Estimating Trade Flows: Trading Partners And Trading Volumes

Estimating Trade Flows: Trading Partners
And Trading Volumes
Elhanan Helpman, Marc Melitz & Yona Rubinstain (2008)
Presented by Camilo A. Ulloa
March 17, 2009

1 Introduction
Gravity equations (Timebergen, 1962):
T Fij : f Economic sizeij; T rade resistenceij
Traditional estimates of trade ‡ows using gravity equations are biased:
– They include only countries with positive trade ‡ows between them.
,! Heckman selection bias.
– They impose symmetry in the trade ‡ows between pairs of countries.
,! Omitted variable bias: fraction of exporting …rms (i.e., extensive
margin of trade).

In this paper, the authors develop a theoretical model of international trade
with productivity heterogeneous …rms (Melitz 2003).
(i) The model is able to predict positive as well as zero trade ‡ows.
(ii) The model can predict positive, but asymmetric, bilateral trade ‡ows.
(iii) The model generates a generalized gravity equation (Anderson and van
Wincoop, 2003)
,! The impact of trade frictions on trade ‡ows can be decomposed:
– Intensive margins: Trade volume per exporter.
– Extensive margins: Number of exporters.

2 A Glance at the data.

Thus, the empirical evidence suggest that:
(i) About 50% of the country pairs do not trade between them.
(ii) The world trade growth from 1970 to 1997 was predominantly due to the
growth of the volume of trade among countries that traded with each other
in 1970.
(iii) Bilateral trading volumes in 1997 of countries that traded in both directions
since 1979 have been much larger than bilateral trading volumes of
countries that did not trade in both directions (35 times larger on average).

3 Theory.
There are J countries, j = 1; 2; :::; J. Each of them consumes and produces
a continuum of products. Country j 0 s preferences are described by
the following utility function (CES).
where:
uj =
" Z
l2Bj
xj (l) dl
– xj (l) : Consumption of product l.
# 1=
; 0 < < 1;
– Bj : Set of products available for consumption in j.
– " = 1=(1 ) : Elasticity of substitution across products.

Then, country j’s demand for product l is:
where
xj (l) = epj (l) "
Yj 1 "
Pj – epj (l) : price of product l in country j
– Pj : country j’s ideal price index:
Pj =
" Z
l2Bj
# 1=(1 ")
epj
1 "
(l)
,! every product has a constant demand elasticity ".
(1)
(2)

Country j has a measure Nj of …rms, each one producing a distinct product,
which is also distinct from the products produced by country-i …rms.
Thus, there are P J j Nj …rms and products in the world economy.
A country- j …rm faces a production cost cja per unit of output.
– a : …rm speci…c unit requirements of country j 0 s inputs (i.e., domestic
…rms only use domestic inputs).
– cj : country speci…c cost of these inputs.
,! Thus, 1=a is the …rm’s productivity level.
It is assumed that a is distributed according to a common CDF for all
countries, G (a), with support [a L; a H], and a H > a L > 0.

A country- j …rm that sells its product in country i faces 2 additional costs:
– A …xed cost of serving country i: cjfij
– A "melting iceberg" transport cost: ij
,! fij and ij are not …rm speci…c.
n
fjj = 0; fij > 0 o
n
jj = 1; ij > 1 o
There is monopolistic competition in …nal products. Thus (1) implies that a
country- j producer with productivity 1=a will set a price: pj (a) = cja=
If the country- j producer of l sells to consumers in country i, then sets a
delivered price (in country i)
cja
(3)
epj (l) = ij

Thus, the operating pro…ts from these sales to country i are:
ij (a) = (1 )
! 1 "
ijcja
Yi cjfij
Pi
,! Since fjj = 0; jj = 1 and 2 (0; 1) =) jj (a) > 0. That is, all
Nj producers sell in country j.
,! Moreover, sales in country i 6= j are pro…table i¤ a aij , where aij
is de…ned by the zero pro…ts condition:
(1 )
! 1 "
ijcjaij
Yi = cjfij
Pi
(4)

Notice that:
(i) Only a fraction G(aij) of country j0s Nj …rms export to country i
=) Bi is smaller than the total set of world products.
(ii) If aij aL =) G(aij) = 0 (i.e. The model can predict zero trade
volumes).
(iii) If aij > a L =) All …rms from country j export to i:

Let de…ne
Vij =
( R aij
a L a 1 " G(aij) for aij > a L
0
Then, the demand function (1) and the pricing rule (3) imply that the
value of country i’s imports from j is
Mij =
! 1 "
cj ij
YiNjVij
Pi
Furthermore the de…nition of the country i’s ideal price index (2) imply:
Pi =
JX
j=1
! 1 "
cj ij
NjVij
Pj
(5)
(6)
(7)

4 Empirical framework
First, a completely parametric approach was implemented and then some
of the parameterization assumptions were progressively relaxed.
Baseline case: 1=a is Pareto distributed, truncated to the support [a L; a H].
G (a) = ak ak L
ak H
ak ; k > (" 1)
L
,! aij < a L for some i j pairs is allowed (i.e., Mij = 0).
,! Mij 6= Mji is also allowed (i.e., Mij = 0 and Mji > 0 or Mij > 0
and Mji = 0).

Under the Pareto distribution assumption, (5) becomes:
where
Vij =
Wij = max
and aij is determined by (4).
k "+1
kaL (k " + 1) a k H ak L
8
<
:
! k "+1
aij
a L
W ij
1; 0
9
=
;
(8)

Rewriting (6) in log-linear form we get
mij = (" 1) ln (" 1) ln cj+nj+(" 1) pi+yi+(1 ") ln ij+ ij
It is assumed that the variable trade costs, ij,.that a¤ect the volume of
…rm-level exports costs are stochastic.
where:
(" 1)
ij D ij exp uij
– Dij : Symmetric distance between i and j (observed trade friction).
– uij i:i:d N 0; 2 u : Country-pair speci…c unmeasured (unobserved)
trade frictions.

Then the log-linear representation of (6) yields the estimating equation
where:
– Constant term:
mij = 0 + j + i dij + wij + uij (9)
0 = (" 1) ln + ln h
k "+1
kaL = (k " + 1) a k H akL – Fixed e¤ect of the exporting country: j = (" 1) ln cj + nj.
– Fixed e¤ect of the importing country: i = (" 1) pi + yi.
– Country-pair speci…c factor which controls for the fraction of …rms that
export from j to i: wij.
The inclusion of this factor is the most important di¤erence with traditional formulations (i.e.,
Anderson and van Wincoop, 2003).
i

Remark 1 Notice that:
(i) If wij is not included, then the coe¢ cient on any trade barrier
(i.e., ) can not be interpreted as the elasticity of a …rm’s trade
with respect to that trade barrier (i.e., distance) =) Omitted variable
bias.
(ii) If country pairs with zero trade ‡ows are excluded, then:
Corr(dij;uij)
> 0
For exmaple, country pairs with large observed trade barriers (i.e., high
dij ) that trade, are likely to have low unobserved trade barriers (i.e.,
high uij). =) Heckman selection bias.

4.1 Firm Selection into Export Markets (Wij : ' aij )
Let de…ne the latent variable
Zij =
(1 ) Picj ij
cjfij
" 1
1 "
YiaL (10)
That is, the ratio of variable export pro…ts for the most productive …rm
(1=a L) to the …xed costs for exports from j to i.
(k "+1)=(" 1)
Notice that equations (4) and (8) imply that: Wij = Z ij
,! a) Wij is a monotonic function of Zij; and b) Positive exports from country
j to i are observed () Zij > 1.
1

It is assumed that the …x trade costs, fij; are stochastic:
where:
fij exp EX;j + IM;i + k ij vij
– EX;j : Country j …xed export costs common across all destinations.
– IM;i : Country i …xed trade barrier imposed on all exporters.
– ij : Any additional country-pair speci…c …xed trade costs.
– vij i:i:d N 0; 2 u : unmeasured (unobserved) trade frictions, possible
correlated with the uij.

Thus, taking into account that (" 1) ln ij ln dij uij, it follows
that (10) can be rewritten in the log-linear form:
where:
zij = 0 + j + i dij ij + ij (11)
– Constant term: 0 = [ln (1 ) + (" 1) ln ( =a L)]
– Exporter …xed e¤ect: j = " ln cj EX;j.
– Importer …xed e¤ect: i = (" 1) pi + yi IM;i
– ij uij + vij i:i:d N 0; 2 with 2 = 2 u + 2 v

Let de…ne the indicator variable
Tij =
(
1 ; country j exports to country i
0 ;
and let ij be the the conditional probability that country j exports to i.
Then a Probit equation can be speci…ed as follows. y
ij = Pr Tij = 1j observed variables
= 0 + j + i d ij ij (12)
where ( ) is de cdf of the unit-normal distribution.
y In order to get a well speci…ed probit without, the equation (11) was divided by the standard
deviation of the error term.

In this context, a consistent estimate for Wij can be obtained as
where
– b Z ij = exp bz ij = exp h
Z ij
exp zij=
cWij = max b Zij 1; 0 (13)
1 bij
i
is the predicted value of
– b ij is the predicted probability of exports from country j to i.
– (k " + 1) = (" 1)

4.2 Consistent Estimation of the Log-Linear Equation
Consistent estimation of (9) requires consistent estimations of:
E h
wijjTij = 1; : i
and E h
uijjTij = 1; : i
The …rst controls for the endogenous number of exporters, and the second
for the selection of country pairs into trading partners.
Notice that (11) implies that both terms depend on
h
= E
In particular, E h
uijjTij = 1; : i
ij
ijjTij = 1; : i
= corr uij; ij ( u= ) ij

Since ij is unit-normal distributed, it follows that ij
estimated from the inverse of the Mills ratio.
b ij = bz ij = bz ij
=) bz ij bz ij + b ij !p E h
z ij jTij = 1; : i
=) bw ij ln n
exp h
bz ij + b ij
1 io
can be consistently
!p E h
wijjTij = 1; : i
Thus, consistent estimation of (9) can be obtained from the transformation
mij = 0 + j + i dij +ln n
exp h
where E h
eijjTij = 1; : i
z Standard Heckman (1979) correction for sample selection.
bz ij + b ij
1 io
+ u b ij +eij
(14)
= 0 and u corr uij; ij ( u= ). z

5 TRADITIONAL ESTIMATES
Table I provides representative traditional estimates of the gravity equation
(i.e., without biases’correction) and the estimates for the …rst stage Probit
equation (12).
– Unidirectional trade value for 158 countries.
– Both equations were estimated for the year 1886 and for an extended
sample period that cover all of the 1980s, adding year …xed e¤ects.
– Additional estimates adding bilateral controls for WTO/GATT membership
are also provided (sample period: 1980s).

Basic results:
– In general, the same variables that have a signi…cant e¤ect on export
volumes from j to i also have a signi…cant e¤ect on the probability
that j exports to i.
,! Strong evidence that traditional gravity equation estimates are biased.
– Common religion strongly a¤ects the formation of trading relationships
(the probability of trading) but has no e¤ect on the volume of trade.
– WTO membership has a strong and signi…cant e¤ect on the formation
of bilateral trading relationships (Subramanian and Wei, 2007).
– The results are not speci…c to 1986. y
An exception is land border that may be due to territorial border con‡icts.
y The 1986 results can be compared with other studies. (i.e., Eaton et al, 2004).

6 TWO-STAGE ESTIMATION
As was described in the previous sections, the estimation of the gravity
equation (14) requires the estimation of a …rst-stage selection equation
(such as the Probit equation 12).
6.1 Identi…cation
Notice that if no additional identi…cation restrictions are imposed (i.e.,
valid excluded variables), then the identi…cation of the parameters in (14)
relays on the normality assumption for the unobserved trade costs.

The economic model suggest that any trade barrier that a¤ect …xed trade
costs but do not a¤ect variable trade costs is a valid excluded variable.
– Country-level data on …rm’s entry regulation costs (Djankov et al, 02).
(i) Regulation cost: f= 1g if the relative cost (% GDP per capita) of
forming a business > median in country i and country j.
(ii) Regulation cost (days and procedures): f= 1g if the sum of the
# of days and procedures to form a business is > median in country
i and country j.
=) There is not available regulation cost data for 42 countries, Moreover,
8 countries export to everyone and Japan imports from everyone in the
remaining sample =) the available observations are reduced to a half.

6.2 Parameterization assumptions.
In principle, the parameterization assumptions (i.e., Pareto cdf for G ( )
and the uij+vij i:i:d N 0; 2 ) may play an important role in the results.
1. Fully parametrized estimation: Probit equation (12) in the …rst step
and NLS in the second step to estimate (14).
2. Semiparametric method: The Pareto assumption for G ( ) was dropped
and reverted and to the general speci…cation for V ij in (5):
– First Stage: Form (4) & (10): ij (zij); can be any increasing
function of zij. Thus, ( bz ij) was approximated with a polynomial
in bz ij, and E h
VijjTij = 1; : i
was directly controlled using ( bz ij).
– Second Stage: using OLS.

3. Nonparametric method: The joint normality assumption for the unobserved
trade costs was relaxed.
– First Stage: Probit equation in (12), but using directly the b ij,
instead of using the normality assumption to recover the bz ij and
b ij. y
The obtained b ij were partioned into a number of bins (50 and
100) with equal observations, and an indicator variable to every bin
was assigned. This procedure allows to approximate an arbitrary
functional form of the b ij in a ‡exible way.
– Second Stage: using NLS.
Mills ratio for the selection correction is not longer used.
y Logit and t-distributions were also considerd and the results were strikingly similar.

Basic Results:
– In general, the benchmark gravity equation leads to upward biased
estimates of the trade frictions in the volume of trade. It confound the
true e¤ect of these frictions with their indirect e¤ect on the proportion
of exporting …rms.
– The Pareto distribution assumption for G ( ) does not seem to a¤ect
the general results.
– The joint normality assumption for the unobserved trade costs does
not seem to a¤ect the general results.

7 AN ALTERNATIVE EXCLUDED VARIABLE
Notice that the results in the table II also suggest that common religion
strongly a¤ects the formation of trading relationships (the probability of
trading) but has no e¤ect on the volume of trade.
In this section, this common religion variable is used as excluded variable
in the estimation of (14).
– First, the three estimation procedures applied to the reduced sample
(i.e., previous section sample) are implemented.
– Then, the estimation is extended to the full sample of countries.
Table III provides the estimation results.

Basic Results:
– Religion is a legitimate excluded variable, and can be used to estimate
the model on the full sample of countries.
– Estimated trade frictions in the benchmark gravity equation are con-
…rmed to be biased.
– By increasing the country coverage it is possible to study how the biases
vary with country pairs’characteristics.

8 ADDITIONAL INSIGHTS
8.1 Decomposing the Biases
To examine the relative importance of the standard gravity equations
biases, two speci…cations of the second-stage equation (14) were estimated
(see Table IV).
1. Correcting the unobserved heterogeneity bias (i.e., adding bz ij = 1 bij
as regressor to the benchmark gravity equation).
2. Correcting the selection bias (i.e., adding b ij as regressor to the benchmark
gravity equation).

,! unobserved heterogeneity bias dominates.

8.2 Evidence on Asymmetric Trade Relationships
In this subsection it is studied whether the predicted asymmetries (…rst
stage estimation) can explain the direction of trade ‡ows and net bilateral
trade balances (see Table V).
1. OLS regression of Tij Tji on b ij b ji.
2. OLS regression of mij mji on the di¤erences in the proportion of
exporting …rms:
– bw ij
bw ji, in the NLS speci…cation.
– b( bz ij) b( bz ji), in the polynomial approximation.

,!The predicted asymmetries have explanatory power for the direction of trade
‡ows and net bilateral trade balances.

8.3 Counterfactuals
Counterfactual experiment: The distance trade cost decreases in 10%
d 0 ij
dij = log 0:9
– The study is focused on the elasticity of the overall trade response for
each country pair:
cm 0 ij mij = d 0 ij dij
(i) These elasticities vary along country pair income. =) Summary
statistics across three groups of country pairs are reported (see Table
VI).
(ii) The elasticity response at the intensive margin is …xed at 0.799 (NLS)
and 0.862 (polynomial approximation) =) the heterogeneity in the
predicted elasticity response is due to the extensive margin.

,! The heterogeneous trade responses show that the variation of these elasticities
is large.
Figure III depicts the distribution of these elasticities across country pairs
group.

,! When trade costs related to distance fall, the response of the extensive
margin of trade is predicted to be larger for less developed countries.