TECHNOMETRICS SUPPLEMENTARY MATERIALS Poserior Confidence Inervals in Linear Calibraion Problems: Calibraing he Thomson Ice Core Inde (Eended Version) J. Huson McCulloch Economics Dearmen Ohio Sae Universi Oc. 3 00 Kewords: Baesian inerval credible inerval diffuse rior aleoclimaolog ogen- 8 Medieval Warm Period Hocke Sick Absrac The auhor is indebed o Sehen McInre Jussi Collin aricians in he OSU Economerics Seminar and Saisics Colloquium and several JASA and NSF referees for helful commens and suggesions. Auhor s e-mail: mcculloch.@osu.edu.
In calibraion roblems an eogenous sae variable and an endogenous resonse variable or ro are boh observed in a se of calibraion observaions. We wish o make inferences abou he unobserved sae variable from an addiional observaion on he resonse variable under a diffuse rior. Hoadle (970) argued ha an informaive rior is required in order o obain a roer oserior disribuion. Huner and Lambo (98) roosed a soluion bu were sharl criicized a he ime. This aer resens a new derivaion of he Huner-Lambo oserior disribuion under a diffuse rior ha mees hese objecions. A he same ime i is shown ha Hoadle s aroach was based on a suble inconsisenc in he alicaion of Baes Rule. The reinsaed Huner-Lambo oserior is alied o he roblem of calibraing he ice core inde of Thomson e al. (003) o insrumenal emeraures. I is found conrar o he famous claim of Gore (006) ha his inde is in fac uninformaive abou he quesion of wheher Medieval Warm Period was warmer or cooler han he resen. I is shown ha he classical confidence inervals roosed b Fieller (954) are a good aroimaion o he oserior confidence inervals when he calibraion sloe coefficien is highl significan relaive o he desired confidence inerval ail robabili. However when he sloe is onl marginall significan he Fieller inervals become increasingl disored and hen meaningless. The roosed oserior confidence inervals siml become wider as he sloe loses significance bu remain bounded. Eensions o mulile roies sequeniall organized daa and rior resricions on he calibraion sloe coefficien are oulined bu no imlemened.
I. Inroducion In he classical calibraion roblem as reviewed b Osborne (99) and Brown (993) an eogenous sae variable i is measured indirecl b an endogenous resonse variable i ha has an affine bu nois relaion o i: ε. () i i i We have a se of calibraion observaions on boh for i n. In he benchmark case considered here i is assumed ha he measuremen errors ε i are i.i.d. Gaussian wih E(ε i i ) 0 and var(ε i ). We wish o make inferences abou he unobserved sae variable for an addiional observaion or observaions ouside he calibraion se for which we have onl he observed resonse. This resonse obes he same rule ' ' ε '. () Such calibraion roblems arise in such diverse fields as hermodnamics harmacolog urolog and chemical analsis (cf. Brown 993) zoolog (du Plessis and van der Merwe 996) and aleoclimaolog (e.g. Mann e al. 999 Kaufman e al. 009). In economerics alicaions could include he reconsrucion of hisorical Gross Naional Produc from fragmenar records (e.g. Romer 989) or he hisorical sandard of living from heigh daa (e.g. Seckel 995). Solving () for ields ' ( ' ε ') / (3) so ha he naural or classical calibraion esimaor of roosed b Eisenhar (939) is ˆ ( ' ˆ) / ˆ (4) I here use and for he reconsrucion values following he nomenclaure of Brown (98).
where ˆ and ˆ are he Ordinar Leas Squares (OLS) esimaes of and from (). The eression in (4) is also he Maimum Likelihood esimaor of (Hoadle 970 Brown 993). The resen aer resens a new derivaion of he Huner and Lambo (HL 98) oserior disribuion for under an uninformaive diffuse rior. The resuling confidence inervals (CIs) or credible inervals as he are ofen called in he Baesian cone are alwas bounded and coniguous and ma readil be consruced wih equal ail robabiliies. The radiional or classical mehod of comuing confidence regions advocaed b mos auhors is based on a heorem of Fieller (954). However his aroach leads o unsaisfacor ses ha ma be unbounded a one or even boh ends and ma even be disconiguous. Even when he are bounded inervals he classical CIs are in mos cases ecessivel skewed awa from he mean of he calibraion values of he sae variable so ha he wo ail robabiliies are unequal. Hoadle (970) aems o consruc a Baesian alernaive bu is unable o do his wihou an unnecessaril informaive rior. The reinsaed Huner-Lambo mehod is alied o he roblem of calibraing he Thomson e al. (003) ice core isooe raio inde o insrumenal emeraure. This inde was made famous in Al Gore s An Inconvenien Truh (006) where i was said o rovide definiive confirmaion ha emeraures during he Medieval Warm Period (MWP) were lower han hose a resen. I is found ha he inde does iml a oin esimae of emeraure ha is uniforml below he 96-990 insrumenal average from he beginning of he series in 00-00 AD unil he 930 s. However he 95% oserior CI for he emeraure anomal icall eends from a leas -.8 C o
3. C so ha he reconsrucion is in fac uninformaive. Alhough his inde does no oin o a MWP i b no means rules one ou conrar o he claim of Gore. The muliro emeraure reconsrucion of Loehle and McCulloch (008) insead shows a saisicall significan MWP relaive o he bimillenial average during mos of 87-036 AD lus 49-65 AD as well as a significanl cool Lile Ice Age (LIA) during mos of 44-746 AD. Secion II of his aer resens he new derivaion of he HL oserior disribuion for he unobserved sae variable conrass i wih he aroach of Hoadle (970) and answers he criicisms ha were raised agains he HL aroach when i was firs ublished. Secion III consrucs a close aroimaion o he Thomson e al. (003) ice core isooe inde from source daa. Secion IV calibraes his inde o global emeraure and comues oserior 50% and 95% CIs. The concluding secion V comares hese inervals o he classical Fieller regions and discusses eensions of he mehod of Secion II. II. Poserior Confidence Inervals for he Classical Calibraion Esimaor The join rior disribuion for ' and he regression arameers ( ) ha governs he enire eercise ma be facored as follows: ( ) ( ) ( ) ( ). (5) Baes Rule imlies ha he disribuion of condiional on and he rue arameers is ( ' ' ) ( ' ' ) ( ' ) (6)
4 so ha in he benchmark case in which he rior densi comonen ( ) ' is diffuse and herefore uninformaive (cf. Zellner 97: 4-53) ( ) ( ) ( ) ( ) ( ) ( ) ( ) / / ' N ~ / / ' ' e ' e ' ' ' ' (7) whence 0 0 0 ' N ~ '. (8) The disribuion of he OLS esimaorsˆ and condiional on he rue arameers is ˆ S S N ~ ˆ ˆ where i i i n s s s s S. Then under he sandard diffuse rior for ( ) (Zellner 97 ch. 3) (9) S S ˆ ˆ N ~ ˆ ˆ whence. (0) s s s s ˆ ˆ N ~ ˆ ˆ S
5 Since ' ( ' ) ' ields ε is indeenden of he regression daa combining (8) wih (0) ˆ s ' ˆ ˆ S ~ N ˆ s s s. () I follows from ' ( ') / ha where RN ( μ Σ) ( ) ˆ s s ' ˆ ˆ ~ RN ˆ s s S () indicaes he disribuion of he raio of wo normall disribued random variables wih mean vecor μ and covariance mari Σ (Fieller 93 Hinckle 969). This disribuion has as secial cases he Cauch disribuion when he numeraor and denominaor boh have mean 0 and he Gaussian disribuion when he denominaor has zero variance. In general i is heav-ailed wih ails inermediae beween hese wo cases and is skewed unless he numeraor or denominaor has mean 0. I ma even be bimodal if he numeraor is large relaive o is sandard deviaion and he denominaor small relaive o is sandard deviaion. See Marsaglia (965) for grahs of he densi in he secial case of zero correlaion. The Cumulaive Disribuion Funcion (CDF) P ( ' ' ˆ ˆ S ) as required for confidence inervals on ma easil be comued in erms of he bivariae normal CDF as shown in Aendi I of he resen aer equaion (6). I ma easil be seen ha condiional on he rue arameers he classical oin esimae ˆ in (4) has he same RN disribuion as () onl wih he OLS esimaes of and relaced b heir rue values:
6 s s s ( ) ˆ S ~ RN. (3) The RN oserior disribuion for in () is herefore aricularl naural and inuiive. Williams (969) noes in he cone of (3) ha he RN disribuion has undefined mean and variance. In fac i lies he domain of aracion of he Cauch disribuion so ha even he Generalized (sable domain of aracion) Law of Large Numbers canno be relied uon o make he disribuion of he average more comac han ha of he conribuions. Care should herefore be eercised in merging mulile classical esimaes b simle averaging as in Loehle and McCulloch (008) or even b GLS as in Brown (98). In racice is unknown and mus be esimaed from he variance of he residuals s s n i e i /( n ) where e i i ˆ ˆ. Under a diffuse and herefore uninformaive rior for log( ) so i (Zellner 97: 4-53) ha ( ) Then ( s ) ~ ( n ) s / χ. ˆ ˆ S s ( n) ˆ ~ BVT ˆ s S n (4) where BVT(μ V ν) reresens he elliical bivariae Suden disribuion wih mean μ covariance mari esimae V and degrees of freedom ν (cf. Zellner 97 ch. 3 aendi B.; Genz 004). I follows from () and (4) ha
7 where RT( Vν ) ( ) ˆ ' s s ˆ ˆ s ~ RT ˆ s n s s S (5) μ reresens he disribuion of he raio of wo elliical bivariae Suden random variables wih mean vecor μ covariance mari esimae V and ν degrees of freedom. Is CDF P( ' ' ˆ ˆ S s ) ma easil be comued in erms of he elliical bivariae Suden CDF as also shown in Aendi I of he resen aer. 3 The resen aer firs uses Baes Rule o derive ( ' ' ) ( ' ' ) under a diffuse rior for ( ' ) from. Then he regression arameers are esimaed using he calibraion daa and he rue values are inegraed ou o obain ( ' ' ˆ ˆ S s ). Hoadle (970: 36-364) likewise considers he use of a diffuse rior for. However he insead firs esimaes he regression arameers o obain he classical Suden disribuion wih n- degrees of freedom for funcion of : as a ( ) ( ) ( ˆ ˆ ) ' ' ˆ ˆ v / s ( n ) / S (6) ( n ) v( ) where ( ) s ( s s s ) v (7) is he esimaed variance of as a funcion of. Hoadle hen noes ha holding consan and considering his eression as a funcion of i has ails ha are asmoicall roorional o / and herefore inegrae o infini when mulilied 3 Huner and Lambo (98) reor he eac formula for he densi of he raio of wo Suden random variables bu hen one mus sill inegrae his densi numericall in order o comue CIs so ha i is easier jus o use he BVT CDF o comue he RT CDF direcl.
8 b a uniform rior. He concludes ha an imroer uniform rior on [] leads o a nonsensical Baes esimaor and herefore rejecs he uniform rior. Brown (98; 993. 98) and Osborne (99) concur wih Hoadle ha a roer rior is required in he case considered here of a single resonse variable. Hoadle consequenl inroduces an informaive rior ha is based on he assumion ha is drawn from he marginal disribuion of he calibraion values and shows ha his leads o he inverse esimaor favored b Kruchkoff (967) based on a calibraion regression in which he indeenden variable is regressed on he deenden variable. Alhough here are alicaions in which such a rior would be jusified he resen aer is rimaril concerned wih he imoran case of an uninformaive diffuse rior. 4 However here is a suble fallac in Hoadle s reasoning: Because he Suden densi (6) is condiioned on he regression daa Hoadle s rior for mus be as well Baes Rule requires no ( ' ' ˆ ˆ S s ) ( ' ' ˆ ˆ S s ) ( ' ˆ ˆ S s ) ( ' ' ˆ ˆ S s ) ( ' ' ˆ ˆ S s ) ( ' ). I is no immediael obvious wh condiioning he rior for ' on he regression daa would make an difference. However consider generalizing he join rior (5) ha governs he enire eercise o be condiioned on an informaion se Ω ha ma or ma no be em: 4 In he case of a sequenial sae variable ha follows a sochasic rocess wih saionar incremens an informaive rior ha deends on he neares of he calibraion sae variables is aroriae as discussed below. Even hen however his rior should be alied before he regression coefficiens are inegraed ou and no aferwards.
9 ( Ω) ( Ω) ( Ω) ( Ω). (8) Since Hoadle s rior for imlicil deends on he regression daa his Ω mus also conain he regression daa in each of he oher erms in (8) including ( Ω). However condiioning he rior for he regression arameers on he regression resuls is inconsisen wih using he same regression resuls o inform he oserior for he arameers. Hoadle s sequence of oeraions is herefore invalid. I has been shown here ha here are no real roblems wih he Baesian aroach under a diffuse rior rovided ha a diffuse rior for he condiional densi ( ' ) ha arises from he facorizaion (5) is alied before he regression arameers are inegraed ou raher han a diffuse rior for he condiional densi ( ' ˆ ˆ S s ) afer he rue regression arameers have been eliminaed. Huner and Lambo (HL 98) derived in densi form eacl he same RT oserior disribuion for as we do above bu using a diffuse rior for η insead of for iself as in he above derivaion. In he same issue of Technomerics Hill (98) Lawless (98) and Orban (98) sharl criicized HL under he assumion ha he were reaing he rior for heir η as being indeenden of ha for he regression arameers. Hill noed ha if he rior for η is uniform he imlici rior for mus be roorional o so ha he riors for η and canno boh be indeenden of he regression arameers and argues ha he laer is he more naural assumion. However indeendence of he riors as insised uon b he HL criics is in fac nowhere required. The full join rior for he eercise ma eiher be eressed in erms of as in (5) or in erms of he HL η as
0 ( ) ( η ) ( ) ( ) η. (9) The assumion ha our ( ) HL assumion ha heir imlici ( ) is consan wih resec o is equivalen o he η is consan wih resec o η. I is immaerial wheher eiher of hese is roorional o or is inverse since an such consan will siml dro ou in he normalizing inegraion when Baes Rule is alied as in (7). 5 III. The Thomson Ice Core Inde The reinsaed Huner-Lambo CI mehod is illusraed b calibraing he millenial ice core isooe raio inde of Lonnie Thomson e al. (003) o recen global insrumenal emeraures. This inde was famousl described b Al Gore in his Nobel Peace Prize-winning An Inconvenien Truh (006: 60-65) as follows: Scienis Lonnie Thomson akes his eam o he os of glaciers all over he world. The dig core drills down ino he ice eracing long clinders filled wih ice ha was formed ear b ear over man cenuries. Lonnie and his eam of eers... can... measure he eac emeraure of he amoshere each ear b calculaing he raio of differen isooes of ogen (ogen-6 and ogen-8) which rovides an ingenious and highl accurae hermomeer.... The hermomeer o he righ measures emeraures in he Norhern Hemishere over he as 000 ears.... [T]he so-called global-warming skeics ofen sa ha global warming is reall an illusion reflecing naure s cclical flucuaions. To suor heir view he frequenl refer o he Medieval Warm Period. Bu as Dr. Thomson s hermomeer shows he vauned Medieval Warm Period (he hird lile red bli from he lef below) was in comared o 5 HL (98: 37) erroneousl acce ha he difference beween heir aroach and ha of Hoadle (970) is ha he assume a uniform rior on heir η whereas Hoadle considers a uniform rior for. In fac he difference is ha HL condiion heir rior for η or equivalenl on he rue arameers whereas Hoadle inaroriael condiions his rior for on he arameer esimaes as noed above.
he enormous increases in emeraure of he las half cenur (he red eaks a he far righ of he char). These global warming skeics... launched a fierce aack agains anoher measuremen of he 000-ear correlaion beween CO and emeraure known as he hocke sick a grahic image reresening he research of Michael Mann and his colleagues. Bu in fac scieniss have confirmed he same basic conclusions in mulile was wih Thomson s Ice core record as one of he mos definiive. (Gore 006: 60-64). McInre and McKirick (003 005) had argued ha he hocke sick reconsrucion of Mann e al. (999) is based on invalid sribark brislecone reering daa as well as imroer use of Princial Comonens Analsis. As i haens he grah ha Gore (006) resens as Dr. Thomson s Thermomeer acuall was he disued hocke sick emeraure reconsrucion for which i was suosed o rovide definiive indeenden confirmaion sliced ogeher wih an insrumenal emeraure record as if he were a single series and had nohing o do wih Thomson s ice core daa. 6 Thomson who served as a member of he Science Advisor Board for An Inconvenien Truh had indeed ublished a similar-looking grah based on decadal averages of ice core ogen isooe raios in Figure 7 of Thomson e al. (003) which is reroduced in Figure below. 6 In fac wo of he series emloed b Mann e al. (999) for he crucial 000-400AD eriod were Quelccaa δ 8 O and reciiaion series so ha he hocke sick is no comleel indeenden of even he rue Thomson ice core inde.
Figure. Thomson e al. (003) Fig. 7 wih original caion. Panel d) of Fig. is he disued Mann e al. hocke sick reconsrucion iself overlain wih a Norhern Hemishere emeraure inde. Gore was suosed o have used anel c) bu misakenl used anel d) of he same figure insead merging is wo lines ino a single grah. 7 The slighl osiive values around 30 AD in anel d) 7 This subsiuion was confirmed b Lonnie Thomson in resonse o a quesion b he auhor a a seminar a Ohio Sae Universi Jan. 008.
3 became he hird lile red bli from he lef reresening he vauned Medieval Warm Period according o Gore. The acual Low Laiude comosie ice core inde in anel c) does urn u sharl in he 0 h cenur much like he insrumenall augmened hocke sick. However he aes a he o of anels a)-c) do no indicae emeraure bu merel comosie Z-scores comued from δ 8 O ice measuremens. Thomson e al. in fac did no calibrae hese Z- scores o emeraure le alone rovide CI s for such a calibraion so ha Dr. Thomson s Thermomeer was no e a hermomeer a all. The resen aer aems o fill his deficienc b calibraing his inde o insrumenal emeraure and comuing confidence inervals. Thomson e al. rovide decadall averaged daa for 5 of he 6 cores used in heir aricle in a sreadshee online a <brc.osu.edu/icecore/climaic-change-003-fig5- able.xls>. Unforunael however he Himalaan comosie Z-score (henceforh HCZ) series is missing along wih he daa for Quelccaa so ha even hough he las column is idenified as 6 core comosie i is in fac based onl on he wo included Andean cores and is definiel no he series loed in anel c) of heir Fig. 7. On March 3 008 I e-mailed Lonnie Thomson and several of his co-auhors requesing he missing series in his sreadshee bu received no rel. Neverheless i is ossible o a leas aroimael reconsruc he 6-core comosie from he five decadal δ 8 O ice series in he sreadshee lus archived annual values for he Quelccaa Summi core wih he hel of formulas embedded in he sreadshee.
4 Figure shows he decadall averaged δ 8 O ice values from he sreadshee for he hree 3 Himalaan cores Gulia Dunde and Dasuou. 8 Each decadal value is idenified b he dae a he o of is inerval. These daes are even mulilies of 0 so ha he decades used for hese hree series are 00-00 0-00 ec. Noe ha onl Dasuou has daa for he final decade of he sud 99-000 (hrough 997). Gulia ends wih 98-990 and Dunde wih 98-987. All hree of hese Himalaan series visuall mach hose in Figure 5 of Thomson e al. (003). Figure. Decadal averages of δ 8 O ice for he hree Himalaan cores: Gulia Dunde and Dasuou. 8 The decadall averaged daa from his sreadshee for he 3 Himalaan cores has been archived a f://f.ncdc.noaa.gov/ub/daa/aleo/icecore/ro/dasuou/dasuou-d8o. <... /dunde/dunde-d8o.> and <... /gulia/gulia-d8o.>. However he decadal Andean daa from he sread shee is no archived here ece inadverenl a he lower righ side of he Gulia file. IPCC AR4 WGI eer reviewer Sehen McInre (006 007) noes ha several inconsisen versions of Gulia and Dunde have been used in he climae lieraure. The resen aer siml uses he version in he sreadshee consruced for Thomson e al. (003).
5 Onl wo of he hree Andean cores used Sajama and Huascarán Core are abulaed in he online sreadshee. Each of hese decadal values is associaed wih wo daes idenified as boom and o decade bu boh are even muliles of 0 saring wih (000 00) (00 00) ec. Evidenl hese Andean cores were averaged over decades beginning wih an even mulile of 0 such as 000-009 00-09 ec. so ha he header o decade refers o he firs ear of he following decade. This inerreaion is reinforced b an annoaion ha he mos recen value for Huascarán Core covers he eriod 990 o 993. The hird Andean core is idenified b Thomson e al. (003) onl as Quelccaa even hough annual δ 8 O ice daa is archived a f.ncdc.noaa.gov for wo differen Quelccaa cores Core and he Summi Core. I was found ha averages of he Summi Core for decades beginning wih even muliles of 0 gave a erfec visual fi o he grah of he Quelccaa series in Fig. 6 of Thomson e al. (003) while decades ending wih even muliles of 0 had obvious differences. Core had obvious differences under eiher inerreaion and clearl was no he source of heir Fig. 6. The average of he wo cores also has obvious differences. The archived annual daa for he Quelccaa Summi Core (Thomson e al. 005) onl eend o 984 e Thomson e al. (003) use daa for Quelccaa hrough 000 based on newer shallow cores (see caion o heir Fig. 7 in Fig. above) ha are no available numericall. Accordingl hese wo values were read visuall off Fig. 6 of Thomson e al. (003) as -7.67 er mil for 980-989 and -6.64 er mil for 990-999.
6 Decadal averages of δ 8 O ice for he hree Andean cores Quelccaa Summi Sajama and Huascarán are shown in Figure 3 below. The Sajama series is a good visual mach o ha in Figure 6 of Thomson e al. (003). The relaive sizes of he local eaks in he Huascarán series since 790 are no quie he same as in Figure 6 of Thomson e al. (003) alhough he general shae of he wo series are ver similar. As noed above Quelccaa is a erfec mach bu onl rovided he Summi Core is used wih decades beginning wih 0. Figure 3 Decadal averages of δ 8 O ice for he hree Andean cores Quelccaa Summi Sajama and Huascarán. The ice-core indices shown in Fig. are comosie Z-scores are suosed o be derived somehow from he decadal δ 8 O ice daa shown in Figs. and 3 above bu
7 Thomson e al. (003) do no rovide deails of how he were comued. Were he δ 8 O ice values averaged and hen Z-scores comued or were he averages comued from Z-scores? Were he means and variances for he Z-scores comued before or afer decadal averaging? Were he comued using each enire series or jus he orion afer 000 AD used for he sud? And were he means and variances comued o he end of each series or onl using he common orions? Alhough he rovided daa sreadshee is incomlee i does conain embedded formulas ha reveal how he Z-scores would have been comued if he daa were resen: Firs each decadall averaged series was convered o Z-scores using is mean and sandard deviaion for all is available decades bu resriced o he eriod of he sud i.e. beginning wih 00-00 for he Himalaan cores and wih 000-009 for he Andean cores. These Z-scores are shown in Figures 4 and 5 below:
8 Fig. 4 Z-scores for he hree Himalaan cores.
9 Fig. 5 Z-scores for he hree Andean cores. Then regional comosie Z-scores were comued b averaging each region s Z- scores as available. The resuling Andean Comosie Z-score series (ACZ) shown in Fig 6 below is hus siml he average of he hree comonen Z-scores for all 00 decades. Unforunael however his series is no a erfec visual mach o Fig. a of Thomson e al. (003) (Fig. a above). Alhough mos of he high frequenc wiggles mach he local high in he 70s and he record low in he 80s are greal aenuaed in he Thomson e al. version. Furhermore alhough he 0h cenur is generall he highes in he Thomson version i is no quie as high as he h-4h cenuries in our emulaion. Perhas Thomson e al. in fac used Quelccaa Core or he average of he
0 wo Quelccaa cores in he consrucion of he inde even hough he Summi core was clearl wha was used in heir Fig. 6. Fig. 6 Andean Comosie Z-score series ACZ. The emulaed Himalaan Comosie Z-score (HCZ) series shown in Fig. 7 below is based on all 3 Himalaan cores for he firs 99 decades (shown in blue) and hen abrul is based onl on Dasuou for he final decade (shown in red) afer he oher wo dro ou. The line segmen connecing he oin reresening he 980s (3 cores) and ha reresening he 990s (Dasuou onl) is also in red. The resuling series is a erfec visual mach o Panel (b) of Fig..
Fig. 7. Himalaan Comosie Z-score series (HCZ) using all hree Himalaan cores 00-990 (in blue) and Dasuou onl 99-000 (in red). The line segmen connecing 98-990 (3 cores) and 99-000 (Dasuou onl) is also in red. Finall he sreadshee formulas show ha wo regional comosies were averaged wih equal weighs o obain a single Low Laiude Comosie Z-Score (LCZ) as shown in Figure 8 below. The final decade as well as he line segmen connecing he 980s (6 cores) o he final decade (4 cores) are loed in red. This series is a ver good visual mach o he 6-core comosie series in Panel (c) of Figure above including in aricular he unrecedenedl high value for he final decade of he 990s. Our relicaion of he series in anel c) in Figure 7 of Thomson e al. (003) is herefore reasonabl successful desie is absence from he daa sreadshee and desie some inconsisencies in he Andean sub-inde.
Fig. 8. Low-Laiude Comosie Z-Scores based on 6 cores for 00-990 (in blue) and on 4 cores for 99-000 (in red). The line segmen connecing 98-990 (6 cores) and 99-000 (4 cores) is also in red. I is no clear wh one would wan o average Z-scores in his manner since i makes inefficien use of he daa. However his aears o be a common racice in aleoclimaolog (see e.g. Kaufman e al. 009). The rimar goal of he resen aer is merel o relicae and calibrae he Thomson e al. (003) comosie ice core series and no o imrove uon i. Secion V below suggess how his daa migh be used more efficienl. I is singular ha alhough he LCZ series in anel (c) of Figure ends on an unrecedenedl high value for he 990s and anels (a) and (b) of Figure above shows
3 ha his is being driven b is HCZ comonen raher han he ACZ comonen none of he individual Himalaan Z-scores shown in Fig. 4 ends on a record high value. This onl comes abou as a saisical arifac however because wo of he Himalaan series dro ou in he las decade leaving onl Dasuou which was running higher or warmer han he oher series o reresen he region b iself. The effec is accenuaed because when HCZ and ACZ are averaged ogeher o obain LCZ he ne weigh on Dasuou suddenl changes from /6 in he firs 99 decades o / in he final decade. Fig. 9 Himalaan Comosie Z-score (HCZ) comued from all 3 Himalaan cores (in blue) and from Dasuou onl (in red). Figs. 9 and 0 show HCZ and LCZ for he full eriod 00-000 AD comued boh from all 6 cores (LCZ6) as well as from onl he 4 cores ha make i o he final
4 decade of he sud eriod (LCZ4). Clearl he 990s are no an warmer han he 940s when eiher HCZ or LCZ is comued on a consisen basis desie he imression given in anels b) and c) of Fig.. Fig. 0 Low-Laiude Comosie Z-score (LCZ) comued from all 6 cores (in blue) and from onl he 4 cores available in he las decade (in red). The changing comosiion of LCZ would no aler is eeced resonse o emeraure if δ 8 O ice were a universall valid and linear indicaor of annual average emeraure aside from a locaion shif o comensae for differences in laiude and/or aliude. However some of he cores such as Quelccaa Summi and Dasuou are srongl correlaed wih annual average global emeraures while ohers such as Sajama and Dunde are no so ha he resonse is no universal. Some of he sies ma
5 be reflecing hisorical variaions in he seasonali or aliude of snow formaion or in reciiaion aerns as he air is ransored o he sie raher han average annual emeraures so ha he correlaion if an wih local or global annual emeraure can onl be deermined emiricall on a sie-b-sie basis. The makeu of he comosie series herefore can make a subsanial difference for is correlaion wih emeraure so ha in fac he final decade of LCZ (based on 4 cores wih / weigh on Dasuou) is a quie differen series han he receding 99 decades (based on 6 cores wih equal weighs on all) and canno be used in is calibraion. IV. Calibraion o Global Temeraure Thomson e al. (003) sae ha Comarison of his [LCZ] ice core comosie wih he Norhern Hemishere ro record (000-000 A.D.) reconsruced b Mann e al. (999) and measured emeraures (856-000) reored b Jones e al. (999) suggess he ice cores have caured he decadal scale variabili in he global emeraure rends. Of course i would be circular o calibrae LCZ o he hocke sick and hen o u i forward as indeenden confirmaion of he hocke sick so onl he comarison o insrumenal emeraures is relevan for our uroses. Alhough Thomson e al. (003) comare heir LCZ favorabl o a Norhern Hemishere insrumenal emeraure inde a global emeraure inde would more aroriae for univariae calibraion given ha half of he si cores are from he Souhern Hemishere. Figure below shows annual averages for 850 009 of he CRUTEM3vGL global land air inde (variance adjused) of Brohan e al. (006) a
6 global sequel o he Jones e al. (999) series o which Thomson e al. comare heir series. 9 Fig.. CRUTEM3vGL global land air emeraure inde annual averages 850-Oc. 009. The widel used CRU series are ariall based on confidenial weaher daa ha has no been made ublicl available (see Me Office 009). Since he are no scienificall relicable he should onl be used wih cauion. However he alernaive GISSem series roduced b NASA/GISS is onl comued back o 880 and hence has hree fewer decades han he CRU series. Preliminar calculaions show ha he 9 Source <h://www.cru.uea.ac.uk/cru/daa/emeraure/cruem3vgl.>. The annual average loed for 009 is incomlee running onl hrough Ocober. However he calibraion onl uses he orion hrough 000.
7 correlaions wih GISSem are much weaker han wih CRUTEM3vGL bu his is more likel due o he smaller samle size han o an difference in behavior since 880. Since half of he Thomson e al. decadal ice core series are based on decades ending in a ear divisible b 0 while he oher half use decades beginning in a ear divisible b 0 i is a maer of indifference which convenion is used for he emeraure series. This aer arbiraril ados he former convenion i.e. 85-860 ec. and reas LCZ as if i were comued consisenl on his basis. Since he 4-core LCZ series LCZ4 does no necessaril have he same relaionshi o emeraure as he 6-core series LCZ6 he wo mus be calibraed searael o emeraure. Table below shows he resuls of regressing he 4 decadal values of LCZ6 for 85-990 on he corresonding decadal averages of he emeraure series. For uroses of comarison he 4-core inde LCZ4 using onl Dasuou for he Himalaan region wih a / weigh as in 99-000 was comued for all 5 os-850 decades 85-000 and also regressed on he emeraure series as shown in he las wo columns of he same able. Table Decadal Regression of 6- and 4- core LCZ on CRUTEM3vGL 85-990 and 85-000 res. LCZ6 LCZ4 ˆ ˆ ˆ ˆ coef. 0.84.794.9.46 s.e. 0.56 0.934 0.95 0.69 -sa 3.90.9 6.5 3.0 (-sa) 0.006 0.079 0.000 0.008 n 4 5 R 0.35 0.45 s 0.363 0.354 s s 0.86 0.083
8 r -0.005-0.007 DW.948.005 I ma be seen ha here is a osiive and weakl significan correlaion beween LCZ6 and insrumenal emeraure (-ailed 0.079) and a osiive and highl significan correlaion beween LCZ4 and emeraure (-ailed 0.008). The firs order serial correlaion coefficiens of he residuals r are esseniall 0 and he Durbin- Wason saisics DW are ver close o so ha osiive serial correlaion is no an issue wih hese regressions a leas no a his decadal frequenc. Fig. CRUTEM3vGL global emeraure inde wih reconsrucions using he 6-core LCZ and he 4-core LCZ along wih oserior median esimaes.
9 Fig. shows he decadal averages of CRUTEM3vGL used for he calibraion along wih he 6- and 4-core reconsruced emeraures using he classical esimaor (4). Also shown are he medians of he oserior disribuion for he 6- and 4- core reconsrucion. In mos cases he classical reconsrucion and oserior median are almos indisinguishable. Since he classical esimaor (4) is much easier o comue and elain i is naural o focus on i as he oin esimae of reconsruced emeraure. I is conjecured ha he classical esimaor coincides wih he oserior mode. Even hough LCZ4 eceeded all revious values of LCZ6 during he decade of he 990s he emeraures reconsruced from i are somewha lower han hose of he 940s using LCZ6. This comes abou because he esimaes of boh he inerce and sloe are higher for LCZ4 han for LCZ6 in Table. Boh hese facors work o reduce he reconsruced emeraure as a funcion of he inde. Figure 3 shows he emeraure reconsrucion for 00-000 using he classical oin esimae (4) raher han he oserior median. The reconsrucion is based on LCZ6 for 00-990 (in blue) and on LCZ4 for 99-000 (in red). The line segmen connecing 98-990 (6 cores) and 99-000 (4 cores) is also in red. Alhough he final decade has a decidedl warm oin esimae i is no as warm as he 940s conrar o he imression given in Figs. and 8.
30 Fig. 3. Reconsruced emeraure oin esimaes using (4) wih LCZ6 for 00-990 (in blue) and LCZ4 for 99-000 (in red). The line segmen connecing 98-990 (6 cores) and 99-000 (4 cores) is also in red. Of course since we have relaivel recise insrumenal emeraures for 850 o he resen he reconsrucion in Fig. 3 is no our bes esimae of global emeraures for hese ears. The full reconsrucion is rovided here and wih is confidence inervals below merel o show how he full Dr. Thomson s hermomeer calibraes o emeraure. The oserior cumulaive disribuion funcions (CDFs) for he LCZ6 reconsrucion for he 99 decades 00-00 o 98-990 were comued in ses of.0
3 from -5.00 o 5.00 C as shown in Fig. 4 below. 0 Because he sloe coefficien is onl weakl significan for his reconsrucion (.079) he ails of he oserior disribuions are eremel heav and slow o converge o 0 and for low and high emeraures. Fig. 4. Poserior CDFs for 99 LCZ6 emeraure reconsrucions 00-00 o 98-990. Fig. 5 below shows he oserior CDF for he LCZ4 reconsrucion comued for he sake of comarison for he 5 calibraion decades 85-860 o 99-000. Since he sloe coefficien in his regression is highl significan ( 0.008) he disribuion is 0 CRU (<h://www.cru.uea.ac.uk/cru/daa/emeraure/>) gives he recision of CRUTEM3vGL ( sandard errors) as aroimael 0.05 C since 95 and abou 0. C in 85 wih gradual imrovemen from 860 o 950 ece for warimes so ha ses of 0.0 C are overkill. Neverheless using Malab s mvcdf as described in Aendi I he rogram for all 00 reconsrucion daes runs in jus a few minues on a desko PC.
3 nearl Suden wih 3 degrees of freedom and he ails converge much more quickl o 0 and for low and high emeraures han wih LCZ6. Fig. 5. Poserior CDF for 5 LCZ4 reconsrucions 85-860 o 99-000. The oserior CDFs in Figs. 4 and 5 were invered b linear inerolaion a robabiliies 0.05 0.5 0.50 0.75 and 0.0975. The resuling 50% and 95% LCZ6 Confidence Inervals (CIs) are loed in blue for 00-990 in Fig. 6 below along wih he classical oin esimaes from Fig. 3. Corresonding values for LCZ4 are loed in red for 99-000. The line segmens connecing 98-990 (6 cores) and 99-000 (4 cores) are also shown in red.
33 Fig. 6. LCZ6 emeraure reconsrucion for 00-990 wih 50% and 95% Confidence Inervals in blue. LCZ4 emeraure reconsrucion for 99-000 wih 50% and 95% Confidence Inervals in red. Line segmens connecing 98-990 (6 cores) and 99-000 (4 cores) are also in red. The arial decade 00-009 was he warmes in he insrumenal record used a 0.64 C relaive o 96-90 0. I ma be seen from Fig. 6 ha emeraures hroughou he eriod 00-850 could have been a leas. C warmer han 96-90 or a leas.7 C colder. The Medieval Warm Period (MWP) or even he Lile Ice Age (LIA) for ha maer herefore could well have been even warmer han he mos recen decade so far as his reconsrucion goes. (Since 00-009 was no used in he calibraion i is no shown in Fig..)
34 An even beer Thomson δ 8 O ice series o calibrae should have been he 7-core inde of Thomson e al. (006). This inde adds a sevenh Himalaan core Puruogangri and eends back 000 ears where hree of he cores sill have daa. B all righs i should suersede he 6-core 000 ear inde of Thomson e al. (003). Unforunael however he suoring daa ses ha accoman ha aer onl abulae he seven cores back 400 ears as five-ear averages so ha he differen subses of he seven cores ha are acive in differen eriods canno be searael calibraed as he mus be. Furhermore he decadall averaged Z-score indices in Daa Se 3 do no mach he averages of he illusraive five-ear average Z-score indices for eiher region ha are abulaed in Daa Se so ha here is no linear relaionshi beween he archived 000- ear comosie Z-score inde and he illusraive 400-ear δ 8 O ice daa on which i is suosed o be based as documened b McCulloch (009a 009b). Three e-mails o Lonnie Thomson and mos of his co-auhors asking for he comlee daa and clarificaion of his inconsisenc received no rel. The resen sud herefore focuses insead on he 000-ear 6-core δ 8 O ice inde of Thomson e al. (003) which is a leas aroimael relicable in erms of is full abulaed decadal comonen series. V. Furher Calibraion Issues This concluding secion discusses a number of furher calibraion issues: i) he classical join samling confidence inervals discussed b Hoadle (970) and Brown (993) ii) efficien muliro calibraion iii) a more owerful sequenial rior aroach; and iv) efficien use of rior informaion abou he sign of he calibraion sloe
35 coefficien. No aem is made a resen o imlemen he roosed soluions o issues ii) iv). i) The radiional or classical alernaive aroach o calibraion CI consrucion is based on Fieller s (954) confidence region for he raio of wo Suden random variables. This aroach firs considers a 00(-γ) CI for as a funcion of using he sandard Suden disribuion (6) for ( ' ' ˆ ˆ S s ) b a air of herbolic funcions of : ( ) / ˆ c ˆ v( ) / ˆ ˆ c v. These CIs bound where c ( γ / ) is he Suden -ailed criical value for es size γ Tν() is he T n sandard Suden CDF wih ν degrees of freedom and v ( ) is as defined in (7). Le ( T ( ˆ ( s ) n / s be he wo-ailed -value of he es saisic for he hohesis ha ˆ 0. If < γ so ha he sloe is significanl differen from 0 a level γ solving he boundaries for as a funcion of he observed ro value ields a quadraic equaion wih wo real roos ξ < ξ where inersecs he wo herbolas and he confidence region is he bounded inerval (ξ ξ ). Bu if γ his inerval becomes eiher (ξ ) (- ξ ) or (- ) deending on he signs of ˆ and where is he mean of he calibraion i -values. And if > γ so ha he sloe is insignificanl differen from 0 a level γ eiher wice inersecs one of he herbolas and he confidence region is he disconiguous se ( ξ ) (ξ ) or else here are no real roos in which case he confidence region is he enire real line. Hoadle (970) drl remarks ha his se ossesses inheren difficulies.
36 Brown (98) rooses ha (ξ ξ ) is a resecable inerval rovided he -es of he hohesis 0 is rejeced. Indeed if sa ˆ > 0 and i were known wih erfec cerain ha has he same sign hen for an < ξ he robabili ha could have been as high as is observed value would be less han or equal o γ/ while for an > ξ he robabili ha could have been as low as is observed value would also be less han or equal o γ/. The hich however is ha here in fac is alwas robabili / ha and ˆ have oosie signs. The Baesian aroach on he oher hand akes full accoun of he ossibili ha and ˆ migh have oosie signs and never gives unbounded CIs le alone disconiguous confidence regions. When > γ as in he LCZ6 eamle above he Baesian CI is siml somewha wider han i oherwise would be as is onl naural. Fig. 7 comares he oserior and Fieller 50% CIs for our LCZ calibraion. Since for LCZ6.079 <<.50 he Fieller 50% region is an inerval and he wo are ver similar. For he final decade based on LCZ4.0008 so ha he similari is even closer. In general he classical CI is a good aroimaion o he Baesian CI when << γ so ha here is onl negligible chance relaive o γ ha and ˆ can have oosie signs. Indeed Huner and Lambo (98) hemselves acuall advocae i as an aroimaion.
37 Fig. 7. Comarison of oserior (lines) and Fieller (shaded) 50% confidence inervals. However when he sloe is onl marginall significan or even insignifican relaive o he desired γ as is ofen he case in aleoclimae cones he eac Baesian inerval is o be referred. Fig. 8 comares he oserior and Fieller 95% confidence regions for our LCZ calibraion. Since for LCZ6 we now have.079 > γ.05 he Fieller region is unreasonabl eiher he enire real line or else a air of semiinfinie inervals. For he final decade however.0008 <<.05 so ha he Fieller 95% region becomes an inerval and is a good aroimaion o he oserior inerval.
38 Fig. 8. Comarison of oserior (lines) and Fieller (shaded) 95% confidence regions. ii) When several roies are available and here is an a riori eecaion ha he will have he same quaniaive relaionshi o he sae variable ne of a locaion shif heir average ma siml be univariael calibraed o he sae variable. However when as in he case of he Thomson e al. (003) ice core daa he individual roies resond unequall o he sae variable and someimes no a all a muliro aroach should be more efficien han arbiraril averaging eiher he raw roies hemselves or heir Z-scores as in Thomson e al. (003 006) and Kaufman e al. (009). Of course one ma no siml disregard he insignifican roies or hose ha give he wrong
39 sign sloe for hen here will be selecion bias ha will invalidae convenional significance ess. Suose here are q roies wih ij being he i-h calibraion-eriod observaion on he j-h ro and j being he reconsrucion-eriod observaion on he j-h ro. Assume firs ha ij ε (0) j j i ij where ε ij ~ N (0 i ) and are indeenden across roies. Le ( ) T j j K nj and T ( n K ) so ha ˆ j ˆ j and s are imlici in j and and ( ) j ˆ j ma be comued under a diffuse rior as in Secion of he e. Then under a diffuse rior for q (...... ) ( ). () q q j j If he regression errors are no indeenden across roies as reliminar calculaions show o be he case wih he Thomson e al. ice core daa hen () is invalid. However if we have q n- roies we ma obain errors ha are uncorrelaed across roies and herefore indeenden under our Gaussian assumion b insead regressing each ro j on he sae variable lus roies... j- during he calibraion eriod i... n as follows: j j j ij j j j i h γ jh ih ε. ij The calibraion OLS esimaes of hese arameers are imlici in he arial calibraion daa... j and. Using he reconsrucion values of he roies define he orhogonal innovaion of ro j and is esimae b
40 j j j h γ jh h for j and j h ˆ ˆ γ j j jh h ˆ. Then ( ˆ... ) ma be comued as in () j j siml b adding he aroriae erms o (6) (0). Equaion () above hen generalizes o q ( ) (... q... q ˆ j... j ). () j Equaion () or () ma be evaluaed numericall eiher b calculaing he RT densiies direcl as in Huner and Lambo (98) or b finie differences from he CDF and hen numericall inegraing he roduc of oserior densiies o normalize and obain CIs. A ransformaion such as Suden wih q- degrees of freedom ma be useful in order o caure he enire ails. When he roies are significanl correlaed wih he sae variable even afer condiioning on he receding roies and if he ell a consisen sor abou he reconsrucion sae variable he muliro oserior disribuion () will end o be much igher abou is mode han he single-ro reconsrucions would be. Bu when he roies are significan e ell an inconsisen sor abou he muliro oserior densi will end o be more sread ou or even mulimodal. CIs comued from his disribuion will hen end o be wide enough o include all or a leas mos of he individual modes. As long as all he roies have he same coverage across observaions he order in which he are arranged should make no difference for he muliro oserior
4 disribuion. However if he have differen coverage as is he case for he calibraion eriod wih wo of he Thomson e al. (003) roies and as is he case during he 000-ear reconsrucion eriod for several of he roies in Thomson e al. (006) i is imoran o arrange hem in decreasing order of coverage so ha all he required condiioning roies are available a each se. If a ro has more coverage in he reconsrucion eriod han he ohers bu less in he calibraion eriod or vice versa i ma unforunael be necessar o give u some daa oins. In he muliro case he oin esimae of ma be aken as he oserior median since if he roies are inconsisen here ma be mulile oserior modes. The GLS esimaor suggesed b Brown (98: eq..6) which generalizes he single-ro classical esimaor in equaion (4) is far simler o comue. However i ma be roblemaic because i is esseniall a weighed average of single ro esimaes each of which has Cauch-like heav ails wih undefined mean. The Law of Large Numbers herefore does no ensure ha he disribuion of he average will be more comac han he disribuions of he conribuing values. When he roies do no have indeenden simle regression errors and q n- as is for eamle he case wih he Kaufman e al. (009) daa () becomes invalid since evenuall zero and even negaive degrees of freedom will be encounered. In his case i ma be aroriae o reduce he dimensionali of he ro daa se b careful use of Princial Comonens Analsis (Preisendorfer 988 Mann e al. 999 McInre and McKirick 005). Such an aroach ma also be referred even when q < n- if n/q is no large.
4 iii). Secion II above reas he roblem of inferring a single reconsrucion value for an observaion ha has no sequenial relaionshi o he calibraion daa. In a ime series or oher sequenial cone however here ma be valuable informaion from wha is known or has been inferred abou he adjacen observaions. In aricular le us suose ha he calibraion sae observaions now wih a subscri... n o indicae ime follow a random walk so ha η (3) where η ~ iid N(0 τ ) and are indeenden of he regression errors. The random walk signal variance ma be esimaed from he calibraion daa b n ˆ τ ( ) /( n ) ~ τ χ n /( n ). Suose we have a ime series of calibraion ro values... n as well as an adjoining ime series of reconsrucion ro values... T wih T ec. as is ordinaril he case in aleoclimae emeraure reconsrucions. We wish o infer he T sae values... T which also follow he random walk (3) wih T ec. Le and reresen he corresonding vecors of reconsrucion values. The random walk imlies ha he disribuion of he sae variable k eriods before or k eriods afer n condiional on he calibraion daa has a normal disribuion wih mean or n and variance kτ and ha he uncondiional disribuion has infinie variance. I herefore moivaes he imroer diffuse rior assumion of Secion II for an isolaed reconsrucion dae ha is so far from he calibraion daa ha
43 he calibraion values become uninformaive. However when he reconsrucion dae is eiher close o he calibraion daa or surrounded b oher reconsrucion daes condiioning on his informaion can greal imrove he recision of he reconsrucion. Condiional on he rue arameer values he reconsrucion ro values and he calibraion sae vecor he oserior disribuions of he are Gaussian wih means and variances deermined b a simle Kalman smooher (e.g. Harve 989 McCulloch 005). This meshes he informaion in a forward Kalman filer ha sars a wih a diffuse rior as in Secion II wih he informaion in a reverse Kalman filer ha sars a T wih a rior ha is governed b and works backwards. Then he rue arameers ma be inegraed ou o find he disribuions of he condiional on he arameer esimaes. To simlif he noaion he following develomen of he Kalman filer and smooher is imlicil condiional on he rue arameer values and τ unil oherwise indicaed. The forward Kalman filer is iniialized wih a diffuse rior as in secion II so ha as in (7) ( ) ~ N ( ) ( / / ) N( c d v ) wih c / d / v. / Now suose ha for an... T he mean of filer densi for - is likewise affine in : (... ) ~ N( c d v ).
44 Then (3) imlies ha he redicive densi for ime is ( ) ( ) ~... τ v d c N. Baes Rule hen imlies ha he filer densi for iself is also affine in : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ~ / / ~............ v d c N v d c N N τ ) wih ( ) ( ) ( ) ( ) ( ) ( ) ( ). / / / / d g g d c g g c v g v v τ The reverse Kalman filer works b backward inducion beginning wih ime T using a rior based on he observed value of T and (3): ( ) ( ) ( ) ~ τ N T T Baes Rule hen imlies ha he reverse filer for T is again affine in : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). / / / / / ; ~ / / ~ τ τ ( ) T T T T T T T T T T T T T T T T T T T T T g d g g c v g v v d c N N N Reasoning analogous o ha for he forward filer hen imlies ha he reverse filer is likewise affine in for all :
45 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). / / / / ; ~... T T d g g d c g g c v g v v v d c N τ Finall merging ogeher he indeenden informaion in ( )... and ( ) ( ) ~... τ T v d c N ields he Kalman smooher ( ) ( ) ( ) ( ) ( ) ( ). / / / ; ~ d G G d D c G G c C v V G v v V V D C N τ τ (4) Smooher oin esimaes ˆ analogous o he classical oinwise esimaor (4) ma be found b evaluaing he mean D C using he arameer esimaes in lace of he arameers hemselves. In order obain valid confidence inervals i remains o inegrae ou he unknown arameers. Unforunael his is no longer deermined b he RN disribuion even condiional on he wo variances since now eners in a comlicaed wa ino C D and V. However he deendence on he inerce is sill simle since i onl aears in he affine eression for he mean. Equaion (9) imlies ( ) ( )( ) ( ) ( ) ˆ / ˆ N ~ ˆ ˆ ρ τ s s s S where. Therefore ( ) / / ρ s s s Noe ha he informaion se { } is equivalen o { } ˆ ˆ since once we know boh and ˆ and add no informaion. Therefore he informaion se ˆ { } ma be aken o imlicil include { } ˆ ˆ.
46 ( ˆ ˆ S τ ) ~ N ( ˆ ( / )( ˆ C D s s ) V D ( ρ ) ( s ). (5) The remaining hree arameers and τ mus be inegraed ou numericall o obain ( ˆ ˆ S s ˆ τ ) using ( τ ˆ s s ˆ τ ) ( ˆ s ) ( s ) ( τ ˆ τ ) and a uniform rior on log(τ ). These inegrals ma be erformed using ernar inegraion as described in Aendi II. Alhough his is a rile inegral i is robabl adequae o use ernar inegraion wih onl a modes number of oins over each arameer e.g. m 0 for and m 4 for he wo variances making a oal of 60 combinaions o evaluae. (In fac he sequenial srucure of he sae variable means ha he ro daa conain some indirec informaion abou and τ ha is no conained direcl in s or τˆ. However he gain from eloiing his indirec informaion is likel o be small.) The resuling sequenial reconsrucion will b consrucion equal he observed sae daa during he calibraion eriod. In order o make his rivial relaionshi clear he calibraion orion should be loed wih a differen color or smbol han he nonrivial orion. A he end of he reconsrucion eriod he reconsrucion will be cenered nearl on wih a ver igh disribuion. Moving back ino he reconsrucion eriod however he oin esimaes will begin o look like a smoohed version of he oinwise reconsrucion wih smooh CIs ha graduall widen. However because he informaion in adjacen ro values is being aken ino accoun he CIs will never become as wide as hose for he oinwise reconsrucion. Siml smoohing he oinwise CIs is herefore no equivalen o comuing a CI for he smoohed