metadata: authors: 0: author: Martin,William J.
1: author: Pham,Cong S.
count: World
docna: 0: docna: Estimating the gravity model when zero trade
            flows are frequent and economically determined
docty: Policy Research Working Paper
subtopic: Rural Settlements,Trade Policy,International Trade and Trade Rules,Trade and Services,Food Security,Agricultural Growth and Rural Development
teratopic: International Economics and Trade
abstracts: cdata!: This paper evaluates the performance of
            alternative estimators of the gravity equation when zero
            trade flows result from economically-based data-generating
            processes with heteroscedastic residuals and
            potentially-omitted variables. In a standard Monte Carlo
            analysis, the paper finds that this combination can create
            seriously biased estimates in gravity models with
            frequencies of zero frequently observed in real-world data,
            and that Poisson Pseudo-Maximum-Likelihood models can be
            important in solving this problem. Standard threshold–Tobit
            estimators perform well in a Tobit-based data-generating
            process only if the analysis deals with the
            heteroscedasticity problem. When the data are generated by a
            Heckman sample selection model, the Zero-Inflated Poisson
            model appears to have the lowest bias. When the data are
            generated by a Helpman, Melitz, and Rubinstein-type model
            with heterogeneous firms, a Zero-Inflated Poisson estimator
            including firm numbers appears to provide the best results.
            Testing on real-world data for total trade  throws up
            additional puzzles with truncated Poisson
            Pseudo-Maximum-Likelihood and Poisson
            Pseudo-Maximum-Likelihood estimators being very similar, and
            Zero-Inflated Poisson and truncated Poisson
            Pseudo-Maximum-Likelihood identical. Repeating the Monte
            Carlo analysis taking into account the high frequency of
            very small predicted trade flows in real-world data
            reconciles these findings and leads to specific
            recommendations for estimators.
repnme: repnme: Estimating the gravity model when zero
            trade flows are frequent and economically determined
display_title: Estimating the gravity model when
            zero trade flows are frequent and economically determined
listing_relative_url: /research/2015/06/24641545/estimating-gravity-model-zero-trade-flows-frequent-economically-determined
topicv3: Impact of Trade m1326020 1192,Agriculture and Food Security m1326205 435,Trade m1326924 698,Trade in Services m1326928 780,Rural Development m1504505 2667
publishtoextweb_dt: 2015-06-19T00:00:00Z
historic_topic: International Economics and Trade
no_of_pages: 61
author_keys: 0: author_key: 94758
1: author_key: 792459
model_extractions: WPS7308 Policy Research Working Paper 7308 Estimating the Gravity Model When Zero Trade Flows Are Frequent and Economically Determined _Will Martin Cong S. Pham_ Development Research Group Agriculture and Rural Development Team June 2015
Policy Research Working Paper 7308

# **Abstract**

This paper evaluates the performance of alternative estimators of the gravity equation when zero trade flows result from economically-based data-generating processes with heteroscedastic residuals and potentially-omitted variables. In a standard Monte Carlo analysis, the paper finds that this combination can create seriously biased estimates in gravity models with frequencies of zero frequently observed in real-world data, and that Poisson Pseudo-Maximum-Likelihood models can be important in solving this problem. Standard threshold–Tobit estimators perform well in a Tobit-based data-generating process only if the analysis deals with the heteroscedasticity problem. When the data are generated by a Heckman sample selection model, the Zero-Inflated Poisson model appears to have the lowest bias. When the data are generated by a Helpman, Melitz, and Rubinstein-type model with heterogeneous firms, a ZeroInflated Poisson estimator including firm numbers appears to provide the best results. Testing on real-world data for total trade throws up additional puzzles with truncated Poisson Pseudo-Maximum-Likelihood and Poisson PseudoMaximum-Likelihood estimators being very similar, and Zero-Inflated Poisson and truncated Poisson Pseudo-Maximum-Likelihood identical. Repeating the Monte Carlo analysis taking into account the high frequency of very small predicted trade flows in real-world data reconciles these findings and leads to specific recommendations for estimators.

This paper is a product of the Agriculture and Rural Development Team, Development Research Group. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The authors may be contacted at w.martin@cgiar.org.

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

# **Economically Determined**

Will Martin and Cong S. Pham *

# JEL: C21, F10, F11, F12, F15

* At the time of writing Will Martin was Research Manager, Development Research Group, the World Bank, and Cong S. Pham was Lecturer, School of Accounting, Economics and Finance, Deakin University, respectively. Martin is now at the International Food Policy Research Institute (IPRI). Their email addresses are w.martin@cgiar.org and cpham@deakin.edu.au. The authors would like to thank João Santos Silva and Silvana Tenreyro for their helpful comments and for access to their data. They also would like to thank Jonathan Eaton, Caroline Freund, Russell Hillberry, Hiau Looi Kee, Daniel Lederman, Douglas Nelson, Guido Porto, Shang-Jin Wei and Tiemen Woutersen for valuable comments on earlier versions of this paper. The usual disclaimer applies.
# **I. Introduction**

The gravity model is now enormously popular for analysis of a wide range of trade questions. This popularity is due to its apparently good performance in representing trade flows and to the strong theoretical foundations provided in papers such as Anderson (1979) and Anderson and van Wincoop (2003). Until recently, the gravity equation was almost invariably estimated using data sets converted into logarithms and truncated to contain only positive trade flows. The problem of zero trade flows was rarely emphasized, partly because trade theories were silent on their causes, partly because of a lack of recognition of their frequency, and partly because of the convenience of the log-linear estimator.<sup>1</sup> Recently, there has been growing recognition among trade economists that zero trade flows do not occur randomly and that using truncated samples in logarithms may yield misleading results. Melitz (2003) and myriad subsequent studies have added firm-level foundations to earlier models of zero trade flows, such as those based on the Tobin (1958) and Heckman (1979) models. Santos Silva and Tenreyro (SST) (2006) raised an important additional concern by pointing to the possibility of serious bias from combinations of heteroscedasticity and nonlinearity in the gravity model. Helpman, Melitz and Rubinstein (HMR) (2008) pointed to another potential source of bias in coefficient estimates—failure to account for the increase in demand for exports as the number of exporting firms increases.

Like SST (2006), we believe that at least part of the process of identifying the right estimator must involve analysis of generated data whose statistical properties are known. In this paper we evaluate different estimators of the gravity equation in the presence of pervasive zero trade flows _and_ under different patterns of heteroscedasticity. Because we are unsure of the process generating real-world data, we test alternative estimators on data sets generated using three different economic models in which the probability of a nonzero trade flow and the volume of trade when trade is nonzero are jointly determined: the threshold-Tobit model of Eaton and Tamura (1994); the Heckman model; and an extended Heckman model of the type used by HMR (2008).

> 1 As shown in Annex A, zero trade flows account for more than 40% of the possible bilateral trade flows in country-level data and more than 60% in U.S. 10-digit product-level export data, and the probability of zero trade flows is strongly linked to the explanatory variables in gravity models. The frequency of zero trade flows is much higher at the more disaggregated levels of product and firm trade that are increasingly being used for analysis.

2
Each of the three data-generating processes used in our Monte Carlo simulations provides a plausible economic representation of observed trade flows, and can be parameterized to represent the frequency of zero trade flows observed in real trade data. The resulting data sets differ from those used by SST (2006) in containing true zero observations, and in being based on economic models that jointly determine whether to trade and, if so, the volume of trade. They are _not_ constant-elasticity models because constant-elasticity models cannot generate finite probability mass at zero trade. Instead of moving from a constant-elasticity model to a model containing zeros by transforming the latent variables from constant-elasticity models with a negative-binomial distribution as in SST (2011), we use limited-dependent variable models whose properties have been thoroughly explored in the literature.

The paper is organized as follows. In the next section we provide a brief review of the relevant literature. Section III discusses the econometric problems associated with zero trade flows. Potential approaches to estimating the gravity equation in the presence of frequent zero trade flows are considered in Section IV. Section V covers the DGPs of our Monte Carlo simulations and the simulation results. Section VI provides an empirical implementation of the gravity estimation using country-level trade data and section VII discusses reconciling Monte Carlo and empirical findings. Finally, the paper concludes in Section VIII.

# **II. Review of Relevant Literature on Estimating the Gravity Model**

Eaton and Tamura (1994) appear to have been the first to provide a systematic approach to estimation of the gravity equation as a limited-dependent model. Their model builds on the Tobin (1958) model, where non-limit observations arise when the latent dependent variable (including the error term) crosses a threshold value.<sup>2</sup> Another popular approach to estimation uses the Heckman (1979) model to deal with the presence of zero trade flows (see, for example, Francois and Manchin (2007) and Lederman and Ozden (2007)). The third approach that we consider builds on the Heckman model to take into account the extensive margin of firm involvement in trade (HMR 2008).

> 2 Note that in the Tobit model the effect of the coefficient estimate on the explanatory variable x cannot be interpreted as the effect of x on the dependent variable y. Rather, as McDonald and Moffitt (1980) correctly point out, it is the effect of x on two components of y: “(1) the change in y for those observations above the limit, weighted by the probability of being above the limit; and (2) the change in the probability of being above the limit, weighted by the expected value of y if above” (pg. 318-319).

3
In an important and influential article, SST (2006) emphasized the potential econometric problems resulting from a combination of heteroscedastic residuals and zero trade flows. Using Monte Carlo simulations, they showed that traditional estimators may yield severely-biased parameter estimates and identified the Poisson Pseudo Maximum Likelihood (PPML) estimator as the preferred estimator for dealing with these problems. The data sets used for the main experiments in their 2006 paper included no zeros (SST 2011, p220), suggesting that their important findings about the bias and variance of standard estimators arise from the combination of nonlinearity and heteroscedasticity highlighted in the paper, rather than from success in dealing with the presence of zero observations. While their paper included some experiments with zeros generated by rounding errors, a process quite different from economic processes generating zeros such as those assumed when applying limited-dependent variable estimators.

The tractable and apparently robust PPML estimator proposed by SST is now widely used in estimation of gravity models (see, for example, Shepherd 2010; Anderson and Yotov 2012). SST show why the normal equations used to solve the PPML estimator should make it more robust than OLS-based estimators when the function is nonlinear and the errors are heteroscedastic, and show using Monte Carlo simulations that its coefficients are much less subject to bias in this situation. Based on their evidence, the case for the PPML estimator seems extremely strong for analysis of nonlinear relationships in models where zero values are infrequent.<sup>3</sup> However, because the tests in their 2006 paper were based on samples containing no zero values, the results obtained cannot be expected to provide a reliable guide to the robustness of the PPML estimator in the presence of economically-determined zero values.

Perhaps a more fundamental problem for reliance on the PPML estimator applied to censored samples (including the zeros) is that the move from the traditional, discrete version of the Poisson distribution to the continuous needed for the gravity equation removes the ability to have positive probability mass at the limit observation (Ilienko 2013, p2). While a discrete version of the Poisson distribution could adequately capture the presence of large shares of limit observations of the type identified by Tobin (1958), the

> 3 Heteroscedasticity in nonlinear models with relatively few zero observations arises in many important applications such as consumption, investment and money demand systems; and representative models for consumer demand, firm costs and profits. The results of SST (2006) appear to deserve much more attention for estimation in this context.

4
continuous version of this distribution is unable to adequately represent these cases. This problem appears worse than the common concern that empirical distributions may be overdispersed relative to the Poisson, in which the mean and the variance are equal. Especially in cases where large fractions of a sample are zero trade flows that are retained in the analysis, the continuous Poisson specification appears likely to be seriously mis-specified. The Zero-inflated Poisson estimator developed by Lambert (1992) and applied to the gravity model by Burger, van Oordt and Linders (2009) might be one way to retain the advantages of the Poisson estimator while allowing for substantial numbers of zeros.

On the theoretical front, there have also been major recent developments. HMR (2008) provide a heterogeneous-firm trade model that both allows for frequent and asymmetric patterns of zeros in bilateral trade data and introduces a number-of-firms variable, correlated with the probability of non-zero trade that affects trade volume through a love-of-variety effect. They extend the Heckman model so that the standard gravity explanatory variables determine not only the probability of positive bilateral trade (i.e. the probability that the most productive firm finds it profitable to export to a foreign destination) but also the fraction of firms that are productive enough to enter foreign markets when positive trade occurs. This overcomes an omitted-variable problem different from that created by the limited-dependent feature of the data.

# **III. Intuition on the Econometric Problems Associated with Zero Flows**

Since Tobin’s famous (1958) paper, it has been known that zero values of the dependent variable can create potentially large limited-dependent variable biases in parameter estimates, even in linear models, if the estimator used does not allow for this feature of the data generating process. Heckman (1979) generalized Tobin’s approach to estimation in the presence of this problem, casting it in the context of estimation using samples with nonrandom sample selection. A nonlinear version of Heckman’s formulation of the problem is presented in a two equation context: where xi and zi are vectors of exogenous regressors while β and α are vectors of parameters, and E(uji) = 0, 5
In Heckman’s formulation, equation (1) is a behavioral equation and equation (2) is a sample selection rule that determines whether observations in equation (1) are observed or not. In our context, this model results in: A key problem for estimation of equation (1) in the presence of sample selection is that its residuals no longer have the properties assumed in standard regression theory, particularly a zero mean. In this situation, standard regression procedures result in biased estimates of the coefficient vector β because they omit relevant explanatory variables.

The Tobit model is a special case of the Heckman model in which the right hand sides of (1) and (2) are identical (Heckman 1979). In this case, a simple diagram (Figure 1) provides important insights into the key problems arising in applying standard approaches to estimation. Our version of the figure includes both the nonlinearity and heteroscedasticity of the gravity equation emphasized by SST (2006) and the limiteddependent variable problem.

6
**Figure 1. Limited-dependent variable bias in the gravity model**

<!-- Start of picture text --> y, y*<br>• • •<br>•<br>• •<br>• •<br>• •<br>•<br>•<br>•<br>• •<br>• • •<br>•<br>•<br>•<br>•<br>• •<br>• •<br>•<br>•<br>• •<br>•<br>•<br>0 • • • • • • • • • • • •<br>x<br> -k<br><!-- End of picture text --> Figure 1 shows the relationship between a latent variable, y<sup>*</sup> , and an explanatory variable, x, with the non-stochastic relationship represented by the solid curve, and individual data points by the bullets. The hollow bullets correspond to any value of y<sup>*</sup> less than or equal to zero. In the censored regression case, the residuals associated with these low values of the latent variable will be replaced by positive residuals that lead to a zero value of the dependent variable, a change represented by the dashed arrows in the diagram. With this model, using a sample containing the zero values—for which E(u)>0—biases the coefficient on the slope and results in an estimate something like the dashed curve (Greene 1981).

Another insight from careful examination of Figure 1 is the difference between the case of censoring shown in the diagram and the case of truncation where all of the observations at the limit (zero in this case) are discarded from the sample. In the censoring case, the error terms on all limit observations but those where y* was precisely zero have been increased. In the case of truncation, observations with negative errors, like the rightmost hollow bullet in the diagram, are more likely to be excluded from the truncated sample and this creates a situation where E(ui)>0. Intuitively, the transformation of the 7
error terms associated with the censored sample seems likely to be greater than that associated with the truncated sample. This suggests that moving from a truncated estimation procedure to one which retains the zero values without changing the estimation procedure may result in greater bias.

The diagram shows another feature of the residuals from application of a regression estimator designed for non-censored problems. Such estimators are likely to find residuals that are both large and serially dependent at both ends of the regression line—note the consistently positive apparent residuals relative to the dashed line in Figure 1 for observations near zero and near the largest observations in the sample. The estimated regression line is likely to be strongly influenced by such extreme observations, particularly when using ordinary least squares (Beggs 1988). However, the implications of this are likely to vary considerably between estimators, given the different weighting implied by the normal equations for the different estimators (SST 2006).

With a little more imagination, Figure 1 can also help visualize a different problem associated with heteroscedasticity even in the absence of the nonlinearity problem identified by SST—a problem associated with mis-measurement of the probability of observations at the limit. Following the notation of Terza (1985), the log-likelihood function for a sample generated by a Tobit-type process is given by L = Σi _di_ log( _f_ ( _yi_ )) + (1- _di_ )log(F( _0_ )) where _di_ is an indicator value taking the value 1 for a non-limit observation; ( _f_ ( _yi_ )) is the density function for non-limit observation i, and F( _0_ ) is the distribution function showing the probabilities of zero observations. If the variance relevant to assessing the value of the distribution function is incorrectly specified—as it would be with a model assuming homoscedasticity—this is clearly likely to change the realized values of the distribution function for limit observations, causing potentially-serious bias in parameter estimates, as pointed out by SST (2014) in their assessment of the HMR (2008) estimator.

# **IV. Potential Approaches to Estimating Gravity Models with Zeros**

In estimating the gravity model when there are many zero observations, some key questions must be confronted:

- i. Which functional form to use for the explanatory variables?

- ii. Whether to truncate or censor the zero observations? and

8
iii. What estimator to use?

There seems to be widespread agreement that the gravity model involves relationships between variables that are nonlinear in levels, with a constant-elasticity functional form almost always used. This implies a relationship: (5) yi = exp(xiβ) + εi where yi represents bilateral trade; xi is a vector of explanatory variables (some of which may be linear, some in logarithms and some dummy variables) for observation i; β is a vector of coefficients; and εi is an error term whose variance, unlike those of equations (1) and (2), need not be constant across observations. As noted by SST, the gravity model has traditionally been estimated after taking logarithms, which allows estimation using linear regression techniques. However, this specification will only involve homoscedastic residuals if εi = exp(xiβ).vi where vi is distributed independently of xi with a zero mean and constant variance. Of course, if this is the true model, the constant-elasticity model estimated in levels will have heteroscedastic errors.

The use of the logarithmic transformation for the dependent variable creates another immediate difficulty when trade is zero, since the log of zero is undefined. The most common response to this problem has been to truncate the sample by deleting the observations with zero trade. This is inefficient, since it ignores the information in the limit observations, and may lead to bias as noted in the previous section. Many studies have replaced the value of imports by the value of imports plus one, allowing the log of the zero values to take a zero value. Others have estimated in levels using nonlinear estimation techniques which allow the zero values to be retained. However, as we have seen from Figure 1, retaining the zero observations without using an estimator that accounts for the limited-dependent nature of the model may lead to more serious bias than simply truncating the sample. What seems to be needed is an approach to estimation that systematically takes into account the information in the limit observations. Once the decision to include the limit observations is taken, a number of other decisions must be confronted. We find it useful to lay out the choices with the decision tree in Figure 2.

At the first stage of the decision tree, analysts must decide between parametric models and semi-parametric models. Semi-parametric models (Chay and Powell 2001) avoid specifying a distribution for the residuals, sometimes at the expense of computational efficiency, and estimate parameters using methods such as Powell’s (1984) Censored Least 9
Absolute Deviations (CLAD) model. While such models have been extended to deal with nonlinear models (Berg 1998), it appears that such applications have been infrequent to date. Certainly, most of the focus in estimation of gravity models has been on the parametric branch of the decision tree.

**Figure 2. Choosing an estimator for the gravity model with limited-dependent trade**

<!-- Start of picture text --> Parametric Semi-Parametric<br>Tobit/Heckman Two-Part<br>Normal<br>Other Normal<br>Residuals<br>Other<br>Residuals<br>2-Step MLE 2-Step MLE<br><!-- End of picture text --> If a parametric approach is taken, the first decision required is whether to adopt a Tobit/Heckman-type model (Amemiya 1984), or a Two-Part model (Jones 2000). The Two-Part model has the desirable feature of allowing the sample selection and the behavioral equations to be estimated independently (Duan et al. 1983). However, this simplification comes at the expense of assuming that these decisions are taken independently, something that seems implausible in a world where decisions on whether to export and on how much to export and how much to ship should they choose to export are taken by individual firms based on the profitability of exports of their particular products to particular markets. In most cases, it seems to us that the variable of interest is the latent variable for the desired level of trade, y<sup>*</sup> , for which the Tobit/Heckman approach seems the most suited.

10
A key argument for the Two-Part model has been a belief that the performance of the sample-selection models is irretrievably compromised by statistical problems, and particularly multicollinearity. Leung and Yu (1996) show that these problems may be less of a problem for practical implementation than was thought based on earlier studies such as Manning, Duan and Rogers (1987), which included insufficient variation in the exogenous variables to mitigate the multicollinearity between these variables and the inverse Mills ratio added to the behavioral equation when estimating using the two-step estimator of the Heckman (1979) sample selection model.<sup>4</sup> Based on a detailed review of the literature on the Heckman correction for sample selection bias, Puhani (2000) found that the full-information maximum likelihood estimator of Heckman’s model generally gives better results than either the two-step Heckman estimator or the Two-Part model, although the Two-Part model is more robust to multicollinearity problems than the other standard estimators. Clearly, these results suggest that the consistency of the data with the assumptions of the Tobin/Heckman models should be examined carefully.

Whichever parametric estimator is chosen, an assumption about the distribution of the residuals must be made. The most common approach is to assume that the residuals are distributed normally. However, alternative assumptions are sometimes used, including the Poisson distribution highlighted by SST (2006) or the Gamma distribution that they also examined. The decision about which distribution to use need not be based solely on judgments about the actual distribution of the residuals. The essence of SST’s recommendation of the PPML estimator was that it is more robust to heteroscedasticity than one based on the normal distribution when the residuals are actually normally distributed.

The first part of the Two-Part approach is the use of a qualitative-dependent model such as Probit to determine whether a particular bilateral trade flow will be zero or positive. The second part is to estimate the relationship between trade values and explanatory variables using only the truncated sample of observations with positive trade (Leung and Yu 1996, p198). Potential estimators for this stage include the standard approach of OLS in logarithms; the nonlinear least squares (NLS) model used by Frankel and Wei (1993);

> 4 Leung and Yu (1996, p201) show that problems with Heckman’s two-step estimator are more likely when there are few exclusion restrictions; a high degree of censoring; limited variability among the exogenous regressors; or a large error variance in the choice equation.

11
and the PPML and Gamma Pseudo-Maximum Likelihood (GPML) estimators discussed by SST. Estimation of the bias of truncated single-equation estimators will therefore help assess the suitability of Two-Part models.

Under the Tobit/Heckman limited-dependent branches of the decision tree, a decision must be made about whether to estimate using two-step estimators of the type proposed by Heckman (1979) or a maximum likelihood approach (see Tobin (1958), Puhani (2000) and Jones (2000)). One concern with the Heckman two-step estimator is that the variable that adjusts for the probability of sample selection<sup>5</sup> may be close to a linear function of the other explanatory variables, resulting in multicollinearity problems (Puhani 2000). A second concern is that this approach introduces heteroscedasticity into the residuals. An alternative, nonlinear, approach to estimating Heckman’s second-step equation is provided by Wales and Woodland (1980, p461). We do not consider this estimator because it performed poorly in their simulations.

The performance of the Heckman-based models appears to depend heavily on whether both equations (1) and (2) are active, and whether there are at least some variables included in equation (1) but excluded from equation (2). Leung and Yu (1996) concluded that this estimator with some excluded variables in the behavioral equation outperformed other estimators for limited-dependent variables.

# **V. Monte Carlo Simulations**

In this section of the paper, we provide a description of the approach we used to create zero observations with frequencies similar to those observed in real-world data and to test the ability of different estimators to retrieve the parameters of these data generating processes. We also followed the approach of SST and of Westerlund and Wilhelmsson (2011) to creating heteroscedastic errors—that is, we posited a range of different types of heteroscedasticity, and tested their implications for the performance of different estimators. Because we know the true parameters, we focus on the bias and imprecision of the coefficient estimates, rather than the goodness of fit measures emphasized by GomezHerrera (2013).

> 5 Which is the inverse Mills ratio for the particular observation (Heckman 1979).

12
We adopted the following specification of the gravity equation from SST (2006).<sup>6</sup> (6) yi = exp(xiβ) + εi =exp(xiβ).ηi = exp(β0 + β1x1i + β2x2i).ηi where ηi ≡1+[εi/exp(xiβ)]; x1 is a binary dummy, designed to mimic variables such as border dummies, that equals 1 with probability 0.4; x2 is a standard normal variable to mimic the behavior of continuous explanatory variables such as the log of distance or income; and the data are randomly generated using β0=0, β1= β2 =1. Following SST, we assumed that ηi is log-normally distributed with mean 1 and variance σi<sup>2</sup> .

To assess the sensitivity of the different estimators to different patterns of heteroskedasticity, we used Santos Silva and Tenreyro’s four cases: Case 4: σi<sup>2</sup> = (exp(xiβ))<sup>-1</sup> +exp(x2i) ; V(yi׀x) = exp(xiβ) + exp(x2i).(exp(xiβ))<sup>2</sup> where Case 1 involves an error term that is homoscedastic when estimated in levels; Case 3 is homoscedastic when estimated in logarithms; Case 2 is an intermediate case between 1 and 3; and Case 4 represents a situation in which the variance of the residual is related to the level of a subset of the explanatory variables, as well as to the expected value of the dependent variable. To help interpret such a large number of results, we also present, in Appendix Table 4, measures of the average absolute bias for each coefficient across the four cases and the average absolute bias across both coefficients, together with averages of their standard errors.

We consider two broad types of data-generating-process (DGP): one based on the Eaton-Tamura model and another based on the Heckman model. We also examine a special case of the Heckman DGP, including an additional variable for the number of products traded, proposed by HMR (2008), whose robustness to heteroscedasticity bias does not appear to have been examined on data sets with known parameters, although SST (2014) raise concerns about potentially serious bias. These DGPs have solid economic foundations and can generate data sets with the required fractions of zero trade values. We considered using the SST (2011) DGP but its economic underpinnings and statistical properties were less obvious to us than was the case for the three more conventional limited-dependent

> 6 These equations correspond exactly to the SST (2006) specification for equations (14) and (15).

13
variable DGPs that we used and, in any event, its properties have already been examined using Monte Carlo techniques. We investigated the Two-Step Method of Moments estimator of Xiong and Chen (forthcoming), but did not find encouraging results with our sample.

# **V.1. ET-Tobit data generating process (DGP)**

For the threshold-Tobit model, we ensured that a significant number of observations would have zero values by adding a negative intercept term, -k, in the levels version of the datagenerating equation, and then transforming all realizations of the latent variable with a value below zero into zero values. We refer to this DGP as the ET-Tobit DGP because it is the model underlying the Eaton and Tamura (1994) estimator. It has the interpretation of introducing a threshold that must be exceeded before trade actually occurs. It differs fundamentally from the rounding approach used by SST (2006) and the approach of setting observations to zero randomly or for particular groups used by Martinez-Zarzoso (2013).

The data generating process for these initial simulations was equation (6) augmented with the negative intercept term, -k, that defines the ET-Tobit model (7) yi<sup>*</sup> = exp(xiβ) + εi – k ≡ exp(xiβ) ).ηi - k = exp(β0 + β1x1i + β2x2i).ηi - k i = yi<sup>*</sup> if yi<sup>*</sup> ≥ 0; yi = 0 if yi<sup>*</sup> < 0 Within these samples, we found that a value for k of 1 provides numbers of zero trade values consistent with the 40-50 percent of zero values frequently observed in analyses of total bilateral trade. As shown in Appendix Table 1, a k value of 1.5 generated higher shares of zeros. The analysis was performed in Stata 9.2, using double precision to minimize numerical errors, with samples of 1000 observations, replicated 10,000 times as in SST (2006).<sup>7</sup>

# _Semi-Parametric Estimators_

Given the pervasive uncertainty about the distribution of the errors, and the relatively poor results obtained using standard limited-dependent variable estimators, it seemed important to investigate the performance of the semi-parametric, Least-Absolute-Deviations (LAD) approach proposed by Powell (1984). Although this model has not, to our knowledge, been

> 7 Our first estimation task was to replicate the simulations of SST (2006). While our results are not exactly the same as theirs because of the stochastic nature of the analysis, they are completely consistent. The replicated simulation results are available upon request.

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applied to gravity models, Paarsch (1984) found the censored version of this model to give satisfactory results in large samples with censored data. Because the estimator for this model in Stata requires linearity, we examined the model in logarithms to make an initial assessment of its suitability. We considered first the truncated LAD estimator based only on the positive observations, and then turned to Powell’s (1984) censored LAD estimator. The results are presented in Table 1.

From the results in Table 1, it appears that the Truncated LAD estimator has quite small bias in Case (3), when the heteroscedasticity is consistent with the functional form adopted for estimation. However, it appears to be very vulnerable to heteroscedasticity. In cases (1), (2) and (4), the bias of the Truncated LAD was typically 20 to 30 percent. The Censored LAD estimator performed worse than the truncated estimator in all cases using k=1, with bias of 0.5 or greater. With k=1.5, the bias averaged 70 percent in cases (1) to (3). Given its overall poor performance we did not pursue the LAD estimator further. _Standard Single-Equation Estimators_ In this section, we first considered the performance of eight parametric estimators that do not account for the limited-dependent nature of the DGP. Despite their failure to account for the limited-dependent nature of the data-generating-process, they might plausibly be the best estimators of the key underlying parameters, just as truncated OLS was widely believed to be the best estimator for limited-dependent variable models for the reasons outlined in Manning, Duan and Rogers (1987). The specific estimators considered were: (i) the traditional truncated OLS-in-logs regression, (ii) its censored counterpart with 0.1 added to the log of output to allow taking logs of the dependent variable, (iii) Truncated nonlinear least squares (NLS) in levels, (iv) Censored NLS in levels, (v) a Poisson PseudoMaximum Likelihood (PPML) estimator, (vi) a truncated Pseudo-Maximum Likelihood estimator, (vii) a Gaussian Pseudo-Maximum Likelihood (GPML) estimator, and (viiii) a Negative Binomial Maximum Likelihood estimator (NBML). We report the results for k=1 in Table 2. Summary results reporting the average absolute bias and standard errors across the four cases, and across the two coefficients, are included in Appendix Table 4.

An important feature of the results for the truncated OLS-in-logarithms model is its apparently strong sensitivity to heteroscedasticity. In Case 3, where the error distribution is homoscedastic in the log-linear model, the bias in the estimates is around 6 percent for both coefficients. However, when we move to cases involving errors that are 15
heteroscedastic in the log-linear equation, the bias of the estimated coefficient on the normally-distributed variable (β2) rises to around 20 percent in cases 1, 2 and 4. For this estimator, unlike many others, the biases in the coefficients are generally similar for the dummy variable, x1 and the normally distributed explanatory variable, x2. The censored OLS model in logarithms (with 0.1 added to allow inclusion of the zeros) produces results that are generally inferior to those obtained from the truncated OLS estimator. Except in Case 4, the biases were smaller for the coefficient estimate on x2 than on x1.

Truncated NLS is the levels counterpart to the traditional OLS-in-logs estimator and the second stage of a two-part estimator. It has lower bias than its logarithmic counterpart in cases 1 and 2, but is inferior in cases 3 and 4. In Case 3, the bias of this estimator is 40 percent for β2, around seven times the bias of truncated OLS in logs. Perhaps the best thing that can be said for the truncated NLS estimator is that it is consistently less biased than the censored NLS regression model. The superiority of the truncated OLS and NLS models over their censored regression counterparts reinforces our intuition that just solving the “zero problem” and adding the zero-valued observations to the sample is likely quite an unhelpful strategy when the data are generated by a process that determines the levels of observed trade and the probability of zero trade together.

The PPML estimator in levels yielded estimates that were strongly biased in all cases. Because this equation was estimated with the dependent variable in levels, the underlying error structure is consistent with the estimator in case 1, but, In this case, the bias in the estimate of β1 was 0.28 and of β2 was 0.25. The bias was very similar for each of the other cases. This confirms SST’s hypothesis that the PPML estimator is robust to heteroscedasticity in the residuals while demonstrating its potential vulnerability to limited-dependent variable bias.

The truncated PPML estimator has much lower bias than the PPML estimator in all cases, again reinforcing our belief that truncated estimators are to be preferred over censored estimators. The truncated PPML, in fact, outperforms truncated OLS in logs in all cases considered, including Case 3 where heteroscedasticity is not a problem for the OLS estimator.

The GPML estimator has very large bias in almost all cases, reinforcing the SST conclusion that this estimator has little role to play in estimation of the gravity model. The NBML estimator is inferior to the two preferred estimators—truncated OLS in logs and 16
truncated PPML—in all except Case 1 with k=1. This good performance seems likely to be an outlier, since the NBML is strongly biased in Case 1 when k=1.5 (see Appendix Table 2).

When the k=1.5 sample is considered (see Appendix Table 2), the superior performance of truncated PPML relative to full-sample, censored PPML is again evident. It has by far the lowest mean absolute bias across the four cases considered and is superior to its nearest competitors (the two OLS in logs models) even in case 3, where the pattern of heteroscedasticity is consistent with the model in logarithms. This result suggests a case for use of the truncated PPML, at least in exploratory analyses. It retains the robustness to heteroscedasticity emphasized by SST (2006), but reduces the vulnerability to the limiteddependent variable bias emphasized in the earlier literature originated by Tobin (1958).

The summary results in Appendix Table 4 give the average absolute bias and standard errors across the four cases and again highlight the superiority of truncated PPML in terms of bias. For both coefficients, it has by far the lowest average absolute bias—less than half that of other estimators in almost every case, and on average across the two coefficients. The standard errors associated with this estimator are almost always among the lowest—although OLS in logs and the NBML have lower standard errors around their highly biased estimates.

# _Limited-Dependent Variable Estimators_

In this section, we considered eight estimators designed specifically for estimation with limited-dependent variables. The estimators considered are: (i) the Eaton-Tamura model with the dependent variable in levels; (ii) the Eaton-Tamura model in logs; (iii) a TobitPoisson model<sup>8</sup> like that of Terza (1985); (iv) the maximum likelihood version of the Heckman model estimated in logs (see Amemiya (1984)); (v) the Heckman (1979) 2SLS estimator in logs; (vi) the ML version of the Heckman estimator in levels; (vii) the 2SLS version of the Heckman model estimated in levels by nonlinear least squares; and (viii) the Zero-Inflated Poisson Maximum likelihood estimator (ZIP) (Lambert 1992).

Key results for cases with k = 1 are presented in Table 3, while results for k=1.5 are presented in Appendix Table 3. The Eaton-Tamura Tobit estimator in levels has quite

> 8 To program the likelihood function in Stata, we needed to replace the factorial function with exp(ln(gamma(y+1))) to allow evaluation with non-integer values of the dependent variable.

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low bias for both coefficients—around three percent—in Case 1. However, it exhibited much larger bias in all other cases. The same model in logarithms suffers from quite large bias (25 percent) in cases 1 and 2 but around 6 percent in Case 3, where the pattern of heteroscedasticity is consistent with the estimator. The average bias across cases was much lower for the log version than for the version in levels. The poor result for the ET estimator in levels is somewhat surprising given that the ET-Tobit model is the data-generating process, and that the ET-Tobit model in levels might be expected to avoid the bias associated with the combination of heteroscedasticity and log-linearization.

The performance of the Poisson-Tobit estimator is much worse than the ET-Tobit in all cases. It yields large bias—20 percent or higher in all but one case—and large standard errors in all cases. Because this model lacks the additive intercept that is central to the ET-Tobit data generating process, it suffers from similar problems to the censored PPML estimator.

The Heckman ML estimator in logs performed poorly in virtually every case. The bias of this estimator was smallest in case (3), where the form of heteroscedasticity is consistent with the estimator, but was still 14 percent for each coefficient. The Heckman 2SLS estimator in logs was worse in all cases, while the Heckman ML in levels was sharply inferior in cases 3 and 4. The Heckman 2SLS estimator in levels had the lowest average absolute bias of any of the limited-dependent variable estimators, although this bias was over 16 percent for β1 in all four cases. The ZIP estimator was uniformly inferior to the Heckman 2SLS estimator.

We had expected the ET-Tobit models to outperform alternative models such as Heckman given that the DGP in this case is ET-Tobit. One potential approach to improving the performance of the ET Tobit model might be to adjust for heteroscedasticity. In Table 4, we report results including the adjustments proposed by Maddala (1985), in which the � � error variance is specified by the process �� ����������� for the log-linear model, with γ and δ are estimated using nonlinear least squares. For the linear-in-levels model, � � the specified error process is �� �����exp������ . These models should capture heteroscedasticity of the types considered in Cases 1 to 3. The resulting estimates were reasonably satisfactory for Cases 1 and 3 using estimators in logs and for Case 1 using estimators in levels. When the true restriction that β0=0 was imposed, the bias in β1 and β2 fell to very low levels. This suggests that it may be important in estimation of the 18
heteroscedasticity-corrected model to use restrictions on the parameter values when prior information or prior testing suggests that these are valid.

A feature of the results for this DGP is that the limited-dependent variable models were, in almost all cases, outperformed in terms of both bias and standard error by the truncated PPML estimator. This suggests that a two-part approach to estimation might be best if the underlying DGP appears to be of the ET-Tobit type, with a first stage used to assess whether trade occurs or not, and the PPML estimator applied in the second stage. This is clearly an imperfect strategy given that the truncated PPML estimator has an average bias across the two coefficients of 7 percent, but the average bias of even the best limited-dependent estimator—the Heckman 2SLS in levels—is around 50 percent higher. The superiority of the truncated PPML estimator over all other estimators appears to be robust to the frequency of zeros in the sample. It also arises in the sample with k=-1.5. It appears worthwhile to pursue limited-dependent-variable estimation only if an appropriate correction for heteroscedasticity can be made—perhaps by following a general to specific approach to estimation in order to introduce as many justified restrictions as is possible.

# **V.2. Heckman sample selection DGPs**

In this section, we examine the performance of the estimators when the data are generated using Heckman Sample Selection models. For this purpose, we first generate the data with frequent zeros based on a standard Heckman sample selection framework. We then construct HMR-type data sets with two sources of omitted variable bias— sample selection and omission of the number of firms engaged in exporting.

# _The Standard Heckman DGP_

For the standard Heckman DGPs, we assume that the selection equation contains at least one variable that is excluded from the behavioral equation. To generate the data for this test, we use equation (6) for the behavioral equation, but add the following sample selection equation: (8) y2i= exp (ziα).η*i ≡ exp (ziα)+ u*2i 19
where ziα ≡ α0+ α1 x1i+ α2 x2i+ α4 x4i; η*i ≡1+[u*2i/exp(ziα)]; x4 is a variable included in the selection equation but excluded from the behavioral equation; and the correlation between ln(η*i) and ln(ηi) is set at 0.5.<sup>9</sup> The dependent variable in (6) is observed when y2i>1.

The results for standard estimators with this DGP are presented in Table 5 while those based on limited-dependent estimators are presented in Table 6. Results for average bias and standard errors across the four cases are summarized in Appendix Table 4. Zero trade flows account approximately for 30% of the observations in this sample.

In Table 5, many of the results are similar to those obtained using Eaton-Tamura data sets. For example, comparison of the truncated and censored versions of each estimator reveals consistently better performance of the truncated versions. Also similarly, the NLS estimators generally have smaller biases and standard errors in cases 1 and 2 while the than in cases 3 and 4. With this DGP, the censored OLS in logs estimator has quite serious bias in case 3, even though the form of the dependent variable is consistent with homoscedasticity.

The Poisson estimator outperforms the standard log-linear estimators in most cases. Specifically, in cases 1 2 and 3 the bias on the estimate for β1 is about 14%. The bias is smaller in case 4 but the standard error is substantially larger. The bias on the—frequently more important—estimate of β2 is substantially lower than for β1. The truncated PPML performs better than the censored PPML in almost all cases, reinforcing our finding using the ET-Tobit DGP. As in the case of the Tobit DGP we find that as the number of zeros increases so does the bias and standard error of the parameter estimates generated by the PPML estimator.<sup>10</sup> The GPML and NBML estimators suffer from very large bias in almost all cases. The standard errors associated with these estimators are also large. This finding reinforces our earlier findings—and those of SST—that these estimators do not appear to perform well for estimation of the gravity model.

When we turn to limited-dependent variable estimators, we find that the ET-Tobit estimators are frequently seriously biased. In some cases, such as the ET-Tobit estimator

> 9 Note that ln(η*i) and ln(ηi) correspond exactly to the two error terms U1i and U2i in Heckman (1979), respectively. The correlation between ln(η*i) and ln(ηi) is assumed to be  0.5 in order to rule out the case of independence between the two error terms in which case, the standard least squares estimator may be used on the selected subsample without introducing bias.

> 10 The simulation results with higher percentages of zero observations are available upon request by the authors.

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in levels in Case 1, this seems related to the mismatch between the data generating process and the more restricted estimator. In Case 3 with this estimator in logs, the bias is again large with the error process homoscedastic. The Heckman estimators generally have lower bias and standard errors than the ET-Tobit estimators, which is not surprising given that the data were generated using a Heckman process. Perhaps somewhat surprisingly, however, the Heckman estimators in logs performed better than those in levels, despite the problems associated with the log transformation. The Poisson-Tobit estimator in levels performed reasonably well, although the Heckman-Maximum Likelihood estimator had lower average bias and smaller standard errors.

As shown in Appendix Table 4, the Zero-inflated Poisson (ZIP) estimator has, on average, substantially lower bias than any of the other estimators considered. The standard errors of this estimator are slightly larger than for the Heckman ML estimator in logs, the estimator with the second-lowest average absolute bias, suggesting that it is important to have a large sample when considering the use of the ZIP estimator<sup>11</sup> . The Heckman ML estimator has lower bias than the ZIP for categorical explanatory variables, but greater bias for the normally-distributed variable. The Heckman 2SLS estimator in logs may also be worth considering if convergence problems with the Heckman ML prove too difficult although its average bias across the four cases is more than twice as high as that for the ZIP estimator. By contrast with the ET-Tobit DGP, the ZIP estimator has clearly lower bias than the truncated PPML model both on average and in all but one case.

# _The HMR Data Generating Process_

To capture the essence of the DGP à la HMR we need a firm number variable that is nonnegative and positively correlated with the probability of nonzero trade. It should also be correlated with the economic mass variable, and take the value zero when there is no trade. We generate synthetic data sets with our four different types of heteroscedasticity for Monte-Carlo analysis using the following two equations and the procedures laid out in more detail in Annex B:

- (10.1)    y1i=exp(xiβ).ηi ≡ exp(β0+β1x1+β2x2+β3x3)+εi

- (10.2)    y2i=exp(ziα).η*i ≡ exp(α0+α1x1+α2x2+α4x4). + u*2i

> 11 Another practical problem with ML estimators with the dependent variable in levels is frequent difficulty with convergence in Stata. In many cases we had to vary initial values to overcome this problem. In other cases, even varying initial values does not lead the ML estimators to convergence, especially when the dependent variable is in levels.

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where x3 is the firm number variable, constructed to be correlated with x2; and all other variables are as defined in equations (6) and (9). Equation (10.2) is the selection equation which determines whether there is nonzero trade. Specifically, if y2i>1 then y1i is positive. Again, as in the standard Heckman sample selection model, we set the correlation between ln(ηi) and ln(η*i) equal to 0.5. Variable x3 in the behavioral equation mimics the behavior of the firm heterogeneity variable in the HMR theoretical framework. It is—for want of better information—assumed to be uniformly distributed<sup>12</sup> . As in the standard Heckman case, x4 is excluded from the behavioral equation.

Finally, we look at some simulation results when the data are generated à la HMR. The first source of potential omitted-variable bias comes from the standard sample selection problem: similar factors determine the volume of trade (y1i) and the probability (Prob(y2i>1)) that trade occurs between two countries. The second source of omitted variable bias results from exclusion of a variable for the number of exporting firms (x3). Like the firm number proxy used by HMR, x3 is uncorrelated with the error term in the trade equation.

From Table 7 and Appendix Table 4, it is clear that the truncated PPML estimator has the lowest average absolute bias of the standard estimators. The truncated OLS estimator in logs has the next lowest average absolute bias, and its standard errors are lower than for the truncated PPML, but the average absolute bias is four times as large as for truncated PPML. The NLS estimators perform extremely poorly, as do the GPML and NBML standard estimators.

The coefficient on x2 is of particular interest when the DGP is based on a Heckman sample selection with omitted variable bias. Since, by construction Cov(x2,x3)=0.5>0, we expect upward bias in the estimated coefficient on x2 when omitted-variable bias is not controlled for.

From Table 8 and Appendix Table 4, it is clear that the ZIP2 estimator with the firm number variable included has by far the lowest average absolute bias. The ZIP estimator without this variable is the next best-performing in terms of bias, with the truncated PPML coming next. The HMR estimator follows in terms of bias, although it has much higher bias on the important β2 coefficient. ZIP2 type estimators assume that

> 12 We also considered the case with a normally distributed variable and reached the same conclusions.

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researchers have available estimates of firm numbers, and have the ability to deal with the potential endogeneity of such a variable. If the available variable for firm numbers suffers from endogeneity, the approaches suggested by Wooldridge (2010, section 18.5) are likely to be useful. One concern with all of the Poisson-based estimators is that they have somewhat higher standard errors than the HMR estimator.

# **VI. Empirical Estimation**

While we are very conscious of the importance of the multilateral resistance terms emphasized by Head and Mayer (2013), and hence concerned about the empirical validity of traditional gravity models, we consider the implications of different estimators for parameter estimates from both traditional (including GDP as explanatory variables) and Anderson-van Wincoop (using country fixed effects) specifications. The results for these analyses are presented in Tables 9 and 10.<sup>16</sup> Since the specification with country fixed effects controls for the multilateral resistance terms the results in Table 10 are our preferred ones.

The results in Table 9 are very informative about the coefficients on GDP and the log of distance. The truncated PPML has very similar but slightly smaller coefficients than the censored PPML. But a more striking feature of the table is that all of the Poisson-based estimators have quite similar coefficients on GDP and distance that are well below (in absolute value) those of traditional estimators such as Truncated OLS and the Heckman estimators. A striking feature of these results is that the ZIP coefficient estimates are identical to those from the truncated PPML estimator. Two important points are noteworthy with respect to this result. First, the fact that the coefficient estimates of the two estimators are the same hides the fact that the marginal effects from each model are different. Second, while the coefficient estimates of the truncated Poisson and the ZIP estimators are the same and equal to the elasticities (i.e. ������������⁄ ��) they are evaluated at different predicted values of the dependent variable.<sup>17</sup> The ZIP2 estimator, which performed the best in our

> 16 Note that our truncated sample contains 8857 observations and our full sample 17028. Both are smaller than SST’s truncated sample (9613) and full sample (18360) because we used common religion as our excluded restriction variable. Data for this variable are not available for all the samples used by SST (2006). For comparison purpose the sample sizes for our truncated OLS and Poisson estimator were kept the same as for our Heckman estimator.

> 17 Specifically, the elasticities are evaluated at ��� �18644.519 for the truncated Poisson and ��� � 1704.81for the ZIP.

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Monte-Carlo tests with an HMR-based DGP, yields coefficient estimates above those of the other Poisson estimators, but still well below those for traditional estimators. The HMR estimator is an outlier in the other direction, with much lower coefficients on GDP, and somewhat lower coefficients on distance, than the PPML-based estimators.

The results for the Anderson-van Wincoop formulation are presented in Table 10 with model estimates when we control for firm heterogeneity à la Helpman, Melitz and Rubinstein. The results on the log of distance variable are very similar with this model to those for the traditional gravity model. The PPML model results for this coefficient are clustered around -0.8, while those for the traditional model are around -1.4. The HMR Model is again an outlier with a coefficient of -1.14.

# **VII. Reconciling Monte Carlo and Empirical Findings**

The contrast between the empirical results and those from the Monte Carlo analysis raises important questions. Why, in particular, is there so little difference between the results from the PPML model and the truncated PPML, and why are the truncated PPML and the ZIP model results identical? One possible explanation lies with the distribution of the continuous explanatory variable used in the Monte Carlo analysis. By using a standard normal variable, we have ensured that, with the unit coefficient used in the Monte Carlo analysis (and assuming a zero value for the 0-1 variable, a relatively small fraction of the Xiβ values have values near zero. For any given distribution of the error term this reduces the frequency of zero observations and requires larger adjustment terms such as the – _k_ in the ET-Tobit model to obtain any given frequency of these observations.

One simple approach—consistent with the Leung-Yu approach of ensuring that the distribution of the explanatory variables in a Monte Carlo analysis is consistent with the parameters of the empirical distribution being studied—is to examine the distribution of a composite, continuous explanatory variable consistent with the empirical data set. To do this for the SST empirical dataset of, we combined the three signature explanatory variables 18 of the SST empirical data set into a composite variable Log[(GDP_Exporter*GDP_Importer)/(Bilateral distance)<sup>1.3</sup> ], where the coefficients on GDP and on distance are based on averages of the estimates in SST (2006, p650). To allow

> 18 We performed the same experiment with the parameters of the PPML model and found qualitatively similar results.

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comparability of the density and distribution functions of this variable and the log standard normal, we rescaled this composite driving variable to have the same mean as the log standard normal. We found that the distribution of this variable was well-represented by a log-normal distribution with mean -5.728896 and standard deviation 3.523514, rather than 0 and 1 as assumed with the log standard normal. To provide a visual comparison of the two distributions, we compare their cumulative distribution functions<sup>19</sup> (CDFs) in Figure 3. The figure reveals that the empirically-based distribution has a much higher fraction of the explanatory variable near zero than the standard normal variable.

**Figure 3. CDFs for the Log Standard Normal and the Empirical Distribution**

<!-- Start of picture text -->
1.2<br>1<br>0.8<br>0.6<br>0.4<br>0.2<br>0<br>1 2 3 4<br>Log Std Normal Empirical Distn<br><!-- End of picture text -->

Note: Generated using MS EXCEL with 50000 data points per distribution at a spacing of 0.0001.

A focus on a couple of data points in Figure 3 highlights the stark differences between the two distributions. With the empirical distribution, 80 (90) percent of the observations are below 0.06 (0.27). With the log standard normal distribution, only 0.3 (11) percent of the

> 19 We had intended to compare their density functions but it proved to be impossible to plot these on the same scale.

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observations lie within this range. The very large fraction of empirical observations near zero has important implications for the limited-dependent data-generating process of interest to us. For any given distribution of errors, the presence of a large share of the observations of the driving variable near zero makes it much more likely that the outcome will be transformed into a zero observation.

We next use a continuous explanatory variable with this distribution and the simplest model previously examined—the ET Tobit model—and repeat our Monte Carlo analysis. When we do this, we find that the absolute value of the _k_ parameter needed to obtain any given share of zeros falls precipitously. Specifically, with a tiny value for k of 0.01, the percentages of zero observations in heteroscedasticity cases 1, 2, 3 and 4 are 71 percent, 64 percent, 53 percent and 64 percent, respectively. The 53 percent share with heteroscedasticity of type 3 is almost exactly the share of zeros observed in the SST data for total trade between 136 countries. It is much higher than the fraction of zeros reported in Table 1 using k values of 1.0 and 1.5 with the log standard normal distribution. Moving to slightly larger values of _k_ would allow us to replicate the shares of zeros observed in studies using disaggregated trade data.

More importantly, we obtain results, shown in Table 11, that have features that are consistent with our empirical findings. The estimated values on the continuous variable are very similar for both the PPML and the truncated PPML models. And the ZIP and the truncated PPML models are identical. Another interesting feature of these results is that the Negative Binomial model—which performed so poorly in our earlier analysis appears to perform very well, although it remains subject to the concerns expressed by SST (2006) about its parameter estimates depending upon the units of measurement.

# **VIII. Conclusions**

The purpose of this paper was to evaluate the performance of different estimators of the gravity equation in the presence of zero trade flows, which are a common feature of trade data both at the aggregate and disaggregate level. We adopt two different data-generating processes in our Monte Carlo simulations— the Tobit and the Heckman sample-selection models. For the Tobit DGP we used Monte Carlo simulations based on the design of Santos Silva and Tenreyro (SST 2006), modified to include a threshold level of trade that must be surmounted before positive trade levels are observed. For the Heckman sample selection 26
DGP, an additional equation determines which observations of the outcome equation are included. We also analyze a data set generated for consistency with the Helpman, Melitz and Rubinstein (2008) model.

In our initial simulations, we follow SST in using Monte-Carlo simulations with a log standard normal distribution of the continuous driving variable. These tests confirm the importance of heteroscedasticity and the prevalence of zeros as sources of bias with many estimators. However, estimators such as the censored PPML that are known to perform well in the presence of heteroscedasticity are found to be potentially susceptible to limiteddependent variable bias when a substantial fraction of the observations are censored.

When the data are generated using an Eaton-Tamura Tobit model, the smallest biases were sometimes found with Eaton-Tamura Tobit estimators consistent with the data generating process, but only when the functional form was consistent with the form of heteroscedasticity, or an appropriate correction had been made to deal with heteroscedasticity. On average across the four cases considered, the truncated PPML had the lowest mean absolute bias, including relative to the explicitly limited-dependent estimators consistent with the data-generating process.

When the data are generated by a Heckman sample-selection DGP, the truncated PPML estimator outperformed all other models that do not explicitly account for the limited-dependent nature of the data set. The Zero-inflated Poisson (ZIP) model appeared to have the lowest bias of the models considered, and particularly for the continuouslydistributed variables that are frequently of greatest interest in a gravity-model context.

Applying a range of estimators to the SST empirical dataset on international trade resulted in a number of surprising findings. In particular, we found very little difference between the PPML estimator and its truncated version. Further, the results from the truncated PPML estimator were identical with those from the ZIP estimator. To try to better understand the relationship between our Monte Carlo results and our empirical findings, we repeated the Monte Carlo analysis with a continuous explanatory variable distributed like a composite of the key explanatory variables in the gravity model, with a much higher fraction of the explanatory variables clustered near zero. This analysis helped explain two key findings in the empirical analysis—that the PPML and truncated PPML estimates were similar, and the truncated PPML and the ZIP were almost identical.

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Our results, in an important sense, restore the SST findings about the importance of heteroscedasticity in the gravity model and in favor of the PPML model over traditional estimators for the gravity model of international trade. However, they also highlight the potential risks of limited-dependent variable bias. They also highlight the importance for Monte-Carlo testing of taking into account the characteristics of the distribution of the driving variables as well as of the residuals. Using a driving variable that is distributed as a log standard normal variable resulted in a relatively qualified recommendation for the PPML estimator as a least-bad option. When we moved to Monte-Carlo testing with a driving variable based on the empirical distribution of the key explanatory variables for trade, we were able to easily explain the large shares of zeros observed in real-world trade data and to validate the recommendation of the PPML as a good estimator for the gravity model, at least for models of aggregate trade flows.

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