classical estimates (bace) approach determinants of long-term growth: a bayesian ... · 2015. 6....

Determinants of Long-Term Growth A Bayesian Averaging ofClassical Estimates (BACE) Approach

By XAVIER SALA-I-MARTIN GERNOT DOPPELHOFER AND RONALD I MILLER

This paper examines the robustness of explanatory variables in cross-countryeconomic growth regressions It introduces and employs a novel approach Bayes-ian Averaging of Classical Estimates (BACE) which constructs estimates by aver-aging OLS coefficients across models The weights given to individual regressionshave a Bayesian justification similar to the Schwarz model selection criterion Of67 explanatory variables we find 18 to be significantly and robustly partiallycorrelated with long-term growth and another three variables to be marginallyrelated The strongest evidence is for the relative price of investment primaryschool enrollment and the initial level of real GDP per capita (JEL O51 O52O53)

Following the seminal work of Roger C Kor-mendi and Philip Meguire (1985) Kevin BGrier and Gordon Tullock (1989) and Robert JBarro (1991) the recent empirical literature oneconomic growth has identified a substantialnumber of variables that are partially correlatedwith the rate of economic growth The basicmethodology consists of running cross-countryor panel regressions of the form

(1) 1 x1 2 x2

n xn

where is the vector of rates of economic

growth is a constant and x1 xn are vec-tors of explanatory variables which vary acrossresearchers and across papers Each paper typ-ically reports a (possibly nonrandom) sample ofthe regressions actually run by the researcherVariables like the initial level of income theinvestment rate various measures of educationsome policy indicators and many other vari-ables have been found to be significantly corre-lated with growth in regressions like (1)

The problem faced by empirical economistsis that growth theories are not explicit enoughabout what variables xj belong in the ldquotruerdquoregression Hence creative theorizing will gen-erate models that ldquopredictrdquo that things like mar-ket distortions distortionary taxes maintenanceof property rights degree of monopolyweather attitudes toward work et ceterashould be included in the growth regressionNote that these potential theories are not mutu-ally exclusive (for example education couldvery well be an important determinant of long-term growth and this does not imply that finan-cial development is unimportant)

The multiplicity of possible regressors is oneof the major difficulties faced by researcherstrying to make sense of the empirical evidenceon economic growth However the problem ishardly unique to the growth literature ldquoartisticrdquoeconomic theory is often capable of suggestingan enormous number of potential explanatoryvariables in any economic field In principle

Sala-i-Martin Department of Economics ColumbiaUniversity New York NY 10027 and Universitat PompeuFabra (e-mail xs23columbiaedu) Doppelhofer Facultyof Economics and Politics University of Cambridge Cam-bridge CB3 9DD UK and Trinity College Cambridge(e-mail gd237camacuk) Miller National EconomicResearch Associates 1166 Avenue of the Americas NewYork NY 10036 (e-mail ronaldmillerneracom) Wethank Manuel Arellano Robert J Barro Steven DurlaufSanket Mohapatra Chris Sims Melvyn Weeks ArnoldZellner an anonymous referee and participants to the CREIEuroconference on Innovation and Growth at UniversitatPompeu Fabra for their comments We also thank ISETR atColumbia University for allowing us to use their computerfacilities Sala-i-Martin acknowledges the NSF Grants Nos20321600079447 and 20336700079447 and Spainrsquos Minis-terio de Ciencia y Tecnologıa through Grant Nos SEC2001-0674 and SEC 2001-0769 for financial support

813

this is strictly a small-sample problem since asthe number of observations becomes large allof the variables which do not belong in theregression will have coefficients that convergeto zero Classical statistics offers us little helpsince it suggests that we include all of the re-gressors and ldquolet the data sort through themrdquo Inmany applications however we do not have theluxury of having a large enough sample size toallow us to draw conclusions on the importanceof potential regressors Cross-country regres-sions provide perhaps the most extreme exam-ple the number of proposed regressors exceedsthe number of countries in the world renderingthe all-inclusive regression computationally im-possible Some empirical economists havetherefore resorted to simply ldquotryingrdquo combina-tions of variables which could be potentiallyimportant determinants of growth and report theresults of their preferred specification Suchldquodata-miningrdquo could lead to spurious inferenceSo how should we proceed

A natural way to think about model uncer-tainty is to admit that we do not know whichmodel is ldquotruerdquo and instead attach probabilitiesto different possible models While intuitivelyappealing this requires a departure from theclassical framework in which conditioning on amodel is essential This approach has recentlycome to be known as Bayesian Model Averag-ing The procedure does not differ from themost basic Bayesian reasoning the idea dates atleast to Harold Jeffreys (1961) although fleshedout by Edward E Leamer (1978) In this paperwe show that this approach can be used in a waythat is well grounded in statistical theory intu-itively appealing easy to understand and easyto implement

The fully Bayesian approach is entirely fea-sible and has been applied to various problemsby a number of authors Examples include Jer-emy C York et al (1995) and Adrian E Rafteryet al (1997) a summary of much of the recentwork can be found in Jennifer A Hoeting et al(1999) In the growth context Carmen Fernan-dez et al (2001) apply techniques from theBayesian statistics literature to the data set ofSala-i-Martin (1997a) A pure Bayesian ap-proach requires specification of the prior distri-butions of all of the relevant parametersconditional on each possible model Under idealconditions elicitation of prior parameters is dif-

ficult and is indeed one of the major reasons forBayesian approaches remaining relatively un-popular When the number of possible regres-sors is K the number of possible linear modelsis 2K so with K large fully specifying priors isinfeasible Thus authors implementing the fullyBayesian approach have used priors which areessentially arbitrary This makes the ultimateestimates dependent on arbitrarily chosen priorparameters in a manner which is extremely dif-ficult to interpret In existing applications of thisapproach the impact of these prior parametershas been neither examined nor explained

To address the issue of fragility of economet-ric inference with respect to modeling choicesLeamer (1983 1985) proposes a sensitivityanalysis of the results with respect to changes inthe prior distribution of parameters In particu-lar Leamer proposes an extreme bounds anal-ysis to identify ldquorobustrdquo empirical relations Fora variable of interest z the extreme bounds ofthe distribution of the associated coefficient es-timate z are calculated as the smallest andlargest value that the coefficient can take onwhen combinations of additional regressors xjenter the regression model (1) When the twoextreme bounds are of opposite sign then vari-able z is labeled ldquofragilerdquo Ross E Levine andDavid Renelt (1992) employ a version of thistest to cross-country data and find that accord-ing to extreme bounds analysis very few (or no)variables are robustly related with growth An-other interpretation however is that the test istoo strong for any variable to pass any oneregression model (no matter how well or poorlyfitting) carries a veto This problem is wellrecognized and solutions have been proposed torestrict attention to better fitting models such asthe reasonable extreme bounds proposed byClive W J Granger and Harald F Uhlig (1990)or to introduce a lower bound on the priorvariance matrix as suggested by Leamer (1985)

In this paper we use the Bayesian approach toaveraging across models while following theclassical spirit of most of the empirical growthliterature1 We propose a model-averaging tech-

1 We build on the work by Sala-i-Martin (1997a b) whoproposed to average estimates of mean and standard devi-ations for variables across regressions using weights pro-portional to the likelihoods of each of the models Sala-i-Martin calculates a likelihood-weighted sum of normal

814 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2004

nique which we call Bayesian Averaging ofClassical Estimates or BACE to determine thesignificance of variables in cross-country growthregressions We show that the weighting methodcan be derived as a limiting case of a standardBayesian analysis as the prior information be-comes ldquodominatedrdquo by the data BACE combinesthe averaging of estimates across models which isa Bayesian concept with classical ordinary least-squares (OLS) estimation which comes from theassumption of diffuse priors This name is chosento highlight the fact that while averaging acrossmodels is an inherently Bayesian idea BACElimits the effect of prior information and uses anapproach otherwise familiar to classicaleconometricians

Our BACE approach has several important ad-vantages over previously used model-averagingand robustness-checking methods firstly incontrast to a standard Bayesian approach thatrequires the specification of a prior distributionfor all parameters BACE requires the specifi-cation of only one prior hyper-parameter theexpected model size k This parameter is easyto interpret easy to specify and easy to checkfor robustness2 Secondly the interpretation ofthe estimates is straightforward for economistsnot trained in Bayesian inference the weightsapplied to different models are proportional tothe logarithm of the likelihood function cor-rected for degrees of freedom (analogous to theSchwarz model selection criterion) Thirdly ourestimates can be calculated using only repeatedapplications of OLS Fourthly in contrast toLevine and Renelt and Sala-i-Martin we con-sider models of all sizes and no variables areheld ldquofixedrdquo and therefore ldquountestedrdquo Fifth wecalculate the entire distribution of coefficientsacross models and do not focus solely on thebounds of the distribution

When we examine the cross-country data

usually used by growth empiricists usingBACE we find striking and surprisingly clearconclusions The data identify a set of 18 vari-ables that are significantly related to economicgrowth They have a great deal of explanatorypower and are very precisely estimated An-other three variables are marginal they wouldbe reasonable regressors if a researcher had astrong prior belief in their relevance The re-maining 46 variables have weak explanatorypower and are imprecisely estimated

The rest of the paper is organized as followsSection I outlines the statistical theory SectionII describes the data set used and Section IIIpresents the main empirical results of the paperThe final section concludes

I Statistical Theory

A Statistical Basics

In classical statistics a parameter has a truethough unknown value so it cannot have adensity because it is not random In the Bayes-ian framework parameters are considered to beuncertain Following is a quick exposition of thebasic reasoning and the language needed forunderstanding our approach A more detailedpresentation of these ideas can be found in DaleJ Poirier (1995) We begin with Bayesrsquo rule indensities

(2) gy fyg

fy

This is true for any random variables y and Inequation (2) g() is the prior density of aparameter vector interpreted as the research-errsquos information about prior to seeing thedata The likelihood function f(y) summarizesthe information about contained in the dataThe vector y is the observed data with priordensity f(y) reflecting our prior opinions aboutthe data g(y) is the density of conditionalon the data and is called the posterior density

ldquoModel averagingrdquo is a special case of Bayesrsquorule Suppose we divide the parameter spaceinto two regions and label them M0 and M1These regions could be what we would usuallycall hypotheses (eg 0 versus 0) orsomething we would usually call models (eg

cumulative distribution functions as the measure of signif-icance of a variable and finds that a number of variables aresignificantly correlated with growth therefore contestingthe pessimistic conclusions reached by Levine and Renelt(1992)

2 In the standard Bayesian sense we can calculate esti-mates for a range of different values of k Thus we can makestatements of the form ldquowhether you think a good modelsize is three regressors or 12 regressors this one particularvariable is importantrdquo

815VOL 94 NO 4 SALA-I-MARTIN ET AL DETERMINANTS OF LONG-TERM GROWTH

1 0 2 0 versus 1 0 2 0) Eachof these has a prior probability specified by theresearcher as P(Mi) These prior probabili-ties summarize the researcherrsquos beliefs concern-ing the relative likelihood of the two regions(models) Given the two regions Bayesrsquo ruleimplies g(y) P(M0)[ f(y)g(M0)f(y)] P(M1)[ f(y)g(M1)f(y)] Rewriting this interms of the posterior probabilities conditionalon the two regions (models) we get

(3) gy PM0yfygM0

fyM0

PM1yfygM1

fyM1

where P(Miy) is the posterior probability of theith region the probability of that region condi-tional on the data Equation (3) says that theposterior distribution of the parameters is theweighted average of the two possible condi-tional posterior densities with the weights givenby the posterior probabilities of the two regionsIn this paper we will be considering linear re-gression models for which each model is a listof included variables with the slope coefficientsfor all of the other possible regressors set equalto zero

B Diffuse Priors

As we explained above fully specifying pri-ors is infeasible when the set of possible regres-sors is large In applications of Bayesian theoryif a researcher is incapable or unwilling to spec-ify prior beliefs a standard remedy is to applydiffuse priors If the parameter space isbounded then a diffuse prior is a uniform dis-tribution When the parameter space is un-bounded as in the usual multiple linearregression model a uniform distribution cannotbe directly imposed and instead we must take alimit as the prior distribution becomes flat Inmany contexts imposing diffuse priors gener-ates classical results in the linear regressionmodel standard diffuse priors and Bayesian re-gression yields posterior distributions identicalto the classical sampling distribution of OLS

We would like to work with diffuse priors butthey create a problem when different regression

models contain different sets of variables Asnoted above when the parameter space is un-bounded we must get results for diffuse priorsby taking a limit of informative priors Theinformative prior must specify prior informa-tion concerning both the vector of slopecoefficients and the error standard deviationThere are no difficulties taking the limit as ourprior information concerning becomes unin-formative so the equations below all reflect adiffuse prior with respect to Equation (4)below gives the ratio of the posterior probabil-ities of two regression models (called the pos-terior odds ratio) with different sets of includedvariables X for M0 and Z for M1

(4)PM0y

PM1y

PM0

PM1

AA XXBB ZZ

12SSE0 Q0

SSE1 Q1T2

where P(Mi) is the prior probability of model ias specified by the researcher This expressionassumes that the marginal prior density for ismultivariate normal with variance-covariancematrices given by A1 under M0 and by B1

under M1 SSEi is the OLS sum of squarederrors under model i T is the sample size andQi is a quadratic form in the OLS estimated andprior parameters that need not concern us hereThis is a textbook expression (see for exampleArnold Zellner 1971) Making the priors dif-fuse requires taking the limit of (4) as A and Bapproach zero so that the variance of our priordensity goes to infinity The mathematical dif-ficulty with this is the factor in (4) with the ratioof the determinants of A and B Both determi-nants approach zero as the variance goes toinfinity so their ratio depends on the rate atwhich each goes to zero Depending on pre-cisely how one parameterizes the matrices onegets different answers when evaluating thislimit3 One limit is the likelihood-weightingmethod of Sala-i-Martin (1997a) If we specifythe prior precision matrices as A gXX and

3 Leamer (1978) provides some intuition for why suchproblems occur but argues in Bayesian spirit that oneshould not be interested in diffuse priors


B gZZ (the g-priors suggested by Zellner1986) and take the limit of (4) as g goes to zerowe get

(5)PM0y

PM1y

PM0

PM1 SSE0

SSE1T2

The second factor on the right-hand side isequal to the likelihood ratio of the two modelsThis weighting is troublesome because modelswith more variables have lower SSErsquos the pos-terior mean model size (average of the differentmodel sizes weighted by their posterior proba-bilities) will always be bigger than the priormodel size irrespective of the data Thus it isnot sensible to use this approach when consid-ering models of different sizes

The indeterminacy of the limit in (4) suggeststhat for fairly diffuse priors the exact specifi-cation of the prior precision matrix which willin practice be arbitrary may generate large dif-ferences in the results There is however an-other limit one can take the limit of (4) as theinformation in the data XX and ZZ becomelarge Instead of taking the limit as the priorbecomes flat we are taking the limit as the databecomes very informative relative to the priorinformation (that is as the prior becomes ldquodom-inatedrdquo by the data) If we assume that thevariance-covariance matrix of the Xrsquos exists andtake the limit of (4) as XX (and ZZ respec-tively) goes to infinity we get4

(6)PM0y

PM1y

PM0

PM1 Tk1 k0 2SSE0

SSE1T2

where ki is the number of included regressors inmodel Mi This provides an approximation tothe odds ratios generated by a wide range ofreasonably diffuse prior distributions Thedegrees-of-freedom correction should be famil-iar since it is the ratio of the Schwarz modelselection criteria for the two models exponen-tiated The similarity to the Schwarz criterion isnot coincidental Gideon Schwarz (1978) usedthe same approximation to the odds ratio to

justify the criterion In our empirical work wewill use the approximation in equation (6)

In order to get weights for different modelswe need the posterior probabilities of eachmodel not the odds ratio We thus normalizethe weight of a given model by the sum of theweights of all possible models ie with Kpossible regressors

(7) PMjy PMj Tkj 2SSEj

T2

i 1

2K

PMi Tki 2SSEi T2

Once the model weights have been calculatedBayesrsquo rule says that the posterior density of aparameter is the average of the posterior densi-ties conditional on the models as shown in (3)for two models A posterior mean is defined tobe the expectation of a posterior distributionTaking expectations with respect to across (3)(with 2K terms instead of only two) gives

(8) Ey j 1

2K

PMjyj

where j E(y Mj) is the OLS estimate for with the regressor set that defines model j InBayesian terms j is the posterior mean condi-tional on model j5 Note that any variable ex-cluded from a particular model has a slopecoefficient with degenerate posterior distribu-tion at zero The posterior variance of is givenby

(9) Vary j 1

2K

PMjyVary Mj

j 1

2K

PMjyj Ey2

Leamer (1978) provides a simple derivation for

4 See Leamer (1978 p 112) equation (416) for a dis-cussion of the limiting argument leading to this expressionThis precise expression arises only if we take the limit usingg-priors For other sorts of priors it is an approximation

5 The difficulty with making the prior diffuse appliesonly to the comparison or averaging of different modelsConditional on one particular set of included variables themean of the Bayesian regression posterior is simply theOLS estimate


(9) Inspection of (9) demonstrates that the pos-terior variance incorporates both the estimatedvariances in individual models as well as thevariance in estimates of the rsquos across differentmodels

We can also estimate the posterior probabil-ity that a particular variable is in the regression(ie has a nonzero coefficient) We will call thisthe posterior inclusion probability for the vari-able and it is calculated as the sum of theposterior model probabilities for all of the mod-els including that variable We will also reportthe posterior mean and variance conditional onthe inclusion of the variable

C Model Size

We have not yet discussed the specificationof the P(Mi)rsquos the prior probabilities attached tothe different models One common approach tothis problem in the statistical literature has beento assign equal prior probability to each possiblemodel While this is sensible for some applica-tions for linear regression with a large numberof potential regressors it has odd and troublingimplications In particular it implies a verystrong prior belief that the number of includedvariables should be large It also implies that theexpected model size which is equal to K2increases with the number of explanatory vari-ables available to researchers We will insteadspecify our model prior probabilities by choos-ing a prior mean model size k with each vari-able having a prior probability kK of beingincluded independent of the inclusion of anyother variables where K is the total number ofpotential regressors6 Equal probability for eachpossible model is the special case in which k K2 In our empirical work we focus on a rela-tively small k on the grounds that most research-ers prefer relatively modest parameterizationsWe examine the robustness of our conclusions

with respect to this hyperparameter in SectionIII subsection B

In order to illustrate further this issue Fig-ure 1 plots the prior probability distributions bymodel size for our baseline model with k 7and with equal probabilities for all models k 33 given the 67 potential regressors we con-sider in our empirical work Note that in thelatter case the great majority of the prior prob-ability is focused on models with many in-cluded variables more than 99 percent of theprior probability is located in models with 25 ormore included variables It is our opinion thatfew researchers actually have such prior beliefsThus while we will calculate results for largemodels (k 22 and 28) below we do notchoose to focus attention on these cases Alter-natively Figure 2 illustrates the weights thatequation (7) gives to models of different sizesstarting with prior model size k 7 a re-searcher would need to observe an adjusted R2

as shown in the figure to be indifferent betweenmodels of different sizes

D Sampling

Equations (7) (8) and (9) all face the prob-lem that they include sums running over 2K

terms for many problems for which model av-eraging is attractive this is an infeasibly largenumber even though each term only requires thecomputation of an OLS regression For ourbaseline estimation with K 67 this wouldmean estimating 148 1020 regressions which

6 In most applications the prior probability of including aparticular variable is not for most researchers independentof the probability of including any other variable For ex-ample in a growth regression if a variable proxying politicalinstability is included such as a count of revolutions manyresearchers would think it less likely that another measuresuch as the number of assassinations be included as wellWhile this sort of interdependence can be readily incorpo-rated into our framework we do not presently pursue thisavenue

FIGURE 1 PRIOR PROBABILITIES BY MODEL SIZE

Note Benchmark with prior model size k 7 and equalmodel probability with k 33


is computationally not feasible As a result onlya relatively small subset of the total number ofpossible regressions can be run

Several stochastic algorithms have been pro-posed for dealing with this issue including theMarkov-Chain Monte-Carlo Model Composi-tion (MC3) technique (David Madigan and Jer-emy C York 1995) stochastic search variableselection (SSVS) (Edward I George and RobertE McCulloch 1993) and the Gibbrsquos sampler-based method of John F Geweke (1994) Wewill take a simpler approach that matches theform of the prior distribution We select modelsby randomly including each variable with inde-pendent sampling probability Ps(i) So long asthe sampling probabilities are strictly greaterthan zero and strictly less than one any valueswill work in the sense that as the number ofrandom draws grows the sampled versions of(7) (8) and (9) will approach their true valuesMerlise Clyde et al (1996) have shown that thisprocedure when implemented with Ps(i) equalto the prior inclusion probability (called by theauthors ldquorandom samplingrdquo) has computationalefficiency not importantly lower than that of theMC3 and SVSS algorithms (for at least oneparticular data set) For the present applicationwe found that sampling models using their priorprobabilities produced unacceptably slow con-

vergence Instead we sampled one set of regres-sions using the prior probability samplingweights and then used the approximate poste-rior inclusion probabilities calculated fromthose regressions for the subsequent samplingprobabilities This ldquostratified samplingrdquo acceler-ates convergence The Technical Appendix(available on the AER Web site at httpww-waeaweborgaercontents) discusses compu-tational and convergence issues in detail andmay be of interest to researchers looking toapply these techniques

As an alternative to model averaging Leamer(1978) suggests orthogonalizing7 the explana-tory variables and estimating the posteriormeans of the effects of the K 1 principlecomponents An advantage of this approach isthe large reduction in the computational burdenwhich is especially relevant for prediction Theproblem however is that with the transforma-tion of the data the economic interpretation ofthe coefficients associated with the originalvariables is lost As we are particularly inter-ested in the interpretation of variables as deter-minants of economic growth we do not followthis approach

II Data

Of the many variables that have been foundto be significantly correlated with growth in theliterature we selected 67 variables using thefollowing criteria First we kept the variablesthat represent ldquostate variablesrdquo of a dynamicoptimization problem Hence we choose vari-ables measured as closely as possible to thebeginning of the sample period (which is 1960)and eliminate all those variables that were com-puted for the later years only This leads to theexclusion of some widely used political vari-ables that were published for the late 1980rsquos (inthis category for example we neglect the widelyused bureaucracy and corruption variables whichwere computed for 1985 only) This is partly doneto deal with the endogeneity problem

The second selection criterion derives fromour need for a ldquobalancedrdquo data set By balanced

7 An orthogonalization of the explanatory variableswould make BACE results invariant with respect to lineartransformations of the data (see also the discussion inLeamer 1985)

FIGURE 2 PRIOR PROBABILITIES P(Mj) AND

CORRESPONDING ADJUSTED R2 NEEDED TO MAKE A

RESEARCHER INDIFFERENT BETWEEN MODELS OF DIFFERENT

SIZES kj

Note Calculated from equation (7) as SSEj [P(Mjy)Tkj2P(Mj)]

2T and R2 [(SSE0 SSEj)SSE0] where P(Mj) arethe prior probabilities for the benchmark case (k 7) andP(Mjy) is the flat posterior probability equal to 168 0015


TABLE 1mdashDATA DESCRIPTION AND SOURCES

Rank Variable Description and source MeanStandarddeviation

Average growth rate of GDPper capita 1960ndash1996

Growth of GDP per capita at purchasing power parities between1960 and 1996 From Alan Heston et al (2001)

00182 0019

1 East Asian dummy Dummy for East Asian countries 01136 031922 Primary schooling in 1960 Enrollment rate in primary education in 1960 Barro and Jong-

Wha Lee (1993)07261 02932

3 Investment price Average investment price level between 1960 and 1964 onpurchasing power parity basis From Heston et al (2001)

924659 536778

4 GDP in 1960 (log) Logarithm of GDP per capita in 1960 From Heston et al (2001) 73549 090115 Fraction of tropical area Proportion of countryrsquos land area within geographical tropics

From John L Gallup et al (2001)05702 04716

6 Population density coastal in1960rsquos

Coastal (within 100 km of coastline) population per coastal area in1965 From Gallup et al (2001)

1468717 5098276

7 Malaria prevalence in 1960rsquos Index of malaria prevalence in 1966 From Gallup et al (2001) 03394 043098 Life expectancy in 1960 Life expectancy in 1960 Barro and Lee (1993) 537159 1206169 Fraction Confucian Fraction of population Confucian Barro (1999) 00156 00793

10 African dummy Dummy for Sub-Saharan African countries 03068 04638

11 Latin American dummy Dummy for Latin American countries 02273 0421512 Fraction GDP in mining Fraction of GDP in mining From Robert E Hall and Charles I

Jones (1999)00507 00769

13 Spanish colony Dummy variable for former Spanish colonies Barro (1999) 01705 0378214 Years open 1950ndash1994 Number of years economy has been open between 1950 and 1994

From Jeffrey D Sachs and Andrew M Warner (1995)03555 03444

15 Fraction Muslim Fraction of population Muslim in 1960 Barro (1999) 01494 0296216 Fraction Buddhist Fraction of population Buddhist in 1960 Barro (1999) 00466 0167617 Ethnolinguistic

fractionalizationAverage of five different indices of ethnolinguistic

fractionalization which is the probability of two random peoplein a country not speaking the same language From WilliamEasterly and Ross Levine (1997)

03476 03016

18 Government consumptionshare 1960rsquos

Share of expenditures on government consumption to GDP in1961 Barro and Lee (1993)

01161 00745

19 Population density 1960 Population per area in 1960 Barro and Lee (1993) 1080735 201444920 Real exchange rate

distortionsReal exchange rate distortions Levine and Renelt (1992) 1250341 417063

21 Fraction speaking foreignlanguage

Fraction of population speaking foreign language Hall and Jones(1999)

03209 04136

22 Openness measure 1965ndash1974

Ratio of exports plus imports to GDP averaged over 1965 to1974 This variable was provided by Robert Barro

05231 03359

23 Political rights Political rights index From Barro (1999) 38225 1996624 Government share of GDP

in 1960rsquosAverage share government spending to GDP between 1960ndash1964

From Heston et al (2001)01664 00712

25 Higher education in 1960 Enrollment rates in higher education Barro and Lee (1993) 00376 0050126 Fraction population in

tropicsProportion of countryrsquos population living in geographical tropics

From Gallup et al (2001)03 03731

27 Primary exports 1970 Fraction of primary exports in total exports in 1970 From Sachsand Warner (1997)

07199 02827

28 Public investment share Average share of expenditures on public investment as fraction ofGDP between 1960 and 1965 Barro and Lee (1993)

00522 00388

29 Fraction Protestant Fraction of population Protestant in 1960 Barro (1999) 01354 0285130 Fraction Hindu Fraction of the population Hindu in 1960 Barro (1999) 00279 01246

31 Fraction population less than15

Fraction of population younger than 15 years in 1960 Barro andLee (1993)

03925 00749

32 Air distance to big cities Logarithm of minimal distance (in km) from New YorkRotterdam or Tokyo From Gallup et al (2001)

43241705 26137627

33 Nominal government GDPshare 1960rsquos

Average share of nominal government spending to nominal GDPbetween 1960 and 1964 Calculated from Heston et al (2001)

0149 00584

34 Absolute latitude Absolute latitude Barro (1999) 232106 16842635 Fraction Catholic Fraction of population Catholic in 1960 Barro (1999) 03283 0414636 Fertility in 1960rsquos Fertility in 1960rsquos Barro and Sala-i-Martin (1995) 1562 0419337 European dummy Dummy for European economies 02159 0413838 Outward orientation Measure of outward orientation Levine and Renelt (1992) 03977 0492239 Colony dummy Dummy for former colony Barro (1999) 075 0435540 Civil liberties Index of civil liberties index in 1972 Barro (1999) 05095 0325941 Revolutions and coups Number of revolutions and military coups Barro and Lee (1993) 01849 02322


we mean an equal number of observations forall regressions Since different variables missobservations for different countries we selectedthe 67 explanatory variables that maximized theproduct of the number of countries with observa-tions for all variables and the number of variables

With these restrictions the total size of thedata set becomes 68 variables (including thedependent variable the annualized growth rateof GDP per capita between 1960 and 1996) for88 countries The variable names their meansand standard deviations are depicted in Table 1Appendix Table A1 provides a list of the in-cluded countries

III Results

We are now ready to conduct our BACEestimation We calculate the posterior distribu-tions for all of the rsquos as well as the posteriormeans and variances given by equations (8) to(9) using the posterior model weights fromequation (7) We also calculate the posteriorinclusion probability discussed in Section Isubsection B Figure 3 shows the posterior den-sities (approximated by histograms) of the co-efficient estimates for four selected variables(the investment price the initial level of GDPper capita primary schooling and the number

TABLE 1mdashContinued


42 British colony Dummy for former British colony after 1776 Barro (1999) 03182 0468443 Hydrocarbon deposits in 1993 Log of hydrocarbon deposits in 1993 From Gallup et al (2001) 04212 4351244 Fraction population over 65 Fraction of population older than 65 years in 1960 Barro and Lee

(1993)00488 0029

45 Defense spending share Average share public expenditures on defense as fraction of GDPbetween 1960 and 1965 Barro and Lee (1993)

00259 00246

46 Population in 1960 Population in 1960 Barro (1999) 2030808 52538386647 Terms of trade growth in

1960rsquosGrowth of terms of trade in the 1960rsquos Barro and Lee (1993) 00021 00345

48 Public education spendingshare in GDP in 1960rsquos

Average share public expenditures on education as fraction ofGDP between 1960 and 1965 Barro and Lee (1993)

00244 00096

49 Landlocked country dummy Dummy for landlocked countries 01705 0378250 Religious intensity Religion measure Barro (1999) 07803 01932

51 Size of economy Logarithm of aggregate GDP in 1960 161505 1820252 Socialist dummy Dummy for countries under Socialist rule for considerable time

during 1950 to 1995 From Gallup et al (2001)00682 02535

53 English-speaking population Fraction of population speaking English From Hall and Jones (1999) 0084 0252254 Average inflation 1960ndash1990 Average inflation rate between 1960 and 1990 Levine and Renelt

(1992)131298 149899

55 Oil-producing countrydummy

Dummy for oil-producing country Barro (1999) 00568 02328

56 Population growth rate1960ndash1990

Average growth rate of population between 1960 and 1990 Barroand Lee (1993)

00215 00095

57 Timing of independence Timing of national independence measure 0 if before 1914 1 ifbetween 1914 and 1945 2 if between 1946 and 1989 and 3 ifafter 1989 From Gallup et al (2001)

10114 09767

58 Fraction of land area nearnavigable water

Proportion of countryrsquos land area within 100 km of ocean orocean-navigable river From Gallup et al (2001)

04722 03802

59 Square of inflation 1960ndash1990

Square of average inflation rate between 1960 and 1990 3945368 11196992

60 Fraction spent in war 1960ndash1990

Fraction of time spent in war between 1960 and 1990 Barro andLee (1993)

00695 01524

61 Land area Area in km2 Barro and Lee (1993) 86718852 18146882962 Tropical climate zone Fraction tropical climate zone From Gallup et al (2001) 019 0268763 Terms of trade ranking Barro (1999) 02813 0190464 Capitalism Degree Capitalism index From Hall and Jones (1999) 34659 1380965 Fraction Orthodox Fraction of population Orthodox in 1960 Barro (1999) 00187 0098366 War participation 1960ndash1990 Indicator for countries that participated in external war between

1960 and 1990 Barro and Lee (1993)03977 04922

67 Interior density Interior (more than 100 km from coastline) population per interiorarea in 1965 From Gallup et al (2001)

433709 880626


of years an economy has been ldquoopenrdquo)8 Notethat in Figure 3 each distribution consists oftwo parts first a continuous part that is theposterior density conditional on inclusion in themodel and second a discrete mass at zero rep-resenting the probability that the variable doesnot belong in the model this is given by oneminus the posterior inclusion probability9 Asdescribed in Section I these densities are

weighted averages of the posterior densitiesconditional on each particular model with theweights given by the posterior model probabil-ities A standard result from Bayesian theory(see eg Leamer 1978 or Poirier 1995) is thatif priors are taken as diffuse by taking the limitof a Student-Gamma prior10 then the posteriorcan be represented by

(10) ti i i

sXX1ii12 tT k

8 The figures for the remaining variables are available inthe Appendix on the AER Web site (httpwwwaeaweborgaercontents)

9 The probability mass at zero is split into seven binsaround zero to make the area of the mass comparable withareas under the conditional density The figure for yearsopen is plotted with a different vertical axis scaling reflect-ing the lower posterior inclusion probability

10 That is a prior in which the marginal prior for theslope coefficients is multivariate student and the marginalprior for the regression error variance is inverted Gamma

FIGURE 3 POSTERIOR DISTRIBUTION OF MARGINAL RELATIONSHIP OF SELECTED VARIABLES WITH LONG-RUN GROWTH

Notes The distribution consists of two parts (1) the ldquohump-shapedrdquo part of the distribution represents the distribution ofestimates conditional on the variable being included in the model (2) the ldquolumprdquo at zero shows the posterior probability thata variable is not included in the regression which equals one minus the posterior inclusion probability Note that the scaleof the vertical axis for the ldquoYears openrdquo variable is larger to reflect the smaller posterior inclusion probability


where s is the usual OLS estimate of the stan-dard deviation of the regression residual Inother words with the appropriate diffuse priorthe posterior distribution conditional on themodel is identical to the classical samplingdistribution Thus the marginal posterior distri-bution for each coefficient is a mixture-t distri-bution In principle these distributions could beof almost any form but most of the densities inFigure 3 look reasonably Gaussian

A Baseline Estimation

This section presents the baseline estimationresults with a prior expected model size k 7The choice of the baseline model size is moti-vated by the fact that most empirical growthstudies include a moderate number of explana-tory variables The posterior model size for thebaseline estimation is 746 which is very closeto the prior model size In Section III subsec-tion B we check the robustness of our results tochanges in the prior mean model size The re-sults are based on approximately 89 millionrandomly drawn regressions11

Table 2 shows the results for the 67 variablesColumn (1) reports the posterior inclusion prob-ability of a variable in the growth regressionVariables are sorted in descending order of thisposterior probability The posterior inclusionprobability is the sum of the posterior probabil-ities of all of the regressions including thatvariable Thus computationally the posteriorinclusion probability is a measure of theweighted average goodness-of-fit of models in-cluding a particular variable relative to modelsnot including the variable The goodness-of-fitmeasure is adjusted to penalize highly parame-terized models in the fashion of the Schwarzmodel selection criterion Thus variables with

high inclusion probabilities are variables thathave high marginal contribution to the goodness-of-fit of the regression model

We can divide the variables according towhether seeing the data causes us to increase ordecrease our inclusion probability relative to theprior probability Since our expected modelsize k equals 7 the prior inclusion probability is767 0104 There are 18 variables for whichthe posterior inclusion probability increases(these are the first 18 variables in Table 2) Forthese variables our belief that they belong inthe regression is strengthened once we see thedata and we call these variables ldquosignificantrdquoThe remaining 49 variables have little or nosupport for inclusion seeing the data furtherreduces our already modest initial assessment oftheir inclusion probability

Columns (2) and (3) show the posterior meanand standard deviation of the distributions con-ditional on the variable being included in themodel That is these are the means and standarddeviations of the ldquohump-shapedrdquo part of thedistribution shown in Figure 3 The true (uncon-ditional) posterior mean is computed accordingto equation (8) while the posterior standard de-viation is the square root of the variance for-mula in equation (9) The true posterior mean isa weighted average of the OLS estimates for allregressions including regressions in which thevariable does not appear and thus has a coeffi-cient of zero Hence the unconditional posteriormean can be computed by multiplying the con-ditional mean in column (2) times the posteriorinclusion probability in column (1)12

If one has the prior with which we began theestimation then the unconditional posteriormean is the ldquorightrdquo estimate of the marginaleffect of the variable in the sense that it is thecoefficient that would be used for forecasting13

The conditional mean and variance are also

11 The total number of possible regression models equals267 which is approximately equal to 148 1020 modelsHowever convergence of the estimates is attained relativelyquickly after 33 million draws the maximum change ofcoefficient estimates normalized by the standard deviationof the regressors relative to the dependent variable issmaller than 103 per 10000 and after 89 million draws themaximum change is smaller than 106 The latter tolerancewas used as one of the convergence criteria for the reportedestimates See technical Appendix (available at the AERWeb site httpwwwaeaweborgaercontents) for furtherdetails

12 Similarly the unconditional variance can be calcu-lated from the conditional variance as follows(14) uncond

2 cond2 cond

2

PosteriorInclusionProb uncond2

13 In a pure Bayesian approach there is not really anotion of a single estimate However for many purposes theposterior mean is reasonable and it is what would be usedfor constructing unbiased minimum mean-squared-errorpredictions


TABLE 2mdashBASELINE ESTIMATION FOR ALL 67 VARIABLES

Rank Variable

Posteriorinclusion

probability(1)

Posterior meanconditional on

inclusion(2)

Posterior sdconditionalon inclusion

(3)

BACEsign

certaintyprobability

(4)

OLSp-value

(5)

OLS signcertainty

probability(6)

Fraction ofregressions

with tstat 2(7)

1 East Asian dummy 0823 0021805 0006118 0999 0505 0999 0992 Primary schooling 1960 0796 0026852 0007977 0999 0155 0999 0963 Investment price 0774 0000084 0000025 0999 0032 0999 0994 GDP 1960 (log) 0685 0008538 0002888 0999 0387 0999 0305 Fraction of tropical area 0563 0014757 0004227 0997 0466 0997 0596 Population density coastal 1960rsquos 0428 0000009 0000003 0996 0767 0996 0857 Malaria prevalence in 1960rsquos 0252 0015702 0006177 0990 0515 0010 0848 Life expectancy in 1960 0209 0000808 0000354 0986 0761 0014 0799 Fraction Confucian 0206 0054429 0022426 0988 0377 0988 097

10 African dummy 0154 0014706 0006866 0980 0589 0980 09011 Latin American dummy 0149 0012758 0005834 0969 0652 0969 03012 Fraction GDP in mining 0124 0038823 0019255 0978 0305 0978 00713 Spanish colony 0123 0010720 0005041 0972 0507 0028 02414 Years open 0119 0012209 0006287 0977 0826 0023 09815 Fraction Muslim 0114 0012629 0006257 0973 0478 0973 01116 Fraction Buddhist 0108 0021667 0010722 0974 0460 0974 09017 Ethnolinguistic fractionalization 0105 0011281 0005835 0974 0991 0974 05218 Government consumption share 1960rsquos 0104 0044171 0025383 0975 0344 0025 077

19 Population density 1960 0086 0000013 0000007 0965 0815 0965 00120 Real exchange rate distortions 0082 0000079 0000043 0966 0835 0034 09221 Fraction speaking foreign language 0080 0007006 0003960 0962 0474 0962 043

22 (Imports exports)GDP 0076 0008858 0005210 0949 0951 0949 06723 Political rights 0066 0001847 0001202 0939 0664 0939 03524 Government share of GDP 0063 0034874 0029379 0935 0329 0935 05825 Higher education in 1960 0061 0069693 0041833 0946 0379 0946 01026 Fraction population in tropics 0058 0010741 0006754 0940 0657 0940 08527 Primary exports in 1970 0053 0011343 0007520 0926 0752 0926 07528 Public investment share 0048 0061540 0042950 0922 0115 0922 00029 Fraction Protestant 0046 0011872 0009288 0909 0715 0091 02930 Fraction Hindu 0045 0017558 0012575 0915 0790 0915 00731 Fraction population less than 15 0041 0044962 0041100 0871 0545 0871 02432 Air distance to big cities 0039 0000001 0000001 0888 0572 0888 01833 Government consumption share deflated

with GDP prices0036 0033647 0027365 0893 0565 0893 005

34 Absolute latitude 0033 0000136 0000233 0737 0484 0263 03735 Fraction Catholic 0033 0008415 0008478 0837 0939 0163 01636 Fertility in 1960rsquos 0031 0007525 0010113 0767 0697 0767 04637 European dummy 0030 0002278 0010487 0544 0768 0456 01938 Outward orientation 0030 0003296 0002727 0886 0882 0886 00139 Colony dummy 0029 0005010 0004721 0858 0770 0858 04440 Civil liberties 0029 0007192 0007122 0846 0740 0846 01541 Revolutions and coups 0029 0007065 0006089 0877 0276 0877 00742 British colony 0027 0003654 0003626 0844 0660 0844 00943 Hydrocarbon deposits in 1993 0025 0000307 0000418 0773 0813 0773 00144 Fraction population over 65 0022 0019382 0119469 0566 0540 0566 02045 Defense spending share 0021 0045336 0076813 0737 0327 0737 02646 Population in 1960 0021 0000000 0000000 0806 0581 0806 00747 Terms of trade growth in 1960rsquos 0021 0032627 0046650 0752 0982 0248 00048 Public education spendingGDP in

1960rsquos0021 0129517 0172847 0777 0322 0777 011

49 Landlocked country dummy 0021 0002080 0004206 0701 0951 0701 00450 Religion measure 0020 0004737 0007232 0751 0910 0249 01851 Size of economy 0020 0000520 0001443 0661 0577 0661 01852 Socialist dummy 0020 0003983 0004966 0788 0916 0212 00053 English-speaking population 0020 0003669 0007137 0686 0543 0314 00754 Average inflation 1960ndash1990 0020 0000073 0000097 0784 0774 0216 00155 Oil-producing country dummy 0019 0004845 0007088 0751 0933 0751 00056 Population growth rate 1960ndash1990 0019 0020837 0307794 0533 0924 0467 02157 Timing of independence 0019 0001143 0002051 0716 0846 0284 011


of interest however From a Bayesian pointof view these have the interpretation of theposterior mean and variance for a researcherwho has a prior inclusion probability equal toone for the particular variable while maintain-ing the 767 inclusion probability for all theother variables In other words if one is certainthat the variable belongs in the regression thisis the estimate to consider It is also compa-rable to coefficient estimates in standardregressions not accounting for model uncer-tainty The conditional standard deviationprovides one measure of how well estimatedthe variable is conditional on its inclusion Itaverages both the standard errors of each pos-sible regression as well as the dispersion ofestimates across models14

From the posterior density we can also esti-

mate the posterior probability that conditionalon a variablersquos inclusion a coefficient has thesame sign as its posterior mean reported incolumn (1)15 This ldquosign certainty probabilityrdquoreported in column (4) is another measure ofthe significance of the variables As notedabove for each individual regression the poste-rior density is equal to the classical samplingdistribution of the coefficient In classical termsa coefficient would be 5-percent significant in atwo-sided test if 975 percent of the probabilityin the sampling distribution were on the sameside of zero as the coefficient estimate So if forexample it just happened that a coefficient wereexactly 5-percent significant in every single re-gression its sign certainty probability would be975 percent Applying a 0975 cutoff to thisquantity identifies a set of 13 variables all ofwhich are also in the group of 18 ldquosignificantrdquovariables for which the posterior inclusion prob-ability is larger than the prior inclusion proba-bility The remaining five have very large signcertainty probabilities (between 0970 and

14 Note that one cannot interpret the ratio of the posteriormean to the posterior standard deviation as a t-statistic fortwo reasons Firstly the posterior is a mixture t-distributionand secondly it is not a sampling distribution However formost of the variables which we consider the posterior dis-tributions are not too far from being normal To the extentto which these are approximately normal having a ratio ofposterior conditional mean to standard deviation around twoin absolute value indicates an approximate 95-percentBayesian coverage region that excludes zero

15 This ldquosign certainty probabilityrdquo is analogous to thearea under the normal CDF(0) calculated by Sala-i-Martin(1997a b)


Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)

58 Fraction land area near navigwater

0019 0002598 0005864 0657 0730 0657 035

59 Square of inflation 1960ndash1990 0018 0000001 0000001 0736 0668 0264 00060 Fraction spent in war 1960ndash1990 0016 0001415 0009226 0555 0904 0445 00061 Land area 0016 0000000 0000000 0577 0950 0577 00162 Tropical climate zone 0016 0002069 0006593 0616 0614 0616 01663 Terms of trade ranking 0016 0003730 0009625 0647 0845 0647 02364 Capitalism 0015 0000231 0001080 0589 0313 0589 00665 Fraction Orthodox 0015 0005689 0013576 0660 0900 0660 00066 War participation 1960ndash1990 0015 0000734 0002983 0593 0868 0593 00267 Interior density 0015 0000001 0000016 0532 0768 0468 000

Notes The left-hand-side variable in all regressions is the growth rate from 1960ndash1996 across 88 countries Apart from the final column allstatistics come from a random sample of approximately 89 million of the possible regressions including any combination of the 67 variablesPrior mean model size is seven Variables are ranked by the first column the posterior inclusion probability This is the sum of the posteriorprobabilities of all models containing the variable The next two columns reflect the posterior mean and standard deviations for the linearmarginal effect of the variable the posterior mean has the usual interpretation of a regression The conditional mean and standard deviationare conditional on inclusion in the model The ldquosign certainty probabilityrdquo is the posterior probability that the coefficient is on the sameside of zero as its mean conditional on inclusion It is a measure of our posterior confidence in the sign of the coefficient The final columnis the fraction of regressions in which the coefficient has a classical t-test greater than two with all regressions having equal samplingprobability


0975) Note that there is in principle no reasonwhy a variable could not have a very highposterior inclusion probability and still have alow sign certainty probability

The results can also be contrasted with theestimates of including all 67 explanatory vari-ables resulting from diffuse prior OLS Firstnote from the p-values reported in column (5)that all but one variable fail to be statisticallysignificant at the 10-percent level Second therank correlation between diffuse prior OLS(ranked by p-values) and BACE (ranked byposterior inclusion probability) equals only043 This implies that BACE and diffuse priorOLS disagree about the relative importance ofseveral important variables Third one can alsocalculate the posterior ldquosign certaintyrdquo that thecoefficient in column (2) takes on the diffuseprior OLS sign Column (6) of Table 2 showsthis probability and we can see that for six of thetop 21 variables BACE gives a very differentprediction of the sign of the effect of a variablethan OLS The disagreement between BACEand OLS concerning the relative ranking ofvariables and their predicted signs could be dueto low number of degrees of freedom andsome correlation among explanatory vari-ables The limited number of observationsdoes not allow us to make reliable inferenceon the relative sign and importance of theexplanatory variables BACE on the otherhand considers models of smaller expectedsize and allows explanatory variables to cap-ture different aspects of the variation in thedependent variable

The final column (7) in Table 2 shows thefraction of regressions in which the variable isclassically significant at the 95-percent level inthe sense of having a t-statistic with an absolutevalue greater than two This statistic was calcu-lated separately from the other estimates16 Thisis reported partly for the sake of comparisonwith extreme bounds analysis results Note thatfor all the variables many individual regres-sions can be found in which the variable is not

significant but even the top variables wouldstill be labeled fragile by an extreme boundstest

Variables Significantly Related to GrowthmdashWe are now ready to analyze the variables thatare ldquosignificantlyrdquo related to growth Not sur-prisingly the top variable is the dummy for EastAsian countries which is positively related witheconomic growth This of course reflects theexceptional growth performance of East Asiancountries between 1960 and the mid-1990rsquosNotice that the dummy is present despite thesignificant positive relationship between thefraction of population Confucian (which isranked 9th in the table) Although the Confu-cian variable can be interpreted as a dummy forEast Asian economies it gains relative to theEast Asia dummy variable as more regressorsare included in the regression with larger priormodel sizes (this is seen in Tables 3 and 5) Theposterior mean coefficient is very precisely es-timated to be positive The sign certainty prob-ability in column (4) shows that the probabilitymass of the density to the right of zero equals to09992 Notice that the fraction of regressionsfor which the East Asian dummy has a t-statisticgreater than two in absolute value is 99 percent

The second variable is a measure of humancapital the primary schooling enrollment ratein 1960 This variable is positively related togrowth and the inclusion probability is 080The posterior distribution of the coefficient es-timates is shown in the first panel of Figure3 Since the inclusion probability is relativelyhigh the mass at zero (which shows one minusthis inclusion probability) is relatively smallConditional on being included in the model a10-percentage-point increase of the primaryschool enrollment rate is associated with a027-percentage-point increase of the growthrate The sign certainty probability for thisvariable is also 0999 and the fraction ofregressions with a t-statistic larger than two is96 percent

The third variable is the average price ofinvestment goods between 1960 and 1964 Itsinclusion probability is 077 This variable isalso depicted graphically in the second panel ofFigure 3 The posterior mean coefficient is veryprecisely estimated to be negative which indi-cates that a relative high price of investment

16 This column was calculated based on a run of 725million regression It was calculated separately so that thesampling could be based solely on the prior inclusion prob-abilities The other baseline estimates were calculated byoversampling ldquogoodrdquo variables for inclusion and thus pro-duce misleading results for this statistic


goods at the beginning of the sample is stronglyand negatively related to subsequent incomegrowth17 The sign certainty probability in col-umn (4) shows that the probability mass of thedensity to the left of zero equals 099 this canalso be seen in Figure 3 by the fact that almostall of the continuous density lies below zero

The next variable is the initial level of percapita GDP a measure of conditional conver-gence The inclusion probability is 069 Thethird panel in Figure 3 shows the posterior dis-tribution of the coefficient estimates for initialincome Conditional on inclusion the posteriormean coefficient is 0009 (with a standarddeviation of 0003) The sign certainty proba-bility in column (4) shows that the probabilitymass of the density to the left of zero equals0999 The fraction of regressions in which thecoefficient for initial income has a t-statisticgreater than two in absolute value is only 30percent so that an extreme bounds test veryeasily labels the variable as not robust None-theless the tightly estimated coefficient and thevery high sign certainty statistic show that ini-tial income is indeed robustly partially corre-lated with growth The explanation is that theregressions in which the coefficient on initialincome is poorly estimated are regressions withvery poor fit so they receive little weight in theaveraging process Furthermore the high inclu-sion probability suggests that regressions thatomit initial income are likely to perform poorly

The next variables reflect the poor economicperformance of tropical countries The propor-tion of a countryrsquos area in the tropics has anegative relationship with income growth Sim-ilarly the index of malaria prevalence hasa negative relationship with growth Table 3shows that the posterior inclusion probability ofthe malaria index falls as models becomelarger indicating that it could act as a catch-allvariable when few explanatory variables areincluded but drops in significance as morevariables explaining different steady statesenter Similarly Table 5 shows the sign cer-

tainty probability falling with model size forthis variable

Another geographical variable that performswell is the density of the population in coastalareas which has a positive relationship withgrowth suggesting that areas that are denselypopulated and are close to the sea have experi-enced higher growth rates

Another measure of health is life expectancyin 1960 (which is related to things like nutritionhealth care and education) Countries with highlife expectancy in 1960 tended to grow fasterover the following four decades The inclusionprobability for this variable is 028

Dummies for Sub-Saharan Africa and LatinAmerica are negatively related to incomegrowth The posterior means imply that thegrowth rate for Latin American and Sub-Saharan African countries were 147 and 128percentage points below the level that would bepredicted by the countriesrsquo other characteristicsThe African dummy is significant in 90 percentof the regressions and the sign certainty proba-bility is 98 percent Although the Latin Ameri-can dummy is only significant in 33 percent ofthe regressions its sign certainty is 97 percent

The fraction of GDP in mining has a positiverelationship with growth and inclusion proba-bility of 012 This variable captures the successof countries with a large endowment of naturalresources While many economists expect thatthe large rents are associated with more politicalinstability or rent-seeking and low growth ourstudy shows that economies with a larger min-ing sector tend to perform better18

Former Spanish colonies tend to grow lesswhereas the number of years an economy hasbeen open has a positive sign Both of thesevariables have inclusion probabilities that in-crease only moderately with larger models inTable 3 indicating that they capture steady-statevariations in smaller models but are relativelyless important in explaining growth when morevariables are included in the regression models

Both the fraction of the population Muslim

17 Once the relative price of investment goods is in-cluded among the pool of explanatory variables the share ofinvestment to GDP in 1961 becomes insignificant and hasthe ldquowrong signrdquo while the other results are unaffected Theestimation results including investment share are availablefrom the authors upon request

18 The regressions that include the fraction of miningtend to have an outlier Botswana which is a country thatdiscovered diamonds in its territory in the 1960rsquos and hasmanaged to exploit them successfully (which implies a largeshare of mining in GDP) and has experienced extraordinarygrowth rates over the last 40 years


TABLE 3mdashPOSTERIOR INCLUSION PROBABILITIES WITH DIFFERENT PRIOR MODEL SIZES

Rank Variable kbar5 kbar7 kbar9 kbar11 kbar16 kbar22 kbar28

Prior inclusion probability 0075 0104 0134 0164 0239 0328 0418




22 (Imports exports)GDP 0063 0076 0085 0099 0131 0181 024023 Political rights 0042 0066 0082 0095 0114 0130 015424 Government share of GDP 0044 0063 0087 0112 0186 0252 029125 Higher education in 1960 0059 0061 0066 0070 0079 0103 013126 Fraction population in tropics 0047 0058 0061 0074 0099 0132 015727 Primary exports in 1970 0047 0053 0065 0072 0104 0137 016228 Public investment share 0023 0048 0096 0151 0321 0525 066929 Fraction Protestant 0035 0046 0055 0061 0083 0120 015630 Fraction Hindu 0028 0045 0059 0077 0126 0179 022731 Fraction population less than 15 0035 0041 0045 0050 0067 0093 012332 Air distance to big cities 0024 0039 0054 0072 0097 0115 014133 Government consumption share deflated with

GDP prices0021 0036 0056 0075 0137 0225 0310

34 Absolute latitude 0029 0033 0040 0042 0059 0086 011535 Fraction Catholic 0019 0033 0042 0056 0104 0163 022336 Fertility in 1960rsquos 0020 0031 0043 0063 0108 0170 023237 European dummy 0020 0030 0043 0049 0094 0148 020138 Outward orientation 0019 0030 0043 0054 0085 0134 017839 Colony dummy 0022 0029 0039 0049 0075 0105 014640 Civil liberties 0021 0029 0037 0044 0069 0106 015541 Revolutions and coups 0019 0029 0038 0056 0106 0188 028242 British colony 0022 0027 0034 0041 0057 0085 011943 Hydrocarbon deposits in 1993 0015 0025 0035 0048 0089 0143 019644 Fraction population over 65 0020 0022 0029 0038 0069 0119 016945 Defense spending share 0016 0021 0027 0033 0049 0073 010246 Population in 1960 0016 0021 0040 0041 0063 0092 011847 Terms of trade growth in 1960rsquos 0015 0021 0026 0033 0051 0068 010448 Public education spendingGDP in 1960rsquos 0014 0021 0027 0037 0063 0102 014149 Landlocked country dummy 0012 0021 0029 0033 0055 0080 010950 Religion measure 0012 0020 0025 0037 0048 0068 009251 Size of economy 0016 0020 0026 0033 0051 0076 010452 Socialist dummy 0012 0020 0024 0032 0054 0091 014453 English-speaking population 0015 0020 0025 0028 0043 0063 008754 Average inflation 1960ndash1990 0015 0020 0024 0030 0043 0064 010055 Oil-producing country dummy 0012 0019 0025 0033 0050 0071 009556 Population growth rate 1960ndash1990 0014 0019 0023 0029 0046 0074 009857 Timing of independence 0014 0019 0024 0031 0048 0076 0099


and Buddhist have a positive association withgrowth where the conditional mean of the lattervariable is almost twice as large (0022) than forthe fraction Muslim (0013) but both are rela-tively small compared to the fraction Confucian(0054) The index of ethnolinguistic fraction-alization is negatively related to growth

Finally the share of government consump-tion in GDP has a negative association witheconomic growth This could be expected be-cause public consumption does not tend to con-tribute to growth directly but it needs to befinanced with distortionary taxes which hurt thegrowth rate Perhaps the real surprise is thenegative coefficient of the public investmentshare Table 2 shows that this variable is notrobust when the prior model size is k 7However we will see later that this is one of thevariables that becomes important in larger mod-els and the sign remains negative

Variables Marginally Related to GrowthmdashThere are three variables that have posteriorprobabilities somewhat lower than their priorprobabilities but nonetheless are fairly preciselyestimated if they are included in the growthregression (that is their sign certainty probabil-ity is larger than 95 percent) These variablesare the overall density in 1960 (which is posi-tively related to growth) real exchange ratedistortions (negative) and the fraction of popu-lation speaking a foreign language (positive)

Variables Weakly or Not Related to GrowthmdashThe remaining 46 variables show little evidenceof any sort of robust partial correlation withgrowth They neither contribute importantly tothe goodness-of-fit of growth regressions asmeasured by their posterior inclusion probabil-ities nor have estimates that are robust acrossdifferent sets of conditioning variables It isinteresting to notice that some political vari-ables such as the number of revolutions andcoups or the index of political rights are notrobustly related to economic growth Similarlythe degree of capitalism measure or a Socialismdummy measuring whether a country was underSocialism for a significant period of time haveno strong relationship with growth between1960 and 199619 This could be due to the factthat other variables capturing political or eco-nomic instability such as the relative price ofinvestment goods real exchange rate distor-tions the number of years an economy has beenopen and life expectancy or regional dummiescapture most of the effect

We also notice that some macroeconomicvariables such as the inflation rate do not appearto be strongly related to growth Other surpris-ingly weak variables are the spending in public

19 We should remind the reader that most of the econo-mies Eastern Europe and the former Soviet bloc are notincluded in our data set (see Appendix Table A1 for a list ofincluded countries)



58 Fraction land area near navig water 0013 0019 0024 0031 0055 0092 014259 Square of inflation 1960ndash1990 0013 0018 0022 0027 0041 0063 010560 Fraction spent in war 1960ndash1990 0010 0016 0019 0024 0039 0060 008761 Land area 0010 0016 0022 0026 0043 0071 010362 Tropical climate zone 0012 0016 0020 0028 0042 0067 010063 Terms of trade ranking 0011 0016 0019 0026 0039 0063 008664 Capitalism 0010 0015 0020 0026 0047 0084 012865 Fraction Orthodox 0011 0015 0020 0025 0036 0059 008366 War participation 1960ndash1990 0010 0015 0019 0025 0040 0060 008967 Interior density 0010 0015 0019 0023 0039 0062 0085

Notes The left-hand-side variable in all regressions is the growth rate from 1960ndash1996 across 88 countries Each columncontains the posterior probability of all models including the given variable These are calculated with the same data but withdifferent prior mean model sizes as labeled in the column headings They are based on different random samples of allpossible regressions using the same convergence criterion for stopping sampling Samples range from around 63 millionregressions for k 5 to around 38 million for k 28


TABLE 4mdashPOSTERIOR MEANS CONDITIONAL ON INCLUSION WITH DIFFERENT PRIOR MODEL SIZES






with GDP prices0030646 0033647 0036617 0037569 0040057 0041811 0043023


1960rsquos0127413 0129517 0128488 0139824 0150386 0156571 0159089

49 Landlocked country dummy 0001509 0002080 0002576 0002740 0002913 0002855 000272550 Religion measure 0004409 0004737 0004924 0004759 0003540 0001831 000041451 Size of economy 0000733 0000520 0000323 0000161 0000014 0000109 000016652 Socialist dummy 0004140 0003983 0003976 0004071 0004318 0004591 000470653 English-speaking population 0004839 0003669 0002854 0002243 0001229 0000485 000031154 Average inflation 1960ndash1990 0000082 0000073 0000064 0000055 0000031 0000007 000000855 Oil-producing country dummy 0003559 0004845 0003855 0003995 0003127 0001992 000099556 Population growth rate 1960ndash1990 0059271 0020837 0021521 0023353 0035501 0036410 000114557 Timing of independence 0001258 0001143 0001034 0000963 0000630 0000360 000012158 Fraction land area near navig water 0002874 0002598 0002666 0002705 0002881 0003575 000368759 Square of inflation 1960ndash1990 0000001 0000001 0000001 0000001 0000000 0000000 000000060 Fraction spent in war 1960ndash1990 0002555 0001415 0001315 0000751 0000211 0000251 000080161 Land area 0000000 0000000 0000000 0000000 0000000 0000000 000000062 Tropical climate zone 0003258 0002069 0001395 0001280 0000922 0001455 000140363 Terms of trade ranking 0004179 0003730 0003079 0002481 0001011 0000251 000089464 Capitalism 0000090 0000231 0000323 0000384 0000564 0000743 000089265 Fraction Orthodox 0007559 0005689 0004019 0003189 0001995 0001679 000191466 War participation 1960ndash1990 0000746 0000734 0000810 0000874 0001009 0001144 000109167 Interior density 0000000 0000001 0000002 0000002 0000003 0000003 0000004


education some measures of higher educa-tion some geographical measures such as theabsolute latitude (the distance from the equa-tor) and various proxies for ldquoscale effectsrdquo suchas the total population aggregate GDP or thetotal area of a country

B Robustness of the Results

Up until now we have concentrated on resultsderived for a prior model size k 7 As dis-cussed earlier while we feel that this is a rea-sonable expected model size it is in some sensearbitrary We need to explore the effects of theprior on our conclusions Tables 3 4 and 5 doprecisely this reporting the posterior inclusionprobabilities the conditional posterior means andsign certainty probabilities for k equal to 5 9 1116 22 and 28 as well as repeating the benchmarknumbers for easy comparison Note that each khas a corresponding value of the prior probabilityof inclusion which is reported in the first row ofTable 3 Thus to see whether a variable improvesits probability of inclusion relative to the prior weneed to compare the posterior probability tothe corresponding prior probability The vari-ables that are important in the baseline case ofk 7 and are not important for other priormodel sizes are shown in italics in Table3 Variables that are not important for k 7but become important with other sizes areshown in boldface

ldquoSignificantrdquo Variables That Become ldquoWeakrdquomdashThe most significant variables show very littlesensitivity to the choice of prior model sizeeither in terms of their inclusion probabilities ortheir coefficient estimates Some of the impor-tant variables seem to improve substantiallywith the prior model size For example for thefraction of GDP in mining the posterior inclu-sion probability rises from 7 percent with k 5to 66 percent with k 22 This suggests thatmining is a variable that requires other condi-tioning variables in order to display its fullimportance20 Both the fraction of Confuciansand the Sub-Saharan Africa dummy are also

variables that appear to do better with moreconditioning variables and have stable coeffi-cient estimates

Although most of the significant variablesremain so five of them tend to lose significanceas we increase the prior model size That is forlarger models the posterior probability declinesto levels below the prior size These variablesare the index of malaria prevalence the formerSpanish colonies the number of years an econ-omy has been open the index of ethnolinguisticfractionalization and the government consump-tion share This suggests that these variablescould be acting as a ldquocatch-allrdquo for various othereffects For example the openness index cap-tures various aspects of the openness of a coun-try to trade (tariff and nontariff barriers blackmarket premium degree of Socialism and mo-nopolization of exports by the government) In-terestingly Table 5 shows the sign certaintyprobabilities of these five variables droppingbelow 095 for relatively large models (k 28)The other 13 variables that were significant inthe baseline model also appear to be robust todifferent prior specifications

ldquoInsignificantrdquo Variables That Become ldquoSig-nificantrdquomdashAt the other end of the scale mostof the 46 variables that showed little partialcorrelation in the baseline estimation are nothelped by alternative priors21 Their posteriorinclusion probabilities rise as k increases whichis hardly surprising as their prior inclusion prob-abilities are rising But their posterior inclusionprobabilities remain below the prior so we areforced to think of them as ldquoinsignificantrdquo

There are three variables that are insignificantin the baseline study but become ldquosignificantrdquowith some prior model sizes These are thepopulation density the fraction of populationthat speak a foreign language (a measure ofinternational social capital and openness) andthe public investment share As mentionedabove the public investment share is particu-larly interesting because it becomes strong forlarger prior model sizes but the sign of thecorrelation is negative That is a larger public

20 Notice that the fraction of GDP in mining is largelydriven by the success of Botswana Once other controlvariables such as a Sub-Saharan dummy are included theMining variable captures Botswanarsquos unusual performance

21 The exception is the public investment share whichhas a posterior inclusion probability greater than the prior inTable 3 and sign certainty probability exceeding 095 inTable 5 for relatively large models with k 16


TABLE 5mdashSIGN CERTAINTY PROBABILITIES WITH DIFFERENT PRIOR MODEL SIZES






GDP prices0870 0893 0912 0921 0933 0941 0943

34 Absolute latitude 0786 0737 0697 0628 0541 0520 057635 Fraction Catholic 0782 0837 0840 0851 0885 0891 089836 Fertility in 1960rsquos 0718 0767 0819 0855 0891 0909 090937 European dummy 0520 0544 0531 0560 0626 0686 074038 Outward orientation 0883 0886 0893 0894 0892 0895 089239 Colony dummy 0866 0858 0852 0853 0840 0821 080540 Civil liberties 0860 0846 0843 0829 0834 0843 084641 Revolutions and coups 0874 0877 0876 0895 0917 0931 093542 British colony 0871 0844 0827 0810 0781 0767 075943 Hydrocarbon deposits in 1993 0684 0773 0816 0844 0873 0893 090044 Fraction population over 65 0547 0566 0632 0677 0787 0847 086445 Defense spending share 0773 0737 0730 0697 0623 0568 054146 Population in 1960 0799 0806 0762 0809 0811 0795 078647 Terms of trade growth in 1960rsquos 0771 0752 0753 0730 0661 0564 052948 Public education spendingGDP in 1960rsquos 0768 0777 0779 0800 0818 0830 083649 Landlocked country dummy 0646 0701 0753 0761 0777 0776 076750 Religion measure 0728 0751 0758 0757 0690 0604 052951 Size of economy 0727 0661 0600 0559 0507 0517 052652 Socialist dummy 0788 0788 0788 0795 0810 0826 082953 English-speaking population 0745 0686 0646 0613 0570 0527 052154 Average inflation 1960ndash1990 0809 0784 0754 0720 0625 0529 053455 Oil-producing country dummy 0704 0751 0718 0726 0675 0610 055256 Population growth rate 1960ndash1990 0595 0533 0530 0532 0562 0573 052857 Timing of independence 0738 0716 0687 0679 0611 0559 051458 Fraction land area near navig water 0666 0657 0668 0675 0700 0748 0761


investment share tends to be associated withlower growth rates Our interpretation of theseresults is that our baseline results are very ro-bust to alternative prior size specifications

IV Conclusions

In this paper we propose a Bayesian Averag-ing of Classical Estimates (BACE) method todetermine what variables are significantly re-lated to growth in a broad cross section ofcountries The method introduces a number ofimprovements relative to the previous literatureFor example we use an averaging method thatis fully justified on Bayesian grounds and we donot restrict the number of regressors in the av-eraged models Our approach provides an alter-native to a standard Bayesian Model Averagingsince BACE does not require the specificationof the prior distribution of the parameters buthas only one hyper-parameter the expectedmodel size k This parameter is easy to inter-pret easy to specify and easy to check forrobustness The interpretation of the BACE es-timates is straightforward since the weights areanalogous to the Schwarz model selection cri-terion Finally our estimates can be calculatedusing only repeated applications of OLS whichmakes the approach transparent and straightfor-ward to implement

Our main results support Sala-i-Martin(1997a b) rather than Levine and Renelt(1992) we find that a good number of economicvariables have robust partial correlation withlong-run growth In fact we find that aboutone-fifth of the 67 variables used in the analysiscan be said to be significantly related to growthwhile several more are marginally related Thestrongest evidence is found for primary school-ing enrollment the relative price of investmentgoods and the initial level of income where thelatter reflects the concept of conditional con-vergence Other important variables includeregional dummies (such as East Asia Sub-Saharan Africa or Latin America) some mea-sures of human capital and health (such as lifeexpectancy proportion of a country lying in thetropics and malaria prevalence) religious dum-mies and some sectoral variables such as min-ing The public consumption and publicinvestment shares are negatively related togrowth although the results are significant onlyfor certain prior model sizes

Finally we show that our results are quiterobust to the choice of the prior model sizemost of the ldquosignificantrdquo variables in the base-line case remain significant for other priormodel sizes Nonlinear relationships betweenexplanatory variables and the dependent vari-able can be readily estimated within the BACEframework





REFERENCES

Barro Robert J ldquoEconomic Growth in a CrossSection of Countriesrdquo Quarterly Journal ofEconomics May 1991 106(2) pp 407ndash43

ldquoDeterminants of Democracyrdquo Jour-nal of Political Economy Pt 2 December1999 107(6) pp S158ndash83

Barro Robert J and Lee Jong-Wha ldquoInterna-tional Comparisons of Educational Attain-mentrdquo Journal of Monetary EconomicsDecember 1993 32(3) pp 363ndash94 [The datafor this paper were taken from the NBERWeb site httpwwwnberorgpubbarrolee]

Barro Robert J and Sala-i-Martin Xavier Eco-

nomic growth New York McGraw-Hill1995

Clyde Merlise Desimone Heather and Parmigi-ani Giovanni ldquoPrediction via OrthogonalizedModel Mixingrdquo Journal of the AmericanStatistical Association September 199691(435) pp 1197ndash208

Easterly William and Levine Ross ldquoAfricarsquosGrowth Tragedy Policies and Ethnic Divi-sionsrdquo Quarterly Journal of Economics No-vember 1997 112(4) pp 1203ndash50

Fernandez Carmen Ley Eduardo and SteelMark F J ldquoModel Uncertainty in Cross-Country Growth Regressionsrdquo Journal ofApplied Econometrics SeptemberndashOctober2001 16(5) pp 563ndash76

TABLE A1mdashLIST OF 88 COUNTRIES INCLUDED IN THE REGRESSIONS

Algeria Mexico NetherlandsBenin Panama NorwayBotswana Trinidad amp Tobago PortugalBurundi United States SpainCameroon Argentina SwedenCentral African Republic Bolivia TurkeyCongo Brazil United KingdomEgypt Chile AustraliaEthiopia Colombia FijiGabon Ecuador Papua New GuineaGambia ParaguayGhana PeruKenya UruguayLesotho VenezuelaLiberia Hong KongMadagascar IndiaMalawi IndonesiaMauritania IsraelMorocco JapanNiger JordanNigeria KoreaRwanda MalaysiaSenegal NepalSouth Africa PakistanTanzania PhilippinesTogo SingaporeTunisia Sri LankaUganda SyriaZaire TaiwanZambia ThailandZimbabwe AustriaCanada BelgiumCosta Rica DenmarkDominican Republic FinlandEl Salvador FranceGuatemala Germany WestHaiti GreeceHonduras IrelandJamaica Italy


Gallup John L Mellinger Andrew and SachsJeffrey D ldquoGeography Datasetsrdquo Center forInternational Development at Harvard Univer-sity (CID) 2001 Data Web site httpwww2cidharvardeduciddatageographydatahtm

George Edward I and McCulloch Robert EldquoVariable Selection via Gibbs SamplingrdquoJournal of the American Statistical Associ-ation September 1993 88(423) pp 881ndash89

Geweke John F ldquoBayesian Comparison ofEconometric Modelsrdquo Working Paper No532 Federal Reserve Bank of Minneapolis1994

Granger Clive W J and Uhlig Harald F ldquoRea-sonable Extreme-Bounds Analysisrdquo Journalof Econometrics AprilndashMay 1990 44(1ndash2)pp 159ndash70

Grier Kevin B and Tullock Gordon ldquoAn Em-pirical Analysis of Cross-National EconomicGrowthrdquo Journal of Monetary EconomicsSeptember 1989 24(2) pp 259ndash76

Hall Robert E and Jones Charles I ldquoWhy DoSome Countries Produce So Much More Out-put Per Worker Than Othersrdquo QuarterlyJournal of Economics February 1999114(1) pp 83ndash116 Data Web site httpemlabberkeleyeduuserschaddatasetshtml

Heston Alan Summers Robert and Aten Bet-tina Penn World table version 60 Center forInternational Comparisons at the Universityof Pennsylvania (CICUP) 2001 DecemberData Web site httppwteconupennedu

Hoeting Jennifer A Madigan David RafteryAdrian E and Volinsky Chris T ldquoBayesianModel Averaging A Tutorialrdquo StatisticalScience November 1999 14(4) pp 382ndash417

Jeffreys Harold Theory of probability 3rd EdLondon Oxford University Press 1961

Kormendi Roger C and Meguire Philip ldquoMac-roeconomic Determinants of Growth Cross-Country Evidencerdquo Journal of MonetaryEconomics September 1985 16(2) pp 141ndash63

Leamer Edward E Specification searches NewYork John Wiley amp Sons 1978

ldquoLetrsquos Take the Con Out of Economet-ricsrdquo American Economic Review March1983 73(1) pp 31ndash43

ldquoSensitivity Analysis Would HelprdquoAmerican Economic Review June 198575(3) pp 308ndash13

Levine Ross E and Renelt David ldquoA SensitivityAnalysis of Cross-Country Growth Regres-sionsrdquo American Economic Review Septem-ber 1992 82(4) pp 942ndash63

Madigan David and York Jeremy C ldquoBayesianGraphical Models for Discrete Datardquo Inter-national Statistical Review August 199563(2) pp 215ndash32

Poirier Dale J Intermediate statistics andeconometrics Cambridge MA MIT Press1995

Raftery Adrian E Madigan David and HoetingJennifer A ldquoBayesian Model Averaging forLinear Regression Modelsrdquo Journal of theAmerican Statistical Association March1997 92(437) pp 179ndash91

Sachs Jeffrey D and Warner Andrew M ldquoEco-nomic Reform and the Process of EconomicIntegrationrdquo Brookings Papers on EconomicActivity 1995 (1) pp 1ndash95

ldquoNatural Resource Abundance andEconomic Growthrdquo CID at Harvard Univer-sity 1997 Data Web site wwwcidharvardeduciddataciddatahtml

Sala-i-Martin Xavier ldquoI Just Ran 2 Million Re-gressionsrdquo American Economic Review May1997a (Papers and Proceedings) 87(2) pp178ndash83

ldquoI Just Ran Four Million RegressionsrdquoNational Bureau of Economic Research(Cambridge MA) Working Paper No 6252November 1997b

Schwarz Gideon ldquoEstimating the Dimension ofa Modelrdquo Annals of Statistics March 19786(2) pp 461ndash64

York Jeremy C Madigan David Heuch I Ivarand Lie Rolv Terje ldquoBirth Defects Registeredby Double Sampling A Bayesian ApproachIncorporating Covariates and Model Uncer-taintyrdquo Applied Statistics 1995 44(2) pp227ndash42

Zellner Arnold An introduction to Bayesianinference in econometrics New York JohnWiley amp Sons 1971

ldquoOn Assessing Prior Distributions andBayesian Regression Analysis with g-PriorDistributionsrdquo in Prem Goel and ArnoldZellner eds Bayesian inference and deci-sion techniques Essays in honor of Bruno deFinetti Studies in Bayesian Econometricsand Statistics series volume 6 AmsterdamNorth-Holland 1986 pp 233ndash43


this is strictly a small-sample problem since asthe number of observations becomes large allof the variables which do not belong in theregression will have coefficients that convergeto zero Classical statistics offers us little helpsince it suggests that we include all of the re-gressors and ldquolet the data sort through themrdquo Inmany applications however we do not have theluxury of having a large enough sample size toallow us to draw conclusions on the importanceof potential regressors Cross-country regres-sions provide perhaps the most extreme exam-ple the number of proposed regressors exceedsthe number of countries in the world renderingthe all-inclusive regression computationally im-possible Some empirical economists havetherefore resorted to simply ldquotryingrdquo combina-tions of variables which could be potentiallyimportant determinants of growth and report theresults of their preferred specification Suchldquodata-miningrdquo could lead to spurious inferenceSo how should we proceed

A natural way to think about model uncer-tainty is to admit that we do not know whichmodel is ldquotruerdquo and instead attach probabilitiesto different possible models While intuitivelyappealing this requires a departure from theclassical framework in which conditioning on amodel is essential This approach has recentlycome to be known as Bayesian Model Averag-ing The procedure does not differ from themost basic Bayesian reasoning the idea dates atleast to Harold Jeffreys (1961) although fleshedout by Edward E Leamer (1978) In this paperwe show that this approach can be used in a waythat is well grounded in statistical theory intu-itively appealing easy to understand and easyto implement

The fully Bayesian approach is entirely fea-sible and has been applied to various problemsby a number of authors Examples include Jer-emy C York et al (1995) and Adrian E Rafteryet al (1997) a summary of much of the recentwork can be found in Jennifer A Hoeting et al(1999) In the growth context Carmen Fernan-dez et al (2001) apply techniques from theBayesian statistics literature to the data set ofSala-i-Martin (1997a) A pure Bayesian ap-proach requires specification of the prior distri-butions of all of the relevant parametersconditional on each possible model Under idealconditions elicitation of prior parameters is dif-

ficult and is indeed one of the major reasons forBayesian approaches remaining relatively un-popular When the number of possible regres-sors is K the number of possible linear modelsis 2K so with K large fully specifying priors isinfeasible Thus authors implementing the fullyBayesian approach have used priors which areessentially arbitrary This makes the ultimateestimates dependent on arbitrarily chosen priorparameters in a manner which is extremely dif-ficult to interpret In existing applications of thisapproach the impact of these prior parametershas been neither examined nor explained

To address the issue of fragility of economet-ric inference with respect to modeling choicesLeamer (1983 1985) proposes a sensitivityanalysis of the results with respect to changes inthe prior distribution of parameters In particu-lar Leamer proposes an extreme bounds anal-ysis to identify ldquorobustrdquo empirical relations Fora variable of interest z the extreme bounds ofthe distribution of the associated coefficient es-timate z are calculated as the smallest andlargest value that the coefficient can take onwhen combinations of additional regressors xjenter the regression model (1) When the twoextreme bounds are of opposite sign then vari-able z is labeled ldquofragilerdquo Ross E Levine andDavid Renelt (1992) employ a version of thistest to cross-country data and find that accord-ing to extreme bounds analysis very few (or no)variables are robustly related with growth An-other interpretation however is that the test istoo strong for any variable to pass any oneregression model (no matter how well or poorlyfitting) carries a veto This problem is wellrecognized and solutions have been proposed torestrict attention to better fitting models such asthe reasonable extreme bounds proposed byClive W J Granger and Harald F Uhlig (1990)or to introduce a lower bound on the priorvariance matrix as suggested by Leamer (1985)

In this paper we use the Bayesian approach toaveraging across models while following theclassical spirit of most of the empirical growthliterature1 We propose a model-averaging tech-

1 We build on the work by Sala-i-Martin (1997a b) whoproposed to average estimates of mean and standard devi-ations for variables across regressions using weights pro-portional to the likelihoods of each of the models Sala-i-Martin calculates a likelihood-weighted sum of normal










(2) gy fyg

fy







(3) gy PM0yfygM0

fyM0

PM1yfygM1

fyM1


B Diffuse Priors




(4)PM0y

PM1y

PM0

PM1

AA XXBB ZZ

12SSE0 Q0

SSE1 Q1T2






(5)PM0y

PM1y

PM0

PM1 SSE0

SSE1T2



(6)PM0y

PM1y

PM0

PM1 Tk1 k0 2SSE0

SSE1T2





T2

i 1

2K

PMi Tki 2SSEi T2


(8) Ey j 1

2K

PMjyj


(9) Vary j 1

2K

PMjyVary Mj

j 1

2K

PMjyj Ey2







C Model Size





D Sampling











II Data







SIZES kj








00182 0019


Wha Lee (1993)07261 02932


924659 536778





1468717 5098276




Jones (1999)00507 00769






03476 03016



01161 00745





03209 04136



05231 03359








07199 02827


00522 00388




03925 00749


43241705 26137627



0149 00584





III Results





(1993)00488 0029


00259 00246





00244 00096





(1992)131298 149899





00215 00095


10114 09767



04722 03802





00695 01524


1960 and 1990 Barro and Lee (1993)03977 04922


433709 880626




(10) ti i i

sXX1ii12 tT k


















2 cond2 cond

2





Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)





with GDP prices0036 0033647 0027365 0893 0565 0893 005


1960rsquos0021 0129517 0172847 0777 0322 0777 011









Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)


0019 0002598 0005864 0657 0730 0657 035

































GDP prices0021 0036 0056 0075 0137 0225 0310




















with GDP prices0030646 0033647 0036617 0037569 0040057 0041811 0043023


1960rsquos0127413 0129517 0128488 0139824 0150386 0156571 0159089




















GDP prices0870 0893 0912 0921 0933 0941 0943




IV Conclusions








REFERENCES














































(2) gy fyg

fy







(3) gy PM0yfygM0

fyM0

PM1yfygM1

fyM1


B Diffuse Priors




(4)PM0y

PM1y

PM0

PM1

AA XXBB ZZ

12SSE0 Q0

SSE1 Q1T2






(5)PM0y

PM1y

PM0

PM1 SSE0

SSE1T2



(6)PM0y

PM1y

PM0

PM1 Tk1 k0 2SSE0

SSE1T2





T2

i 1

2K

PMi Tki 2SSEi T2


(8) Ey j 1

2K

PMjyj


(9) Vary j 1

2K

PMjyVary Mj

j 1

2K

PMjyj Ey2







C Model Size





D Sampling











II Data







SIZES kj








00182 0019


Wha Lee (1993)07261 02932


924659 536778





1468717 5098276




Jones (1999)00507 00769






03476 03016



01161 00745





03209 04136



05231 03359








07199 02827


00522 00388




03925 00749


43241705 26137627



0149 00584





III Results





(1993)00488 0029


00259 00246





00244 00096





(1992)131298 149899





00215 00095


10114 09767



04722 03802





00695 01524


1960 and 1990 Barro and Lee (1993)03977 04922


433709 880626




(10) ti i i

sXX1ii12 tT k


















2 cond2 cond

2





Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)





with GDP prices0036 0033647 0027365 0893 0565 0893 005


1960rsquos0021 0129517 0172847 0777 0322 0777 011









Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)


0019 0002598 0005864 0657 0730 0657 035

































GDP prices0021 0036 0056 0075 0137 0225 0310




















with GDP prices0030646 0033647 0036617 0037569 0040057 0041811 0043023


1960rsquos0127413 0129517 0128488 0139824 0150386 0156571 0159089




















GDP prices0870 0893 0912 0921 0933 0941 0943




IV Conclusions








REFERENCES







































(3) gy PM0yfygM0

fyM0

PM1yfygM1

fyM1


B Diffuse Priors




(4)PM0y

PM1y

PM0

PM1

AA XXBB ZZ

12SSE0 Q0

SSE1 Q1T2






(5)PM0y

PM1y

PM0

PM1 SSE0

SSE1T2



(6)PM0y

PM1y

PM0

PM1 Tk1 k0 2SSE0

SSE1T2





T2

i 1

2K

PMi Tki 2SSEi T2


(8) Ey j 1

2K

PMjyj


(9) Vary j 1

2K

PMjyVary Mj

j 1

2K

PMjyj Ey2







C Model Size





D Sampling











II Data







SIZES kj








00182 0019


Wha Lee (1993)07261 02932


924659 536778





1468717 5098276




Jones (1999)00507 00769






03476 03016



01161 00745





03209 04136



05231 03359








07199 02827


00522 00388




03925 00749


43241705 26137627



0149 00584





III Results





(1993)00488 0029


00259 00246





00244 00096





(1992)131298 149899





00215 00095


10114 09767



04722 03802





00695 01524


1960 and 1990 Barro and Lee (1993)03977 04922


433709 880626




(10) ti i i

sXX1ii12 tT k


















2 cond2 cond

2





Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)





with GDP prices0036 0033647 0027365 0893 0565 0893 005


1960rsquos0021 0129517 0172847 0777 0322 0777 011









Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)


0019 0002598 0005864 0657 0730 0657 035

































GDP prices0021 0036 0056 0075 0137 0225 0310




















with GDP prices0030646 0033647 0036617 0037569 0040057 0041811 0043023


1960rsquos0127413 0129517 0128488 0139824 0150386 0156571 0159089




















GDP prices0870 0893 0912 0921 0933 0941 0943




IV Conclusions








REFERENCES







































(5)PM0y

PM1y

PM0

PM1 SSE0

SSE1T2



(6)PM0y

PM1y

PM0

PM1 Tk1 k0 2SSE0

SSE1T2





T2

i 1

2K

PMi Tki 2SSEi T2


(8) Ey j 1

2K

PMjyj


(9) Vary j 1

2K

PMjyVary Mj

j 1

2K

PMjyj Ey2







C Model Size





D Sampling











II Data







SIZES kj








00182 0019


Wha Lee (1993)07261 02932


924659 536778





1468717 5098276




Jones (1999)00507 00769






03476 03016



01161 00745





03209 04136



05231 03359








07199 02827


00522 00388




03925 00749


43241705 26137627



0149 00584





III Results





(1993)00488 0029


00259 00246





00244 00096





(1992)131298 149899





00215 00095


10114 09767



04722 03802





00695 01524


1960 and 1990 Barro and Lee (1993)03977 04922


433709 880626




(10) ti i i

sXX1ii12 tT k


















2 cond2 cond

2





Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)





with GDP prices0036 0033647 0027365 0893 0565 0893 005


1960rsquos0021 0129517 0172847 0777 0322 0777 011









Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)


0019 0002598 0005864 0657 0730 0657 035

































GDP prices0021 0036 0056 0075 0137 0225 0310




















with GDP prices0030646 0033647 0036617 0037569 0040057 0041811 0043023


1960rsquos0127413 0129517 0128488 0139824 0150386 0156571 0159089




















GDP prices0870 0893 0912 0921 0933 0941 0943




IV Conclusions








REFERENCES








































C Model Size





D Sampling











II Data







SIZES kj








00182 0019


Wha Lee (1993)07261 02932


924659 536778





1468717 5098276




Jones (1999)00507 00769






03476 03016



01161 00745





03209 04136



05231 03359








07199 02827


00522 00388




03925 00749


43241705 26137627



0149 00584





III Results





(1993)00488 0029


00259 00246





00244 00096





(1992)131298 149899





00215 00095


10114 09767



04722 03802





00695 01524


1960 and 1990 Barro and Lee (1993)03977 04922


433709 880626




(10) ti i i

sXX1ii12 tT k


















2 cond2 cond

2





Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)





with GDP prices0036 0033647 0027365 0893 0565 0893 005


1960rsquos0021 0129517 0172847 0777 0322 0777 011









Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)


0019 0002598 0005864 0657 0730 0657 035

































GDP prices0021 0036 0056 0075 0137 0225 0310




















with GDP prices0030646 0033647 0036617 0037569 0040057 0041811 0043023


1960rsquos0127413 0129517 0128488 0139824 0150386 0156571 0159089




















GDP prices0870 0893 0912 0921 0933 0941 0943




IV Conclusions








REFERENCES










































II Data







SIZES kj








00182 0019


Wha Lee (1993)07261 02932


924659 536778





1468717 5098276




Jones (1999)00507 00769






03476 03016



01161 00745





03209 04136



05231 03359








07199 02827


00522 00388




03925 00749


43241705 26137627



0149 00584





III Results





(1993)00488 0029


00259 00246





00244 00096





(1992)131298 149899





00215 00095


10114 09767



04722 03802





00695 01524


1960 and 1990 Barro and Lee (1993)03977 04922


433709 880626




(10) ti i i

sXX1ii12 tT k


















2 cond2 cond

2





Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)





with GDP prices0036 0033647 0027365 0893 0565 0893 005


1960rsquos0021 0129517 0172847 0777 0322 0777 011









Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)


0019 0002598 0005864 0657 0730 0657 035

































GDP prices0021 0036 0056 0075 0137 0225 0310




















with GDP prices0030646 0033647 0036617 0037569 0040057 0041811 0043023


1960rsquos0127413 0129517 0128488 0139824 0150386 0156571 0159089




















GDP prices0870 0893 0912 0921 0933 0941 0943




IV Conclusions








REFERENCES










































00182 0019


Wha Lee (1993)07261 02932


924659 536778





1468717 5098276




Jones (1999)00507 00769






03476 03016



01161 00745





03209 04136



05231 03359








07199 02827


00522 00388




03925 00749


43241705 26137627



0149 00584





III Results





(1993)00488 0029


00259 00246





00244 00096





(1992)131298 149899





00215 00095


10114 09767



04722 03802





00695 01524


1960 and 1990 Barro and Lee (1993)03977 04922


433709 880626




(10) ti i i

sXX1ii12 tT k


















2 cond2 cond

2





Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)





with GDP prices0036 0033647 0027365 0893 0565 0893 005


1960rsquos0021 0129517 0172847 0777 0322 0777 011









Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)


0019 0002598 0005864 0657 0730 0657 035

































GDP prices0021 0036 0056 0075 0137 0225 0310




















with GDP prices0030646 0033647 0036617 0037569 0040057 0041811 0043023


1960rsquos0127413 0129517 0128488 0139824 0150386 0156571 0159089




















GDP prices0870 0893 0912 0921 0933 0941 0943




IV Conclusions








REFERENCES








































III Results





(1993)00488 0029


00259 00246





00244 00096





(1992)131298 149899





00215 00095


10114 09767



04722 03802





00695 01524


1960 and 1990 Barro and Lee (1993)03977 04922


433709 880626




(10) ti i i

sXX1ii12 tT k


















2 cond2 cond

2





Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)





with GDP prices0036 0033647 0027365 0893 0565 0893 005


1960rsquos0021 0129517 0172847 0777 0322 0777 011









Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)


0019 0002598 0005864 0657 0730 0657 035

































GDP prices0021 0036 0056 0075 0137 0225 0310




















with GDP prices0030646 0033647 0036617 0037569 0040057 0041811 0043023


1960rsquos0127413 0129517 0128488 0139824 0150386 0156571 0159089




















GDP prices0870 0893 0912 0921 0933 0941 0943




IV Conclusions








REFERENCES








































(10) ti i i

sXX1ii12 tT k


















2 cond2 cond

2





Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)





with GDP prices0036 0033647 0027365 0893 0565 0893 005


1960rsquos0021 0129517 0172847 0777 0322 0777 011









Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)


0019 0002598 0005864 0657 0730 0657 035

































GDP prices0021 0036 0056 0075 0137 0225 0310




















with GDP prices0030646 0033647 0036617 0037569 0040057 0041811 0043023


1960rsquos0127413 0129517 0128488 0139824 0150386 0156571 0159089




















GDP prices0870 0893 0912 0921 0933 0941 0943




IV Conclusions








REFERENCES

















































2 cond2 cond

2





Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)





with GDP prices0036 0033647 0027365 0893 0565 0893 005


1960rsquos0021 0129517 0172847 0777 0322 0777 011









Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)


0019 0002598 0005864 0657 0730 0657 035

































GDP prices0021 0036 0056 0075 0137 0225 0310




















with GDP prices0030646 0033647 0036617 0037569 0040057 0041811 0043023


1960rsquos0127413 0129517 0128488 0139824 0150386 0156571 0159089




















GDP prices0870 0893 0912 0921 0933 0941 0943




IV Conclusions








REFERENCES







































Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)





with GDP prices0036 0033647 0027365 0893 0565 0893 005


1960rsquos0021 0129517 0172847 0777 0322 0777 011









Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)


0019 0002598 0005864 0657 0730 0657 035

































GDP prices0021 0036 0056 0075 0137 0225 0310




















with GDP prices0030646 0033647 0036617 0037569 0040057 0041811 0043023


1960rsquos0127413 0129517 0128488 0139824 0150386 0156571 0159089




















GDP prices0870 0893 0912 0921 0933 0941 0943




IV Conclusions








REFERENCES












































Rank Variable

Posteriorinclusion

probability(1)


inclusion(2)


(3)

BACEsign


(4)

OLSp-value

(5)

OLS signcertainty

probability(6)


with tstat 2(7)


0019 0002598 0005864 0657 0730 0657 035

































GDP prices0021 0036 0056 0075 0137 0225 0310




















with GDP prices0030646 0033647 0036617 0037569 0040057 0041811 0043023


1960rsquos0127413 0129517 0128488 0139824 0150386 0156571 0159089




















GDP prices0870 0893 0912 0921 0933 0941 0943




IV Conclusions








REFERENCES



































































GDP prices0021 0036 0056 0075 0137 0225 0310




















with GDP prices0030646 0033647 0036617 0037569 0040057 0041811 0043023


1960rsquos0127413 0129517 0128488 0139824 0150386 0156571 0159089




















GDP prices0870 0893 0912 0921 0933 0941 0943




IV Conclusions








REFERENCES























































with GDP prices0030646 0033647 0036617 0037569 0040057 0041811 0043023


1960rsquos0127413 0129517 0128488 0139824 0150386 0156571 0159089




















GDP prices0870 0893 0912 0921 0933 0941 0943




IV Conclusions








REFERENCES























































GDP prices0870 0893 0912 0921 0933 0941 0943




IV Conclusions








REFERENCES







































IV Conclusions








REFERENCES






































REFERENCES






































classical estimates (bace) approach determinants of long-term growth: a bayesian ... · 2015. 6....

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