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\author{Simon J. Evenett\thanks{$\,\,\,$Department of Economics, 75 Hamilton Street,
New Brunswick NJ 08901-1248; email: evenett@fas-econ.rutgers.edu.} \\
%EndAName
Rutgers University and The Brookings Institution \and Wolfgang Keller\thanks{%
$\,\,\,\,$Department of Economics, 1180 Observatory Drive, Madison, WI
53706; email: wkeller@ssc.wisc.edu, web: www.ssc.wisc.edu/\symbol{126}%
wkeller.} \\
%EndAName
University of Wisconsin and NBER}
\title{On Theories Explaining the Success of the Gravity Equation\thanks{
\thinspace \thinspace First version: November 1996.\thinspace We have
benefited from comments at seminar presentations at Berlin (Humboldt),
Columbia, Geneva (Graduate Institute), Michigan, Penn, Toronto, and
Wisconsin, as well as from participants at the 1997 NBER Summer Institute,
the August 1997 EEA meeting at Toulouse, the September 1997 ERWIT/CEPR
conference in Helsinki, and the Empirical Trade Conference 1996 at Purdue
University. Special thanks go to Don Davis, Mahmoud El-Gamal, Gordon Hanson,
and Elhanan Helpman. All remaining errors are our own.}}
\date{May 1998}
\maketitle

\begin{abstract}
\noindent We analyze two main theories of international trade, the
Heckscher-Ohlin theory and the Increasing Returns trade theory, by examining
whether they can account for the empirical success of the so-called Gravity
Equation. Since versions of both models can generate this prediction, we
tackle the model identification problem by conditioning bilateral trade
relations on factor endownment differences and the share of intra-industry
trade, because only for large factor endowment differences does the
Heckscher-Ohlin model generate specialization of production and the Gravity
Equation, and it predicts inter-, not intra-industry trade. There are three
major findings: First, little production is perfectly specialized due to
factor endowment differences, making the perfect specialization version of
the Heckscher-Ohlin model an unlikely candidate to explain the empirical
success of the Gravity Equation. Second, increasing returns are important
causes for perfect product specialization and the Gravity Equation,
especially among industrialized countries. Third, to the extent that
production is not perfectly specialized across countries, we find support
for both Heckscher-Ohlin and Increasing Returns models. Based on these
findings, we argue that both models explain different components of the
international variation of production patterns and trade volumes, with
important implications for productivity growth, labor, and macro economics.

\textbf{Keywords}: Gravity equation, Heckscher-Ohlin trade theory,
increasing returns trade theory, model identification, resampling,
non-parametric analysis.

\textbf{JEL Classification}: F11, F12, C14
\end{abstract}

\newpage\setcounter{page}{1}

\section{Introduction}

The so-called Gravity Equation of trade predicts that the volume of
bilateral trade is positively related to the product of the countries' gross
domestic products (GDPs) and negatively related to trade barriers between
trade partners. Empirical research has found that various versions of the
Gravity Equation well describe the variation in the volume of trade across
country-pairs as well as over time (see Leamer and Levinsohn 1995).

Since Anderson (1979) it has been increasingly recognized that the gravity
equation prediction can be derived from very different structural models,
including Ricardian models, Heckscher-Ohlin (H-O) models, and increasing
returns to scale (IRS) models.\footnote{%
Bergstrand (1990), Deardorff (1998), Eaton and Kortum (1997), Helpman and
Krugman (1985), Hummels and Levinsohn (1995), Leamer (1992), and Markusen
and Wigle (1990).} When consumers have both identical homothetic preferences
and access to the same goods prices, a sufficient condition for obtaining a
gravity equation is perfect product specialization, in the sense that each
commodity is produced in only one country. The three types of trade models
differ in the way product specialization is obtained in equilibrium:
Technology differences across countries in the Ricardian model, factor
proportions outside the vector space-of-diversification in the H-O model,
and increasing returns at the firm level in the IRS model. Indeed, Deardorff
(1998) has recently argued that the gravity prediction per se cannot be used
to test any of these trade theories. Yet, the gravity prediction
constitutes, perhaps along with the H-O-Vanek factor service trade
prediction (see recent work by Trefler 1995, Davis et al. 1997), the most
important result regarding the volume of international trade. Therefore,
major insights are to be gained if it could be determined which model
generates gravity-like trade volumes in a given sample of data, a problem we
refer to as a model identification issue. In addition, it would allow us to
assess the suggestion by Hummels and Levinsohn (1995, 828) that a theory
other than IRS is responsible for the empirical success of the gravity
equation.

In this paper, first we address this identification problem by noting that,
on the one hand, in a constant returns (CRS) H-O world, bilateral factor
proportions differences must be very large in order to ensure the economies
lie outside a common space of diversification and to generate product
specialization (see Figure 1). On the other hand, when product
specialization is the result of IRS, the gravity prediction can be obtained
even when there are no factor proportions differences. This difference has
the following implication for the type of trade in the two models: In the
H-O model, trade is exclusively inter-industry trade, defined as trade in
goods with different factor intensities. However for the IRS model at least
some, and potentially all, trade is intra-industry trade. Consequently, in
the analysis below, we will take samples with low shares of intra-industry
trade in total trade together with large factor proportions differences as
those where a model of H-O-based product specialization might be behind the
Gravity Equation; and samples with high shares of bilateral intra-industry
trade (irrespective of factor proportions differences) as those where
IRS-based product specialization might drive the Gravity Equation.

This might suggest a country-by-country reconciliation of the perfect
specialization models of the H-O and IRS-based trade theories. Along these
lines, the perfect specialization H-O theory would be responsible for the
gravity prediction's success in explaining the variation in bilateral trade
flows among countries with large factor proportions differences and low
shares of intra-industry trade (the so-called `North-South' trade). At the
same time, the IRS theory would account for the success of the gravity
prediction in trade between countries where product differentiation and
intra-industry trade is prevalent (the so-called `North-North' trade.)
However, no such reconcilation emerges from our analysis of a large and
heterogeneous set of bilateral trade relations in the year 1985. We present
evidence that perfect product specialization due to factor proportions
differences is not a major part of explaining the success of the gravity
prediction. In contrast, there is robust evidence that IRS-based theories of
trade are an important reason why the gravity equation fits trade flows
among industrialized countries well.

Secondly, the paper assesses the empirical relevance of the IRS-based trade
models in general, with their far-reaching implications for productivity
growth, labor, and open macroeconomics. This is more important now than ever
before, as it has been demonstrated that there is no need to resort to IRS
models to explain intra-industry trade (Davis 1995), or large trade volumes
between countries with similar factor endowments (Chipman 1992, Davis 1997).
Even at an empirical level, Helpman's (1987) work showing that key
implications of the IRS-based model are consistent with OECD countries'
trade data was followed up by Hummels and Levinsohn (1995), who repeated
Helpman's analysis with a set of non-OECD countries, the trade between which
was not expected to contain much IRS-based trade. They showed that many
correlations found by Helpman continue to hold in their non-OECD sample. To
take Hummels and Levinsohn's result as evidence against the IRS-based trade
models would be wrong, however, because their finding is just an expression
of the model identification problem. In this paper, we account for both
Helpman's (1987) and Hummels and Levinsohn's (1995) findings by showing that
the former obtained his results because IRS-based trade is prevalent among
OECD countries, whereas the latter found a similar correlation among
non-OECD countries because there is some (but not perfect) product
specialization driven by factor proportions differences.

We present strong evidence that the volume of international trade is
determined by the extent of product specialization, which in turn is due to
IRS and factor proportions differences. Where there is little or no two-way
trade between nations, we find that a H-O model that predicts imperfect
specialization better explains trade flows than a H-O model with perfect
specialization. Also where there is two-way trade among nations,
international trade flows are better explained by a model that incorporates
both factor proportions differences and IRS than by a model where IRS alone
generates product specialization. These findings highlight the significance
of both factor proportions differences and IRS\ as determinants of the
extent of specialization and international trade flows. It parallels recent
results by Antweiler and Trefler (1997), who nest IRS and CRS models in the
H-O-Vanek factor service trade expression to estimate scale parameters from
these input demand equations. These authors find that although a majority of
industries seems to be well-characterized by the assumption of CRS, there is
evidence for IRS in a number of sectors.\footnote{%
Also see Davis and Weinstein (1996, 1997) who test for the presence of IRS
by exploiting the `home-market' effect prediction (strong idiosyncratic
demand for a certain good means a country will be exporting the good in the
presence of IRS\ and transport costs); their results so far are mixed.}

There are several important caveats to our approach: First of all, in our
analysis of the gravity equation we focus on the proportionality of the
volume of trade to the countries' GDPs rather than its relationship to trade
resistance. This corresponds to the fact that to date there is relatively
little theory on what form the gravity equation takes in the presence of
transport costs.\footnote{%
It includes, however, Anderson (1979), Bergstrand (1990), and Deardorff
(1998).} Although we take up the influence of trade resistance in section
4.4 below and conclude that our major findings are not sensitive to that
omission, we plan to address the issue more comprehensively in future work.

Our approach relies on identifying intra-industry trade with IRS-based
trade. Therefore, the second caveat is that instead of IRS\ we cannot rule
out that Ricardian technology differences are what is really behind
intra-industry trade (see Figure 1).\footnote{%
Also for that reason, we cannot rule out that trade within an industry is
driven by specialization across countries due to external effects at the
sub-industry level, even though there are CRS at the product level.} To make
progress on this issue requires product-level estimates of production
technologies across countries, which are unavailable. Thirdly, we note that
also in the H-O model it is possible to generate trade in goods of identical
factor intensity if one follows Armington (1969) and assumes that products
are differentiated by country of production. A direct test of the Armington
assumption would require data on people's perceptions of the differences (if
any) between domestic and foreign-produced goods. We are not aware of data
allowing us to test this assumption. In addition, by relying on the
assumption of identical homothetic preferences across countries throughout,
we do not give any role to demand as, for instance, emphasized by Markusen
(1986). Lastly, we ignore trade imbalances in the analysis below. Those
effects play, as in Helpman (1987) and Hummels and Levinsohn (1995), a
quantitatively very minor role in what follows.

The remainder is as follows. In the following section, we derive the gravity
equation predictions of four different trade models, and describe in more
detail how we attempt to identify a particular model in our empirical
analysis. Section 3 discusses the data set employed in this study, whereas
section 4 presents the key empirical results. Section 5 concludes. The
sensitivity of our results is discussed in the appendix.

\section{Derivation of the Estimating Equations}

\subsection{The Gravity Equation with Perfect Product Specialization}

Throughout the paper, we assume balanced trade, zero trade and transport
costs, and that consumers in all countries have identical homothetic
preferences. First, we consider a typical model as laid out in Helpman and
Krugman (1985, Ch. 8.1), where there are two countries, $c=i,$ $j,$ and two
goods, $g=X,Z$ (see Appendix A for more details). Both $X$ and $Z$ come in
many symmetric differentiated varieties which are identically produced with
increasing returns to scale. With preferences valuing product variety, both
countries will demand all foreign varieties according to the countries' GDP
as a share of world GDP, denoted $s^c$. Given that IRS leads to perfect
product specialization for every variety, country $i$'s imports from $j,$
denoted $M^{ij},$ will be 
\begin{equation}
M^{ij}=s^i\left[ p_xn_x^jx^j+n_z^jz^j\right] .  \label{mij}
\end{equation}
Here, $n_g^c$ is the number of varieties of good $g$ in country $c,$ and $p_x
$ is the relative price of good $x$. Country $j$'s imports from $i$, $M^{ji},
$ will be 
\begin{equation}
M^{ji}=s^j\left[ p_xn_x^ix^i+n_z^iz^i\right] .  \label{mji}
\end{equation}
The terms in the hard brackets of (\ref{mij}) and (\ref{mji}) are equal to
the GDP of country $j$ (denoted $Y_j$) and country $i$ (denoted $Y_i$)
respectively. Substitution of $Y_i$ and $Y_j$ yields 
\begin{equation}
M^{ij}=s^iY^j=\frac{Y^iY^j}{Y^w}=s^jY^i=M^{ji}.  \label{ge}
\end{equation}
Equation (\ref{ge}) is the gravity prediction referred to above,\footnote{%
This gravity prediction, as well as those below, is readily generalized for
the case that tradable goods make up a share $\lambda $, $0\leq \lambda \leq
1$ of GDP which is constant across all countries. In that case, one obtains $%
M^{ij}=M^{ji}=\lambda \frac{Y^iY^j}{Y^w}$.} and hence, this IRS\ trade model
is potentially a candidate to explain the success of the gravity equation.

The equation (\ref{ge}) is very general, as its derivation does not require
assumptions on factor price equalization, factor endowment differences
across countries, or factor intensities in the production of goods $X$ and $%
Z $ (Helpman and Krugman 1985). Equation (\ref{ge}) holds whenever there is
perfect product specialization in equilibrium, all consumers face the same
goods prices and have identical homothetic preferences, and trade is
balanced. Therefore, if instead one assumes that goods $X$ and $Z$ are
homogeneous and produced with CRS, one can still derive (\ref{ge}) when
perfect product specialization is obtained in equilibrium. For the following
exposition, we assume without loss of generality that country $i$ is
capital-abundant and good $X$ capital-intensive. Consider a model where
factor endowments between the countries differ by so much that each country
specializes in the production of one good. Let $k^x$ and $k^z$ denote the
equilibrium capital-labor ratios which are employed in industry $X$ and $Z,$
respectively, in the integrated equilibrium (see Helpman and Krugman 1985,
Ch.1). Then, if factor proportions in countries $i$ and $j$ lie outside that
range, 
\begin{equation}
\frac{K^i}{L^i}\geq k^x>k^z\geq \frac{K^j}{L^j},  \label{fdifl}
\end{equation}
at least one country will specialize in the production of only one good (if
the capital-abundant country $i$ specializes, it produces the
capital-intensive good $X,$ and if the labor-abundant country $j$
specializes, it does so producing the labor-intensive good $Z,$ by
comparative advantage). For the generic case--obtained with sufficiently
similar country size and no strong taste bias--, both countries specialize
in producing only their comparative-advantage good. In that case, if $X^c$
and $Z^c$ denote the production of good $X$ and $Z,$ respectively, we have $%
X^i=X^w$ and $Z^j=Z^w$; further, it is clear that the value of good $X-$%
production in country $i$ is equal to its GDP, i.e., $p_xX^i=Y^i,$ and
similarly, $Z^j=Y^j.$ This means that 
\begin{equation}
\begin{array}{ccc}
M^{ij}=s^iZ^j=s^iY^j=\frac{Y^iY^j}{Y^w}, &  & M^{ji}=s^jp_xX^i=s^jY^i=\frac{%
Y^iY^j}{Y^w},
\end{array}
\label{hosp}
\end{equation}
which is identical to the gravity equation (\ref{ge}) above.

As discussed by Helpman and Krugman (1985), this analysis generalizes to the
multi-sector, multi-factor, and multi-country settings as long as
specialization is the equilibrium outcome in all sectors of all countries.
In the IRS-based model from equation (\ref{mij}), if there more countries
and/or more goods, country $i$ will import the same share $s^i$ of total
production of each, in the same way country $i$ imports that share of
country-$j$'s varieties $X$ and $Z$. Product specialization for all goods
due to factor proportions differences will not necessarily occur when there
are more than two countries (or goods), but only two factors of production.
With more than two factors, factor-specificity can generate product
specialization. In an extreme case where the production of any good in any
country requires inputs that are specific to that good in that country (such
as technological know-how), equation (\ref{ge}) will reappear. Therefore,
for an appropriately defined metric of factor endowment differences, also
the H-O rationale for the gravity equation generalizes to a multi-sector,
multi-country, and multi-factor setting.

So far our discussion has left open the question of whether one can derive
gravity-type import volume predictions for the case where at least some
goods are produced in more than one country. While product specialization
has recently emphasized to be an important phenomenon of the patterns of
world production (e.g., Haveman and Hummels 1997), no doubt there are many
commodities which are produced in several countries. We now turn to two of
those trade models with imperfect product specialization for the $2\times
2\times 2$ case: the simplest, textbook Heckscher-Ohlin model, as well as
its generalization to include a IRS sector due to Helpman (1981).

\subsection{Gravity Equations with Imperfect Product Specialization}

First, we derive a gravity equation for the generalization of the H-O model
where one sector ($Z$) produces a homogeneous good under CRS, whereas a
second sector ($X)$ produces a differentiated good under IRS (Helpman 1981;
see also Helpman and Krugman 1985, Ch.7,8). There are two countries ($i$ and 
$j$) and two factors ($K$ and $L$). For endowments inside the factor price
equalization set, the volume of bilateral trade, defined as the sum of a
country's exports and imports, is given by 
\begin{equation}
VT^{ij}=s^jp_xX^i+s^ip_xX^j+\left( Z^j-s^jZ^w\right) ,  \label{hekr}
\end{equation}
where the first term on the right hand side gives country $i$'s exports ($%
M^{ji}$), and the remaining two terms are country $i$'s imports ($M^{ij}$).%
\footnote{%
With zero trade costs, $Z$ being homogeneous and produced under CRS, and
factor price equalization, consumers are indifferent with regard to $Z$'s
country of origin and producers are indifferent with respect to who buys it.
Therefore, underlying equation (\ref{hekr}) is a ``minimum trade-volume''
assumption, that only a country's excess demand for a good is imported from
abroad. It is well-known that the H-O model determines only net, but not
gross, trade flows.}

Let $\gamma ^c$ be the share of good $Z$ in country $c$'s GDP, $\gamma ^c=%
\frac{Z^c}{p_xX^c+Z^c}$. It is easy to show that the assumption that country 
$i$ is capital-abundant implies that $0\leq \gamma ^i<\gamma ^j.$ The
following reproduces a result from Keller (1998).

\begin{proposition}
If good $X$ is capital-intensive and produced under IRS, good $Z$
labor-intensive and produced under CRS, and country $i$ is relatively
capital-abundant, then country $i$'s imports from country $j$ are given by
the following gravity equation: 
\begin{equation}
M^{ij}=\left( 1-\gamma ^i\right) \frac{Y^iY^j}{Y^w}.\,  \label{hyb}
\end{equation}
\end{proposition}

\TeXButton{Proof}{\proof}With balanced trade, $M^{ij}=M^{ji}$; $%
M^{ji}=s^jp_xX^i,$ and using the definition of $\gamma ^i$ gives $%
M^{ji}=s^j\left( 1-\gamma ^i\right) Y^i=\left( 1-\gamma ^i\right) \frac{%
Y^iY^j}{Y^w}.$ \TeXButton{End Proof}{\endproof}\smallskip\ 

This proposition states that for any value $\gamma ^{i}>0,$ the level of
bilateral imports is lower than in the case where both goods are
differentiated (compare \ref{hyb} to the gravity equation \ref{ge}).
Furthermore, as the share of homogeneous good production in GDP declines,
the predicted level of imports rises, and in the limit, as $\gamma
^{i}\rightarrow 0,$ the generalized gravity equation (\ref{hyb}) reverts
back to the simple gravity equation (\ref{ge}) above. In a sense, therefore,
the volume of trade is higher, the lower is the share of homogeneous goods
in GDP.\footnote{%
Note that this finding is in part due to H-O reasons, because $\gamma ^i$ is
inversely related to country $i$'s capital-labor ratio. A decline in $\gamma
^i$ therefore implies an increase in the volume of imports due to an
increase in the countries' factor proportions differences (triggering a
production response a la Rybczynski). What we emphasize is that for given
factor proportions differences, the more product specialization there is the
higher is the level of imports; see below.} It is also interesting to note
that except for country $j$'s GDP, only the characteristics of the
capital-abundant country $i$ enter the bilateral imports prediction (\ref
{hyb}).

Second, we show the particular form of the gravity equation in the case of a
simple $2\times 2\times 2$ H-O model; that is, now both good $Z$ and good $X$
are homogeneous and produced under CRS. The volume of trade between
countries $i$ and $j$ is given by 
\begin{equation}
VT^{ij}=p_x\left( X^i-s^iX^w\right) +\left( Z^j-s^jZ^w\right) ,  \label{hovt}
\end{equation}
where, according to the H-O theorem, the capital-abundant country $i$
exports the capital-intensive good $X$ and imports the labor-intensive good $%
Z.$\footnote{%
The ``minimal trade-volume'' rule has again been used to derive (\ref{hovt}).%
} For this model, the following proposition, reproduced from Keller (1998),
states the corresponding gravity-type relationship.

\begin{proposition}
If both goods are homogeneous and produced under CRS, with country $i$ being
relatively capital-abundant and good $X$ being relatively capital-intensive,
then country$\,i$'s imports from country $j$ are given by the following
gravity equation: 
\begin{equation}
M^{ij}=\left( \gamma ^j-\gamma ^i\right) \frac{Y^iY^j}{Y^w}.  \label{ho}
\end{equation}
\end{proposition}

\TeXButton{Proof}{\proof}See Appendix B.

The gravity equation in the imperfect specialization H-O case, equation (\ref
{ho}), depends not only on the product of the GDP's in the familiar way, but
also on $\gamma ^j$ and $\gamma ^i,$ which are characteristics of both
countries. Note that as the capital-labor ratios in the two countries
converge, so do $\gamma ^j$ and $\gamma ^i.$ In the limit, when the factor
proportions in $i$ and $j$ are equal, we have that $\gamma ^j=\gamma ^i,$ in
which case equation (\ref{ho}) gives the familiar result that there is no
trade in a H-O model when factor proportions are identical across countries.
Equation (\ref{ho}) includes the volume of imports prediction of the
multi-cone H-O model given in (\ref{hosp}) as a special case, because as
factor proportions differences between $i$ and $j$ increase, the share of
GDP derived from good $Z$ in country $j,$ $\gamma ^j,$ approaches one,
whereas the share of good $Z$ in the GDP of country $i,$ $\gamma ^i,$ tends
to zero. Indeed, when $\gamma ^j=1$ and $\gamma ^i=0,$ equation (\ref{ho})
reverts to $M^{ij}=\frac{Y^iY^j}{Y^w},$the gravity equation for the perfect
specialization model (see \ref{hosp}).\footnote{%
This assumes the generic case where \textit{both} countries specialize in
production; see above.} Also, we can rewrite equation (\ref{ho}) in the
following way: 
\[
M^{ij}=\left( \gamma ^j-\gamma ^i\right) \frac{Y^iY^j}{Y^w}=\left[ \left(
1-\gamma ^i\right) -\left( 1-\gamma ^j\right) \right] \frac{Y^iY^j}{Y^w}.
\]
Therefore, if we denote the volume of imports prediction for the case where
the production of both goods is specialized across countries, equations (\ref
{ge}) and (\ref{hosp}), by $M_S$; the generalized H-O case with
specialization for one good, but not the other good (equation \ref{hyb})
with $M_G$; and the H-O case in which both countries produce all goods,
equation (\ref{ho}), with $M_H,$ the following inequalities hold, ceteris
paribus: 
\begin{equation}
M_S>M_G>M_H.  \label{rank}
\end{equation}
This confirms that the bilateral volume of imports is higher, the more
product specialization there is.

\subsection{Model Identification}

In the preceding section, we have derived the specific form of the gravity
equation of trade for four models: (1) Multi-cone H-O: factor endowment
differences lead to perfect product specialization; (2) Pure IRS: IRS leads
to perfect product specialization; (3) Generalized H-O: with one good
produced using IRS technology and perfectly specialized, the other good
being produced with CRS technology and not specialized, and (4) Uni-cone
H-O: where both goods are produced using CRS technology and are not
specialized. If we restrict ourselves to $2\times 2\times 2$ models, then $%
M_S$ is the import prediction for models (1) and (2), $M_G$ is the
prediction for model (3), and $M_H$ is the prediction for model (4).

It is unlikely, however, that any of the observed trade flows are solely
determined by any one of these four archetypal models. First of all, the
data comes from a world with more sectors, countries, and factors than our $%
2\times 2\times 2$ models. Secondly, there may be positive amounts of
IRS-based trade even between countries with the lowest recorded shares of
IRS-based trade. Similarly, we expect there to be some
factor-proportions-based trade even among countries where the share of
IRS-based trade is highest.\footnote{%
Note that this point does not rule out the generalized H-O model (3).}
Observed bilateral trade among countries is likely to be the result of the
combination of the determinants of trade flows formalized in the four models
considered here, and perhaps by other determinants which we have not
addressed. However, in different circumstances (such as different
distributions of factor endowments across trading partners) we expect
different trade models to account for different proportions of the observed
trade flows. Our inferences are based on whether each of the four trade
models actually performs better in the very sample(s) (which are identified
using explicitly stated criteria) where one expects the theory to perform
better.

Consider a cross-section of country-pairs where there is little (or no)
product specialization due to IRS, but the absolute difference between the
two countries' factor proportions, denoted $FDIF$, differs from one pair to
another.\footnote{%
We treat from now on a country-pair observation drawn from the actual
multilateral trading world as resembling the two-country world observations
of models (1)-(4). There is an empirical and a theoretical concern, only the
first of which applies to the models with perfect product specialization.
First, trade between nations is often not balanced. Trade imbalance effects,
however, are too small to affect our qualitative results. Second, if a
homogeneous good is produced in two or more countries, then bilateral trade
volumes may become indeterminate (this led to the ''minimal-trade volume
rule'' above). We cannot be sure that, in a multi-country model, bilateral
trade volumes are determined by the same rule. For a given rule, though, we
conjecture that the two-country predictions are closely related to the
bilateral predictions from a multilateral model, and plan on working to show
this rigorously in the future.} From equation (\ref{fdifl}), we expect 
\textit{ceteris paribus} more product specialization in a pair where $FDIF$
is larger relative to another country-pair where factor proportions
differences are smaller. Consequently, we expect the gravity prediction (\ref
{ge}), i.e., prediction $M_S,$ to be more accurate for the pair where $FDIF$
is larger, compared to the other with the lower value of $FDIF$. This
observation allows to identify the H-O motivation for product specialization
and the gravity prediction: If factor proportions differences are important
in explaining the success of the gravity equation, then the prediction $M_S$
should fit better in samples where the observed factor proportions
differences $FDIF$ are higher.\footnote{%
This result holds \textit{ceteris paribus}: As discussed above, perfect
specialization in homogeneous good production can be obtained not only
through large factor proportions differences. Our empirical strategy
accounts for that, see below.}

Moreover, we can test the multi-cone H-O model against the uni-cone H-O
model for different values of factor proportions differences: If product
specialization is principally the result of factor proportions differences,
then the uni-cone H-O model should be preferred to the multi-cone H-O model
in samples where the observed factor proportions differences $FDIF$ are
lower.

We will employ the index proposed by Grubel and Lloyd (1975) to control for
the extent of IRS-based trade. Trade due to IRS and product differentiation
can result in a country simultaneously importing and exporting varieties of
a particular product (intra-industry trade).\footnote{%
We define intra-industry trade as trade in goods with identical factor input
requirements; for our empirical analysis, though, intra-industry trade is
taken as two-way trade of goods in the same four-digit SITC class. The two
concepts need not be the same in a deterministic sense, but our analysis
remains valid as long as they are the same on average; see below.} The
index, denoted $GL^{ij},$ measures the share of intra-industry trade in the
total trade 
\begin{equation}
GL^{ij}=1-\left( \frac{\sum_g\left| M_g^{ij}-M_g^{ji}\right| }{\sum_g\left(
M_g^{ij}+M_g^{ji}\right) }\right) ,\text{\thinspace \thinspace \thinspace }%
0\leq GL^{ij}\leq 1.  \label{gl}
\end{equation}
In the extreme case where a given good $g$ is either exported or imported
(no intra-industry trade), the Grubel-Lloyd index will be equal to zero.
With positive amounts of intra-industry trade, it will be between zero and
one;\footnote{%
In the pure IRS model (1), the maximum value of $GL^{ij}=1$ is obtained if
the share of GDP derived from producing $X$-varieties is the same in both
countries; in the $2\times 2\times 2$ model, this will happen if factor
proportions are identical in the two countries.} it is generally higher when
the share of intra-industry trade in total trade is higher.

The share of intra-industry trade is not a perfect indicator for the share
of trade based on IRS. Finger (1975) has argued that intra-industry trade is
found because products which are actually different are classified to the
same industry. Clearly, a high degree of disaggregation is desirable when
studying intra-industry trade. Recently, researchers have emphasized the
importance of trade in intermediate goods (vertical differentiation) in
accounting for a high values of $GL$ (Greenaway, Hine, and Milner 1994, and
Fontagne, Freudenberg and Peridy 1997); also this can in principle be due to
IRS \textit{or} other reasons. Moreover, a significant part of trade which
is classified as inter-industry trade might be based on IRS, such as the
example of wide-bodied aircraft exports from the U.S. to most other
countries.\footnote{%
The only important producer of such aircraft (made by Boeing) outside the
U.S. is the European joint venture Airbus, so that Boeing exports to any
country other than the Airbus maker countries are classified as
inter-industry trade. This is an example given in Krugman (1994).} At the
macro level, the presumption that $GL$ falls with larger factor proportions
differences across countries has generally been confirmed in the literature.%
\footnote{%
See Helpman (1987), Bergstrand (1990), and Fontagne, Freudenberg, Peridy
(1997); the results in Hummels and Levinsohn\ (1995) are mixed. The
presumption can be established rigorously in the two country, two factor,
two industry case, see Helpman (1981).} At the micro level, the evidence on
whether $GL$ correlates positively with economies of scale at the industry
level is mixed,\footnote{%
An example of a recent study is Fontagne, Freudenberg and Peridy (1997), who
find a strong positive correlation of the share of two-way trade and a proxy
of IRS at the industry-level, but this is not true for all recent studies,
see their references, and also Krugman (1994).} which in part is due to
obtaining good measures which can identify IRS. After discussing the issue
in some detail, Krugman (1994, 23) concludes that although $GL$ does not
exactly measure the share of international trade which is due to IRS, this
true share might be higher \textit{or} lower. In this study, we use the $GL$
index to infer the importance of IRS-based trade\ in total trade using a
methodology which accounts for the fact that $GL$ is an imperfect measure of
that.

The Grubel-Lloyd index is employed to identify the samples where the pure
IRS model is likely to determine trade flows. Specifically, we expect that
the IRS model accounts for the performance of the gravity equation in those
samples where the bilateral Grubel-Lloyd indices are larger, indicating that
a larger proportion of bilateral trade is two-way trade in perfectly
specialized differentiated products. In section 4.1.2 we examine whether in
fact the prediction $M_S$ (of the IRS\ model) is more accurate in samples
with higher Grubel-Lloyd indices.

We emphasize that prediction $M_S$ is common to both the multi-cone H-O and
the pure IRS models. However, if we were to find that the prediction $M_S$
is less at odds with the data in samples with higher Grubel-Lloyd indices,
it would be incorrect to interpret this finding as evidence in favor of the
multi-cone H-O theory.\footnote{%
This is so unless there was a perfect positive correlation across country
pairs between the observed Grubel-Lloyd indices and factor endowment
differences--which there is not.} Furthermore, we can test the generalized
H-O model against the pure IRS model. We expect the latter model to be
preferred to the former model in samples where there is a greater share of
intra-industry trade, as measured by the Grubel-Lloyd index.

In the following section, we briefly discuss the data which will be used.

\section{Data}

We have developed a cross-sectional data set for fifty-eight countries for
the year of 1985. The data set includes all countries with both GDPs above 1
billion US dollars and where internationally comparable capital-per-worker
estimates (measured in US\ dollars) are available. These fifty-eight
countries account for 67\% of world imports and 79\% of world GDP in 1985.
The countries are listed in Table 1A. The data set includes nearly all
industrialized countries, but relatively few of the less developed
countries. This reflects the paucity of capital stock estimates available
from the latter.

The source for data on GDP and capital-per-worker is the current Summers and
Heston (Version 5.6) database (see Summers and Heston 1991 for a description
of this database). Both variables are in internationally comparable
(purchasing power adjusted), real US dollars for the year 1985.\footnote{%
One reason of using data for 1985 is that this is the benchmark year for
inflation and PPP adjustments in the Summers and Heston dataset.} Each
country's GDP and capital-per-worker data are reported in Table 1A. The
trade data comes from the NBER World Bilateral Trade Database, see Feenstra
et al. (1997). The overall imports of country $i$ from country $j$ ($M^{ij}$%
) are immediately available from this source. We have used the values
reported by the importing country, as these are known to be more accurate
than trade reported by the exporting country. With fifty-eight countries, we
have $58\times 57=3306$ bilateral import relations.

The bilateral Grubel-Lloyd indices $GL^{ij}$ are also derived from data in
Feenstra et al. (1997). These indices are computed using all goods at the
four-digit SITC classification; that is, we do not confine ourselves to the
manufacturing sector as is frequently done when Grubel-Lloyd indices are
calculated. The Grubel-Lloyd index (\ref{gl}) can only be computed for
country-pairs where there are positive amounts of trade. This is the case
for 87\% of all bilateral relations, so that our sample consists of $2870$
observations.\footnote{%
In addition to the $2870$ bilateral observations for which we can compute
the Grubel-Lloyd index, the NBER database reports positive levels of trade
at an aggregate level for three more pairs: Kenya/Sierra Leone,
Madagascar/Ireland, and Zambia/Guatemala. We have not been able to track
down the reason for this inconsistency, but do not expect that it will
affect the results significantly.} The number of bilateral trade relations
with partner countries varies by country from a low of 27 for Nepal to the
maximum of 57 for most industrialized countries. Furthermore, the number of
different goods classes traded with a partner country varies from country to
country. At one end, Mauritius trades on average 36 types of goods with its
partners, whereas at the other end, the U.S. trades on average 332 types of
goods with its trade partners.

Keeping these facts in mind, the average Grubel-Lloyd indices we compute are
presented in Figure 2. Across each of their respective trading partners,
Bolivia has the lowest average Grubel-Lloyd index in this sample, with a
value of 0.0006; and the U.K. has the highest value of 0.1495. Each nation's
average Grubel-Lloyd index is shown in Table 1A, along with national GDP per
capita, the number of bilateral relations the average is computed from, and
the average number of industry classes where there is positive trade. In
Table 1B we show the correlation among the variables of Table 1A. All
correlations are positive; in particular, the correlation of the average
Grubel-Lloyd index with GDP per capita is 0.78, and the correlation between
the average Grubel-Lloyd index and the average number of industry classes
traded is 0.93.

In Figure 3, we plot the cumulative distribution of bilateral Grubel-Lloyd
indices (there are 1435 distinct indices, corresponding to 1435
country-pairs in our sample). The distribution is very skewed: 44\% of all
indices are equal to 0, and 78\% of all (1120 country-pairs) have a value of
0.05 or less. In the empirical analysis below, we will treat these 1120
country-pairs as trade relations where there is little or no IRS-based
trade, whereas the remaining 315 country-pairs with Grubel-Lloyd indices
above 0.05 are taken as those where IRS-based trade is present .\footnote{%
The threshold level of the Grubel-Lloyd index of 0.05 is based on our
priors. In Appendix C we show that our main findings do not depend on this
choice of threshold level.} It is important to note that this separation of
the data set into 1120 and 315 country-pairs does not mean that IRS-based
trade must account for a small share of international trade; in fact, the
315 country-pairs with Grubel-Lloyd indices above 0.05 account for 87.1 \%
of all of the imports by the 1435 country-pairs.

We are concerned about heteroskedasticity in our sample. To that end, let
the total imports of country $i$ and total exports by country $j$ be denoted
by $\bar M^i$ and $\bar X^j,$ respectively. Then, we scale all the import
values $M^{ij}$ and the GDP term $\frac{Y^iY^j}{Y^w}$ by $\psi ^{ij}=(\bar
M^i\times \bar X^j)^{0.5}$ and assume that this creates a scalar variance in
regressions based on the Gravity Equation prediction. Finally, the gravity
prediction gives a relationship between two endogenous variables, so the
results might be affected by simultaneity bias (as discussed by Saxonhouse
1989 and Harrigan 1996). However, while in a world with positive levels of
production of all goods in all countries\footnote{%
This and other conditions need to be satisfied for factor endowments to be
linearly related to GDPs with equal coefficients across countries; see,
e.g., Harrigan (1996).} factor endowments might be valid instruments for
countries' GDPs, here we are especially concerned with perfect product
specialization across countries. Because this makes factor endowments much
less desirable instruments, we have decided to leave the gravity prediction
un-instrumented.

\section{Empirical Results}

\subsection{Perfect Specialization Models}

\subsubsection{Heckscher-Ohlin-Based Product Specialization (Low-GL Sample)}

We first consider the bilateral pairs where the computed Grubel-Lloyd ($%
GL^{ij}$) indices are 0.05 or less. As discussed above, those are the
country-pairs in which we expect only negligible amounts of IRS-based trade
to be present. We assume that the expected value of the true share of trade
in perfectly specialized goods rises with the measured bilateral factor
endowment differences$.$ Specifically, the true share of trade in perfectly
specialized goods (denoted $^{*}FDIF^{ij})$ is determined by the observed
level of factor proportion differences ($FDIF^{ij}$) plus a confounding
variable: 
\begin{equation}
FDIF^{ij}=\text{ }^{*}FDIF^{ij}+\theta ^{ij},\forall ij,  \label{theta}
\end{equation}
where we assume that $\theta ^{ij}\sim N(0,\sigma _\theta ^2),$ $\sigma
_\theta ^2>0$. The confounding variable $\theta ^{ij}$ accounts for two
effects: First, it captures measurement error in the underlying
capital-labor ratio data. Second, it picks up changes in the share of
product-specialized trade which are due to effects other than changes in
factor proportions differences; the latter could be due, e.g., to Ricardian
technology differences, bilateral distance, demand or relative country size
for that pair $ij.$ With $E\left[ \theta ^{ij}\right] =0,\forall ij,$ we
assume that \textit{on average} the effect of these influences, and all
other determinants of the share of trade in perfectly specialized goods, is
equal to zero.

Essentially, we want to examine how the estimated parameter in the gravity
regression (\ref{ge}) varies for subsamples of the data as $FDIF$ changes.
However, for the reasons given above, $FDIF^{ij}$ is only a noisy signal of
the share of perfectly specialized trade in total trade in country-pair $ij.$
Therefore, even if the relationship between the gravity equation estimate
and the true index were strong and monotonic, $FDIF^{ij}$ need not vary
monotonically with the gravity equation estimate. We are interested to see
whether there is a robust relationship between $FDIF^{ij}$ and the gravity
equation parameter. The following approach, which combines elements of
resampling (see Efron 1982) and non-parametric regression analysis (e.g.,
H\"ardle 1990), suits this need.

We first create, using (\ref{theta}), artificial distributions for $FDIF$
which are centered on the data we have collected from the Summers and Heston
dataset. Second, for a specified number $K$ of equally sized $FDIF$ classes,
we collect all the observations which fall into the $k$th class, $k=1,...,K$%
. Third, from these $K$ sets of observations, we compute $K$ sets of import
parameters corresponding to the gravity predictions of the models under
consideration.

To implement this, we rank the 2240 country-pair observations under
consideration (those with $GL^{ij}<0.05$) by the log of the factor
proportions difference we obtained from the Penn World Tables. Then, for a
given Monte-Carlo replication $r,r=1,...,R,$ we draw a $\left( 2240\times
1\right) $ vector of $\theta ^{ij}$ and add this to the factor proportions
differences column. This is our factor proportions differences vector for
that replication, $FDIF^{ij}(r).$ In the following step, we sort the data on
the column $FDIF^{ij}(r)$. The $k$th class, $k=1,..,K,$ class of
observations of this replication is denoted $M_k^{ij}(r)$ and $\frac{\left(
Y^iY^j\right) _k}{Y^w}(r)$. We repeat this for the following replications $%
r+1,r+2,...,R$, collecting all bilateral relations falling into a given
group $k$ across all $R$ replications. Finally, with these sets of data on
imports and GDP's, we run the following least squares regression 
\begin{equation}
y_k^{ij}(R)=\alpha _k\,x_k^{ij}(R)+\epsilon _k^{ij},\forall k=1,...,K,
\label{reg}
\end{equation}
where $y_k^{ij}(R)$ is the set of all $M_k^{ij}(r)$, and $x_k^{ij}(R)$ is
the set of all $\frac{\left( Y^iY^j\right) _k}{Y^w}(r)$ across all
replications which fall into percentile class $k,$ where $\epsilon _k^{ij}$
has mean zero.\footnote{%
Note that $\epsilon _k^{ij}$ is not independent, because the resampling
procedure leads by design to observations being repeated in a given class $k$%
, as in standard bootstrapping techniques. The variance on $\theta ^{ij}$
which we assume determines the degree of smoothing, related to the choice of
the bandwidth in non-parametric regression analysis. What is different here
is that we apply the smoothing to the identifying variable $FDIF,$ not to
the variables entering the regression equation.}

Recall that the model under consideration here is the multi-cone H-O model
where all trade is in (homogeneous) specialized products. The imports
prediction is given by (\ref{ge}): $M^{ij}=\frac{Y^iY^j}{Y^w}.$ Therefore,
in the multi-cone H-O model, $\alpha _k$ is only one parameter, predicted to
equal 1 when all goods are tradable.

Figure 4 shows the result for $K=5$ classes, with average values of $FDIF$
rising from class $FDIF=1$ to $FDIF=5$.\footnote{%
The graphs are based on simulations with $R=15,$ $K=5,$ $\sigma _\theta
^2=5. $ We have experimented with different choices on these parameters, but
that did not qualitatively change the results noted in the text. The log of $%
FDIF$ has a mean of 9.31 and a standard error of 1.21 in this sample of low-$%
GL$ country-pairs.} The line indicates how the estimated value of $\alpha _k$
varies as $FDIF$ changes. First, note that the estimated values are all
around $0.024$, which is much smaller than the predicted value of $\alpha =1$%
.\footnote{%
The gravity equation regressions conducted here correspond to the bilateral
trade volume expressions we derived above; we include therefore none of the
other variables frequently found in empirical gravity regressions, such as
distance or adjacency dummies. Also note that there is no constant in
equation (\ref{ge}). Empirical gravity regressions are typically done in
logs, estimating the elasticity of imports with respect to the GDP term;
from our results, this elasticity ranges between $0.44$ and $0.47$ for the
five $FDIF$ classes, which is somewhat lower than the standard results in
the range of $0.7$ to $1.0.$ One of the reasons for our lower elasticity
estimate is likely to be that, contrary to most gravity regressions, we
constrain the elasticity to be the same for both exporting as well as
importing country GDP.} Recall, however, that we are interested primarily in
how $\alpha $ varies across $FDIF$ class, not whether we can reject the
hypothesis of $\alpha =1.$ One sees that the value of $\alpha _k$ does not
vary systematically with $k,$ the index of the $FDIF$-class. This implies
that as there is more product specialization due to factor proportions
differences in the sample, the estimated parameter does not move closer to
its theoretically predicted value. This is inconsistent with product
specialization due to factor proportions differences being an important
component in explaining the success of the gravity equation.\footnote{%
The standard errors of the estimates $\alpha _k$ for the multi-cone H-O
model are as follows; FDIF=1: 0.0008, FDIF=2: 0.0008, FDIF=3: 0.0008,
FDIF=4: 0.0007, FDIF=5: 0.0007. Using these standard errors and the
estimated parameters, therefore, $\alpha _5$ is not significantly larger
than the $\alpha _1$ in a statistical sense ($\alpha _1=0.0242+1.96\times
0.0008$ gives $0.0258,$ which is larger than the point estimate of $\alpha _5
$, with $0.0245$).}

We now turn to examining the Pure IRS model.

\subsubsection{IRS-Based Product Specialization (High-GL Sample)}

Our approach here is analogous to the one used above. We assume that the
actual share of IRS-based trade in total trade (denoted $^{*}GL^{ij}$) in a
given country-pair equals the calculated Grubel-Lloyd index (denoted $GL^{ij}
$) plus a confounding variable: 
\[
GL^{ij}=\text{ }^{*}GL^{ij}+\chi ^{ij},\forall ij,
\]
where $\chi ^{ij}\sim N(0,\sigma _\chi ^2),$ with $\sigma _\chi ^2>0.$ As
noted above, the Grubel-Lloyd index imperfectly captures the share of
IRS-based trade in total trade. Therefore, $\chi $ captures not only
measurement error due to aggregation and the existence of vertically
differentiated products, but also the extent of two-way trade that is due to
comparative advantage. With $E\left[ \chi ^{ij}\right] =0,$ though, we
assume that, \textit{on average}, the calculated Grubel-Lloyd index
determines the share of IRS-based trade in total trade.

In this subsample, there are 630 country-pair observations accounting for
87\% of all imports. We classify the 630 observations into $K=5$ classes,
with the Grubel-Lloyd indices rising from class $K=1$ to class $K=5.$ The
model considered here is the pure IRS model, where both $X$ and $Z$ are
produced under economies of scale. The model implies a volume of imports
prediction of $M^{ij}=\frac{Y^iY^j}{Y^w}.$ The regression is again 
\begin{equation}
y_k^{ij}(R)=\alpha _k\,x_k^{ij}(R)+\epsilon _k^{ij},\forall k=1,...,K,
\label{glreg}
\end{equation}
with $x_k^{ij}(R)$ is equal to the set of all $\frac{\left( Y^iY^i\right) _k%
}{Y^w}(r)$ in this percentile class $k$.

Figure 5 reports the results of this resampling experiment.\footnote{%
This is based on choosing $R=70,$ $K=5,$ and $\sigma _\chi ^2=0.15;$ the
noted features are not sensitive to this choice. The average (standard
deviation) of $GL$ is equal to 0.199 (0.133) for these high$-GL$ country
pairs.} The line shows for the pure IRS model how the estimated value of $%
\alpha _k$ varies as the Grubel-Lloyd index increases. The parameter is
estimated to be between 0.03 to 0.13, which is higher than the comparable
parameter estimates for the multi-cone H-O model. Second, the estimate rises
in samples with higher values of the Grubel-Lloyd index; the higher the
share of IRS-based trade, the closer is the estimated parameter to the
theoretical value of $\alpha _k=1$. Therefore, this finding is consistent
with IRS-based trade theory being the reason why the gravity prediction
`works' empirically. Note, however, that the parameter is never anywhere
close to 1, the theoretically predicted value when all goods are tradable.%
\footnote{%
The standard errors of the estimates $\alpha _k$ for the Pure IRS model are
as follows; GL=1: 0.0006, GL=2: 0.0007, GL=3: 0.0009, GL=4: 0.0012, GL=5:
0.0022. This means that $\alpha _5$ is also significantly larger than $%
\alpha _1$ in a statistical sense.}

Summarizing, from our analysis of the H-O and IRS-perfect specialization
models, we find that there is no evidence suggesting that perfect product
specialization due to H-O reasons is important in explaining the success of
the gravity equation; in contrast, there is some evidence supporting the
Pure IRS model's trade volume prediction. However, the trade volume
prediction of both of these models holds only if all goods are perfectly
specialized. We now turn to estimating the import volume predictions of the
two above models which incorporate imperfectly specialized production, the
Generalized H-O model with import prediction $M_G,$ and the H-O model with
import prediction $M_H.$

\subsection{Imperfect-Specialization Models}

\subsubsection{Heckscher-Ohlin-Based Product Specialization (Low-GL Sample)}

We return to the sample of bilateral pairs where the computed Grubel-Lloyd ($%
GL^{ij}$) indices are 0.05 or less, and adopt an analogous approach by
estimating gravity equations for samples which differ by their average value
of $FDIF,$ indicating the extent of product specialization due to factor
proportions differences. In the uni-cone H-O model, all trade is in
imperfectly specialized products; its import prediction $M_H$ is equal to $%
M^{ij}=\left( \gamma ^j-\gamma ^i\right) \frac{Y^iY^j}{Y^w}$ if country $i$
is capital abundant and $X$ is capital-intensive, and $M^{ij}=\left( \gamma
^i-\gamma ^j\right) \frac{Y^iY^j}{Y^w}$ if country $j$ is capital-abundant.
Therefore, we estimate (up to) fifty-eight parameters $\gamma ,$ the share
of the labor-intensive good $Z$ in the countries GDP's.

The regression is analogous to (\ref{reg}), 
\[
y_k^{ij}(R)=\alpha _k\,x_k^{ij}(R)+\epsilon _k^{ij},\forall k=1,...,K, 
\]
with $x_k^{ij}(R)$ equal to the set of all $\left( \gamma ^j-\gamma
^i\right) \frac{Y^iY^j}{Y^w}(r)$ (or $\left( \gamma ^i-\gamma ^j\right) 
\frac{Y^iY^j}{Y^w}(r)$)$,$ depending on which country is capital-abundant
and which labor-abundant) in this class $k$.

Figure 6 shows the average of the estimated differences $\gamma ^j-\gamma ^i,
$ that is the difference in the share of the labor-intensive good in the
labor-abundant country from that in the capital-abundant country. This
average difference is estimated to be in order of 0.039 to 0.051.
Furthermore, the average difference increases as the average value of factor
proportions differences rises in the sample. This is what the theory (in the 
$2\times 2\times 2$ case) predicts: Larger differences in factor endowment
ratios are mapping into larger differences in the shares of GDP derived from
a given good.\footnote{%
The resampling experiment is identical to that for the other model in the
Low-GL\ sample, see footnote 22. The average standard errors of the
estimates $\alpha _k$ for the uni-cone H-O model are as follows; FDIF=1:
0.015, FDIF=2: 0.014, FDIF=3: 0.016, FDIF=4: 0.013, FDIF=5: 0.011. Based on
the parameter estimates and these standard errors, the true $\alpha _5$ is
only marginally higher than the true $\alpha _1$ at standard levels of
statistical significance. Compared to the multi-cone H-O experiment where
one parameter is estimated, here, we estimate $58$ parameters; therefore, in
order to obtain comparable standard errors, the number of replications
should now be higher--leading to lower estimated standard errors--than in
the multi-cone H-O case above. The limitations imposed by the computer
hardware we use have prevented this estimation.}

\subsubsection{IRS-Based Product Specialization (High-GL Sample)}

In the high-GL subsample, there are 630 country-pair observations. As above
when examining the Pure IRS model, we classify the observations into $K=5$
classes, with the Grubel-Lloyd indices rising from class $K=1$ to class $K=5.
$ The model considered here is the generalized H-O model (where the
capital-intensive $X$ sector in IRS, whereas the labor-intensive $Z$ sector
is assumed to be labor-intensive.) It implies imports at a level $%
M^{ij}=(1-\gamma ^i)\frac{Y^iY^j}{Y^w}$ if country $i$ is capital-abundant
relative to $j$ , and $M^{ij}=(1-\gamma ^j)\frac{Y^iY^j}{Y^w}$ otherwise.
The regression is analogous to (\ref{reg}), with the set of the $(1-\gamma
^i)\frac{Y^iY^j}{Y^w}(r)$ (or $(1-\gamma ^j)\frac{Y^iY^j}{Y^w}(r),$ if
country\thinspace $j$ is relatively capital-abundant) for a given class $k$
as the independent variable.

Figure 7 reports the results of this resampling experiment.\footnote{%
This is based on choosing the same $R=70,$ $K=5,$ and $\sigma _\chi ^2=0.15$
as in the above analysis of the Pure IRS model.} The line shows the average
estimate of the share of the IRS sector in the capital-abundant country, $%
(1-\gamma ^i$), across the five different classes. This share is estimated
to be between $0.43$ and $0.78,$ and tends to rise as the Grubel-Lloyd index
rises. Note that from country-pair to country-pair there are changes as to
which of the two, exporter or importer, is the relatively capital-abundant
country. Therefore, a non-structural explanation of the average of the
estimated $\left( 1-\gamma ^i\right) $ is that it gives the average size of
the share of the differentiated goods sector. According to this
interpretation, we estimate a higher GDP share for the IRS-sector in samples
with higher values of the Grubel-Lloyd index. This is consistent with the
generalized H-O theory, where a higher share of intra-industry trade is
driven by IRS and product differentiation.\footnote{%
The average standard errors of the estimates $\alpha _k$ for the Generalized
H-O model are as follows; GL=1: 0.054, GL=2: 0.065, GL=3: 0.105, GL=4:
0.227, GL=5: 0.837. With these standard errors, we estimate $\alpha _1$ to
be lower than $0.78,$ our point estimate for $\alpha _5,$ in a statistical
sense. Note, however, that one cannot reject the hypothesis that $\alpha _5=0
$ at standard significance levels. This is so because the generalized H-O
model fits relatively badly for samples with high $GL$-values, and the pure
IRS model fits increasingly better; see below.}

Summarizing, from our analysis of imperfect-specialization models, we find
supportive evidence for factor-proportions driven product specialization
both from the Generalized H-O model as well as from the uni-cone H-O model.

\subsection{Comparing Perfect and Imperfect-Specialization Models: Model
Selection Criteria}

We now employ a widely-used model selection criteria to compare the
performance of both the multi-cone H-O and uni-cone H-O models for the
low-GL sample on the one hand, and the Pure IRS and the Generalized H-O for
the high GL-sample on the other. The Amemiya prediction criterion (denoted
as $APC$) is defined as 
\[
APC=\frac{e^{\prime }e}{v-w}\left( 1+\frac wv\right) , 
\]
where $e^{\prime }e$ is the residual sum of squares, $v$ is the number of
observations, and $w$ is the number of estimated parameters.\footnote{%
We have also considered the Akaike Information Criterion; using that gives
very similar results as with the Amemiya criterion, so we report only the
latter.} The $APC$ criterion penalizes more heavily losses of degrees of
freedom than the adjusted $R^2$ criterion; lower values indicate better
model performance.

Figure 8 shows that the uni-cone H-O model is preferred to the multi-cone
H-O model across all $FDIF$ classes. This figure also shows the relative
performance of the multi-cone to the uni-cone H-O model across FDIF classes,
computed as the ratio of the $APC$ criteria of the two models for a given
FDIF class (the vertical axis on the right). There is not much change of the
relative performance of the multi-cone model across FDIF classes; if
anything, the multi-cone H-O model performs relatively best where factor
proportions differences are smallest ($FDIF=1$), and relatively worst where
they are largest ($FDIF=5$), which is the\textit{\ opposite} of what one
would expect if factor proportions differences would be important in driving
trade in perfectly specialized products.

The corresponding picture for the Pure IRS and the Generalized H-O model is
shown in Figure 9, for different values of $GL,$ indicating a different
share of IRS-based trade in total trade. The generalized H-O is preferred to
the pure IRS model for all five classes. As in the previous subsection, the
model with imperfect specialization performs better. Still, there is an
important difference relative to the samples with predominantly H-O-based
trade: the relative performance of the generalized H-O model is better in
samples with lower Grubel-Lloyd indices. That means that although the model
in which all goods are produced with IRS is never as successful as the
generalized H-O model, the pure IRS model puts in its best relative
performance where it should (namely, in the sample with the highest value of
the Grubel-Lloyd index).

Overall, we take these results as evidence which, first of all, confirms
that perfect product specialization due to H-O reasons is unlikely to be an
empirically important determinant of international trade volumes, whereas
IRS appears to be, especially for highly-industrialized countries. Second,
perfect specialization of production occurs only for a limited range of
goods, and factors leading to imperfect product specialization, such as
(limited) factor proportions differences, are important in accounting for
the volume of international trade.

\subsection{Is the Effect of Distance Driving the Results?}

One potential problem with the above analyses is that they do not control
for differences across $FDIF$- or $GL$-classes in the average distance
between countries. It is well-established empirically that bilateral trade
volumes fall as bilateral distance increases (the second pillar of the
gravity equation), raising the possibility that the different estimated
parameters are driven by differences in bilateral distance among trade
partners.

Even though the import volume predictions presented above do not incorporate
the effect of trade costs, at an empirical level it is important to see
whether our results depend crucially on the fact that we omit the bilateral
distance variable from our gravity regressions. There are two aspects to
this question: First, is the average bilateral distance varying \textit{%
between} $FDIF$- or $GL$-classes, and if so, how does this affect our
findings? Second, does the fact that our analysis ignores the variation of
bilateral distance across country-pairs \textit{within} a given $FDIF$- or $%
GL$-class explain our findings?

It turns out that the average bilateral distance between trade partners does
not vary much between $FDIF$-classes, whereas it does change between $GL$%
-classes.\footnote{%
For $FDIF$-classes, it hovers around 8800 miles, with a maximum average of
8840 miles for $FDIF$=5 and a minimum average of 8760 miles for $FDIF$=2;
for $GL$-classes, it falls from about $6300$ miles for $GL$=1 to circa $3000$
for $GL$=5. We would like to thank Jon Haveman at Purdue University for
making the distance data available at his webpage (
http://intrepid.mgmt.purdue.edu/Jon/Data/TradeData.html\#Gravity ); the data
gives the Great Circle distance between capital cities, as the crow flies.}
However, the analysis in Appendix $D$ shows that controlling for between-
and within-class effects related to bilateral distance differences in
empirical gravity regressions leaves our results unaffected. While this
cannot settle the question of how structural bilateral import relations look
in the presence of trade costs, it means that our earlier findings are
robust to incorporating distance effects.

\section{Conclusions}

We have compared the accuracy of the perfect specialization versions of the
Heckscher-Ohlin model and the Increasing Returns-based model, both of which
predict the gravity equation. In addition, we have demonstrated that, as far
as their predictions for bilateral trade volumes are concerned, each of
these perfect specialization models is a limiting case of a model with
imperfect specialization. Our empirical strategy has exploited two factors
to identify which models might explain trade flows in a given sample. First,
in the multi-cone H-O model large factor proportions differences are
required to induce product specialization, whereas product specialization in
the pure IRS model occurs for arbitrary differences in factor proportions.
Secondly, there is no intra-industry trade in the multi-cone H-O model,
whereas there is intra-industry trade, and possibly \textit{only}
intra-industry trade, in the Pure IRS model.

There are three major findings: First, little production is perfectly
specialized due to factor proportions differences, making the perfect
specialization version of the H-O model an unlikely candidate to explain the
empirical success of the Gravity Equation. Secondly, increasing returns are
important causes for perfect product specialization and the Gravity
Equation, especially among industrialized countries. Thirdly, models of
imperfect specialization better explain the variation of bilateral trade
flows than perfect specialization models. Factor proportion differences are
important determinants of trade flows within the context of imperfect
specialization models only, whereas increasing returns is a cause of product
specialization both in models with imperfect as well as with perfect
specialization of production.

In the light of this paper, there is no reason to believe that the results
by Hummels and Levinsohn (1995) throw doubt on the empirical relevance of
IRS trade theory. Our results suggest that the H-O model with imperfect
specialization is likely to account for the regression results that these
authors found in their sample where little intra-industry trade was
expected. More generally, the data is supportive of several predictions of
IRS-based trade theory, and we take our results as indicating that IRS-based
trade plays, in conjunction with trade based on factor proportions
differences, an important role in determining world trade flows.

Several important determinants of trade flows are missing from our analysis.
First of all, we do not allow for preferences to differ across countries,
even though other studies have found that countries tend to
overproportionately demand home produced goods (or, exhibit a 'home bias').
This appears to be one reason of why standard trade models predict trade
volumes which are far larger than what is found in the data (Trefler 1995).
Secondly, a related point is that we do not consider explicitly models with
transport costs and trade barriers. We know that the latter are empirically
relevant and can also lead to a home bias. However, CRS production of the
same good in two or more countries in the presence of transport costs is
inconsistent with factor price equalization; moreover, as emphasized by the
recent economic geography literature (e.g. Davis and Weinstein 1996), CRS
and IRS models might behave differently in the presence of transport cost
and differences in demand across countries.

Thirdly, we have already noted that part of the product specialization which
we attribute to factor proportions differences or IRS might actually be due
to technology differences across countries. A fourth point is that we
estimate two-country models with data from a multi-country world. The
problem of developing multi-lateral trade predictions arises from the
well-known indeterminacy of bilateral trade flows when there are imperfectly
specialized homogeneous goods produced with CRS. Traditionally, researchers
have instead considered the H-O-Vanek equation which predicts net factor
service trade of a country with the rest of the world. However, as Trefler
(1996) has shown, generally, the H-O-Vanek equation holds only in cases when
also the simplest gravity equation with perfect product specialization
holds. It is clear, though, and our results confirm that imperfect product
specialization is empirically important. Therefore, we hope to make further
progress by extending this work in the context of the gravity equation to a
multilateral world with trade costs.

\newpage\pagestyle{empty}

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\newpage\ \appendix

\section{The Pure IRS Model}

(See Helpman and Krugman 1985). Consider two countries, $i$ and $j$, two
goods, $X$ and $Z,$ and two factors, $K$ (capital) and $L$ (labor), the
latter being immobile across countries. The goods $X$ and $Z$ come in many
differentiated varieties which are identically produced with increasing
returns to scale. Assume that countries have identical homothetic
preferences, and that consumers value all varieties symmetrically in the
Dixit-Stiglitz (1977) CES fashion. In equilibrium, any product variety will
only be manufactured by one producer (perfect product specialization).
Producers of differentiated products behave monopolistically competitively,
and free entry determines the number of varieties (and firms) in
equilibrium. Let $n_g^c$ be the number of good $f$ varieties produced in
country $c,$ $g=X,Z,$ and $c=i,j$, $s^c$ the share of country $c$ in world
spending and $x^c$($z^c$) the equilibrium quantity of a type $X$ (type $Z$)
variety. Let $Y^c$ denote a country's GDP, and world GDP is given by $%
Y^w=Y^i+Y^j.$ Finally, choose good $Z$ as numeraire and let $p_x$ denote the
relative price of a good $X$ variety.

With balanced trade--so that $s^c=\frac{Y^c}{Y^w},\forall c$--and zero trade
and transport costs, both countries will demand all foreign varieties
according to the countries' GDP as a share of World GDP. Given that the
varieties produced in country $j$ are available in country $i$ only through
imports, country $i$'s imports from $j,$ denoted $M^{ij},$ will be 
\[
M^{ij}=s^i\left[ p_xn_x^jx^j+n_z^jz^j\right] , 
\]
whereas country $j$'s imports from $i$, $M^{ji},$ will be 
\[
M^{ji}=s^j\left[ p_xn_x^ix^i+n_z^iz^i\right] . 
\]
The last two equations are shown as (\ref{mij}) and (\ref{mji}) in the text.

\section{Proof of Proposition 2}

See Keller (1998) for the following result and further discussion. Starting
from equation (\ref{hovt}), 
\[
VT^{ij}=p_x\left( X^i-s^iX^w\right) +\left( Z^j-s^jZ^w\right) ,
\]
country $i$'s imports from $j$ are $M^{ij}=\left( Z^j-s^jZ^w\right) .$ With
balanced trade, $M^{ij}=M^{ji}=p_x\left( X^i-s^iX^w\right) ;$ using the
definition of $\gamma $ leads to 
\[
M^{ij}=\left( 1-\gamma ^i\right) Y^i-s^i\left( 1-\gamma ^i\right)
Y^i-s^i\left( 1-\gamma ^j\right) Y^j.
\]
Because $s^j=1-s^i,$ this gives 
\[
M^{ij}=s^j\left( 1-\gamma ^i\right) Y^i-s^i\left( 1-\gamma ^j\right)
Y^j=s^js^iY^w\left( \gamma ^j-\gamma ^i\right) =\left( \gamma ^j-\gamma
^i\right) \frac{Y^iY^j}{Y^w}.
\]
The last term on the right hand side is expression (\ref{ho}) given in the
text. \TeXButton{End Proof}{\endproof}

\section{Alternative Prior Beliefs on the Presence of IRS-based Trade in
Relation to the Grubel-Lloyd Index}

Here we briefly review the sensitivity of our results with respect to the
assumption that there is little to no IRS-based trade for Grubel-Lloyd
values of $GL^{ij}<0.05$. First, we compare the multi-cone H-O (M1) to the
uni-cone H-O model (M4) for two alternative assumptions: (a) No IRS-based
trade for $GL^{ij}<0.075,$ and (b) No IRS-based trade for $GL^{ij}<0.033.$
Here, $\emptyset (M1)$ means, e.g., the average import parameter for model 1.%
\footnote{%
For these experiments, $R=10$ and $\sigma _\theta ^2=2.$} 
\[
\fbox{$
\begin{tabular}{ll|l|l|l|l|l}
&  & FDIF=1 & FDIF=2 & FDIF=3 & FDIF=4 & FDIF=5 \\ \hline
& $\emptyset ($M1) & \multicolumn{1}{|c|}{0.033} & \multicolumn{1}{|c|}{0.021
} & \multicolumn{1}{|c|}{0.019} & \multicolumn{1}{|c|}{0.019} & 
\multicolumn{1}{|c}{0.017} \\ 
Case (a) & $\emptyset ($M4) & \multicolumn{1}{|c|}{0.062} & 
\multicolumn{1}{|c|}{0.049} & \multicolumn{1}{|c|}{0.038} & 
\multicolumn{1}{|c|}{0.038} & \multicolumn{1}{|c}{0.044} \\ 
& $\frac{APC(M1)}{APC(M4)}$ & \multicolumn{1}{|c|}{1.28} & 
\multicolumn{1}{|c|}{1.2} & \multicolumn{1}{|c|}{1.13} & 
\multicolumn{1}{|c|}{1.2} & \multicolumn{1}{|c}{1.21} \\ 
&  & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{}
& \multicolumn{1}{|c|}{} & \multicolumn{1}{|c}{} \\ \hline
&  & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{} & \multicolumn{1}{|c|}{}
& \multicolumn{1}{|c|}{} & \multicolumn{1}{|c}{} \\ 
& $\emptyset ($M1) & \multicolumn{1}{|c|}{0.03} & \multicolumn{1}{|c|}{0.022}
& \multicolumn{1}{|c|}{0.022} & \multicolumn{1}{|c|}{0.02} & 
\multicolumn{1}{|c}{0.02} \\ 
Case (b) & $\emptyset ($M4) & \multicolumn{1}{|c|}{0.045} & 
\multicolumn{1}{|c|}{0.055} & \multicolumn{1}{|c|}{0.042} & 
\multicolumn{1}{|c|}{0.048} & \multicolumn{1}{|c}{0.033} \\ 
& $\frac{APC(M1)}{APC(M4)}$ & \multicolumn{1}{|c|}{1.16} & 
\multicolumn{1}{|c|}{1.08} & \multicolumn{1}{|c|}{1.17} & 
\multicolumn{1}{|c|}{1.25} & \multicolumn{1}{|c}{1.33}
\end{tabular}
$} 
\]
According to the table, one finds irrespective of the $GL$ cut-off point
that the import parameter is non-increasing as FDIF rises in the multi-cone
model M1. This parallels the findings in the main text. Furthermore, the
ratio of the Amemiya Prediction Criterion (APC) does not vary systematically
for either case (a) or case (b). Again this is identical to what we find
with a $GL=0.05$ cut-off, and it is inconsistent with perfect product
specialization driven by factor proportions differences being a major
element in explaining the success of the gravity equation. The average
import parameter for the uni-cone H-O model, however, increases
monotonically with FDIF for the $GL=0.05$ cut-off point discussed in the
text, whereas now, the relation between the average parameter estimate and
FDIF is non-monotonic. In case (a), for instance, it first falls and then
rises with FDIF. Note that case (a) includes country-pairs with a higher $GL$%
-value than that of $0.05$ as discussed in the text. The finding of high
values of $\emptyset (M4)$ for FDIF=1 and FDIF=2 is consistent with
country-pairs with relatively high $GL$ values (which have relatively low
FDIF values) leading to relatively high import parameter estimates.
Therefore, this finding need not contradict the earlier result in the text
as much as it underlines the importance of the model identification issue.
This point cannot explain the pattern of $\emptyset (M4)$ in case (b),
though.

Second, comparing the pure IRS (M2) with the generalized H-O model (M3) for
the same cases (a) and (b), we find the following: \footnote{%
The following results are based on $R=10$ and $\sigma _\chi ^2=0.1.$} 
\[
\fbox{$
\begin{tabular}{ll|l|l|l|l|l}
& \multicolumn{1}{l|}{} & \multicolumn{1}{l|}{GL=1} & \multicolumn{1}{l|}{
GL=2} & \multicolumn{1}{l|}{GL=3} & \multicolumn{1}{l|}{GL=4} & 
\multicolumn{1}{l}{GL=5} \\ \hline
& \multicolumn{1}{l|}{$\emptyset ($M2)} & \multicolumn{1}{|c}{0.077} & 
\multicolumn{1}{|c}{0.081} & \multicolumn{1}{|c}{0.087} & 
\multicolumn{1}{|c}{0.109} & \multicolumn{1}{|c}{0.131} \\ 
Case (a) & \multicolumn{1}{l|}{$\emptyset ($M3)} & \multicolumn{1}{|c}{0.287}
& \multicolumn{1}{|c}{0.298} & \multicolumn{1}{|c}{0.304} & 
\multicolumn{1}{|c}{0.323} & \multicolumn{1}{|c}{0.358} \\ 
& \multicolumn{1}{l|}{$\frac{APC(M2)}{APC(M3)}$} & \multicolumn{1}{|c}{1.41}
& \multicolumn{1}{|c}{1.28} & \multicolumn{1}{|c}{1.28} & 
\multicolumn{1}{|c}{1.22} & \multicolumn{1}{|c}{1.24} \\ 
& \multicolumn{1}{l|}{} & \multicolumn{1}{|c}{} & \multicolumn{1}{|c}{} & 
\multicolumn{1}{|c}{} & \multicolumn{1}{|c}{} & \multicolumn{1}{|c}{} \\ 
\hline
& \multicolumn{1}{l|}{} & \multicolumn{1}{|c}{} & \multicolumn{1}{|c}{} & 
\multicolumn{1}{|c}{} & \multicolumn{1}{|c}{} & \multicolumn{1}{|c}{} \\ 
& \multicolumn{1}{l|}{$\emptyset ($M2)} & \multicolumn{1}{|c}{0.026} & 
\multicolumn{1}{|c}{0.031} & \multicolumn{1}{|c}{0.042} & 
\multicolumn{1}{|c}{0.075} & \multicolumn{1}{|c}{0.145} \\ 
Case (b) & \multicolumn{1}{l|}{$\emptyset ($M3)} & \multicolumn{1}{|c}{0.323}
& \multicolumn{1}{|c}{0.33} & \multicolumn{1}{|c}{0.332} & 
\multicolumn{1}{|c}{0.351} & \multicolumn{1}{|c}{0.377} \\ 
& \multicolumn{1}{l|}{$\frac{APC(M2)}{APC(M3)}$} & \multicolumn{1}{|c}{2.43}
& \multicolumn{1}{|c}{2.0} & \multicolumn{1}{|c}{1.88} & \multicolumn{1}{|c}{
1.48} & \multicolumn{1}{|c}{1.24}
\end{tabular}
$} 
\]
The table shows that for comparing the pure IRS with the generalized H-O
model, we obtain the same results with a cut-off values of $GL=0.075$ (case
(a)) and $GL=0.033$ (case (b)) as with the value of $GL=0.05$ as employed in
the text: First, the import parameter estimate of the pure IRS model is
rising as GL is rising. Secondly, also the average import parameter of the
generalized H-O model is rising as GL is rising. Thirdly, the generalized
H-O model is less preferred for higher values of GL.

Summarizing, five out of six inferences made in the text continue to hold
even if different reasonable prior beliefs on the presence of IRS-based
trade in relation to the $GL$ index are adopted. With different prior
beliefs, the importance of factor proportions differences in determining
trade volumes is evidenced primarily by the our findings in support of the
generalized H-O model M3.

\section{The Effect of Distance}

\subsection{Between-Class Effects}

For the between-class effects, we focus on the High-GL sample ($GL^{ij}>0.05$%
), as the average bilateral distance across $FDIF$-classes varies little. We
conduct a two-dimensional resampling analysis analogous to the earlier ones,
smoothing both the Grubel-Lloyd and distance variables: 
\[
\begin{array}{c}
GL^{ij}=\text{ }^{*}GL^{ij}+\chi ^{ij}, \\ 
\\ 
DIST^{ij}=\text{ }^{*}DIST^{ij}+\pi ^{ij},\forall ij,
\end{array}
\]
where $\pi \sim N(0,\sigma _\pi ^2),\sigma _\pi ^2>0,$ and $DIST^{ij}$
denotes the bilateral distance between $i$ and $j$.\footnote{%
Here, $R=50,$ $\sigma _\chi ^2=0.3,$ and $\sigma _\pi ^2=1.5.$}

In Figure A1, we have plotted the average estimated $(1-\gamma ^i$), denoted 
$\alpha _{km},$ for $K\times M$ classes of $GL$ ($K=5$ classes) and $DIST$ ($%
M=5$ classes), where the latter denotes the average bilateral distance in a
sample. It is clear from Figure A1 that we estimate a lower value of $\alpha
_{km}$, the higher is the average bilateral distance. The correlation
between $GL$ and $\alpha _{km}$ remains positive. However, this still does
not determine whether there is an independent effect associated with $GL,$
because the average distance for $(GL=1,DIST=1)$ is much higher than for $%
(GL=5,DIST=1),$ i.e., $GL$ and $DIST$ co-vary. Therefore, we have taken the $%
25$ values of $\alpha _{km},$ $GL_{km},$ and $DIST_{km}$, and run the
following least-squares regression: 
\begin{equation}
\alpha _{km}=\beta _0+\beta _1GL_{km}+\beta _2dist_{km}+\mu _{km},
\label{areg}
\end{equation}
where $\mu _{km}$ is assumed to be mean zero and constant variance $\sigma
_\mu ^2,$ and $dist$ denotes the log of the variable $DIST$. We find the
following parameters (standard errors) for $\beta _1$ and $\beta _2:$ 0.174
(0.064), and $-0.084$ (0.011), respectively. In sum, higher values of the
Grubel-Lloyd index are associated with higher import parameter estimates
even when differences in the average distance between trade partners across
subsamples are controlled for.

Figure A2 shows the analogous picture for the Pure IRS model. The import
parameter estimate rises both with the Grubel-Lloyd index rising and the
average bilateral distance falling; for maximum values of $GL$ and minimum
values of $DIST$ (the combination $GL=5,$ $DIST=1$), a value of $\alpha
_{51}=0.26$ is estimated, whereas for the minimum values of $GL$ and maximum
values of $DIST,$ we estimate $\alpha _{15}=0.03$. Because the resampling
experiments are identically designed, the same co-variance of distance and
the Grubel-Lloyd index is present as above for the Generalized H-O model
analysis. We run again the least squares regression (\ref{areg}), this time
with the data generated by the pure IRS model underlying Figure A2. The
estimated coefficients (standard errors) are: $\beta _1=0.08$ (0.02) and $%
\beta _2=-0.02$ (0.003). Also here, the estimated import parameter is higher
as the Grubel-Lloyd index rises, even when the effects of distance are
controlled for. This suggests that our earlier interpretation of the results
is robust.

\subsection{Within-Class Effects}

To see whether the omission of the distance variable is important in
obtaining our results, we use the same resampling approach as in the text,
and then run, instead of (\ref{reg}) or the analogous regressions, the
following gravity regression: 
\begin{equation}
M^{ij}=\alpha \left( \frac{Y^i\times Y^j}{Y^w\times dist^{ij}}\right)
+\epsilon ^{ij},  \label{memp}
\end{equation}
where $\alpha $ is the import parameter of the respective model (a constant
for the multi-cone H-O and the pure IRS models, $(1-\gamma ^i)$ for the
generalized H-O model if country $i$ is capital-abundant, and $\left( \gamma
^j-\gamma ^i\right) $ for the uni-cone H-O model if $i$ is capital-abundant
relative to country $j$). Equation (\ref{memp}) corresponds in levels to the
gravity equation which is often run in logs, with a common-parameter
restrictions imposed. The results of (\ref{memp}) are given in Table A1: 
\[
\fbox{$
\begin{tabular}{c|c|ccc|ccc|c|c}
\multicolumn{10}{l}{Table A1: The Effect of Including a Distance Variable}
\\ \hline\hline
\multicolumn{3}{c}{Low-GL Sample} &  &  & \multicolumn{1}{c}{} &  & 
\multicolumn{3}{c}{High-GL Sample} \\ \hline
& Multi-cone H-O & Uni-cone H-O &  &  & \multicolumn{1}{c}{} &  &  & Pure IRS
& Gener. H-O \\ \hline
\multicolumn{1}{l|}{$FDIF=1$} & $%
\begin{array}{c}
0.281 \\ 
(0.014)
\end{array}
$ & $%
\begin{array}{c}
0.302 \\ 
(0.292)
\end{array}
$ & \multicolumn{1}{l}{} & \multicolumn{1}{l}{} & \multicolumn{1}{|l}{} & 
\multicolumn{1}{l}{} & \multicolumn{1}{l|}{$GL=1$} & $%
\begin{array}{c}
0.324 \\ 
(0.010)
\end{array}
$ & $%
\begin{array}{c}
3.386 \\ 
(0.873)
\end{array}
$ \\ \cline{1-3}\cline{8-10}
\multicolumn{1}{l|}{$FDIF=2$} & $%
\begin{array}{c}
0.239 \\ 
(0.016)
\end{array}
$ & $%
\begin{array}{c}
0.418 \\ 
(0.318)
\end{array}
$ & \multicolumn{1}{l}{} & \multicolumn{1}{l|}{} & \multicolumn{1}{|l}{} & 
\multicolumn{1}{l}{} & \multicolumn{1}{l|}{$GL=2$} & $%
\begin{array}{c}
0.403 \\ 
(0.012)
\end{array}
$ & $%
\begin{array}{c}
3.502 \\ 
(1.075)
\end{array}
$ \\ \cline{1-3}\cline{8-10}
\multicolumn{1}{l|}{$FDIF=3$} & $%
\begin{array}{c}
0.206 \\ 
(0.009)
\end{array}
$ & $%
\begin{array}{c}
0.427 \\ 
(0.194)
\end{array}
$ & \multicolumn{1}{l}{} & \multicolumn{1}{l|}{} & \multicolumn{1}{|l}{} & 
\multicolumn{1}{l}{} & \multicolumn{1}{l|}{$GL=3$} & $%
\begin{array}{c}
0.544 \\ 
(0.015)
\end{array}
$ & $%
\begin{array}{c}
3.601 \\ 
(1.489)
\end{array}
$ \\ \cline{1-3}\cline{8-10}
\multicolumn{1}{l|}{$FDIF=4$} & $%
\begin{array}{c}
0.202 \\ 
(0.009)
\end{array}
$ & $%
\begin{array}{c}
0.456 \\ 
(0.196)
\end{array}
$ & \multicolumn{1}{l}{} & \multicolumn{1}{l|}{} & \multicolumn{1}{|l}{} & 
\multicolumn{1}{l}{} & \multicolumn{1}{l|}{$GL=4$} & $%
\begin{array}{c}
0.655 \\ 
(0.017)
\end{array}
$ & $%
\begin{array}{c}
3.782 \\ 
(2.127)
\end{array}
$ \\ \cline{1-3}\cline{8-10}
\multicolumn{1}{l|}{$FDIF=5$} & $%
\begin{array}{c}
0.199 \\ 
(0.010)
\end{array}
$ & $%
\begin{array}{c}
0.670 \\ 
(0.173)
\end{array}
$ & \multicolumn{1}{l}{} & \multicolumn{1}{l|}{} & \multicolumn{1}{|l}{} & 
\multicolumn{1}{l}{} & \multicolumn{1}{l|}{$GL=5$} & $%
\begin{array}{c}
1.004 \\ 
(0.023)
\end{array}
$ & $%
\begin{array}{c}
5.057 \\ 
(3.584)
\end{array}
$
\end{tabular}
$}
\]
The entries give average import parameter estimates for the four models (in
brackets are the estimated standard errors). It is clear that the estimated
parameters average are, with the inclusion of the distance variable,
different from those in the text. The \textit{patterns} of how the estimates
change across classes, however, are the same in both cases: the estimate of $%
\alpha $ is non-increasing for the multi-cone H-O model as FDIF increases,
whereas the average $\alpha $ rises with class index for the other three
models.

Building on the same resampling approach, we have also estimated an equation
where the distance variable enters additively, 
\begin{equation}
M^{ij}=\alpha \left( \frac{Y^i\times Y^j}{Y^w}\right) +\beta
DIST^{ij}+\epsilon ^{ij},  \label{mreg2}
\end{equation}
as well as an equation which includes an overall constant: 
\begin{equation}
M^{ij}=\alpha \left( \frac{Y^i\times Y^j}{Y^w}\right) +\beta DIST^{ij}+\beta
_0+\epsilon ^{ij}.  \label{mreg3}
\end{equation}
For both equation (\ref{mreg2}) and equation (\ref{mreg3}), we obtain the
same patterns for the average import parameter as given in Table A1 and in
the main text. Overall, this suggests that the findings are robust to
including the distance variable into the analysis as an empirical
specification.

\end{document}
