Controlling for Heterogeneity in Gravity Models
of Trade and Integration
I-Hui Cheng and Howard J. Wall
been common to instead specify the augmented model using per capita income, which captures the same effects.3Whichever specification of the augmented model is used, the purpose is to allow for non-homothetic preferences in the importing country and to proxy for the capital/labor ratio in the exporting country (Bergstrand, 1989).
The gravity model of trade has been used widely as a baseline model for estimating the impact of a variety of policy issues, including regional trading groups, currency unions, political blocs, patent rights, and various trade distortions.4 Typically, these events and policies are modeled as deviations from the volume of trade predicted by the baseline gravity model and, in the case of regional integration, are captured by dummy
vari-I
n the 1860s, H. Carey first appliedNewtonian physics to the study of human behavior, and the so-called “gravity equa-tion” has since been widely used in the social sciences. In economics, gravity model studies have achieved empirical success in explaining various types of inter-regional and international flows (including labor migration, commuting, customers, hospital patients, and international trade). The gravity model of inter-national trade was developed independently by Tinbergen (1962) and Pöyhönen (1963). In its basic form, the amount of trade between countries is assumed to be increasing in their sizes, as measured by their national incomes, and decreas-ing in the cost of transportation between them, as measured by the distance between their econ-omic centers.1Following this work, Linnemann (1966) included population as an additional measure of country size, employing what we will call the augmented gravity model.2It has also
This paper compares various specifications of the gravity model of trade as nested versions of a general specification that uses bilateral country-pair fixed effects to control for heterogeneity. For each specification, we show that the atheoretical restrictions used to obtain them from the general model are not supported statistically. Because the gravity model has become the “workhorse” base-line model for estimating the effects of international integration, this has important empirical implications. In particular, we show that, unless heterogeneity is accounted for correctly, gravity models can greatly overestimate the effects of integration on the volume of trade.
Federal Reserve Bank of St. Louis Review, January/February 2005, 87(1), pp. 49-63.
1
For examples see McCallum (1995), Helliwell (1996), and Boisso and Ferrantino (1997).
2
For uses of the augmented gravity model, see Oguledo and MacPhee
(1994), Boisso and Ferrantino (1997), and Bayoumi and Eichengreen (1997).
3
Examples of the augmented model with per capita income include Sanso, Cuairan, and Sanz (1993), Frankel and Wei (1998), Frankel, Stein, and Wei (1995, 1998), and Eichengreen and Irwin (1998).
4
See Aitken (1973), Brada and Mendez (1983), Bikker (1987), Sanso, Cuairan, and Sanz (1993), McCallum (1995), Helliwell (1996), Frankel (1997), Wei and Frankel (1997), Bayoumi and Eichengreen (1997), Mátyás (1997), Frankel and Wei (1998), Frankel, Stein, and Wei (1998), Smith (1999), and Rose (2000).
I-Hui Cheng is an assistant professor at the National University of Kaohsuing, Taiwan. Howard J. Wall is an assistant vice president at the Federal Reserve Bank of St. Louis. The authors thank Martin Sola, Ron Smith, Jim Dunlevy, and Rob Dittmar for their comments and suggestions. Kristie M. Engemann provided research assistance.
ables. The continuing popularity of the gravity model is highlighted by Eichengreen and Irwin (1998, p. 33), who call it the “workhorse for empiri-cal studies of [regional integration] to the virtual exclusion of other approaches.”
The perceived empirical success of the gravity model has come without a great deal of analysis regarding its econometric properties, as its empiri-cal power has usually been stated simply on the basis of goodness of fit (i.e., a relatively high R2).5 The lack of attention paid to the empirical prop-erties of the model is despite the fact that the strength of any baseline model lies in the accuracy of its estimates. Recently, though, several papers have argued that standard cross-sectional methods yield biased results because they do not control for heterogeneous trading relationships. Because of this, these papers introduced fixed effects into the gravity equation. Fixed-effects models allow for unobserved or misspecified factors that simul-taneously explain trade volume between two countries and, for example, the probability that the countries will be in the same regional integra-tion regime (Mátyás, 1997; Bayoumi and
Eichengreen, 1997; Cheng, 1999; Wall, 2002, 2003; Coughlin and Wall, 2003).6Gravity models with fixed effects have also been used by Glick and Rose (2001) and Pakko and Wall (2001) to estimate the trade effects of currency unions; by Wall (2000) and Millimet and Osang (2004) to estimate the effects of borders on trade; by Egger (2002) to calculate trade potentials; and by Wall (1999) to estimate the costs of protection.
Although the arguments underlying the use of fixed effects as a solution to unobserved hetero-geneity are roughly the same in all of these papers, there is little agreement about how to actually specify the fixed effects. For example, Cheng (1999) and Wall (1999) propose two fixed effects for each pair of countries, one for each direction of trade. In Glick and Rose (2001), each pair of
countries has only one fixed effect. In Mátyás (1997), each country has two fixed effects, one as an exporter and one as an importer. The purpose of this paper is to evaluate the various fixed-effect specifications in terms of the econometric appro-priateness of their underlying assumptions. Specifically, we show (i) how the standard pooled-cross-section specification and other fixed-effects specifications are special cases of the Cheng (1999) and Wall (1999) specification and (ii) that the restrictions to obtain them cannot be supported empirically. To underscore the importance of getting the fixed-effects specification right, we illustrate how the choice of specification has significant implications when estimating the effects of integration on trade volume.
A STATISTICAL OVERVIEW
This section briefly sets out the various forms of the gravity model that have been used to esti-mate bilateral trade flows. These models are restricted versions of a general gravity model, which has a log-linear specification but places no restrictions on the parameters. In the general model, the volume of trade between countries i and j in year t can be characterized by
(1) lnX
ijt= α0+ αt+ αij+ ββ′ijtZijt+ εijt, t = 1,…,T,
where Xijtis exports from country i to country j in year t and Z′ijt= [zit, zjt…] is the 1 ×k vector of
gravity variables (gross domestic product [GDP], population, and distance). The intercept has three parts: one common to all years and country pairs,
α0; one specific to year t and common to all pairs,
αt; and one specific to the country pairs and
com-mon to all years, α
ij. The disturbance term, εijt,
is assumed to be normally distributed with zero mean and constant variance for all observations. It is also assumed that the disturbances are pair-wise uncorrelated.
Obviously, because (1) has only one observa-tion, it is not useful for estimation unless restric-tions are imposed on the parameters. The standard single-year cross-section model (CS) imposes the restrictions that the slopes and intercepts are the same across country pairs, that is, αij= 0 and
ββijt= ββt,
5
See Sanso, Cuairan, and Sanz (1993) for an examination of the predictive power of various specifications of the augmented gravity model. Also, see Oguledo and MacPhee (1994) for a survey of pre-1990 empirical results.
6
Soloaga and Winters (2001) also recognize this problem, but their solution is to estimate yearly gravity models and to calculate the effects of integration as the differences in the predicted trade vol-umes over time.
(CS) lnX
ijt= α0+ αt+ ββ′tZijt+ εijt, t = 1,…,T,
where α0and αtcannot be separated. Assuming that all the classical disturbance-term assumptions hold, the CS model is estimated by ordinary least squares (OLS) for each year.
The other standard estimation method is a pooled-cross-section model (PCS), which imposes the further restriction on the general model that the parameter vector is the same for all t, ββ1= βββ2= ... = ββT= ββ, although it normally allows the inter-cepts to differ over time:
(PCS) lnXijt= α0+ αt+ ββ′Zijt+ εijt, t = 1,…,T. This is estimated by OLS using data for all avail-able years.
Nearly all estimates of the gravity model of trade use either the CS or the PCS model, which, as we show below, both provide biased estimates. To address this bias, we remove the restriction that the country-pair intercept terms equal zero, although we maintain the restriction that the slope coefficients are constant across country pairs and over time. Specifically, we estimate the fixed-effects (FE) model of Cheng (1999) and Wall (1999): (FE) lnX
ijt= α0+ αt+ αij+ ββ′Zijt+ εijt, t = 1,…,T.
Note that the country-pair effects are allowed to differ according to the direction of trade (i.e., αij≠
αji). The FE model is a two-way fixed-effects model
in which the independent variables are assumed to be correlated with αijand is a classical regres-sion model that can be estimated using LSDV (least squares with a dummy variable for each of the country pairs).
As mentioned above, others have proposed alternative fixed-effects models to handle country-pair heterogeneity, each of which can be modeled as a restricted version of the FE model above. The symmetric fixed-effects (SFE) model of Glick and Rose (2001) differs from FE only in that it imposes the restriction that the country-pair effects are symmetric (i.e., αij= αji).
In the Bayoumi and Eichengreen (1997) model, call it DFE, the differences in the dependent and independent variables are used to eliminate the fixed variables, including the country-pair dum-mies and distance. As with the FE specification,
this model allows for the most general fixed effects possible. But rather than estimating the fixed effects using LSDV, it eliminates them by subtract-ing them out. Specifically,
(DFE) ∆lnXijt= γ0+ γt+ ββ′∆Zijt+ µijt, t = 1,...,T, where ∆is the difference operator and γ0+ γt=
αt– αt –1. In this model the intercept has two parts:
γ0is the change in the period-specific effect that is common across years and γtis the change that is specific to year t.
When there are no time dummies, such a differencing model yields results identical to a model with dummy variables to control for fixed effects. However, with time dummies it is neces-sary to impose restrictions on the time effects to avoid collinearity, which in turn makes the DFE estimation a restricted form of the FE estimation. If the collinearity restriction is that the first time dummy in the DFE model is equal to zero, this is equivalent to restricting the common component of the change in the period-specific effects as equal to the difference in the first two period-specific effects (i.e., γ0= α2– α1). If, instead, the collinear-ity restriction is that the sum of the time dummies in the DFE model is zero, this is equivalent to restricting the common component as equal to the difference between the first and last time dummies (i.e., γ0= αT– α1).
Mátyás (1997) proposes (XFE)
lnXijt= α0+ αt+ θi+ ωj+ ββ′Zijt+ εijt, t = 1,…,T, as the correct specification of the gravity model, where the country-specific effect is θiwhen a country is an exporter and is ωjwhen it is an importer. Note that in this specification, distance, contiguity, and language are eliminated because they are fixed over time, even though they are not collinear with the country-specific effects. This model is a special case of the FE model in that it has a unique value for each trading pair’s intercept, with the restrictions that a country’s fixed effect as an exporter or importer is the same for all of its trading partners. This imposes cross-pair restric-tions on the intercepts—that is, one of the compo-nents of the intercept for Germany-to-Canada trade
must be the same as one of the components of the intercept for Germany-to-France trade. These restrictions do not change the coefficient estimates very much but, as we show below, lead to biased and rather large residuals, indicating inaccurate in-sample predictions of trade flows.
STANDARD RESULTS
This section presents regression results for the augmented version of the standard PCS model.7 The data set is a balanced panel with 3,188 obser-vations (797 unidirectional country pairs in each of four years: 1982, 1987, 1992, and 1997).8We
included observations of non-zero trade between countries listed in all of the relevant World Bank
World Development Reports as being
upper-middle or high income during these years. Also, we excluded countries that were identified as high-income oil exporters. The result is a manageable data set that is fairly representative of the literature, which typically includes only OECD members or industrialized countries. Descriptions of the data and their sources are provided in the data appendix.
In the augmented version of the gravity model, the gravity variables are the countries’ GDPs, their populations, and the distance between them. Thus, the augmented PCS model assumes that, in a given year, trade flows from exporting country 7
Because the results for the single-year CS model do not differ sub-stantially from those for the PCS model, we do not present them here. However, they are available upon request.
8
Fixed-effects estimation is sometimes criticized when applied to
Table 1
Regression Results for Models Using Pooled Data Dependent Variable = Log of Real Exports
Pooled Unrestricted
cross-section FE model Restricted FE models
PCS FE SFE DFE XFE
Intercept 6.852* (0.546) — — 0.209* (0.028) — Origin GDP 0.617* (0.038) 0.122* (0.023) 0.213* (0.025) 0.098* (0.029) 0.122* (0.055) Destination GDP 0.511* (0.035) 0.208* (0.027) 0.117* (0.024) 0.258* (0.029) 0.208* (0.054) Origin population 0.141* (0.038) –0.390 (0.298) 0.935* (0.268) –0.482 (0.344) –0.390 (0.565) Destination population 0.214* (0.038) 2.313* (0.319) 0.989* (0.268) 1.906* (0.344) 2.313* (0.584) Distance –1.025* (0.023) — — — — Contiguity –0.125 (0.085) — — — — Common language 1.075* (0.072) — — — — 1987 0.077 (0.067) 0.199* (0.029) 0.199* (0.038) — 0.199* (0.063) 1992 0.014 (0.068) 0.357* (0.043) 0.357* (0.053) –0.040 (0.029) 0.357* (0.093) 1997 0.051 (0.064) 0.482* (0.058) 0.481* (0.070) –0.064* (0.028) 0.482* (0.122) Observations 3,188 3,188 3,188 2,391 3,188 Parameters 11 804 408 7 63 Log-likelihood –5,163.27 –1,663.07 –2,863.46 –1,979.64 –4,704.08 R–2 0.690 0.954 0.916 0.050 0.768
NOTE: All non-dummy variables are in logs. White-corrected standard errors are in parentheses; * denotes significance at the 5 percent level. For the DFE model, all variables are in differences from the previous year.
data pooled over consecutive years on the grounds that dependent and independent variables cannot fully adjust in a single year’s time. To avoid this, we left five years between our observations.
i to importing country j can be estimated using9
(2)
lnXijt= α0+ αt+ β1lnYit+ β2lnYjt+ β3lnNit +β4lnNjt+ δ1lnDij+ δ2Cij+ λLij+εijt ,
where α0is the portion of the intercept that is common to all years and trading pairs, αtdenotes the year-specific effect common to all trading pairs, Yiand Yjare the two countries’ GDPs, Ni and Njare their populations, Dijis the distance between them, Cijis a contiguity dummy, and Lij is a common-language dummy. Note that our estimation omits the dummy for 1982 to avoid collinearity.
Because trade flows are expected to be posi-tively related to national incomes, and negaposi-tively
related to distance, β1, β2, and δ2are expected to be positive and δ1is expected to be negative. The signs expected for population coefficients are not as unambiguous, and the literature has not tended to find a consistent sign for β3or β4.10Because Lij is meant to capture cultural and historical similar-ities between the trading pairs, which are thought to increase the volume of trade, λis expected to be positive. Finally, we take the time dummies as indicators of the extent of “globalization,” which we define as the purported common trend toward greater real trading volumes, independent of the sizes of the economies.
The regression results for PCS are reported in the first column of Table 1. The signs of the coefficients on distance, common language, and the countries’ GDPs are as expected and are statis-tically significant. Only the negative coefficient on 10
See Oguledo and MacPhee (1994).
–8 –6 –4 –2 0 2 4 6 0 100 200 300 400 500 600 700 800 Country Pair Residual
Figure 1
PCS Residuals by Country Pair
9
Note that the regression could be suitably rearranged to instead obtain the augmented model with per capita income.
the contiguity dummy of PCS is not as expected, although it is not statistically different from zero. Perhaps surprisingly, the coefficients on the time dummies do not indicate a trend toward globalization.
According to the estimates of the PCS model, (i) an increase in a country’s GDP will lead to a less-than-proportional increase in its imports and exports and (ii) a country will export 103 percent more to a market that is half as distant as another otherwise-identical market and 108 percent more to a country with the same first language. Finally, we take the fact that the time dummies are not statistically different from zero to mean that global-ization, as defined above, was not an important factor in increasing trade over the period.
Despite the supposed empirical success that we have replicated, there is a severe problem with the standard PCS model. This is clear from Figure 1, which plots the residuals for the PCS model for the 797 unidirectional country pairs in our data set, ordered by the pairs’ average resid-uals. If the PCS estimation were unbiased, there would be no discernable pattern in Figure 1 because the average residual for each country pair would be zero. The residuals for 544 of the coun-try pairs, however, always have the same sign. In other words, the PCS model consistently misesti-mated the volume of trade for at least 68 percent of the country pairs.
THE GRAVITY MODEL WITH
COUNTRY-PAIR FIXED EFFECTS
The Model
Standard cross-section estimates of the gravity model yield biased estimates of the volume of bilateral trade because there is no heterogeneity allowed for in the regression equations. With such heterogeneity, a country would export different amounts to two countries, even though the two export markets have the same GDPs and are equi-distant from the exporter. This can be because there are historical, cultural, ethnic, political, or geographic factors that affect the level of trade and are correlated with the gravity variables (GDP, population, distance). If so, then estimates that
do not account for these factors will suffer from heterogeneity bias.
Some studies using the PCS model have, to some extent, tried to control for this by including things such as whether trading partners share a common language, have had a colonial history, or are in military alliance. However, cultural, histori-cal, and political factors are often difficult to observe, let alone quantify. This is why we control for these factors using a simple fixed-effects model that assumes that there are fixed pair-specific fac-tors that may be correlated with levels of bilateral trade and with the right-hand-side variables. It is in this sense that fixed-effects modeling is a result of ignorance: We do not have a good idea which variables are responsible for the heterogeneity bias, so we simply allow each trading pair to have its own dummy variable.
We assume that the gravity equation for a country pair may have a unique intercept and that it may be different for each direction of trade (i.e.,
αij≠αji). However, we retain the assumptions of
the PCS model that the slope coefficients are constant over time and across trading pairs. The Cheng (1999) and Wall (1999) specification of the augmented FE is
(3)
lnXijt= αij+ αt+ β1lnYit+ β2lnYjt+ β3lnNit +β
4lnNjt+ εijt ,
where α
ijis the specific “country-pair” effect
between the trading partners. The country-pair intercepts include the effects of all omitted vari-ables that are cross-sectionally specific but remain constant over time, such as distance, contiguity, language, and culture. Using the pooled data described above, we have 797 country-pair intercepts.
Because there is a long-standing problem with determining the appropriate measure of economic distance to capture transportation and information costs (see Head and Mayer, 2001, for a review of the issue), an added benefit of the fixed-effects model is that it eliminates the need to include distance in the regression. The most common method for measuring distance is to do as we have done and simply measure it between the centers (often assumed to be the capital cities) of the two
countries. There are problems with this, such as the implicit assumptions that overland transport costs are the same as those over sea and that all overland/oversea distances are equally costly. To provide just one example, Los Angeles is about 1,300 km farther from Tokyo than is Moscow, but the economic distance between Tokyo and Los Angeles is certainly much lower than that between Tokyo and Moscow. Our FE approach eliminates the need to include a distance variable, as it con-trols for all variables that do not change over time.
Another difficulty with standard measures of economic distance is the common assumption that the capital city, or any other single point in the country, is a useful proxy for the economic center. While this may be useful for small coun-tries with one major city, it is wide of the mark for countries like Canada and the United States, which have major cities thousands of miles apart on different oceans and which serve as centers for trade with completely different countries. By using Washington, D.C., or Ottawa to measure dis-tance between the United States or Canada and its Pacific trading partners is to overstate distance by the entire breadth of the North American con-tinent. As the United States has the highest GDP and the highest volume of trade, the mismeasure-ment of economic distance can bias the estimation of the coefficients on the other variables in the gravity model.
Another advantage of our approach is that it removes the problem of controlling for contiguity. Although it is potentially important, as a great deal of trade can occur from people crossing the border to make everyday purchases, it is accounted for only sometimes. Even when it is accounted for with a dummy variable, as we do above, it still assumes that all contiguity is equivalent and time invariant in terms of its effect on trade. Consider-ing that Canada and the United States, China and Russia, and Argentina and Chile are all equiva-lently contiguous pairs, this is difficult to abide by.
The Results
Table 1 reports the estimation results for the augmented version of the FE model. Note that, for comparison with the PCS results, the year dum-mies are measured relative to that of 1982. Also,
the estimates of the country-pair intercepts are omitted for space considerations. According to the results for the FE model, (i) an increase in a country’s GDP will lead to a less-than-proportional increase in its imports and exports and (ii) global-ization has increased the real volume of trade by 48 percent between 1982 and 1997.
A comparison of the results of the FE and PCS models shows that allowing for trading-pair heterogeneity lowers the estimated income elastic-ities of trade, greatly increases the absolute value of the coefficients on the countries’ populations, and greatly increases the estimated role of global-ization. It is obvious from the results that restrict-ing the country-pair effects to zero, as the PCS model does, has statistically significant effects on the results, as is easily confirmed by a likelihood ratio test.11Note also that the residuals from the FE estimation across country pairs (Figure 2) have no discernible pattern.
Therefore, because the PCS model is a restricted form of the FE model and the restrictions are not supported statistically, we conclude that the FE model is the preferred specification of the gravity model. In short, there is no statistical support for imposing the parameter restrictions required by the standard procedures for estimating the gravity model of trade. In the absence of any economic arguments for believing that the inter-cepts of the gravity equation are the same across trading pairs, we conclude that the FE model is the more appropriate specification.
Oddly, Wei and Frankel (1997, p. 125) reject the inclusion of country-pair dummies a priori on the basis that doing so would undermine their efforts at estimating the effects of variables that are constant over the sample period. Presumably, their worry is that, because these variables are subsumed into the country-pair effects, they are hidden from analysis. This is unfounded because the effects of these variables are easily estimated by regressing them on the country-pair effects from the FE model. Specifically, where the estimates of the 797 country-pair effects are denoted as αˆ
ij,
and including the log of distance and the contiguity 11
and language dummies as independent vari-ables, we obtain
αˆ
ij= 10.408 – 1.236lnDij– 0.746Cij+ 1.565Lij.
(0.405) (0.049) (0.246) (0.198) The numbers in parentheses are White-corrected standard errors and the R–2= 0.158. According to these results, all three variables are statistically significant determinants of the country-pair effects. Inexplicably, though, the coefficient on the con-tiguity dummy is negative, as in the PCS results. Note that these estimates are quite different from those obtained from the PCS model, in which estimates of the effects of time-invariant factors suffer from the same heterogeneity bias as the time-variant factors. So, far from undermining estimation efforts, it is instead necessary to control for country-pair heterogeneity to obtain unbiased estimates of the importance of time-invariant factors.
ALTERNATIVE FIXED-EFFECTS
SPECIFICATIONS
As discussed earlier, others have used less-general fixed-effects specifications. The first of these, the SFE model of Glick and Rose (2001), is simply (3) with the restriction that αij= αji. Because our data set does not have the entire set of country pairs for both directions, this does not mean that there are exactly one-half as many country-pair effects as the FE model, although it is close. Our FE estimation had 797 country-pair effects, whereas our SFE estimation has 401. The results of the SFE estimation are in Table 1 and indicate that the symmetry restriction on the country-pair effects has a statistically significant effect on the results. Each of the coefficients on the gravity variables is very different from what we obtain with the FE model, although the coefficients on the year dum-mies are nearly identical. Also, a likelihood ratio
–8 –6 –4 –2 0 2 4 6 0 100 200 300 400 500 600 700 800 Country Pair Residual
Figure 2
test easily rejects the null hypothesis that the restrictions do not have a statistically significant effect on the estimation.12This means that the FE model is preferred statistically to the SFE model.
Taking the time difference of (3), the DFE model of Bayoumi and Eichengreen (1997) is (4)
∆lnXijt= γ0+ γt+ β1∆lnYit+ β2∆lnYjt+ β3∆lnNit +β4∆lnNjt+ µijt ,
where the intercept is as defined in the statistical overview, γ0+ γt= αt– αt –1. To prevent collinearity, we set the time dummy for 1987 equal to zero, meaning that other time dummies are measured relative to it. In terms of the more-general FE model, this is equivalent to restricting the common component of the change in the period-specific effects as equal to the difference in the first two period-specific effects (i.e., γ0= α2– α1).13The empirical results are presented in Table 1.
The results for the FE and DFE models are similar in terms of the signs and order of magni-tude of the coefficients. Nonetheless, the FE and DFE results differ enough to reject the restrictions needed to obtain the DFE model. This can be con-firmed easily by a likelihood ratio test. Therefore, given that the restrictions that DFE imposes on the time dummies are not justified on any econ-omic or statistical grounds, our results indicate that they should not be imposed.
The third alternative to the FE model, XFE, is (5)
lnXijt= α0+ αt+ θi+ ωj+ β1lnYit+ β2lnYjt+
β3lnNit+β4lnNjt+ εijt ,
where the fixed effect when a country is an exporter is θiand when it is an importer is ωj. One way to prevent perfect collinearity in estimating (5) is to impose the restrictions that one of the θs and one of the ωs is zero. Because each θiand ωj comprise part of many αijs, this is the same as imposing a series of cross-pair restrictions on the
αijs. From the empirical results summarized in the
last column of Table 1, it seems that the coeffi-cients are the same as those from the FE model. In fact, the coefficients are not the same, but the differences are so small that they appear only beyond the seventh decimal places provided by STATA. More importantly, though, the standard errors from the XFE model are much larger. Con-sequently, the FE model is preferred to the XFE model on the basis of any standard goodness-of-fit criteria. As with the other restricted fixed-effects specifications, a likelihood ratio test easily rejects the null hypothesis that the arbitrary restrictions imposed by XFE are not statistically benign.
THE EFFECTS OF INTEGRATION
As we discuss in our introduction, the gravity model has become the primary tool for estimating the effects of regional integration on trade volumes. Up to this point, we have omitted integration vari-ables in order to focus on the importance of con-trolling for country-pair heterogeneity when estimating gravity models. We now introduce integration into our model and demonstrate the striking effect that heterogeneity bias has on the results. We would also like to alleviate the valid concern that the heterogeneity bias we detected above was due to our implicit assumption that regional integration is uncorrelated with the inde-pendent variables.The most common and straightforward method for estimating the effects of integration in a gravity model is to include dummy variables for each integration regime in place during the sample period (see, for example, Frankel, 1997). Each of these dummies takes the value of 1 for an observation for which the two countries are mem-bers of the regime, with the expectation that the coefficients on these dummies are positive. We include five such dummy variables in our model, one each for the European trading bloc, the North American trading bloc, the South American trad-ing bloc (Mercosur), the Australia–New Zealand Closer Economic Relations (CER), and the Israel– United States Free Trade Agreement (FTA). 12
This is with LR = 2,400.78 and χ2(395) = 442.34 at the 5 percent level.
13
The alternative assumption that the sum of the year dummies is zero means that γ
0= αT– α1and yields the same results except for the
Although there has been some deepening of trade integration in the European bloc, the primary change over the period was an expansion in the number of countries covered under the customs union. The formation of the European Community (EC) predates our data set, and Portugal and Spain joined in 1986. The 12 countries of the EC renamed themselves the European Union (EU) in 1992, but this had relatively little effect on internal trade policy, as it was already nearly unfettered under the EC. Expansion of the bloc came in 1994 with the European Economic Area (EEA), which extended the free trade zone to include Austria, Iceland, Finland, Norway, and Sweden. To capture the effect of this trading bloc, our European bloc dummy variable takes the value of 1 when trade is between members of the EC or EU for 1982, 1987, 1992 and when trade is between members of the EEA for 1997.
The Canada–United States Trade Agreement of 1988 established a North American trading bloc that included only Canada and the United States. The North American Free Trade Agreement (NAFTA) expanded the free trade zone in 1994 to include Mexico. We ignore NAFTA’s relatively mild deepening of Canada–United States integra-tion and focus on it instead as an extension of the free trade bloc to Mexico. Our North American bloc dummy takes the value of 1 for trade between the United States and Canada for 1992 and between Mexico, Canada, and the United States for 1997.
The third significant trade bloc during the period was Mercosur, which came into force in 1995, reducing trade barriers between Argentina, Brazil, Paraguay, and Uruguay. Our Mercosur dummy takes the value of 1 for trade between any two of these countries in 1997. The Australia– New Zealand CER was formed in 1983, so its dummy variable is equal to 1 for trade between the two countries for all years but 1982. Similarly, the Israel–United States FTA entered into force in 1985, so its dummy variable is equal to 1 for trade between the two countries for 1987, 1992, and 1997.
We include these trade bloc dummies in the PCS and FE models and report the empirical
results in Table 2. Note that inclusion of these dummies makes little difference for the PCS model. Nonetheless, a likelihood ratio test rejects the null hypotheses that including the trade bloc dummies in the PCS model does not alter the results to a statistically significant extent.14 Simi-larly, the results for the FE model are also not dramatically different when the trade bloc dum-mies are included, although the null hypothesis that the inclusion of these variables has no statisti-cally significant effect on the results is rejected.15
Both models find modest effects on trade from the European trade bloc. The PCS estimates say that the bloc had a statistically insignificant effect, but the FE estimates say that it had a statisti-cally significant effect of 8.2 percent (e0.079– 1 = 0.082). The larger differences between the two models are in the estimated effects of the other trade blocs. The PCS model suggests a 172 percent increase in trade between North American coun-tries because of their trading bloc, whereas the FE model suggests that the bloc led to only a 34 per-cent increase in trade. For Mercosur, the PCS model estimates an increase in trade of 23 percent that is far from being statistically significant, whereas the FE model estimates a statistically significant effect of 61 percent. The PCS model also estimates the effects of the Australia–New Zealand CER and the Israel–United States FTA as increases in intra-bloc trade of about 300 and 400 percent, respectively. The FE model, however, finds a statistically significant effect of –12 percent for the Australia–New Zealand CER and a statisti-cally insignificant effect of –7.3 percent for the Israel–United States FTA.
These results highlight how allowing for unobserved or unmeasurable heterogeneity can alter gravity model estimates. Specifically, the fact that the estimated effects of the trade blocs change when country-pair heterogeneity is allowed for means that there are pair-specific effects that are correlated with the level of trade between pairs of countries and with the likelihood that the pair 14
This is with LR = 23.6 and χ2(5) = 11.07 at the 5 percent level.
15
will enter a trading bloc.16In particular, the lower estimated effect of the Israel–United States FTA using the FE model indicates that there is some-thing special about the relationship between the United States and Israel that makes them trade relatively more with each other than the gravity variables would predict, and which led them to sign a trade agreement. Suppressing this pair-specific effect, as the PCS model does, mistakenly
suggests that it is the FTA that is responsible for the high trade volume, rather than the special relationship. Similarly, our results suggest for the Australia–New Zealand CER and the North American bloc that the high levels of intra-bloc trade can be attributed to cultural and geographic proximity not completely captured by the language and distance variables, and not primarily to the blocs themselves.
For the sake of comparison, we also estimated the effects of integration using the three alternative fixed-effects specifications. As shown in Table 2, the point estimates of the effects of the blocs on 16
We should note that if we regress the estimated fixed effects from this estimation against distance, contiguity, and language, the results do not differ substantially from those obtained above, which used the estimated fixed effects without controlling for regional integration.
Table 2
Regression Results with Integration Dummies Dependent Variable = Log of Real Exports
Pooled Unrestricted
cross-section FE model Restricted FE models
PCS FE SFE DFE XFE
Intercept 6.756* (0.581) — — 0.208* (0.029) — Origin GDP 0.618* (0.038) 0.125* (0.023) 0.217* (0.026) 0.098* (0.029) 0.176* (0.049) Destination GDP 0.512* (0.035) 0.212* (0.028) 0.121* (0.025) 0.258* (0.029) 0.263* (0.050) Origin population 0.138* (0.039) –0.316 (0.315) 1.009* (0.297) –0.476 (0.351) 0.884†(0.529) Destination population 0.210* (0.038) 2.386* (0.339) 1.063* (0.298) 1.912* (0.351) 3.575* (0.561) Distance –1.013* (0.028) — — — — Contiguity –0.145†(0.087) — — — — Common language 1.050* (0.073) — — — — European bloc 0.059 (0.062) 0.079†(0.041) 0.079 (0.051) 0.021 (0.048) 1.196* (0.059) North American bloc 1.000* (0.165) 0.294* (0.140) 0.295†(0.177) 0.204 (0.227) 2.675* (0.216) Mercosur 0.203 (0.436) 0.475* (0.172) 0.475* (0.166) 0.365 (0.228) 4.133* (0.246) Australia–N.Z. CER 1.370* (0.113) –0.125* (0.044) –0.124 (0.120) –0.158 (0.393) 3.981* (0.174) Israel–U.S. FTA 1.642* (0.102) –0.076 (0.296) –0.076 (0.126) 0.024 (0.393) 0.426* (0.168) 1987 0.067 (0.067) 0.189* (0.031) 0.189* (0.040) — 0.020 (0.062) 1992 0.0005 (0.069) 0.338* (0.047) 0.338* (0.058) –0.040 (0.029) 0.027 (0.089) 1997 0.022 (0.069) 0.437* (0.069) 0.436* (0.085) –0.070* (0.029) –0.220†(0.119) Observations 3,188 3,188 3,188 2,391 3,188 Parameters 16 809 413 12 68 Log-likelihood –5,151.48 –1,657.12 –2,860.65 –1,977.80 –4,427.03 R–2 0.692 0.954 0.916 0.050 0.805
NOTE: All non-dummy variables are in logs. White-corrected standard errors are in parentheses; * and †denote significance at the 5 and 10 percent levels, respectively. For the DFE model, all variables are in differences from the previous year.
trade are nearly identical between the FE and SFE models. Nonetheless, because the standard errors from the SFE estimates are larger, one would con-clude from them that the effects of the European bloc and the Australia–New Zealand CER were statistically no different from zero, even though the FE estimates indicate their statistical significance.
Estimates using the DFE model are also not dramatically different from those using the FE model. Again, though, the larger standard errors mean that the estimated effects are further from standard levels of statistical significance. Indeed, the DFE estimates indicate that none of the trading blocs had a statistically significant effect on trade between members. This occurs because the DFE model imposes restrictions on the time dummies, thereby leading to the misestimation of the effects of regional integration regimes, the expansions of which have a significant trend component.
The XFE model provides estimates of the effects of integration that are dramatically different from those provided by any of the other models. Specifically, it suggests that the European bloc led to an increase in trade of 230 percent, that the North American bloc led to a 1,350 percent increase in trade, and that Mercosur and the Australia–New Zealand CER led to increases in trade of greater than 5,000 percent.
CONCLUSIONS
The objective of this paper is to compare ways that heterogeneity has been allowed for when using the gravity model to estimate bilateral trade flows. Our empirical analysis shows first that standard pooled-cross-section methods for estimat-ing gravity models of trade suffer from estimation bias due to omitted or misspecified variables. It also shows that the problem is eliminated using the two-way fixed-effects model of Cheng (1999) and Wall (1999) in which country-pair and period dummies are used to reflect the bilateral relation-ship between trading partners. The fixed effects capture those factors such as physical distance, the length of the border (or contiguity), history, culture, and language that are constant over the span of the data and that are correlated with the volume of bilateral trade.
We show that alternative fixed-effects models proposed by Glick and Rose (2001), Mátyás (1997), and Bayoumi and Eichengreen (1997) are special cases of our model and that the restrictions neces-sary to obtain these special cases are not supported statistically. Also, because these restrictions have little or no economic support, we argue that they should not be imposed. As the gravity model has become the “workhorse” of empirical studies of the effects of integration, we also compare the vari-ous specifications in this regard. We conclude that the country-pair fixed-effects model is preferred statistically to all other specifications and show that estimates of the effects of integration on trade can differ a great deal across the specifications.
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DATA APPENDIX
DEFINITIONS OF VARIABLES
Real Exports, measured in millions of U.S. dollars, from World Trade Flows, 1980-1997
(see Feenstra, 2000). Deflated using CPI-U-RS from the Bureau of Labor Statistics.
Real Gross Domestic Product is in millions of U.S. dollars at market prices from the World Bank’s World Development Indicators 1999 CD-ROM. Deflated using CPI-U-RS from the Bureau of Labor
Statistics.
Population in thousands of inhabitants from the World Bank’s World Development Indicators 1999
CD-ROM.
Distance, expressed in kilometers, is the great circle distance between geographic centers, using the
Haversine formula. Coordinates from the CIA’s The World Factbook 2000.
Contiguity is equal to 1 if two trading partners share a border. From the CIA’s The World Factbook 2000.
Common Language is equal to 1 if two trading partners share a common first language. From the
CIA’s The World Factbook 2000.
European Bloc is equal to 1 when both countries are members of the EC for 1982 or 1987, the EU
for 1992, or the EEA for 1997.
North American Bloc is equal to 1 for Canada–United States trade for 1992 and 1997, and for
Canada-Mexico and United States–Mexico trade for 1997.
Mercosur is equal to 1 in 1997 for trade between Argentina, Brazil, Paraguay, and Uruguay.
Australia–New Zealand CER is equal to 1 in 1987, 1992, and 1997 for trade between Australia and
New Zealand.
Israel–United States FTA is equal to 1 in 1987, 1992, and 1997 for trade between Israel and the
United States.
THE 29 COUNTRIES INCLUDED IN THE DATA SET
Argentina, Australia, Austria, Belgium-Luxembourg, Brazil, Canada, Denmark, Finland, France, Germany, Greece, Hong Kong, Ireland, Israel, Italy, Japan, South Korea, Mexico, the Netherlands, New Zealand, Norway, Portugal, Singapore, Spain, Sweden, Switzerland, the United Kingdom, Uruguay, and the United States.
