Structural Gravity with Dummies Only: Constrained ANOVA-type Estimatioon of Gravitty Panel Data Models Peter Egger and Sergey Nigai

Transcription

1 Structu ural Gravity with Dummies Only: Const rained ANOVA-type Estimation of Gravit ty Panel Data Models Peter Egger and Sergey Nigai

2 Structural Gravity with Dummies Only: Constrained ANOVA-type Estimation of Gravity Panel Data Models Peter Egger ETH Zurich CEPR, CESifo, GEP, WIFO Sergey Nigai ETH Zurich, CESifo March, 2014 Abstract The measurement of trade costs and their effects on outcome is at the heart of a large quantitative literature in international economics. The majority of the recent significant contributions on the matter employs exporter and importer fixed country-specific effects and specifies bilateral trade costs as a function of observable variables to estimate cross-sectional, multi-country gravity models. In such models, exporter and importer country-specific effects have a structural interpretation and can be represented as nonlinear functions of bilateral trade costs and of each other. We show that measurement errors about bilateral trade costs lead to bias in the estimates of the countryspecific fixed effects as structural model variables. This paper proposes a relatively simple remedy to this problem: estimation of a three-way fixed effects model with panel data which can be estimated on as few as two years and does not require any information on determinants of bilateral trade. We dub this approach a constrained analysis-ofvariance gravity model as it estimates the variation in bilateral trade flows (aggregate or sector-level) as a function of exporter-time, importer-time, and exporter-importer fixed effects with a suitable set of nonlinear structural model constraints on them. This model leads to largely different estimates of trade costs and country-specific structural variables from parameterized cross-sectional models and outperforms the latter in a real data example as well as in Monte Carlo simulations. We also illustrate that the change in bilateral trade flows over, e.g., a six-year period is extremely well predicted by the change in bilateral trade costs alone when estimating them by the advocated approach in comparison to traditional models. Keywords: Gravity models; Structural general equilibrium models; Fixed effects estimation; Panel econometrics JEL-codes: F14: C23. Corresponding author: ETH Zurich, Department of Management, Technology, and Economics, Weinbergstr. 35, 8092 Zurich, Switzerland; egger@kof.ethz.ch; ETH Zurich, Department of Management, Technology, and Economics, Weinbergstr. 35, 8092 Zurich, Switzerland; nigai@kof.ethz.ch. Egger acknowledges funding by Czech Science Fund (GA ČR) through grant number P402/12/0982. The authors acknowledge numerous helpful comments on earlier versions of the manuscript by Richard Baldwin, Joseph Francois, Marcelo Olarreaga and participants at the 5th Workshop in International Trade at Villars, the 2nd Meeting on International Economics at University of Castellón, a research workshop at CAGE at University of Warwick, a research seminar at University of Glasgow, a research seminar at the IES at Charles University, and at the Annual Conference of the Austrian Economic Association. 1

3 1 Introduction The gravity equation is undoubtedly the most popular tool in empirical international economics. Main reasons for this are its straightforward implementation and its close fit to the data. However, while ad-hoc estimation is easy, it provides little value in terms of predicting how trade (or other economic outcomes) would change in response to a counterfactual shock of, e.g., trade costs. On the other hand, structural gravity models tend to have less explanatory power than many ad-hoc models do. The reason is that they impose nonlinear constraints, which may lead to a loss of explanatory power. In particular, we show that a parametrization of trade costs as is commonly pursued in structural as well as ad-hoc gravity work leads to sizable measurement error which is exacerbated by the nonlinear model constraints in structural models. As a consequence, both the estimates of trade costs as well as country-specific fundamental variables may be biased, and so may be the associated comparative static effects. We propose an empirical model which rests on two pillars a large set of fixed effects and a set of constraints so that the model is structural which overcomes this fundamental problem. In contrast to structural gravity models with calibrated trade costs, this approach is capable of delivering standard errors to trade costs, country-specific fundamentals, and even comparative static effects. Estimating structural gravity equations with fixed effects has become standard in the recent literature (see Head and Mayer, 2013, for an overview). Following Harrigan (1996), Eaton and Kortum (2002), and Feenstra (2004), researchers now control for exporter- and importer-specific variables via fixed effects and parameterize the unobservable bilateral trade costs as a function of observable country-pair-specific variables such as geographical distance and a set of indicator variables reflecting other aspects of cultural, geographical, historical, or political proximity (contiguity and common official language are prominent examples thereof). This approach is commonly judged to provide a good fit to data on aggregate (and even sector-level) bilateral trade flows, the estimates of trade costs are used for counterfactual analysis, and the estimated country-specific constants are interpreted as estimates of structural model variables in the observed benchmark equilibrium (see Anderson and Yotov, 2010, 2012; Fally, 2012). 1 This paper demonstrates that there is a nontrivial gap between the data and structural model predictions of bilateral trade flows in traditional approaches based on country-specific fixed effects and parameterized trade costs. By design, this gap roots in the measurement error of countrypair-specific factors. The latter may be interpreted as a combination of iceberg trade costs and preference parameters, since the two concepts exert an isomorphic impact on bilateral trade. The paper shows in a Monte Carlo simulation analysis, where the true model parameters are known to the researcher, that even a random measurement error of bilateral trade costs of a magnitude as 1 Dekle, Eaton, and Kortum (2007) and Arkolakis, Costinot, and Rodríguez-Clare (2012) propose calculating comparative static effects of trade costs on trade flows and welfare based on observed data on which the model structure is imposed. However, such quantitative analysis can not inform us about changes in specific trade costs, since we do not know how specific elements (such as free trade area membership) relate to total trade costs. Hence, structural estimation will remain an important strategy. 2

4 found in real-world data leads to biased estimates of the endogenous, country-specific fundamentals. Hence, we may speak of a case of biased (and inconsistent) country-specific fixed effects as estimates of country-specific structural variables. The root of the bias is the combination of measurement error about bilateral trade costs (or preferences) and the nonlinear empirical model structure. A consequence of this problem are biased model predictions and comparative static effects of trade costs. The paper proposes a remedy to the aforementioned problems by way of a structural constrained analysis-of-variance model. This model requires three ingredients only: panel data on bilateral exports or imports; a full set of fixed exporter-time, importer-time, and country-pair effects; and a full set of structural constraints so that the fixed effects estimator becomes a structural model. 2 It provides estimates of structural variables consistent with a widely-used class of isomorphic models of international trade (see Arkolakis, Costinot, and Rodríguez-Clare, 2012) such as the ones of Eaton and Kortum (2002), Anderson and van Wincoop (2003), Melitz (2003), or Bergstrand, Egger, and Larch (2013). Bilateral trade flows are specified as an exponential function of exporter-time, importer-time, and exporter-importer-specific fixed effects (see Santos Silva and Tenreyro, 2006). In short panels, where trade costs can be safely assumed to be time-invariant, this estimation strategy reduces the unexplained variation in bilateral trade flows drastically. However, unlike model calibration, it the approach does not assume that the measurement error about trade flows themselves (and, consequently, of the theoretical model) is zero which is inconsistent with the notion of statistical discrepancy in GDP accounts as is the case with calibrated models. As a consequence, one may obtain not only model-consistent parameters but also standard errors (including ones on comparative static effects of trade costs). The virtues of the proposed approach are the following. First, it does not require observable measures of trade costs (or preferences) such as cultural, geographical, historical and political proximity or distance to be orthogonal or even log-additive (or exponentially-multiplicative) to the unobservable component of bilateral trade costs. Hence, observable measures of time-invariant proximity or distance in a wide sense may be generally endogenous. Second, the approach can be readily used with either log-transformed or exponential-family models of bilateral trade. Third, despite well-known incidental parameters problem, the advocated model can be estimated consistently. The reason is that the country-pair component in bilateral trade flow panel data is dominant and imposition of structural model constraints appears to help removing the incidental parameter bias to a large extent as is illustrated in a Monte Carlo simulation based on a model world. The Monte Carlo analysis also shows that even a random, log-additive (exponential-multiplicative) measurement error of bilateral trade costs of a magnitude consistent with reality leads to biased bilateral trade costs and biased country-specific fundamentals as measured by exporter and importer fixed 2 Earlier research has proposed three-way fixed effects which did not impose general equilibrium constraints on the fixed effects (see Baltagi, Egger, and Pfaffermayr, 2003; Baldwin and Taglioni, 2006). Such models are linear in parameters whereas the proposed model is nonlinear in parameters due to the structural constraints. Hence, earlier estimates could not have been used to obtain structurally consistent measures of bilateral trade costs and country-specific fundamentals of bilateral trade. 3

5 effects in a cross section. The direction and magnitude of the bias of country-specific fundamentals in the simulations is similar to the one found in the data. Finally, the proposed approach outperforms structural cross-section models in out of sample prediction. We show this by taking the proposed panel data model and estimating it on two-year panels in and We take the change in trade cost estimates and then predict the corresponding change in trade flows in general equilibrium (where country-time fixed effects are endogenous and change in response to changes in trade costs), using an approach with parameterized trade imbalances consistent with Dekle, Eaton, and Kortum (2007). Based on this analysis, we find that changes in trade costs or preferences alone predict about 70% of the variance in changes of trade flows over the six-year period between 2000 and Crosssectional, parameterized structural gravity models as commonly estimated predict only about 10% of the change in trade flows by a change in trade costs, leaving a large fraction of the change unexplained. This evidence on the importance of changes in trade costs or preferences for trade is also consistent with a much larger variation of changes in trade across countries and country pairs than would be attributable to changes, say, in employment or population and in technological progress or changes in average productivity of exporters across country pairs due to a change at the extensive country-pair margin of trade. The remainder of the paper is organized as follows. Section 2 outlines the general empirical framework, discusses the fundamental problem with estimating structural gravity equations from crosssectional data with parameterized trade costs, and emphasizes the advantages of the proposed panel model approach. Section 3 illustrates the issue in an application of the proposed approach in comparison to traditional frameworks with data on bilateral trade among 31 OECD countries in the period Section 4 estimates the measurement error about trade costs by the gap between parameterized trade costs as in cross-sectional models and estimated ones as in the proposed model and analyzes the consequences of a measurement error of that magnitude on trade costs and country-specific variable estimates in a Monte Carlo simulation exercise. It illustrates that such a measurement error leads to biased country fixed effects in cross-sectional models, whose size and direction compares well with insights from the comparison of various models estimated on real-world data. Section 5 illustrates how well the proposed model can predict trade flows out of the sample in based on estimated changes in trade costs alone, and how largely this differs from traditional models. Section 6 outlines two straightforward extensions, and the last section provides a brief conclusion. 2 Structural gravity models with fixed effects 2.1 Some general notation on trade costs and country-pair-specific preferences In the class of bilateral trade models which dominates empirical and quantitative work in international economics, country-pair-specific trade costs and country-pair-specific preferences have an 4

6 isomorphic impact on trade flows. Since these factors cannot be discerned empirically, we will speak of them as trade costs and generally denote all ad-valorem trade costs on imports of country i from j at time s in logs as τ ijs. The direct elasticity of bilateral imports of i from j at s, X ijs with respect to trade costs is β<0. The interpretation of β depends on the underlying theoretical model. In Ricardian models of comparative advantage, it is a measure of the dispersion of technology across firms in the exporter country, and in the new trade theory models based on the monopolistic competition it reflects the degree of competition and the elasticity of demand for (and substitution between) differentiated varieties of products contained in the aggregate. 3 We could think of βτ ijs as the imports in log differences which are forgone by the presence of τ ijs 0. We will use δ ijs βτ ijs for the sake of brevity. Since δ ijs and τ ijs are unobserved, they are typically parameterized as a log-additive (weighted) function of observable variables such as log bilateral distance and indicator variables reflecting cultural, economic, geographical, historical, or institutional proximity or distance. With a few exceptions such as tariffs, trade agreement membership, or currency union membership, the observable variables underlying the parametrization of τ ijs in the literature are time-invariant, and even the time-varying measures are relatively invariant over short time spans. In general, we may state ( K ) δ ijs = o ijs + χ ijs = α k o k,ijs +χ ijs. (2.1) k=1 } {{ } o ijs where o ijs is the (parameter-)weighted sum of log observable trade costs, and χ ijs is measurement error (see Eaton and Kortum, 2002, for such a specification). This paper is concerned with problems emerging from a difference between δ ijs (true trade costs) and o ijs (specified/estimated trade costs) in structural models. 2.2 Model outline In a class of new trade models often used in quantitative work, aggregate bilateral demand (imports) of country i from country j at time s are characterized by a gravity equation of the following generic form: X ijs =exp(ζ js + δ ijs + μ is + u ijs ), (2.2) where u ijs is a disturbance term which has an unknown cumulative distribution function which we denote by F u (0,σu). 2 The terms ζ js and μ is are exporter-time and importer-time-specific factors, respectively. Their interpretation depends on the underlying theoretical model. For instance, exporter-time-specific factors are associated with the average level of technology in the source country (e.g., in Eaton and Kortum, 2002; Chaney, 2006; Alvarez and Lucas, 2007; Helpman, 3 As already pointed out in the introduction, δ ijs could also measure the influence of preferences of households in i towards goods from j at time s. In that sense, preferences and trade costs display an isomorphic impact on bilateral trade. 5

7 Melitz, and Rubinstein, 2008), the number of exporters (see Krugman, 1980; or Bergstrand, Egger, and Larch, 2013), the endowment with goods of the exporting country (see Anderson and van Wincoop, 2003), and to factor costs per efficiency unit (in all models except for endowment-economy approaches). Importer-time-specific factors are proportional to the size of demand in the importer (aggregate factor endowments and average factor income) and to the ideal price index of goods in the importing country in all aforementioned isomorphic model types. 2.3 Gravity models based on cross-sectional data Most quantitative studies based on a generic demand equation as in (2.2) employ cross-section data in estimation (see Eaton and Kortum, 2002; Anderson and van Wincoop, 2003; Baier and Bergstrand, 2009; Bergstrand, Egger, and Larch, 2013). In this case, time index s in (2.1)and in (2.2) disappears. True trade costs δ ij are distributed as F δ (δ, σδ 2 ) and the observable and unobservable components are distributed as F o (o, σo)andf 2 χ (χ, σχ). 2 We leave it open for now whether o ij is correlated with χ ij (i.e., whether some observable trade costs are endogenous) ( or not. In general K ) and particularly with cross-sectional data, δ ij is commonly replaced by o ij = k=1 α ko k,ij in (2.1), and χ ij is ignored and subsumed in u ij. We argue and illustrate on the basis of panel data that models with parameterized trade costs involve a relatively large measurement error component χ ij. Moreover, it will become clear below that ignorance of this measurement error leads to bias not only of the estimate of δ ij(s) but also of the estimates of ζ j(s) and μ i(s) as well as of comparative static effects of trade costs. Cross-sectional models of (2.2) have the generic form K h X ij =exp ζj h + αk h o k,ij +μ h i + u h ij }{{} k=1 }{{} o h ij u ij +χ h ij, (2.3) where all such models (potentially inadequately) assume that o h ij = δ ij with h {C, A, S}, and C refers to a model which assumes that true trade costs are symmetric and that intra-trade costs are unity, δ ij = δ ji, while models h {A, S} assume that true trade costs may be asymmetric, δ ij δ ji, or non-unity for intra-trade, respectively. Model A does so by including exporter-specific (or importer-specific) binary indicators which are zero whenever i = j (domestic sales) and unity for a given exporter (or importer) otherwise (see Eaton and Kortum, 2002; Waugh, 2010). Model S does so by including country-specific binary indicators which are zero whenever i j (foreign sales) and unity for a given exporter (or importer) otherwise (see Anderson and Yotov, 2010). In any case, the assumption of symmetric trade costs alone may induce quantitatively large errors in the structural estimates (see Balistreri and Hillberry, 2007), so that we would expect σ 2 χ A σ 2 χ C 6

8 and σ 2 χ S σ 2 χ C. We argue and will illustrate that this is indeed the case. 2.4 Gravity and panel data We propose an alternative model to eliminate the measurement error in trade costs or preferences using panel data and country-pair fixed effects by running the following version of (2.2): X ijs =exp ζp js + δij P + μ P is + u P ijs }{{} u ijs +χ P ijs u ijs, (2.4) where δij P = op ij δ ijs is assumed and estimated using country-pair fixed effects with δjj P =0. Estimating (2.4) is possible as long as s>1. With panel data covering a large number of years, allowing for rolling windows of shorter periods of 2-5 years may be advisable to avoid a bias of δij P as an estimate of δ ijs (see Baldwin and Taglioni, 2006). Even over longer periods of time, country-pair fixed effects contribute the lion s share to the explanation of the variance in bilateral trade flows (see Baltagi, Egger, and Pfaffermayr, 2003) so that trade costs can be safely modeled as being constant, at least within reasonably short time windows, such that δ ijs δij P = χp ijs 0.4 We propose estimating (2.4) by imposing general equilibrium constraints using Poisson pseudomaximum likelihood estimation. In general, this strategy would work with any form of exponentialfamily models (see Nelder and Wedderburn, 1972; Cameron and Trivedi, 2005; and Santos Silva and Tenreyro, 2006). Estimates of ζjs P, δp ij,andμp is are sufficient to characterize the benchmark general equilibrium, and the estimates can then be used to conduct comparative static experiments. 2.5 Structural constraints and estimation bias The model in (2.4) receives a structural interpretation only when complying with a set of constraints. Here, we show that when χ ijs χ ij is large, all country-specific effects will be subject to potentially large bias and cannot be interpreted as to have a structural interpretation anymore. Let us start with the general equilibrium identity (i.e., the payments constraint) of any structural gravity model J J = +D js, (2.5) X ijs i=1 }{{} Total sales of j in s i=1 X jis }{{} Total expenditures of j in s 4 In general, since the number of δ P ij to be estimated is almost proportional to N(N 1), there may be an incidental parameter bias. However, this problem is small in the present context for two aforementioned reasons. First, there is a set of nonlinear constraints which reduces the problem. Second, the variance in the data due to the country-pair dimension is large so that the pair-specific fixed effects may be estimated more precisely than with a small country-pair-specific variation. We will illustrate in a simulation section that this is indeed the case. 7

9 where D js is an exogenous trade balance deficit parameter of country j in s (see Dekle, Eaton, and Kortum, 2007). The latter is often set to zero in applications, but this does not have to be the case. Substituting the deterministic part of trade flows from (2.2) in (2.5) obtains: [ J ] [ J ] exp (ζ js + δ ijs + μ is ) = exp (ζ is + δ jis + μ js ) + D js, (2.6) i=1 i=1 which is equivalent to: [ J ] i=1 exp(ζ js )= exp (ζ is + δ jis + μ js )+D js J i=1 exp (δ ijs + μ is ) (2.7) or [ J j=1 exp(μ is )= exp (ζ ] is + δ jis + μ js ) D is J j=1 exp (ζ. (2.8) js + δ ijs ) It is immediately clear, when substituting δ ij(s) by o ij(s) from (2.1) in (2.7) and (2.8), that the measurement error χ ijs may lead to a significant bias in the estimation of ζ js and μ is,andtheir further interpretation as structural variables of the gravity model. The reason for this is that the right-hand sides of (2.7) and (2.8) are nonlinear in χ ijs so that the bias results from Jensen s inequality. There are two more structural constraints that the estimates of ζ js, μ is,andδ ijs are commonly assumed to satisfy. (i) The level of log trade costs must be nonnegative: τ ijs 1 δ ijs 0. (2.9) (ii) In some work (as the proponents of Models {C, A, S} above), the marginal transaction costs to intranational sales are assumed to be zero: τ ijs =1 δ ijs =0. (2.10) Notice that, unless constraints (2.7)-(2.10) are lax (i.e., fulfilled by an unconstrained model), the parameters of interest (ζ js, μ is,andδ ji ) in (2.4) will be biased. Moreover, if the estimate of δ jis is biased, the bias will feed into μ is and ζ js through (2.7) and (2.8). We will illustrate below that this is likely the case with models that parameterize trade costs and simply replace δ ijs by o ijs (in panel or cross section, such as Models C, A, ands). 8

10 2.6 Comparative static effects of trade costs in general equilibrium Let us use Δ to denote relative changes of all variables between a counterfactual and a benchmark (observed or estimated) situation. If the underlying variable is in levels, as, e.g., X ij(s), Δ refers to a ratio of the underlying variable. If the underlying variable is in logs, as, e.g., ζ j(s), δ ij(s),orμ i(s), Δ refers to a (log)difference. What is of interest here are effects of specific changes of estimated trade costs, Δô h ij(s) Δδ ij(s) withh {C, A, S} and Δˆδ ij(s) P Δδ ij(s), each of which are taken as estimates of Δδ ij(s).thetermsδˆζ j(s) h and Δˆμh i(s) can be recovered from the following set of 2J equations: x ij = [ J ] exp(δˆζ j h ) 1 β X lj = l=1 X ij k=j k=1 X, Δˆμ i = ik ( l=j l=1 x lj exp k=j k=1 x lk exp exp(δˆζ i h) 1 β k=j k=1 X ki ( ), (2.11) k=j k=1 x ik exp Δˆζ k h +Δˆδ ik h ) Δˆζ j h +Δˆδ lj h [ ] k=j ( ) exp(δˆζ h Δˆζ k h +Δˆδ lk h l ) 1 β X kl + D l, (2.12) k=1 where β is the (direct) elasticity of trade (with respect to ad-valorem trade costs). Clearly, (2.7) and (2.8) as well as (2.11) and (2.12) indicate that the country(-time)-specific effects as well as their changes while often treated as constant not only in estimation but in comparative static analysis are generally endogenous to trade costs (and measurement error) as captured by pair-specific fixed effects in the proposed approach. 3 Estimation results In this section, we compare Models {C, A, S, P } in terms of their fit to the data and consistency with the structural constraints. For that, we use data on 31 OECD countries. More precisely, we use bilateral exports of manufactures (measured inclusive of cost, insurance, and freight) from trade statistics, construct domestic sales by using manufacturing production together with total manufacturing exports, and parameterize the observable trade cost function in terms of six binary variables reflecting sextiles of the distance distribution 5 and six binary variables reflecting economic, geographical, institutional, and historical similarities: land adjacency, common language, colonial relationship, legal origin, regional trade agreement, and common currency. 6 Naturally, the sample 5 Hummels and Hillberry (2008) point out that the parameter on log distance may vary across different intervals. See already Eaton and Kortum (2002) for an approach that is similar to the one adopted here. 6 The data on bilateral trade flows in manufacturing (and production) come from the OECD s STAN Database. The data-set includes all OECD countries as of 2013 except for Belgium, Iceland, and Luxembourg. We leave them out either due to data unavailability and/or because of special treatment by the OECD statistics (for example Belgium and Luxembourg tend to be bundled together). We also had to impute production output of the manufacturing sector for Australia, Chile, and Turkey from the data on value added from the World Bank s World Development Indicators database. All data on observable bilateral trade cost variables employed here come from the Centre d Études Prospectives et d Informations Internationales. 9

11 Table 1: Trade Cost Function Parameter Estimates in Cross-section Models (s = 2005) Model C Model A Model S coeff. (α k ) s.e. coeff. (α k ) s.e. coeff. (α k ) s.e. Land adjacency α Common language α Colonial relationship α Legal origin α Regional trade agreement α Common currency α Distance [0, 375km) α Distance [375km, 750km) α Distance [750km, 1500km) α Distance [1500km, 3000km)α Distance [3000km, 6000km) α Distance [6000km, ) α Standard errors are robust to an unknown form of heteroskedasticity. of countries could be easily extended, however, since we account for unbalanced trade this is of minor importance, here. For illustration, we estimate the parameters for the period with panel regression. Together with the calculated intranational trade flows or sales, this gives a sample with =1, 922 observations. To highlight the differences between the proposed panel approach and the traditional cross-sectional estimates, we take the year 2005 as our reference period. Accordingly, cross-sectional regressions are estimated using = 961 observations for s = Both the cross-section and panel models obtain estimates of parameters and trade flows for s = 2005, and the latter may be compared with the data. In all regressions, we include a grand constant and use ζ 1s = 0 as a numéraire, whereby the constant measures ζ 1s and the other exporter fixed effects measure deviations from it. Notice that there are as many balance-ofpayments constraints as there are importers and time periods. Clearly, this flows from the fact that the importer-time-specific variables are fully determined by ζ js and δ ij. In the panel regression we recover 2N 2 (N + 1) fixed effects for ζjs P and μp is and (N 1) N fixed effects for δp ij,after imposing the aforementioned constraints. In cross-sectional regression, we recover 2N fixed effects for ζj h and μ h i and twelve coefficients on bilateral trade costs variables for Models h {C, A, S} plus N exporter-specific trade cost variables ξij h for Models h {A, S}. In Table 1, we report the estimated trade cost function for Models {C, A, S} and parameters on land adjacency (α 1 ), common language (α 2 ), colonial relationship (α 3 ), legal origin (α 4 ), regional trade agreement membership (α 5 ), common currency (α 6 ) and on six variables that capture potentially different effects of log distance (α 7, α 8, α 9, α 10, α 11, α 12 ), at the bottom. To ensure that the poor fit of cross-sectional models at relatively low trade volumes is not explained by that phenomenon, we specify six different distance intervals with o k,ij =1fork =7,..., 12 if the distance between i and j falls into the interval [0, 375km), [375km, 750km), [750km, 1500km), [1500km, 3000km), [3000km, 6000km), [6000km, + ), respectively. 10

12 The reported pseudo-r 2 value is calculated for each model as the squared correlation between (log) observed and (log) predicted trade flows in each panel of the figure. Figure 1: Fit of the models for year s = Second, we examine the overall fit of the four discussed models to the data. Since we estimate the parameters by Poisson pseudo-maximum likelihood on the exponential model version, we may not compute a regular R 2 but resort to graphical inspection and pseudo-r 2 measures. A comparison of the estimated log-linear index underlying the exponential model and the log-transformed data is provided in Figure 1. 7 Apparently, the fit of the four models to the data varies considerably with the specification of trade costs. Unsurprisingly, Model C exhibits the lowest pseudo-r 2 = Hence, the assumption of log trade costs to be proportional to a weighted sum of log bilateral geographical distance and the aforementioned six indicator variables (land adjacency, common language, colonial relationship, legal origin, regional trade agreement, and common currency) as well as the assumption of symmetry of trade costs (τ ijs = τ jis ) likely leads to high measurement error in trade costs. As we pointed out in Section 2.3, Models A and S produce the same predictions of trade flows. Both models predictions fit the data better than Model C as they partially account for the heterogeneity in trade costs and 7 Naturally, zero trade flows are excluded from the analysis, here. However, this is not a problem for two reasons: (i) they are almost non-existent with trade flows among OECD countries, and (ii) a proportional number of zeros is taken care of in the estimated model, and the log-transformation purely serves the exposition. 11

13 reduce the variance of the measurement error term. However, there are still considerable deviations of the predictions from the data which result in a pseudo-r 2 = The model based on panel data, on the other hand, almost entirely eliminates the wedge between the observed data and the model s prediction with a pseudo-r 2 1. All four models fit the data well for large trade flows. An important difference between the advocated model based on panel data and the parameterized trade-cost models based on cross-sectional data is that the latter three largely overpredict small and medium-sized trade flows. In order to scrutinize on this difference, we divide the data in ten deciles, d = {1,...,10}, and calculate the pseudo-r 2 and the variance of the residuals for each decile. As we are unable to distinguish between the measurement error in trade costs, χ ijs, and the residual term, u ijs, without making further assumptions, we calculate the variance of the composite error term, which we dub σχ+u. 2 We report the results in Table 2. Table 2: Estimation Results (s = 2005) Decile of Model C Model A Model S Model P X ij,2005 Pseudo-R 2 σχ+u 2 Pseudo-R 2 σχ+u 2 Pseudo-R 2 σχ+u 2 Pseudo-R 2 σχ+u 2 d= d= d= d= d= d= d= d= d= d= Reported pseudo-r 2 values are calculated as the squared correlation between (level) observed trade flows and (level) predicted trade flows. The estimated bilateral trade costs for Model P, δ p ij, are reported in Tables 9 and 10 in the Appendix. There are two main striking insights from Table 2. First, the models based on cross-sectional data match the trade data in levels well only in the highest decile, d = 10, where the corresponding pseudo-r 2 values are close to unity. In some deciles, the pseudo-r 2 is lower than 1%, pointing to an extremely poor fit of the models to the data at those deciles. This issue shows also in the differences in the error variance terms, σ 2 = σ 2 u h χ h +u, across deciles.8 For the lowest decile, d =1, the calculated variance of the error terms of Model C and each Models A and S are almost 4,000 and 250 times, respectively, larger than the variance of the error term of Model P. The differences become smaller for higher deciles. However, even for the highest decile, d = 10, the variances of the error terms of Model C and Model {A,S} are larger than the corresponding variance of Model P by factors of almost 62 and 4, respectively. We argue that the main reason for such large differences between the models lies in the specification of bilateral trade costs. 8 The fact that σ 2 u h is larger for bigger deciles across all models indicates that the data are heteroskedastic. 12

14 Table 3: Estimation Results (s = 2005) Model C Model A Model S Model P ˆζC j s.e. ˆμ C j s.e. ˆζA j s.e. ˆμ A j s.e. ˆξA ij s.e. ˆζS j s.e. ˆμ S j s.e ˆξS ij s.e ˆζP j s.e. ˆμ P j AUS AUT CAN CHE CHL CZE DEU DNK ESP EST FIN FRA GBR GRC HUN IRL ISR ITA JPN KOR MEX NLD NOR NZL POL PRT SVK SVN SWE TUR USA Standard errors are robust to an unknown form of heteroskedasticity. 13

15 We interpret the difference in σ 2 between Model P and the cross-sectional models as evidence of u h a large measurement error about trade costs. As argued before, the bias in δ ij(s) is expected to show in the estimates of ζ j(s) and μ i(s) that are estimated as country-year-specific fixed effects in Model P and, with cross-sectional data, as country-specific effects in Models {C, A, S}. Hence, in the corresponding differences in estimates across models are solely attributable to the impact of differences in the underlying estimates of δ ij. Table 3 summarizes point estimates and standard errors about ζ js and μ is for year s = 2005 across all models and confirms that the country-specific fundamentals differ largely across them. Notice that the fixed country-specific effects differ between Model A and Model S since the two models imply a different implicit normalization, however, the predictions of trade flows between them are identical. We conjecture that in cross-sectional models the measurement error about trade costs induces considerable bias in the structural gravity model estimates of not only trade costs but also exporter- and importer-specific characteristics. 4 Monte Carlo simulations 4.1 Estimating the size of the measurement error In this section, we will illustrate that the measurement issues emerging from data as the ones underlying Section 3 are fundamental. For this, we generate cross-sectional data in line with general equilibrium and perform Monte Carlo simulations in a multi-country world where the parameters are known. What is important for the sensitivity of results is the magnitude and nature of the measurement error. By the former we mean the relative size of χ ij in δ ij. This could be measured by the variance ratio σχ/σ 2 δ 2,whereσ2 δ measures the variance of total trade costs δ ij. To shed light on the latter, we regress (log) trade costs as predicted when using Model P on the OECD data in the previous section, ˆδ ij P, net of (log) trade costs as predicted by Models {C, A, S} on a constant as follows: ˆδ ij P ô h ij = constant h χ + χ h ij for h = {C, A, S}, (4.1) ( where ô h ij K ) k=1 ˆαh k o k,ij.thetermχ h ij is the residual to this regression, and the variance ratio ˆσ 2 /ˆσ 2 is a model-specific estimate of σ χ h δ χ/σ 2 2 P δ. We report the results in Table 4: Table 4: Moments of measurement errors in estimated trade costs Model C Model A Model S Coeff. Std.err. Coeff. Std.err. Coeff. Std.err. constant χ ˆσ 2 χ h ˆσ 2 χ /ˆσ 2 h δ P Due to the inclusion of a constant, χ h ij for h = {C, A, S} is centered around zero. The variance of 14

16 the measurement error is relatively highest in Model S with ˆσ 2 = and lowest in Model A χ S with ˆσ 2 = Table 4 also suggests that the ratio of the variance of the measurement errors χ A of Models C, A, ands to the estimated trade costs of Model P equals , , and , respectively. Hence, the variance of trade costs is quite dramatically underestimated in all models with parameterized trade costs (by 39% in Model S and by almost 68% in Model A). Figure 2: Distribution of χ ij The distributions of the residuals ˆχ h ij based on the regressions in (4.1) are displayed in Figure 2 for the three models h {C, A, S}. It suggests that these distributions can be approximated relatively well by normal distributions (represented by the solid Gaussian curves in the figure). We use the findings from Table 4 and Figure 2 to inform the data-generating process underlying the Monte Carlo simulations. In particular, we choose the variance of total trade costs to be σδ 2 = and we assume a measurement error of trade costs which is distributed normally with mean zero and a variance of σχ 2 = We set σχ 2 =ˆσ 2 which is the minimum among the three crosssectional models considered. Moreover, we assume that intranational trade costs are zero and that χ A the minimum of δ ij is zero, in line with the empirical analysis in Section The true nonstochastic model In order to simulate the model, we need to inform it regarding country-specific fundamentals. Specifically, we recast the model in (2.11)-(2.12) as follows in terms of structural variables. Clearly, for the issue of interest and due to the isomorphism of Eaton and Kortum s model with an endowment-economy model as in Anderson and van Wincoop (2003) or a multi-country Krugmantype model as in Bergstrand, Egger, and Larch (2013), the choice of exact micro-foundations is not crucial here. Let the total endowment of labor in any country be identical to exp(10) and let the first moment of the productivity distribution across firms in the model in any country be unity. 15

17 Then, we can generate trade flows according to the following system of non-linear equations: μ i =ln exp(ζ i) 1 β exp(10) l=j l=1 exp (ζ l + δ il ) ; X ij =exp(ζ j + δ ij + μ i )= J X ij = j=1 exp (ζ j + δ ij ) l=j l=1 exp (ζ l + δ il ) ( ) exp(ζ i ) 1 β exp(10), (4.2) ( ) exp(ζ i ) 1 β exp(10), (4.3) where we assume β = 6.5 which is consistent with Costinot, Donaldson, and Komunjer (2012) who use productivity data to estimate β for OECD countries. Due to the assumptions of homogeneous productivity and labor endowments, trade costs are the only source of heterogeneity across countries and country-pairs. Then, suppose that an econometrician only relies on the observable component of trade costs, o ij, and that she inadequately estimates the following model: ( K ) X ij =exp ζ j + ˆα k h o k,ij +μ i + u 0 ij }{{} k=1 }{{} ô ij u ij +χ ij, (4.4) where o ij are drawn randomly according to the processes described in the next section. Hence, the measurement error in trade costs is the difference between ô ij and δ ij. Based on the estimated counterpart to (4.4), we may predict ˆζ j and ˆμ i and compare them to the true exporter and importer fixed effects obtained from the system in (4.2)-(4.3). We are interested in the size of the differences between true and estimated fixed effects, ζ j ˆζ j and μ i ˆμ i, and whether they can be systematically explained by the country-specific average of the measurement error in trade costs, χ ij = δ ij ô ij. 4.3 Two stochastic processes: exogenous versus endogenous observable trade costs o ij Throughout the analysis, we assume three different values for the number of countries, J {50, 100, 200}, which appear plausible in view of the typical application of cross-sectional gravity models. In the first experiment, we assume that observable trade costs are exogenous, whereby σδ 2 = σ2 o +σχ 2 and σo 2 = σδ 2 σ2 χ = In this experiment, we draw for each configuration J {50, 100, 200} the true trade costs, δ ij, only once so that they are fixed in repeated samples. Then, we draw 1, 000 independent vectors of measurement error with elements χ ij from a normal distribution with mean zero and variance σχ 2 and produce trade costs that are observable to the researcher, o 1,ij. Finally, we estimate all parameters ζ j, μ i,andô ij ˆα 1 o 1ij in 1, 000 regressions per configuration of the world and summarize the results. In an alternative experiment, we examine whether a correlation between observable trade costs 16

18 o 1,ij and unobserved trade costs χ ij (i.e., endogeneity) would change the results. This is important to consider, since the usually applied parametrization of the trade cost function assumes that the function (in logs) is linear in its arguments, and that all its elements are uncorrelated with the unobservable component. Clearly, the strategy proposed in this paper does not rely on such an assumption. In fact, δ ij may be an arbitrary parametric or even nonparametric function of the observable and unobservable trade cost components. There is a host of empirical studies pointing to the endogeneity of various elements in the trade cost function, ranging from distance (see Egger and Pfaffermayr, 2004) over common language (see Egger and Lassmann, 2012) to trade agreement membership (see Baier and Bergstrand, 2007). For that experiment, we assume a moderate average correlation coefficient of 0.10 between o 1,ij and χ ij and randomly draw the two components (true trade costs and measurement error) from the following bivariate normal distribution: ( ( )) σo 2 0.1σ δ σ χ (o 1,ij,χ ij ) δ, 0, 0.1σ δ σ χ σχ 2 where δ corresponds to the average of ˆδ ij P in the previous section, and σ oσ χ is the product of standard deviations of ˆδ ij P and χ ij. As in the previous case, we draw δ ij and χ ij and solve the model using otherwise identical parameters as in the previous experiment. We generate 1,000 independent draws of the observable o 1,ij use each of them to estimate the respective ô ij ˆα 1 o 1,ij as in equation (4.4). 9 (4.5) 4.4 Simulation results We start discussing our simulation results assuming exogenous trade costs and 200 generic countries. With this configuration, each regression is based on 40,000 data points for each of the 1,000 draws that we perform. In this section, we aim at calculating the bias induced by the measurement error in trade costs in both the estimates of true trade costs and the remaining model parameters. For that we proceed in several steps: Step 1. Use the simulated data to estimate the following regression for each draw k: X ij (k) =exp(ζ j (k)+β o (k)o ij (k)+μ i (k)+u ij ), (4.6) where o ij are simulated observable trade costs that include the measurement error. Step 2. Obtain fitted values of composite exporter and importer fixed effects and trade costs as: ˆf ij (k) =ˆζ j (k)+ˆμ i (k) andˆδ ij (k) = ˆβ o (k)o ij (k). For brevity and convenience, we lump the two effects together because for accurate predictions of trade flows the explicit distinction between them is irrelevant. 9 Hence, in contrast to the previous set of simulations, δ ij is now not fixed in repeated samples. 17

19 Step 3. Compute the average bias and the root mean squared error for b ij = {ln(x ij ),f ij,δ ij } as follows: 1,000 1 N BIAS(b ij )= 1, 000 N 2 (b ij ˆb ij (k)); RMSE(b ij )= 1,000 N (b ij ˆb ij (k)) 2 1, 000 N 2. (4.7) k=1 i,j=1 k=1 n=1 Hence, BIAS(b ij )andrmse(b ij ) calculate bias and root means squared error for trade flows, exporter-importer fixed effects and trade costs for cross-sectional models, respectively. In Section 3, we showed that cross-sectional models with parameterized trade costs tend to result in an upward bias of trade flows. Here, we confirm this observation via simulations. In Table 5, we report bias and root mean squared errors under trade cost measurement error for several quantities of interest. Table 5: Simulation Results ln(x ij) δ ij f ij o ij Countries BIAS RMSE BIAS RMSE BIAS RMSE N = exog. N = N = N = endog. N = N = First, Table 5 suggests that the main results are not sensitive to a particular design of the datagenerating process. Regardless of whether one considers exogenous or endogenous observable trade costs, the exporter and importer-specific fixed effects estimates are biased, and the bias is systematically related to the measurement error in trade costs. Notice that the sum of country-specific fixed effects f ij is upward-biased while δ ij is downward-biased. Altogether, there is considerable bias in the prediction of (log) total trade flows, ln(x ij )=f ij + δ ij. Figure 3 illustrates the sources of the bias in terms of b ij = {ln(x ij ),f ij,δ ij } for one draw of an experiment with exogenous observable trade costs and 50 countries. In the figure, we plot the country-specific average bias in b ij against the average bias in predicted trade flows. Since, by design, E(ln X ij )=E(f ij + δ ij ), the average bias in f ij + δ ij isthesameastheoneinln(x ij ). In any case, an advantage of portraying the results as averages per country is that it is obvious that the biases in δ ij and f ij add up to the one in ln(x ij ). 10 Obviously, there is a systematic relationship between the country-specific average bias in δ ij and f ij and the average bilateral bias in trade flows. Because of the measurement error in trade costs, estimates based on observable trade costs tend to lead to upward-biased trade costs and to downward-biased combined country-specific fixed effects, where the former bias dominates the 10 Clearly, since the models are nonlinear, the center of the data may generally differ from the center of the regression, here. 18

20 Figure 3: Sources of bias in cross-sectional models: Monte Carlo latter, such that in total predicted bilateral trade flows are upward-biased (this can be seen from the negative values of the bias in ln(x ij ) on the abscissa of the figure. 4.5 Model P and the incidental parameters bias One problem of the proposed framework could be that the incidental parameters problem (i.e., that the number of parameters δ ij to be estimated is almost proportional to N(N 1)) mitigates the advantages of nonlinearly-constrained fixed effects estimation relative to relying on observable trade costs. In this section, we will illustrate that this is not the case. The reason is that the variance in the country-pair-specific, time-invariant trade cost and preference effects is large so that the variance in δ ij dominates everything and the individual parameters can be estimated relatively well. The incidental parameters problem would be pertinent if the importance of trade costs relative to importer and exporter fixed effects on the one hand and to the remainder (idiosyncratic) error on the other hand were small. However, as we showed in Section 3 the opposite is true. The variance of total trade costs is much larger than the variance of the fixed country-time effects and of the remainder disturbances together, and it alone explains a large share of trade. We illustrate through simulations that trade cost estimates are not biased severely due to the incidental parameters problem and that Model P provides unbiased estimates not only of trade costs but also of country-time-specific fixed effects and bilateral trade flows even in the presence of small unobservable, exogenous, idiosyncratic shocks to trade flows. To inform the simulation procedure, we use the results from model Model P and calculate the variance of the idiosyncratic error shock, u ijs, from the data on trade flows, X ijs, and the model s prediction, Xp ijs,fortheyears 2005 and 2006 as follows: ( ) X ijs û ijs =ln X p. (4.8) ijs 19

21 The variance of this term is We use it in the simulation procedure as follows: Step 1. Draw an N 2 1 vector of trade costs as δ ij N( δ, σ δ ) and specify its elements being the same for two periods: δ ijs = δ ij. (4.9) Step 2. Draw a 2N 1 vector of exporter-time-specific fundamentals, ζ is : ζ i,1 = ζ i,andζ i,2 = ζ i + abs(v i ), whith v i N(0, 0.01), (4.10) wherethevalueofζ i is taken from the experiments with exogenous trade shocks (in the absence of time) in the previous subsection. Step 3. Draw a 2N 2 1 vector of idiosyncratic shocks to trade flows: u ijs N(0, ), (4.11) where the latter do not affect ζ is or δ ij. We use these simulated data to estimate Model P, predict trade costs, and examine whether this model permits estimating δ ij, combined country-time-specific effects f ijs = ζ js + μ is, and trade flows ln(x ijs ). To keep our simulation exercise close to the empirical exercise in Section 3, we simulate the model for 31 countries. Figure 4: Model P and Monte Carlo simulations Figure 4 illustrates the bias for one draw with exogenous trade costs and 31 countries. In the left panel, we plot estimated country-pair fixed effects, ˆδ ij, against true trade costs, δ ij. In the right panel of Figure 4, we plot b ij1 = {ln(x ij1 ),f ij1,δ ij } for the first year in the generated data against the bias in trade flows in that year, akin to the cross-section data underlying Figure 3. As with real-world data, the estimated trade costs deviate only to a small extent from their respec- 20

22 tive true values. The correlation between the true and estimated trade costs is 0.99, illustrating that the incidental parameters problem is negligible in this example with 50 countries. The right panel of the figure suggests two insights. First, trade costs and exporter-plus-importer country-year fixed effects are predicted without systematic bias. Unlike in Figure 3, there is no obvious systematic relationship between ˆf ij1 or ˆδ ij and the bias in b ij for both variables is centered at zero. This is in contrast to Figure 3 where there was a clear systematic bias. Second, the difference between the estimates and true values of the exporter- and importer-specific effects is orthogonal to the difference in predicted and true trade flows. Hence, we conclude that Model P recovers estimates that are not subject to estimation bias. We also conduct Monte Carlo simulations with 50 countries where we perform 1,000 draws. The calculated BIAS(b ij ) (and RMSE(b ij ) in parentheses) for trade costs, country-time fixed effects and trade flows are negligible relative to the cross-section results and amount to (0.4742), (0.4726), and (0.1311), respectively. 5 Estimated trade cost changes as an out-of-sample predictor of observed changes in trade flows In Section 3 we have demonstrated that the advocated Model P fits the data considerably better than models based on cross-sectional estimates. In this section, we conduct a specific kind of outof-sample prediction exercise to provide further evidence for this claim. Specifically, we estimate all cross-sectional Models {C,A,S} on data for the year 2000 in addition to the year 2005, and we estimate the panel Model P for the period in addition to With all models, we predict the year 2005 as the reference (benchmark) equilibrium, being based on estimated trade costs and country-specific fixed effects (ˆζ js h, ˆμh is, h {C, A, S, P }) for that year according to (2.4) and (2.7)-(2.8). Then, we change the estimated trade costs for each model to the respective level as estimated for 2000 and let ˆζ js h and μh is of 2005 adjust endogenously to the new counterfactual equilibrium, ˆζ h js and μh is, which is consistent with estimated trade costs of 2000 and in line with (2.11)-(2.12). For simplicity, denote log changes between the year 2005 and 2000 in any variable by Δ (i.e., the change is the log of a variable in 2000 minus the log of the variable in Accordingly, ΔX ij is the decline in logs of bilateral exports from 2005 to 2000 in the data, while Δ ˆX ij h are the model-specific out-of-sample back-casts due to changes in trade costs only. The model predictions are based on Δˆτ ij h whichisδˆτ ij P =Δˆδ ij P /β with Model P and Δˆτ ij h =Δôh ij /β with Models {C, A, S}. The terms Δˆζ j h and Δˆμh i can be recovered as outlined in (2.11)-(2.12) in Subsection 2.6. Assuming that β = 6.5, following Costinot, Donaldson, and Komunjer (2012), we obtain the results for Δ ln τij h from the respective estimates of Δˆτ ij h as summarized in Table 6 for all models h {C, A, S, P }. Obviously, there are large differences in the calculated changes in trade costs. For instance, Model S 21

23 Table 6: Calculated changes in trade costs ˆτ ij h = exp(ˆδ ij/β) h from 2005 to 2000 with models h = {C, A, S, P }. Δˆτ ij C Δˆτ ij A Δˆτ ij S Δˆτ ij P Mean Variance implies that the average change in trade costs between 2000 and 2005 was positive but insignificant and amounted to less than 1%. Model C and Model A imply higher and more heterogeneous changes in trade costs than Model S. In Figure 5, we plot Δˆτ ij h corresponding to Table 6 in Models {C, A, S} against Model P. The correlation between Δˆτ ij P and Δˆτ ij h in all Models {C, A, S} is extremely weak. However, several bits Figure 5: Estimated changes in trade costs between 2005 and 2000 of information suggest that the panel results regarding trade cost changes are much more plausible than the ones obtained from cross-sectional structural models. The first piece of evidence relates to European integration in the time period considered in the analysis. For instance, the trade cost changes predicted by Model P are consistent with intensive trade integration between Eastern European countries and the European Union incumbent countries between 2000 and For example, Czech Republic, Estonia, Hungary, Poland, Slovakia, and Slovenia joined the European Union in The estimates of Model P imply that these countries experienced the largest reductions in trade costs in that period within the considered country sample. The estimated reductions in trade costs for these countries were 17.16%, 15.16%, 15.16%, 16.37%, 22.83%, and 12.87%, respectively. These changes are substantially higher than the average reduction in trade costs of 10.96%. The corresponding estimates are 6.34%, 6.09%, 5.89%, 5.79%, 5.99%, 6.28% in Model C ; 16.87%, 13.56%, 13.37%, 15.81%, 19.31%, 14.38% in Model Model A; and %, 14.82%, -2.01%, -1.9%, -0.66%, 1.2% in Model S. We report all estimates for the year 2000 in the Appendix. - Moreover, with estimated changes in ˆδ ij h and induced changes in ˆζ j h and ˆμ h i, we may compute induced changes in log exports from 2005 to 2000 due to changes in ˆδ ij h only, Δ ˆX ij h. In Figure 6, we contrast Δ ˆX ij h with observed changes in log bilateral exports ΔXh ij. Comparing the four 22