987), Sanso, Cuairan, and Sanz (1993), Oguledo and
MacPhee (1994), McCallum (1995), Helliwell (1996), Wei and Frankel (1997), Bayoumi and Eichengreen
(1997), Mátyás (1997), Frankel and Wei (1998), Frankel, Stein, and Wei (1998), and Smith (1999).
2
simply on the basis of goodness of fit; i.e. a relatively high R 2 .6 The lack of attention paid to the
empirical properties of the model is despite the fact that the strength of any baseline model lies in
the accuracy of its estimates. The aim of this paper is to begin to fill the gap regarding the
empirical estimation of gravity models of trade. In particular, we demonstrate that standard
methods for estimating the gravity model produce biased estimates, tending to overestimate trade
between low-trade countries, and to underestimate it between high-trade countries. We argue
that the primary source of this bias is the failure of standard methods to account for the pairwise
heterogeneity of bilateral trade relationships.
The solution we propose uses simple panel data methods to allow for the intercepts of the
gravity equation to be specific to each trading pair. We demonstrate how with this empirical
model the correlation between the residuals and the volume of trade disappears. To illustrate the
empirical significance of our findings, we apply our model to the question of the effects of
regional integration on trade volumes. We find that standard methods find a strong negative
relationship between membership in a trading bloc and intra-bloc trade, whereas this
counterintuitive finding is eliminated when heterogeneity is controlled for.
Section 2 briefly sets out the various statistical models we examine. Section 3 presents
standard empirical results for the basic and augmented gravity models, and illustrates the inherent
estimation bias. In Section 4 we offer a solution to this, namely using a panel with fixed effects
to estimate the bilateral trade relationship. In Section 5 we compare our model to alternatives
6
See Sanso, Cuairan, and Sanz (1993) for an examination of the predictive power of various specifications of the
augmented gravity model. Also see Oguledo and MacPhee (1994) for a survey of pre-1990 empirical results.
3
which also control for heterogeneity. Section 6 illustrates the importance of controlling for
heterogeneity when estimating the effects of trade blocs. Concluding remarks are provided in
Section 7.
2. A Statistical Overview
This section briefly sets out the various forms of gravity models that have been
constructed to estimate bilateral trade flows. These models can be considered as restricted
versions of a general gravity model, which has a log-linear specification,7 but places no
restrictions on the parameters. In the general model, the volume of trade between countries i and
j in year t can be characterized by
ln X ijt = α 0 + α t + α ij + ′ijt Z ijt + ε ijt ,
t = 1,…,T;
(1)
where X ijt is exports from country i to country j in year t, and Z ′ijt = [ z it z jt ... ] the 1 × k row
vector of gravity variables (GDP, population, and distance). The intercept has three parts, one
which is common to all years and country pairs, α 0 , one which is specific to year t and common
to all pairs, α t , and one which is specific to the country pairs and common to all years, α ij . The
disturbance term ε ijt is assumed to be normally distributed with zero mean and constant variance
for all observations, i.e. ε ijt ~ IN (0, σ t2 ) , E (ε ijt , ε ij′t ) = 0 and E (ε ijt , ε ijt −1 ) = 0 . It is also assumed
that the disturbances are pairwise uncorrelated.
Obviously, because (1) has only one observation, it is not useful for estimation unless
7
Sanso, Cuairan, and Sanz (1993) conclude that the log-linear specification, while not optimal, is a fair and ready
approximation of the optimal form.
4
restrictions are imposed on the parameters. The standard single-year cross section model (CS)
imposes the restrictions that the slopes and intercepts are the same across country pairs; i.e. that
α ij = 0 and
=
ijt
t
,
ln X ijt = α 0 + α t + ′t Z ijt + ε ijt ,
t = 1, ..., T;
(CS)
where α 0 and αt cannot be separated. Assuming that all the classical disturbance-term
assumptions hold, the CS model is estimated by ordinary least squares (OLS) for each year.
The other standard estimation method is a pooled cross-section model (PCS), which
imposes the further restriction on the general model that the parameter vector is the same for all t,
1
=
2
= ... =
T
= , although it normally allows for the intercepts to differ over time;
ln X ijt = α 0 + α t + ′Z ijt + ε ijt ,
t = 1, ..., T.
(PCS)
This is estimated by OLS using data for all available years.
Virtually all estimates of the gravity model of trade use either the CS or the PCS model,
which, as we show below, provide biased estimates. We attribute this to heterogeneity bias due
to the restriction that the parameters are the same for all country-pairs. To address this, we
remove the restriction that the country-pair intercept terms equal zero, although we maintain the
restriction that the slope coefficients are constant across country pairs and over time.
Specifically, we consider a fixed effects model (FE)
ln X ijt = α 0 + α t + α ij + ′Z ijt + ε ijt ,
t = 1, ..., T.
(FE)
Note also that in the FE model the country-pair effects are allowed to differ according to the
direction of trade, i.e. αij ≠ α ji . The FE model is a two-way fixed effects model in which the
5
independent variables are assumed to be correlated with α ij , and is a classical regression model
which can be estimated using OLS.
Bayoumi and Eichengreen (1997) and Mátyás (1997) have also proposed models to
handle country-pair heterogeneity, each of which can be modeled as a restricted version of the FE
model. In the Bayoumi and Eichengreen (BE) model the differences in the dependent and
independent variables are used to eliminate the fixed variables, including the country-pair
dummies and distance. Specifically,
∆ ln X ijt = γ 0 + γ t + ∆Z ijt + µ ijt ,
t = 1, ..., T;
(BE)
ZKHUH LVWKHGLIIHUHQFHRSHUDWRUDQG γ 0 + γ t = α t − α t −1 . In this model the intercept has two
parts: γ 0 is the change in the period-specific effect that is common across years, and γ t is the
change that is specific to year t. As is well known, when there are no time dummies, such a
differencing model should yield results identical to a model with dummy variables to control for
fixed effects. However, with time dummies it is necessary to impose restrictions on the time
effects so as to avoid collinearity, which in turn makes the BE model a restricted form of the FE
model. If the collinearity restriction is that the first time dummy in the BE model is equal to
zero, this is equivalent to restricting the common component of the change in the period-specific
effects as equal to the difference in the first two period-specific effects, i.e. γ 0 = α 2 − α1 . If
instead the collinearity restriction is that the sum of the time dummies in the BE model is zero,
this is equivalent to restricting the common component as equal to the difference between the
first and last time dummies, i.e. γ 0 = α T − α1 .
6
Mátyás (1997) proposes
ln X ijt = α 0 + α t + θ i + ω j + ′Z ijt + ε ijt ,
t = 1, ..., T;
(M)
as the correct specification of the gravity model, where the country-specific effect when a country
is an exporter is θ i , and when it is an importer is ω j . Note also that in this specification,
distance, contiguity, and language are eliminated because they are fixed over time, even though
they are not collinear with the country-specific effects. This model is a special case of the FE
model in that it imposes arbitrary restrictions on the country-pair effects; i.e. because
α ij = θ i + ω j and α ik = θ i + ω k ; it must also be true that α ij = α ik − ω k + θ j . These cross-pair
restrictions do not change the coefficient estimates, but instead lead to odd residuals, and greatly
inaccurate predictions of trade flows.
3. An Overview of the Standard Empirical Results
This section presents regression results for the basic and the augmented versions of the
standard empirical models, CS and PCS. The data set is a balanced panel with 2110
observations, and includes countries with many different levels of economic development and
performance for the period 1991-1995. It includes observations on exports from 22 countries to
116 destination countries, although we do not have data on all possible pairs. Descriptions of the
data and their sources are provided in the Data Appendix.
3.1. Single-year cross-section data
In the augmented version of the gravity model, the gravity variables are the countries’
7
GDPs, their populations, and the distance between them. Thus, the augmented CS model (CSa)
assumes that in a given year trade flows from exporting country i to importing country j can be
estimated using: 8
ln X ij = α + β1 ln Yi + β 2 ln Y j + β 3 ln N i + β 4 ln N j + δ1 ln Dij + δ 2 C ij + λLij + ε ij ;
(2)
where Yi and Y j are the two countries’ GDPs; N i and N j are their populations, Dij is the
distance between their economic centers (their capital cities); Cij is a contiguity dummy; and Lij
is a common-language dummy. As trade flows are expected to be positively related to national
incomes, and negatively related to distance, it is expected that β1 , β 2 , and δ 2 are positive, and
that δ1 is negative. Also, estimation typically yields a negative sign for β 3 , which would
indicate that exported goods tend to be capital-intensive. It is also common to obtain a negative
sign for β 4 , which would indicate that traded goods tend to have income-elastic demands.
Finally, because Lij is meant to capture cultural and historical similarities between the trading
SDLUVZKLFKDUHWKRXJKWWRLQFUHDVHWKHYROXPHRIWUDGH LVH[SHFWHGWREHSRVLWLYH
The basic version of the gravity model does not include the populations of the two
countries, so it can be viewed as a special case of the augmented model in which the coefficients
on population are restricted to zero. Thus, the basic CS model (CSb) assumes that bilateral trade
can be estimated with the following regression:
ln X ij = α + β1 ln Yi + β 2 ln Y j + δ1 ln Dij + δ 2 C ij + λLij + ε ij .
The expected signs for the coefficients are as in the augmented model.
Note that because ln ( per capita incom e i ) = ln Yi − ln N i , the regression could be suitably rearranged
to instead obtain the augmented model with per capita income.
8
8
(3)
Table 1 reports the results for the five yearly cross-sections of the CSa and CSb models.
For each year, the coefficients on the GDPs of the two countries are statistically significant and
have the expected sign for CSa and CSb. For CSa, only the coefficient on the destination
populations has the usual negative sign, although the positive sign for the coefficient on origin
population is not statistically different from zero for any year. For all years for CSa and CSb the
coefficient on distance has the expected sign and is statistically significant. Despite having the
expected sign, none of the coefficients on contiguity are statistically significant at the 5% level.
The coefficient on the common-language dummy is positive and statistically significant for every
year. Importantly for assessing the appropriateness of pooling the data over the five years, the
result differ little from year to year for either version.
Comparing the two versions of the CS model, for CSa, exports are less elastic with
respect to origin GDP, and more elastic with respect to destination GDP. However, for none of
the years is there a startling distinction between the two versions, as they have almost identical
R 2 s and log-likelihoods. In fact, for none of the years does a likelihood ratio test reject the null
hypothesis that the CSa and CSb models are statistically the same.9
3.2. Pooling cross-section and time-series data
The other standard estimation method is to pool cross-sectional and time series data so as
to increase the number of observations without greatly increasing the number of variables. The
regression equation for the augmented version of the pooled cross-section model (PCSa) is:
This is with a critical value of 5.99 at the 5% level, and χ 2 ( 2 ) = 2[Log-likelihood CSa − Log-likelihood
CSb].
9
9
ln X ijt = α 0 + α t + β1 ln Yit + β 2 ln Y jt + β 3 ln N it + β 4 ln N jt + δ1 ln Dij + δ 2 C ij + λLij + ε ijt ;
(4)
where α 0 is the portion of the intercept that is common to all years and trading pairs, and αt
denotes the year-specific effect common to all trading pairs. Note that we omit the dummy for
1991 so as to avoid collinearity. We suppress the regression equation for the basic version of the
model (PCSb), as it is the same as (4) except for the restriction that β 3 = β 4 = 0 . The expected
signs for the coefficients are the same as for the CS models, except that the PCS models have
time dummies to consider. We take the time dummies as an indicator of the extent of
“globalization”, which we define as the common trend towards greater real trading volumes,
independent of the sizes of the economies.
The regression results for PCSa and PCSb are reported in the first two columns of Table
2. Unsurprisingly, the results are similar to those from the single-year cross-sections. The CS
and PCS models yield roughly the same elasticities on GDPs and distance, and have roughly the
same predictive power as measured by R 2 . Note, however, that because of the large number of
observations relative to the CS models, the coefficients on the countries’ populations and on
contiguity are statistically significant. Comparing the two versions of the PCS model, although
PCSa and PCSb yield very similar results, a likelihood ratio test rejects the null hypothesis that
they are statistically the same.10 We therefore conclude that the augmented version of the gravity
model is preferred to the basic model when using a pooled cross-section.
According to the estimates of the preferred PCSa model: (i) a 10% rise in a country’s
10
This is with a critical value of 5.99 at the 5% level, and χ2(2) = 14.94.
10
GDP should be associated with a 6.2% rise in its exports and an 8.5% rise in its imports, all else
constant; (ii) exports tend to be labor-intensive and income elastic (luxury goods), as indicated by
WKHSRVLWLYHVLJQIRU 3DQGWKHQHJDWLYHVLJQIRU 4; and (iii) a country will export 82% less to a
market that is twice as distant as another otherwise-identical market, 20% more to a country that
is contiguous, and 70% more to a country with the same first language. Finally, we take the fact
that our time dummies are not statistically different from zero to mean that globalization, as
defined above, was not an important factor in increasing trading volumes during the period.
The general conclusion from our estimation of standard gravity models is that they yield
stable results that do not differ a great deal over the sample period, nor between the basic and
augmented versions. We also conclude that the augmented version is preferred statistically,
although the basic version provides predictions of trade volumes that are nearly as accurate.
Finally, the estimates we obtain are not greatly out of line with those obtained previously in the
literature.
One should be cautious before concluding that there are no empirical problems with these
standard methods. This is clear from the upper-left panel of Figure 1, which plots the residuals
for the PCSa model. The strong positive relationship between the residuals and the level of
exports indicates that the PCSa model tends to underestimate the level of trade when the actual
level is high, and overestimates it when the actual level is low. To our knowledge, this bias has
not been recognized in the literature. Because the gravity model is used to establish baseline
levels of trade, it is important to have unbiased estimates the coefficients on the gravity variables,
11
even if one is not interested in these coefficients themselves. It is difficult to argue that a model
can be useful for establishing a baseline when it yields such obviously biased predictions.
4. The Model with Pairwise Heterogeneity
a. The model
As we describe in the previous section, standard cross-section estimates of the gravity
model yield biased estimates of the volume of bilateral trade. One possible source of this bias is
that heterogeneity is not allowed for by the regression equations. With such heterogeneity a
country may export different amounts to two countries, even though the two export markets have
the same GDPs and are equidistant from the exporter. This can be because there can be
historical, cultural, ethnic, political, or geographic factors that affect the level of trade, and are
correlated with the gravity variables (GDP, population, distance). If so, then estimates that do
not account for these factors will suffer from heterogeneity bias.
Various studies have to some extent tried to control for this by including things such as
whether trading partners share a common language, have had a colonial history, are in military
alliance, etc. However, cultural, historical, and political factors are often difficult to observe, let
alone quantify. This is why we will control for these factors using a simple fixed-effects model
that assumes that there are fixed pair-specific factors that may be correlated with levels of
bilateral trade and with the right-hand-side variables.
We assume that the gravity equation for a country pair may have a unique intercept, and
12
that it may be different for each direction of trade (i.e. α ij ≠ α ji ). However, we retain the
assumptions of the PCS model that the slope coefficients are constant over time and across
trading pairs. Our specification of the augmented gravity model with fixed effects (FEa) is:
ln X ijt = αij + αt + β1 ln Yit + β 2 ln Y jt + β 3 ln N it + β 4 ln N jt + ε ijt ;
(5)
where α ij is the specific “country-pair” effect between the trading partners. The basic version
(FEb) is the same as this, except for the constraint that β 3 = β 4 = 0 . The country-pair intercepts
include the effects of all omitted variables that are cross-sectionally specific but remain constant
over time, such as distance, contiguity, language, culture, etc. Using the pooled data described
above, we have 422 country-pair intercepts.
Because there is a long-standing problem with determining the appropriate measure of
economic distance so as to capture transportation and information costs, an added benefit of the
fixed effects model is that it eliminates the need to include distance in the regression. The most
common method for handling distance is to do as we have above and simply measure it between
the economic centers (assumed to be the capital cities) of the two countries. There are obvious
problems with this, such as the implicit assumptions that overland transport costs are the same as
those over sea, and that all overland/oversea distances are equally costly. To provide just one
obvious example, Los Angeles is about 1300 kms further from Tokyo than is Moscow, but it is
difficult to believe that the economic distance between Tokyo and Los Angeles is not much
lower than that between Tokyo and Moscow. Our fixed-effects approach eliminates the need to
include a distance variable, as it controls for all variables that do not change over time.
13
Another difficulty with standard measures of economic distance is the simple assumption
that the capital city is a useful proxy for the economic center. While this may be useful for small
countries with one major city, it is wide of the mark for countries like Canada and the US, which
have major cities thousands of miles apart on different oceans, and which serve as centers for
trade with completely different countries. By using Washington, DC or Ottawa to measure
distance between the US or Canada and its Pacific trading partners is to overstate distance by the
entire breadth of the North American continent. As the US has the highest GDP and the highest
volume of trade, the mis-measure of economic distance can bias the estimation of the coefficients
on the other variables in the gravity model.11
Another advantage of our approach is that it removes the problem of controlling for
contiguity. Although it is clearly important, as a great deal of the trade can occur from people
crossing the border to make everyday purchases, it is accounted for only sometimes. Even when
it is accounted for with a dummy variable as we do above, it still assumes that all contiguity is
equivalent in terms of its effect on trade. Considering that Canada and the US, China and Russia,
and Argentina and Chile are all equivalently contiguous pairs, this is difficult to abide by.
b. The results
The middle columns of Table 2 report the estimation results for the augmented and basic
versions of the fixed-effects model (FEa and FEb). Note that for comparison with the pooled
11
As a practical matter this mis-measure of distance is magnified by the fact that data sets tend to have
proportionally more data on US trade. For example, in our data set 700 of the 2110 observations (33%) have the
US as either the importing or exporting country.
14
cross-section results, the year dummies are measured relative to that of 1991. Also, the estimates
of the country-pair intercepts are omitted for space considerations. The difference between the
augmented and basic versions of the FE model is not glaring, but is nonetheless clear statistically.
A likelihood ratio test rejects the null hypothesis that the two models are statistically the same.12
In other words, it rejects the restriction that the coefficients on population are zero (as in FEb).
We therefore conclude that FEa is the preferred version of the FE model.
According to the results for the preferred FEa model: (i) a 10% rise in a country’s GDP
should be associated with a 2.9% rise in exports and a 5.2% rise in imports; (ii) exports tend to
be labor-intensive, as indicated by a positive sign of origin population, and income-elastic
(normal non-luxury goods), as indicated by the negative sign on destination population; and (iii)
globalization increased the real volume of trade by nearly 20% between 1991 and 1995.
Comparing the results of the FEa and PCSa models, allowing for trading-pair
heterogeneity, as in the FEa model, lowers the estimated income elasticities of trade, greatly
increases the absolute value of the coefficients on the countries’ populations, and greatly increase
the estimated role of globalization. It is obvious from the results that restricting the country-pair
effects to zero, as does the PCSa model, has significant effects on the results, and this is easily
confirmed by a likelihood ratio test. Further, as shown by the upper-right panel of Figure 1, there
is no obvious correlation between the residuals and the log of exports, indicating that the FEa
model does not suffer from the estimation bias exhibited by the PCSa model. It is also obvious
12
This is with a critical value of 5.99 at the 5% level, and χ2(2) = 46.42.
15
from Figure 1 that the residuals from the FEa model tend to be much smaller than those from the
PCSa model.
To summarize, because the PCSa model is a restricted form of the FEa model, and the
restrictions are not supported statistically, we conclude that the FEa model is the preferred
specification of the gravity model. Also, the FEa model does not exhibit the obvious
heterogeneity bias of the PCSa model, and is preferred on the basis of traditional measures of
goodness of fit in that it provides a higher R 2 and a lower sum of squared residuals. Note
though that this improved statistical performance arises from the FEa model having 422 more
independent variables than does the PCSa model. Nonetheless, on the basis of having smaller
values for the Akaike Information Criteria and the Amemiya Probability Criteria, which have
more severe penalties for the number of parameters, the FEa model is still easily preferred.
In short, there is no statistical support for imposing the parameter restrictions required by
the standard procedures for estimating the gravity model of trade. In the absence of any
economic arguments for believing that the intercepts of the gravity equation are the same across
trading pairs, we conclude that the fixed effects model is the more appropriate specification.
Oddly, Wei and Frankel (1997, p.125) reject the inclusion of country-pair dummies a
priori on the basis that doing so would undermine their efforts at estimating the effects of
variables that are constant over the sample period. Presumably their worry is that because these
variables are subsumed into the country-pair effects they are hidden from analysis. This is
unfounded because the effects of these variables are easily estimated by regressing them on the
16
country-pair effects from the FE model. Specifically, where the estimates of the 422 country-pair
effects are denoted as α̂ ij , and including the log of distance and the contiguity and language
dummies as independent variables, we obtain
αˆ ij = 4.45 − 1.04 ln Dij + 1.01C ij − 0.51Lij .
(2.57) (0.296)
(1.19)
(0.63)
The numbers in parentheses are standard errors, and the R 2 = 0.049. According to these results,
only the distance variable is a statistically significant determinant of the country-pair effects.
Contiguity and a common language do not therefore appear to be important determinants of the
volume of bilateral trade. Further, the low R 2 indicates that very little of the country-pair effects
are explained by the variables traditionally included in the standard cross-section models. Note
that these estimates are quite different from those obtained from the PCSa model, in which
estimates of the effects of time-invariant factors suffer from the same heterogeneity bias as the
time-variant factors. So, far from undermining estimation efforts, it is instead necessary to
control for country-pair heterogeneity to obtain unbiased estimates of the importance of timeinvariant factors. Curiously, though, the coefficient on the distance variable is practically the
same as obtained from the PCSa model. It is not possible to tell if this is a coincidence, or if it is
due to the distance variable being uncorrelated with the other independent variables. However,
because the distance variable is so suspect, we would not want to push this result in either case.
5. Alternatives with Heterogeneity
As discussed earlier, two other papers have proposed empirical models for dealing with
17
heterogeneity, and these models can be regarded as restricted forms of the FE model. Using
many of the same arguments we use to argue for the FE model, Bayoumi and Eichengreen (1997)
propose estimating the gravity equation in first differences. This method of handling
heterogeneity is nearly identical to ours, except in the restrictions that it imposes on the effects of
time. An additional difference between their model and our FEa model is that they use their
independent variables are the products of the GDPs and populations, therefore imposing the
additional restrictions that β1 = β 2 and β 3 = β 4 . Because we wish to focus on their treatment of
heterogeneity only, we will not estimate the model under those restrictions. However, as is clear
from the results, these additional restrictions are easily rejected statistically.
Taking the time difference of (5), the model that we will estimate (BEa) is
∆ ln X ijt = γ 0 + γ t + β1∆ ln Yit + β 2 ∆ ln Y jt + β 3 ∆ ln N it + β 4 ∆ ln N jt + µ ijt ;
(6)
where the intercept is as defined in Section 2, γ 0 + γ t = α t − α t −1 . To prevent collinearity, we set
the time dummy for 1992 equal to zero, meaning that other time dummies are measured relative
to it. In terms of the more-general FEa model, this is equivalent to restricting the common
component of the change in the period-specific effects as equal to the difference in the first two
period-specific effects, i.e. γ 0 = α 2 − α1 . 13 The empirical results are presented in Table 2.
The results for the FEa and BEa models are very similar in terms of the signs and order of
magnitude of the coefficients. Also, as illustrated by the lower-left panel of Figure 1, the BEa
model is not subject to the obviously biased residuals of the PCSa model. Nonetheless, the FEa
13
The alternative assumption that the sum of the year dummies is zero means that
same results except for the time dummies and the constant.
18
γ 0 = α T − α1 , and yields the
and BEa results differ enough to reject the restrictions needed to obtain BEa model. This is
confirmed by a likelihood ratio test. Further, in terms of fitting the data, the FEa model is
preferred in terms of the sum of squared residuals, the Akaike Information Criteria, and the
Amemiya Probability Criteria. As for the further restrictions in Bayoumi and Eichengreen (1997)
that β1 = β 2 and β 3 = β 4 , our results indicate that this would clearly have significant effects on
the estimates, and therefore should not be imposed.
The other alternative to the FEa model is due to Mátyás (1997), who proposes using the
specification
ln X ijt = α 0 + α t + θ i + ω j + β1 ln Yit + β 2 ln Y jt + β 3 ln N it + β 4 ln N jt + ε ijt ;
(7)
where the effect when a country is an exporter is θ i , and when it is an importer is ω j . This
model is a restricted form of the FEb model in that it imposes arbitrary cross-pair restrictions on
the country-pair effects; α ij = α ik − ω k + θ j .
The empirical results, summarized by the last column of Table 2, show that the coefficients
are identical to those from the FEa model, although their standard errors are very large. In fact,
they are large enough to reject the statistical significance of all but the coefficient on destination
GDP. The lower-right panel of Figure 1 plots the residuals of this model against the log of
exports, and shows that this model has very peculiar results. The wide dispersion of the residuals
also indicates a very poor fit relative to FEa. Further, a likelihood ratio test easily rejects the null
hypotheses that the cross-pair restrictions do not change the results in a statistically important
way. So, although this model eliminates the bias in the estimates of the coefficients on the
19
independent variables, it has little predictive power.
6. Implications for Estimating the Effects of Integration
As we discuss in the Introduction, the gravity model has become the primary tool for
estimating the effects of regional integration on trade volumes. Up to this point, we have omitted
integration variables in order to focus on the importance of controlling for country-pair
heterogeneity when estimating gravity models. However, now that we have established that the
FEa model is statistically preferred to standard cross-sectional methods, we introduce integration
into our model and demonstrate the striking effect that heterogeneity bias has on the results. We
would also like to alleviate the legitimate concern that the heterogeneity bias we detected above
was due to our implicit assumption that regional integration is uncorrelated with the independent
variables.
The most common and straightforward method for estimating the effects of integration in
a gravity model is to include dummy variables for each integration regime in place during the
sample period. Each of these dummies takes the value of 1 for each observation for which the
two countries are members of the regime, with the expectation that the coefficients on these
dummies are positive. We include three such dummy variables in our model, one each for the
European trading bloc, the North American trading bloc, and the South American trading bloc
(MERCOSUR).
Although there has been some deepening of trade integration in the European bloc, the
20
primary change over the period was an expansion in the number of countries covered under the
customs union. The twelve countries of the European Community (EC) renamed themselves the
European Union (EU) in 1992, but this had relatively little effect on internal trade policy, as it
was already nearly unfettered under the EC. Expansion of the bloc came in 1994 with the
European Economic Area (EEA), which extended the free trade zone to include Austria, Iceland,
Finland, Norway, and Sweden. To capture the effect of this trading bloc, our European bloc
dummy variable takes the value of 1 when trade is between members of the EC or EU for 199193, and between members of the EEA for 1994-95.
The North American trading bloc included only Canada and the United States for 199193, under the Canada-US Trade Agreement of 1988. The North American Free Trade Agreement
(NAFTA) expanded the free trade zone in 1994 to include Mexico. For present purposes, we
will ignore NAFTA’s relatively mild deepening of US-Canada integration, and focus instead on
it as an extension of the free trade bloc to Mexico. To capture the effects of North American
integration, our North American bloc dummy takes the value of 1 for trade between the US and
Canada for 1991-95, and between Mexico, Canada, and the US for 1994-95.
The third significant trade bloc during the period was MERCOSUR, which came into
force in 1995, reducing trade barriers between Argentina, Brazil, Paraguay, and Uruguay. Our
MERCOSUR dummy takes the value of 1 for trade between any two of these countries in 1995.
We include these three trade bloc dummies in the PCSa and FEa models, and report the
empirical results in Table 3. Note that inclusion of these dummies makes little difference for the
21
PCSa model. Nonetheless, a likelihood ratio test rejects the null hypotheses that including the
trade bloc dummies in the PCSa model does not alter the results to a statistically significant
extent.14 The results for the FEa model are also not dramatically different when the trade bloc
dummies are included. In fact, the null hypothesis that the inclusion of these variables has no
statistically significant effect on the results cannot be rejected.15
The dramatic change in the empirical results is in the comparison of the FEa and PCSa
models. The negative effects for all three trade blocs in the PCSa model are certainly contrary to
expectations. Not only are the coefficients on the European bloc and MERCOSUR dummies
statistically different from zero, they are also very large. The results suggest that membership in
the European trade bloc decreases trade with another bloc member by roughly the same as would
occur if the distance between the countries doubled. The predicted decrease in trade due to
membership in MERCOSUR is three time this.
Although these predictions have interesting implications regarding the future of the world
trading system, they do not hold up when the estimation allows for heterogeneity. Using the FEa
model, all three trade blocs have positive effects on the volume of trade, although none are
statistically different from zero. However, this is likely due to the simplistic nature of the
dummies, which do not account for trade diversion effects, rather than to the absence of real
effects (Cheng and Wall, 1999). For our present purposes though, the dummies we use are useful
to illustrate how the FEb model at least brings the results into the realm of believability.
7KLVLVZLWKDFULWLFDOYDOXHRIDWWKHOHYHODQG 2(3) = 79.38.
7KLVLVZLWKDFULWLFDOYDOXHRIDWWKHOHYHODQG 2(3) = 4.12.
14
15
22
7. Conclusions
The objective of this paper is to argue that heterogeneity needs to be allowed for when
using the gravity model to estimate bilateral trade flows. Our empirical analysis indicates that
standard methods for estimating gravity models of trade suffer from heterogeneity bias due to
omitted or misspecified variables. To address the problem we adopt a two-way fixed-effects
model in which country-pair and period dummies are used to reflect the bilateral relationship
between trading partners. The fixed effects capture those factors such as physical distance, the
length of border (or contiguity), history, culture, language, etc., that are constant over the span of
the data, and which are correlated with the volume of bilateral trade.
We show that existing empirical models are special cases of our model, and that the
restrictions necessary to obtain these special cases are not supported statistically. We conclude
that the preferred specification of the gravity model takes into account the heterogeneity of the
country pairs using country-pair dummies, and includes the populations of the two countries. As
the gravity model has become the “workhorse” of empirical work on the effects of integration,
we also check the importance of allowing for heterogeneity when doing this work. Our results
suggest that the results when heterogeneity is allowed for can differ wildly from when it is not.
23
Data Appendix
1. Definitions of variables
Volume of Exports, measured in millions of US dollars, from the International Monetary Fund’s
Direction of Trade Statistics. Bilateral Exchange Rates are from the IMF’s International
Financial Statistics. Both are downloaded from Datastream’s IMF database, and deflated
using the $-deflator from the World Tables 97 CD-ROM.
Gross Domestic Product is in millions of 1987 US dollars, and Population is in thousands of
inhabitants. Both are from the World Bank’s World Tables 97 CD-ROM.
Distance, expressed in kilometers, is the distance between capital cities, obtained from John
Haveman’s web site at ftp://intrepid.mgmt.purdue.edu/pub/Trade.Data/dist.txt, and from
http://www.indo.com/distance/.
Contiguity is equal to 1 if two trading partners share common border.
Common Language is equal to 1 if two trading partners share a common first language.
European Bloc is equal to 1 when both countries are members of the EC for 1991, the EU, for
1992-93, or the EEA for 1994-95.
North American Bloc is equal to 1 for Canada-US trade for all years, and Canada-Mexico and
US-Mexico trade for 1994-95.
MERCOSUR is equal to 1 in 1995 for trade between Argentina, Brazil, Paraguay, and Uruguay.
2. Countries included in data set
22 Exporters: Argentina, Australia, Canada, China, Finland, France, Hong Kong, Italy, Japan,
Kenya, South Korea, Malaysia, The Netherlands, Philippines, Portugal, Singapore, Spain,
Sweden, Switzerland, Thailand, United Kingdom, United States
116 Importers: Albania, Algeria, Angola, Argentina, Australia, Austria, Bahamas, Bahrain,
Bangladesh, Barbados, Belgium, Bolivia, Botswana, Bulgaria, Burkina Faso, Burundi, Brazil,
Brunei, Cameroon, Canada, Central African Republic, Chad, Chile, China, Colombia, Congo,
Costa Rica, Czech Republic, Denmark, Djibouti, Dominican Republic, Ecuador, Egypt, El
Salvador, Ethiopia, Finland, France, Gabon, Gambia, Germany, Ghana, Greece, Guinea, GuineaBissau, Guyana, Haiti, Honduras, Hong Kong, Hungary, India, Indonesia, Ireland, Israel, Italy,
Jamaica, Japan, Kenya, S. Korea, Kuwait, Lebanon, Luxembourg, Macao, Madagascar, Malawi,
Malaysia, Mauritania, Mexico, Moldova, Mongolia, Morocco, Mozambique, Namibia,
Netherlands, New Zealand, Nicaragua, Nigeria, Norway, Oman, Pakistan, Panama, Papua New
Guinea, Paraguay, Peru, Philippines, Poland, Portugal, Qatar, Romania, Russia, Rwanda, Saudi
Arabia, Senegal, Sierra Leone, Singapore, Solomon Islands, South Africa, Spain, Sri Lanka,
Suriname, Swaziland, Sweden, Switzerland, Syria, Tanzania, Thailand, Togo, Tunisia, Turkey,
Uganda, United Kingdom, United States, Uruguay, Venezuela, Yemen, Zambia, Zimbabwe
24
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26
Table 1: Regression results for single-year cross-section; augmented and basic versions; 1991-95
dependent variable = log of exports
1991
1992
1993
1994
1995
CSa
CSb
CSa
CSb
CSa
CSb
CSa
CSb
CSa
CSb
constant
-5.154*
(1.058)
-4.954*
(0.997)
-5.380*
(1.068)
-5.184*
(1.014)
-4.913*
(1.049)
-4.943*
(0.993)
-4.798*
(1.073)
-4.797*
(1.014)
-4.686*
(1.099)
-4.694*
(1.036)
origin GDP
0.635*
(0.050)
0.670*
(0.040)
0.641*
(0.051)
0.671*
(0.041)
0.609*
(0.050)
0.658*
(0.039)
0.612*
(0.053)
0.663*
(0.041)
0.607*
(0.055)
0.662*
(0.042)
destination GDP
0.853*
(0.046)
0.805*
(0.034)
0.828*
(0.046)
0.784*
(0.034)
0.840*
(0.046)
0.807*
(0.034)
0.858*
(0.046)
0.820*
(0.034)
0.857*
(0.046)
0.817*
(0.034)
origin population
0.073
(0.063)
0.061
(0.064)
0.099
(0.064)
0.102
(0.065)
0.106
(0.066)
dest. population
-0.075
(0.055)
-0.069
(0.055)
-0.044
(0.055)
-0.055
(0.056)
-0.060
(0.056)
distance
-0.819*
(0.081)
-0.823*
(0.079)
-0.765*
(0.078)
-0.771*
(0.080)
-0.811*
(0.081)
-0.806*
(0.080)
-0.849*
(0.082)
-0.847*
(0.081)
-0.850*
(0.083)
-0.849*
(0.081)
contiguity
0.160
(0.316)
0.151
(0.316)
0.212
(0.320)
0.204
(0.320)
0.208
(0.318)
0.200
(0.319)
0.236
(0.322)
0.225
(0.323)
0.188
(0.323)
0.175
(0.324)
common language
0.704*
(0.169)
0.701*
(0.169)
0.729*
(0.171)
0.725*
(0.171)
0.740*
(0.170)
0.736*
(0.171)
0.661*
(0.172)
0.658*
(0.172)
0.662*
(0.173)
0.660*
(0.173)
log-likelihood
-709.37
0.642
-710.92
0.641
-714.69
0.625
-715.90
0.624
-712.71
0.635
-714.19
0.634
-717.65
0.641
-719.33
0.640
-718.88
0.638
-720.71
0.637
R2
All variables except for the contiguity and language dummies are in logs. Standard errors are in parentheses. * denotes significance at
5% level. The basic version (CSa) is a form of the augmented version (CSa) with the coefficients on the populations restricted to zero.
There are 422 observations for each year.
27
Table 2: Regression results for models using pooled data; 1991-95;
dependent variable = log of exports
pooled cross-section
PCSa
PCSb
fixed effects
FEa
FEb
alternatives w/heterog.
BEa
Ma
constant
-5.007*
(0.477)
-4.931*
(0.453)
-
-
0.034†
(0.020)
-4.505
(10.887)
origin GDP
0.621*
(0.023)
0.664*
(0.018)
0.293*
(0.066)
0.323*
(0.063)
0.195*
(0.083)
0.293
(0.263)
destination GDP
0.847*
(0.020)
0.807*
(0.015)
0.520*
(0.054)
0.506*
(0.054)
0.427*
(0.066)
0.520*
(0.214)
origin population
0.088*
(0.029)
1.762*
(0.685)
2.293†
(1.195)
1.769
(2.732)
dest. population
-0.061*
(0.024)
-2.014*
(0.352)
-1.458*
(0.587)
-2.016
(1.403)
distance
-0.819*
(0.036)
-0.819*
(0.036)
contiguity
0.200
(0.142)
0.190
(0.143)
common language
0.700*
(0.076)
0.696*
(0.076)
1992
-0.005
(0.091)
-0.004
(0.091)
0.030
(0.019)
0.021
(0.018)
1993
0.027
(0.091)
0.028
(0.091)
0.050*
(0.023)
0.035*
(0.018)
-0.023
(0.022)
0.050
(0.094)
1994
0.022
(0.091)
0.022
(0.091)
0.099*
(0.029)
0.073*
(0.019)
0.023
(0.021)
0.099
(0.114)
1995
0.056
(0.091)
0.056
(0.091)
0.198*
(0.035)
0.164*
(0.022)
0.080*
(0.021)
0.198
(0.138)
2110
10
-3582.57
0.638
3.405
1.764
3686.24
2110
430
131.01
0.987
0.283
0.078
109.11
2110
428
107.80
0.986
0.304
0.080
111.54
1688
8
-331.93
0.073
0.403
0.088
146.45
2110
147
-2950.43
0.788
2.935
1.102
2024.71
observations
parameters
log-likelihood
2110
12
-3575.10
0.641
R2
Akaike Info. Crt.
3.400
Amemiya Prob. Crt. 1.755
sum of sqd. resids.
3660.23
0.030
(0.076)
All non-dummy variables are in logs. Standard errors are in parentheses. * and † denote significance
at 5% and 10% levels. For the BEa model all variables are in differences from the previous year.
28
Table 3: Regression results with integration dummies; 1991-95;
dependent variable = log of exports
pooled cross-section
PCSa
fixed effects
FEa
constant
-3.572* (0.490)
-
origin GDP
0.649* (0.023)
0.312* (0.067)
destination GDP
0.870* (0.020)
0.524* (0.054)
origin population
0.054† (0.028)
1.864* (0.690)
destination population
-0.083* (0.024)
-1.948* (0.356)
distance
-1.022* (0.041)
contiguity
0.025 (0.144)
common language
0.622* (0.075)
European bloc
-1.112* (0.115)
0.064 (0.044)
North American bloc
-0.062 (0.397)
0.157 (0.233)
MERCOSUR
-3.165* (0.924)
0.166 (0.202)
1992
-0.006 (0.089)
0.027 (0.019)
1993
0.027 (0.089)
0.047* (0.024)
1994
0.104 (0.089)
0.087* (0.030)
1995
0.149† (0.089)
0.181* (0.037)
2110
15
-3524.73
0.657
3.355
1.678
3489.61
2110
433
132.97
0.987
0.284
0.078
108.91
observations
parameters
log-likelihood
R2
Akaike Info. Crt.
Amemiya Prob. Crt.
sum of sqd. resids.
All non-dummy variables are in logs. Standard errors are in parentheses. * and †
denote significance at 5% and 10% levels.
29
Figure 1: Plots of Residuals; Various Models
Pooled Cross-Section (PCSa)
-3
6
4
2
0
-2
-4
-6
-8
2
7
Fixed Effects (FEa)
12
-3
6
4
2
0
-2
-4
-6
-8
2
log of exports
-3
2
7
12
log of exports
Restricted Time Effects (BEa)
6
4
2
0
-2
-4
-6
-8
7
Restricted Fixed Effects (Ma)
12
-3
log of exports
6
4
2
0
-2
-4
-6
-8
2
7
log of exports
30
12