On the Specification of the Gravity Model of Trade: Zeros, Excess Zeros and Zero-inflated Estimation

TL;DR: In this article, modified Poisson fixed-effects estimations (negative binomial, zero-inflated) are proposed to overcome the bias created by the logarithmic transformation, the failure of the homoscedasticity assumption, and the way zero values are treated.

Abstract: Conventional studies of bilateral trade patterns specify a log-normal gravity equation for empirical estimation. However, the log-normal gravity equation suffers from three problems: the bias created by the logarithmic transformation, the failure of the homoscedasticity assumption, and the way zero values are treated. These problems normally result in biased and inefficient estimates. Recently, the Poisson specification of the trade gravity model has received attention as an alternative to the log-normality assumption (Santos Silva and Tenreyro, 2006). However, the standard Poisson model is vulnerable for problems of overdispersion and excess zero flows. To overcome these problems, this paper considers modified Poisson fixed-effects estimations (negative binomial, zero-inflated). Extending the empirical model put forward by Santos Silva and Tenreyro (2006), we show how these techniques may provide viable alternatives to both the log-normal and standard Poisson specification of the gravity model of trade.

Summary

1. The Gravity Model of Trade and the Log-Normal Specification

• This basic model can easily be augmented to include other variables, such as whether countries i and j share borders, have the same language, or are member of a regional integration agreement (Feenstra, 2004) .
• Taking logarithms of both sides of the equation and adding a random disturbance term, the multiplicative form (1) can be converted into a linear stochastic form, yielding a testable equation: EQUATION where ij  is assumed to be independent and identically distributed (i.i.d.) .
• The terminology used in the field of regional science reflects that the model does not take into account the constraints that the estimated bilateral outflows should add up to the total outflows, and that the estimated bilateral inflows should add up to the total inflows.
• This has resulted in formulations of the gravity model that derive from general equilibrium modeling of bilateral trade patterns (Bröcker, 1989; Eaton and Kortum, 2002; Anderson and Van Wincoop, 2003; Feenstra, 2004) .

2. Problems with the Log-Normal Specification of the Gravity Model

• Until now, the log-normal formulation of the gravity model has been the most commonly used economic tools to investigate international bilateral trade flows.
• From a methodological point of view, there are some serious problems with this gravity model specification.
• Flowerdew and Aitkin (1982) specifically point to (1) the bias created by the logarithmic transformation, (2) the failure of the assumption that all error terms have equal variance, and (3) the sensitivity of research results to zero-valued flows.
• Using bilateral trade data from 1990 from the World Trade Database (WTDB), Haveman and Hummels (2004) reported that in 58% of the cases, trade in a specific good originates from fewer than 10% of all countries.
• By tradition, the most common strategies to circumvent the "zero problem" in the analysis of trade flows are to omit all zero-valued trade flows or arbitrarily add a small positive number (usually 0.5 or 1) to all trade flows in order to ensure that the logarithm is well-defined (Linders and De Groot, 2006) .

3.1. From a Log-Normal to a Poisson Specification

• Given the problems with the log-normal specification, the use of alternative regression techniques could be more appropriate in the context of the gravity model.
• As will be shown later in this article, this class of models is, from both a theoretical and empirical point of view, a viable alternative to standard Poisson and its log-normal counterpart.
• By applying the Poisson specification to the fixed effects specification of the gravity model of trade (Andersen and Van Wincoop, 2003) , the authors state that the observed volume of trade between countries i and j has a Poisson distribution with a conditional mean ) ( that is a function of the independent variables (3).
• In contrast to the log-normal specification, the Poisson specification of the gravity model does not face the problems outlined in the previous section.

3.2. Overdispersion and the Negative Binomial Specification

• The conditional variance is most often higher than the conditional mean, which means that the dependent variable is overdispersed.
• According to Greene (1994) , an important reason why the authors frequently find more variation than expected is the presence of unobserved heterogeneity consistent, yet inefficient, estimation of the dependent variable, which is exemplified by spuriously large z-values and spuriously small p-values due to downward biased standard errors (Gourieroux et al., 1984; Cameron and Trivedi, 1986) .
• The expected value of the observed trade flow in the negative binomial regression model is the same as for in Poisson regression model (Long, 1997) , but the variance here is specified as a function of both the conditional mean ) ( and a dispersion parameter (α), thereby incorporating unobserved heterogeneity into the conditional mean.
• In other words, an additional error term has been added to the negative binomial regression model.
• A likelihood ratio test of α can be employed to test whether the negative binomial distribution is preferred over a Poisson distribution (Cameron and Trivedi, 1986 ).

3.3. Excess Zeros and the Hurdle and Zero-Inflated Specification

• A related problem stems from the excessive number of zeros in the data, meaning that the number of zeros is greater than the Poisson or negative binomial distribution predicts.
• It is, according to Greene (1994) , important to separate these two issues into different processes underlying the deficiencies of the Poisson model.
• There is a difference between scientists who do not write any papers or and hence do not receive citations, and scientists who do write papers but are still not cited.
• Hence, the profitability of trade, which reflects the trade potential, can be separated from the volume of trade as stemming from two different processes.

4.1. Data and Variables

• To compare the different specifications, the authors focus on trade patterns for a set of 138 countries in the period 1996-2000 (UN COMTRADE database; Feenstra et al., 2005) .
• There are potentially 138x137 = 18906 individual trade flows between the 138 countries of origin (the exporters) and the 138 countries of destination (the importers).
• The authors use averagely yearly exports expressed in millions of US dollars as an indicator of the bilateral trade volume, such that each pair of countries yields two observations, each country being both an exporter and importer.
• As can be derived from Figure 1 , the frequency distribution of the volume of trade across trade flows strongly deviates from normality (skewness=37.57, kurtosis=1906).
• In fact, over 50% of all bilateral trade flows takes the value zero.

INSERT FIGURE 1 ABOUT HERE

• Barriers of physical distance, culture, institutional frameworks and economic policy still yield considerable costs to international trade (Anderson and Van Wincoop, 2004) .
• The authors have included a variety of explanatory variables in the gravity equation, which affect trade patterns by increasing or decreasing the transactional distance between countries.
• More specifically, the authors distinguish between tangible and intangible trade barriers (Andersen and Van Wincoop, 2004; Linders et al., 2008) .
• Table 1 provides summarized statistics of the variables included in the gravity equation.

4.2. Empirical Results

• The authors include zero flows in the gravity equation using the different specifications that they distinguished above.
• First, the authors present results for the log-normal specification, estimated by OLS.
• Overall, it can be inferred that, in line with the trade literature, most variables have the expected sign and are highly statistically significant.
• This is consistent with the average estimate of distance decay of -0.9 found in the trade literature (Disdier and Head, 2008) .
• The variables describing cultural and economic proximity of countries, such as common language, having ever been in a colonial relationship, and having a free trade agreement, all positively affect the volume of bilateral trade.

INSERT TABLE 2 AND 3 ABOUT HERE

• Because the logarithmic transformation of the gravity model suffers from Jensen"s inequality, potentially severe bias due to unobserved heterogeneity, and cannot deal with zero values in a straightforward way, alternative estimation techniques may be more appropriate.
• With respect to the ZIPPML model ( 5), the authors find that geographical distance, common language, and institutional distance in particular affect the probability of trade, which can be derived from the logit part of the model.
• Compared to the PPML estimator, the regression coefficients estimated by ZIPPML in the Poisson part of the model are similar, while the regression coefficients estimated by NBPML and ZINBPML differ substantially from the effects under PPML.

4.3. Model Comparison among OLS, Poisson and Modified Poisson Estimations

• After comparing the effect size estimates between OLS, Poisson and modified Poisson estimators, the authors want to assess the choice of correct model specification explicitly.
• As the use of the log-normal specification can be refuted on theoretical grounds, and as the available goodness-of-fit statistics to compare the OLS with the Poisson and modified Poisson specifications are rather limited, the authors predominantly focus on the comparison between the Poisson and modified Poisson estimations.
• Probably the most popular way to compare the goodness-of-fit of different models is by comparing the estimated and observed values of the dependent variable (e.g., Bergkvist and Westin, 1998; Martinez-Zarozo et al., 2007) .
• In particular, the NBPML and ZINBPML estimators tend to overpredict the volume of medium and large trade flows.
• PPML and ZIPPML do not only outperform NBPML and ZINBPML, but also OLS .

INSERT FIGURE 2A AND 2B ABOUT HERE

• The good performance of PPML and ZIPPML based on the comparison between the linearly predicted volume of trade and the observed volume of trade is also reflected in the Stavins and Jaffe Goodness-of-Fit statistic (Stavins and Jaffe, 1990 ; see also Martinez-Zarozo et al., 2007) .
• The Stavins and Jaffe goodness-of-fit statistics are based on the Theil inequality coefficient (Theil"s U), which usually ranges from 0 to 1 (Theil, 1958) .
• This would indeed confirm that the PPML and ZIPPML models provide a more accurate forecast.
• The modified Poisson models such as the NBPML, ZIPPML and ZINBPML have the advantage of imposing fewer restrictions on variance and allowing more heterogeneity.
• Figure 3 shows these for PPML, NBPML, ZIPPML, and ZINBPML for all observed bilateral trade between 0 and 20 million (about 75% of the sample).

INSERT FIGURE 3 ABOUT HERE

• Examining more formal statistics concerned with comparing the observed and predicted distributions (see Table 3 ), the likelihood ratio test of overdispersion (α) and the To summarize, the different goodness-fit statistics do not all point to the same conclusion.
• On the one hand, PPML and ZIPPML perform the best when comparing the expected and observed values of the dependent variable.
• On the other hand, NBPML and ZINBPML perform the best when comparing the expected and observed probabilities, thereby taking into account the model improvement by introducing less restrictive variance assumptions (Liu and Cela, 2008) .
• Overall, it can be inferred that ZIPPML performs the best on average, as rated by both criteria.
• It has a reasonable fit of estimated trade, can include zero flows, and accounts for different types of zero flows, correcting for excess zeros and the overdispersion that results from that.

5. Discussion and Conclusion

• The renewed and extended interest in the correct econometric specification of the gravity model of trade fosters the discussion on the estimation techniques applied (Santos Silva and Tenreyro, 2006; Martinez-Zarzoso et al., 2007) .
• Three problems often encountered when analyzing bilateral trade data using the conventional log-normal specification of the gravity model of trade are (1) the bias created by the logarithmic transformation (Jensen"s inequality), (2) the failure of the homoscedasticity assumption, and (3) the way zero-valued trade flows are treated.
• From a theoretical point of view, these specifications of the gravity model do not bring about the problems of the log-normal formalism, and zero-inflated models allow for the possibility to detach the trade probability from the trade volume.
• On the one hand, the Poisson model (PPML) and the zero-inflated Poisson model perform the best when comparing the estimated and observed values of the dependent variable, and even outperform OLS in their example.
• Institutional barriers are identified as intangible trade barriers, although in principle some of the costs related to institutions are directly observable (for example, legal costs).

Figures

Table 1: Statistics of Variables used in the Gravity Equation

Table 2: OLS and Average Yearly Trade from 1996-2000

Table 3 Poisson and Modified Poisson PML on Average Yearly Trade from 1996-2000.

Full text

Accepted and forthcoming in Spatial Economic Analysis
ERIM REPORT SERIES RESEARCH IN MANAGEMENT
ERIM Report Series reference number
ERS-2009-003-ORG
Publication
January 2009
Number of pages
42
Persistent paper URL
http://hdl.handle.net/1765/14614
Email address corresponding author
mburger@ese.eur.nl
Address
Erasmus Research Institute of Management (ERIM)
RSM Erasmus University / Erasmus School of Economics
Erasmus Universiteit Rotterdam
P.O.Box 1738
3000 DR Rotterdam, The Netherlands
Phone: + 31 10 408 1182
Fax: + 31 10 408 9640
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Bibliographic data and classifications of all the ERIM reports are also available on the ERIM website:
www.erim.eur.nl
On the Specification of the Gravity Model of Trade: Zeros,
Excess Zeros and Zero-Inflated Estimation
Martijn J. Burger, Frank G. van Oort, and Gert-Jan M. Linders

ERASMUS RESEARCH INSTITUTE OF MANAGEMENT
REPORT SERIES
RESEARCH IN MANAGEMENT
ABSTRACT AND KEYWORDS
Abstract
Conventional studies of bilateral trade patterns specify a log-normal gravity equation for
empirical estimation. However, the log-normal gravity equation suffers from three problems: the
bias created by the logarithmic transformation, the failure of the homoscedasticity assumption,
and the way zero values are treated. These problems normally result in biased and inefficient
estimates. Recently, the Poisson specification of the trade gravity model has received attention
as an alternative to the log-normality assumption (Santos Silva and Tenreyro, 2006). However,
the standard Poisson model is vulnerable for problems of overdispersion and excess zero flows.
To overcome these problems, this paper considers modified Poisson fixed-effects estimations
(negative binomial, zero-inflated). Extending the empirical model put forward by Santos Silva
and Tenreyro (2006), we show how these techniques may provide viable alternatives to both the
log-normal and standard Poisson specification of the gravity model of trade.
Free Keywords
international trade, distance, gravity model, modified Poisson models
Availability
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Academic Repository at Erasmus University (DEAR), DEAR ERIM Series Portal
Social Science Research Network (SSRN), SSRN ERIM Series Webpage
Research Papers in Economics (REPEC), REPEC ERIM Series Webpage
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The electronic versions of the papers in the ERIM report Series contain bibliographic metadata
by the following classification systems:
Library of Congress Classification, (LCC) LCC Webpage
Journal of Economic Literature, (JEL), JEL Webpage
ACM Computing Classification System CCS Webpage
Inspec Classification scheme (ICS), ICS Webpage

1
On the Specification of the Gravity Model of Trade:
Zeros, Excess Zeros and Zero-Inflated Estimation
Martijn J. Burger
Corresponding author: Department of Applied Economics, Erasmus University Rotterdam,
P.O. Box 1738, 3000 DR Rotterdam. Tel: +31 (0)10 4089579. Fax: +31 (0)10 4089141.
E-mail: mburger@few.eur.nl.
Frank G. van Oort
Department of Economic Geography, Utrecht University and Environmental Assesment
Agency. E-mail: f.vanoort@geo.uu.nl.
Gert-Jan M. Linders
Department of Spatial Economics, Free University Amsterdam.
E-mail: glinders@feweb.vu.nl.
Abstract
Conventional studies of bilateral trade patterns specify a log-normal gravity equation for
empirical estimation. However, the log-normal gravity equation suffers from three problems:
the bias created by the logarithmic transformation, the failure of the homoscedasticity
assumption, and the way zero values are treated. These problems normally result in biased
and inefficient estimates. Recently, the Poisson specification of the trade gravity model has
received attention as an alternative to the log-normality assumption (Santos Silva and
Tenreyro, 2006). However, the standard Poisson model is vulnerable for problems of
overdispersion and excess zero flows. To overcome these problems, this paper considers
modified Poisson fixed-effects estimations (negative binomial, zero-inflated). Extending the
empirical model put forward by Santos Silva and Tenreyro (2006), we show how these
techniques may provide viable alternatives to both the log-normal and standard Poisson
specification of the gravity model of trade.
Keywords: International trade, distance, gravity model, modified Poisson models
JEL Classification: C13, C21, F15

2
1. The Gravity Model of Trade and the Log-Normal Specification
Spatial interaction patterns, such as international trade, migration or commuting flows, can be
predicted and elucidated with an analogy to Newton‟s law of universal gravitation. The
gravity model, which has been used in modern economics since Isard (1954), Ullman (1954),
and Tinbergen (1962), hypothesizes that the gravitational force between two objects is
directly proportional to the product of the masses of the objects and inversely proportional to
the geographical distance between them. Over the years, this model has become popular in
international economics when analyzing the pattern of trade flows between countries
(Eichengreen and Irwin, 1998; Overman et al., 2004).
i
In its most elementary form, the
gravity model can be expressed as
12
3
ij
ij
ij
MM
IK
d



, (1)
where I
ij
is the interaction intensity or the volume of trade between countries i and j, K is a
proportionality constant, M
i
is the mass of the country of origin (in applications to bilateral
trade patterns usually reflected by the country‟s GDP), M
j
is the mass of the country of
destination, d
ij
is the physical distance between the two countries, β
1
is the potential to
generate flows, β
2
is the potential to attract flows, and β
3
is an impedance factor reflecting the
distance decay in trade. This basic model can easily be augmented to include other variables,
such as whether countries i and j share borders, have the same language, or are member of a
regional integration agreement (Feenstra, 2004).

3
Taking logarithms of both sides of the equation and adding a random disturbance term, the
multiplicative form (1) can be converted into a linear stochastic form, yielding a testable
equation:
1 2 3
ln ln ln ln ln ,
ij i j ij ij
I K M M d
   
    
(2)
where
ij

is assumed to be independent and identically distributed (i.i.d.). Equation (2) is in
the trade literature better known as the traditional or empirical gravity model (e.g.,
Eichengreen and Irwin, 1998) and in the field of regional science as the unconstrained gravity
model (e.g., Fotheringham and O‟Kelley, 1989; Sen and Smith, 1995). The terminology used
in the field of regional science reflects that the model does not take into account the
constraints that the estimated bilateral outflows should add up to the total outflows, and that
the estimated bilateral inflows should add up to the total inflows.
Recently, the international trade literature has shown a renewed interest in the theoretical
foundations of the gravity model. This has resulted in formulations of the gravity model that
derive from general equilibrium modeling of bilateral trade patterns (Bröcker, 1989; Eaton
and Kortum, 2002; Anderson and Van Wincoop, 2003; Feenstra, 2004). One of the key
insights in the recent contributions to this field is that the traditional specification of the
gravity model suffers from omitted variable bias, as it does not take into account the effect of
relative prices on trade patterns. As shown by Anderson and Van Wincoop (2003), bilateral
trade intensity not only depends on bilateral trade costs (affected by spatial distance, language
differences, trade restrictions, and the like), but also on GDP-share average weighted
multilateral trade costs indices or “multilateral resistance terms” (affecting the prices of