## Abstract

We investigate whether the gravity model (GM) can explain the statistical properties of the International Trade Network (ITN). We fit data on trade flows with a GM using alternative estimation techniques and we build GM-based predictions for the weighted topological properties of the ITN, which are then compared to the observed ones. Our exercises show that the GM: (i) may replicate part of the weighted-network structure only if the observed binary architecture is kept fixed; (ii) is not able to explain higher-order statistics that, like clustering, require the knowledge of triadic link-weight topological patterns, even if the binary structure perfectly replicates the observed one; (iii) performs very badly when asked to predict the presence of a link, or the level of the trade flow it carries, whenever the binary structure must be simultaneously estimated.

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## Notes

- 1.
See for example Li et al. (2003), Serrano and Boguñá (2003), Garlaschelli and Loffredo (2004, 2005); Garlaschelli et al. (2007), Serrano et al. (2007), Bhattacharya et al. (2007, 2008), Fagiolo et al. (2008, 2009, 2010), Reyes et al. (2008), Fagiolo (2010), Barigozzi et al. (2010a, b), De Benedictis and Tajoli (2011).

- 2.
For example, Abeysinghe and Forbes (2005) show that bilateral trade can only explain a small fraction of the impact that an economic shock originating in a given country can have on another one, which is not among its direct-trade partners. Similarly, Dees and Saint-Guilhem (2011) report that countries that do not trade very much with the US are largely influenced by its dominance over other trade partners linked with the US More generally, (Ward and Ahlquist (2012), p. 2) argue that “as intuition would suggest and recent theoretical advance has formalized, bilateral trade is not independent of the production, consumption, and trading decisions made by firms and consumers in third countries”.

- 3.
See Bhattacharya et al. (2008) and Garlaschelli and Loffredo (2004) for exceptions. See also Squartini et al. (2011a, b) for an alternative approach employing null random models that are able to predict whether observed properties of the ITN are statistically meaningful or simply the result of “constrained” randomness.

- 4.
Defined as the ratio between \(L(t)\) (existing trade partnerships) and \(N(t)\cdot [N(t)-1]\) (all possible trade partnerships).

- 5.
- 6.
See Ward and Ahlquist (2012) for an attempt to account for network dependencies in the GM specification.

- 7.
ZINB estimates turn out to be very similar to ZIP ones. No dramatic differences are detected between cross-section and panel-data analyses. Similarly, the introduction of country fixed effects do not alter our results below in any crucial ways. Note also that we employ the same set of regressors in both stages of ZIP and ZINB estimates, as listed in Table 2. Reducing the set of regressors in the first stage does not dramatically change our main results. The whole set of estimation results is available from the authors upon request.

- 8.
Whenever a variable resulted not significant we decided to keep it among the regressors anyway to preserve comparison between estimation techniques.

- 9.
From now on, we suppress time labels for the sake of notational convenience and we refer to a cross-section sequence of estimations.

- 10.
In our simulations, we typically employ samples of 10,000 independent matrix realizations. Our results are robust to different sample sizes.

- 11.
Notice that, in principle, one could have used directly the expected values implied by the fitted model (either PPML or ZIP, see Eqs. 3 and 4) to build a single instance of the predicted ITN and compare its properties with the observed ITN. However, by correctly sampling from the implied distributions, one can have a better idea of the variability of predictions around their expected values.

- 12.
These are labelled

*cycle*(if \(i\) exports to \(j\), who exports to \(h\), who exports to \(i\)),*in*(if both \(j\) and \(h\), who are trade partners, exports to \(i\)),*out*(if both \(j\) and \(h\), who are trade partners, imports from \(i\)) and*mid*(if \(i\) imports from \(h\) and exports to \(j\), and \(j\) and \(h\) are trade partners). - 13.
In our exercises, we are implicitly assuming that the expected value of any network statistics (given the implied probability distributions of the estimation method employed) can be replaced by the statistic computed on expected values of links and weights, and that expected values of ratios are equal to ratios of expected values. In fact, Squartini et al. (2011a, b) show that such assumptions do not lead to dramatic

prediction biases, as long as distinct pairs of binary and weighted links are independent, which is indeed the case if we use a well-specified GM.

- 14.
- 15.
Our exercises show that predicted uPPML probabilities for the event that a link is present are all very high and close to unity. Therefore, in the majority of all simulations, the predicted binary structure is close to that of a full graph. Conversely, ITN density ranges from 0.40 to 0.50 (see Table 1), meaning that slightly less than a half of possible trade relationships are present.

- 16.
Among all possible correlations of directed statistics with node in- and out-strength we have selected only those economically more relevant. For example, we have focused on the correlation coefficient between \(ANNS^{out,in}\) and \(NS^{out}\) (and not that between \(ANNS^{out,in}\) and \(NS^{in}\)) because one is much more interested in understanding whether a country that

*exports*more, in turn exports to countries that imports more, rather than knowing whether a country that*imports*more, in turn exports to countries that imports more. - 17.
We focus here only on undirected measures. All main results hold also for directed network statistics.

- 18.
KS-tests almost always rejects the null hypothesis that observed and Bernoulli-Logit simulated (total, in and out) degree distributions are the same.

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## Acknowledgments

The Authors gratefully acknowledge financial support received by the research project “The international trade network: empirical analyses and theoretical models” (www.tradenetworks.it) funded by the Italian Ministry of Education, University and Research (Scientific Research Programs of National Relevance 2009).

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Dueñas, M., Fagiolo, G. Modeling the International-Trade Network: a gravity approach.
*J Econ Interact Coord* **8, **155–178 (2013). https://doi.org/10.1007/s11403-013-0108-y

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### Keywords

- International Trade Network
- Gravity equation
- Weighted network analysis
- Topological properties
- Econophysics

### JEL Classification

- F10
- D85