Workshop for empirical trade analysis December 2015 Bangkok, Thailand Cosimo Beverelli (WTO) Rainer Lanz (WTO)
Content a. What is the gravity equation? b. Naïve gravity estimation c. Theoretical foundations d. Mistakes to avoid e. Estimating theoretically-founded gravity equation f. Recent theoretical developments g. References 2
a. What is the gravity equation? Econometric model (ex-post analysis) Models many social interactions (migration, tourism, trade, FDI) For decades, social scientists have been using a modified version of Isaac Newton's Law of Gravitation to predict movement of people, information, and commodities between cities and even continents The gravity model takes into account the size of two places and their distance. Since larger places attract people, ideas, and commodities more than smaller places and places closer together have a greater attraction, the gravity model incorporates these two features Initially, no theoretical foundations Why so popular? High explanatory power (R 2 between 0.65 and 0.95) Easy access to relevant data Estimation standards and benchmarks clearly established 3
a. What is the gravity equation? The good news: X =Y i Y j /d really stable relationship The bad news: To amaze your friends with another allegedly important trade effect, develop a new proxy for trade costs and use a really big dataset; success is not guaranteed, but you're likely to find significance (standard errors involve the inverse of the square root of number of observations) and you'll have loads of fun in any case" 4
Newton s universal law of gravitation and the gravity specification in trade Newton F G M M i 2 D j Where F is the attraction force, G is the gravitational constant, M is mass, D is distance, i and j index point masses Gravity Y X G Y i j T Where X = exports from country i to j or total trade; Y=economic size (GDP, POP) and T= Trade costs 5
The effects of size and distance More bilateral trade the larger the countries Less bilateral trade the more the distance between them A and B are predicted to trade more than C and D C A D B 6
Proxies for trade costs Distance Contiguity Common language Colonial links Common currency Island, landlocked country Institutions, infrastructure, migration flows, Tariffs are very often omitted Main reason: endogeneity issue Do tariffs reduce trade or does more trade induce tariff reductions? (reverse causality problem) 7
b. Naïve gravity estimation Naïve estimation of the gravity regression is ln( Trade ) 1 ln( GPDi ) 2 ln( GDPj ) 3 ln( dist ) This regression fits the data very well R-squared of 0.7 in cross-section data ( dataset) However this naïve version can lead to very biased results Serious omitted variable bias: any i- or j- characteristic that correlates both with trade and GDP ends up in the error term. The basic OLS assumption of orthogonality between the error term and the explanatory variables is violated 8
c. Theoretical foundations Deardorff (1998) I suspect that just about any plausible model of trade would yield something very like the gravity equation Baldwin and Taglioni (2006) The emergency of the new trade theory in the late 1970s and early 1980s started a trend where the gravity model passed from having too few theoretical foundations to having too many 9
Major contributions Anderson (1979) Armington hypothesis (goods differentiated by country of origin) Bergstrand (1990) Monopolistic competition Price indices are those used in practice, not suggested by the theory Anderson and Van Wincoop (2003) Monopolistic competition No complete account of price indices => practical way to estimate gravity coefficients in cross country Helpman et al. (2008) Trade model with heterogeneous firms: structural treatment of zeros See also Head and Mayer (2013) 10
Deriving the gravity equation a. Step 1: the (Dixit-Stiglitz) demand function x p Yj 1 Pj (1) where i indexes exporter, j indexes importer LHS = nominal demand by j s consumers for i s goods Y j is j s nominal income p is imports price P j p i 1 1 1 (2) is the ideal CES price index in j, σ > 1 is the elasticity of substitution across varieties 11
Deriving the gravity equation (ct d) (1) can be rewritten in terms of value: p x T Y j p P j 1 (3) Equation (3) could be estimated directly, but researchers often lack good data on trade prices b. Step 2: adding the pass-through equation p p t i (4) Where p i is the producer price in country i and t are all trade costs 12
Deriving the gravity equation (ct d) c. Step 3: Market clearing condition (aggregate supply equals aggregate demand) to eliminate the nominal price i T i Where Y W is world nominal income and θ i Y i /Y W is the share of i s nominal income in world nominal income Using (3) and (4) into (5) we obtain: Y i j Y Y W 1 1 i p i i j T Y W (5) (6) In this expression, i j t Pj is the multilateral resistance (openness of i s exports to world markets) 1 j 1 1 (7) 13
Deriving the gravity equation (ct d) We now want to get rid of producer prices Substitute (6) back into (3) and use (4) to obtain: T YY i Y W j ip j 1 (8) Theoretically-founded gravity equation Major contribution of Anderson and Van Wincoop (2003): bilateral trade is determined by relative trade costs Using (6) into (2), one can derive the price index as: P j j i 1 i 1 1 Equations (7) and (9) can be solved for all P s and Ω s (9) 14
Deriving the gravity equation (ct d) Anderson and Van Wincoop (2003) show that if trade costs are symmetric (τ = τ ji ), Ω i = P i and the gravity equation (9) becomes: X YY i Y W j t i PP j 1 (10) Taking dist as proxy for t, rewrite (10) to look like as Newton s physical law of gravity: bilateral YY i j trade G ( dist ) 1 Where G (1/P i P j ) 1-σ is not a constant as it is in the physical world, but rather specific (also t if we have time variation) 15
Deriving the gravity equation (ct d) In the gravity literature it is in general assumed that trade costs take the form: t d 1 exp 2cont 3lang 4ccol 5col 6llock 7 PTA Where d is bilateral distance, and cont, lang, ccol, col, llock, RTA are dummy variables denoting respectively whether the two countries have a common border, common language, common colonizer, whether one was a colony of the other at some point in time, whether one of the two is a landlocked country, whether the two countries are member of a PTA 16
d. Mistakes to avoid Gold medal mistake: estimating the gravity equation omitting the unconstant term G Omitted variable bias, since Ω i and P j (in error term) include t (in matrix of explanatory variables) Other common mistakes include: Silver medal mistake: averaging the reciprocal trade flows (exports from i to j and exports from j to i) before taking logs (theory tells that one should take the average of logs, not the log of the average, of X and X ji ) Bronze medal mistake: deflating the nominal flows by US price index Gravity is an expenditure function allocating nominal GDP into nominal imports. If use US GDP deflator, need to include time dummies to control for inflation 17
e. Estimating theoretically-founded gravity equation Log-linearization of (8) yields: ln( x ) ln( Y ) ln( Y i j ) ln Y W 1 ln 1 ln i 1 ln Pj Multilateral trade resistance (MTR) terms Ω i and P j not observable Three ways to take MTR into account: 1. Use an iterative method to solve MTR as function of observables (see Anderson and Van Wincoop, 2003) or linear approximation of MTR (Baier and Bergstrand, 2009) 2. Proxy MTR using remoteness REM (trade/gdp weighted average distances from the rest of the world) dist REM i GDP /GDP 3. Fixed Effects (FE) j j W 18
Fixed effects estimation Importer (exporter) fixed effect is a (0, 1) dummy that denotes the importer (exporter) Fixed effects control for unobserved characteristics of a country, i.e. any country characteristic that affect its propensity to import (export) They are used to proxy each country remoteness They do not control for unobserved characteristics of pair of countries (need country-pair fixed effects for this) Cross section vs. panel datasets 19
Fixed effects estimation in a cross section In cross country analysis MTRs are fixed. Therefore, using country fixed effects yields consistent estimation x g ) ( i j 1 Where g is a function π i denotes the set of exporter dummies χ j denotes the set of importer dummies τ are bilateral trade costs There are 2n dummies Total observations = n(n-1) It is impossible to estimate the coefficient for GDP and other countryspecific variables (they are perfectly collinear with the country fixed effect) 20
Fixed effects estimation in a panel At the very minimum, we can now include time dummies (γ) and estimate coefficients on GDPs: x t g GDP ) ( i j t 1 t 1GDPit 2 jt t Here There are 2n country-specific dummies Total observations = n(n-1)t It is still not possible to estimate time-invariant country-specific characteristics (e.g. island, landlockedness ) 21
Fixed effects estimation in a panel (ct d) We can include pair fixed effects (μ) to control for all possible timeinvariant bilateral frictions: x t g GDP ) ( t 1 t 1GDPit 2 jt t We cannot estimate coefficient for distance, common border, common language (they do not vary over time) Still, MTRs vary over time. So the gold standard specification is: x t g ) ( it jt 1 t t (11) This specification uses pair, importer-time and exporter-time fixed effects and identifies the effect of bilateral time-varying variables This effect can be given a causal interpretation, as explained by Baier and Bergstrand (2007) 22
How to proceed? Sensitivity analysis, test the robustness of the results to alternative specifications of the gravity equations Report the results for the different equations estimated 23
Interpretation of results Most of the variables are expressed in natural logarithms, so coefficients obtained from linear estimation can be read directly as elasticities The elasticity of trade to distance, for instance, is usually between -1 and - 1.5, so a 10 per cent increase in distance between two countries cuts their trade, on average, by 10 to 15 per cent Elasticities with respect to importing-country GDPs are also typically unitary, suggesting unitary income elasticities of imports at the aggregate level 24
Interpretation of results (ct d) The coefficients for the dummies (e.g. common border) are not elasticities They need to be transformed as follows to be interpreted as elasticities: p exp( ) 1 where p is the % change in the dependent variable and δ is the estimated coefficient of the dummy variable To derive this formula: Consider that lnx (1) is the predicted value of trade when the dummy = 1 while lnx (0) is the value of trade when dummy = 0 The difference lnx (1) - lnx (0) = δ X (1) / X (0) =exp(δ), which in turn implies that the percentage change in trade value due to the dummy switching from 0 to 1 is: X (1) -X (0) / X (0) = exp(δ)-1 You can do better and estimate p = exp [δ] and (almost) unbiased exp [ 1 2 var δ ] 1, which is consistent 25
f. Recent developments Recently, Helpman et al. (2008) (HMR) derive a gravity equation from an heterogeneous firms model of trade The importance of this derivation relates to three issues that previous models of trade could not explain Zero-trade observations Asymmetric trade flows The extensive margin of trade: more countries trade over time 26
Incidence of zero trade 27
Extensive margin of trade Trade growth in the extensive margin is minor 28
How to handle zero trade data? a. Traditionally When taking logs, zero observations are dropped from the sample. Then, the OLS estimation is run on positive values Take the log(1+x ), but then use Tobit estimation as the OLS would provide biased results (distribution is censored at zero: all observations possibly negative are identified as equal to zero) 29
How to handle zero trade data? (ct d) b. More recently (ct d) An approach is to use (Pseudo) Poisson maximum likelihood (ML) estimator. This method can be applied on the levels of trade, thus estimating directly the non-linear form of the gravity model and avoiding dropping zero trade Santos Silva and Tenreyro (2006) highlight that, in the presence of heteroskedasticity (as usual in trade data), the PPML is a robust approach This approach has been used in a number of estimation of gravity equations See the log of gravity webpage for details, FAQs, Stata codes and references 30
The gold standard with PPML With PPML, the gold standard gravity equation (11) is written as: xt ) exp( it jt 1 t t 31
The importance of intra-national trade flows To ensure consistency with the gravity theory, estimations should include, if possible, intra-national trade data (Heid, Larch and Yotov, 2015) 32
Sluggish adjustment to trade policy changes It is natural to expect that the adjustment of trade flows to trade policy changes may not be instantaneous Accordingly, Trefler (2004) criticizes trade estimations pooled over consecutive years To avoid this critique, researchers have used panel data with 3- or 4- or 5- years intervals Olivero and Yotov (2012) show that gravity estimates using 3- or 5-years intervals are very similar and confirm that estimations performed with panel samples pooled over consecutive years produce suspicious trade cost parameters 33
Applications Estimate / evaluate the impact of trade barriers: Direct estimation: influence of RTAs, tariffs, exchange rate volatility Estimate parameters of trade model () Measuring border effects Proxies of trade costs: influence of distance, cultural proximity (language, colonial links, migrations, etc...) 34
Meta-analysis (Head and Mayer 2013) 35