COMPARING THEORIES OF ENDOGENOUS PROTECTION: BAYESIAN COMPARISON OF TOBIT MODELS USING GIBBS SAMPLING OUTPUT Kishore Gawande* Abstract —Bayesian inference and model comparisons are easily per- formed quite accurately using Gibbs sampling, even if (1) the likelihood is analytically intractable and (2) nonstandard prior probability density functions (pdfs) are required. In this study Bayesian model comparisons are performed among five competing theories of endogenous protection. Tariff and nontariff barrier data from 1983 between the United States and five OECD partner countries—Japan, France, Germany, Italy, and the United Kingdom—are used in the analysis. Posterior odds based on two priors show special-interest models to be more likely than other models in determining U.S. protection. I. Introduction T HE OBJECTIVE of this paper is twofold. The first is to perform an econometric model comparison to assess the validity of five political-economic theories of protection. In order to do such a comparison it is important to employ the correct methodology. A Bayesian model comparison is the relevant methodology in this context. The second objective of this paper is to perform Bayesian comparisons of tobit models using a recently developed powerful and flexible computational methodology, namely, the Gibbs sampler with data augmentation. The estimations in this paper are an application to the tobit model of the theoretical methods developed in Tanner and Wong (1987), Gelfand and Smith (1990), and Chib (1995). The econometric assessment of the validity of political- economic theories of protection has generally involved the examination of the size and sign of coefficients on a list of variables that represent each theory. Based on such an analysis, previous studies have found support for a range of theories. Pincus (1975) finds support for the Olson–Stigler theory of special-interest-group behavior; Caves (1976) finds support for the adding-machine model, which empha- sizes political influence through voting strength; Ray (1981) finds evidence in support of these models of political self-interest, but draws different inferences about compara- tive advantage. The influential study by Baldwin (1985) finds support for both these and other theories that empha- size political altruism, such as the status-quo model based on Corden’s (1974) conservative social utility function. More recently, using U.S. nontariff barrier coverage ratios during the 1980s, Trefler (1993) shows comparative advantage and political self-interest to be important, and Gawande (1995, 1996, 1997) finds the presence of a retaliatory component in U.S. protection, support for the Olson–Stigler special- interest model, and support for Corden’s model of the public interest. It may be argued, and it is probably true, that each theory of political economy has some truth to it in the sense that at least one case may be found to support that theory. Even in econometric studies the strict hypothesis that ‘‘theory i is not valid’’is usually not difficult to reject. What is interesting, but has not been investigated, is the impor- tance of one political economy model relative to another. This paper seeks to fill that void in the literature using a carefully constructed nontariff barrier (NTB) data set for the United States that spans 4-digit SIC industries and five developed partner countries from 1983. Bilateral U.S. NTBs on imports of 4-digit industries from Japan and a bloc of four European Community (EC) countries (France, Germany, Italy, and the United Kingdom) are analyzed separately. In the area of empirical political economy, a wide range of prior beliefs about the models are entertained. Since the link between the formal theoretical model and its empirical representation in an econometric specification by a set of variables is usually fragile, prior beliefs are an important determinant of the choice of the set of variables used to represent a model. Hence prior belief plays an understated but important role in making inferences. However, it is quite infrequently the case that the results reported in studies explicitly incorporate these prior beliefs, with the result that the reader cannot gauge the level of uncertainty about the inferences other than from the reported asymptotic standard errors. The Bayesian methodology allows the formal incor- poration of prior information, and is therefore the relevant tool here. In Bayesian inference the data likelihood is combined with the prior information to compute posterior parameter densities which provide the basis for inference. However, the analysis of posterior densities has been hampered by two shortcomings. First, if the likelihood function is analytically intractable, Bayesian analysis must resort to computational methods. There is a growing consensus, given the availabil- ity of fast computing, in favor of the use of Gibbs sampling. Second, little flexibility could be allowed in the specification of prior information. As demonstrated here by using an inequality-restricted prior probability density function (pdf), the specification of prior information need no longer be restricted to the straight jacket of ‘‘conjugate’’prior pdfs. The main result of the paper is a pairwise comparison of five political economy models using posterior odds ratios where prior beliefs about the relative importance of two models are formally incorporated into the analysis. The analysis is based on three priors about the model parameters. The first is a diffuse or uninformative prior, which represents the belief of those who are uncommitted or indifferent between models. Estimates based on that prior are domi- nated by the data, so they are usually close to maximum- likelihood estimators. The second prior takes the normal- gamma form.Were the model a linear one, this ‘‘conjugate’’ prior would result in a posterior pdf of the same form with Received for publication January 9, 1996. Revision accepted for pub- lication October 30, 1996. * University of New Mexico. I am grateful to the anonymous referees, for their useful suggestions which have improved the paper considerably. 3 128 4 r 1998 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology
Transcript
COMPARING THEORIES OF ENDOGENOUS PROTECTION: BAYESIANCOMPARISON OF TOBIT MODELS USING GIBBS SAMPLING OUTPUT
Kishore Gawande*
Abstract—Bayesian inference and model comparisons are easily per-formed quite accurately using Gibbs sampling, even if (1) the likelihood isanalytically intractable and (2) nonstandard prior probability densityfunctions (pdfs) are required. In this study Bayesian model comparisonsare performed among five competing theories of endogenous protection.Tariff and nontariff barrier data from 1983 between the United States andfive OECD partner countries—Japan, France, Germany, Italy, and theUnited Kingdom—are used in the analysis. Posterior odds based on twopriors show special-interest models to be more likely than other models indetermining U.S. protection.
I. Introduction
THE OBJECTIVE of this paper is twofold. The first is toperform an econometric model comparison to assess the
validity of five political-economic theories of protection. Inorder to do such a comparison it is important to employ thecorrect methodology. A Bayesian model comparison is therelevant methodology in this context. The second objectiveof this paper is to perform Bayesian comparisons of tobitmodels using a recently developed powerful and flexiblecomputational methodology, namely, the Gibbs samplerwith data augmentation. The estimations in this paper are anapplication to the tobit model of the theoretical methodsdeveloped in Tanner and Wong (1987), Gelfand and Smith(1990), and Chib (1995).
The econometric assessment of the validity of political-economic theories of protection has generally involved theexamination of the size and sign of coefficients on a list ofvariables that represent each theory. Based on such ananalysis, previous studies have found support for a range oftheories. Pincus (1975) finds support for the Olson–Stiglertheory of special-interest-group behavior; Caves (1976)finds support for the adding-machine model, which empha-sizes political influence through voting strength; Ray (1981)finds evidence in support of these models of politicalself-interest, but draws different inferences about compara-tive advantage. The influential study by Baldwin (1985)finds support for both these and other theories that empha-size political altruism, such as the status-quo model based onCorden’s (1974) conservative social utility function. Morerecently, using U.S. nontariff barrier coverage ratios duringthe 1980s, Trefler (1993) shows comparative advantage andpolitical self-interest to be important, and Gawande (1995,1996, 1997) finds the presence of a retaliatory component inU.S. protection, support for the Olson–Stigler special-interest model, and support for Corden’s model of the publicinterest. It may be argued, and it is probably true, that eachtheory of political economy has some truth to it in the sense
that at least one case may be found to support that theory.Even in econometric studies the strict hypothesis that‘‘theory i is not valid’’ is usually not difficult to reject. Whatis interesting, but has not been investigated, is the impor-tance of one political economy model relative to another.This paper seeks to fill that void in the literature using acarefully constructed nontariff barrier (NTB) data set for theUnited States that spans 4-digit SIC industries and fivedeveloped partner countries from 1983. Bilateral U.S. NTBson imports of 4-digit industries from Japan and a bloc of fourEuropean Community (EC) countries (France, Germany,Italy, and the United Kingdom) are analyzed separately.
In the area of empirical political economy, a wide range ofprior beliefs about the models are entertained. Since the linkbetween the formal theoretical model and its empiricalrepresentation in an econometric specification by a set ofvariables is usually fragile, prior beliefs are an importantdeterminant of the choice of the set of variables used torepresent a model. Hence prior belief plays an understatedbut important role in making inferences. However, it is quiteinfrequently the case that the results reported in studiesexplicitly incorporate these prior beliefs, with the result thatthe reader cannot gauge the level of uncertainty about theinferences other than from the reported asymptotic standarderrors. The Bayesian methodology allows the formal incor-poration of prior information, and is therefore the relevanttool here.
In Bayesian inference the data likelihood is combinedwith the prior information to compute posterior parameterdensities which provide the basis for inference. However,the analysis of posterior densities has been hampered by twoshortcomings. First, if the likelihood function is analyticallyintractable, Bayesian analysis must resort to computationalmethods. There is a growing consensus, given the availabil-ity of fast computing, in favor of the use of Gibbs sampling.Second, little flexibility could be allowed in the specificationof prior information. As demonstrated here by using aninequality-restricted prior probability density function (pdf),the specification of prior information need no longer berestricted to the straight jacket of ‘‘conjugate’’ prior pdfs.
The main result of the paper is a pairwise comparison offive political economy models using posterior odds ratioswhere prior beliefs about the relative importance of twomodels are formally incorporated into the analysis. Theanalysis is based on three priors about the model parameters.The first is a diffuse or uninformative prior, which representsthe belief of those who are uncommitted or indifferentbetween models. Estimates based on that prior are domi-nated by the data, so they are usually close to maximum-likelihood estimators. The second prior takes the normal-gamma form. Were the model a linear one, this ‘‘conjugate’’prior would result in a posterior pdf of the same form with
Received for publication January 9, 1996. Revision accepted for pub-lication October 30, 1996.
* University of New Mexico.I am grateful to the anonymous referees, for their useful suggestions
which have improved the paper considerably.
3 128 4 r 1998 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology
nice analytic properties. The third prior is of the normal-gamma form, but imposes inequality restrictions on a subsetof parameters. This nonstandard prior is the best representa-tion of my prior beliefs, and its use demonstrates theflexibility allowed by the Gibbs sampling method.
Model comparisons using posterior odds show somemodels to be far more likely than others. A strong conclusionto emerge from the study is that the models of politicalself-interest—the Olson–Stigler model of special-interest-group behavior and Caves’ adding-machine model of votingstrength—are far more influential in determining U.S. NTBprotection against its OECD partners than models of publicinterest and political altruism. Notably, however, inferencesabout the models differ across instruments of protection andin the way NTB protection data are generated acrosspartners.
The paper proceeds as follows. In section II the methodol-ogy of the Gibbs sampler, especially as it is applied to thecomputations of Bayes estimates and marginal likelihoods inthe tobit model, is described. In section III alternativepolitical economy models are described and the estimatingequation is motivated. In section IV three sets of priors areenumerated and Bayes estimates (from tobit models) usingthe Gibbs sampler are presented for each set of priors. Insection V marginal likelihoods are computed from the Gibbsoutput and posterior odds constructed for the comparison ofalternative political-economic models. Section VI con-cludes.
II. Methodology: Applying the Gibbs Sampler in theTobit Model
The Gibbs sampler, described in the seminal paper byGelfand and Smith (1990), has proved an able vehicle tocarry the computational load required in Bayesian studies.Many computational methods have been advanced, buteither they are too specialized to the problem analyzed, andso are not widely applicable, or they suffer from the need tointegrate numerically in high dimensions, which is not oftenpossible to do accurately enough. While numerical MonteCarlo integration techniques have been shown to be feasible,they have been superseded by Gibbs sampling as the methodof choice due to its conceptual and computational simplicity.Tanner (1993) provides a nice intuitive introduction to theGibbs sampler with several illuminating examples. The dataaugmentation method of Tanner and Wong (1987) hasopened the way for Bayesian inference in a wide variety ofnonlinear econometric models currently in use, especiallymodels with censoring and truncation. In order to describethe application of this idea to the tobit model we follow theexposition in Chib and Greenberg (1996). Consider the tobitmodel onn identically and independently distributed (iid)observations, where observationyi is generated by
yi* , N(xi8b, s2), yi 5 max (0,yi
* ), 1 # i # n (1)
whereN denotes normality,xi8(1 3 k) are fixed explanatoryvariables, andb and s2 are parameters whosemarginaldistributions are the issue of interest. LetC be the set ofcensored observations andC8 the uncensored set. Then thelikelihood function forb ands2 is
pi[C
31 2 F1xi8b
s 24 pi[C8
(s21) exp 321
2s2(yi 2 xi8b)24 (2)
whereF is the cumulative distribution function (cdf) of thestandard normal random variable. Suppose prior informationis incorporated in the prior densities
b 0s2 , N(b0, s2B021)
s2 , IG1g0
2,g0s0
2
2 2(3)
whereN denotes normality,IG denotes the inverse gammadensity, and the parameters of the prior densities, sub-scripted by 0, are presumed known. The Gibbs samplingmethodology (described below) requires the generation of asample from the conditional posterior density. However, thetobit likelihood function after multiplying by the priordensity is difficult to simplify into tractable conditionaldensities for generating the Gibbs sample. The ‘‘dataaugmentation’’ idea of Tanner and Wong (1987) has beenusefully employed by Chib (1992) to get around thecensoring problem within the Gibbs sampling framework.1
The idea is to augment theparameterspace by the latentdata corresponding to the censored observations. Supposewe have available the vectoryC* 5 5yi* 6, i [ C. Definey*:(n 3 1) as follows:
y* 5 5yi* if i [ C
yi, if i [ C8.
Then the Gibbs sampling algorithm described below can beapplied to thethreeblocksb, s2, andyC* with the respectiveconditional densitiesf [b 0 y, yC* , s2], f [s2 0 y, yC* , b], andf [ yC* 0 y, b, s2]. The first two of these conditional densitiesare simple extensions of their linear-model versions:2
f [b 0 y, yC* , s2] 5 N(b̂, (s22B0 1 s22X8X)21),
where
b̂ 5 (B0 1 X8X)21(B0b0 1 X8y*)
1 For an application of data augmentation in the tobit model, but using theoutput from the expectation–maximization algorithm, see Gawande (1995).
2 See, for example, Zellner (1971) or Leamer (1978) for the analytic formof the conditional densities when these ‘‘conjugate’’ priors are combinedwith the data likelihood from a linear model.
129BAYESIAN COMPARISON OF TOBIT MODELS USING GIBBS SAMPLING OUTPUT
and
f [s2 0 y, yC* , b] 5 IG1g0 1 n
2,g0s0
2 1 d
2 2 (4)
where d 5 ( y* 2 Xb)8( y* 2 Xb), while the conditionaldensity of the latent data is the product of independentdistributions,
pi[C
f [ yi* 0 yi 5 0, b, s2].
Here
f [ yi* 0 yi 5 0, b, s2] 5 TN(2`,0](xi
8b, s2), i [ C (5)
whereTN(2`,0] denotes the truncated normal density withsupport (2`, 0]. The truncated normal distribution is compu-tationally tractable, and it is easy to simulate draws from it.The Gibbs algorithm for the tobit model can be simplydescribed as follows:
1. Specify starting values, say maximum-likelihood esti-mates, (b(0), s2(0)) and seti to 0.
2. SimulateyC* (i11) from f [ yC* 0 y, b, s2].Simulates2(i11) from f [s2 0 y, yC* (i11), b(i )].Simulateb(i11) from f [b 0 y, yC* (i11), s2(i11)].
3. Seti 5 i 1 1 and go to step 2.
A Markov chain 5b(0), s2(0)6, 5b(1), s2(1), yC* (1)6, 5b(2), s2(2),yC* (2)6, . . . is thus produced, where each subsequent item ofthe chain is simulated using the full conditional densities,with the conditioning elements revised during a full itera-tion. After discarding the firstl realizations of the sequence(where l varies depending on the application), the nextMrealizations can be used for computing the characteristics ofthe marginal distribution ofb ands2, or even approximatingthe densities themselves.
Our objective is to compare two modelsM1 and M2
explaining the same left-hand-side variable but using differ-ent sets of regressorsX1 andX2. Why the Gibbs output is souseful in enabling tobit model comparisons will be seenbelow. For comparing models, we first need to compute themarginal density ofy for each model,
f(y0Mi ) 5 ee f(y0b, s2, Mi ) f(b, s2 0Mi ) db ds2 (6)
wheref ( y0b, s2, Mi), f (b, s2 0Mi) are, respectively, the datalikelihood and the prior density for modelsMi, i 5 1, 2. Theintegration in equation (6) produces a weighted average ofthe data density where the prior density provides theweights. However, the integration in equation (6) is analyti-
cally impossible for a large class of priors and likelihoods,certainly for the tobit models analyzed here. It is alsocomputationally nontrivial in general. Chib (1995) providesa simple way of computing the marginal density as aby-product of the Gibbs sampling output. Chib’s idea is torewrite Bayes’ formula for the posterior parameter density(suppressing the index for modeli )
f(b, s 0 y) 5f(y0b, s2) f(b, s2)
f(y)
as
f(y) 5f(y0b, s2) f(b, s2)
f(b, s2 0 y). (7)
Equation (7) is an identity in (b, s2) and so may beevaluated at any point. Following Chib (1995) a high-density point such as the posterior mean in used. Noting thatthe posterior pdf in the denominator can be written asf (s2 0 y) 3 f (b 0 y, s2), the Gibbs sampling output can be usedto estimate the logarithm of this marginal density as
The numerator in equation (7) is computed from the firstthree terms on the right-hand side of equation (8) evaluatedat the posterior parameter meansb* and s2* from the MGibbs realizations. The denominator in equation (7) isevaluated as themean pdfover theM Gibbs realizations.Since we are sampling over three vector blocks (b, s2, y*),the evaluation of the denominator is done in two steps. Thefirst step consists of generatingM Gibbs realizations fromthe full conditional pdfs and computing the posterior ordi-nate lnf̂ (s2* 0 y) at s2* as described in equation (9). Thesecond step consists of generating a new set ofM Gibbsrealizations from thereducedconditional pdfs, where theconditioning is on the values2* computed from the firstMrealizations, and computing the posterior ordinate lnf̂ (b* 0y, s2*) at b* (from the second run) ands2*.
The posterior odds ofM1 relative toM2 are given by theproduct of the ratio of their marginal data densities, called
130 THE REVIEW OF ECONOMICS AND STATISTICS
the Bayes factor, with the prior odds ratio, where the priorodds ratio is specified by the researcher,
P(M1 0 y)
P(M2 0 y)5
f(y0M1)
f(y0M2)3
P(M1)
P(M2). (11)
The numerator and the denominator, respectively, of theBayes factor are computed from the Gibbs output for eachmodel by exponentiating equation (8). The posterior oddsthus computed are the basis for the model comparisonsperformed in this paper.
First the maximum-likelihood and Bayes estimates arereported. From an inspection of the Bayes estimates, thosemodels for which the theoretical predictions clearly runcounter to the estimates are dropped for the purpose ofmodel comparisons. That is, if the estimates are of the wrongsignandstatistically significant, the model is presumed notat all likely. The posterior odds of acceptable models arethen reported. In order to investigate the sensitivity of themodel comparison results to changes in the prior, estimatesand posterior odds from two priors are reported: an informa-tive normal-gamma prior whereb 0s2 is normally distributedand s2 is distributed as an inverse gamma pdf, and aninformative normal-gamma prior but with inequality restric-tions on a subset of parameters. These priors closely reflectmy personal beliefs, and I hope they agree with yours.
III. Econometric Specification and Choice of Variables
The starting point for the econometric specification isRobert Baldwin’s (1985) view of the world, where a tradingnation’s NTBs have three components—a self-interestedpolitical component, which is a response to protectionistpressures, that is substantially influenced by the lobbyingefforts of private agents, an altruistic political componentthat is influenced by welfare-oriented motives of the govern-ment, and a comparative (dis)advantage component. To thisis added a fourth, based on the theoretical development inRichard Baldwin (1990)—a retaliatory component thatserves as a strategic deterrent against undesirable protection-ist policies of its partners. Following Robert Baldwin’sframework, the following specification is employed in theeconometric analysis:
U.S. NTBs on goodi against countryj, Ni, j, are determinedby a self-interested political component whose variables arerepresented by the vectorx1i, j, an altruistic political compo-nent represented byx2i, j, the theory of comparative costs/comparative advantage represented byx3i, j, and an offen-
sive component,g0N*i, j, designed to thwart foreign NTBs.Country-effect dummy variables are included inDi j, j 51, . . . , J. A feature that separates this study from earlierwork is a separate analysis of the determinants of bilateralUS–EC4 NTBs and US–Japan NTBs, enabling comparisonsand contrasts across partner blocs. The EC4 bloc pools4-digit SIC data across France, Germany, Italy, and theUnited Kingdom. (For the US–Japan runs, the subscriptj inequation (12) is redundant.) The errors are assumed homo-skedastic across countries and goods.a1, a2, a3, andg0 arepresumed to be different for US–EC4 NTBs and US–JapanNTBs. The study investigates the political economy of twodistinct disaggregated NTBs—price NTBs (e.g., countervail-ing duties, antidumping duties) and quantitative NTBs (e.g.,quotas, variable entrance requirements (VERs))—in addi-tion to post–Tokyo round ad valorem tariffs.
Cross-industry trade and manufacturing data from 1983 atthe 4-digit SIC level of detail are employed in the construc-tion of the exogenous variables in equation (12). The level ofNTB protection is measured as coverage ratios, that is, thepercentage of imports covered bysomeNTB. Ad valoremtariffs are aggregated up to the 4-digit SIC level from thetariff line post–Tokyo round rates. It should be noted thatunlike NTB data, tariffs do not include partner-specificlevels. Corporate lobbying expenditures are constructedfrom campaign contributions over the four congressionalelection cycles between 1977 and 1984. Data on corporatePAC contributions from Federal Election Commission tapesare concorded into cross-industry data, as described in thedata appendix. Other industry characteristics variables areconstructed from the 1982 Census of Manufacturing andmany annual surveys of manufactures. A description of thevariables included in the empirical analysis is provided intable 1. Details on the construction of variables are given in adata appendix.
Table 2 depicts the association of regressors with theunderlying theory, together with some descriptive statistics.The special-interest-group model associated with Olson(1965), Stigler (1971), Brock and Magee (1978), Findlayand Wellisz (1982), and Grossman and Helpman (1994)suggests measures of special-interest pressure. The concen-tration ratio (CONC4) and the measures of scale economies(SCALE) have traditionally been used as proxies for special-interest pressures on the assumption that protection wouldlead to larger gains in industries with higher concentration orscale economies. In addition to these proxies, a more directmeasure of pressure by corporate campaign contributions(PACCVA83), which is presumed to be positively related tothe level of protection, is also included. Corporate PACcontributions are scaled by industry value added to preventspurious scale effects. The adding-machine model in Caves(1976) focuses on the voting strength of the industry andsuggests that number of employees (NE82) and degree ofunionization (UNION) in that industry are both positivelyrelated to the level of NTBs. The number of states in which
131BAYESIAN COMPARISON OF TOBIT MODELS USING GIBBS SAMPLING OUTPUT
production is located (REPRST) is another measure of thespread and, consequently, of voting power. Further, thismodel predicts that industries with a large number ofunconcentrated firms are more likely to receive protectionthan a concentrated industry so that the level of protection isexpected to be negatively associated with the concentrationratio CONC4. The special-interest-group and adding-machine models fall under the category of political modelsmotivated by self-interest, and variables that represent themare collected inx1.
The status-quo and social-justice models fall under thecategory of altruistic political models and are represented byvariables collected inx2. The status-quo model associatedwith Corden (1974) focuses on maintaining the status quo,and has been used by Robert Baldwin with some success inexplaining tariff cuts during the Tokyo round. This modelsuggests that swings in average earnings (EARN7982) willbe moderated through protection, and sectors whose marketsexperience large changes in import penetration (MPEN7982),perhaps due to a loss in comparative advantage or adverseexchange rate fluctuations, will receive more protection forthe same reason. Also, the level of tariff protection (TAR)will be positively related to the level of NTB protection,
indicating that higher NTBs compensate for the tariff cuts inthe Tokyo round. The social-justice model that has beenoffered as an explanation as to why industries without muchpolitical clout such as apparel and textiles have beensuccessful in obtaining protection suggests that industrieswith high labor intensity (LABINT82), high proportion ofunskilled workers (P UNSK), and low employment growth(NEGR82) will likely obtain high levels of protection.
The comparative-cost/comparative-advantage model sug-gests that NTBs are positively related to import penetration(Mi, j/CONS) and negatively to exports (Xi, j /CONS). Since itis widely acknowledged that U.S. industries have a compara-tive advantage in skill-intensive industries, those with a highproportion of scientists (P_SCI) and managers (P_MAN) areexpected to require less protection. Other control variablesincluded inx3, in addition to the comparative-cost variables,address the concerns of incorporating the effects of realexchange rates (RERs) into the cross-sectional analysis.From 1981 till 1984 the United States experienced anextended period of RER appreciation. The rise in importpenetration (and the lowering of exports), and thus protec-tionist pressures in industries, particularly those with a highabsolute RER elasticity of imports (and/or exports), mayhave been due to such an appreciation. Hence theabsolutevalue of both, the RER elasticity of imports (MELAST) andthe RER elasticity of exports (XELAST), are expected to be
TABLE 1.—DESCRIPTIONS OFVARIABLES USED IN ECONOMETRIC ANALYSIS
Variable Description
Ni,j U.S., all NTB coverage of imports of goodi frompartnerj (ratio)
Pi,j U.S., price NTB coverage of imports of goodi frompartnerj (ratio)
Qi,j U.S., quantitative NTB coverage of imports of goodifrom partnerj (ratio)
Ni, j* Partnerj, all NTB coverage of its imports of goodi
from U.S. (ratio)PACCORP Corporate PAC spending by industry, per election cycle,
1977–1984 ($100 million)PACCVA83 PACCORP/value added (1983) ($100 million/$ billion)SCALE Measure of industry scale: value added per firm, 1982
($ billion/firm)CONC4 Four-firm concentration ratio, 1982NE82 Number of employees, 1982 (million persons)UNION Fraction of employees unionized, 1981REPRST Number of states in which production is located, 1982
(scaled by 100)EARN7982 Change in average earnings per employee between
1979 and 1982 ($ million/year)TAR Ad valorem tariff rate (post–Tokyo round implementa-
tion)P_UNSK Fraction of employees classified as unskilled, 1982LABINT82 Labor intensity: share of labor in value added, 1982NEGR82 Growth in employment, 1982–1982Mi, j /CONS Penetration of U.S. consumption of goodi by imports
from partnerjXi, j /CONS U.S. exports of goodi to partnerj, scaled by consump-
tionMPEN7982 IMP/CONS(1982)2 IMP/CONS(1979),IMP 5 total
industry importsP_SCI Fraction of employees classified as scientists and engi-
neers, 1982P_MAN Fraction of employees classified as managerial, 1982MELAST Absolute real-exchange-rate elasticity of importXELAST Real-exchange-rate elasticity of exportsDj, j 5 1, . . . , 4 Four-partner country dummies for US–EC4 runs
Note: Data are for 1983 unless indicated otherwise.
TABLE 2.—VARIABLES REPRESENTINGPOLITICAL ECONOMY
THEORIES—EXPECTED SIGNS AND DESCRIPTIVE STATISTICS
Other control variables MELAST 2 21.007 0.286XELAST 1 1.417 0.504
Constant Dj, j 5 1, . . . , 4
Notes: (1) Cross-industry 4-digit SIC level data pooled across the 5 partners: France, Germany, Italy,Japan, and United Kingdom. Number of observations 3233 5 5 1615.
(2) The adding-machine model also includesCONC4(common to the special-interest-group model),but with a negative expected sign.
132 THE REVIEW OF ECONOMICS AND STATISTICS
positively related to the level of protection during thisperiod. Finally, the aim of using retaliatory NTBs, asopposed to employing them for purely protectionist pur-poses, is to deter undesirable foreign trade policy at mini-mum domestic cost. Following Gawande (1995),Ni, j* isincluded as an exogenous variable to capture any retaliationcomponent that may exist.
The dependent variableNi, j is measured only when ittakes a positive value. Theoretically there exist argumentsfor export taxes and import subsidies. With intraindustrytrade in intermediate goods, which characterize a large partof the trade among the countries in the present analysis,there is certainly the possibility of subsidizing imports.More commonplace is the case where a foreign exportsubsidy (which is not included in the construction of theNTB measure) is not countervailed in the United States,which effectively works as a negative NTB. Hence thedependent variable is truncated at zero, requiring the use of atobit specification.3
IV. Enumeration of Priors and Some PreliminaryInferences from Bayes Estimates
A. Priors
Table 3 sets out in detail the functional forms of threepriors for which model comparisons are performed. Prior 1is an uninformative prior which presumes ignorance aboutthe coefficientsb and the regression variances2, and onethat lets the data evidence dominate the posterior outcome.The functional form used to represent ignorance is Jeffrey’sdiffuse prior,f (b, s2) 5 1/s.4 The more informative priors
are represented by priors 2 and 3. Prior 2 takes thenormal-gamma form withb 0s2 distributed normally withprior meanb0 and prior variances2B0
21 (or prior precisions22B0), ands2 is distributed as an inverse gamma randomvariable with prior parameters (g0/2, s0
2/2) (whereg0 is theprior value for the degrees of freedom andg0s0
2 the priorvalue for the sum of squared errors).5 Priors 2 and 3 areinformative aboutb but presume very little knowledge abouts2—g0 is set to 2 ands0
2 to 0.5, values easily dominated bythe data.6
The informative priors reflect a view of the world inwhich trade policy is largely influenced by special-interestgroups, comparative costs, and the possibility of retaliationto influence bilateral policy on trade. Prior means andvariances are expressed in elasticities which is a convenientunit-free medium, and are converted to actual units at thevariable means. I expect the elasticity of the retaliationcoefficient onNi, j* to be between 0 and 1 with about a 65%probability. This is an enumeration of my belief, based onGawande (1995) and Richard Baldwin’s (1990) theory thatsome form of retaliation does discourage foreign NTBs. Theshare of imports in consumption, I believe, is a majordeterminant of NTBs. The demonstration of injury, whichoften precedes the granting of NTB protection in the UnitedStates, is to a large extent based on showing a decliningmarket share that has gone to a foreign competitor, com-bined with some evidence on unfair pricing. Hence my priorbelief that the elasticity of NTBs with respect to importpenetration (Mi, j /CONS) lies between 0.25 and 0.75 with a
3 The single-equation tobit employed here may be seen as a firstapproximation to the more appropriate simultaneous tobit, as in Trefler(1993). Bayes estimation for the simultaneous tobit requires integration ink dimensions, wherek is the number of parametes and, as such, is anontrivial computation.
4 We cannot use an improper prior to represent ignorance since themarginal density ofy does not exist for such a prior.
5 This would be a conjugate prior if the data were normal, that is, theposterior pdf has the same normal-gamma form as the prior pdf. However,the likelihood function for censored data does not allow us the analyticalniceties from employing conjugate priors. With data augmentation of thecensored part ofy, however, analytical results can be convenientlyemployed.
6 If either of these hyperparameters were set to zero, the prior wouldbecome improper. I therefore set them to small numbers that are easilydominated by the data and yet there remains a proper prior.
TABLE 3.—ENUMERATION OF PRIORS
Type of Prior Functional Form Elasticities (Mean, Variance)Inequality Restrictions
Notes: (1) Prior means and variances given in terms of (unit-free) elasticities are converted into actual units (i.e.,b0 andB 021 ) at the variable means.
(2) b0 is prior mean ands22B0 is prior precision matrix of coefficients. Their prior covariance iss2B 021.
(3) For the informative prior the prior covariance is diagonal with prior variances of coefficients on the diagonal.(4) The hyperparameters of the prior pdf fors are set tog0 5 2 ands0
2 5 0.5 (so that ‘‘prior sum of squared errors’’ (s02 5 1). These values reflect practically negligible information abouts, and are easily dominated
by the data. With improper priors the marginal likelihood does not exist, and therefore model comparisons with the diffuse prior (which is improper) are not performed.(5) In the runs with the informative priors, rather than use0B0 0 in the definition of the prior pdf forb 0s2, we use0B0 1 X8X 0 2 0X8X 0 . This prevents the factor0B0 0 from dominating the model comparisons (see also
footnote 14).
133BAYESIAN COMPARISON OF TOBIT MODELS USING GIBBS SAMPLING OUTPUT
probability of about 65%.7 Although poor export perfor-mance may affect the demand for protection, its effect is lessdirect than that of imports. I expect the elasticity of NTBswith respect toXi, j /CONSto be between20.50 and 0 with aprobability of about 65%. Finally, corporate campaigncontributionsPACCVA83,I believe, do influence the grant-ing of protection. All other variables I treat as doubtful,although in their case I am perfectly willing to let the dataevidence dominate my skepticism. In deciding whether yourpriors are closer to the diffuse prior (prior 1) or my prior(priors 2 and 3), note that posteriors based on the diffuseprior are expected to be very close to the maximum-likelihood estimator.
A third prior, which is probably the best enumeration ofmy priors,8 imposes strict inequality restrictions on fourparameters, but otherwise is the same as the normal-gammaprior in prior 2. Thusb 0s2 is normally distributed with priormeanb0 and prior variances2B0
21, but the coefficients onNi, j* , (Mi, j /CONS), andPACCVA83are each restricted to benonnegative and the coefficient onXi, j /CONSis restricted tobe nonpositive. The use of this prior also serves to demon-strate the simplicity with which complicated and nonstand-ard forms of priors can be handled using Gibbs sampling.
B. Conditional Distributions for Gibbs Sampling
For each of the three priors the censored data areaugmented from the conditional pdff [yi* 0yt 5 0, b, s2] 5TN(2`,0](x8ib, s2), i [ C, whereC is the set of censoredobservations, ands2 is simulated conditionally fromf [s2 0y, yC* , b] 5 IG [(g0 1 n)/2, (g0s0
2 1 d)/2], with n being thenumber of observations andd 5 ( y* 2 Xb)8( y* 2 Xb ).9
Where the conditionals differ is in the (conditional) genera-tion of b. For prior 1 the conditional pdff [b 0 y, yC* , s2] 5N(b̂, (s22X8X8)21), whereb̂ 5 (X8X)21Xy8*, while for prior2 the pdf takes the same form but nowb̂ 5 (B0 1 X8X)21
(B 0b0 1 X8y*).10 For prior 3 the support ofb: (k 3 1) is thesubset ofRk given by the coefficient restrictions in table 3.Conditional generation ofb for prior 3 is done using thesimple method of Gelfand et al. (1992) through theunivari-
ate generation of each restricted parameter from its fullconditional distribution.11
For the analysis with priors 1, 2, and 3 the size of theGibbs sample isM 5 3000 after dropping the first 500realizations since the initial realizations may be highlyautocorrelated. The Bayes estimates use everylth realizationwith l 5 5 (so the effective Gibbs sample size is 601) toreduce autocorrelation even further. The sample autocorrela-tions are low, usually below 0.05. Since prior 1 is improper,its marginal pdfs do not exist, and hence the posterior oddscannot be computed using this prior. For model comparisonsfrom prior 2 the marginal pdfs are computed fromM 5 3000realizations,eachfrom the full and reduced conditionals, asexplained above. For model comparison from prior 3 themarginal likelihoods are computed at the end of an addi-tional (11 kT) reduced conditional runs—one run forbN
plus kT runs, one for each restricted coefficient (see Chib(1995, sec. 2.1.3))—each withM 5 1000.
C. Bayes Estimates: Some Preliminary Inferences
Tables 4 and 5 present estimates from the US–EC4 runs(n 5 3233 4 5 1292) and the US–Japan runs (n 5 323),respectively.12 First consider the Bayes estimates for vari-ables explaining ad valorem tariffs in the first two columnsof table 4.13 The Bayes estimates from the diffuse prior(prior 1) are almost identical to the maximum-likelihoodestimates (which are not reported here for brevity). Since theBayes estimates based on the informative prior (prior 2) areclose to those from the diffuse prior, and therefore themaximum-likelihood estimates, it implies that data evidenceoverwhelms my prior information in prior 2. The same isgenerally true of the estimates based on price and quantita-tive NTB data, but since the data evidence is weaker here(e.g.,s2 is higher than in the tariff runs, even though NTBs,such as tariffs, lie between 0 and 1), prior information,especially the strong prior information in prior 3, doesinfluence some estimates importantly, as will be seen to betrue for the US–Japan runs.
Based on the size and sign of coefficients, the US–EC4runs in table 4 indicate that every political-economy model
7 Grossman and Helpman’s (1994) theory, on the other hand, postulates apositive relationship between protection and value added per dollar ofimports, which is the converse of my prior. Perhaps evidence of that maylead me to reconsider my priors, but evidence has usually pointed to apositive relationship between import penetration and protection.
8 The elicitation of precise priors is usually costly, and sometimes notpossible at all. There usually exists some fuzziness and imprecision inspecifying priors. Hence I use both priors 2 and 3 to convey my‘‘probable’’ priors. If the inferences from both are similar, then they arerobust to the imprecision in my mind about my priors. See also Leamer(1978) and Gawande (1995).
9 See Devroye (1986) for simulating from the truncated normal andinverse gamma distributions. The inverse gamma here reduces simply tothe inverse of a chi-squared random variable.
10 It is readily seen that given data augmentation of the censored data,these conditional pdfs have means that are matrix-weighted averages of theprior and least-squares estimates with the prior precision and dataprecision matrices, respectively, providing the weights (for the diffuseprior the conditional mean is simply the least-squares vector).
11 Denote the unrestricted set of parameters bybN: (kN 3 1) and therestricted set bybT: (kT 3 1), wherebT 5 (bT1, b T2, . . . ,bTkT
). Then, withnormal-gamma priors, (bN 0bT , s2, y*) is multivariate normal with theappropriate conditional mean and variance (see, e.g., Chow (1983, pp.8–13)). Similarly, for each restricted coefficient, (bTi 0bN , bTi ( j Þ i ), s2, y*)is distributed as a truncated univariate. The individual conditionals areused to simulate each restricted coefficient using the algorithm in Devroye(1986, p. 38): if U is a uniform (0, 1) variate and the unrestrictedconditional pdf ofbTi is Fi, thenbTi restricted to the interval [a, b] can bedrawn asF i
21[Fi (a) 1 U (Fi (b) 2 Fi (a))]. This is in fact the algorithmused to augment the censoredy observations.
12 All estimation was done using GAUSS v. 2.1. I am grateful to Sid Chibfor providing the core part of the code.
13 For ad valorem tariffs, the analysis with prior 3 is not presentedbecause the positivity restriction onPACCVAR83and the negativerestriction onMi, j /CONSare in conflict with the data information (thelikelihood function), and the Gibbs sample cannot be generated. That is,the strict prior restrictions are not likely at all.
134 THE REVIEW OF ECONOMICS AND STATISTICS
except perhaps the social-justice model finds support fromthe data. Each set of variables representing a theory has atleast one variable for which the estimate has the expectedsign and a large (posterior)t-value. The model of retaliationis clearly supported by tariffs as well as price and quantita-tive NTBs, where the retaliation coefficient is both statisti-cally and economically significant. The special-interest-group model finds significant support: firm scale (SCALE)and firm concentration (CONC4) corroborate similar find-ings by Robert Baldwin for tariff data, whereas corporatePAC contributions (PACCVA83) and firm concentration areseen to be influential determinants of price NTBs. Theadding-machine model finds support from the estimate onthe number employed (NE82) in the case of price andquantitative NTBs and from the estimate on geographicaldispersion in the case of tariff data, validating Caves’ (1976)theory that voting power in terms of numbers is an importantdeterminant of protection. The negative coefficient onchange in earnings (EARN7982) is proof of the validity ofCorden’s status-quo model for U.S. tariffs. The positivecoefficient on ad valorem tariffs (TAR) indicates that thesame industries that were earlier protected by tariffs nowreceive price NTB protection, thus undermining the multilat-eral cuts from the Tokyo round, which is in line with theprediction from Corden’s status-quo model that to preventdamage to industries from a sudden removal of protection,
the most highly protected industries will be supported insome alternative way, here by NTBs. The social-justicemodel does not receive support from the tariff and priceNTB data, but the negative estimate on proportion unskilled(P_UNSK) does impart the model some validity fromquantitative NTB data. The model of comparative costs issupported by the data. Bilateral exports (Xi,j/CONS) has theexpected sign and larget-values. Surprisingly, the sign onimport penetration is not unambiguously positive. Industrieswith high skill levels measured by the proportion ofscientists and engineers (P_SCI) do not receive protectionlargely because, as is widely believed, the United States hasa comparative advantage in the production of skill-intensivegoods.
Determinants of U.S. quantitative NTBs against the EC4countries are different from factors that determine U.S. priceNTBs or tariffs. The Bayes estimates in the last threecolumns of table 4 show little support for the status-quomodel, whereas tariffs and price NTBs do support the modelstrongly. Quantitative NTBs are determined in part bysocial-justice considerations, but not tariffs or price NTBs.As indicated by the shaded cells, some estimates arestatistically significant but run counter to the model’spredictions. For the purpose of model comparisons, we willdrop the shaded cells from consideration to put the models in
Notes: (1) Prior 1 is diffuse prior, prior 2 is normal-gamma prior, and prior 3 is normal-gamma prior with inequality restrictions (see table 3). Bayes estimates based on Gibbs sampling. Size of Gibbs run is 3000.Every 5th realization chosen so that effective sample is 601. Numerical standard errors are generally below 0.05 except for coefficients witht’s below 0.5
(2) ** and * indicate that posterior0 t 0 $ 2 and 2. 0 t 0 . 1.5, respectively.(3) Shaded cells indicate statistical significance (posterior0 t 0 . 1.5) but wrong sign.(4) T 5 1292,k 5 23, degree of truncation for5TARIFF, PRICE, QUANT6 5 53.7%, 85.5%, 93.0%6.(5) Frequentist measures of fit for the tobit ML runs5TARIFF, PRICE, QUANT6: likelihood-ratio statistic5 5605.4, 205.6, 171.46; Maddala’sR2 5 50.374, 0.147, 0.1246; McFadden’sR2 5 50.154, 0.203, 0.2936;
Cragg–Uhler’sR2 5 50.405, 0.271, 0.3406.(6) ForTARIFFruns, all four partner dummies are positive with posterior0 t 0 . 2. ForPRICEandQUANTruns, all four partner dummies are negative with0 t 0 . 2.
135BAYESIAN COMPARISON OF TOBIT MODELS USING GIBBS SAMPLING OUTPUT
their best light. However, if a majority of variables, say, twoof three, representing a theory are of the wrong sign, we seethe model as not being likely relative to other models anddrop it from consideration.
Now consider the estimates from US–Japan data in table5. While the determinants of tariffs are the same as in table 4(this is not surprising since the tariff data vary only acrossindustries, not partners, unlike the NTB data), the determi-nants of US–Japan NTBs differ from factors that determineUS–EC4 NTBs. The adding-machine model and the status-quo model receive little support from US–Japan NTB data.The social-justice model is rejected outright by U.S. quanti-tative NTBs against Japan and fares no better with tariff orprice NTB data. The special-interest model, the comparative-cost model, and the model of retaliation can each claim somesupport from the US–Japan runs. Some important messagesconcerning the influence of priors are in evidence in table 5.Prior information is constant across all models and data, butwhile that is dominated by the large amount of data evidencein the pooled US–EC4 runs (1292 observations relative to323 for the US–Japan runs), prior information significantlycontributes to the posterior estimates here. For example, inthe last two columns for quantitative NTBs, (1) the retalia-tory coefficientNi, j* shows up with high posteriort’s relativeto the estimate using the diffuse prior, and (2) the coefficienton PACCVA83reverses sign and shows up with a hightwhen prior 3 is used.
V. Model Comparisons of Endogenous ProtectionTheories
Traditional hypothesis tests are not a satisfactory way toproceed here because each political economy model prob-ably does have some degree of truth to it, and the strict nullhypothesis that ‘‘modeli has no validity’’ has a high a priorprobability of being rejected. Therefore, a pairwise compari-son of competing political economy models on the basis ofposterior odds, which quantify how likely one model isrelative to another, is performed here. The posterior oddsdefined in equation (11) also serve to update prior beliefs(summarized in prior odds) based on the data evidence(summarized in the Bayes factor).
In tables 6 and 7 posterior odds are used to make severalmodel comparisons from the US–EC4 and US–Japan runs.14
14 A factor that weighs importantly in the computation of the marginaldata density (8) is the (logarithm of) the ratio (0B0 0 / 0 B0 1 X8X 0 )1/2 (seeZellner (1971, pp. 309–310)), the information in the prior pdf relative toinformation in the posterior pdf. In some cases this influences thecomputation of posterior odds unduly. Consider the comparison, usingprice NTB data, of the full modelF with the model with the CC variablesomitted,F\CC. For modelF (23 variables) 0.53 ln 0B0 0 5 225.6 while itis 217.9 for F \CC (20 variables). The corresponding values of 0.53ln ( 0B0 0 / 0 B0 1 X8X 0 ) are252.2 forF and247.0 forF\CC. SinceB0 isdiagonal, the wide disparity is caused by the inclusion in the full model ofvariables that have very low prior precision (P_SCI, P_MAN), which‘‘spuriously’’ lowers the marginal likelihood of the full model so that whenthe CC variables are dropped, the restricted model actually looks betterthan the full model. Therefore in the computations that follow0B0 0 is
Notes (1)–(3): see table 4.(4) T 5 323,k 5 20, degree of truncation for5TARIFF, PRICE, QUANT6 5 53.7%, 72.5%, 77.1%6.(5) Frequentist measures of fit for tobit ML runs5TARIFF, PRICE, QUANT6: likelihood-ratio statistic5 5151.0, 34.08, 179.26; Maddala’sR2 5 50.373, 0.100, 0.4266; McFadden’sR2 5 50.053, 0.105, 0.5306;
Cragg–Uhler’sR2 5 50.404, 0.158, 0.6576.
136 THE REVIEW OF ECONOMICS AND STATISTICS
The first five rows compare the full model with abridgedversions of the full model, where sets of variables represent-ing the theories are dropped for a theory at a time. The firstrow thus compares the full model (F ) with one that excludesforeign NTBs on U.S. exports (F\R), that is, one where thetheory of retaliation is disregarded. My prior odds are thatthis abridged model is only one-third as likely as the fullmodel. In table 6 the high posterior odds in favor of theretaliation model, especially from the US–EC4 quantitativeNTB data, attest to the validity of this model. The lastcolumn, for example, indicates that the full model (the term‘‘model’’ taken to include the incorporation of prior 3) is5.76 3 106 times as likely as the abridged model. Theselarge numbers for the odds of one model relative to anotherare quite commonplace (see, e.g., the application in Zellner(1971, p. 314)), especially with data sets containing a largenumber of observations.15 The model of special-interest-group pressure (SI) has strong validity across all types ofprotection, including tariffs. The other model of politicalself-interest, the adding-machine model (AM), while notapplicable to tariffs (since two of the three variablesrepresenting this model have the contrary sign, it is notconsidered likely a priori) finds strong empirical supportfrom the NTB data. The posterior odds for the status-quomodel (SQ) give mixed messages. While the SQ model is
seen to be overwhelmingly favored by tariff data, especiallydue to the highly statistically significant estimate onEARN7982,it receives only lukewarm support from theprice NTB data. (Personally, the results do succeed inmaking me revise my even prior odds which neither favornor oppose the SQ model into odds that favor it, but not ashigh as the tariff results dictate.) Certainly, model compari-sons using more recent NTB data would be valuable inupdating my beliefs about the SQ model. The other model ofpolitical altruism, the social-justice model (SJ) is not muchfavored by the data. The comparative-cost model (CC) isseen as a driving force behind tariffs and quantitative NTBson EC imports. While that would be a reasonable conclusionabout US–Japan NTBs, it is surprising for EC countries, forthey are considered quite similar to the United States.US–EC4 trade is therefore largely intraindustry in contrastto the Heckscher–Ohlin nature of US–Japan trade. Quitepossibly, quantitative NTBs are erected mainly against thoseEC4 imports that are due to comparative advantage reasons,while price NTBs are not.
The methodology of classical nonnested tests is some-what ambiguous. There are at least three different asymp-totic methods to do nonnested model comparisons, eachbased on a different philosophy.16 In contrast, the Bayesianmethod is straightforward. If two (nonnested) models areentertained, the posterior probability of modelM1 is
P(M1 0 y) 5P(y0M1) 3 P(M1)
P(y0M1) 1 P(y0M2). (13)
Hence the posterior odds ofM1 relative toM2 are exactly thesame as given by equation (11). For example, in order todiscriminate between the two models of political self-interest, their posterior odds are simply computed by
replaced by (0B0 1 X8X 0 2 0 X8X 0 ). A justification is that since the ratio0X8X 0 / 0 B0 1 X8X 0 is the measure of data precision relative to posteriorprecision, 12 ( 0X8X 0 / 0 B0 1 X8X 0 ) is a measure of prior relative toposterior information. This is just (0B0 1 X8X 0 2 0X8X 0 )/ 0B0 1 X8X 0justifying use of the numerator. For modelF, 0.5 3 ln ( 0B0 1 X8X 0 20X8X 0 ) 5 26.5 while it is 29.0 forF\CC, while the corresponding numbersfor 0.5 3 ln ( 0B0 1 X8X 0 2 0X8X 0 )2ln ( 0 B0 1 X8X 0 )] are 20.023 and20.112, correctly indicating that the prior information in the two models isnot very far apart and the difference is even smaller when considered inrelation to thir posterior information.
15 A quick comparison of the posterior odds from the US–EC4 andUS–Japan tariff runs shows this. Since the US–EC4 data set is four times aslarge as the US–Japan data set, the odds from the former are in theneighborhood of the odds from the US–Japan runs raised to the fourthpower.
16 They are, for example, the classic Cox test, the Davidson–Mackinnonjoint J-test and its extensions, and the Mizon–Richard encompassing test.
TABLE 6.—MODEL COMPARISONS—POSTERIORODDS FROM US–EC RUNS
7. F: F \ (SI < AM) 15:1 Same as 2 1.353 106 2.103 107 1.253 106 6.853 105
8. F: F \ (SQ< SJ) 1:1 4.913 10 42 1.053 103 518.0 — —9. F \SI: F \AM 1.5:10 — 0.714 0.047 2.37 0.424
10. F \SQ: F \SJ 1:1 1.123 10243 0.252 0.262 — —
Notes: (1) Posterior odds based on priors onb ands2 in table 3. Posterior odds given by (prior odds)3 (Bayes factor) where the Bayes factor equals the ratio of marginal likelihoods. Marginal likelihoods computedusing equation (8).
(3) For the following models posterior odds are computed after dropping variables with wrong sign and0 t 0 . 1.5 (shaded in table 4): (model: variables dropped [runs]):SI: PACCVA83[TAR]; SJ: P_UNSK[TAR];AM: UNION [Q, both priors];SJ: LABINT82[P, both priors];CC: P_MAN[Q, both priors].
(4) The following models are not compared since at least two variables have the wrong sign and0 t 0 . 1.5:AM for TARruns;SQfor Q runs;SJfor Q runs.
137BAYESIAN COMPARISON OF TOBIT MODELS USING GIBBS SAMPLING OUTPUT
dividing the posterior odds forF: F\AM by the posteriorodds forF: F\SI. Consider the odds from the price NTB runscomparing the (nonnested) SI and AM models in row 9 oftable 6.17 While the posterior odds based on prior 2essentially state that they are both almost equally likely, theodds from prior 3 (which favors the SI model) indicates thatthe model without the SI variables is only 0.047 times aslikely as the model without the AM variables, that is, SIoutweighs AM by the odds of 1047:1, or about 21:1.Quantitative NTB data would slightly favor the AM model,were it not for the large prior odds in favor of the SI model.(Personally, the posterior odds would have me revise myprior odds into one that weighs each model equally regard-ing the incidence of NTBs, but it will take similar resultsfrom the US–Japan runs to convince me to do so.) From theresults reported, it is easy to perform other nonnestedcomparisons. A comparison of rows 7 and 8 shows that thetwo models of political self-interest together (SI < AM)dominate the two models of political altruism (SQ< SJ) inthe granting of NTB protection to U.S. industries from EC4imports. Tariff data, however, seems to be generated more byaltruistic forces underlying the SQ model.
Now consider the posterior odds from US–Japan runsreported in table 7.18 They afford quite different inferencesfrom the US–EC4 runs. The main reason is that priorinformation weighs more strongly since, withn 5 323, thereare fewer data points. Also, the posterior odds estimates aresmaller numbers than in the EC4 runs since the sample sizehere is only a quarter of that sample. The Japan runs againunderscore the relevance of the Olson–Stigler SI model and
also the role of comparative costs in determining protection.However, the AM model of political self-interest is seen tohave much less applicability in explaining US–Japan NTBs.The comparison between SI and AM in row 9 indicates thatSI is preferred over AM across all modes of protection. Thetariff data rule in favor of SI by the overwhelming odds of1600:1, the price NTB data by 25:1 (prior 2), and thequantitative NTB data by 15:1 (prior 2). Based on prior 3,which is biased toward SI, those odds from the NTB data areeven higher at 70:1. This is in contrast to results from theUS–EC4 runs, where both models were seen to be fairlyequally likely. (Personally, the results do not provide astrong enough basis for me to revise my prior odds of 6.7:1in favor of the SI model. While tariff data are still seen asstrongly favoring the AM model, US–Japan NTB dataclearly do not encourage support for the models of politicalaltruism.) The odds of the NTB data, especially quantitativeNTBs, being generated by the models of political self-interest (SI < AM) far outweighs the odds of the NTB databeing generated by the models of political altruism(SQ< SJ). The tariff data, on the other hand, greatly supportthe models of public concern, particularly the SQ model.
From a Bayesian perspective three observations arenotable. First, the results are fairly robust across the twopriors (except for the conflict between the inequality-restricted prior and the tariff data). Second, the data evidenceoverwhelms the prior evidence in the US–EC4 runs. How-ever, the prior odds do make a difference in the more subtlemodel comparisons between, say, the SI and AM models.Third, when the strong inequality-restricted prior conflictswith the data, it shows up in the computation of the posteriorodds. For example, in the US–Japan runs, while the inequal-ity-restricted prior expectedly strengthens the case for theretaliatory model (R) and the political self-interest models(SI) from the price and quantitative NTB runs, it unexpect-edly weakens the case for the comparative cost model (CC).This is because the positivity restriction on the importpenetration ratio is somewhat in conflict with the dataevidence.
17 The modelF \SI is one with SI variables excluded. If this has higherposterior odds than the modelF\AM, with AM variables excluded, thenthat is evidence in favor of the AM model.
18 Computation Notes: On a 75-MHz Pentium with 16-Mb RAM thefollowing are the run times. With the normal-gamma prior andm 5 3000(for the full as well as the reduced conditional runs) after discarding thefirst 500 realizations, the full Japan model took approximately 25 minutes,and the full EC4 model took 100 minutes. With the parameter-restrictedprior the run times were substantially higher, since each run required asmany reduced conditional runs as the number of parameter restrictionsplus 1. The full Japan model took 75 minutes, and the full EC model took 5hours to run.
TABLE 7.—MODEL COMPARISONS—POSTERIORODDS FROM US–JAPAN RUNS
8. F: F \ (SQ< SJ) 1:1 5.433 107 15.18 4.350 Same as 4 Same as 49. F \SI: F\AM 1.5:10 6.253 1024 0.040 0.014 0.064 0.014
10. F \SQ: F\SJ 1:1 5.113 1028 0.151 0.280 — —
Notes (1), (2): see table 6.(3) For the following models posterior odds are computed after dropping variables with wrong sign and0 t 0 . 1.5 (shaded in table 5): (model: variables dropped [runs]):SJ: P_UNSK[P, both priors];AM: NE82,
CONC4[TAR].(4) SJis not compared forQ runs since two out of three variables have wrong sign and0 t 0 . 1.5.
138 THE REVIEW OF ECONOMICS AND STATISTICS
VI. Summary and Conclusion
This paper demonstrates that Bayesian inference andmodel comparisons are easily performed with a high degreeof accuracy and confidence by employing the Gibbs sam-pling methodology, even if (1) the likelihood is analyticallyintractable, as is the case for the tobit model, and (2)adequate representation of prior information requires non-standard prior pdfs, as in the case of the inequality-restrictedprior employed here. Chib (1995) has shown that the Gibbsoutput can be used to compute posterior odds for comparingeconometric models. Using Chib’s method, posterior oddsratios are computed in order to perform a pairwise compari-son of five theoretical models explaining protection usingbilateral US–EC and US–Japan NTB coverage data from1983 across 4-digit SIC industries.
Bayes estimates show that each model of protection hassome validity in the sense that at least some subset ofvariables representing each theory is shown to have thecorrect sign. However, a comparison of models usingposterior odds shows some models to be more likely thanothers. A strong conclusion to emerge from the study is thatthe models of political self-interest—the Olson–Stiglermodel of special-interest-group behavior (SI) and Caves’adding-machine model of voting strength (AM)—are farmore influential in determining U.S. NTB protection againstits OECD partners than models of public interest andpolitical altruism. Within the self-interest category, the SImodel outweighs the AM model upon using US–Japan data,while both models are approximately equally likely for theUS–EC4 NTBs. Evidence from the NTB data in favor ofeither of the public-interest models—the status-quo modeland the social-justice model—is at best weak, although thetwo together are shown to exert some weight in thedetermination of US–EC4 price NTBs. However, evidencefrom tariff data strongly supports Corden’s status-quo model.The model of comparative costs based on comparativeadvantage considerations is found to have strong empiricalsubstance. Inference about the models differ across instru-ments of protection and in the way NTB protection data aregenerated across the two sets of partners. Theoretical modelsexplaining these differences would further enrich the litera-ture on endogenous protection.
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DATA APPENDIX
Aggregation and Measurement of NTBs
For data construction see Leamer (1990). The data used here are froman UNCTAD and World Bank project on nontariff barriers to trade.
139BAYESIAN COMPARISON OF TOBIT MODELS USING GIBBS SAMPLING OUTPUT
Other Variables
The sample accounts for over 80% of manufacturing sales. In thefollowing COMTAP refers to theCompatible Trade and ProductionDatabase,1968–1986; CM refers to the 1982Census of Manufactures;ASM refers to the 1983Annual Survey of Manufactures;and CPS refers tothe 1983Current Population Survey.For countries other than the UnitedStates, bilateral and total trade and production (the latter required to obtaindomestic consumption) were constructed using 1983 figures from COMTAP.These data are at the ISIC level, which was concorded into the SITC (r1)level and then into the 4-digit SIC level. For the United States, bilateral andtotal (across all partners) imports and exports are aggregated up from tariffline data. Political Action Committee (PAC) campaign contribution dataare from the Federal Election Commission (FEC) tapes for the electioncycles 1977–1978, 1979–1980, 1981–1982, and 1983–1984. Since PACsare associated with individual firms, the variablePACCVA83is constructed
as follows. Using COMPUSTAT tapes, firms are classified into 3- or4-digit SIC industries. Where firm coverage is incomplete in COMPU-STAT, PACs are classified into 2-digit SIC industries using Weinberger andGreavey (1984) and replicated at the 4-digit level. The PAC contributionsare then scaled by value added. Value added is from ASM.REPRSTisconstructed from theGeographic Area Seriesof the COM. Earnings andemployment (forEARN7982, LABINT82) are also from ASM, as arecapital stock figures.NE82andCONC4are taken from CM. Division ofworkers by skill class (used forP_SCI, P_MAN, P_UNSK) is from CPS.UNION is from Kokkelenberg and Sockell (1985).MELASTandXELASTare replicated from the 2-digit figures in Ceglowski (1989). In convertingimport and export data from ISIC to 4-digit SIC, 112 industries of the 4354-digit industries under consideration were dropped due to concordanceproblems. Hence the sample consists of 323 observations for the US–Japanruns and 3233 4 5 1292 observations for the US–EC4 runs.
