SIMULATIONS MODELS FOR INTERNATIONAL TRADE
GRAVITY EQUATIONS FOR INTERNATIONAL TRADE MODELS
Paris-Dauphine / September 2015
DOCUMENT 1:
CLASS SCRIPT: BASIC DEFINITIONS AND CONCEPTS
Ramón Mahía – UAM
(Based on the material provided y UNCTAD-WTO)1
DOCS AND SOURCES OF INTEREST
Apart from the main reference mentioned in the footnote, OTHER RELEVANT WORKS for those who want
to go further in this specific topic are:
-
Head, K. (2003), “Gravity for beginners”, mimeo, University of British Columbia.
http://vi.unctad.org/tda/background/Introduction%20to%20Gravity%20Models/gravity.pdf
-
A really complete and in depth work about gravity equations and their use for empirical exercises
(almost 70 pages!).
Head, K., & Mayer, T. (2013). Gravity equations: Workhorse, toolkit, and cookbook. Handbook of
international economics, 4.
http://strategy.sauder.ubc.ca/head/papers/headmayer_revised.pdf
-
Another “accessible” example of a STATA Gravity – Model exercise:
http://www.agrodep.org/sites/default/files/Technical_notes/AGRODEP-TN-05_2.pdf
-
For those seeking for technical details about some of the main on the econometrical side of gravity
equations:
The Gravity Model in International Trade AGRODEP Technical Note TN -04. Luca Salvatici.
http://www.agrodep.org/sites/default/files/Technical_notes/AGRODEP-TN-04-2_1.pdf
-
Seminal work on the econometric implications of Jensen’s inequality in the context of gravity
equation:
1
IMPORTANT NOTE: The content of this document, and specially the exercise section, is
based on the document prepared by UNCTAD-WTO entitled “A Practical Guide to Trade
Policy Analysis. (Chapter 3. Analyzing bilateral trade using the gravity equation). To access
the on-line version of this UNCTAD-WTO doc, visit the WEB page:
http://vi.unctad.org/tpa/index.html
1
Silva, J. S., & Tenreyro, S. (2006). The log of gravity. The Review of Economics and statistics,
88(4), 641-658.
http://personal.lse.ac.uk/tenreyro/jensen08k.pdf
-
Seminal paper about the role of Multilateral Trade Resistance term in gravity equation:
Anderson, J. E. and van Wincoop, E. (2003), “Gravity with gravitas: a solution to the border
puzzle”, American Economic Review 93: 170–92.
http://www.econ.ku.dk/Nguyen/teaching/Anderson%20van%20Wincoop%202003%20Gravitas.
pdf
1.- SOME BASIC THEORY AND DEFINITIONS
Foundations of a Gravity Model for international trade:
-
Following Newton expression for masses it states that, broadly constructed, trade (attraction)
between two countries is proportional to their size (size of the economy) and inversely proportional
to their “distance”.
-
Even before that someone did find a theoretical basis, and surprisingly enough, simple gravity
equations did (and do) a pretty successful job at explaining bilateral trade with just the size of the
economies2 and their distance.
-
Gravity approach is, in fact, widely used to infer trade flow potentials, or to estimate the effects on
trade of institutions such as customs unions, monetary agreements, exchange rate mechanism, ethnic
ties, linguistic identity, international borders
-
The basic expression is currently formulated as:
“the trade flow from country i to country j (Xij) is proportional to the product of the two
countries’ GDPs (Yi , Yj) and inversely proportional to their distance, Dij, broadly construed to
include all factors that might create trade resistance”.
𝛼
𝛼
𝛼
𝑋𝑖𝑗 = 𝛼0 𝑌𝑖 1 𝑌𝑗 2 𝐷𝑖𝑗3 𝜀𝑖𝑗
o
o
o
o
o
o
-
α0: Coefficient (intercept) not depending on “i” or “j”
Xij: Exports from “i” to “j” (or imports of “j” from “i” )
Yi: Exporter factors (GDPi, for example…)
Yj: Importer factors (GDPj, for example...)
Dij:distance/trade barriers of exporter “i” to enter / reach market “j”
εij: random term (stating the random nature of the eq, in contrast to Physics determinism)
Newcomers feel OK with the link between GDP’s and trade3 but one of the common questions of
newcomers is: Is really the physical distance an explanatory variable for trade?. Yes, it is. Some
2
Puede comprobarse posteriormente, usando los fichero de ejemplo, como existe una muy notable relación entre el tamaño de las importaciones
y el PIB del exportador (de origen)
twoway (scatter lgdp_exporter limports) (lfit lgdp_exporter limports) if importer =="USA" & year==2000
3
Try to represent relationship between GDP of exporter and imports for a given country using the data of the example 1 for OECD countries to
see how clear it looks (twoway (scatter lgdp_exporter limports) (lfit lgdp_exporter limports) if importer =="USA" & year==2000
2
EMPIRICAL studies clearly illustrate that, overall, distance matter SO MUCH because it measures
somehow a lot of relevant trade resistance issues: transport costs, transaction costs,
perishability/losses of goods during transport, synchronization costs, communication costs, cultural
distance….
-
Apart from being intuitive and empirically true (at least apparently), some important studies have
been able to provide THEORETICAL foundations for the gravity model/equation4 deriving gravity
equations valid to represent a variety of different trade theoretical models. This theoretical
foundations are critical to understand the right specification of the empirical model, the “correct”
signs and size of estimated coefficients, and the proper way of conducting correct econometrical
inferences. For example, the theoretical background is able to explain some of the following model
features:
o
o
o
o
-
According to the theoretical background, elasticities of GDP’s and distance should be around
1
According to that theory behind, the gravity equation SHOULD follow a multiplicative function
According to the theory derivation, the gravitation equation MUST also include a Multilateral
Resistance Term OR “remoteness” term (will talk about it later on)
……and many more interesting facts that frame and condition empirical exercises
Some almost evident - additional - main features of the equation:
o
o
Multiplicative form (with crucial empirical/econometrical implications)
Flexible “nature”:
 Unit of observations: “i” and “j” might represent countries, regions, provinces, or
even firms and …
 “x” might represent TOTAL trade, share of trade, sectoral trade, product trade,…
 …or even a NON TRADE variable: migration, commuting, passengers, trips, fleets…
 …..BUT…be carefull when dealing with disaggregated data because some critical
issues appear when dealing with empirics
 A time subindex “t” might be added to every term
An introductory note to the importance of Multilateral Trade Resistance terms (MTR) in gravity
equations. (Anderson and van Wincoop’s (2003))
-
The basic idea of the MTR is: Two neighbor countries will trade more if they are isolated from the rest
of the world (and near). The authors (among others) showed that “The gravity equation tells us that
bilateral trade, after controlling for size, depends on the bilateral trade barriers between “i” and “j”
BUT RELATIVE to the product of their Multilateral Resistance Indices”: bilateral “ij” barrier relative to
average trade barriers that both regions face with all their trading partners.
-
Good example of the idea found at Head, k (2003): “The importance of remoteness in actual trade
patterns can be illustrated by comparing trade between Australia and New Zealand with trade
4
See an intuitive sketch in the text Head, K. (2003), “Gravity for beginners” of British Columbia University.
http://vi.unctad.org/tda/background/Introduction%20to%20Gravity%20Models/gravity.pdf
3
between Austria and Portugal. The distance between each pair’s major cities is approximately the
same: Lisbon–Vienna and Auckland–Canberra both happen to be 1430 miles apart. Furthermore the
product of their GDP’s are similar (Australia–New Zealand is only 20% smaller). Hence, omitting
remoteness, the gravity equation would predict that Austria–Portugal trade would be slightly larger.
In fact, however, in 1993 Australia–New Zealand trade was nine times greater than Austria–Portugal
Trade”.
-
-
-
It is easy to see why higher multilateral resistance of the importer “j” raises its trade with “i”.
Price/Cost advantage: For a given bilateral barrier between “i” and “j”, higher barriers between “j”
and its other trading partners will reduce the relative price of goods from “i” and raise imports from
“i”.
Higher multilateral resistance of the exporter also raises trade: Higher trade barriers faced by an
exporter will lower the demand for its goods and therefore its supply price “pi”. For a given bilateral
barrier between “i” and “j”, this raises the level of trade between them.
Empirical implication: All this means that in the gravity equation, we have to include MTR (or the
inverse, that said, easiness to trade) for EITHER “i” and “j”. Following Anderson and van Wincoop’s
derivation, the gravity equation turns out to be:
1−𝜎
𝑋𝑖𝑗 =
𝑌𝑖 𝑌𝑗 𝑡𝑖𝑗
(
)
𝑌 Π𝑖 Ρ𝑗
Where, in the “resistance to trade” term we find:
o
o
o
tij: (bilateral trade resistance measured as one plus the tariff equivalent of overall trade
costs of imports of “j” from “i”)
Πj: Ease of access of exporter “i”
Ρj: Ease of access of importer “j”
Πj and Ρj are low if the countries are “remote” from world markets (physically or in
terms of high trade protection).
o
o
o
-
Xij: Exports from “i” to “j” (or imports of “j” from “i” )
Yi/Yj/Y: GPD measures for “i”, “j” and the world
σ: elasticity of substitution
We will see how to implement this terms in the empirical estimations but, What if we omit MRT?. The
empirical problem is that this MRT’s are obviously highly correlated with bilateral trade barriers (tij).
Omitting MTR induce potentially severe bias in the coefficients of the distance and border variables.
2.- INTRODUCTION TO EMPIRICAL ESTIMATION OF A GRAVITY MODEL
-
Basic expression:
o
Expression: we come back to the Anderson and van Wincoop’s formula using now a
generalized linear expression such as:
4
𝑙𝑛𝑋𝑖𝑗 = 𝑎0 + 𝑎1 𝑙𝑛𝑌𝑖 + 𝑎2 𝑙𝑛𝑌𝑗 + 𝑎3 𝑙𝑛𝑡𝑖𝑗 + 𝑎4 lnΠ𝑖 + 𝑎5 lnΡ𝑗 + 𝜀𝑖𝑗
Where:


“a’s” = elasticities (as ever in a log-log expression)
a3=1-σ (been σ the elasticity of substitution5)
Attention to some basic data handling issues:


-
“Xij” represents only the flow from “i” to “j”; there might be another observation for
trade from “j” to “i”. We shouldn’t aggregate both averaging the reciprocal trade
flows
GDP’s and trade flows (X) are measured in nominal terms (not real terms)6
Measuring “observables”:
o
“Xij”: “ij” trade flow
 ATTENTION:
- Xij represents only ONE flow from “i” to “j” (exports, for example) so we will
have a different second observation for the model containing “Xji (exports
from “j” to “i”) we shouldn’t aggregate both “ij” and “ji” averaging the
reciprocal trade flows7
- FOB or CIF values? One might understand that it depends on if we are
measuring Xit as exports from “i” to “j” or imports of “j” from “i” but NO:
Basically, FOB are the best option (if available)8.
o
Yij: Size of “I” and “j, usually GDP’s
 ATTENTION: GDP’s and trade flows (X) should be measured in nominal terms (not
real terms)9
o
tij (resistance terms) could contain both BARRIERS and INCENTIVES to trade between “i” and
“j”
o
“ij” Natural barriers and/or costs related barriers:
5
I know that you know but, just in case, remember that this term refers to the degree of trade elasticity of substitution between domestic and
traded goods. The higher the elasticity, the greater the effect of resistance term to trade.
6
“Gravity is an expenditure function allocating nominal GDP into nominal imports; therefore inappropriate deflation probably creates biases
via spurious correlations”. UNCTAD-WTO A Practical Guide to Trade Policy Analysis
7
We will have one observation for each flow. Imagine the case of trade between a big and small country: one of “ij” or “ji” might be very high
and the other one might be very low. The equation could explain both attending to relative sizes of “I” and “j” (as exporter and importer) but it
could never explain the average.
8
Using CIF data may lead to simultaneous equation biases, as the dependent variable includes costs that are correlated with the right hand
side variables for distance and other trade costs.
9
“Gravity is an expenditure function allocating nominal GDP into nominal imports; therefore inappropriate deflation probably creates biases
via spurious correlations”. UNCTAD-WTO A Practical Guide to Trade Policy Analysis
5


o
(dij): Bilateral distance “ij” as proxy for transport costs (different means of distance
calculation between countries10)
Others (dummies 0/1): (ISij) one of the two being islands, (LLij) one of the two being
landlocked countries…
Incentives “ij”:
 (CBij) Common border or adjacency
 Existence of Bilateral, Regional or Multilateral Trade Agreement (TAij) (Usually with
a dummy, although some caveats should be made11)
 Lower information costs:
- Common language (CLij)
- Cultural / Historical ties (had been colonies of each other or a common
colonizer) (COLij)
This vector tij usually takes the theoretical based form of (for the previous variables):
𝛿
𝑡𝑖𝑗 = 𝑑𝑖𝑗1 · exp⁡(𝛿2 𝐼𝑆𝑖𝑗 + 𝛿3 𝐿𝐿𝑖𝑗 + 𝛿4 𝐶𝐵𝑖𝑗 + 𝛿5 𝑇𝐴𝑖𝑗 + 𝛿6 𝐶𝐿𝑖𝑗 + 𝛿7 𝐶𝑂𝐿𝑖𝑗 )
-
How to deal with non-observable nature of MTR’s:
1.- To use a nonlinear iterative method to estimate MTR effects (Anderson and van
Wincoop’s)12.
2.- To use a linear estimator adding fixed effects (country, or country x time):
2.1.- With a cross – section dataset:

2.1.a.- Country dummies for countries “i” and “j”: (Ii) and (Ij).
𝑙𝑛𝑋𝑖𝑗 = 𝑎0 + 𝑎1 𝐼𝑖 + 𝑎2 𝐼𝑗 + 𝑎3 ln⁡(𝑡𝑖𝑗 ) + 𝜀𝑖𝑗


ATTENTION if we allow country dummies in a cross-section approach, we
cannot estimate/identify parameters for other country specific variables
(Such as GPDi, or GDPj !!) so this approach is only valid when the interest is
on bilateral / country pair coefficients (such as the effect of a common border
or the impact of a bilateral Trade Agreement, for example)
2.1.b.- To proxy MTR with remoteness - like indexes13.
10
Both in terms of distance measurement choice (Euclidean, great-circle, orthodromic…) and in terms of among “where” to measure
(economic “poles”, capitals, main ports,…)
11 A dummie is not able to capture those trade agreements with asymmetrical benefits/conditions for its members, for different products or
simply with a progressive effect. A single dummy suppose that the “treatment” effect is the same for all the countries participating in this
agreement.
12
For details see section 3.2. (Iterative structural estimation) in “Gravity Equations: Workhorse,Toolkit, and Cookbook Keith Head and Thierry
Mayer” http://www.cepii.fr/PDF_PUB/wp/2013/wp2013-27.pdf
13
See section 3.1. (Proxies for multilateral resistance term) in “Gravity Equations: Workhorse,Toolkit, and Cookbook Keith Head and Thierry
Mayer” http://www.cepii.fr/PDF_PUB/wp/2013/wp2013-27.pdf
6
For example, to compute (GDP weighted) average distance of trader “i” to all
countries/regions other than “j” (and vice versa).
𝑅𝐸𝑀𝑖 = ∑𝑚≠𝑗


𝑑𝑖𝑚
⁄𝑌 or
𝑚
𝑅𝐸𝑀𝑖 = ∑m≠𝑗
𝑑𝑖𝑚
𝑌𝑚 ⁄𝑌
This is the only option if we are interested in country – specific variables
(country specific dummies would preclude the estimation of such
parameters)
The major drawback relates to the use of distance. It is difficult to find a good
way of measuring that distance “ij” and, even if we manage, it might be a
simplistic way of measuring MRT’s.
2.2.- With a panel dataset:
Panel Datasets are a good solution. Apart from controlling for MTR terms, the use of standard
Panel Data estimations help us to control, in addition, for non - observable bilateral / country pair
heterogeneity (bilateral / country pair fixed effects)…..

2.2.a.- TIME INVARIANT Country dummies for countries “i” and “j” (assuming MTR
constant over time).
𝑙𝑛𝑋𝑖𝑗𝑡 = 𝑎0 + 𝑎1 ln⁡(𝐺𝐷𝑃𝑖𝑡 ) + 𝑎2 ln⁡(𝐺𝐷𝑃𝑗𝑡 ) + 𝑎3 𝐼𝑖 + 𝑎4 𝐼𝑗 + 𝑎5 ln⁡(𝑡𝑖𝑗 ) + (𝑎6 𝐼𝑡 )+𝜀𝑖𝑗




We need to ponder if the MTR time invariant assumption is plausible14
In that case, that allows the estimation of OTHER country specific factors
(such as GDP’s) but only under the assumption that this country effects vary
over time.
REMEMBER THAT country pair time invariant effects cannot be estimated
with Fixed Effects panel data estimator if our interest relies on other time
invariant variables (such as being landlocked, distance, common borders,
common language, trade agreement,….). In this case, the use of Random
Effects is an option if we want to get time invariant coefficients (at the risk,
as ever, of bias in the rest of coefficients)
2.2.b.- NON TIME INVARIANT Country dummies for countries “i” and “j” and period
“t” (allowing MTR change over time): (Iit) and (Ijt).

ATTENTION, if we allow time varying country dummies, we cannot estimate
parameters for other time varying country specific variables (such as GDP, for
example,…) = so again this approach is only valid when the interest is on
bilateral / country pair coefficients
𝑙𝑛𝑋𝑖𝑗𝑡 = 𝑎0 + 𝑎1 𝐼𝑖𝑡 + 𝑎2 𝐼𝑗𝑡 + 𝑎3 ln⁡(𝑡𝑖𝑗 ) + (𝑎4 𝐼𝑡 )+𝜀𝑖𝑗
14
Yet with some exceptions, MRT is only constant within short periods of time.
7
3.- OTHER EMPIRICAL (more advanced) ISSUES.
-
There are a handful of relevant issues (sometimes critical):
o
o
o
o
o
o
o
o
-
How to deal with zero trade values (Xij=0)15
Log – Log form inducing bias in the (highly probable) presence of heteroskedasticity
Some extra-cautions when working with disaggregated data such as sectors, firms,…
Endogeneity in gravity equation: causation between trade and trade policy could be reversed
when, for example in the case of the signature of a FTA, there exist a selection of countries
based on intensity of trade, and not the other way round.
Spatial correlation
Critical omitted variables biasing coefficients systematically (migration, for example, that
might be related to trade and other exogenous variables)
Statics nature: several authors have proposed a dynamic gravity equation in place of the
traditional static gravity equation
…..and some others….
How to deal with zeros:
o
Reasons for zeros:
 Real zero trade
 Value rounded to zero trade
 Missing values for sometimes unknown reasons (non reported by countries, error in
dissemination or manipulation,….)
….
The problem is that, in many times, the researcher doesn’t know what proportions
of different types of “zeroes” does he have in his dataset.
o
Alternatives:




15
Substitute zeros by small numbers (1, for example): only appropriate if real zero or
almost zero trade BUT arbitrary level generate unexpected impact on parameters
Simply truncate the sample to avoid zero cases (delete “ij” cases with zeroes): only
“reasonable” if zeros are true missings and thus randomly distributed across “n” and
“t”BUT obviously bias selection problem
To control for the “selection bias” using a Heckman procedure (Heckman 2- stages
least squared estimation that introduces in the specification the inverse of the so
called Mills ratio). However, that method requires “instrumental” variables that may
explain the selection (zero or positive trade) but not the value of positive trade.
Not to use logs, and estimate the model in levels:
 With a linear estimator = NO: theoretical foundation of the gravity equation
implies multiplicative form
Remember that we are using a log-log model
8

-
With a non-linear estimator:
 (Pseudo) Poisson maximum likelihood (ML) estimator applied to the levels
of trade estimating directly the non- linear form16.
 Tobit (that allows a significant proportion of zeros) on the log of trade plus
a constant: the critique is that Tobit applies for left-censored zeros only
when those “zeroes” have the economical meaning of zeros (no trade or
almost no trade because of prohibitive trade costs, for example)
Log inducing bias in presence of heteroskedasticity
The Gravity model is basically defined as:
𝛼
𝛼
𝛼
𝑋𝑖𝑗 = 𝛼0 𝑌𝑖 1 𝑌𝑗 2 𝐷𝑖𝑗3 𝜀𝑖𝑗
Where the random term εij have the standard property:
𝐸[𝜀𝑖𝑗 /𝑌𝑖 𝑌𝑗 𝐷𝑖𝑗 ] = 1
The model is then usually linearized taking the following simple form:
𝑙𝑛𝑋𝑖𝑗 = 𝛼0 + 𝛼1 𝑙𝑛𝑌𝑖 + 𝛼2 𝑙𝑛𝑌𝑗 + 𝛼3 𝑙𝑛𝐷𝑖𝑗 + ln(𝜀𝑖𝑗 )
The problem is that, as Jansen’s inequality states:
E[ln(ε)]≠lnE[ε]
On the contrary, the expectation of a log of a random variable E[ln(ε)] not only depends on the mean
of that variable E[ε] but also on higher order moments (such as the variance) V[ε]. Then when the
random term in the original (nonlinear) model presents heteroskedasticity,
𝑉[𝜀𝑖𝑗 ] = 𝜎𝑖𝑗 = 𝑓(𝑌𝑖 , 𝑌𝑗 , 𝐷𝑖𝑗 )
Then the expected value of its logarithm E[ln(ε)], that is also a function of σij, is also a function of the
explanatory variables:
𝐸[𝑙𝑛(𝜀𝑖𝑗 )] = 𝑓(𝜎𝑖𝑗 ) = 𝑓(𝑌𝑖 , 𝑌𝑗 , 𝐷𝑖𝑗 )
The result is therefore that some of the parameters in the log model might be biased and inconsistent.
A way of solving this problem is to assume a pattern of heteroskedasticity where conditional variance
depends on conditional mean, that is, a Poisson distribution for the endogenous variable where
E[yi/X]=V[yi/X]. Then a PPML (Poisson pseudo- maximum likelihood estimator) traditionally used for
count data can be used to consistently estimate the model. One interesting extra - point is that this
16
Silva, J. S., & Tenreyro, S. (2006). The log of gravity. The Review of Economics and statistics, 88(4), 641-658.
9
estimator is applied for the model in LEVELS so no logs are needed and thus, at the same time, we fix
the problem of zeros.
4.- EMPIRICAL EXERCISES USUALLY COVERED WITH GRAVITY MODELING
-
Simulate scenarios for international trade adjustments:
o
o
Related to regular changes in main explanatory variables (such as GDP’s or tariffs)
Related to particular events / policy issues: Free Trade Agreement (Trade creation Vs Trade
Diversion), Currency unions, Political federations, and other similar trade distortions.
-
Quantify trade potential between two countries. Basically, a gravity equation is estimated using data
only of those countries that may have supposedly reached their trade potential (for example those
belonging to a free trade area) and then, after some adjustments, the estimated equation is applied
to pairs of countries out of the sample: the real trade is then compared with predicted (potential)
trade.
-
Measure bilateral trade resistance / cost. A gravity equation can also be used “in reverse” to measure
bilateral trade costs at an aggregate level. The idea is to solve a theoretical gravity equation for the
trade costs term instead of trade flows and to express these costs as a function of the observable
trade data and this allows to estimate the tariff equivalent of non tariff barriers
10