DYT3
1. Introduction
Theil and Latinen (1979) described qaxi■uI likelih叩d(JL) procedures for the RotterdaJL AOdelunder tTO different conditions. For ntlJerical eェatples ,
in his b00k,Theil(1976) used DtltCh and Ds data Thick are shoTn in Table 5.2
(the htch data) and in Table 7.3(the Us data). To illustrate JL procedures.
They used the saJLe data･ The J)utch data consists of coIstlLPtion data for 4 coL‑
qodities(F叩d , Beyerages. Durables. and EeIainder) in the ⅣetherlatLd fro1 1922 to 1963 ezcept for the period of TOrld Tar.The Us data consists of annual ob‑
ser▼ations on lealt( Beef . Pork, Chicken.and LaJb) in Dhited States 1950‑1972.
7e used these data for the nulerical exa叩les･ The first thele in this paper
is to estiJate the rotterda. 10dels uJlder the defferent conditiotLS by bayesiaJI
yay yia YarkoyIChaizI Ionte Carlo(lC暮C)Jethod. For bayesiaJl iJIference, Te adopt vague prior畠On the lOdel paraIeters･ so that the estilation results coIpamble
yith the rL estiJateS by Theil aLnd tAtiJ)en (1979)are obtailled.
For coJJOdity i(申,... I, the ntlJLber of couodities) aJld period t(=1,.. T, the
saIP.le size after lagging), the absolute price yersion ofthe Rotterdat JlOdel is
‑T･･t Dqit ‑LLiDQt+弓2Tj Dpjt 'e;t･ i‑1･････t‑1･‑Tq^
For the htch data,t=4.T=31, az)d for the us'data. n=4, T=22.
Restrictions on slutsky coefficients are
甘oIOgeneity constmints･弓3T9 ‑0･ i‑1･ ･･.･
syltetry COnStraints 2Tj = ZTJ･･･ i･j=1････.
The relatiye price yersioJ) is
〜
育;I Dq;I =LLiDQ.'写uj (Dp,･t ‑S.pkDp.. )'e;i i‑1･････t‑1･‑T･
There
2Tj= uj ‑ ￠LLi L! and弓u).‑め LLi
l /￠‥the incole elasticity of the larginal utility of incote.
hder conditions of block‑independence･もff‑diagonal elelentS Of 【 Vj ] yanish･
For the dutch data, asstlIPtioJl Of preference independence is relaxed to that of
block‑ independence Tith three blocks;Food, 8eyerages and Durables/ReIainder.
Then･luj ] is restricted as
≡
｢｣
恥ノ
二ll
‰ =恥 (3)
le assu■e the error ter] EI‑〔e托j folloTS十ultinoz･Ial distribution Tith 0 zLean aJ)d yariance Jatrix E2 and E:s are iJ)dependent oyer tiJre. For the Vs data,
Theil and Latinen(1978) specifiedE2 to be proportinal to slutsky Aatrix L'ZT･)･l l
uhder the theory of ratiotLal ratldo) behayior. b this paper.Te adopt an alter‑
natiye specificatioz) as folloTS:
yhere
E2 ‑A A
A :unhoT7)e ParaIeter. 1>o
A‑[aジ】
(1‑両) fori=j
苛j for iキj(4)
(5)
W; :ayafage budget share of i‑‑th coIIOdity･
breafter･T̲e. Tort Tith the precision latriI 0 rather than the coyariance Q so that O=Af2
Final task in this paper is to coJIPre the Rotterdal LOde)sunder the dif‑
ferent constraints. For coTlYemience, Te distigTish the工Odels as folloTS;
the absolt)te price yersion Tith no restriction by 10del(0), AOdel (0)under ho10getLety COnStraints (1) by todel (1),
10del (1) uJIder sy■JLetry COnStraiTItS (2)● by ■odel(2),
the relatiye price yersiotl tmder block‑independenc (3) by JlOdel(3)
alld
todel(2) tlTlder the constraiTltS (4) by Todel(4).
IJI his book,Theil(1976) has eyaluated the oyerall goodness of fit by the ay‑
erage inforlation inacctlraCy aJId for the dtltch data, he coJLpared )odel(3) Tith
‑ 148‑
dLOdel (2) in terJl Of the ayerage itlfortation inaccuracy(the one corrected for
degrees of freedo)). In this paper. for lOdel coIparison. Te use the fractional
bayes factor (FBF ) propsed by O hgeJ)(1995). For caluclation of the FBF. JC暮C
tecniques are the indispensable tools.
Appendix sul■arizes ■athelatical supple■ents for the bayesian colptltatioTI
used in this paper.
2. Bayesian lnferenceand the Results
ln this paper,all of the JOdelsare TritteJl in Jatriz forl as
y.‑‡β +Et ,or Y.‑乙0 +Et,
there Yt and Etare 4 × 1vector stJCh that
Yt=[T;tDq;e]Jor Yc=lT;I Dqit /Tも] and Et=le.･t】 for t=1,..,T
∫
To estipate the JIOdel paraJLeterS β or 0 ,one of the equatioZ)S Shotlld be de‑
leted fro■ the eqtlatioJl SySteh Other‑ise yariance )atriI Of E becotes singtllar
･ Thus, Te delted 4th coJk)dity(EeJlinder for the Dutch data. LzLJLb for the Vs data)
･so that Yf is reduced to 3× 1 Vector. After deleting 4th colJLOdity,the re‑
gression for■ats for q.odel(0)〜JlOdel (4) Tis Setup(see.lppendi王Jl). CoJISequently I to estilate the lOdel para■eters iJicluding eli)inated para)eters, the additional
I
cotIplJtation Tie (8. 5. 1) and (8. 5. 2) aT･e required̲
AdoptiJ)g a Vague prior density on the para■eter yectorand ▼ariaJ)Ce tatri王
of error terlS,1arginal posterior density p (0 I Data)are obtained frol the likelih00d function. In the bayesian fraJeTOk.the point estiJate Of 0 is giyen by lean Value of p (OIData)(posterior leai ofO ).
2. 1. EstiJ[ation for暮Odel(0) and JLOdel(1)
beta Aatrix Xtof 斗odel(0) is abbreyiated as‑xtl‑I3 @ (写或) that is the spe‑
cial feature of AOdel(0)･ Xodel(1) has sate feature･ Ⅰn this case, p(βl Data) can be eyaluated eェactly and posterior inferences atx)tlt SPeCific regressiotL CO‑
efficient βジare ･ade fro･miyariate Student s t‑distribution (see, AppendiL D) Then,bayesian esti'ate of垂is giyen by the p,sterior lean Of卑in (D･5) that
is the sate as OLS estilate and the standard eror of the posterior Ieanis giy‑
en in (J) 5). The ilpOrtant POitlt is that the denoIinator(degrees of freedot) in
(D.5) is T‑(tI+i+I), not T.
EstiJLation restJlts of todel(1) are suJLJLarized iJl the upperpart of Table 1.1(for the I)tltCh data) and table 2.1(for the Us data). The figures in paren‑
thes are the staJ)dard errors of posterior ■eanS. ColpariJ)g Tith table 5. 6 and table 7.5 ill Theil(1976). the point estiIatesare pair■ise sate eェept the staJl‑
dard erros. The discrepaJ)Cies in the stazldard erros are oTed to the differez)ces
in ‑°eg. of freedo1‑.
IolOgetLeity test. There are tTO Tars for testing the hyptheses of deJand
holqgeneity. As shoTn in AppendiェD. the conyentiotLal bayesiatL teStSare based on
the qP region in (D. 9) that areJnuLericaly equYaletLt tO the Laitinen s exact
test for ho暮喝eAeity.i.e.Let demte the test statistic in (D.ll) by VNanda ‑ 0･05. IfVN <F".A, then hol喝eneity for all coLdities are accepted̲ The al‑
tematiye Tar for ho■ogeJleity test is to cotPre lOdel(0)Tith ■del(1)by use of the FBF.
Testillg the hoLOgeneity of i‑co■▲didy is ■ade frol the tlniYariate sttldent
也
t‑disfribution in (D･7)･The basic statistic is b･･ ‑写2Tj･ ･Let b‑(bt･b王･b3)･
Using test statictic机in (D･10)･If机<t,･bJ.,P･LOgeneit7 0f i th codity is accepted. The yalues ofⅦand q; are
For the Dutch data(tzQ.A.* ‑2. 064, F3,1V,AL.t ‑3. 01)
bi q.I
Food 0.042 1.702
8eyerages 0̲012 1.101 (6) DtLrables ‑0.046 ‑1.857
VH‑0. 972
For the Us data(tJt,0.0,f =2. 131,屯′sp.I =3. 29)
b ･. q.･
Beef ‑0.016 ‑1.802 Pork ‑0.002 ‑0.181 ChickizI 0̲025 3.173
(7)
VH‑1. 024
Thus, for both血ta, the hoLOgeneity of the etLtire todelare supported by the
ゐta.althoghthe ho■ogeneity of ●chickiJ)'is doutfu1.
Sy山etry test lJnder the hoJLOgeneity
Since the sp･etry constrai･,ts (丑β ‑0)are the lin飽r restrictions betTeeZL dif‑
ferent co･IOdites･the恥rginal pstrior detLSity of O ‑Eβ is not ayailable for otu use in the encfLj∴ forl･ If Te use the asy･totic density ･ as Sho.A
in (D. 12), resultaLnt test is mtericaly equiyalent to asyltOtic Chive test
. The alternatiye Tay tO the cotLYentional test is to coJL騨Ire tOdel(1) Tith del
(2) by use of the FBF.
‑150‑
2. 2ー Esti叫tion of lOdel(2) by Gibbs saIpler
Although Te CaJI‑t eYaltlate the posterior leans Of 0 e王aCtly. by using the tC‑
XC techiques･ it s pssible to estiTate these posterior teals Tith the required
precisiotL. Joint psterior detLSity p ( 0. 0IY) of Iodel(2)is giyen itL (E. 1).
The GibbS saJLPler algorithl for draTingO frot p (0,◎l y)is described in Appendix El･ Te generated ll,000 drats fro･ p (0,Oty).The first 1.000 draTS are discarded and the neェt 10.000 drats are used for the estilation of the pos‑
terior ･eans･Let I q･. ;i‑1･･･Nl denote the realized sequence of the j‑th elel
̲ JJ>
lent ofO ･posteri0‑ean ofOJ･ are eSti･ated by salple ･ean q‑ EPq/N･
Using these 10. 000 drats,Te Caluculated the posteror tehs. Besides teans. re‑
lated statistics are deriyed frol the sequence. e. g. NSE, CJ), 95Xcredibility izlter‑
yal etc. For assesing the acctlracy of the estiJLateS, USE are calucIJlated by
using (E.5)and (E.6). Te caluculated CD for detectitLg the conyergence of the
sequences鵬ere私‑1･ 000 and N8=5000･ The 95X iTIter▼als are given by using 2. 5th and 97･ 5th percentile苧Of 10,000 drats. All of the statistcs are suLtarized ill
Table. 1.2(for the batch data)and Table 2.2(for the Vs data).For all regression coefficients, the Values of NSE are qtLite sTall and according to CD yaltleS, re‑
alized sequences are supposed to be stationary. As cotpred Tith Table 5.6 aJld
Table 7. 5 in Theil(1976). the pint estilateS Of lodel paraJLeterS are PairTise yery close, but the discrepncies arise for the intez,yal estilateS aS folloTS;95先inter▼al reduced fro■
Table 7. 5 in Theil(1976)
ParaJLeter : ■argiJIal share
Beef L 596, . 788]
Pooh l. 113. . 285]
Chikitl l. 018, . 100]
LaIb l.026, . 074]
95X credibility iJ)ter▼al in
JTable 2.2 (the Vs data)
[.579, .810】
[.094, .299]
[.011, .108] (8) [.020, .077】
The iTIterYals at left side are shorter in Tidth than the ones atright side for
each co■10dity. The saJe discrepancies are seen betTeen Table 1.3 aLnd Table 1
in Theil and LaitineJI(1979).The sotlrCe Of these dicrepaJlCies is, e.g. tJ)e Stan‑
dard erros in Table 7. 5 are the appro王itate erros based the sptotic ti(eory̲
The negatiye se■i‑definiteness of slutsky Iatrix ( CozLCa▼ity constraints).
The negatiye seJi‑definiteness is the 3rd restrictions for slut軸tatri叫〕･
A coJIOn Tay testing this concayity constraints is to coIPIte the eigemyalues
of【25]･ All of the nob‑zero eigenyalues mSt be negatiye for the concayity conditions tO hold･ Sop any騨,Sitiye eigenyalues JIeanS that the coTICaYity condi‑
tions is rejected. JCXC tethod Till be the only feasible lay tO CaltlCtllate the
probability that the concayity restrictions hold. Dsing the Gibbs saJpler,Te generated 20･ 000 drats fro‑odel(2)･ For each draT･ the eigenyalues of 【2Tj･ ] are caluculated･ After deleting any draT Tith positiye eigetL▼ales, Te COnPIJtedthe proprtion satisfying the eigenyaltle test. Besides of it, the estiIateS of
l3g restricted by the concayity conditionsare obtained by ayeragi喝the dra･
Thick satisfied the eigenyalue test.
The proprtiotL that the concayity codition hold is 0̲ 9995 (for the甘さ
data). aJId 0. 6401 (for the htch data). The restricted estiJrateS Of
tlzLder the coIICaYity conditio▼JlS aS folloTS̲･
Restricted estitates (for the DtLtCh data)