Estimating Gravity Equation Models in the Presence of. Sample Selection and Heteroskedasticity

Transcription

1 Estimating Gravity Equation Models in the Presence of Sample Selection and Heteroskedasticity Bo Xiong Sixia Chen Abstract: Gravity models are widely used to explain patterns of trade. However, two stylized features of trade data, sample selection and heteroskedasticity, challenge the estimation of gravity models. We propose a Two-Step Method of Moments (TS-MM) estimator that deals with both issues. The Monte-Carlo experiments show that the TS-MM estimates are resistant to various combinations of sample selection and heteroskedasticity. Moreover, the TS-MM estimator performs reasonably well even when the data generating process deviates from the TS-MM assumptions. We revisit the world trade in 1990 to illustrate the usefulness of the proposed model, with emphasis on the identification of the extensive margin of trade. Keywords: gravity equation, heteroskedasticity, zeros, sample selection, Two-Step Method of Moments, extensive margin, market access JEL classification: F10, C10 Bo Xiong (the corresponding author), boxiong@ucdavis.edu, is a postdoctoral researcher at the Agricultural Issues Center, University of California at Davis. Sixia Chen, SixiaChen@westat.com, is a senior survey statistician at the Westat corporation, Rockville, Maryland. The authors thank John Beghin and Joseph Herriges at Iowa State University for discussions, and the Westat editorial team for assistance. The authors also appreciate the comments from an anonymous reviewer. 0

2 1. Introduction The gravity equation model has been a long-time workhorse in international trade since Tinbergen (1962). It posits that bilateral trade flow can be explained by the two countries income levels, geographic distance, and various other factors affecting the cost of trade (such as import tariffs, non-tariff measures, contiguity condition, historical tie, and religion similarity). In addition to its empirical success in fitting trade data well, the gravity equation model has gained further recognition recently because of the development of its microeconomic foundations. 1 Anderson and van Wincoop (2003) derive a full specification of the gravity equation model from maximizing the utility of a representative consumer with Constant Elasticity of Substitution (CES) preferences. Most importantly, they emphasize the role of countries multi-lateral trade resistance terms in a cross-sectional gravity analysis. 2 Novy (2013) shows that the gravity equation can also emerge from a translog demand system. Deardorff (1998) argues that the gravity equation is compatible with the Heckscher-Ohlin trade theory as well. Evenett and Keller (2002) report that both the Heckscher-Ohlin theory and the monopolistic-competition trade theory can lead to the gravity equation and that each provides unique insights into trade patterns. Feenstra, Markusen, and Rose (2001) review how various trade theories interfere with the gravity model and provide evidence in favor of the monopolistic-competition theory and the reciprocal-dumping theory. Building upon the new trade theory of heterogeneous firms, Helpman, Melitz, and Rubinstein (2008) (HMR hereafter) propose a generalized gravity model in which 1 See Anderson (2011) for a review of the developments of the gravity model. 2 Baldwin and Taglioni (2006) extend the analysis by suggesting ways to control for multi-lateral trade resistance terms when time series data are available. 1

3 firms face fixed costs of exporting. Their model predicts that only the most productive firms overcome the fixed cost of trade and penetrate foreign markets. Empirically, they find that trade liberalization promotes international trade mainly through inducing more firms to participate in the world market. 3 With its microeconomic foundations well developed, the gravity model now faces challenges in its application. In particular, there is no consensus in the literature on how to estimate a gravity model in the presence of the two stylized features of trade data: sample selection and heteroskedasticity. On one hand, zeros are common in trade data. For example, nearly half of the aggregate bilateral trade flows were zero in 1986 (HMR), as well as in 1990 (Santos Silva and Tenreyro, 2006). The treatment of zeros in the estimation is critical for at least two reasons. First, the improper handling of zeros causes the classical selection bias (Heckman, 1979). Second, from a policy perspective, the modeling of zeros allows the impact evaluation of policy initiatives on international market accessibility. The promotion of participation is particularly relevant when the policy initiatives are aimed to benefiting small or mediumsized exporters from the developing world. For example, Shepherd (2010) shows that the reduction in export costs, tariffs, and transport costs can encourage developing countries to export to more foreign markets. Besedes and Prusa (2011) argue that developing countries are more likely to experience long-run export growth if they penetrate foreign markets and survive there for more than a year. Bergin and Lin (2008) find that currency unions facilitate trade predominantly through inducing more exporters to trade. 3 Santos Silva and Tenreyro (2013) document that the HMR model might be mis-specified, but their investigation does not result in an alternative specification. 2

4 Heteroskedasticity is another common feature of trade data. The data sample of a gravity analysis usually consists of trade records sourced from multiple national customs. Therefore, the varying quality of data can cause heteroskedasticity in regressions. In particular, Santos Silva and Tenreyro (2006) (SST hereafter) show that estimates from log-linearized gravity models are severely biased when the true model takes the multiplicative form and data exhibits heteroskedasticity. The problems of sample selection and heteroskedasticity have prompted researchers to devise appropriate estimators for the gravity model. Consequently, two camps of ideas emerge. One camp focuses on the economics of zero trade records. The new trade theory, pioneered by Melitz (2003) and developed by several others such as Chaney (2008) and HMR, attributes the absence of trade to firms selection behavior: zero trade flow is observed when no firm in the potential exporting country finds it profitable to bear the fixed costs of trade and sell to the destination market. Therefore, zeros should be modeled in a selection process, which is compatible with the Heckman sample selection model (Heckman, 1979), or, to a lesser degree, the E.T.-Tobit model (Eaton and Tamura, 1994). The Heckman model can be implemented via a two-step procedure. First, the selection equation captures zeros and explains why bilateral trade occurs at all. Second, the outcome equation, based on the gravity equation, characterizes the volume of trade conditioned on trade taking place. The HMR model extends the Heckman model in that the inverse Mill s ratio, which corrects for the selection bias in the second stage, takes a particular functional form to account for firms entry and exit behavior. 4 The E.T.-Tobit model is a maximum likelihood estimator in which zeros are realized when potential 4 More specifically, the choice of the functional form depends on the distribution of firms productivity. Without prior knowledge of the distribution, HMR recommends a non-parametric method to correct for the selection bias. 3

5 trade flows are too small to be observed. Both the Heckman-type model and the E.T.- Tobit model deliver rich comparative statics. For example, one can decompose the effect of trade liberalization into the intensive margin (the intensification of pre-existing trade) and the extensive margin (the creation of new trade partnership). 5 Nevertheless, both models are subject to the potential bias due to the logarithmic transformation. In pursuit of a proper estimator for the gravity model, the other camp advocates estimating the gravity equation in its multiplicative form with models originally devised for count data. In particular, SST proposes the Poisson Pseudo Maximum Likelihood (PPML) estimator to accommodate zeros and heteroskedasticity. Without the practice of log-linearization, the PPML method permits zero outcomes. Moreover, the PPML estimates are robust to heteroskedasticity because the second or higher moment conditions are absent from the estimation procedure. 6 However, Martin and Pham (2008) note that the PPML estimates are severely biased when zeros are not random outcomes. 7 Some variants of the PPML estimator are also proposed in the literature. For example, Burger, van Oort, and Linders (2009) consider the Negative Binomial Pseudo Maximum Likelihood estimator, the Zero Inflated Poisson Pseudo Maximum Likelihood estimator, and the Zero Inflated Negative Binomial Pseudo Maximum Likelihood estimator. Although with merits of their own (e.g., permitting over-dispersion and excessive zeros), the variants of PPML fail to recover the elasticities in the gravity 5 Throughout the paper, we refer to the extensive margin of trade as a new trade partnership at the country level. Alternatively, the extensive margin can refer to the newly entered firms (HMR), or the newly traded varieties (Hummels and Klenow, 2005), or the newly reached consumers (Arkolakis, 2010). We omit these dimensions because we deal with country-level data. 6 Westerlund and Wilhelmsson (2011) extend the cross-sectional analysis in SST to a panel setting and further confirm the robustness of the PPML estimator. 7 Santos Silva and Tenreyro (2011) reply that the PPML estimator can satisfactorily accommodate a high frequency of zeros. But the self-selection issue remains unaddressed. 4

6 model. 8 Therefore, we exclude all variants of the PPML estimator from the rest of the discussion. We contribute to the estimation of gravity models by proposing a Two-Step Method of Moments (TS-MM) estimator that simultaneously deals with sample selection and heteroskedasticity. The estimator works as follows. In the first step, we characterize the participation decision to trade by a selection process and predict the probability of trade accordingly. This allows us to investigate the extensive margins of trade and assess the impact of trade policies on market accessibility. In the second step, we capture positive trade flows by a gravity equation in its multiplicative form, with the potential selection bias corrected. We draw statistical inferences and policy implications based on the point estimates from the method of moments, coupled with heteroskedasticityresistant standard errors (White, 1980). Our Monte-Carlo experiments show that the TS-MM estimates are resistant to various patterns of sample selection and heteroskedasticity. Moreover, we find that the TS-MM estimator performs reasonably well even when the model assumptions do not strictly hold. In addition, we re-examine the world trade in 1990 to illustrate how the proposed estimator sheds light on the two margins of trade. The rest of this article is organized as follows. Section 2 introduces the TS-MM estimator and discusses its properties. Section 3 designs various experiments and evaluates the performance of several estimators for the gravity model. Section 4 provides an application with real data. Section 5 concludes the presentation. 8 More specifically, the estimates from the variants are sensitive to the unit of measurement of the dependent variable. For instance, measuring trade flows in thousands of dollars instead of dollars would result in different estimates. See the log of gravity website for more discussions of the pitfalls of the variants of PPML. 5

7 2. The Gravity Equation and the Two-Step Method of Moments Estimator country i, The gravity equation approach to trade posits that country j s import from M, is characterized by φ ϕ γ (1) M = YY D = exp( Xβ ), i j where Y i and Y j denote country i and j s characteristics (e.g., GDP, population, remoteness to the rest of the world); D includes any trade cost terms specific to the pair of countries (e.g., applied tariff rates, geographic distance, contiguity condition, historical colonial relationship, religion similarity, and the existence of a preferential trade agreement); 9 φ,ϕ, andγ are the elasticities to be estimated. Simple algebra leads to the last term in equation (1), where X is a row vector containing all explanatory variables in their logarithmic scales, and β is a column vector of all parameters. The application of the gravity model requires a stochastic version of (1), which we pursue next. Motivated by the new trade theory, we explicitly account for the absence of trade by introducing a selection process. Specifically, our empirical gravity model is as follows: (2a) (2b) M = exp( X β ) + µ, * d = Z α + υ, * (2c) d I( d * > 0), = (2d) M = dm. * 9 See Anderson and van Wincoop (2004) for more discussions of trade costs. 6

8 In the system (2a)-(2d), j in the absence of fixed cost of trade. 10 Variable * M is the notional trade flow from country i to country * d is the latent variable for the participation decision d, which equals 1 if country j imports from country i and 0 otherwise. Variable Z contains all factors potentially affecting the fixed cost of trade, and α is the associated vector of parameters. Variable M is the observed trade flow, which is a product of the participation decision and the notional trade volume. As in Heckman (1979), we assume that µ and υ are two idiosyncratic terms governed by a bivariate normal distribution. Specifically, u 0 σ, 11 σ12 N υ 0 σ21 σ, where 22 σ = σ. The correlation between the two observables accounts for omitted variables contributing to both the fixed and variable costs of trade. Note that heteroskedasticity is admissible because σ 11 varies across pairs of countries. The system (2a)-(2d) is appealing in three aspects. First, the underlying gravity equation (2a) is expressed in its multiplicative form and therefore free from the logarithmic transformation. As shown later, β can be estimated consistently regardless of the heteroskedastic patterns in (2a). Second, (2b) and (2c) explain zero outcomes and allow the investigation of incentives to participate. Note that Z incorporates all variables, which may or may not in X, that affect the fixed cost of trade. 11 Third, the characterization of observed trade flows in (2d) facilitates the decomposition of the 10 The concept of the notional trade is similar to the desired amount of trade as in Ranjan and Tobias (2007). 11 For instance, HMR examines how institutional factors such as days to start business affect firms decisions to trade. 7

9 extensive and intensive margins of trade. For example, the elimination of import tariffs fosters international trade, M, either through inducing more countries to participate in the world market, d, or through enlarging the size of pre-existing trade, Following Maddala (1986), we estimate system (2a)-(2d) via a two-step * M, or both. procedure. In the first step, we estimate (2b)-(2c) using a standard Probit model. That is, the probability of country j importing from country i is derived as: (3a) Pr( d = 1) = Φ ( Z θ ), where θ = α / σ22. Defining the extensive margin of trade as the probability of trade in its logarithmic scale, we compute the marginal effect through the extensive margin by differentiating (3a). Specifically, an exogenous change in z leads to a change in the extensive margin of the magnitude (3b) ln(pr( d = 1)) z = ˆ θ λ, z where λ = φ( Z ˆ θ) Φ ( Z ˆ θ) is the inverse Mill s ratio as in Heckman (1979). In the second step, we characterize the size of trade conditioned on trade occurring. The bivariate normality of the error structure allows us to derive the conditional trade volume as: (4a) EM ( d = 1) = exp( Xβ ) + ωλ, where ω = σ12 / σ22. Intuitively, (4a) states that the size of trade follows a gravity 8

10 equation augmented by an additional term correcting for the selection bias. In the special case where σ 12 = 0, (4a) reduces to the PPML specification as in SST. 12 We estimate the second-stage equation (4a) via the method of moments and construct the heteroskedasticity-consistent variance covariance estimates as in White (1980). Specifically, the point estimates of [ β', ω ']' solve the following system of equations: (4b) ( M exp( X β ) ωλ ) Ω= 0, where p p p M p is a column vector containing all positive outcomes, X p and λ p are subsets of X and λ where outcomes are positive, and Ω= [ X, λ ]'. The method of moments has two practical merits. First, it delivers consistent estimates as long as (4a) is correctly specified. In particular, the solution to (4b) does not involve p p σ 11. Therefore, the resulting estimates are robust to heteroskedasticity. Second, when endogeneity is a concern, the generalized method of moments can be readily adopted. Defining the intensive margin of trade as the conditional trade volume in its logarithmic scale, we compute the marginal effect through the intensive margin by differentiating (4a). Specifically, an exogenous change in x leads to a change in the intensive margin of the magnitude (4c) 2 x( '( Z ) Φ( Z ) ) x θ φ θ θ λ βλ ln( EM ( d = 1)) x = βx + ω, exp( X β ) + ωλ where ϕ '( ) is derivative of the normal density function. Intuitively, (4c) states that the exogenous change affects the size of pre-existing bilateral trade through two channels. 12 Nevertheless, (4a) maintains that the PPML specification only holds for the subsample of positive outcomes. 9

11 Besides the direct impact through β x, there is an indirect impact through the changes of incentives to participate, as captured by the second term on the right hand side of (4c). The overall marginal effect, if the factor of interest affects both the fixed and variable costs of trade, is the sum of its effects through both margins of trade. Or, the overall marginal effect is computed as (5) ln EM ( ) x = ln(pr( d = 1)) / x + ln( EM ( d = 1)) x. ex t ensive margin int ensive margin We now compare the TS-MM estimator with the alternatives in the literature, i.e., the Heckman, the E.T.-Tobit, the HMR, and the PPML estimators. The treatment of zeros in the TS-MM estimator is similar to that in the Heckman, or the E.T.-Tobit, or the HMR model: all zeros are attributed to countries self-selection to no trade. However, the TS- MM model differs from the three models in that it characterizes trade flows in levels, rather than in logarithmic scales. Therefore, the TS-MM estimates remain consistent even when trade date exhibit heteroskedasticity. As a practical merit, the identification of the TS-MM model does not require an excluded variable as in the Heckman or the HMR model because the inverse Mill s ratio is collinear with X β but not with exp( X β ). 13 p p Compared to the PPML estimator which uses a single process to explain both positive and zero trade records, the TS-MM model accounts for the absence of trade with a selection process and allows the identification of both the extensive and the intensive margins of trade. From a policy perspective, the investigation of the marginal effects through the extensive margin enables researchers to evaluate the impacts of policy 13 In other words, the exponential functional form facilitates the identification in the second stage of the TS- MM model. 10

12 initiatives on market accessibility, which is of great importance to international development. Table 1 summarizes the advantages and disadvantages of various estimators for the gravity model. Note that the E.T.-Tobit model has the same merits and drawbacks as the Heckman model. However, the implementation of the E.T.-Tobit model requires full information of the likelihood function, leaving the resulting estimates vulnerable to heteroskedasticity. Therefore, we consider the E.T.-Tobit model weakly dominated by the Heckman model and exclude it from the rest of the discussion. Table 1. Advantages and disadvantages of several estimators for the gravity model Estimator Zeros admissible? Log-linearization? (heteroskedasticity) Two margins? (self-selection) Trun-OLS no yes no PPML yes no no Heckman yes yes yes E.T.-Tobit yes yes yes TS-MM yes no yes Note: Trun-OLS denotes the ordinary least square estimator applied to the subsample of positive trade flows. 3. The Monte-Carlo Experiments We design two sets of experiments to assess the performance of the TS-MM estimator relative to alternative models. First, we verify that the TS-MM estimates are resistant to various patterns of sample selection and heteroskedasticity when the conditional mean (4a) is correctly specified. Second, we allow mis-specifications in the data generating processes and further evaluate the performance of competing estimators When the true model is known We generate data according to the system (2a)-(2d). As in Westerlund and Wilhelmsson (2011), we consider two explanatory variables. Variable x 1 is drawn from a 11

13 normal distribution with mean 1 and variance 0.1. This continuous variable can be interpreted as the income level of the importing country in the gravity model. We assign β 1 = 1 and α 1 = 0.5 for x 1 to indicate that larger economy imports more and has a higher propensity to source from abroad. The other explanatory variable x2 is a dummy variable taking the value of 1 with probability 0.4. This dummy variable can be seen in the gravity model as an indicator whether the two countries are engaged in a preferential trade agreement. We assign β 2 = 1 and α 2 = 0.5 for x 2 to suggest that preferential trade agreements promote international trade through both margins of trade. We set β 0 = 1 as the intercept in (4a). To allow different degrees of sample selection, we consider two values for the intercept in (4b): (a) α 0 = 0.35, and (b) α 0 = The issue of sample selection is potentially more severe in the latter case because zero outcomes are more likely. In particular, with σ 22 = 0.05, the proportion of zero outcomes in the sample is about 15 per-cent in case (a) and 50 per-cent in case (b). forms for σ 11k To permit various patterns of heteroskedasticity, we consider three functional, where k denotes a specific observation: (i) homoskedastic errors, or σ 11k = 0.1; (ii) heteroskedastic errors where the variance is proportional to the mean, or σ 11k = 0.1mk, where mk = exp( β0+ β1x1 k + β2x2k) ; (iii) super-heteroskedastic errors where the variance is a quadratic function of the mean, or 2 11k 0.1( mk 0.5 mk) σ = +. Finally, we set σ 12 = 0.05, so that the correlation of the two idiosyncratic terms is about 0.7 in case (i), 0.6 in case (ii), and 0.5 in case (iii). 12

14 In summary, we evaluate the performance of the TS-MM estimator, relative to its alternatives, under six different scenarios. More specifically, to assess the impact of sample selection, we generate data samples that either contain (a) few zeros or (b) many zeros. To investigate the effects of heteroskedasticity, we generate data featured with (i) homoskedasticity, or (ii) heteroskedasticity, or (iii) super-heteroskedasticity. For each of the six scenarios, we generate a sample of 1000 observations (i.e., k = 1,2, ) and apply each estimator to the simulated data. We iterate this procedure for 1000 times and report the biases and variances of ˆ β 1 and 2 ˆβ in Table 2. 13

15 Table 2. Simulation results when the data generating process follows TS-MM Estimator Few zeros (15 per-cent) Many zeros (50 per-cent) Bias Variance Bias Variance Homoskedasticity PPML Heckman TS-MM Heteroskedasticity PPML Heckman TS-MM Super-heteroskedasticity PPML Heckman TS-MM Note: The results for the coefficient of the binary variable are in Italic. 14

16 We discuss the performance of each estimator in Table 2. Because unity is the true value of β 1 and β 2, the reported biases can be interpreted as biases in percentage terms. When zero outcomes are relatively few in the sample, the PPML estimates of β 1 and β 2 are biased upward by less than 8% and 20%. The PPML model overestimates the two coefficients because it co-finds the effects through β and α. That is, without distinguishing the extensive margin from the intensive margin, the PPML model reports the overall marginal effects. When zero outcomes account for half of the sample, the PPML model overestimates β 1 by nearly 40% and β 2 by more than 100%. This is because the effect through α, or the extensive margin, takes a larger share in the overall impact as zero records become more prevalent. 14 Nevertheless, the PPML estimates are fairly stable across different heteroskedastic patterns, as claimed in SST. As shown in Table 2, the Heckman model delivers fairly consistent estimates when zero outcomes are relatively few. This is because the Heckman procedure is in line with the data generating process (2a)-(2d), except for the choice of the functional form in the second stage. However, when zero outcomes become more frequent, the Heckman estimates of β 1 and β 2 are biased by more than 10% and 20% respectively. The reason is that the choice of the functional form in the second stage (in particular, how the inverse Mill s ratio interfaces with the main equation) becomes more important as self-selection is more severe in the sample. The performance of the TS-MM estimator in Table 2 is satisfactory. In all six scenarios, the bias of the estimate of either coefficient is below 5%. The variances of the 14 Martin and Pham (2008) also report that the PPML estimates are severely biased when data contain frequent zero outcomes. 15

17 estimates are also small, although they become slightly larger as more zero outcomes reduce the degree of freedom in the second stage estimation. These results confirm that the TS-MM estimator is capable of accommodating various patterns of sample selection and heteroskedasticity When the true model is unknown However, the true data generating process may deviate from the system (2a)-(2d). To further evaluate the performance of the TS-MM model when it is subject to specification errors, we conduct another set of Monte-Carlo experiments in which the underlying data generating process stems from either the Heckman or the PPML specifications. The Heckman model differs from the system (2a)-(2d) in that the gravity equation is expressed in its logarithmic scales. In Appendix A, we describe the Heckman data generating process and the parameter assignment in detail. We apply all three estimators to the simulated data samples and report the biases and variances of estimates in Table 3. 16

18 Table 3. Simulation results when the data generating process follows Heckman Estimator Few zeros (15 per-cent) Many zeros (50 per-cent) Bias Variance Bias Variance Homoskedasticity PPML Heckman TS-MM Heteroskedasticity PPML Heckman TS-MM Super-heteroskedasticity PPML Heckman TS-MM Note: The results for the coefficient of the binary variable are in Italic. 17

19 We discuss the performance of each estimator in Table 3, where the data is generated as in the Heckman procedure. We find that the PPML model delivers biased estimates, because it fails to distinguish the impact through the intensive margin from that through the extensive margin. This problem is more evident as the proportion of zero outcomes increases. The Heckman model leads to fairly consistent estimates. Specifically, the magnitudes of biases are well below 1.5% across all six scenarios. This is consistent with the fact that the Heckman model is true model governing the data generating process. The performance of the TS-MM model is dominated by that of the Heckman procedure, but the resulting estimates are fairly close to the true values. As shown in Table 3, the bias of the TS-MM estimate is generally below 10%. However, the magnitude of the bias increases in the degree of heteroskedasticity. This is because heteroskedastic errors cause the second stage of the TS-MM model to be mis-specified as one takes the exponential transformation of the data. Next we consider another data generating process that resembles the PPML experiments in SST. In Appendix B, we describe the specification and the parameter assignment in detail. Note that our experimental design departs from SST in that the thresholds for outcomes to be observable are relatively high. As a result, the mean of the observed outcome differs significantly from the mean of the latent outcome which strictly follows the PPML specification. We report in Table 4 the simulation results from all three estimators. 18

20 Table 4. Simulation results when the data generating process follows PPML Estimator Few zeros (15 per-cent) Many zeros (50 per-cent) Bias Variance Bias Variance Homoskedasticity PPML Heckman TS-MM Heteroskedasticity PPML Heckman TS-MM Super-heteroskedasticity PPML Heckman TS-MM Note: The results for the coefficient of the binary variable are in Italic. 19

21 We discuss the performance of each estimator in Table 4 in turn. In general, the PPML estimates are biased because the mean of the dependent variable, inclusive of zeros, does not take the form of exp( X β ) after the censoring takes place (see Appendix B for technical details). This problem is more severe as the censoring threshold rises. In particular, when zero outcomes account for half of the sample, the PPML estimates of β 1 and β 2 are biased by more than 25% and 100% respectively. As shown in Table 4, the biases of the Heckman estimates are below 5% when zero outcomes are relatively few in the sample. However, the estimates are less consistent when zero outcomes are prevalent. The reason is that, while addressing the issue of selfselection properly, the Heckman model is subject to the specification error inherent in the logarithmic transformation of the dependent variable. The TS-MM estimator performs reasonably well in Table 4. Specifically, the biases of all but one TS-MM estimates are below 10%. Moreover, the performance of the TS-MM model does not vary significantly across different patterns of sample selection and heteroskedasticity. These results suggest that the TS-MM estimator is fairly resistant to specification errors. In summary, we evaluate the performance of the TS-MM model, among other estimators for the gravity model, under various data generating processes. The simulation results show that the TS-MM estimates are robust to various patterns of sample selection and heteroskedasticity. Furthermore, we find evidence that the TS-MM model recovers the parameters to a satisfactory degree even if it is subject to specification errors. In addition, we find that the Heckman model is a viable candidate in most of the scenarios, but the PPML estimates are severely biased when self-selected zero outcomes prevail. 20

22 4. An Empirical Application In this section we illustrate the usefulness of the TS-MM estimator by re-visiting world trade in 1990 as in SST. In particular, we shed light on the policy implications from the identification of the extensive margin of trade. The data set covers aggregate bilateral trade records between 136 countries. Among the 18,360 (=136*135) observations, 48 per cent are zeros. The explanatory variables include the geographic distance, the border dummy variable, the common language dummy variable, the colonial tie dummy variable, and the Regional Trade Agreement (RTA) dummy variable. As in Anderson and van Wincoop (2003), we include both the importers and exporters fixed effects in the estimation to control for the multi-lateral trade resistance terms. 15 We also apply the PPML and the Heckman models to the data. The former replicates the results in SST and the latter provides another set of benchmark results. We expect bilateral trade to be larger or more likely between countries that are geographically closer, or share a common border, a common language, a colonial relationship, or engage in a regional trade agreement. We compute the marginal effects using the estimated coefficients and present them in Table Due to the cross-sectional nature of the data, all country-specific characteristics, such as income, population, and remoteness to the rest of the world are subsumed into the fixed effects. 21

23 Table 5. Regression results from the PPML, Heckman, and TS-MM models PPML Heckman Probit TS-MM 2 nd stage 1 st stage a 2 nd stage Overall Effects Intensive Margin Extensive Margin Intensive Margin (1) (2) (3) (4) (5) ln( dist) b -0.75*** (0.04) -1.35*** (0.03) -1.22*** (0.06) -0.77*** (0.04) border 0.37*** (0.09) 0.16 (0.12) 0.16 (0.13) 0.35*** (0.09) language 0.38*** (0.09) 0.41*** (0.06) 0.46*** (0.06) 0.42*** (0.09) colony 0.08 (0.13) 0.67*** (0.07) 0.32*** (0.06) 0.04 (0.13) RTA 0.38*** (0.08) 0.29*** (0.10) 1.59*** (0.19) 0.38*** (0.08) IMR c n.a (0.06) n.a (849.66) Importers yes yes yes yes fixed effects Exporters yes yes yes yes fixed effects Ramsey test d Accepted Rejected n.a. Accepted Note: a. The 1 st stage estimation is shared by the Heckman and the TS-MM models; b. The distance variable is exponentially transformed in the 1 st stage estimation, in order to make the Heckman model identifiable; c. Inverse Mill s ratio is calculated from the 1 st stage estimation; d. The auxiliary regressor in the Ramsey test is the squared fitted value as in SST. Heteroskedasticity-resistant standard errors are in parentheses. *, **, and *** denote the significance levels at 10, 5, and 1 per-cent respectively. 22

24 The PPML estimates, shown in column (2) of Table 5, replicate the results in SST. All estimated coefficients bear the expected signs and all are statistically significant except for the colonial tie. Note that the estimated coefficients can be interpreted as the overall marginal effects because the PPML model co-finds the effects through the two margins of trade. The first-stage Probit estimation, shared by the Heckman and the TS-MM models, explains countries participation decision. We present in column (4) of Table 5 the marginal effects through the extensive margins of trade, as defined by equation (3b). Interestingly, we find that countries sharing a border are not more likely to trade with each other than otherwise. The second-stage estimation of the Heckman model, shown in column (3) of Table 5, characterizes the size of trade conditioned on trade occurring. The estimated coefficient of the border dummy variable suggests that contiguity does not help explain the trade volumes. In addition, the inverse Mill s ratio is found statistically insignificant, indicating that the issue of self-selection is not severe for this particular data set. This result also allows us to interpret the estimated coefficients in the second-stage as the marginal effects through the intensive margins of trade. The estimation results from the second stage of the TS-MM model are available in column (5) of Table 5. As in the Heckman procedure, the TS-MM estimate of the coefficient of the inverse Mill s ratio suggests that the potential selection bias is not statistically significant. Therefore, the estimated coefficients in the second stage can be read as the marginal effects via the intensive margins of trade (see equation (4c)). 23

25 According to the TS-MM estimates, countries with common borders trade more with each other, but countries with colonial ties do not trade significantly more. Since the results in Table 5 vary greatly across estimators, it is important to select the most appropriate estimation method for the gravity analysis. To this end, we conduct the Ramsey specification test (Ramsey, 1969) for all three models and report the test results at the bottom of Table 5. We find evidence that the Heckman model is misspecified. Although the PPML model survives the specification test, we consider it weakly dominated by the TS-MM model because the PPML method fails to disentangle the two margins of trade. Therefore, using both the statistical test and economic reasoning, we conclude that the TS-MM model is the most appropriate estimator for our case study. We finally draw policy implications based on the TS-MM estimates. We find that the elasticity of distance is about -1.99(= ), more than doubling the magnitude in SST. 16 In terms of the border effect, our finding echoes with Anderson and van Wincoop (2003) in that a common border enlarges the size of trade by more than 30 percent. While a colonial tie fosters trade primarily through creating trade partnership, a common language promotes both the likelihood and volume of trade. We also find that regional trade agreements facilitate trade through two channels. In addition to improving market accessibility, RTAs enlarge the volume of trade between countries by nearly 40 per-cent Conclusions 16 Nevertheless, the distance effect is within the range reported by Disdier and Head (2008). 17 Baier and Bergstrand (2007) report a RTA effect of a similar magnitude. 24

26 A vexing issue in the gravity model of international trade is the estimation in the presence of two stylized features of trade data: sample selection and heteroskedasticity. We contribute to the literature by proposing a Two-Step Method of Moments (TS-MM) estimator that deals with both problems. The novel estimator works as follows. In the first step, the estimator explains why trade takes place at all and sheds light on the extensive margin of trade. In the second step, the estimator characterizes the volumes of trade by a gravity equation in its multiplicative form, with the potential selection bias properly corrected for. The TS-MM estimator is shown robust to various patterns of sample selection and heteroskedasticity. As a practical advantage, it allows researchers to distinguish the extensive margin of trade from the intensive margin of trade. Therefore, the proposed method is particularly useful in the impact evaluation of policy initiatives aimed at broadening market access for exporters from the less or least developed nations. Several extensions of the proposed estimator are possible. For instance, the application to firm-level trade data might offer new perspectives to the interplay of firm characteristics and market accessibility. The proposed estimator is also a fine candidate for other models with constant-elasticity features, such as the wage earnings models in labor economics. 25

27 References Anderson, J. E., The Gravity Model, Annual Review of Economics (2011), Anderson, J. E., and E. van Wincoop, Gravity with Gravitas: A Solution to the Border Puzzle, American Economic Review 93 (2003), Anderson, J. E., and E. van Wincoop, Trade Costs, Journal of Economic Literature 42 (2004), Arkolakis, C., Market Penetration Costs and the New Consumers Margin in International Trade, Journal of Political Economy 118 (2010), Baier, S. L., and J. H. Bergstrand, Do Free Trade Agreements Actually Increase Members' International Trade? Journal of International Economics, 71 (2007), Baldwin, R., and D. Taglioni, Gravity for Dummies and Dummies for Gravity Equations, NBER Working Papers 12516, National Bureau of Economic Research, Inc. (2006). Bergin, P. R., and C.-Y. Lin, Exchange Rate Regimes and the Extensive Margin of Trade, NBER Chapters, in: NBER International Seminar on Macroeconomics (pp ), National Bureau of Economic Research, Inc. (2008). Besedes, T., and T. J. Prusa, The Role of Extensive and Intensive Margins and Export Growth, Journal of Development Economics 96 (2011), Burger, M., F. van Oort, and G.-J. Linders, On the Specification of the Gravity Model of Trade: Zeros, Excess Zeros and Zero-inflated Estimation, Spatial Economic Analysis, 4 (2009),

28 Chaney, T., Distorted Gravity: The Intensive and Extensive Margins of International Trade, The American Economic Review 98 (2008), Deardorff, A., Determinant of Bilateral Trade: Does Gravity Work in a Neoclassical World? in J. Frankel (Eds.), The Regionalization of the World Economy. (Chicago: University of Chicago Press, 1998). Disdier, A.-C., and K. Head, The Puzzling Persistence of the Distance Effect on Bilateral Trade, Review of Economics and Statistics 90 (2008): Eaton, J., and A. Tamura, Bilateralism and Regionalism in Japanese and U.S. Trade and Direct Foreign Investment Patterns, Journal of the Japanese and International Economies 8 (1994), Evenett, S. J., and W. Keller, On Theories Explaining The Success Of The Gravity Equation, Journal of Political Economy 110 (2002), Feenstra, R. C., J. R. Markusen, and A. K. Rose, Using the Gravity Equation to Differentiate among Alternative Theories of Trade, The Canadian Journal of Economics 34 (2001), Heckman, J. J. Sample Selection Bias as a Specification Error, Econometrica, 47 (1979), Helpman, E., M. Melitz, and Y. Rubinstein. Estimating Trade Flows: Trading Partners and Trading Volumes, The Quarterly Journal of Economics, 123 (2008), Hummels, D., and P. J. Klenow, The Variety and Quality of a Nation's Exports, American Economic Review 95 (2005),

29 Maddala, G. S., Limited-Dependent and Qualitative Variables in Econometrics, (Cambridge: Cambridge University Press, 1986). Martin, W., and C. S. Pham, Estimating the Gravity Model When Zero Trade Flows are Frequent, Economics Series 2008, Deakin University, Faculty of Business and Law, School of Accounting, Economics and Finance. (2008). Melitz, M. J. The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity, Econometrica 71 (2003), Novy, D. "International Trade without CES: Estimating Translog Gravity." Journal of International Economics 89 (2013), Ramsey, J. B. "Tests for Specification Errors in Classical Linear Least-Squares Regression Analysis," Journal of the Royal Statistical Society. Series B (Methodological) (1969), Ranjan, P., and J. L. Tobias, Bayesian Inference for the Gravity Model, Journal of Applied Econometrics 22 (2007), Santos Silva, J. M. C., and S. Tenreyro. The Log of Gravity, Review of Economics and Statistics 88 (2006), Santos Silva, J. M.C., and S. Tenreyro. Further Simulation Evidence on the Performance of the Poisson Pseudo-Maximum Likelihood Estimator, Economic Letters 112 (2011), Santos Silva, J. M., and Tenreyro, S. Trading Partners and Trading Volumes: Implementing the Helpman Melitz Rubinstein Model Empirically, Oxford bulletin of economics and statistics (2013). 28

30 Shepherd, B., Geographic Diversification of Developing Country Exports, World Development 38 (2010), Tinbergen, J., Les Données Fondamentales d'un Plan Régional, Revue Tiers Monde, 3 (1962), Westerlund, J., and Wilhelmsson, F. Estimating the Gravity Model without Gravity Using Panel Data, Applied Economics 43 (2011), White, H., A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity, Econometrica 48 (1980),

31 Appendix A. The data generating process that follows the Heckman specification In this appendix we describe the data generating process that produces the data samples for Table 3. Specifically, the data are generated according to the Heckman specification: (A1) M = β + x β + x β + µ, * ln( k) 0 1k 1 2k 2 k (A2) d = α + x α + x α + υ, * k 0 1k 1 2k 2 k (A3) d k = Id >, * ( k 0) (A4) * ln( Mk) dk ln( Mk) =. (A1) is the gravity equation after the logarithmic transformation. (A2)-(A3) characterizes the decision to trade. (A4) states that the observed trade, in logarithmic scale, is the product of the participation decision and the log-linearized gravity equation. As in (2a)-(2d), we alter the variance-covariance structure of µ k and υk to allow various patterns of heteroskedasticity, and we change the intercept in (A2) to consider different degrees of sample selection. The explanatory variables and the values of parameters are identical to their counterparts in the subsection 3.1. where the TS-MM specification is the underlying data generating process. 30

32 Appendix B. The data generating process that follows the PPML specification In this appendix we describe the data generating process that produces the data samples for Table 4. Specifically, the data are generated according to the PPML specification in SST: (B1) M = exp( β + x β + x β ) ε, * k 0 1k 1 2k 2 k (B2) M k * * M k if M k > c =, 0 otherwise where c is a parameter below which the outcome is zero and ε k is an idiosyncratic term governed by a lognormal distribution with mean unity. The explanatory variables and values of parameters are identical to their counterparts in the subsection 3.1. We alter the variance structure of ε k to allow various patterns of heteroskedasticity, and we change the value of c to consider different degrees of sample selection. Specifically, Table B reports the parameter assignment leading to the six scenarios in Table 4. Table B. Scenario construction when system (B1)-(B2) is the underlying process Parameter assignment Heteroskedasticity * (i) homoskedasticity Var( M k ) = 0.1 (ii) heteroskedasticity * Var( M k) = 0.1exp( X kβ ) (iii) super-heteroskedasticity * 2 Var( M ) = 0.1[exp( X β) + 0.5exp ( X β)] Sample selection (a) zeros account for 15% c = 0.75 (b) zeros account for 50% c = 1.25 k k k 31