THE LOG OF GRAVITY

THE LOG OF GRAVITY

J.M.C.Santos Silva and Silvana Tenreyro*

Abstract—Although economists have long been aware of Jensen’s in-equality,many econometric applications have neglected an important implication of it:under heteroskedasticity,the parameters of log-linearized models estimated by OLS lead to biased estimates of the true elasticities.We explain why this problem arises and propose an appropri-ate estimator.Our criticism of conventional practices and the proposed solution extend to a broad range of applications where log-linearized equations are estimated.We develop the argument using one particular illustration,the gravity equation for trade.We?nd signi?cant differences between estimates obtained with the proposed estimator and those ob-tained with the traditional method.

I.Introduction

E CONOMISTS have long been aware that Jensen’s in-

equality implies that E(ln y) ln E(y),that is,the expected value of the logarithm of a random variable is different from the logarithm of its expected value.This basic fact,however,has been neglected in many economet-ric applications.Indeed,one important implication of Jen-sen’s inequality is that the standard practice of interpreting the parameters of log-linearized models estimated by ordi-nary least squares(OLS)as elasticities can be highly mis-leading in the presence of heteroskedasticity.

Although many authors have addressed the problem of obtaining consistent estimates of the conditional mean of the dependent variable when the model is estimated in the log linear form(see,for example,Goldberger,1968;Man-ning&Mullahy,2001),we were unable to?nd any refer-ence in the literature to the potential bias of the elasticities estimated using the log linear model.

In this paper we use the gravity equation for trade as a particular illustration of how the bias arises and propose an appropriate estimator.We argue that the gravity equation, and,more generally,constant-elasticity models,should be estimated in their multiplicative form and propose a simple pseudo-maximum-likelihood(PML)estimation technique. Besides being consistent in the presence of heteroskedas-ticity,this method also provides a natural way to deal with zero values of the dependent variable.

Using Monte Carlo simulations,we compare the perfor-mance of our estimator with that of OLS(in the log linear speci?cation).The results are striking.In the presence of heteroskedasticity,estimates obtained using log-linearized models are severely biased,distorting the interpretation of the model.These biases might be critical for the compara-tive assessment of competing economic theories,as well as for the evaluation of the effects of different policies.In contrast,our method is robust to the different patterns of heteroskedasticity considered in the simulations.

We next use the proposed method to provide new esti-mates of the gravity equation in cross-sectional https://www.wendangku.net/doc/4f14117389.html,ing standard tests,we show that heteroskedasticity is indeed a severe problem,both in the traditional gravity equation introduced by Tinbergen(1962),and in a gravity equation that takes into account multilateral resistance terms or?xed effects,as suggested by Anderson and van Wincoop(2003). We then compare the estimates obtained with the proposed PML estimator with those generated by OLS in the log linear speci?cation,using both the traditional and the?xed-effects gravity equations.

Our estimation method paints a very different picture of the determinants of international trade.In the traditional gravity equation,the coef?cients on GDP are not,as gen-erally estimated,close to1.Instead,they are signi?cantly smaller,which might help reconcile the gravity equation with the observation that the trade-to-GDP ratio decreases with increasing total GDP(or,in other words,that smaller countries tend to be more open to international trade).In addition,OLS greatly exaggerates the roles of colonial ties and geographical proximity.

Using the Anderson–van Wincoop(2003)gravity equa-tion,we?nd that OLS yields signi?cantly larger effects for geographical distance.The estimated elasticity obtained from the log-linearized equation is almost twice as large as that predicted by PML.OLS also predicts a large role for common colonial ties,implying that sharing a common colonial history practically doubles bilateral trade.In con-trast,the proposed PML estimator leads to a statistically and economically insigni?cant effect.

The general message is that,even controlling for?xed effects,the presence of heteroskedasticity can generate strikingly different estimates when the gravity equation is log-linearized,rather than estimated in levels.In other words,Jensen’s inequality is quantitatively and qualitatively important in the estimation of gravity equations.This sug-gests that inferences drawn on log-linearized regressions can produce misleading conclusions.

Despite the focus on the gravity equation,our criticism of the conventional practice and the solution we propose ex-tend to a broad range of economic applications where the equations under study are log-linearized,or,more generally, transformed by a nonlinear function.A short list of exam-ples includes the estimation of Mincerian equations for wages,production functions,and Euler equations,which are typically estimated in logarithms.

Received for publication March29,2004.Revision accepted for publi-

cation September13,2005.

*ISEG/Universidade Te′cnica de Lisboa and CEMAPRE;and London

School of Economics,CEP,and CEPR,respectively.

We are grateful to two anonymous referees for their constructive

comments and suggestions.We also thank Francesco Caselli,Kevin

Denny,Juan Carlos Hallak,Daniel Mota,John Mullahy,Paulo Parente,

Manuela Simarro,and Kim Underhill for helpful advice on previous

versions of this paper.The usual disclaimer applies.Jiaying Huang

provided excellent research assistance.Santos Silva gratefully acknowl-

edges the partial?nancial support from Fundac?a?o para a Cie?ncia e

Tecnologia,program POCTI,partially funded by FEDER.A previous

version of this paper circulated as“Gravity-Defying Trade.”

The Review of Economics and Statistics,November2006,88(4):641–658

?2006by the President and Fellows of Harvard College and the Massachusetts Institute of Technology

The remainder of the paper is organized as follows. Section II studies the econometric problems raised by the estimation of gravity equations.Section III considers constant-elasticity models in general;it introduces the PML estimator and speci?cation tests to check the adequacy of the proposed estimator.Section IV presents the Monte Carlo simulations.Section V provides new estimates of both the traditional and the Anderson–van Wincoop gravity equa-tion.The results are compared with those generated by OLS,nonlinear least squares,and tobit estimations.Section VI contains concluding remarks.

II.The Econometrics of the Gravity Equation

A.The Traditional Gravity Equation

The pioneering work of Jan Tinbergen(1962)initiated a vast theoretical and empirical literature on the gravity equa-tion for trade.Theories based on different foundations for trade,including endowment and technological differences, increasing returns to scale,and Armington demands,all predict a gravity relationship for trade?ows analogous to Newton’s law of universal gravitation.1In its simplest form, the gravity equation for trade states that the trade?ow from country i to country j,denoted by T ij,is proportional to the product of the two countries’GDPs,denoted by Y i and Y j, and inversely proportional to their distance,D ij,broadly construed to include all factors that might create trade resistance.More generally,

T ij??0Y i?1Y j?2D ij?3,(1) where?0,?1,?2,and?3are unknown parameters.

The analogy between trade and the physical force of gravity,however,clashes with the observation that there is no set of parameters for which equation(1)will hold exactly for an arbitrary set of observations.To account for devia-tions from the theory,stochastic versions of the equation are used in empirical studies.Typically,the stochastic version of the gravity equation has the form

T ij??0Y i?1Y j?2D ij?3?ij,(2)where?ij is an error factor with E(?ij?Y i,Y j,D ij)?1, assumed to be statistically independent of the regressors, leading to

E?T ij?Y i,Y j,D ij???0Y i?1Y j?2D ij?3.

There is a long tradition in the trade literature of log-linearizing equation(2)and estimating the parameters of interest by least squares,using the equation

ln T ij?ln?0??1ln Y i??2ln Y j

??3ln D ij?ln?ij.

(3)

The validity of this procedure depends critically on the assumption that?ij,and therefore ln?ij,are statistically independent of the regressors.To see why this is so,notice that the expected value of the logarithm of a random variable depends both on its mean and on the higher-order moments of the distribution.Hence,for example,if the variance of the error factor?ij in equation(2)depends on Y i, Y j,or D ij,the expected value of ln?ij will also depend on the regressors,violating the condition for consistency of OLS.2 In the cases studied in section V we?nd overwhelming evidence that the error terms in the usual log linear speci-?cation of the gravity equation are heteroskedastic,which violates the assumption that ln?ij is statistically indepen-dent of the regressors and suggests that this estimation method leads to inconsistent estimates of the elasticities of interest.

A related problem with the analogy between Newtonian gravity and trade is that gravitational force can be very small,but never zero,whereas trade between several pairs of countries is literally zero.In many cases,these zeros occur simply because some pairs of countries did not trade in a given period.For example,it would not be surprising to ?nd that Tajikistan and Togo did not trade in a certain year.3 These zero observations pose no problem at all for the estimation of gravity equations in their multiplicative form. In contrast,the existence of observations for which the dependent variable is zero creates an additional problem for

1See,for example,Anderson(1979),Helpman and Krugman(1985), Bergstrand(1985),Davis(1995),Deardoff(1998),and Anderson and van Wincoop(2003).A feature common to these models is that they all assume complete specialization:each good is produced in only one country. However,Haveman and Hummels(2001),Feenstra,Markusen,and Rose (2000),and Eaton and Kortum(2001)derive the gravity equation without relying on complete specialization.Examples of empirical studies framed on the gravity equation include the evaluation of trade protection(for example,Harrigan,1993),regional trade agreements(for example, Frankel,Stein,&Wei,1998;Frankel,1997),exchange rate variability(for example,Frankel&Wei,1993;Eichengreen&Irwin,1995),and currency unions(for example,Rose,2000;Frankel&Rose,2002;and Tenreyro& Barro,2002).See also the various studies on border effects in?uencing the patterns of intranational and international trade,including McCallum (1995),and Anderson and van Wincoop(2003),among others.

2As an illustration,consider the case in which?ij follows a log normal distribution,with E(?ij?Y i,Y j,D ij)?1and variance?ij2?f(Y i,Y j,D ij).The error term in the log-linearized representation will then follow a normal distribution,with E[ln?ij?Y i,Y j,D ij]??1

2

ln(1??

ij

2),which is also a function of the covariates.

3The absence of trade between small and distant countries might be explained,among other factors,by large variable costs(for example, bricks are too costly to transport)or large?xed costs(for example, information on foreign markets).At the aggregate level,these costs can be best proxied by the various measures of distance and size entering the gravity equation.The existence of zero trade between many pairs of countries is directly addressed by Hallak(2006)and Helpman,Melitz,and Rubinstein(2004).These authors propose a promising avenue of research using a two-part estimation procedure,with a?xed-cost equation deter-mining the cutoff point above which a country exports,and a standard gravity equation.Their results,however,rely heavily on both normality and homoskedasticity assumptions,the latter being the particular concern of this paper.A natural topic for further research is to develop and implement an estimator of the two-part model that,like the PML estimator proposed here,is robust to distributional assumptions.

THE REVIEW OF ECONOMICS AND STATISTICS 642

the use of the log linear form of the gravity equation.

Several methods have been developed to deal with this

problem[see Frankel(1997)for a description of the various

procedures].The approach followed by the large majority of

empirical studies is simply to drop the pairs with zero trade

from the data set and estimate the log linear form by OLS.

Rather than throwing away the observations with T ij?0, some authors estimate the model using T ij?1as the dependent variable or use a tobit estimator.However,these

procedures will generally lead to inconsistent estimators of

the parameters of interest.The severity of these inconsis-

tencies will depend on the particular characteristics of the

sample and model used,but there is no reason to believe that

they will be negligible.

Zeros may also be the result of rounding errors.4If trade

is measured in thousands of dollars,it is possible that for

pairs of countries for which bilateral trade did not reach a

minimum value,say$500,the value of trade is registered as

0.If these rounded-down observations were partially com-

pensated by rounded-up ones,the overall effect of these

errors would be relatively minor.However,the rounding

down is more likely to occur for small or distant countries,

and therefore the probability of rounding down will depend

on the value of the covariates,leading to the inconsistency

of the estimators.Finally,the zeros can just be missing

observations that are wrongly recorded as0.This problem is

more likely to occur when small countries are considered,

and again the measurement error will depend on the covari-

ates,leading to inconsistency.

B.The Anderson–van Wincoop Gravity Equation Anderson and van Wincoop(2003)argue that the tradi-tional gravity equation is not correctly speci?ed,as it does not take into account multilateral resistance terms.One of the solutions for this problem that is suggested by those authors is to augment the traditional gravity equation with exporter and importer?xed effects,leading to

T ij??0Y i?1Y j?2D ij?3e?i d i??j d j,(4)

where?0,?1,?2,?3,?i,and?j are the parameters to be estimated and d i and d j are dummies identifying the exporter and importer.5

Their model also yields the prediction that?1??2?1, which leads to the unit-income-elasticity model

T ij??0Y i Y j D ij?3e?i d i??j d j,

whose stochastic version has the form

E?T ij?Y i,Y j,D ij,d i,d j???0Y i Y j D ij?3e?i d i??j d j.(5)As before,log-linearization of equation(5)raises the prob-

lem of how to treat zero-value observations.Moreover,

given that equation(5)is a multiplicative model,it is also

subject to the biases caused by log-linearization in the

presence of heteroskedasticity.Naturally,the presence of

the individual effects may reduce the severity of this prob-

lem,but whether or not that happens is an empirical issue.

In our empirical analysis we provide estimates for both

the traditional and the Anderson–van Wincoop gravity equa-

tions,using alternative estimation methods.We show that,

in practice,heteroskedasticity is quantitatively and qualita-

tively important in the gravity equation,even when control-

ling for?xed effects.Hence,we recommend estimating the

augmented gravity equation in levels,using the proposed

PML estimator,which also adequately deals with the zero-

value observations.

III.Constant-Elasticity Models

Despite their immense popularity,empirical studies in-

volving gravity equations still have important econometric

?aws.These?aws are not exclusive to this literature,but

extend to many areas where constant-elasticity models are

used.This section examines how the deterministic multipli-

cative models suggested by economic theory can be used in

empirical studies.

In their nonstochastic form,the relationship between the

multiplicative constant-elasticity model and its log linear

additive formulation is trivial.The problem,of course,is

that economic relations do not hold with the accuracy of

physical laws.All that can be expected is that they hold on

average.Indeed,here we interpret economic models like the

gravity equation as yielding the expected value of the

variable of interest,y?0,for a given value of the explan-

atory variables,x(see Goldberger,1991,p.5).That is,if

economic theory suggests that y and x are linked by a

constant-elasticity model of the form y i?exp(x i?),the function exp(x i?)is interpreted as the conditional expecta-tion of y i given x,denoted E[y i?x].6For example,using the notation in the previous section,the multiplicative gravity

relationship can be written as the exponential function exp

[ln?0??1ln Y i??2ln Y j??3ln D ij],which is interpreted as the conditional expectation E(T ij?Y i,Y j,D ij). Because the relation y i?exp(x i?)holds on average but not for each i,an error term is associated with each obser-vation,which is de?ned asεi?y i?E[y i?x].7Therefore,the stochastic model can be formulated as

4Trade data can suffer from many other forms of errors,as described in Feenstra,Lipsey,and Bowen(1997).

5Note that,throughout the paper,T ij denotes exports from i to j.

6Notice that if exp(x i?)is interpreted as describing the conditional median of y i(or some other conditional quantile)rather than the condi-tional expectation,estimates of the elasticities of interest can be obtained estimating the log linear model using the appropriate quantile regression estimator(Koenker&Bassett,1978).However,interpreting exp(x i?)as a conditional median is problematic when y i has a large mass of zero observations,as in trade data.Indeed,in this case the conditional median of y i will be a discontinuous function of the regressors,which is generally not compatible with standard economic theory.

7Whether the error enters additively or multiplicatively is irrelevant for our purposes,as explained below.

THE LOG OF GRA VITY643

y i?exp?x i???εi,(6) with y i?0and E[εi?x]?0.

As we mentioned before,the standard practice of log-linearizing equation(6)and estimating?by OLS is inap-propriate for a number of reasons.First of all,y i can be0,in which case log-linearization is infeasible.Second,even if all observations of y i are strictly positive,the expected value of the log-linearized error will in general depend on the covariates,and hence OLS will be inconsistent.To see the point more clearly,notice that equation(6)can be expressed as

y i?exp?x i???i,

with?i?1?εi/exp(x i?)and E[?i?x]?1.Assuming for the moment that y i is positive,the model can be made linear in the parameters by taking logarithms of both sides of the equation,leading to

ln y i?x i??ln?i.(7)

To obtain a consistent estimator of the slope parameters in equation(6)estimating equation(7)by OLS,it is neces-sary that E[ln?i?x]does not depend on x i.8Because?i?1?εi/exp(x i?),this condition is met only ifεi can be written asεi?exp(x i?)v i,where v i is a random variable statistically independent of x i.In this case,?i?1?v i and therefore is statistically independent of x i,implying that E[ln?i?x]is constant.Thus,only under very speci?c con-ditions on the error term is the log linear representation of the constant-elasticity model useful as a device to estimate the parameters of interest.

When?i is statistically independent of x i,the conditional variance of y i(andεi)is proportional to exp(2x i?).Although economic theory generally does not provide any informa-tion on the variance ofεi,we can infer some of its properties from the characteristics of the data.Because y i is nonnega-tive,when E[y i?x]approaches0,the probability of y i being positive must also approach0.This implies that V[y i?x],the conditional variance of y i,tends to vanish as E[y i?x]passes to0.9On the other hand,when the expected value of y is far away from its lower bound,it is possible to observe large deviations from the conditional mean in either direction, leading to greater dispersion.Thus,in practice,εi will generally be heteroskedastic and its variance will depend on exp(x i?),but there is no reason to assume that V[y i?x]is proportional to exp(2x i?).Therefore,in general,regressing ln y i on x i by OLS will lead to inconsistent estimates of?.

It may be surprising that the pattern of heteroskedasticity and,indeed,the form of all higher-order moments of the conditional distribution of the error term can affect the consistency of an estimator,rather than just its ef?ciency. The reason is that the nonlinear transformation of the dependent variable in equation(7)changes the properties of the error term in a nontrivial way because the conditional expectation of ln?i depends on the shape of the conditional distribution of?i.Hence,unless very strong restrictions are imposed on the form of this distribution,it is not possible to recover information about the conditional expectation of y i from the conditional mean of ln y i,simply because ln?i is correlated with the regressors.Nevertheless,estimating equation(7)by OLS will produce consistent estimates of the parameters of E[ln y i?x]as long as E[ln(y i)?x]is a linear function of the regressors.10The problem is that these parameters may not permit identi?cation of the parameters of E[y i?x].

In short,even assuming that all observations on y i are positive,it is not advisable to estimate?from the log linear model.Instead,the nonlinear model has to be estimated.

A.Estimation

Although most empirical studies use the log linear form of the constant-elasticity model,some authors[see Frankel and Wei(1993)for an example in the international trade literature]have estimated multiplicative models using non-linear least squares(NLS),which is an asymptotically valid estimator for equation(6).However,the NLS estimator can be very inef?cient in this context,as it ignores the het-eroskedasticity that,as discussed before,is characteristic of this type of data.

The NLS estimator of?is de?ned by

???arg min

b

?

i?1

n

?y i?exp?x i b??2,

which implies the following set of?rst-order conditions:?

i?1

n

?y i?exp?x i????exp?x i???x i?0.(8)

These equations give more weight to observations where exp(x i??)is large,because that is where the curvature of the conditional expectation is more pronounced.However, these are generally also the observations with larger vari-ance,which implies that NLS gives more weight to noisier observations.Thus,this estimator may be very inef?cient, depending heavily on a small number of observations.

If the form of V[y i?x]were known,this problem could be eliminated using a weighted NLS estimator.However,in

8Consistent estimation of the intercept would also require E[ln?i?x]?0. 9In the case of trade data,when E[y i?x]is close to its lower bound(that

is,for pairs of small and distant countries),it is unlikely that large values of trade are observed,for they cannot be offset by equally large deviations in the opposite direction,simply because trade cannot be negative. Therefore,for these observations,dispersion around the mean tends to be small.

10When E[ln y i?x]is not a linear function of the regressors,estimating equation(7)by OLS will produce consistent estimates of the parameters of the best linear approximation to E[ln y i?x](see Goldberger,1991,p.

53).

THE REVIEW OF ECONOMICS AND STATISTICS 644

practice,all we know about V [y i

?x ]is that,in general,it goes to 0as E [y

i

?x ]passes to 0.Therefore,an optimal weighted

NLS estimator cannot be used without further information on the distribution of the errors.In principle,this problem

can be tackled by estimating the multiplicative model using

a consistent estimator,and then obtaining the appropriate

weights estimating the skedastic function nonparametri-

cally,as suggested by Delgado (1992)and Delgado and

Kniesner (1997).However,this nonparametric generalized

least squares estimator is rather cumbersome to implement,

especially if the model has a large number of regressors.

Moreover,the choice of the ?rst-round estimator is an open

question,as the NLS estimator may be a poor starting point

due to its considerable inef?ciency.Therefore,the nonpara-

metric generalized least squares estimator is not appropriate

to use as a workhorse for routine estimation of multiplica-

tive models.11Indeed,what is needed is an estimator that is

consistent and reasonably ef?cient under a wide range of

heteroskedasticity patterns and is also simple to implement.

A possible way of obtaining an estimator that is more

ef?cient than the standard NLS without the need to use

nonparametric regression is to follow McCullagh and

Nelder (1989)and estimate the parameters of interest using

a PML estimator based on some assumption on the func-

tional form of V [y i

?x ].12Among the many possible speci?-cations,the hypothesis that the conditional variance is

proportional to the conditional mean is particularly appeal-

ing.Indeed,under this assumption E [y i

?x ]?exp(x i ?)?

V [y i

?x ],and ?can be estimated by solving the following set

of ?rst-order conditions:?i ?1n

?y i ?exp ?x i ??

??x i ?0.(9)

Comparing equations (8)and (9),it is clear that,unlike

the NLS estimator,which is a PML estimator obtained

assuming that V [y i

?x ]is constant,the PML estimator based

on equation (9)gives the same weight to all observations,rather than emphasizing those for which exp(x i

?)is large.This is because,under the assumption that E [y i

?x ]?V [y i ?x ],

all observations have the same information on the parame-ters of interest as the additional information on the curvature

of the conditional mean coming from observations with

large exp(x i

?)is offset by their larger variance.Of course,this estimator may not be optimal,but without further

information on the pattern of heteroskedasticity,it seems

natural to give the same weight to all observations.13Even if E [y i ?x ]is not proportional to V [y i ?x ],the PML estimator based on equation (9)is likely to be more ef?cient than the NLS estimator when the heteroskedasticity increases with the conditional mean.The estimator de?ned by equation (9)is numerically equal to the Poisson pseudo-maximum-likelihood (PPML)estimator,which is often used for count data.14The form of equation (9)makes clear that all that is needed for this estimator to be consistent is the correct speci?cation of the conditional mean,that is,E [y i ?x ]?exp(x i ?).Therefore,the data do not have to be Poisson at all—and,what is more important,y i does not even have to be an integer—for the estimator based on the Poisson likelihood function to be consistent.This is the well-known PML result ?rst noted by Gourieroux,Monfort,and Trognon (1984).The implementation of the PPML estimator is straight-forward:there are standard econometric programs with commands that permit the estimation of Poisson regression,even when the dependent variables are not integers.Because the assumption V [y i ?x ]?E [y i ?x ]is unlikely to hold,this estimator does not take full account of the heteroskedastic-ity in the model,and all inference has to be based on an Eicker-White (Eicker,1963;White,1980)robust covariance matrix estimator.In particular,within Stata (StataCorp.,2003),the PPML estimation can be executed using the following command:poisson export i ,j ln ?dist ij ?ln Y i ln Y j ?other variables ?ij ,robust where export (or import )is measured in levels.Of course,if it were known that V [y i ?x ]is a function of higher powers of E [y i ?x ],a more ef?cient estimator could be obtained by downweighting even more the observations with large conditional mean.An example of such an esti-mator is the gamma PML estimator studied by Manning and Mullahy (2001),which,like the log-linearized model,as-sumes that V [y i ?x ]is proportional to E [y i ?x ]2.The ?rst-order conditions for the gamma PML estimator are given by ?i ?1n ?y i ?exp ?x i ˇ???exp ??x i ˇ??x i ?0.In the case of trade data,however,this estimator may have an important drawback.Trade data for larger countries (as gauged by GDP per capita)tend to be of higher quality (see Frankel &Wei,1993;Frankel,1997);hence,models assuming that V [y i ?x ]is a function of higher powers of E [y i ?x ]might give excessive weight to the observations that 11A nonparametric generalized least squares estimator can also be used to estimate linear models in the presence of heteroskedasticity of unknown form (Robinson,1987).However,despite having been proposed more than 15years ago,this estimator has never been adopted as a standard tool by researchers doing empirical work,who generally prefer the simplicity of the inef?cient OLS,with an appropriate covariance matrix.12See also Manning and Mullahy (2001).A related estimator is proposed by Papke and Wooldridge (1996)for the estimation of models for fractional data.13The same strategy is implicitly used by Papke and Wooldridge (1996)in their pseudo-maximum-likelihood estimator for fractional data models.14See Cameron and Trivedi (1998)and Winkelmann (2003)for more details on the Poisson regression and on more general models for count data.

THE LOG OF GRA VITY 645

are more prone to measurement errors.15Therefore,the

Poisson regression emerges as a reasonable compromise,

giving less weight to the observations with larger variance

than the standard NLS estimator,without giving too much

weight to observations more prone to contamination by

measurement error and less informative about the curvature

of E[y i?x].16

B.Testing

In this subsection we consider tests for the particular

pattern of heteroskedasticity assumed by PML estimators,

focusing on the PPML estimator.Although PML estimators

are consistent even when the variance function is misspeci-

?ed,the researcher can use these tests to check if a different

PML estimator would be more appropriate and to decide

whether or not the use of a nonparametric estimator of the

variance is warranted.

Manning and Mullahy(2001)suggested that if

V?y i?x???0E?y i?x??1,(10) the choice of the appropriate PML estimator can be based on

a Park-type regression(Park,1966).Their approach is based

on the idea that if equation(10)holds and an initial consis-

tent estimate of E[y i?x]is available,then?1can be consis-tently estimated using an appropriate auxiliary regression.

Speci?cally,following Park(1966),Manning and Mullahy

(2001)suggest that?1can be estimated using the auxiliary

model

ln?y i?y?i?2?ln?0??1ln y?i?v i,(11) where y?i denotes the estimated value of E[y i?x].Unfortu-nately,as the discussion in the previous sections should have made clear,this approach based on the log-linearization of equation(10)is valid only under very restrictive conditions on the conditional distribution of y i. However,it is easy to see that this procedure is valid when the constant-elasticity model can be consistently estimated in the log linear form.Therefore,using equation(11)a test for H0:?1?2based on a nonrobust covariance estimator provides a check on the adequacy of the estimator based on the log linear model.

A more robust alternative,which is mentioned by Man-

ning and Mullahy(2001)in a footnote,is to estimate?1

from

?y i?y?i?2??0?y?i??1??i,(12) using an appropriate PML estimator.The approach based on equation(12)is asymptotically valid,and inference about?1 can be based on the usual Eicker-White robust covariance matrix estimator.For example,the hypothesis that V[y i?x]is proportional to E[y i?x]is accepted if the appropriate con?-

dence interval for?1contains1.However,if the purpose is to test the adequacy of a particular value of?1,a slightly simpler method based on the Gauss-Newton regression(see Davidson&MacKinnon,1993)is available.

Speci?cally,to check the adequacy of the PPML for which?1?1and y?i?exp(x i??),equation(12)can be expanded in a Taylor series around?1?1,leading to

?y i?y?i?2??0y?i??0??1?1??ln y i?y?i??i.

Now,the hypothesis that V[y i?x]?E[y i?x]can be tested against equation(10)simply by checking the signi?cance of the parameter?0(?1?1).Because the error term?i is unlikely to be homoskedastic,the estimation of the Gauss-Newton regression should be performed using weighted least squares.Assuming that in equation(12)the variance is also proportional to the mean,the appropriate weights are given by exp(?x i??),and therefore the test can be performed by estimating

?y i?y?i?2/?y?i??0?y?i

??0??1?1??ln y?i??y?i??*i(13) by OLS and testing the statistical signi?cance of?0(?1?1) using a Eicker-White robust covariance matrix estimator.17 In the next section,a small simulation is used to study the Gauss-Newton regression test for the hypothesis that V[y i?x]?E[y i?x],as well as the Park-type test for the hypothesis that the constant-elasticity model can be consistently esti-mated in the log linear form.

IV.A Simulation Study

This section reports the results of a small simulation study designed to assess the performance of different meth-ods to estimate constant-elasticity models in the presence of heteroskedasticity and rounding errors.As a by-product,we also obtain some evidence on the?nite-sample performance of the speci?cation tests presented above.These experi-ments are centered around the following multiplicative model:

15Frankel and Wei(1993)and Frankel(1997)suggest that larger countries should be given more weight in the estimation of gravity

equations.This would be appropriate if the errors in the model were just the result of measurement errors in the dependent variable.However,if it is accepted that the gravity equation does not hold exactly,measurement errors account for only part of the dispersion of trade data around the gravity equation.

16It is worth noting that the PPML estimator can be easily adapted to deal with endogenous regressors(Windmeijer&Santos Silva,1997)and panel data(Wooldridge,1999).These extensions,however,are not pur-sued here.

17Notice that to test V[y i?x]?E[y i?x]against alternatives of the form V[y i?x]??0exp[x i(???)],the appropriate auxiliary regression would be

?y i?y?i?2/?y?i??0?i??0?x i?i??i*,

and the test could be performed by checking the joint signi?cance of the elements of?0?.If the model includes a constant,one of the regressors in the auxiliary regression is redundant and should be dropped.

THE REVIEW OF ECONOMICS AND STATISTICS 646

E?y i?x????x i???exp??0??1x1i??2x2i?,

i?1,...,1000.(14)

Because,in practice,regression models often include a mixture of continuous and dummy variables,we replicate this feature in our experiments:x1i is drawn from a standard normal,and x2is a binary dummy variable that equals1with a probability of0.4.18The two covariates are independent, and a new set of observations of all variables is generated in each replication using?0?0,?1??2?1.Data on y are generated as

y i???x i???i,(15) where?i is a log normal random variable with mean1and variance?i2.As noted before,the slope parameters in equa-tion(14)can be estimated using the log linear form of the model only when?i2is constant,that is,when V[y i?x]is proportional to?(x i?)2.

In these experiments we analyzed PML estimators of the multiplicative model and different estimators of the log-linearized model.The consistent PML estimators studied were:NLS,gamma pseudo-maximum-likelihood(GPML), and PPML.Besides these estimators,we also considered the standard OLS estimator of the log linear model(here called simply OLS);the OLS estimator for the model where the dependent variable is y i?1[OLS(y?1)];a truncated OLS estimator to be discussed below;and the threshold tobit of Eaton and Tamura(1994)(ET-tobit).19

To assess the performance of the estimators under differ-ent patterns of heteroskedasticity,we considered the four following speci?cations of?i2:

Case1:?i2??(x i?)?2;V[y i?x]?1.

Case2:?i2??(x i?)?1;V[y i?x]??(x i?).

Case3:?i2?1;V[y i?x]??(x i?)2.

Case4:?i2??(x i?)?1?exp(x2i);V[y i?x]??(x i?)?exp(x2i)?(x i?)2.

In case1the variance ofεi is constant,implying that the NLS estimator is optimal.Although,as argued before,this case is unrealistic for models of bilateral trade,it is included in the simulations for completeness.In case2,the condi-tional variance of y i equals its conditional mean,as in the Poisson distribution.The pseudo-maximum-likelihood esti-mator based on the Poisson distribution is optimal in this situation.Case3is the special case in which OLS estimation of the log linear model is consistent for the slope parameters of equation(14).Moreover,in this case the log linear model not only corrects the heteroskedasticity in the data,but,

because?i is log normal,it is also the maximum likelihood

estimator.The GPML is the optimal PML estimator in this

case,but it should be outperformed by the true maximum

likelihood estimator.Finally,case4is the only one in which

the conditional variance does not depend exclusively on the

mean.The variance is a quadratic function of the mean,as

in case3,but it is not proportional to the square of the mean.

We carried out two sets of experiments.The?rst set was

aimed at studying the performance of the estimators of the

multiplicative and the log linear models under different

patterns of heteroskedasticity.In order to study the effect of

the truncation on the performance of the OLS,and given

that this data-generating mechanism does not produce ob-

servations with y i?0,the log linear model was also estimated using only the observations for which y i?0.5 [OLS(y?0.5)].This reduces the sample size by approxi-

mately25%to35%,depending on the pattern of heteroske-

dasticity.The estimation of the threshold tobit was also

performed using this dependent variable.Notice that,al-

though the dependent variable has to cross a threshold to be

observable,the truncation mechanism used here is not equal

to the one assumed by Eaton and Tamura(1994).Therefore,

in all these experiments the ET-tobit will be slightly mis-

speci?ed and the results presented here should be viewed as

a check of its robustness to this problem.

The second set of experiments studied the estimators’

performance in the presence of rounding errors in the

dependent variable.For that purpose,a new random vari-

able was generated by rounding to the nearest integer the

values of y i obtained in the?rst set of simulations.This

procedure mimics the rounding errors in of?cial statistics

and generates a large number of zeros,a typical feature of

trade data.Because the model considered here generates a

large proportion of observations close to zero,rounding

down is much more frequent than rounding up.As the

probability of rounding up or down depends on the covari-

ates,this procedure will necessarily bias the estimates,as

discussed before.The purpose of the study is to gauge the

magnitude of these biases.Naturally,the log linear model

cannot be estimated in these conditions,because the depen-

dent variable equals0for some observations.Following

what is the usual practice in these circumstances,the trun-

cated OLS estimation of the log-linear model was per-

formed dropping the observations for which the dependent

variable equals0.Notice that the observations discarded

with this procedure are exactly the same that are discarded

by OLS(y?0.5)in the?rst set of experiments.Therefore,

this estimator is also denoted OLS(y?0.5).

The results of the two sets of experiments are summa-

rized in table1,which displays the biases and standard

errors of the different estimators of?obtained with10,000

replicas of the simulation procedure described above.Only

results for?1and?2are presented,as these are generally the

parameters of interest.

18For example,in gravity equations,continuous variables(which are all

strictly positive)include income and geographical distance.In equation

(14),x1can be interpreted as(the logarithm of)one of these variables.

Examples of binary variables include dummies for free-trade agreements,

common language,colonial ties,contiguity,and access to water.

19We also studied the performance of other variants of the tobit model,

?nding very poor results.

THE LOG OF GRA VITY647

As expected,OLS only performs well in case 3.In all

other cases this estimator is clearly inadequate because,

despite its low dispersion,it is often badly biased.More-

over,the sign and magnitude of the bias vary considerably.

Therefore,even when the dependent variable is strictly

positive,estimation of constant-elasticity models using the

log-linearized model cannot generally be recommended.As

for the modi?cations of the log-linearized model designed

to deal with the zeros of the dependent variable—ET-tobit,

OLS(y ?1),and OLS(y ?0.5)—their performance is also

very disappointing.These results clearly emphasize the

need to use adequate methods to deal with the zeros in the

data and raise serious doubts about the validity of the results

obtained using the traditional estimators based on the log

linear model.Overall,except under very special circum-

stances,estimation based on the log-linear model cannot be

recommended.

One remarkable result of this set of experiments is the

extremely poor performance of the NLS estimator.Indeed,

when the heteroskedasticity is more severe (cases 3and 4),this estimator,despite being consistent,leads to very poor results because of its erratic behavior.20Therefore,it is clear that the loss of ef?ciency caused by some of the forms of heteroskedasticity considered in these experiments is strong enough to render this estimator useless in practice.In the ?rst set of experiments,the results of the gamma PML estimator are very good.Indeed,when no measure-ment error is present,the biases and standard errors of the GPML estimator are always among the lowest.However,this estimator is very sensitive to the form of measurement error considered in the second set of experiments,consis-tently leading to sizable biases.These results,like those of the NLS,clearly illustrate the danger of using a PML estimator that gives extra weight to the noisier observations.As for the performance of the Poisson PML estimator,the results are very encouraging.In fact,when no rounding error is present,its performance is reasonably good in all 20Manning and Mullahy (2001)report similar results.T ABLE 1.—S IMULATION R ESULTS

UNDER D IFFERENT F ORMS OF H ETEROSKEDASTICITY Estimator:Results Without Rounding Error Results with Rounding Error ?1

?2?1?2Bias S.E.Bias S.E.

Bias S.E.Bias S.E.Case 1:V [y i ?x ]?1

PPML

?0.000040.0160.000090.0270.018860.0170.020320.029NLS

?0.000060.008?0.000030.0170.001950.0080.002740.018GPML

0.012760.0680.007540.0820.109460.0960.093380.108OLS

0.390080.0390.355680.054————ET-tobit

?0.478550.030?0.477860.032?0.499810.030?0.499680.032OLS(y ?0.5)

?0.164020.027?0.154870.038?0.221210.026?0.213390.036OLS(y ?1)?0.402370.014?0.376830.022?0.377520.015?0.349970.024

Case 2:V [y i ?x ]??(x i ?)

PPML

?0.000110.0190.000090.0390.021900.0200.023340.041NLS

0.000460.0330.000660.0570.002620.0330.003600.057GPML

0.003760.0430.002110.0620.132430.0730.113310.087OLS

0.210760.0300.199600.049————ET-tobit

?0.423940.028?0.423160.033?0.455180.028?0.455130.033OLS(y ?0.5)

?0.178680.026?0.172200.043?0.244050.026?0.238890.040OLS(y ?1)?0.423710.015?0.399310.025?0.394010.016?0.368060.028

Case 3:V [y i ?x ]??(x i ?)2

PPML

?0.005260.091?0.002280.1300.023320.0910.028120.133NLS

0.23539 3.0660.07323 1.5210.23959 3.0820.07852 1.521GPML

?0.000470.041?0.000290.0830.171340.0680.144420.104OLS

0.000150.032?0.000030.064————ET-tobit

?0.319080.044?0.321610.058?0.364800.043?0.367890.056OLS(y ?0.5)

?0.344800.039?0.346140.064?0.410060.037?0.412000.060OLS(y ?1)?0.518040.021?0.500000.038?0.485640.022?0.465970.040

Case 4:V [y i ?x ]??(x i ?)?exp(x 2i )?(x i ?)2

PPML

?0.006960.103?0.006470.1440.020270.1040.018560.146NLS

0.351397.5160.08801 1.8270.356727.5210.09239 1.829GPML

0.003220.057?0.001370.0830.128310.0850.102450.129OLS

0.132700.039?0.125420.075————ET-tobit

?0.299080.049?0.427310.063?0.343510.047?0.462250.060OLS(y ?0.5)

?0.392170.042?0.413910.070?0.451880.040?0.461730.066OLS(y ?1)?0.514400.021?0.580870.041?0.486270.022?0.560390.044THE REVIEW OF ECONOMICS AND STATISTICS

648

cases.Moreover,although some loss of ef?ciency is notice-able as one moves away from case2,in which it is an optimal estimator,the biases of the PPML are always small.21Moreover,the results obtained with rounded data suggest that the Poisson-based PML estimator is relatively robust to this form of measurement error of the dependent variable.Indeed,the bias introduced by the rounding-off errors in the dependent variable is relatively small,and in some cases it even compensates the bias found in the?rst set of experiments.Therefore,because it is simple to im-plement and reliable in a wide variety of situations,the Poisson PML estimator has the essential characteristics needed to make it the new workhorse for the estimation of constant-elasticity models.

Obviously,the sign and magnitude of the bias of the estimators studied here depend on the particular speci?ca-

tion considered.Therefore,the results of these experiments

cannot serve as an indicator of what can be expected in

other situations.However,it is clear that,apart from the

Poisson PML method,all estimators will often be very

misleading.

These experiments were also used to study the?nite-

sample performance of the Gauss-Newton regression

(GNR)test for the adequacy of the Poisson PML based on

equation(13)and of the Park test advocated by Manning

and Mullahy(2001),which,as explained above,is valid

only to check for the adequacy of the estimator based on the

log linear model.22Given that the Poisson PML estimator is

the only estimator with a reasonable behavior under all the

cases considered,these tests were performed using residuals

and estimates of?(x i?)from the Poisson regression.Table 2contains the rejection frequencies of the null hypothesis at

the5%nominal level for both tests in the four cases

considered in the two sets of experiments.In this table the

rejection frequencies under the null hypothesis are given in

bold.

In as much as both tests have adequate behavior under the

null and reveal reasonable power against a wide range of

alternatives,the results suggest that these tests are important

tools to assess the adequacy of the standard OLS estimator

of the log linear model and of the proposed Poisson PML

estimator.

V.The Gravity Equation

In this section,we use the PPML estimator to quantita-

tively assess the determinants of bilateral trade?ows,un-

covering signi?cant differences in the roles of various

measures of size and distance from those predicted by the logarithmic tradition.We perform the comparison of the two techniques using both the traditional and the Anderson–van Wincoop(2003)speci?cations of the gravity equation. For the sake of completeness,we also compare the PPML estimates with those obtained from alternative ways re-searchers have used to deal with zero values for trade.In particular,we present the results obtained with the tobit estimator used in Eaton and Tamura(1994),OLS estimator applied to ln(1?T ij),and a standard nonlinear least squares estimator.The results obtained with these estimators are presented for both the traditional and the Anderson–van Wincoop speci?cations.

A.The Data

The analysis covers a cross section of136countries in 1990.Hence,our data set consists of18,360observations of bilateral export?ows(136?135country pairs).The list of countries is reported in table A1in the https://www.wendangku.net/doc/4f14117389.html,rma-tion on bilateral exports comes from Feenstra et al.(1997). Data on real GDP per capita and population come from the World Bank’s(2002)World Development Indicators.Data on location and dummies indicating contiguity,common language(of?cial and second languages),colonial ties(di-rect and indirect links),and access to water are constructed from the CIA’s(2002)World Factbook.The data on lan-guage and colonial links are presented on tables A2and A3 in the appendix.23Bilateral distance is computed using the great circle distance algorithm provided by Andrew Gray (2001).Remoteness—or relative distance—is calculated as the(log of)GDP-weighted average distance to all other countries(see Wei,1996).Finally,information on preferen-tial trade agreements comes from Frankel(1997),comple-mented with data from the World Trade Organization.The list of preferential trade agreements(and stronger forms of trade agreements)considered in the analysis is displayed in table A4in the appendix.Table A5in the appendix provides

21These results are in line with those reported by Manning and Mullahy (2001).

22To illustrate the pitfalls of the procedure suggested by Manning and Mullahy(2001),we note that the means of the estimates of?1obtained

using equation(11)in cases1,2,and3(without measurement error)were 0.58955,1.29821,and1.98705,whereas the true values of?1in these cases are,respectively,0,1,and2.

23Alternative estimates based on Boisso and Ferrantino’s(1997)index of language similarity are available,at request,from the authors.

T ABLE2.—R EJECTION F REQUENCIES AT THE5%L EVEL

FOR THE T WO S PECIFICATION T ESTS

Case

Frequency

GNR Test Park Test

Without Measurement Error

10.91980 1.00000

20.05430 1.00000

30.581100.06680

40.491000.40810

With Measurement Error

10.91740 1.00000

20.14980 1.00000

30.57170 1.00000

40.47580 1.00000

THE LOG OF GRA VITY649

a description of the variables and displays the summary

statistics.

B.Results

The Traditional Gravity Equation:Table 3presents the

estimation outcomes resulting from the various techniques

for the traditional gravity equation.The ?rst column reports

OLS estimates using the logarithm of exports as the depen-

dent variable;as noted before,this regression leaves out

pairs of countries with zero bilateral trade (only 9,613

country pairs,or 52%of the sample,exhibit positive export

?ows).

The second column reports the OLS estimates using

ln(1?T ij )as dependent variable,as a way of dealing with

zeros.The third column presents tobit estimates based on

Eaton and Tamura (1994).The fourth column shows the

results of standard NLS.The ?fth column reports Poisson

estimates using only the subsample of positive-trade pairs.

Finally,the sixth column shows the Poisson results for the

whole sample (including zero-trade pairs).

The ?rst point to notice is that PPML-estimated coef?-

cients are remarkably similar using the whole sample and

using the positive-trade subsample.24However,most coef-

?cients differ—oftentimes signi?cantly—from those ob-tained using OLS.This suggests that in this case,heteroske-dasticity (rather than truncation)is responsible for the differences between PPML results and those of OLS using only the observations with positive exports.Further evi-dence on the importance of the heteroskedasticity is pro-vided by the two-degrees-of-freedom special case of White’s test for heteroskedasticity (see Wooldridge,2002,p.127),which leads to a test statistic of 476.6and to a p -value of 0.That is,the null hypothesis of homoskedastic errors is unequivocally rejected.Poisson estimates reveal that the coef?cients on import-er’s and exporter’s GDPs in the traditional equation are not,as generally believed,close to 1.The estimated GDP elas-ticities are just above 0.7(s.e.?0.03).OLS generates signi?cantly larger estimates,especially on exporter’s GDP (0.94,s.e.?0.01).Although all these results are conditional on the particular speci?cation used,25it is worth pointing out that unit income elasticities in the simple gravity framework are at odds with the observation that the trade-to-GDP ratio decreases with increasing total GDP,or,in other words,that smaller countries tend to be more open to international trade.2624The reason why truncation has little effect in this case is that observations with zero trade correspond to pairs for which the estimated value of trade is close to zero.Therefore,the corresponding residuals are also close to zero,and their elimination from the sample has little effect.25

This result holds when one looks at the subsample of OECD countries.It is also robust to the exclusion of GDP per capita from the regressions.26Note also that PPML predicts almost equal coef?cients for the GDPs of exporters and importers.T ABLE 3.—T HE T RADITIONAL G RAVITY E QUATION

Estimator:

OLS OLS Tobit NLS PPML PPML Dependent Variable:ln(T ij )

ln(1?T ij )ln(a ?T ij )T ij T ij ?0T ij Log exporter’s GDP 0.938**

1.128** 1.058**0.738**0.721**0.733**(0.012)

(0.011)(0.012)(0.038)(0.027)(0.027)Log importer’s GDP 0.798**

0.866**0.847**0.862**0.732**0.741**(0.012)

(0.012)(0.011)(0.041)(0.028)(0.027)Log exporter’s GDP per capita 0.207**

0.277**0.227**0.396**0.154**0.157**(0.017)

(0.018)(0.015)(0.116)(0.053)(0.053)Log importer’s GDP per capita 0.106**

0.217**0.178**?0.0330.133**0.135**(0.018)

(0.018)(0.015)(0.062)(0.044)(0.045)Log distance ?1.166**

?1.151**?1.160**?0.924**?0.776**?0.784**(0.034)

(0.040)(0.034)(0.072)(0.055)(0.055)Contiguity dummy 0.314*

?0.241?0.225?0.0810.2020.193(0.127)

(0.201)(0.152)(0.100)(0.105)(0.104)Common-language dummy 0.678**

0.742**0.759**0.689**0.752**0.746**(0.067)

(0.067)(0.060)(0.085)(0.134)(0.135)Colonial-tie dummy 0.397**

0.392**0.416**0.0360.0190.024(0.070)

(0.070)(0.063)(0.125)(0.150)(0.150)Landlocked-exporter dummy ?0.062

0.106*?0.038?1.367**?0.873**?0.864**(0.062)

(0.054)(0.052)(0.202)(0.157)(0.157)Landlocked-importer dummy ?0.665**

?0.278**?0.479**?0.471**?0.704**?0.697**(0.060)

(0.055)(0.051)(0.184)(0.141)(0.141)Exporter’s remoteness 0.467**

0.526**0.563** 1.188**0.647**0.660**(0.079)

(0.087)(0.068)(0.182)(0.135)(0.134)Importer’s remoteness ?0.205*

?0.109?0.032 1.010**0.549**0.561**(0.085)

(0.091)(0.073)(0.154)(0.120)(0.118)Free-trade agreement dummy 0.491**

1.289**0.729**0.443**0.179*0.181*(0.097)

(0.124)(0.103)(0.109)(0.090)(0.088)Openness ?0.170**

0.739**0.310**0.928**?0.139?0.107(0.053)

(0.050)(0.045)(0.191)(0.133)(0.131)Observations 9613

183601836018360961318360RESET test p -values

0.0000.0000.2040.0000.9410.331THE REVIEW OF ECONOMICS AND STATISTICS

650

The role of geographical distance as trade deterrent is

signi?cantly larger under OLS;the estimated elasticity is ?1.17(s.e.?0.03),whereas the Poisson estimate is ?0.78(s.e.?0.06).This lower estimate suggests a smaller role for

transport costs in the determination of trade patterns.Fur-

thermore,Poisson estimates indicate that,after controlling

for bilateral distance,sharing a border does not in?uence

trade ?ows,whereas OLS,instead,generates a substantial

effect:It predicts that trade between two contiguous coun-

tries is 37%larger than trade between countries that do not

share a border.27

We control for remoteness to allow for the hypothesis that

larger distances to all other countries might increase bilat-

eral trade between two countries.28Poisson regressions

support this hypothesis,whereas OLS estimates suggest that

only exporter’s remoteness increases bilateral ?ows be-

tween two given countries.Access to water appears to be

important for trade ?ows,according to Poisson regressions;

the negative coef?cients on the land-locked dummies can be

interpreted as an indication that ocean transportation is

signi?cantly cheaper.In contrast,OLS results suggest that

whether or not the exporter is landlocked does not in?uence

trade ?ows,whereas a landlocked importer experiences

lower trade.(These asymmetries in the effects of remote-

ness and access to water for importers and exporters are

hard to interpret.)We also explore the role of colonial

heritage,obtaining,as before,signi?cant discrepancies:

Poisson regressions indicate that colonial ties play no role in

determining trade ?ows,once a dummy variable for com-

mon language is introduced.OLS regressions,instead,gen-

erate a sizeable effect (countries with a common colonial

past trade almost 45%more than other pairs).Language is

statistically and economically signi?cant under both estima-

tion procedures.

Strikingly,in the traditional gravity equation,preferential-

trade agreements play a much smaller—although still sub-

stantial—role according to Poisson regressions.OLS esti-

mates suggest that preferential trade agreements raise

expected bilateral trade by 63%,whereas Poisson estimates

indicate an average enhancement below 20%.

Preferential trade agreements might also cause trade di-

version;if this is the case,the coef?cient on the trade-

agreement dummy will not re?ect the net effect of trade agreements.To account for the possibility of diversion,we include an additional dummy,openness,similar to that used by Frankel (1997).This dummy takes the value 1whenever one (or both)of the countries in the pair is part of a preferential trade agreement,and thus it captures the extent of trade between members and nonmembers of a preferen-tial trade agreement.The sum of the coef?cients on the trade agreement and the openness dummies gives the net creation effect of trade agreements.OLS suggests that trade destruc-tion comes from trade agreements.Still,the net creation effect is around 40%.In contrast,Poisson regressions pro-vide no signi?cant evidence of trade diversion,although the point estimates are of the same order of magnitude under both methods.Hence,even when allowing for trade diversion effects,on average,the Poisson method estimates a smaller effect of preferential trade agreements on trade,approximately half of that indicated by OLS.The contrast in estimates suggests that the biases generated by standard regressions can be substantial,leading to misleading inferences and,perhaps,erroneous policy decisions.We now turn brie?y to the results of the other estimation methods.OLS on ln(1?T ij )and tobit give very close estimates for most coef?cients.Like OLS,they yield large estimates for the elasticity of bilateral trade with respect to distance.Unlike OLS,however,they produce insigni?cant coef?cients for the contiguity dummy.They both generate extremely large and statistically signi?cant coef?cients for the trade-agreement dummy.The ?rst method predicts that trade between two countries that have signed a trade agree-ment is on average 266%larger than that between countries without an agreement.The second predicts that trade be-tween countries in such agreements is on average 100%larger.NLS tends to generate somewhat different estimates.The elasticity of trade with respect to the exporter’s GDP is signi?cantly smaller than with OLS,but the corresponding elasticity with respect to importer’s GDP is signi?cantly larger than with OLS.The estimated distance elasticity is smaller than with OLS and bigger than with Poisson.Like the other methods,NLS predicts a signi?cant and large effect for free-trade agreements.It is noteworthy that all methods,except the PPML,lead to puzzling asymmetries in the elasticities with respect to importer and exporter characteristics (especially remoteness and access to water).To check the adequacy of the estimated models,we performed a heteroskedasticity-robust RESET test (Ramsey,1969).This is essentially a test for the correct speci?cation of the conditional expectation,which is performed by checking the signi?cance of an additional regressor con-structed as (x ?b )2,where b denotes the vector of estimated parameters.The corresponding p -values are reported at the bottom of table 3.In the OLS regression,the test rejects the hypothesis that the coef?cient on the test variable is 0.This

27The formula to compute this effect is (e b i

?1)?100%,where b i is

the estimated coef?cient.28To illustrate the role of remoteness,consider two pairs of countries,(i,j )and (k,l ),and assume that the distance between the countries in each pair is the same (D ij ?D kl ),but i and j are closer to other countries.In this case,the most remote countries,k and l,will tend to trade more between each other because they do not have alternative trading partners.See Deardoff (1998).T ABLE 4.—R ESULTS OF THE T ESTS FOR T YPE OF H ETEROSKEDASTICITY

(p -V ALUES )

Test (Null Hypothesis)Exports ?0Full Sample

GNR (V [y i ?x ]??(x i ?))0.1440.115Park (OLS is valid)0.0000.000

THE LOG OF GRA VITY 651

means that the model estimated using the logarithmic spec-

i?cation is inappropriate.A similar result is found for the

OLS estimated using ln(1?T ij )as the dependent variable

and for the NLS.In contrast,the models estimated using the

Poisson regressions pass the RESET test,that is,the RESET

test provides no evidence of misspeci?cation of the gravity

equations estimated using the PPML.With this particular

speci?cation,the model estimated using tobit also passes

the test for the traditional gravity equation.

Finally,we also check whether the particular pattern of

heteroskedasticity assumed by the models is appropriate.As

explained in section III B,the adequacy of the log linear

model was checked using the Park-type test,whereas the

hypothesis V [y i ?x ]??(x i ?)was tested by evaluating the

signi?cance of the coef?cient of (ln y ?i )?i in the Gauss-

Newton regression indicated in equation (13).The p -values

of the tests are reported in table 4.Again,the log linear

speci?cation is unequivocally rejected.On the other hand,

these results indicate that the estimated coef?cient on (ln y ?i )?i is insigni?cantly different from 0at the usual 5%level.

This implies that the Poisson PML assumption V [y i ?x ]??0E [y i ?x ]cannot be rejected at this signi?cance level.

The Anderson–van Wincoop Gravity Equation:Table 5

presents the estimated coef?cients for the Anderson–van

Wincoop (2003)gravity equation,which controls more

properly for multilateral resistance terms by introducing

exporter-and importer-speci?c effects.As before,the col-

umns show,respectively,the estimated coef?cients obtained

using OLS on the log of exports,OLS on ln(1?T ij ),tobit,

NLS,PPML on the positive-trade sample,and PPML.Note

that,because this exercise uses cross-sectional data,we can

only identify bilateral variables.29

As with the standard speci?cation of the gravity equation,we ?nd that,using the Anderson–van Wincoop (2003)speci?cation,estimates obtained with the Poisson method vary little when only the positive-trade subsample is used.Moreover,we ?nd again strong evidence that the errors of the log linear model estimated using the sample with posi-tive trade are heteroskedastic.With this speci?cation,the two-degree-of-freedom special case of White’s test for het-eroskedasticity leads to a test statistic of 469.2and a p -value of 0.Because we are now conditioning on a much larger set of controls,it is not surprising to ?nd that most coef?cients are sensitive to the introduction of ?xed effects.For example,in the Poisson method,although the distance elasticity remains about the same and the coef?cient on common colonial ties is still insigni?cant,the effect on common language is now smaller and the coef?cient on free-trade agreements is larger.The results of the other estimation methods are generally much more sensitive to the inclusion of the ?xed https://www.wendangku.net/doc/4f14117389.html,paring the results of PPML and OLS for the positive-trade subsample,the following observations are in order.The distance elasticity is substantially larger under OLS (?1.35versus ?0.75).Sharing a border has a positive effect on trade under Poisson,but no signi?cant effect under OLS.Sharing a common language has similar effects under the two https://www.wendangku.net/doc/4f14117389.html,mon colonial ties have strong effects under OLS (with an average enhancement effect of 100%),whereas Poisson predicts no signi?cant effect.Finally,the 29Anderson and van Wincoop (2003)impose unit income elasticities by using as the dependent variable the log of exports divided by the product of the countries’GDPs.Because we are working with cross-sectional data and the model speci?cation includes importer and exporter ?xed effects,income elasticities cannot be identi?ed,and there is no need to impose restrictions on them.Still,the estimation of the PML models could be performed using as the dependent variable the ratio of exports to the product of the GDPs.This would downweight the observations with large values of the product of the GDPs,implicitly assuming that the variance of the error term is proportional to the square of this product.This is contrary to what is advocated by Frankel and Wei (1993)and Frankel (1997),who suggest that larger countries should be given more weight in the estimation of gravity equations because they generally have better data.In any case,whether this should be done or not is an empirical question,and the right course of action depends on the pattern of heteroskedasticity.With our data,using the ratio of exports to the product of GDPs as the dependent variable leads to models that are rejected by the speci?cation tests.Therefore,the implied assumptions about the pattern of heteroskedasticity are not supported by our data.Hence,we use exports as the dependent variable of the gravity equation,and not the ratio of exports to the product of GDPs.T ABLE 5.—T HE A NDERSON –VAN W INCOOP G RAVITY E QUATION

Estimator:

OLS OLS Tobit NLS PPML PPML Dependent variable:ln (T ij )

ln (1?T ij )ln (a ?T ij )T ij T ij ?0T ij Log distance ?1.347**

?1.332**?1.272**?0.582**?0.770**?0.750**(0.031)

(0.036)(0.029)(0.088)(0.042)(0.041)Contiguity dummy 0.174

?0.399*?0.2530.458**0.352**0.370**(0.130)

(0.189)(0.135)(0.121)(0.090)(0.091)Common-language dummy 0.406**

0.550**0.485**0.926**0.418**0.383**(0.068)

(0.066)(0.057)(0.116)(0.094)(0.093)Colonial-tie dummy 0.666**

0.693**0.650**?0.736**0.0380.079(0.070)

(0.067)(0.059)(0.178)(0.134)(0.134)Free-trade agreement dummy 0.310**

0.1740.137** 1.017**0.374**0.376**(0.098)

(0.138)(0.098)(0.170)(0.076)(0.077)Fixed effects Yes

Yes Yes Yes Yes Yes Observations 9613

183601836018360961318360RESET test p -values

0.0000.0000.0000.0000.5640.112THE REVIEW OF ECONOMICS AND STATISTICS

652

two techniques produce reasonably similar estimates for the

coef?cient on the trade-agreement dummy,implying a trade-

enhancement effect of the order of 40%.

As before,the other estimation methods lead to some

puzzling results.For example,OLS on ln(1?T

ij )now

yields a signi?cantly negative effect of contiguity,and under

NLS,the coef?cient on common colonial ties becomes

signi?cantly negative.

To complete the study,we performed the same set of

speci?cation tests used before.The p -values of the het-

eroskedasticity-robust RESET test at the bottom of table 5

suggest that with the Anderson–van Wincoop (2003)spec-

i?cation of the gravity equation,only the models estimated

by the PPML method are adequate.The p -values of the tests

to check whether the particular pattern of heteroskedasticity

assumed by the models is appropriate are reported in table

6.As in the traditional gravity equation,the log linear

speci?cation is unequivocally rejected.On the other hand,

these results indicate that the estimated coef?cient on (ln y ?

i )

?y ?i is insigni?cantly different from 0at the usual 5%level.

This implies that the Poisson PML assumption V [y i ?x ]?

?0E [y

i ?x ]cannot be rejected at this signi?cance level.To sum up,whether or not ?xed effects are used in the

speci?cation of the model,we ?nd strong evidence that

estimation methods based on the log-linearization of the

gravity equation suffer from severe misspeci?cation,which

hinders the interpretation of the results.NLS is also gener-

ally unreliable.In contrast,the models estimated by PPML

show no signs of misspeci?cation and,in general,do not

produce the puzzling results generated by the other meth-

ods.30

VI.Conclusions

In this paper,we argue that the standard empirical meth-

ods used to estimate gravity equations are inappropriate.

The basic problem is that log-linearization (or,indeed,any nonlinear transformation)of the empirical model in the presence of heteroskedasticity leads to inconsistent esti-mates.This is because the expected value of the logarithm of a random variable depends on higher-order moments of its distribution.Therefore,if the errors are heteroskedastic,the transformed errors will be generally correlated with the covariates.An additional problem of log-linearization is that it is incompatible with the existence of zeros in trade data,which led to several unsatisfactory solutions,including truncation of the sample (that is,elimination of zero-trade pairs)and further nonlinear transformations of the depen-dent variable.We argue that the biases are present both in the traditional speci?cation of the gravity equation and in the Anderson–van Wincoop (2003)speci?cation,which includes country-speci?c ?xed effects.To address the various estimation problems,we propose a simple Poisson pseudo-maximum-likelihood method and assess its performance using Monte Carlo simulations.We ?nd that in the presence of heteroskedasticity the standard methods can severely bias the estimated coef?cients,cast-ing doubt on previous empirical ?ndings.Our method,instead,is robust to different patterns of heteroskedasticity and,in addition,provides a natural way to deal with zeros in trade data.We use our method to reestimate the gravity equation and document signi?cant differences from the results obtained using the log linear method.For example,income elastici-ties in the traditional gravity equation are systematically smaller than those obtained with log-linearized OLS regres-sions.In addition,in both the traditional and Anderson–van Wincoop speci?cations of the gravity equation,OLS esti-mation exaggerates the role of geographical proximity and colonial ties.RESET tests systematically favor the Poisson PML technique.The results suggest that heteroskedasticity (rather than truncation of the data)is responsible for the main differences.Log-linearized equations estimated by OLS are of course used in many other areas of empirical economics and econo-metrics.Our Monte Carlo simulations and the regression out-comes indicate that in the presence of heteroskedasticity this practice can lead to signi?cant biases.These results suggest that,at least when there is evidence of heteroskedasticity,the Poisson pseudo-maximum-likelihood estimator should be used as a substitute for the standard log linear model.REFERENCES Anderson,J.,“A Theoretical Foundation for the Gravity Equation,”American Economic Review 69(1979),106–116.Anderson,J.,and E.van Wincoop,“Gravity with Gravitas:A Solution to the Border Puzzle,”American Economic Review 93(2003),170–192.Bergstrand,J.,“The Gravity Equation in International Trade:Some Microeconomic Foundations and Empirical Evidence,”this RE -

VIEW ,69(1985),474–481.

30It is worth noting that the large differences in estimates among the various methods persist when we look at a smaller subsample of countries that account for most of world trade and,quite likely,have better data.More speci?cally,we run similar regressions for the subsample of 63countries included in Frankel’s (1997)study.These countries accounted for almost 90%of the world trade reported to the United Nations in 1992.One advantage of this subsample is that the number of zeros is signi?-cantly reduced.Heteroskedasticity,however,is still a problem:The null hypothesis of homoskedasticity is rejected in both the traditional and the ?xed-effects gravity equations.As with the full sample,PPML generates a smaller role for distance and common language than OLS,and,unlike OLS,PPML predicts no role for colonial ties.In line with the ?ndings documented in Frankel (1997),the OLS estimated coef?cient on the free-trade-agreement dummy is negative in both speci?cations of the gravity equation,whereas PPML predicts a positive and signi?cant effect (slightly bigger than that found for the whole sample).These results are available—on request—from the authors.T ABLE 6.—R ESULTS OF THE T ESTS FOR T YPE OF H ETEROSKEDASTICITY (p -V ALUES )

Test (Null Hypothesis)Exports ?0Full Sample

GNR (V [y i ?x ]??(x i ?))0.1000.070Park (OLS is valid)0.0000.000

THE LOG OF GRA VITY 653

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APPENDIX

T ABLE A1.—L IST OF C OUNTRIES

Albania Denmark Kenya Romania

Algeria Djibouti Kiribati Russian Federation

Angola Dominican Rep.Korea,Rep.Rwanda

Argentina Ecuador Laos P.Dem.Rep.Saudi Arabia

Australia Egypt Lebanon Senegal

Austria El Salvador Madagascar Seychelles

Bahamas Eq.Guinea Malawi Sierra Leone

Bahrain Ethiopia Malaysia Singapore

Bangladesh Fiji Maldives Solomon Islands

Barbados Finland Mali South Africa

Belgium-Lux.France Malta Spain

Belize Gabon Mauritania Sri Lanka

Benin Gambia Mauritius St.Kitts and Nevis

Bhutan Germany Mexico Sudan

Bolivia Ghana Mongolia Suriname

Brazil Greece Morocco Sweden

Brunei Guatemala Mozambique Switzerland

Bulgaria Guinea Nepal Syrian Arab Rep.

Burkina Faso Guinea-Bissau Netherlands Tanzania

Burundi Guyana New Caledonia Thailand

Cambodia Haiti New Zealand Togo

Cameroon Honduras Nicaragua Trinidad and Tobago Canada Hong Kong Niger Tunisia

Central African Rep.Hungary Nigeria Turkey

Chad Iceland Norway Uganda

Chile India Oman United Arab Em.

China Indonesia Pakistan United Kingdom

Colombia Iran Panama United States

Comoros Ireland Papua New Guinea Uruguay

Congo Dem.Rep.Israel Paraguay Venezuela

Congo Rep.Italy Peru Vietnam

Costa Rica Jamaica Philippines Yemen

Co?te D’Ivoire Japan Poland Zambia

Cyprus Jordan Portugal Zimbabwe

T ABLE A2.—C OMMON O FFICIAL

AND S ECOND L ANGUAGES English

French Spanish Dutch Australia

Belgium-Lux.Argentina Belgium-Lux.Bahamas

Benin Belize Netherlands Barbados

Burkina Faso Bolivia Suriname Belize

Burundi Chile Brunei

Cameroon Colombia German Cameroon

Canada Costa Rica Canada

Central African Rep.Dominican Rep.Austria Fiji

Chad Ecuador Germany Gambia

Comoros El Salvador Switzerland Ghana

Congo Dem.Rep.Eq.Guinea Guyana

Congo Rep.Guatemala Greek Hong Kong

Co ?te D’Ivoire Honduras India

Djibouti Mexico Cyprus Indonesia

Eq.Guinea Nicaragua Greece Ireland

France Panama Israel

Gabon Paraguay Hungarian Jamaica

Guinea Peru Jordan

Haiti Spain Hungary Kenya

Lebanon Uruguay Romania Kiribati

Madagascar Venezuela Malawi

Mali Malaysia

Mauritania Arabic Italian

Maldives

Mauritius Algeria Malta

Morocco Bahrain Italy Mauritius

New Caledonia Chad Switzerland New Zealand

Niger Comoros Nigeria

Rwanda Djibouti Lingala Oman

Senegal Egypt Pakistan

Seychelles Israel Congo Dem.Rep.Panama

Switzerland Jordan Congo Rep.Papua New Guinea

Togo Lebanon Philippines

Tunisia Mauritania Russian Rwanda

Morocco Seychelles

Malay Oman Mongolia Sierra Leone

Saudi Arabia Russian Federation Singapore

Brunei Sudan South Africa

Indonesia Syria Swahili Sri Lanka

Malaysia Tanzania St.Helena

Singapore Tunisia Kenya St.Kitts and Nevis

United Arab Em.Tanzania Suriname

Portuguese Yemen Tanzania

Chinese Trinidad and Tobago

Angola Turkish Uganda

Brazil China United Kingdom

Guinea-Bissau Cyprus Hong Kong United States

Mozambique Turkey Malaysia Zambia

Portugal Singapore Zimbabwe THE REVIEW OF ECONOMICS AND STATISTICS

656

THE LOG OF GRA VITY657

T ABLE A3.—C OLONIAL T IES

United Kingdom France Spain

Australia Mauritius Algeria Argentina

Bahamas New Zealand Benin Bolivia

Bahrain Nigeria Burkina Faso Chile

Barbados Pakistan Cambodia Colombia

Belize Saint Kitts and Nevis Cameroon Costa Rica

Cameroon Seychelles Central African Rep.Cuba

Canada Sierra Leone Chad Ecuador

Cyprus South Africa Comoros El Salvador

Egypt Sri Lanka Congo Eq.Guinea

Fiji Sudan Djibouti Guatemala

Gambia Tanzania Gabon Honduras

Ghana Trinidad and Tobago Guinea Mexico

Guyana Uganda Haiti Netherlands

India United States Laos Nicaragua

Ireland Zambia Lebanon Panama

Israel Zimbabwe Madagascar Paraguay

Jamaica Mali Peru

Jordan Mauritania Venezuela

Kenya Morocco Portugal

Kuwait Niger

Malawi Senegal

Angola

Malaysia Syria

Brazil

Maldives Togo

Guinea-Bissau

Malta Tunisia

Mozambique

Oman

Vietnam

T ABLE A4.—P REFERENTIAL T RADE A GREEMENTS IN1990

EEC/EC CARICOM CACM

Belgium Bahamas Costa Rica

Denmark Barbados El Salvador

France Belize Guatemala

Germany Dominican Rep.Honduras

Greece Guyana Nicaragua

Ireland Haiti

Italy Jamaica Bilateral Agreements

Luxembourg Trinidad and Tobago

Netherlands St.Kitts and Nevis

EC-Cyprus Portugal Suriname

EC-Malta

EC-Egypt Spain

United Kingdom SPARTECA

EC-Syria

EC-Algeria

Australia EC-Norway EFTA

New Zealand EC-Iceland Iceland Fiji EC-Switzerland

Norway Kiribati Canada–United States

Switzerland Papua New Guinea Israel–United States

Liechtenstein Solomon Islands

CER PATCRA

Australia Australia

New Zealand Papua New Guinea

T ABLE A5.—S UMMARY S TATISTICS

Variable

Full Sample

Exports ?0Mean Std.Dev.Mean Std.Dev.Trade

172132.21828720328757.72517139Log of trade

——8.43383 3.26819Log of exporter’s GDP

23.24975 2.3972724.42503 2.29748Log of importer’s GDP

23.24975 2.3972724.13243 2.43148Log of exporter’s per capita GDP

7.50538 1.639868.09600 1.65986Log of importer’s per capita GDP

7.50538 1.639867.98602 1.68649Log of distance

8.785510.741688.694970.77283Contiguity dummy

0.019610.138650.023610.15185Common-language dummy

0.209700.407100.212840.40933Colonial-tie dummy

0.170480.376060.168940.37472Landlocked exporter dummy

0.154410.361350.107670.30998Landlocked importer dummy

0.154410.361350.114010.31784Exporter’s remoteness

8.946540.263898.903830.29313Importer’s remoteness

8.946540.263898.907870.28412Preferential–trade-agreement dummy

0.025050.156290.044520.20626Openness dummy 0.563730.495940.657960.47442

THE REVIEW OF ECONOMICS AND STATISTICS

658

地球物理勘查名词术语

中华人民共和国国家标准 GB XXXX--XX 地球物理勘查名词术语 Terms Of geophysical exploration 1 主题内容及适用范围 本标准规定了地球物理勘查(包括重力勘查、磁勘查、电勘查、地震勘查、测井及核物探)中常用的、主要的、本学科专有的名词术语。 本标准适用于地球物理勘查工作的语言和文字交流。 2 基本术语 2.1 地球物理勘查geophysical exploration 运用物理学的原理、方法和仪器以研究地质情况或寻查埋藏物的一类勘查。 同义词物探；地球物理勘探：(勘探地球物理；地球物理探矿) 注：1.取决于使用场合，该术语可附加后缀“法”或“学”。 2.根据具体情况，可以使用“航空物探”，“海洋物探”，“地面物探”，“地下物探”，“深部物探”，“区域物探”， “工程物探”，“环境物探”，……等术语。 2.2 正常场normal field 物理场的相对平稳部分。 2.3 异常anomaly 物理场对正常场的偏离。 2.3.1 理论异常theoretical anomaly 正演所获得的异常。 同义词计算异常 2.4 物性physical properties 岩(矿)石或其它探测对象的物理性质。 2.5 异向性系数coefficient of anisotropy 描述介质垂直层理(片理、节理等)方向与平行层理方向的物性差异的一种参数。 同义词(各向异性系数;非各向同性系数) 2.6 地球物理正演geophysical direct problem 根据地质体或其它探测对象的几何参数和物理参数计算地球物理场值。 同义词物探正演 2.7 地球物理反演geophysical inversion 根据地球物理场值，计算地质体或其它探测对象的几何参数和物性参数。 同义词物探反演 国家技术监督局XXXX—XX—XX批准 XXXX—XX—XX实施

#基于单片机的多功能电子万年历设计

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(完整word版)沪教牛津版初中英语单词表 编辑整理： 尊敬的读者朋友们： 这里是精品文档编辑中心，本文档内容是由我和我的同事精心编辑整理后发布的，发布之前我们对文中内容进行仔细校对，但是难免会有疏漏的地方，但是任然希望（(完整word版)沪教牛津版初中英语单词表）的内容能够给您的工作和学习带来便利。同时也真诚的希望收到您的建议和反馈，这将是我们进步的源泉，前进的动力。 本文可编辑可修改，如果觉得对您有帮助请收藏以便随时查阅，最后祝您生活愉快业绩进步，以下为(完整word版)沪教牛津版初中英语单词表的全部内容。

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目录 摘要 (3) 第一章引言 (4) 1.1 设计任务 (4) 1.2 设计目的 (4) 1.3 设计思路 (4) 1.3.1 方案论证 (4) 1.3.2 芯片的选择 (5) 1.3.3 显示模块选择方案和论证 (5) 1.3.4 时钟信号的选择方案和论证 (5) 1.3.5 最终方案 (6) 第二章硬件系统的设计 2.1原理图设计 (6) 2.2温度感应电路 (7) 2.3 复位电路部分 (7) 2.4液晶显示电路 (7) 2.5时钟信号电路 (8) 2.6 AT89C52原理及说明 (8) 2.6.1引脚功能 (9) 第三章软件系统的设计. 3.1系统程序流程图 (9) 3.2系统具体程序代码 (10) 第四章系统调试 (23) 4.1 软件调试 (23) 4.2 硬件调试 (23) 第五章设计心得 (23) 元件清单表 (24) 致谢 (24) 参考文献 (24)

基于可调式电子万年历与温度显示的设计 重庆三峡学院应用技术学院 5人 摘要：本文介绍了一种基于单片机的可调的电子万年历和温度显示。该设计主要由五个模块组成：微处理器（单片机），温度传感器，控制调节按键，实时时钟模块及显示模块。温传感器器主要由DS18B20来完成，它负责把采集到的温度传给单片机。实时时钟模块主要由DS1302构成，它负责产生始终数据送给单片机，微处理器芯片AT89C52来完成DS18B20，DS1302，按键传来的数据进行处理，并送与显示模块（LCD1602）进行显示。 该系统的电路简单，所用的元件较少，成本低，且测量精度和可靠性较高。可以测量-55°到+125°的温度和显示年，月，日，星期，时，分，秒，并且可通过按键调节时间。 关键词单片机；万年历；温度；AT89C52；LCD1602，DS1302，DS18B20

THE LOG OF GRAVITY

THE LOG OF GRAVITY J.M.C.Santos Silva and Silvana Tenreyro* Abstract—Although economists have long been aware of Jensen’s in-equality,many econometric applications have neglected an important implication of it:under heteroskedasticity,the parameters of log-linearized models estimated by OLS lead to biased estimates of the true elasticities.We explain why this problem arises and propose an appropri-ate estimator.Our criticism of conventional practices and the proposed solution extend to a broad range of applications where log-linearized equations are estimated.We develop the argument using one particular illustration,the gravity equation for trade.We?nd signi?cant differences between estimates obtained with the proposed estimator and those ob-tained with the traditional method. I.Introduction E CONOMISTS have long been aware that Jensen’s in- equality implies that E(ln y) ln E(y),that is,the expected value of the logarithm of a random variable is different from the logarithm of its expected value.This basic fact,however,has been neglected in many economet-ric applications.Indeed,one important implication of Jen-sen’s inequality is that the standard practice of interpreting the parameters of log-linearized models estimated by ordi-nary least squares(OLS)as elasticities can be highly mis-leading in the presence of heteroskedasticity. Although many authors have addressed the problem of obtaining consistent estimates of the conditional mean of the dependent variable when the model is estimated in the log linear form(see,for example,Goldberger,1968;Man-ning&Mullahy,2001),we were unable to?nd any refer-ence in the literature to the potential bias of the elasticities estimated using the log linear model. In this paper we use the gravity equation for trade as a particular illustration of how the bias arises and propose an appropriate estimator.We argue that the gravity equation, and,more generally,constant-elasticity models,should be estimated in their multiplicative form and propose a simple pseudo-maximum-likelihood(PML)estimation technique. Besides being consistent in the presence of heteroskedas-ticity,this method also provides a natural way to deal with zero values of the dependent variable. Using Monte Carlo simulations,we compare the perfor-mance of our estimator with that of OLS(in the log linear speci?cation).The results are striking.In the presence of heteroskedasticity,estimates obtained using log-linearized models are severely biased,distorting the interpretation of the model.These biases might be critical for the compara-tive assessment of competing economic theories,as well as for the evaluation of the effects of different policies.In contrast,our method is robust to the different patterns of heteroskedasticity considered in the simulations. We next use the proposed method to provide new esti-mates of the gravity equation in cross-sectional https://www.wendangku.net/doc/4f14117389.html,ing standard tests,we show that heteroskedasticity is indeed a severe problem,both in the traditional gravity equation introduced by Tinbergen(1962),and in a gravity equation that takes into account multilateral resistance terms or?xed effects,as suggested by Anderson and van Wincoop(2003). We then compare the estimates obtained with the proposed PML estimator with those generated by OLS in the log linear speci?cation,using both the traditional and the?xed-effects gravity equations. Our estimation method paints a very different picture of the determinants of international trade.In the traditional gravity equation,the coef?cients on GDP are not,as gen-erally estimated,close to1.Instead,they are signi?cantly smaller,which might help reconcile the gravity equation with the observation that the trade-to-GDP ratio decreases with increasing total GDP(or,in other words,that smaller countries tend to be more open to international trade).In addition,OLS greatly exaggerates the roles of colonial ties and geographical proximity. Using the Anderson–van Wincoop(2003)gravity equa-tion,we?nd that OLS yields signi?cantly larger effects for geographical distance.The estimated elasticity obtained from the log-linearized equation is almost twice as large as that predicted by PML.OLS also predicts a large role for common colonial ties,implying that sharing a common colonial history practically doubles bilateral trade.In con-trast,the proposed PML estimator leads to a statistically and economically insigni?cant effect. The general message is that,even controlling for?xed effects,the presence of heteroskedasticity can generate strikingly different estimates when the gravity equation is log-linearized,rather than estimated in levels.In other words,Jensen’s inequality is quantitatively and qualitatively important in the estimation of gravity equations.This sug-gests that inferences drawn on log-linearized regressions can produce misleading conclusions. Despite the focus on the gravity equation,our criticism of the conventional practice and the solution we propose ex-tend to a broad range of economic applications where the equations under study are log-linearized,or,more generally, transformed by a nonlinear function.A short list of exam-ples includes the estimation of Mincerian equations for wages,production functions,and Euler equations,which are typically estimated in logarithms. Received for publication March29,2004.Revision accepted for publi- cation September13,2005. *ISEG/Universidade Te′cnica de Lisboa and CEMAPRE;and London School of Economics,CEP,and CEPR,respectively. We are grateful to two anonymous referees for their constructive comments and suggestions.We also thank Francesco Caselli,Kevin Denny,Juan Carlos Hallak,Daniel Mota,John Mullahy,Paulo Parente, Manuela Simarro,and Kim Underhill for helpful advice on previous versions of this paper.The usual disclaimer applies.Jiaying Huang provided excellent research assistance.Santos Silva gratefully acknowl- edges the partial?nancial support from Fundac?a?o para a Cie?ncia e Tecnologia,program POCTI,partially funded by FEDER.A previous version of this paper circulated as“Gravity-Defying Trade.” The Review of Economics and Statistics,November2006,88(4):641–658 ?2006by the President and Fellows of Harvard College and the Massachusetts Institute of Technology

电子万年历设计

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摘要： 本文介绍了基于AT89C51单片机的多功能电子万年历的硬件结构和软硬件设计方法。系统以AT89C51单片机为控制器，以串行时钟日历芯片DS1302记录日历和时间，它可以对年、月、日、时、分、秒进行计时，还具有闰年补偿等多种功能。万年历采用直观的数字显示，可以在LED上同时显示年、月、日、周日、时、分、秒，还具有时间校准等功能。此万年历具有读取方便、显示直观、功能多样、电路简洁、成本低廉等诸多优点，具有广阔的市场前景。。 关键词：AT89C51单片机，DS1602时钟芯片，LCD1602显示屏。串口通信。一：引言 本设计的基于单片机控制的电子万年历，具有年、月、日、星期、时、分、秒的显示等功能，实现过程就是由主控制发送信息给DS1302时钟芯片再由时钟芯片反馈给单片机，再由主控制器传送给LCD1602显示屏显示信息。并且可以在键盘设置模块输入修改时间，当键盘设置时间、日期时，单片机主控制根据输入信息，通过串口通信传送给DS1302时钟芯片，DS1302芯片读取当前新信息产生反馈传送给单片机，然后单片机根据控制最后输送显示信息到LCD1602液晶显示屏模块上显示。 二：硬件设计: 2.0.硬件的设计总框图 2.1 DS1032时钟电路 DS1302的引脚排列,其中Vcc1为后备电源，VCC2为主电源。在主电源关闭的情况下，也能保持时钟的连续运行。DS1302由Vcc1或Vcc2两者中的较大者供电。当Vcc2大于Vcc1＋0.2V时，Vcc2给DS1302供电。当Vcc2小于Vcc1时，DS1302由Vcc1供电。X1和X2是振荡源，外接32.768kHz晶振。芯片如图。 DS1302的内部主要由移位寄存器、指令和控制逻辑、振荡分频电路、实时时钟以及RAM组成。每次操作时，必须首先把CE置为高电平。再把提供地址和命令信息的8位装入移位寄存器。数据在SCLK的上升沿串行输入。无论是读周期还是写周期发生，也无论传送方式是单字节还是多字节，开始8位将指定内部何处被进行访问。在开始 8个时钟周期把含有地址信息的命令字装入移位寄存

简单解释一下android 中layout_gravity和gravity_华清远见

简单解释一下android 中layout_gravity和gravity 本篇文章的目的就是为大家简单解释android 中layout_gravity和gravity，希望看完对大家有帮助。 相信很多学习了android的人，都知道布局中存在两个很相似的属性：android :layout_gravity和android:gravity。一般的都知道， android :layout_gravity 是表示该组件在父控件中的位置 android:gravity 是表示该组件的子组件在自身中的位置 例子： 如上的代码显示效果如下： 经管很明了，但是某些时候也会出现失效的情况，如下：

常用金融术语

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