Estimating Trade Elasticities Using Product-Level Data

Transcription

1 Estimating Trade Elasticities Using Product-Level Data (Very preliminary, Please do not cite) Juyoung heong, Do Won Kwak, and Kam Ki Tang bstract This paper estimates trade cost (tariff) elasticities using bilateral tariff data at the HS6-digit level for 47 countries from 22 to 2. It extends the Helpman et al. (28) model to incorporate firms fixed costs of exporting that vary at the pair-product level. It uses a multi-way error component model framework to account for high-dimensional unobserved factors at the most disaggregate level (5,3 products) including firm heterogeneity and product level multilateral resistance. Furthermore, a novel use of the within estimator is introduced because conventional within estimator in a multi-way error component model is inconsistent with unbalanced data. The empirical results show that there is substantial upward bias in the estimates of trade cost elasticities in the literature. Properly accounting for zero trade flows and firm heterogeneity using more disaggregated data provide significantly smaller estimates of trade cost elasticities, which imply much larger welfare gains from trade. JEL ode: 23, F4 Keywords: Trade ost Elasticity, Tariff, Unobserved Firm Heterogeneity, Within Estimator School of Economics, University of Queensland, QLD 472, ustralia; j.cheonguq.edu.au School of Economics, University of Queensland,QLD 472, ustralia; d.kwakuq.edu.au School of Economics, University of Queensland, QLD 472, ustralia; kk.tanguq.edu.au

2 Introduction Trade elasticities are key parameters to quantify trade costs and welfare gains from trade. rkolakis et al. (22) show that for a range of trade models, gains from trade can be consistently estimated using two variables, the import-penetration ratio (or domestic share) and the trade elasticity with respect to variable trade cost obtained from a gravity equation. The import-penetration ratio data are regularly compiled by national statistical agencies and readily available across countries. On the contrary, the trade elasticity is not directly observable and, thus, needs to be estimated. The precise estimation of trade elasticities is difficult however, because of data limitations. In this paper, we approximate the trade elasticities using the responsiveness of imports to tariffs changes. We focus on identifying trade elasticities from the changes in tariffs because tariffs suffer less from measurement errors than other variable trade cost elements like transportation cost. 2 Many studies have attempted to empirically estimate trade elasticities (e.g. nderson (979), Harrigan (993), Hummels (2), aier and ergstrand (2), roda and Weinstein (26) and ergstrand et al. (23)). Our estimates for the trade cost elasticities are most comparable to those from roda and Weinstein (26) for the elasticities of substitution, which is typically interpreted as the trade cost elasticities. roda and Weinstein (26) acknowledge that they must make simplifying assumptions on the demand and supply structure to empirically implement their method; as a result, their model does not directly address factors such as firm heterogeneity and other unobserved determinants of firms export market entry decisions. Our estimates are transparent in controlling for those factors. Furthermore, unlike this study most of these studies estimate the elasticities using either prod- rkolakis et al. (22) note that the perfect competition models in nderson and van Wincoop (23) and Eaton and Kortum (22) and the monopolistic competition models in Melitz (23) are all in a broad class of models sharing Dixit Stiglitz preferences, one factor of production, linear cost functions, complete specialization, and iceberg trade costs. They show that if three macro-level restrictions hold, then all four models will share a common estimator of the gains from trade which depends only on the import-penetration ratio and a gravity-equation-based estimate of the trade-cost elasticity of trade flows (of which sv is a measure of in many models). (ergstrand et al. (23), p.) 2 We assume a tariff to be a form of variable trade costs, and changes of tariff will have the identical effect on trade flows as equal changes in any other sources of variable costs such as transportation costs due to fungibility of capital. Given that transportation cost and other costs are typically measured with serious errors while tariffs are measured more precisely in relative sense, focusing trade costs changes form tariff changes is beneficial, especially when they are measured at very disaggregated product level. 2

3 uct or firm-level data for a single exporting country or country-level data for multiple country-pairs (abbreviated as pairs hereafter) due to the limited availability of disaggregate data. On the one hand, using data for a single country cannot account for unobserved heterogeneity at the importer level and that could potentially cause severe bias. On the other hand, using aggregate data is even more problematic because two potentially serious biases could arise in the estimations of trade elasticities. Firstly, at any given time, tariffs vary substantially across products for each country, but data aggregation omits the information on substitution between products, causing severe bias in the trade elasticity estimation. 3 Secondly, unfiltered bilateral trade data typically features a large proportion of zero observations but aggregation can hide these zero flows which has special implication. Information on zero trade flows at the product level is naturally lost with aggregate data and firm s self-selection of exporting at the product level could not be addressed as a result. Previous studies also point out problems associated with zeros observations. For instance, vw (24) and roda and Weinstein (26) illustrate that ignoring zero trade flows in estimating trade cost elasticities in the gravity equation using aggregate data can cause upward bias. Helpman et al. (28) (denoted as HMR hereafter) show that zero trade flows in aggregate data are due to firms self-selection out of foreign/export markets, and suggest a two-stage procedure to account for the bias caused by their self-selection. However, their method is difficult to implement with product-level data, because the exclusion restriction at their first stage estimation require a variable that could determine firms export market entry decision but could not affect the volume of exports. 4 In this paper, we propose a method to estimate trade cost (tariff) elasticities using product level data that can avoid biases from various sources; pile-ups at zero trade flows, heterogeneous pair-product fixed costs of exporting, and product-level multilateral resistance terms (MRTs). For highly disaggregated product-level bilateral trade data, it is crucial to have an estimator that could 3 nderson and van Wincoop (24) (denoted as vw (24) hereafter) demonstrate with a numerical example that the elasticity of substitution at the more aggregated level is entirely irrelevant. (p. 727). They advise that: one should choose elasticities at a sufficiently disaggregated level at which firms truly compete. 4 Recent theoretical studies like haney (28) and Krautheim (22) pay attention to the role of fixed costs in heterogeneous firms entry decision in new export markets, which is empirically supported by Koenig et al. (2). 3

4 deal with the source of zero observations in unbalanced data. This is because product-level bilateral trade data could have more than half of the observations being zero, as countries typically trade only specific products with individual partners. We extend the idea from the HMR approach in controlling for firms fixed costs of exporting to a product level model using various within transformations for unbalanced data in multi-way error components models. Our proposed estimator is attractive because first, it allows us to account for various unobserved factors without the requirement for any exclusion restrictions and second, unlike the conventional within estimator it is consistent even when being directly applied to multiple error components with high dimensional indexes (e.g. importer-product-year, exporter-product-year, importer-exporter-product) for unbalanced data (Wansbeek and Kapteyn (989); altagi et al. (2); Davis (22)). We algebraically show that our proposed within estimator has following main properties with unbalanced data. Firstly, it can remove only one error component completely without causing any inconsistency. Second, the order of demeaning matters for the magnitude of inconsistency in the sense that only the first ordered demeaning could remove one error component completely, while the second and further ordered demeanings could remove error components partially and cause inconsistency. Third, using partially balanced data we could obtain certain within estimators that could remove more than one error component completely (partially balanced data is defined as the data are balanced in only two or three dimensions out of four) data. Therefore, the within estimators with partially balanced data could avoid these biases from unbalanced data as well as computational difficulty in dealing with high-dimensional unobserved factors. Our estimator directly accounts for unobserved product-level heterogeneity using multiple high-dimensional dimensional fixed effects. This paper contributes to the literature in the following ways. Firstly, we construct a new bilateral tariff dataset for 47 countries and about 5,3 products at the HS6-digit level, from 22 to 2 as they are not readily available. Few studies have used product level tariff data for multiple countries to estimate trade cost elasticity. 5 For the case of trade flows, about 7% of the observa- 5 Exceptions are Hummels (2) which uses year 992 data of sectoral imports of six countries and aliendo and Parro (24) which estimate trade elasticities for 2 sectors using year 993 data for 3 countries. 4

5 tions, however, are of a zero value, meaning that many pairs trade only a limited range of products. Disaggregated product level trade data (e.g, SIT 4 or 5 digits, HS6-digit) for multiple countries trading partners are publicly available from various data sources such as WITS and Uomtrade but typically only positive flows are used in the estimation. 6 Secondly, factors important for firms foreign market entry decision, including pair-product level fixed costs and pair-year specific fixed costs, are explicitly addressed following HMR and haney (28). In our dataset, tariffs and trade flows vary across four dimensions, namely: importing country, exporting country, product, and year. This allows us to directly control for (i) unobserved demand factors such as consumer tastes using importer-product-year fixed effects, (ii) unobserved supply factors such as productivity changes using exporter-product-year fixed effects, (iii) unobserved pair-product-specific fixed costs of exporting (such as information costs) using importer-exporter-product fixed effects, and (iv) unobserved pair-time-specific fixed costs (such as political alliance) using importer-exporter-year fixed effects. To our knowledge, our empirical strategy is the most comprehensive in terms of accounting for these four distinct unobserved factors that are important to the estimation of trade elasticities. Our proposed novel use of the within estimator with partially balanced data could also avoid inconsistency and computational difficulties associated with using the conventional within estimator and using a large number of dummy variables for high-dimensional unobserved factors respectively. s a result, we can perform regression analyses with product-level data while avoiding omitted variable bias (OV) caused by product-level MRTs and unobserved fixed costs which arise from information barriers, interestgroup lobbying, and government red-tape, to name just a few. 7 Our empirical results show that ignoring zero observations and any one of the demand factors, supply factors, product-level MRTs or fixed costs at the disaggregated level could overestimate the trade elasticities (i.e. the absolute values of the estimates are smaller than their true values ). 6 See roda and Weinstein (26), aier et al. (24), and Dutt et al. (23) for examples. 7 nderson and Yotov (2 and 22) show importance of product level MRTs to accounting for the demand and supply factors. The HMR and haney (28) show the importance of controlling for fixed costs that is pair-product and pair-time varying unobserved factors which are related to fixed costs due to bureaucratic corruption and inefficiency of trade that is not specific to a sector. 5

6 It implies that welfare gains from trade are much larger when measured with the trade cost elasticity estimated using product-level data including zero observations with proper controls. s an illustration, we provide some examples of welfare effects using trade costs elasticity and importpenetration ratio measures following rkolakis et al. (22) and compare our results and those from previous studies. Furthermore, our results imply that substantial heterogeneity exist in the elasticity estimates across products. We examine elasticity heterogeneity using different market structures of products and input quality in skill levels. The rest of the paper is organized as follows. Section 2 describes the HMR model to introduce an estimable gravity equation at the product level data using multi-way error components model. Section 3 describes the data. In section 4, we explain our empirical model framework and provide the estimation results. The last section provides concluding remarks and policy implications. Simulation and sensitivity analysis for our proposed estimator results are provided in the ppendix. 2 Gravity Model 2. Gravity Model at Sectoral Level In this section, the gravity model is derived for sectoral (or product) level estimations. We adopt the HMR model but extend to multiple sectors, k (k =,2,...K). For aggregate data, K = ; for HS 2-digit data, K = 98; and for HS 6-digit data, K 5,4. s in haney (28) and rkolakis et al. (22), we further assume that consumer s utility function is the obb-douglas for products and Dixit-Stiglitz for varieties within a product. representative consumer in country j ( j =,2,...J) has the following utility function at time t: U jt = K k= ˆ h2h jkt x jkt (h) (s k )/s k dh! qk s k /(s k ) () where x jkt (h) is a differentiated variety h of product k available in j at time t, H i jt is the set of 6

7 varieties available for product k in j s country at time t, s k is the elasticity of substitution between varieties of product k, which is assumed to vary across products with s k > and q k is the share of income Y jt on goods from sector k with K k q k =. lthough consumption can change over time as Y jt changes, for simplicity we assume no borrowing or lending so that the consumer is maximizing consumption period by period instead of in an intertemporal manner. There is also no dynamic in the production side unlike lessandria et al. (24). Then, with country j s income at time t, Y jt which equals the expenditure at time t we obtain country j s demand for variety h of product k at time t from (): x jkt (h)= p jkt (h) s ke jkt P s k jkt (2) where p jkt (h) is the price of variety h of product k in country j at time t, E jkt = q k Y jt is the expenditure by j on goods from sector k from all origins at time t, and P jkt is the price index associated with varieties form sector k in j at time t where P jkt = "ˆ h2h jkt p jkt (h) s k dh # /( sk ) (3) Using firms standard markup pricing equation in the monopolistic competition model and following HMR (28), we essentially have the same outcome for i s imports of product k from j as HMR (28) for each product k in each time t. T ijkt = c jkt t sk ijkt E ikt jktv ijkt (4) a k P it where T ijkt is the import of product k by country i from country j at time t, jkt is the number of all firms in sector k in j at time t, V ijkt is a monotone function of the fraction of j s firms that export to i in sector k (which is based on firms relative productivity and fixed export costs to i) at time t. c jkt is a measure of average productivity of firms in sector k in j at time, but different firms have 7

8 different productivity based on a kt, t ijkt is the variable trade cost in sector k exported from j to i, and ( s k ) is the trade elasticity with respect to variable trade cost for product k. s in HMR, we also have (P ikt ) s k = J j= c jkt t sk ijkt jktv ijkt (5) a k V ijkt = = ˆ aijkt a Lkt (a kt ) s k dg(a kt ) ga g s k+ Lt (g s k + )(a g H kt a g L kt ) W ijkt (6) W ijkt = max ( aijkt g sk + a Lkt, ) (7) In the baseline specification, HMR assume that firm productivity /a kt is truncated Pareto distributed to the support [a Lkt, a Hkt ] so that G(a kt )=(a g kt a g Lkt )/(ag Hkt a g Lkt ),g > (s k ). P ikt is corresponding to i s MRTs in product k at time t as introduced by nderson and van Wincoop (23), but it additionally takes into account the impact of firm heterogeneity with the fixed costs of exporting on prices in product k. The model can be solved for the following estimable equation: lnt ijkt = (s k )lna k (s k )lnc jkt + ln jkt +(s k )lnp ikt +lne ikt (s k )lnt ijkt + lnv ijkt (8) P ikt, and jkt and c jkt are unobserved, but they can be accounted for by using exporter-sectorspecific factors and importer-sector-specific factors. Thus, with W ijkt, being a monotonic function 8

9 of V ijkt, we can obtain the following estimable equation: lnt i jtk = (s k )lnt ijkt + lnw ijkt + u ikt + v jkt (9) where u ikt =(s k )lnp ikt +lne ikt +(s k )lna, and v jkt = ln jkt (s k )lnc jkt +ln(ga g s k+ Lt ) ln(g s k + ) ln(a g H kt a g L kt ). Here we depart from HMR and further assume that lnw ijkt could be decomposed into a number of additively separable components: lnw ijkt = w 2ijk + h 2i jt + u 2ikt + v 2 jkt + e 2ijkt () where w 2ijk represents time-invariant pair-sector specific fixed costs, h 2i jt is pair-time specific random shock to fixed costs, u 2ikt is importer-sector-time specific random shock to fixed costs, v 2 jkt is exporter-sector-time specific random shock to fixed costs, and e 2ijkt is all remaining random elements that could affect fixed costs of firm s export market entry decision. These five factors determine the fraction of firms that export from j to i in sector k at time t. Likewise, we further assume that non-tariff-related variable trade costs can be decomposed into five additively separable factors such that: lnt ijkt = ln( +tari f f ijkt )+w 3ijk + h 3i jt + u 3ikt + v 3 jkt + e 3ijkt () where w 3ijk + h 3i jt + u 3ikt + v 3 jkt + e 3ijkt is the linear approximation of non-tariff-related variable trade costs. Substituting () and () into (4), we obtain the following equation: lnt i jtk = a (s k ) ln( +tari f f ijkt )+w ijk + h i jt + u ikt + v jkt + e ijkt (2) where w ijk w 2ijk (s k )w 3ijk, h i jt h 2i jt (s )h 3i jt, u ikt u ikt + u 2ikt (s k )u 3ikt, v jkt v jkt + v 2 jkt (s k )v 3 jkt, and e ijkt e 2ijkt (s k )e 3ijkt +(s s k )h 3i jt with s is the 9

10 average of s k. s w ijk, h i jt, u ikt, and v jkt are unobservable, we model them using fixed effects. Here (s k ) can be consistently estimated if we can assume ov(ln( +tari f f ijkt ),e ijkt w ijk,h i jt,u ikt,v jkt )= (3) That is, we assume that once we have controlled for all i jt-, ijk-, ikt-, and jkt-varying factors, the remaining trade costs factors that could vary in all four dimensions of i, j, k and t simultaneously (i.e. e ijkt ) are uncorrelated with tari f f ijkt. One possible scenario in which this assumption is violated is that, when a country lowers its tariffs on certain imports from a specific source country as a results of, for instance, trade agreement negotiations, it might provide assistance to the affected industries or erect non-tariff barriers on those products as a compensation (Ray, 98; Limao and Tovar, 2). However, as long as the assistance or non-tariff barriers are industry specific but not industry and source country specific, they would be controlled for by u ikt. lso, the consistency condition (3) is not conditional on T ijkt > unlike the HMR method. In order to directly incorporate zero trade flows in the estimation, we replace lnt ijkt in (9) with ln(t ijkt + ). 2.2 Self-Selection and Firm Heterogeneity key insight of HMR is that zero trade flows commonly observed in bilateral trade data are results of firms decision of self-selection into (or out of) foreign market and the decision is not uniform across country due to firm heterogeneity. 8 HMR suggest a two-stage procedure to account for self-selection and firm heterogeneity for aggregate data. The first-stage estimation requires an exclusion restriction variable that affects firms entrance into a foreign market but not their trade volume. 9 In order to extend the HMR approach to estimate product level models, the first stage 8 s we use product level data, we assume that firms self-selection of exporting occurs at pair-product level for the rest of paper. 9 HMR first consider bilateral regulation measure as the exclusion restriction variable. esides strong assumption of excludability in the model for trade volume, a practical limitation of this exclusion restriction is data availability, which prompts HMR to consider an alternative variable of an index of common religion (between any pair). Data on common religion has been available only for sporadic periods until Maoz and Henderson (23) construct a world

11 estimation requires an exclusion restriction variable that affects firms entrance into individual product markets in a foreign country but not their trade volumes in each of those markets. However, finding an exclusion restriction that varies over pair-product, importer-product-time, and/or exporter-product-time is extremely difficult due to data limitation. To avoid data problems, we propose an estimator that allows us to by-pass the first-stage estimation in the HMR s two-stage procedure while still controlling for self-selection and firm heterogeneity. We assume that unobserved factors associated with self-selection and firm heterogeneity could be approximated by four additively separable factors, w ijk + h i jt + u ikt + v jkt, then we can write an estimable equation using a multi-way error components model. Furthermore, it could be estimated by the within estimator that treat unobserved factors w ijk + h i jt + u ikt + v jkt as fixed effects. Therefore, when using disaggregate panel data, our estimator is superior to the two-stage HMR method in several ways. First, it avoids data limitations as mentioned above. Second, fixed effects are free from measurement errors and validity assumption associated with exclusion restriction (e.g. the religion variable). ext, besides pair-time unobserved factors associated with self-selection and firm heterogeneity, it allows us to additionally control for pair-product, importerproduct-time, and exporter-product-time unobserved factors associated with self-selection and firm heterogeneity using fixed effects.. Furthermore, those multi-dimensional fixed effects account for other unobserved factors as well. For instance, importer (exporter)-product-time fixed effects additionally control for producttime varying MRTs and exporter s supply (importer s demand), which are crucial to obtain unbiased estimates, but again very difficult to collect at fairly dissagregate product level. religion dataset for every five years from 945 to 2. While not to depreciate the value of this improved dataset on religion, we should be cautious about the measurement errors due to the challenging nature in measuring them at the first place.

12 3 Data We construct two sets of data. The first data set is HS 6-digit import data from U OMTRDE for 47 countries from 22 to 2 for about 5,3 products. The second data set is bilateral tariff data from TRIS of UTD for the same sample. More detailed explanations of the data are provided in the ppendix. For each pair of country and any product with available tariffs, unreported entries of trade flows are replaced with zero. In our data only positive observations are reported. ecause every country in our sample imports from other sample countries for all sample periods in aggregate level, we presume that unreported entries at product level are zeros. onceptually, balanced data mean that for all pairs of countries (ij), products (k) and time (t), observations are available for both the dependent variable (y) and all explanatory variables (X). In practice, bilateral trade data can never be truly balanced. This is because, data is naturally unbalanced because countries do not conduct international trade with themselves so that T ii is not defined for all i. Furthermore, there are some cases where we do not observe a country s tariff rate for a specific product for entire sample years. For example, in the cases of rgentina, no tariffs are reported for about 2 out of 5,3 products. We leave them as missing to be truthful to the data. Finally, some product codes are removed, combined with others, or newly introduced and these products could induce missing data over time. We define partially balanced data to deal with this type of missing data. esides balanced data, we also consider unbalanced and partially balanced data (i.e. only part of dimension is balanced among four dimensions) for the comparison purpose in evaluating our estimator and conventional within estimator. The use of the unbalanced data in most of previous studies is inevitable as they use only positive trade flows, i.e. zero trade flows are not imputed in These two types of missing data together make up only less than 3% of all data. ecause the proportion of missing is very small, we regard the data as (almost) balanced in practical sense if there are not other sources of missing data. Extensive simulation experiments are performed with various data generating processes to examine the effect of missing data for our proposed within estimators. Simulation results show that there is no size bias of the estimators. Results can be found in Kwak et al. (22) for three-way error components model and in the appendix for four-way error components model. 2

13 estimations. The degree of unbalancedness depends on the level of data aggregation, increasing up to about 7% for HS6-digit data in our sample. We show that the size bias of conventional within estimator increases with missing proportion in unbalanced data so that more care is required in using within estimator with disaggregate data to avoid bias. 4 Empirical Framework 4. Gravity Equation with 4-way Error omponents Our estimation equation is obtained from (2) in Section 2. : ln(t ijkt + ) = a + a ln( +tari f f ijkt )+w ijk + h i jt + u ikt + v jkt + e ijkt (4) where ln(t ijkt + ) is log trade flows, tari f f ijkt is a tariff that importing country i imposes on product k from exporting country j at time t. w ijk, h i jt, u ikt and v jkt are pair-product, pairtime, importer-product-time, and exporter-product-time heterogeneity, respectively and e ijkt is an idiosyncratic error term. Our primary object of interests is to obtain a consistent estimate of a =( s), the elasticity of trade costs. This specification implies that, in contrast to the theoretical model in section 2, here we assume the trade elasticity, s k, is homogeneous across all products so that s k = s. We relax this assumption later to adherent to the theoretical model. To control for unobserved factors, we adopt fixed effects following standard approach in the studies.. However, unlike in 3-way error components (3WE) model followed by most of studies in the literature (e.g., aier and ergstrand (27); Magee (28); aier et al. (24) ) using fixed effects in the 4-way error components (4WE) model may not be trivial depending on degree of balancedness of data. Using the dummy variable method is computationally challenging and infeasible in most statistical package while within estimator could be inconsistent unless data are 3

14 balanced. In this sub-section, we discuss within estimators for different degrees of balancedness of data and suggest a method to adopt in our estimations. 4.. Within Estimator for alanced Data s provided in the ppendix, with balanced data and conditional exogeneity assumption on e ijkt, the conventional LSDV (Least Squared Dummy Variables) estimator to eq.(??) is numerically equivalent to the pooled Least Squared (LS) estimator (See Wooldridge (29); brevaya (23) for two dimensional panel data cases) in the following equation with the within transformed variables: ỹ ijkt = X ijkt a + ẽ ijkt (5) where multiple transformations that remove all four unobserved factors is defined as follows: ỹ ijkt = y ijkt ȳ ijk ȳ ij t ȳ i kt ȳ jkt +ȳ ij + ȳ i k + ȳ i t + ȳ jk + ȳ j t + ȳ kt (6) ȳ i ȳ j ȳ k ȳ t + ȳ In eq.(6) a dot in the subscript denotes the averaging dimension. For instance, ȳ ijk = T ijk T t= s ijkty ijkt, where s i jt is an indicator for observability of both y ijkt and X ijkt (i.e. s ijkt = if both y ijkt and X ijkt are observed, and zero otherwise), and T ijk = T t= s ijkt, so that T ijk = T for balanced Multi-way error components model are typically estimated using dummy variables (e.g. ntweiler (2); altagi et al. (2); Davis (22) among others) or within transformations of variables to remove all or some of these error components (e.g. Matyas and alazsi (22) and Kwak et al. (22)). However, an extension of these methods to estimate product level data are very difficult. Firstly, the dummy variables method confronts computation difficulties arising from the large size of the data set. ntweiler (2); altagi et al. (2); Davis (22) propose estimators that can be applied to unnested and unbalanced bilateral trade data; in practice, however, implementation of their method requires to specify the transformation matrix and to generate too many dummy variables. s a result, we are not able to find any empirical study of gravity models implementing this type of estimators. On the other hand, the within-transformation based estimators generally are biased with unbalanced panel data although they do not suffer from computational difficulty. 4

15 data but T ijk < T for unbalanced data. 2 We denote averaging over other dimension as follows: ȳ i kt = n i jt n j= s i jty i jt, n ikt = n j= s ijkt,ȳ jkt = m jkt m i= s ijkty ijkt and m jkt = m i= s ijkt,ȳ i k = T ik T t= n ikt n j= s ijkty ijkt. We can similarly define the rest of averaging terms in eq.(6). The four error components, w ijk, h i jt, u ikt, and v jkt, can be removed by demeaning over the dimensions of t, k, j, and i, respectively. Our estimation strategy is to remove all four error components by using four distinct within transformations. Four successive demeanings need to be applied and these give ỹ ijkt as the dependent variable in eq.(6). lso note that the order of demeaning in t, k, j, and i does not matter for balanced data but it matters for the bias size in unbalanced data. The transformations of X and e are denoted as X i jt and ẽ i jt, respectively, could be similarly defined as in (6). We call eq.(6) the four-way within transformation as it removes all 4WEs. The four-way within estimator is consistent only when the data is balanced and conditional exogeneity assumption on e ijkt is satisfied; for unbalanced data it is inconsistent. 3 Our sample includes 47 largest countries countries in terms of GDP in 27 but excludes some countries such as India that do not have tariffs data at least for one partner country. Therefore, for each country at a given year, both tariff data and trade flows data imputed with zeros are available for HS 6-digit products for all partner countries. Therefore, tariff data are balanced for pair and product but not year. This is because each year variety of products vary over some years. For instance, in 27 reclassification occurs for some HS 6-digit products. In the process, some product codes are removed, combined with others, or newly introduced. Therefore, data are not balanced in strict sense. We name this partially balanced data and provide further discussion on this in section Using selection indicator notation is very powerful to show our algebraic results as it manifests the sources of bias for the conventional within estimators in unbalanced data. 3 We assume conditional exogeneity of idiosyncratic error terms throughout the paper. For the within estimator in (5), non-zero terms remain after the within transformation. Moreover, these terms are not mean-zero and do not go to zero as the sample size increases. These terms are all zero if s ijkt = for all observations as in the case of balanced data, but once s ijkt becomes random non-zero terms, additional terms remain after transformation and cause inconsistency. See Matyas and alazsi (22); Kwak et al. (22) for the detailed discussion of within transformation for 3WE models and Monte arlo evidence. 5

16 4..2 Within Estimator for Unbalanced Data In general, the within estimator that removes multiple error components in unbalanced data is inconsistent. ias arises when eq.(6) is applied to unbalanced data and its size depends upon which of error components are the source of bias and the order of demeaning. Unlike the case of balanced data, the values of the last eleven terms in eq.(6) (i.e. all terms that are averaging two or more dimensions ȳ ij,ȳ i k,ȳ i t,ȳ jk,...ȳ... ) now depend on the order of averaging. For these eleven terms, we use superscripts to denote the order of averaging over various dimensions. For instance, ȳ jt i k is obtained by averaging over the j-dimension first and the t-dimension second, i.e. ȳ jt i k = T ik T t= n ikt n j= s ijkty ijkt, while ȳ tj i k is obtained the reversed order of averaging over the two dimensions, i.e. ȳ tj i k = n ik n j= T ijk T t= s ijkty ijkt. The mean values depend on the order of averaging, ȳ jt i k 6= ȳtj i k unless by coincidence. This is because s ijkt depends on all i, j, k, and t. s a result, Tik T t= n ikt n j= s ijkt and n ikt n j= T ik T t= s ijkt become distinctively different values. Similarly, the mean values of the other eleven terms in eq.(6) depend upon the order of averaging. Finally, for the ȳ (i.e. ȳ ijkt,ȳ i jtk,ȳ it jk,ȳ itk j,ȳ ikt j,ȳ jk jt,...), there are 24 distinct sequences of four dimensional averaging and their values are different each. For a 4WE model, there are a total of 24 different orders of averaging and thus 24 different transformations based on eq.(6). For any unbalanced data, we use the property of the within estimator that one of the error components in eq.(??) can be completely removed by the within transformation without causing any bias. For our purpose, the important implication is that t-dimensional demeaning, if performed very first, can remove the unobserved factor w ijk : s ijkt w ijk T ijk T t= s ijkt w ijk = s ijkt w ijk w ijk T ijk T t= s ijkt = s ijkt w ijk w ijk T ijk T ijk = s ijkt w ijk w ijk where first equality hold because w ijk does not depend on t. Thus, for observed value, s ijkt =, s ijkt w ijk T ijk T t= s ijkt w ijk = s ijkt w ijk w ijk = (7) 6

17 Similarly, one error component can be removed by the within transformation such that k- dimensional demeaning, i-dimensional demeaning, and j-dimensional demeaning, if performed very first, can remove h i jt, u ikt, and v jkt, respectively. Obviously, since only one of these four demeanings can be performed first, three unobserved factors cannot be completely removed and, thus, potentially become the sources of bias in the estimations. Furthermore, less bias is retained if it is an earlier demeaning dimension that takes out the error component that causes inconsistency. 4 For instance, suppose that h i jt is the only unobserved factor correlated with tari f f ijkt in equation (??). Let us consider demeaning over product and four classes of demeaning orders: k, k, k, k. Here k indicates demeaning over the k dimension first (e.g. ki jt, k jti, kt ji etc.), k demeaning over the k dimension second, and so forth. Within estimators of each of these demeaning order have different magnitudes of bias due to omitted variable h i jt. We have already proven that the class k can completely remove OV due to h i jt. Simulation results in the ppendix show that absolute magnitude of OV due to h i jt for other classes increases as the demeaning order for k increases, i.e. k will cause no bias, k will cause the second smallest bias and k the largest bias. corollary of these findings is that, within transformations can remove one error component completely and for the other error components only partially. However, compared to the estimates from pooled OLS, the absolute magnitude of the bias from each source is reduced from the within transformations Within Estimator for Partially alanced Data Our sample is balanced in pair and product dimension but unbalanced in time dimension. In this case, we can still design within transformations that could take out multiple error components completely. y investigating the sources of biases and how biases arise from each demeaning of within transformation, we could obtain a consistent within estimator using partial balanced of data, s ijkt = s t. Our main insight to obtain a consistent estimator is that we remove w ijk as the first order 4 The ppendix provides some algebraic results. 5 Matyas and alazsi (22) suggest a solution is to remove one error component using within transformation, and then model the remaining error components using dummy variables. However, in the case of using pair-product-level panel data, there are still way too many dummy variables to be feasibly estimate even using their method. 7

18 t-dimensional demeaning (note that t is not balanced dimension) and remove u ikt, and v jkt as j and i dimensional demeanings (note that data are balanced at pair level). Let us consider transformations that remove w ijk, u ikt, andv jkt completely. The demean ordering we use as an illustration is i- j-t. First, first order demeaning over dimension i can remove v jkt as shown in section ow, we consider u ikt with double demeanings over dimensions i and j. s t u ikt i= s t u ikt j= (s t u ikt i= s t u ikt ) = s t u ikt s t = s t u ikt s t = s t u ikt s t i= i= i= u ikt s t u ikt u ikt s t u ikt + s t u ikt s t u ikt + s t Finally, we consider w ijk with triple demeanings over dimensions i, j, and t. +s t j= i= i= j= u ikt u ikt = i= j= u ikt s t w ijk i= s t w ijk j= (s t w ijk i= s t w ijk ) T T t= [s t w ijk i= s t w ijk j= (s t w ijk i= s t w ijk )] = s t w ijk s t w ijk T T t= = s t w ijk s t w ijk + s t i= w ijk s t s t + s t i= i=w ijk + T w ijk s t i=w ijk + w ijk + s t j= T t= s t w ijk + s t j= w ijk j= j= i= w ijk w ijk j= T j= j= i= i= w ijk Therefore, for s t =,we can rewrite the last equation as follows: T t= w ijk s t j= i= w ijk 8

19 w ijk i= w ijk j=w ijk + j= i=w ijk w ijk + i=w ijk + and we obtain zero because all terms are canceled out each other. w ijk j= j= i= w ijk = In sum, using s t = and triple demeanings over i- j-t (or it could be shown for demeanings over j-i-t), we algebraically show that within transformations can remove w ijk, u ikt, andv jkt completely. Thus, for all our estimations in results section, we obtain elasticity estimates from using within transformations over i- j-t and dummy variables for h i jt unless indicated otherwise. ote that number of dummy variables for h i jt is = 3,42 and no computational difficulty arises with less than 4, explanatory variables. We provide extensive simulation results for within estimators with triple transformations and partially balanced data, s t = s ijkt in the ppendix. 4.2 Estimation 4.2. Trade Elasticities with Disaggregate data Table shows various estimation results of using HS6-digit disaggregate data. In column (), we only use non-zero observation while in columns (2)-(5) zero observations are included, and various columns differ in term of the controls of unobserved factors. t the HS6-digit level of disaggregation, about 67% of trade flow observations are zeros. The inclusion of such large number of zero observations dramatically reduces the trade elasticity estimate from -3.6 down to -.7. The reason for this is that almost all country pairs trade only a limited number of HS6-digit level products at any time. For instance, countries that do not have oil reserves cannot export crude oil, and developing countries do not have skilled labour to export high-technology products. When zero trade flows are excluded from the estimation, the trade elasticity estimate is determined entirely by the response of the intensive product margin of trade to the reduction in tariffs. ccording to column (), a % reduction in tariffs is expected to increase the flows of existing trade link between two countries by 3.6%. When zero trade flows are included, 9

20 the estimate is determined by the response of both the intensive and the extensive product margins of trade. Even if some firms are responsive to reduction in variable trade costs and start to export to new market, the initial volume is likely to be small compared to the export volume to established markets. s a result, a % reduction in tariff only increase the average trade flow per product by a much smaller extent of.7%. However, it should be noted that column (2) has not properly accounted for the product-specific unobserved heterogeneity including product-specific fixed costs and MRTs, and therefore suffers from omitted variable bias. s explained in Section 2, we assume these unobserved factors can be collectively decomposed into h i jt, u ikt, v jkt and w ijk. We start with controlling for i jt-varying unobserved factors in column (3), which reduces the trade elasticity estimate marginally to In column (4) we further control for ikt- and jkt-varying unobserved factors. The elasticity estimate reduces from -.68 to In column (5) we have the full set of controls for unobserved heterogeneity at the product level; the additional control for ijk-varying factors further reduces the elasticity estimate to comparison of columns (2) to (5) shows that the progressively more comprehensive control of unobserved heterogeneity reduces the estimation bias. In particular, the absolute value of the elasticity estimate is reduced by nearly 4% from -.7 in column (2) to -.43 in column (5). This indicates that OV is as sizable as aggregation bias. ut bias from omitting zero observations is by far the largest amongst the three sources of bias. rkolakis et al. (22) show that for a range of trade models, including the rmington model and new trade models with micro-foundation like Eaton and Kortum (22) and Melitz and Ottaviano (28), the welfare gains from trade (compared to autarky) can be simply measured using two statistics, the share of expenditure on domestic goods, l and the elasticity of imports with respect to variable trade cost, f. Using their welfare change formula to evaluate the welfare change in US s year 2 compared to utarky, which is ( l /f ) where l =.93, they illustrate that the percentage change in real income needed to compensate a representative consumer for going back to autarky is.7 percent to.4 percent depending if the trade elasticity are ranged from -5 to - as surveyed in nderson and van Wincoop (24). If we use the estimate obtained from 2

21 Table : Trade elasticities: HS 6-digit () (2) (3) (4) (5) Disaggregate: HS6 Elasticity (-s) -3.66*** -.75*** -.68*** -.57**** -.427*** (.354) (.23) (.25) (.268) (.) ij, it, jt Yes Yes Yes Yes Yes ijt o o Yes Yes Yes ikt, jkt o o o Yes Yes ijk o o o o Yes Model log-linear o. of obs. 6,6,28 8,58,824 otes: luster (pair) robust standard errors are reported in parentheses. *, **, *** denote statistical significance at the %, 5%, % levels, respectively. We used balanced data and the within estimator. non-zero observations as in column (), US s gains from trade in year 2 is implied to be 2 percent, slightly higher shown in rkolakis et al. (22) still seems to be small. When we use the estimate obtained from data including zero observations with proper product-level controls as in column (5), the implied US s gains from trade in year 2 increase to 6 percent. This simple numerical example serves the purpose that, to evaluate the welfare impact of trade liberalization, it is paramount to have an unbiased estimate of trade elasticity The Extensive Margin Given that the elasticity estimate is substantially smaller when the extensive product margin of trade is added to the intensive product margin of trade, it will be interesting to know that how the extensive product margin of trade itself responses to change in variable trade costs. We use the following model to estimate the effect on the probability of starting import. I(T ijkt > ) = F(a + a tari f f ijkt + w ijk + h i jt + u ikt + v jkt ) (8) 2

22 Table 2: Probability of starting new product trade () (2) (3) Disaggregate: HS6 Probability *** -.38*** (.9) (.23) (.) ijt Yes Yes Yes ikt, jkt o Yes Yes ijk o o Yes Model linear probability model o. of obs. 8,58,824 otes: luster (pair) robust standard errors are reported in parentheses. *, **, *** denote statistical significance at the %, 5%, % levels, respectively. We used balanced data and the within estimator. where I is an indicator and F( ) is normal DF. We estimate (8) using linear probability model that allows to directly control four multi-way error components. Table 2 shows the probability of a country starting to export a new product to a partner country in response to a tariff cut by its partner for that product. Given this is a probability model, the tariff enters the model in level terms instead of in logarithmic terms. The estimated probability varies depending on what type of unobserved heterogeneity being accounted for. In column (3) where the control for unobserved heterogeneity is most comprehensive, it is found that on average there is a close to 3.8% chance that a country will start importing a new product from a partner country if the latter cuts the tariff on that product by percentage point. t least in a one-year horizon, the response of the extensive trade margin to changes in variable trade cost is rather small Products from Different Market Structures In this section, we decompose the products into three groups using the classifications in Rauch (999) and examine whether goods traded on organized exchanges have larger trade elasticity. Rauch (999) classifies goods into three groups: commodities, reference priced goods and differentiated goods. ommodities are those goods traded on organized exchanges. Reference priced goods are the goods that reference prices of them being published in trade journals, and the rest are 22

23 differentiated goods. 6 In our sample, 5.%, 3.7% and 64.2% of the observations are commodities, reference priced and differentiated goods, respectively. roda and Weinstein (26) use a model of import demand and supply equations and find the commodities have higher elasticity of substitution than the other goods. However, as they point out, commodities do not necessarily imply perfectly substitutable goods. They said, For example, although tea is classified by Rauch as a commodity, it is surely quite differentiated. Similarly, it is hard to see why a commodity like dried, salted, or smoked fish would be more homogeneous than a reference priced good like fresh fish or a differentiated good like frozen fish. Table 4 shows the results. 7 In column (), commodity goods have the largest trade elasticity, followed by referenced priced goods, and then differentiated goods. This is consistent with the expectation in Rauch (999) and finding in roda and Weinstein (26). However, it can be noticed that the trade elasticity of differentiated goods is of an unexpected positive sign. This could be due to fact that we have not fully accounted for all product-level unobserved heterogeneity and therefore the estimate is biased. When we progressively control for more unobserved factors, surprisingly the relative trade elasticities of the three types of goods reverse. In column (3), differentiate goods now have the largest elasticity and of a correct sign, while commodities goods have the smallest and insignificant one. The reverse of relative magnitude can be explained as the following. The world markets for commodities are more integrated (via organized market trading for instance). s a result, changes in tariffs on a product between a pair has greater effects on the prices of that product elsewhere, and the feedback (or general equilibrium) effect on the trade between the pair will also be stronger. In the nderson and van Wincoop (23) framework, the relative prices of goods in the rest of the world for a pair is denoted as MRTs. In our modeling setting, productlevel MRTs are captured by ikt and jkt fixed effects. Therefore, once we have controlled for iktand jkt-unobserved heterogeneity, we eliminate this feedback effect. ecause the feedback effect could be larger for goods sold in more organized market, its removal may reverse the relative 6 See Rauch (999) for details. 7 We use the conservative definition from Rauch (999). Using his liberal definition yields very similar results. 23

24 Table 3: Trade elasticities heterogeneity according to homogeneity of goods with unobserved heterogeneity () (2) (3) Disaggregate: HS6 Differentiated goods.483*** *** (.69) (.366) (.43) Reference priced goods -.444*** -.59* -.3*** (.5) (.28) (.8) ommodity goods -.887*** (.27) (.46) (.83) ijt Yes Yes Yes ikt, jkt o Yes Yes ijk o o Yes Model log-linear o. of obs. 5,835,89 otes: Some sectors that were not categorized in these three groups are dropped in the estimations. luster (pair) robust standard errors are reported in parentheses. *, **, *** denote statistical significance at the %, 5%, % levels, respectively. We used balanced data and the within estimator. trade elasticities of the three types of products in column (3) as compared to column () Products of Different Skill-Intensity Firms markups depend on trade elasticity. Higher trade elasticity yields lower markups. Skill intensity of products may indicate the firms market power, where greater market power sets higher markups. Firms produce higher skill intensity products, for example, contract lenses, may have more market power compared firms producing lower skill intensity products, for example toilet papers. In this section, we divide the products based on skill-intensity based on the classification of asu (2). 8 In our sample, 25.5%, 34.8%, 6.7% and 23% are minerals, resource and low-skill intensive manufacturing (e.g., toilet paper), medium-skill intensive manufacturing (e.g., air conditioner) and high-skill intensive manufacturing (e.g., microscopes), respectively. Table 5 shows the results. In column (), without proper controls at product level, the trade elasticity estimates 8 See ppendix for details of the classification. 24