Economics Letters 66 (000) 5 31 www.elsevier.com/ locate/ econbase A note on te proper econometric specification of te ravity equation Peter Eer* Austrian Institute of Economic Researc, P.O. Bo 91, A-1103 Vienna, Austria Received 8 February 1999; accepted 13 July 1999 Abstract Tis paper seds some lit on te problems associated wit random effects ravity approaces. Aruments for te superiority of a fied effects model are iven bot alon intuitive and econometric lines based on a Hausman test. 000 Elsevier Science S.A. All rits reserved. Keywords: Gravity equation; Panel econometrics JEL classification: C33; F14; F15 1. Introduction Amon te ue empirical literature on ravity models publised in te last decade most studies ave been done wit a cross-section metodoloy. However, a panel framework reveals several advantaes over cross-section analysis: On te one and panels allow to capture te relationsips between te relevant variables over a loner period and to identify te role of te overall business cycle penomenon (in cross-section researc one usually employs data averaes over a certain period 1 to lower te influence of outliers ). On te oter and, trou a panel approac one is able to disentanle te time invariant country-specific effects. Above all, one sould take into account tat te interpretation of te estimated coefficients is crucially different from tat of cross-section analysis. In a panel framework one cecks for cross-section deviations and is tus able to interpret te parameters *Tel.: 143-1-798-601/ 08; fa: 143-1-798-9386. E-mail address: eer@wifo.ac.at (P. Eer) 1 Note tat from cross-section parameters we only et valid predictions of te comparative statics if we are in te equilibrium (Scmalensee, 1989). Offside a unique equilibrium te estimated parameters would deviate from tose out of a panel analysis. In suc circumstances te estimated sin of te coefficients could be wron in te etreme case. Panels also allow to draw on te time dimension and do not need te assumption of identical steady-states in levels across roups. 0165-1765/ 00/ $ see front matter 000 Elsevier Science S.A. All rits reserved. PII: S0165-1765(99)00183-4
6 P. Eer / Economics Letters 66 (000) 5 31 as elasticities of te influence of independent variables on te dependent one (witin interpretation). In cross-section analysis in many cases one is tempted to interpret te coefficients in te same way wic is conceptually wron, as in fact tey sould be read as a composite of witin and between effects (see Hsiao, 1986). Neverteless, so far only a few autors in tis field investiated a panel framework (Baldwin, 1994; Matyas, 1997, 1998). But it seems not clear weter one sould apply a random (REM) or a fied effects model (FEM). Lookin at some of te latent variables tat one would arue to stand beind te country-specific and time invariant eport and import effects will sed some lit on te problem. Fied effects are due to omitted variables tat are specific to cross-sectional units (eport and import effects) or to time periods (Hsiao, 1986). Some of te main forces beind te fied eport effects sould be tariff policy measures and eport drivin or impedin environmental variables. Te former can be tout of as averae tariff or non-tariff barriers (tariffs, taes, duties, bureaucratic leal requirements, etc.) eiter on te eport side of te reporter or on te import side of te wole sample of partner countries. Te latter include size of country, access to transnational infrastructure networks, eorapical and istorical determinants (e.., te relatively important role of trade relations between te CEECs because of former membersip in COMECON, etc.). As most of tese effects are not random but (e.., because of pat dependencies, membersip in supranational oranisations, etc.) deterministically associated wit certain istorical, political, eorapical and oter facts, a FEM would be te rit coice from tis intuitive point of view. Anoter arument wic favours te FEM is based on te problem of sample selection. In many applications te ravity model is used to calibrate interation effects and, tus, to project trade flows between EU or OECD countries and te Central and Eastern European Countries (CEECs). In tat case one is not interested in te estimation of typical trade flows between a randomly drawn sample of countries but between an 3 e ante predetermined selection of nations. One would like to know, ow te typical trade relations between, e.., a CEEC and a EU member country would look like if tey followed te pattern of a typical relationsip between EU countries. Under suc circumstances te FEM would be te rit coice, since te sample is eaustive. I sow tat also because of pure econometrical reasons preference is iven to te FEM over te REM. As te teoretical content of te ravity equation was criticised (Deardorff, 1995) for bein derivable from any plausible model of trade, I coose a specification wic is associated as closely as possible wit a Heckscer Olin model under product differentiation. Te followin section briefly introduces te econometric specification and te Hausman-test procedure, Section 3 provides information on te database and estimation results, Section 4 contains te conclusions.. A model wit time and country effects Matyas (1997) arued tat te correct ravity specification is a tree-way model. One dimension is time (reflectin te common business cycle or lobalisation process over te wole sample of Wile Baldwin (1994) employs a REM, Matyas does not ive preference to te FEM over te REM or vice versa. 3 Tis is wat Matyas (1998) as in mind wen notin tat for lare country samples (e.., wen one s interest lies in te eneral evaluation of te effects of transportation costs and oter variables on bilateral trade volumes) one sould treat te country specific (import and eport) effects as non-observable random variables.
P. Eer / Economics Letters 66 (000) 5 31 7 countries) and te oter two dimensions of roup variables are time invariant eport and import country effects. Accordin to Helpman and Kruman (1985) and Helpman (1987) an endowment based 3 3 model is cosen, were one of te two oods is differentiated and te oter is omoeneous. Te two factors of production are te stock of capital and te labor force (proied by population). In suc a framework te total volume of trade of eac country could be defined as te sum of inter- and intra-industry trade volumes. Te correspondin reduced form equation to estimate te world volume of trade in suc a model reads X 5 b 1 b RLFAC 1 b GDPT 1 b SIMILAR 1 b DIST 1 a 1 1 d 1 u (1) ijt 0 1 ijt ijt 3 ijt 4 ij i j t ijt were X is te lo of country i s eports to country j in year t. b is te constant. ijt 0 U Kjt Kit RLFACijt 5 ln] ln] N N jt it U measures te distance between te two countries in terms of relative factor endowments. Tis variable could take a minimum value of 0 (equality in relative factor endowments). Accordin to teory, te larer tis difference, te ier is te volume of interindustry (and overall) trade, and te lower te sare of intra-industry trade, GDP ]]]]] GDPit ]]]]] jt F S D S DG it jt it jt SIMILARijt 5 ln 1 GDP 1 GDP GDP 1 GDP captures te relative size of two countries in terms of GDP. Tis inde is bounded between 0 (absolute diverence in size) and 0.5 (equal country size). Te larer tis measure and, tus, te more similar two countries in terms of GDP, te ier te sare of intra-industry trade. It is also clear tat te total volume of trade sould be ier, te larer te overall economic space GDPTijt 5 ln(gdpit 1 GDP jt) of te two countries for iven relative size and factor endowments. DISTij is te lo of te distance variable wic is a proy for transportation costs. Lookin at te factor bo representation for suc a model witout transport costs, we would associate GDPT wit te lent of te diaonal of te bo, SIMILAR wit te location of te consumption point alon tis diaonal, and RLFAC as a measure of distance between te endowment point and te consumption point alon te relative factor price 4 line. dt reflects te time effect wic is due to all countries, ai and j are te country specific fied effects. Accordin to Baltai (1995) and Greene (1995) Hausman s statistic for testin random versus fied effects is applied. Terefore, one as initially to compute te (feasible) GLS (FGLS) reressors. Tis is done by splittin up te total variance into its tree components (sˆ 1 ˆ s 1 ˆ s m). Te first term (s ˆ ) is equivalent to te variance from te FEM (witin roup variance) and te oter two components are parts of te between-variances for te eport and import country factor. Tere are now tree ways to estimate tese components wic are equivalent if bols is consistent. (1) One can run te roup means estimations to et te variance components and furtermore te weits to construct te FGLS estimator. () Alternatively one can start directly from te OLS estimator to fiure 4 It is not tested for te randomness of time-effects, owever, as te overall cycle, te eneral development of openness, or watever is measured be tat factor, enerally sould not be treated as random.
8 P. Eer / Economics Letters 66 (000) 5 31 out ˆ s and ˆ s m.(3)sˆ and ˆ s m can also be based on te sample variance of te fied effects from te FEM. Te latter possibility, owever, is only available if one as initially fitted te FEM but uarantees positive estimates of ˆ s and ˆ s m. Te variance components are used to calculate te correspondin weits needed for te variables in te REM (see Greene, 1995, p. 313, or Baltai, 1995, p. 3). Weter te REM or te FEM is te econometrically more appropriate setup stronly depends on te correlation of te individual effects wit te reressors. However, it is a basic assumption in te REM tat tere is no suc correlation. If some variables are omitted, te REM may suffer from tat. Te Hausman statistic tests for te ortoonality of te random effects and te reressors, tis is tus a test for misspecification. Te test statistic is asymptotically distributed as central. A sinificant test statistic reveals a i importance of roup-specific effects and teir correlation wit te rit-and variables and is an econometric arument at and tat underpins te importance to control for permanently unobserved differences across roups. In suc a case te random-effects estimates are sinificantly inconsistent (see Hsiao, 1986, p. 49), but under te null-ypotesis tey are bot efficient and consistent. 3. Data and empirical results Te data series cover a period of 1 years (1985 96). All variables are in constant prices and dollars wit 1990 as te base year. Bilateral eport data were taken from OECD Statistics of Forein Trade. GDP, population, and ross fied capital formation (GFCF) are obtained from OECD National Accounts. Eport price indices are taken from te OECD Economic Outlook and GDP deflators come from te OECD and te WIFO database. Te distance variable is measured in miles between capitals and was computed in te followin way (see Scumacer, 1997) D 5 r? ar cos[sin(w )? sin(w ) 1 cos(w )? cos(l l )]. ij i j i j i Were r is te eart radius in miles, wi and wj are radian measures of te parallel of latitude of te two countries capitals, and (l j l i) is te radian measure of te difference in meridians of te two countries capitals. Capital stocks ave been calculated accordin to te perpetual inventory metod: K 5 5*(GFCF 1 GFCF ) 1984 1983 1984 Furtermore I assumed all countries capital stocks to depreciate at a constant rate of 10%. So te capital stock of te followin years becomes K 5 0.9? K 1 GFCF. t t1 t Nominal capital stocks were converted to real numbers by te use of GDP deflators. Te country sample contains all 15 EU member countries, were Belium and Luembur were treated as a sinle country. Note tat te commonly used setup of te ravity equation is unbalanced
P. Eer / Economics Letters 66 (000) 5 31 9 sui eneris, because no country is eportin to itself. Tus, even in te case of equal roup sizes te panel would be unbalanced. As we are just testin for te randomness of te two country-dimensions te witin and between transformations reduce to te two-way case (see Wansbeek and Kapteyn, 5 1989; Baltai, 1995). Because of te unbalancedness of our data set we come up wit 184 data points for te estimation (Table 1). Note tat te OLS estimation was sown as it ad to be estimated for te Larane multiplier test. As we are about to test weter te country specific (eport and import) effects sould be modeled by a FEM and not a REM, time effects are treated as fied for all estimations (also for OLS). From te Larane multiplier test statistic we see tat te assumption of OLS tat tere is no roupwise eteroscedasticity is rejected. In te OLS and te REM estimations all te coefficients ave te 6 epected sin and are ily sinificant. Te results for te FEM empasize tat relative country size (SIMILAR) and distance in relative factor endowments (RLFAC) do not vary muc in te time dimension and are captured by te country-specific fied effects. However, in te FEM we observe no sinificant influence of tose two variables. It sould be noted tat tis obviously is not so for distance. In all te estimations te scalin variable (GDPT) and distance (DIST) eibit major influence. Te likeliood ratio tests in te FEM reveal tat a lot of information is comin from country-effects and, tus, out of te cross-section. Te restriction of time-effects to be zero is also rejected in bot te OLS and te FEM estimations. Te ily sinificant Hausman statistic in our case is driven by differences bot between te variance-covariance matrices of te models and also te parameter estimates. It demonstrates tat te FEM is consistent, but REM (FGLS) is not. 4. Conclusions Most of te contributions to te empirical ravity literature made use of cross-section data. Wan and Winters (1991) and Hamilton and Winters (199) followed tis line as well as Collins and Rodrik (1991). A panel framework as many advantaes vis-a-vis ` te cross-section approac. First of all it allows to disentanle country-specific and time-specific effects. Te present paper demonstrates tat te proper econometric specification of a ravity model in most applications would be one of fied country and time effects. Tis was demonstrated by te Hausman -test and was motivated by te eplanation of country effects as widely predetermined because of eorapical, istorical, or political contets. 5 Matyas (1998) provides a solution for te estimation of te variance components from te OLS residuals in a tree-way unbalanced ravity panel model. 6 My understandin of epected sin is te followin: In te 33 model of te aforementioned type it is clear tat rowin economic space (sum of GDPs) c.p. is associated wit a rowin trade volume. Te same olds true for a ceteris paribus cane in relative factor endowments wic increases te volume of trade wenever te possibility of a factor intensity reversal is ecluded (see Breuss and Eer, 1998). Te analysis of te impact of c.p. canin relative country size on te volume of trade is not so easy. Tere te effect depends on ow and weter at all relative factor endowments are adjusted. For equally endowed countries te result is an increase in te volume of trade wic is due to an increase of intra-industry trade. All of wic olds true for endowment situations witin te factor price equalisation reion.
30 P. Eer / Economics Letters 66 (000) 5 31 Table 1 Estimation results a FEM REM OLS b t b t b t RLFAC 0.03 0.9 0.06.1 0.14 3.6 GDPT 0.8 4.8 1.01 6.0 1.39 71.4 SIMILAR 0.0 0.7 0.34 1.1 0.55 1.1 DIST 1.08 48.6 1.13 49. 1.3 50.7 CONST. 19.64 13. 5.41 1. 7.47 1.9 N 184 184 184 ] R 0.95 0.75 0.89 s 0.40 0.43 0.59 ŝ 0.47 ŝ m 0.64 b LR-X 178.9 (14) c LR-M 893.9 (14) d LR-T 16.1 80.8 (1) (1) e Hausman 86.1 (16) f LM 6.4 (8) a Note: country and time effects are not reported. b Likeliood ratio test, Greene (1997, p. 161): fied eport effects. c Likeliood ratio test: fied import effects. d Likeliood ratio test: fied time effects. e ˆ ˆ 1 Hausman statistic: (b b )9Var[b ] Var[b ]j (bˆ lsdv ls lsdv ls lsdv b ls), Greene (1997, p. 633). f Breusc Paan Larane multiplier test, Baltai (1995), p. 6: Testin for random effects. Note, tat te test was computed for te averae year: LM 5 18/(M 1) F O 1/1O u Y 1/1OOu 1 SS mdds md m m G and LM 5 18/(X 1) F O 1/1O u Y 1/1OOu SS mdds md m m G wit LM5LM 1LM. As we observe 1 years, te correspondin residuals and residual squares are divided by tis number 1 to obtain time averaes. X and M are te roup sizes for eporters and importers, eac 14 in our case. Derees of freedom in parentesis. Sinificant at 1%. Acknowledements I wis to tank R. Kunst, R. Winter-Ebmer, A. Winters, and te participants of a researc seminar of te University of Linz, Department of Economics for teir useful comments. I am indebted to W.
P. Eer / Economics Letters 66 (000) 5 31 31 Koler, L. Matyas, and M. Pfaffermayr for elpful discussions. Of course, any remainin errors are my own. References Baldwin, R., 1994. In: Towards an Interated Europe, CEPR, London. Baltai, B., 1995. In: Econometric Analysis of Panel Data, Wiley, Cicester. Breuss, F., Eer, P., 1998. How Reliable are Estimations of East-West Trade Potentials based on Cross-Section Gravity Analyses? Revised Workin Paper, Austrian Institute of Economic Researc, Vienna. Collins, S., Rodrik, D., 1991. In: Eastern European and te Soviet Union in te World Economy, Institute of International Economics, Wasinton, DC. Deardorff, A.V., 1995. Determinants of Bilateral Trade: Does Gravity Work in a Neoclassic World, NBER Workin Paper 5377. Greene, W.H., 1995. Limdep, Version 7.0, User s Manual, Econometric Software, Bellport. Greene, W.H., 1997. In: Econometric Analysis, 3rd ed., Prentice-Hall International, London. Hamilton, C.B., Winters, L.A., 199. Openin up international trade wit Eastern Europe. Economic Policy 14, 77 116. Helpman, E., 1987. Imperfect competition and international trade: evidence from fourteen industrial countries. Journal of te Japanese and International Economies 1 (1), 6 81. Helpman, E., Kruman, P.R., 1985. In: Market Structure and Forein Trade. Increasin Returns, Imperfect Competition, and te International Economy, MIT Press, Cambride, MA. Hsiao, C., 1986. In: Analysis of Panel Data, Cambride University Press, Cambride, MA. Matyas, L., 1997. Proper econometric specification of te ravity model. Te World Economy 0 (3), 363 368. Matyas, L., 1998. Te Gravity Model: some econometric considerations. Te World Economy 1 (3), 397 401. Scmalensee, R., 1989. Inter-industry studies of structure and performance. In: Scmalensee, R., Willi, R.D. (Eds.), Handbook of Industrial Oranization, Vol., Nort-Holland, Amsterdam. Scumacer, D., 1997. In: Perspektiven des Außenandels zwiscen West- und Osteuropa: Ein disareierter Gravitationsansatz, Deutsces Institut fur Wirtscaftsforscun, Berlin. Wan, Z.K., Winters, L.A., 1991. Te Tradin Potential of Eastern Europe, CEPR Discussion Paper 610. Wansbeek, T.J., Kapteyn, A., 1989. Estimation of te error components model wit incomplete panels. Journal of Econometrics 41 (3), 341 361.