metadata: authors: 0: author: Arvis,Jean Francois
count: World
docna: 0: docna: Integrating gravity: the role of scale
            invariance in gravity models of spatial interactions and trade
docty: Policy Research Working Paper
subtopic: Trade Policy,International Trade and Trade Rules,Trade and Services,Energy Demand,Energy and Environment,Energy and Mining,Transport Services
teratopic: Law and Development,Transport,Macroeconomics and Economic Growth,International Economics and Trade
abstracts: cdata!: This paper revisits the ubiquitous
            bi-proportional gravity model and investigates the reasons
            why different theoretical frameworks may lead to the same
            empirical formula. The generic gravity equation possesses
            scale invariance symmetries that constrain possible
            theoretical explanations based on optimal allocation
            principles, such as neoclassical or probabilistic
            frameworks. These constraints imply that a representative
            consumer's utilities must be separable, and that an
            entropy model is the only consistent maximum likelihood
            allocation of a matrix of flows between origin and
            destination. The paper explores the feasibility of wider
            classes of non-scale invariant gravity equations, where
            gravity is no longer bi-proportional by including nonlinear
            interactions between trade costs and fundamental country
            factors such as economic size. It shows that such extensions
            are feasible but that they do not result in a significant
            improvement in the explanatory power of the empirical analysis.
repnme: repnme: Integrating gravity: the role of scale
            invariance in gravity models of spatial interactions and trade
display_title: Integrating gravity: the role of
            scale invariance in gravity models of spatial interactions
            and trade
listing_relative_url: /research/2013/01/17206376/integrating-gravity-role-scale-invariance-gravity-models-spatial-interactions-trade
topicv3: Impact of Trade m1326020 1192,Energy and Extractives m1326595 507,Transport m1326661 135,Trade m1326924 698,Trade in Services m1326928 780
publishtoextweb_dt: 2013-01-01T00:00:00Z
historic_topic: Law and Development,Transport,Macroeconomics and Economic Growth,International Economics and Trade
no_of_pages: 23
author_keys: 0: author_key: 452197
model_extractions: WPS6347 Policy Research Working Paper 6347

# Integrating Gravity

The Role of Scale Invariance in Gravity Models of Spatial Interactions and Trade _Jean-François Arvis_ The World Bank Poverty Reduction and Economic Management Network January 2013
Policy Research Working Paper 6347

# **Abstract**

This paper revisits the ubiquitous bi-proportional gravity model and investigates the reasons why different theoretical frameworks may lead to the same empirical formula. The generic gravity equation possesses scale invariance symmetries that constrain possible theoretical explanations based on optimal allocation principles, such as neoclassical or probabilistic frameworks. These constraints imply that a representative consumer’s utilities must be separable, and that an entropy model is the only consistent maximum likelihood allocation of a matrix of flows between origin and destination. The paper explores the feasibility of wider classes of non-scale invariant gravity equations, where gravity is no longer biproportional by including nonlinear interactions between trade costs and fundamental country factors such as economic size. It shows that such extensions are feasible but that they do not result in a significant improvement in the explanatory power of the empirical analysis.

This paper is a product of the International Trade Department, Poverty Reduction and Economic Management Network. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank. org. The author may be contacted at jarvis1@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# **Integrating Gravity: The role of scale invariance in gravity models of spatial interactions and trade**

Jean-François ARVIS<sup>1</sup> The World Bank

**Keywords:** Gravity models, international trade, trade costs

**JEL Codes** : C61, F12

EPOL, TRAN

> 1 <u>Jarvis1@worldbank.org, International Trade Department</u>
>

MM. F = GDP * GDP. X_a = —_*_+ D* ag a
How trade gravity modeling emerges from trade theory has been the object of intense investigation following the influential work by Anderson and Wincoop (2003), itself following a much earlier proposal by Anderson himself (Anderson 1979)and Bergstrand. As noted by several observers, the problem is that there are almost too many successful explanations (De Benedictis & Taglioni 2010)(James E. Anderson, 2011). However, as recently observed by Novy (2012), at the core of most neoclassical explanations of trade gravity is the notion that a representative consumer will source his or her imports according to destination, following the Armington hypothesis, with his or her preferences derived from a CES utility.

This paper essentially reverses the problem. It does not try to explain the structure of the gravity equation from an economic model but rather looks at how the formal structure of gravity constrains the set of models that can be used to generate the gravity equation. The approach followed amounts to an “integration” of the gravity model, where the utility of a problem is deduced or integrated from the specification of the demand functions.

Sections 2 and 3 discuss how neo-classical models or probabilistic derivations explain gravity as an optimal allocation, the utility of a representative consumer, or the likelihood/entropy of flows that are maximized under budget or market constraints. Section 4 describes how scale invariance of the generic bi-proportional model limits the functional form of the objective function to be optimized to essentially the ones that are used in the literature (i.e. CES). Sections 5 and 6 look at how to construct alternative models where the scale invariance is broken, and gravity is no longer strictly bi-proportional because the impact of trade costs on flows is no longer independent from size.

# **2 The generic gravity equation and its scale invariances**

The generic gravity model takes the general bi-proportional form describing flows between origin i and destination j where is a factor that represents the push potential of origin i is a factor that represents the pull potential of destination j is the impedance representing the intensity of the interaction between i and j The potentials A and B incorporate information about the "size" of origin and destination. Potentials are either fixed effects or exogenously determined from country variable explaining the “size” of origin and destination.

3
Cc. i <!-- Start of picture text --> B<br><!-- End of picture text --> B <!-- Start of picture text --> Bp<br><!-- End of picture text --> Bp Ki = exp(—fic,,) <!-- Start of picture text --> K,=(+e, y?<br><!-- End of picture text --> K,=(+e, y? o <!-- Start of picture text --> C5 log” +c,) log” +c,) +c,)<br><!-- End of picture text --> <!-- Start of picture text --> Beo<br><!-- End of picture text --> C5 log” +c,) log” +c,) +c,) Beo <!-- Start of picture text --> Log(X,,) = 4,Log(GDP) + a,Log(GDP) = 4,Log(GDP) + a,Log(GDP) 4,Log(GDP) + a,Log(GDP) + a,Log(GDP) a,Log(GDP)<br>SS _ i i<br>I<br>+ bLog(GDPpercap,) + b,Log(Log(GDPpercap,) + b,Log( + b,Log( b,Log( GDPpercap ,,) + c,domestic efficiency variables for i + c,domestic efficiency variables for i c,domestic efficiency variables for i efficiency variables for i variables for i for i i<br>+iO fj<br>tt Sos ro’<br>Ht<br>+d, Log(distance,,) + d,other separation variables + d,bilateral dummies Log(distance,,) + d,other separation variables + d,bilateral dummies + d,other separation variables + d,bilateral dummies d,other separation variables + d,bilateral dummies separation variables + d,bilateral dummies variables + d,bilateral dummies + d,bilateral dummies d,bilateral dummiesbilateral dummies dummies<br>lH YS q_“ ue oq_9~$soewrnN’<br>if<br><!-- End of picture text --> Log(X,,) = 4,Log(GDP) + a,Log(GDP) = 4,Log(GDP) + a,Log(GDP) 4,Log(GDP) + a,Log(GDP) + a,Log(GDP) a,Log(GDP) SS _ i i I + bLog(GDPpercap,) + b,Log(Log(GDPpercap,) + b,Log( + b,Log( b,Log( GDPpercap ,,) + c,domestic efficiency variables for i + c,domestic efficiency variables for i c,domestic efficiency variables for i efficiency variables for i variables for i for i i +iO fj tt Sos ro’ Ht +d, Log(distance,,) + d,other separation variables + d,bilateral dummies Log(distance,,) + d,other separation variables + d,bilateral dummies + d,other separation variables + d,bilateral dummies d,other separation variables + d,bilateral dummies separation variables + d,bilateral dummies variables + d,bilateral dummies + d,bilateral dummies d,bilateral dummiesbilateral dummies dummies lH YS q_“ ue oq_9~$soewrnN’ if
# **_Scale invariance_**

The gravity specification of bilateral flows is trivially invariant by the scaling transformations.

or in the case of exponential dependence on costs where the and are arbitrary real values.

The possibility of such a transformation is obviously consubstantial to the bi-proportional standard gravity structure, and tied to the fact that trade costs and country size factors act independently: trade costs have the same effects on bilateral flows independently irrespective of the size of the partners. The scale invariance means that if a partner i is further away but bigger, the corresponding trade flows are unchanged: twice as big potential yields the same bilateral flows when the corresponding bilateral costs are increased by .

This phenomenon where rescaling of potential is compensated by a shift in trade costs is not just an algebraic observation, but carries some economic significance. Obviously the inverse problem (inverse gravity) of determining trade costs from trade flows is ambiguous and requires additional assumptions to resolve the ambiguity—e.g., that the diagonal trade costs with oneself is zero or unit impedance or (Novy 2009), (Arvis & Shepherd, 2011). It means also that trade costs are to some extent not directly observable, although individual components of trade costs dependent on directly measurable factors such as distance or transportation cost are.

In the context of trade economics, bilateral trade costs include two categories:

- Bilateral costs that depend on both origin and destination (group III in the above loglinear gravity equation)

- Endogenous costs that represent the thickness of the borders at origin and destination and do not depend on the country pair, typically represented in log-linear trade gravity regressions by country specific dummies or non-dimensional variables where country size does not intervene explicitly.

5
Pj C.. y P= +c, Pi; P; U=(—,...—) m m m= )'X;Py
Xx.. y <!-- Start of picture text --> dU dU<br>x, =—-——/ =—-——/<br>5 Op.yy Om<br><!-- End of picture text --> dU dU x, =—-——/ =—-——/ 5 Op.yy Om U=) P. 2a(4)TMa(4)TM<sup>iC)</sup> o x... y X, © a,p," a,p," <!-- Start of picture text --> U=) P.<br>2a(4)TMa(4)TM iC)<br><!-- End of picture text --> <!-- Start of picture text --> x...<br>y<br><!-- End of picture text --> <!-- Start of picture text --> X, © a,p," a,p,"<br><!-- End of picture text --> N; DIN,SpSp =N, N. 4 <!-- Start of picture text --> DIN,SpSp =N,<br>4<br><!-- End of picture text --> <!-- Start of picture text --> N.<br><!-- End of picture text --> DAN, =N, IL;! T14:: <!-- Start of picture text --> L =) log, != )/N, log N,-N,<br>a] y<br><!-- End of picture text --> <!-- Start of picture text --> DINEf]_Ef]_ 7_c_F =Constant_c_F =ConstantF =Constant =Constant<br><!-- End of picture text --> _ 7_c_F =Constant_c_F =ConstantF =Constant =Constant DINEf]_Ef]_
log N,, = 4, +b. — fre, <!-- Start of picture text --> B<br><!-- End of picture text --> B L1=)/N, log N,—N, + Bc,N, 7 <!-- Start of picture text --> B<br><!-- End of picture text --> B o <!-- Start of picture text --> o<br><!-- End of picture text --> <!-- Start of picture text --> B<br><!-- End of picture text --> B <!-- Start of picture text --> B<br><!-- End of picture text --> B
Probability of buying fromi_ X, 4, Ky Probability of buying fromj X, 4; Ky <!-- Start of picture text --> X<br>x,<br><!-- End of picture text --> X x, Lm €¢i,j U (Dg >Popo) U akCom”?Py, Pak Px
<!-- Start of picture text --> ‘Op,<br><!-- End of picture text --> <!-- Start of picture text --> Xe<br>XxX, U,<br><!-- End of picture text --> Xe Uy XxX, U, ‘Op, <!-- Start of picture text --> u=FO si’)<br>; m<br><!-- End of picture text --> Si f(Q=aq" F(q) = a, exp(-q) <!-- Start of picture text --> »%,, = X, = X, X,<br><!-- End of picture text --> »%,, = X, = X, X, xX, = X; a =a,+b+pe, =a,+b+pe,pe, <!-- Start of picture text --> a =a,+b+pe, =a,+b+pe,pe,<br><!-- End of picture text --> c,, clog X + fixed effects
<!-- Start of picture text --> OL<br>~—— =a =a +b +f logx ,<br><!-- End of picture text --> OL ~—— =a =a +b +f logx , L=f'X log X_, + lmear combination of total of X in rows and columns <!-- Start of picture text --> B<br><!-- End of picture text --> B U(consumer j) = exp(>,a log p,+ B>» logp,logp,) i kJ#j X =ABK'. 7 af 7 K', = das 275 ogp,) Py a, k#j
- The bilateral impedance depends not only on the bilateral trade costs but also of other trade costs of the importer.

- The size of the exporting economy matters: K1 is sensitive to the share coefficient. of the exporter, with the direction effect depending on the sign of : if the latter is positive, the "impedance" is higher for smaller exporter.

- The translog model breaks the formal symmetry between origin and destination.

- Transslog is essentially a nonlinear extension of the Cobb-Douglass utility, which corresponds to a CES of one, while In the context of trade the typical observed CES is much larger (seven).

We propose below a simpler alternative symmetry-breaking extension of gravity, which departs more smoothly from standard gravity.

# **Two-dimensional allocation: Modified gravity equation**

In the case of two-dimensional symmetric gravity, the simplest approach to symmetrybreaking "gravity" consists in adding interaction terms between costs and flows in the Lagrangian yielding the gravity model, so that the scale invariance does not hold. Indeed, the original Wilson’s Lagrangian also reads as a weighted average of a linear combination of and the trade costs (in the following angle brackets stands for weighted averages ) which is obviously invariant by the scaling transformation , and Therefore, a minimalist way to break the scaling symmetry is to add to the Lagrangian a nonlinear multiplicative second-order interaction between trade costs and logarithm of flows, the most natural choice being their covariance (which is invariant by translation of the log flows and costs): 12
log X, + Be, +¥(c, —{c) log X,, — (log X)) = a, + b, log X, =a,+ b, — Be, —y(a,+ be, +(log X) c_—(c) c= y * 1+y(6,—(e)) <!-- Start of picture text --> Y<br><!-- End of picture text --> Y (log X) X) <!-- Start of picture text --> (log X) X)<br><!-- End of picture text --> <!-- Start of picture text --> dist.<br>y<br><!-- End of picture text --> dist. y log X,, =a,t b. — Plog dist, —yt
1. Poisson regression for the model without interaction ( )

2. Poisson regression for the model with interaction ( ) 3. OLS for the model without interaction ( ) 4. OLS for the model with interaction ( ) The graphs in Annex 2 plot for Poisson and OLS the predicted values for both standards and modified gravity.

# **Table 1 results**

|20710 observations|1|2|3|4|
|---|---|---|---|---|
|R2/ pseudo R2|nd|nd|0.749|0.820|
|Log distance coeff ( β<br>−<br>)|-0.837|-0.802|-1.739|0.837|
|Wald Chi2/ F|68035|55339|175|266|
|,, z statistics|-34.0|-35.9|-77.4|24.2|
|,, standardized “beta”<br>coefficient|-1.008|-0.965|-2.093|1.008|
|Interaction coeff ( γ<br>−)||0.114||0.298|
|,, z statistics||16.1||89.4|
|,, standardized “beta”<br>coefficient||0.013||0.034|

The regression results suggests the following conclusions

1. As expected the Poisson regression has the most consistent result with trade spatial interaction data (Silva & Tenreyro 2006)(Arvis & Shepherd 2013)

2. The improvement in fit provided by the model (R2 or pseudo-R2) is relatively small in both models (also visually apparent in the shape of distributions in Annex 2).

3. In Poisson, the coefficient :

- has a small standardized (“beta”) value meaning that the effect of the non-linear covariance term is small as compared with the classical log-linear impact of distance

- is negative and significant, which means a relatively small effect of stronger suppression of small flows.

- the predicted values for the trade flows are relatively closed in each models

- the distance coefficients are close in both Poisson model

4. The OLS regressions has observed by various authors are less robust than the Poisson ones 14
5. The OLS results (coefficient predicted value) are less consistent between standard and modified gravity model:

- Loose fit between predicted and actual value of trade

- The coefficient of log-distance in the modified gravity turns positive with a relatively higher (than in Poisson) coefficient of interaction, which suggests an issue of multicolinearity better addressed in the Poisson estimate.

# **7  Conclusions**

The use of symmetry principles is not very popular in economics, although the quest for symmetry is central to scientific understanding in other discipline (physics, chemistry). Hence rather than a restriction they are a desirable property that make the description of a phenomenon consistent, less dependent upon extra assumption or parameters, and likely more analytically tractable (the parsimony principle). Scale invariance in gravity constrains the possible explanation to essentially the known one. This should be considered more as a desirable property than a strong restriction.

Possible scale symmetry breaking alternatives are indeed feasible, where trade costs and size of trade flows interact, instead of the standard bi-proportional structure. However even with the simplest modification, modified gravity equations add parameters and complexity including nonlinear equations to estimate econometrically. They do not seem to provide a major improvement over standard gravity. Thus, given the high quality of fit of modern gravity models, especially when using Poisson Quasi Maximum Likelihood estimation, there does not appear to be a very strong case for using an alternative to the bi-proportional standard gravity equation. This result provides additional support for ongoing and recent efforts to better understand and measure the origin and nature of bilateral impedances or trade costs within the standard model.

15