Variance Stabilizing Power Transformation for Time Series

Transcription

1 Journal of Modern Appled Sascal Meods Volume 3 Issue Arcle Varance Sablzng Power Transformaon for Tme Seres Vcor M. Guerrero Insuo Tecnológco Auónomo de Méxco, guerrero@am.mx Rafael Perera Insuo Tecnológco Auónomo de Méxco, rafael.perera@publc-eal.oxford.ac.uk Follow s and addonal works a: p://dgalcommons.wayne.edu/jmasm Par of e Appled Sascs Commons, Socal and Beavoral Scences Commons, and e Sascal Teory Commons Recommended Caon Guerrero, Vcor M. and Perera, Rafael (004) "Varance Sablzng Power Transformaon for Tme Seres," Journal of Modern Appled Sascal Meods: Vol. 3 : Iss., Arcle 9. DOI: 0.37/jmasm/ Avalable a: p://dgalcommons.wayne.edu/jmasm/vol3/ss/9 Ts Regular Arcle s broug o you for free and open access by e Open Access Journals a DgalCommons@WayneSae. I as been acceped for ncluson n Journal of Modern Appled Sascal Meods by an auorzed edor of DgalCommons@WayneSae.

2 Journal of Modern Appled Sascal Meods Copyrg 004 JMASM, Inc. November, 004, Vol. 3, No., /04/$95.00 Varance Sablzng Power Transformaon for Tme Seres Vcor M. Guerrero Rafael Perera Deparmen of Sascs Insuo Tecnológco Auónomo de Méxco A confdence nerval was derved for e ndex of a power ransformaon a sablzes e varance of a me-seres. Te process sars from a model-ndependen procedure a mnmzes a coeffcen of varaon o yeld a pon esmae of e ransformaon ndex. Te confdence coeffcen of e nerval s calbraed roug a smulaon. Key words: Auocorrelaon, coeffcen of varaon, confdence nerval, model assumpons Inroducon Appled model-based sascal analyss usually requres some assumpons o be sasfed by e daa under sudy. Wen workng w meseres, covarance-saonary s ofen requred o begn e modelng process. Terefore s reasonable o look for a varance sablzng ransformaon a wll make e daa ge closer o fulfllng s assumpon. Wn e forecasng area, recall de Brun and Franses (999) concluson a daa ransformaons sould be consdered pror o forecasng. Tere are wo approaces o searc for e ransformaon. () Selec e ransformaon before acually buldng a sascal model for e me seres, or () decde wc ransformaon o use durng e model buldng process. In e laer approac bo model form and parameer esmaon nerac w e searc for e ransformaon. Vcor M. Guerrero s Professor of Sascs a Insuo Tecnológco Auónomo de Méxco. s researc neress are n e analyss and forecasng of economc me seres. e graefully acknowledges fnancal suppor from Asocacon Mexcana de Culura, A. C., n carryng ou s work. Emal m a: guerrero@am.mx. Rafael Perera graduaed from e Unversy of Oxford and s employed a e Deparmen of Prmary eal Care a Oxford Unversy. Emal m a: rafael.perera@publc-eal.oxford.ac.uk. In e former, e scale were e analyss sould be carred ou s fxed before aempng o buld a sascal model. Ts approac allows e analys o selec a ransformaon wou condonng on or nerferng w a gven model. Terefore s called model-ndependen. Te focus n s arcle s on a modelndependen meod a s useful o selec a power ransformaon a bes sablzes e varance of a me seres varable Z > 0, for =,,N. Suc a meod was proposed by Guerrero (993) as a ool o be employed wen e analys wans o use e power λ ransformaon famly: T(Z )= Z f λ 0 and T(Z )= log(z ) f λ=0 or wen usng s Box-Cox verson: Z (λ) =(Z λ -)/ λ f λ 0 and Z (λ) =log(z ) f λ=0. One of e mos mporan works a proposed e second approac for coosng a ransformaon s e exbook by Box and Jenkns (976). Tey suggesed usng e Box- Cox ransformaon n order o valdae no only e consan varance assumpon, bu all e underlyng assumpons of an Auo-Regressve Inegraed Movng Average (ARIMA) model by esmang e ransformaon ndex (λ) ogeer w e model parameers. Cen and Lee (997) proposed a Bayesan meod o coose e value of λ for a gven model srucure. Tose works are suppored by sound sascal eory, aloug n pracce ey presen e problem a e model form may depend on e ransformaon seleced. In fac, Goureroux and Jasak (00) ave sown a e 357

3 358 VARIANCE STABILIZING POWER TRANSFORMATION FOR TIME SERIES auocorrelaons (ence e ARIMA model srucure) cange as a funcon of e nonlnear ransformaon employed. Terefore, fxng e model form before selecng e ransformaon ndex could be napproprae n some cases. An advanage of e second approac for coosng e ransformaon s a a measure of varably, as well as a reference dsrbuon, can be obaned for e esmaed ransformaon ndex. Tus, s possble o dscrmnae among dfferen alernave values of λ based on à pror consderaons. For nsance, decdng weer e daa sould be analyzed n e orgnal scale (λ = ) or n logarms (λ = 0), can be performed on e bass of e daa a and. Ts does no appen w e frs approac because no model form and no reference dsrbuon exs a wll suppor e decson on an emprcal bass. Ts fac can be consdered a drawback of s approac. In s arcle, we consder s problem and work ou a feasble soluon by means of a confdence nerval for e rue λ value. In e followng secon a summary of Guerrero s (993) meod s presened a produces a pon esmae of e ndex λ by mnmzng a coeffcen of varaon. Ten, a confdence nerval s derved for e rue value of λ. Approxmae expressons for some sample momens nvolved n e calculaons are provded, and a reference dsrbuon for e rue coeffcen of varaon employed by e meod s suggesed. Some small sample smulaons are used o calbrae e confdence coeffcen of e nerval and o ge an nsg no e performance of e procedure. Nomnal confdence levels are relaed o realzed levels and, useful emprcal resuls are obaned. A secon s devoed o llusrae e use of e meod roug some emprcal applcaons. Tese examples elp o undersand ow e meod works n pracce. Selecon of e Transformaon Guerrero (993) proposed wo meods for selecng e power ransformaon ndex λ. Underlyng ese meods s e eorecal resul a saes a e coce of e ransformaon ndex sould be done n suc a var Z / λ / E Z olds vald way a [ ( )] [ ( )] c = for all and some consan c > 0. To use s resul, s necessary o esmae bo e mean and e varance nvolved. In appled me seres analyss ere s usually only one observaon a eac me, erefore var ( Z ) canno be esmaed and a resul canno be appled drecly. In order o operaonalze e resul, work w e observaons grouped no subseres. Ts enables e calculaon of pars of sample means and sandard devaons, for example, ( Z, S ) for =,...,, and en searc for e λ value a produces λ S / Z = c for =,, () for some consan c>0. Te elemens n s equaon are gven by Z Z / R and = R r = = R r= ( Z,r Z ) /( R ), r S, were Z, r denoes e r observaon of subseres. Te subseres { Z,,..., Z,r,..., Z,R }, for =,...,, are formed by groupng R consecuve observaons of e orgnal seres { Z : =,..., N}, ryng o keep omogeney beween e subseres. For s o appen ey mus be equal-szed. Terefore, some number (n) of observaons, w 0 n<r, wll ave o be lef ou of e calculaons, leavng R=(N-n)/. Te subseres sze mus be cosen appropraely, and be equal o e leng of e seasonaly, f suc an effec s presen n e seres. Te proposed meods semmed from wo emprcal nerpreaons of equaon (). Te frs one led o mnmzng e coeffcen of λ varaon of S / Z as a funcon of λ. Ts meod s no lnked o a formal sascal model and erefore no assumpons need o be valdaed o be appled correcly n pracce. Te second emprcal nerpreaon led o a meod based on a smple lnear regresson n logarms. Te assumpon of zero error auocorrelaon a underles s meod needs careful aenon as s seldom vald wen workng w me seres. Tus, e man meod, because of s robusness agans volaon of assumpons, s e one a mnmzes relave varaon. We sall concenrae on a meod.

4 GUERRERO & PERERA 359 A Confdence Inerval for λ To be able o make nferences abou λ, esmaed as e mnmzer of e coeffcen of varaon, we requre a reference sascal dsrbuon. To ge suc a dsrbuon we sar by assumng a e random varables λ W = S / Z for =,,, can be represened by a Movng Average model of W λ µ = and order. Ta s ( ) a W ( λ ) µ = a θa for =,,, w { a } a zero-mean we nose Gaussan process, µ>0 and θ a consan parameer suc a θ < o E λ =, ensure s nverble. Tus, [ W ( )] µ var[ W ( λ )] = σ and [ W ( λ ), W ' ] = ρ ' ± corr f =, and zero oerwse, w ρ = θ /( + θ ) ( 0.5, 0.5). Suc a model makes sense because λ s obaned n suc a way a W,..., W are approxmaely consan, bu a slg auocorrelaon srucure s expeced n e process { W ( λ )} gven a Z and S are calculaed from me seres observaons. Ts assumpon was valdaed by e smulaons repored below as e expeced beavor was observed. For e sake of smplcy, do no wre µ, σ and ρ even oug ese parameers are funcons of λ. Te sample counerpars of µ and σ wll be denoed as m W / and = = [ W m] /( ) ( ) se/m = = se so a CV λ = s e sample coeffcen of varaon. In wa follows we sall derve an approxmae dsrbuon for CV(λ), from wc a confdence nerval for e rue λ value can be obaned. Several proposals may be found n e leraure o oban e dsrbuon, ence confdence nervals, for a Normal coeffcen of varaon (see Vangel, 996, and e references eren), bu none of em allows for auocorrelaon n e observaons. We frs apply e Teorem n Appendx (known as e Dela Meod) o e bvarae case, w X = se, X = m and g ( X, X ) = X / X, o ge E CV λ E se / E m and [ ( )] ( ) ( ) var CV E ( se) var( se) var( m) cov( se, m). E ( m) + E ( se) E ( m) E( se) E( m) Ten, evaluae eac erm n s expresson as ndcaed n Appendx, so a E (m) = µ, var(m) E(se) = σ [ + ρ( ) / ] /, { ρ / / [ ( ) ]} / σ, [ ( ) ] var(se) σ / and cov( se, m) 0. ence, and E [ CV( λ )] σ µ ρ ( ) / σ var CV ( λ ) ( ) µ σ ρ ρ ( -) + + µ ( ) [ ( ) ] σ / µ were e las approxmaon follows from e fac a σ/µ mus be close o zero, snce λ s cosen o accompls a goal. I s clear a E[CV(λ)] σ / µ as and a s a decreasng funcon of ρ. In fac, wen ρ 0 we observe a E[CV(λ)]< σ / µ for all, and e oppose occurs wen ρ<0. Smlarly, s easy o see a var[cv(λ)] 0 as. Because e varance of CV(λ) s proporonal o e square of s mean, e logarm becomes an adequae varancesablzng power ransformaon (see Guerrero, 993, eq. 4). In urn, assume a (rougly) log[cv(λ)]~n(η,δ²). From e Lognormal dsrbuon, E[CV(λ)]=exp(η+δ²/) and var[ CV ] = [ exp( δ ) ] exp( δ + η). Tus, solve for η and δ², o ge log η = log{ E[ CV ]} δ / ( σ / µ ) + log{ - ρ/ -/[ ( -)]} / δ /

5 360 VARIANCE STABILIZING POWER TRANSFORMATION FOR TIME SERIES and log [ CV ] + [ CV ] var δ = log E ( ρ / ) log{ ρ / / [ ( ) ]} I s known a log CV α Pr δ σ Pr CV µ η z α ρ / / [ ( ) ] / ( ρ / ) exp ( δz ) = α w z α e 00α upper percenle of e un Normal dsrbuon. Te prevous asseron leads us o an approxmae 00(-α)% confdence nerval for e rue coeffcen of varaon. Because, ere s a one-o-one correspondence beween coeffcen of varaon and λ value, follows a an approxmae 00(-α)% confdence nerval for λ s gven by CI α ( ) λ = / ( ρ/ ) [ ] σ CV λ: exp δ µ ρ/ / ( ) ( z ) α. () In order for s confdence nerval o be useful n pracce, esmae CV as e mnmum sample coeffcen of varaon, denoed as CV ( λˆ ). Smlarly, use e esmaed frs-order auocorrelaon coeffcen, ρ= ˆ ( λˆ) W ( ˆ + ) W ( ˆ λ) m = W m λ m = To apprecae numercally e effec a α, and ρ ave on e leng of e confdence nerval, some calculaons are presened n Table for seleced values of ose consans. Ts able sows values of e funcon exp( δz α ) f ( α,, ρ) ( ρ / ) / /{ ρ / / [ ( ) ]} wc s e expandng facor of CV a defnes e leng of CI α. I s clear a f ( α,,ρ) ges smaller as: () α ges larger, () ges larger (n fac, f ( α,, ρ) as ) and/or () ρ moves from posve o negave values. Te frs wo of ese conclusons ave a clear nerpreaon n erms of confdence and sample sze. Te rd as no clear explanaon, bu sould be borne n mnd wen ryng o undersand wy wo smlar suaons, dfferng only n e sgn of ρ, wll yeld dfferen resuls (especally wen α and are small). In praccal applcaons, ypcally 6, so a ρ sould no be expeced o be e decsve facor n defnng e sze of e confdence nerval, bu we sould be aware of s poenal relevance. In order o beer undersand ow e meod works, n Fgure e grap s presened of CV(λ) agans λ for e Sales Daa a wll be consdered as an llusrave example below. Observe a e confdence nerval s obaned by slcng e curve produced by e coeffcen λ of varaon of e varable S / Z, for =,...,, as a funcon of λ. Te mnmum of s curve yelds CV( λˆ ) and e requred confdence nerval s bul by projecng on e orzonal axs e pons were e curve CV λˆ f α,, ρˆ, for a gven α value. reaces ( ) ( ) and an esmae of δ, say δˆ, can be obaned by usng ρˆ n place of ρ. Keep n mnd a e nerval () was derved from several approxmaons, n suc a way a e acual confdence level may dffer from e nomnal level and calbraon s requred.

6 GUERRERO & PERERA 36 Table : Expandng facor of CV(λ) as a funcon of α, and ρ. α \ρ Meodology Te confdence nerval for λ was derved from several approxmaons a may cause e acual confdence level o dffer from s nomnal level. In order o calbrae e confdence coeffcen, a small smulaon sudy based on e followng wo model specfcaons was conduced. Fgure. 95% Confdence nerval for λ bul from CV( λˆ ) for e Sales Daa. ) Z = T + S + I, were T =, S = δ D w D, q = f q, q q= = + q for = 0,,,- and 0 oerwse. { } ( ) ( σ = E Z λ ) and E ( Z ) T + S I ndependen ( 0, ) =. D, q = N σ w ) Z ( + φ) Z φz + a θa { } 0, σ = ( ) ( λ ) E Z and E( Z ) = ( φ) φ / φ =, were a ndependen ( ) [ ] ( ). N σ w Te frs one s a seasonal model w seasonaly leng R=. Te parameer values for e seasonal effecs were cosen as δ =, δ = 4, δ 3 = 5, δ 4 = 0, δ 5 =, δ 6 =, δ 7 = 3, δ 8 = 3, δ =, δ 0 = 0, δ =, 9 δ = so a q = δq = 0. Te sample szes were of e form N=, w =6,, 0, 30. Te second s an ARIMA(,,) model w nal values Z 0 =, and Z = w parameer values φ = 0. 7 and θ = In s case, e subseres sze was aken as R = 4 and e sample szes were N=4, 48, 80, 0, so a e values of became agan 6,, 0 and 30. Anoer exercse was carred ou w e laer model and R=3, and sample szes N=8, 36, 60, 90 o ge e same values for as before. For bo models, λ=0,0.5, was employed; us, wen λ ere s nonconsan varance, because depends on e mean of e seres. Jennngs (987) suggeson abou e way a smulaon sudes sould be repored was followed n order o provde nformaon no only on coverage raes bu also on bas. In Table, some resuls are presened from e smulaons for e seasonal model. Smlarly, Tables 3 and 4 sow e correspondng resuls for e nonseasonal model, w R=4 and R=3, respecvely.

7 36 VARIANCE STABILIZING POWER TRANSFORMATION FOR TIME SERIES Table : Observaons below\above of a 00(-α)% nomnal confdence nerval for λ and acual α, w Model and R= (000 samples). λ Acual α α z nom average z ac \5 9\3 4\ \9 5\6 4\ \3 6\6 6\ \55 5\50 45\ \58 56\53 49\ \6 \ 3\ \6 \ 3\ \5 35\3 8\ \7 36\5 9\ \48 6\50 59\ \48 63\53 60\ \ 9\4 3\ \3 9\4 5\ \0 8\6 9\ \0 3\6 9\ \39 56\45 5\ \4 57\47 57\ \ \ 8\ \ 4\ 8\ \7 37\5 30\ \8 40\6 33\ \35 76\5 60\ \36 8\8 6\ Table 3: Observaons below\above of a 00(-α)% nomnal confdence nerval for λ and acual α, w Model and R = 4 (000 samples). λ Acual α α z nom average z ac \9 7\ 9\ \3 4\38 0\ \3 4\46 3\ \45 4\8 4\ \46 4\87 43\ \3 0\0 7\ \4 0\ 7\ \9 \6 0\ \9 \63 0\ \4 0\3 37\ \48 \9 38\ \0 \9 4\ \0 \0 4\ \4 0\56 0\ \5 0\58 0\ \ 9\ 34\ \ 0\ 35\ \0 \7 \ \0 \7 3\ \3 \66 6\ \3 \68 8\ \0 4\7 37\ \ 7\4 39\

8 GUERRERO & PERERA 363 In ese ables, e nomnal confdence levels of e nervals were seleced by ral and error. Ta s, we ncreased e confdence level by an amoun of uns and looked for e levels a yeld acual coverage raes of 99%, 95% and 90%, wc are e mos commonly used n pracce. Te acual α values were obaned by averagng over e dfferen coverage raes obaned for λ=0,0.5,. Te group sze R= was used for e monly seasonal seres because s s e usual pracce. Tere s no commonly acceped value for nonseasonal me seres. For nsance, Guerrero s (993) advce was o employ R= n order o mnmze e loss of nformaon by groupng. owever, w s coce e esmaon of varably requred s very poor and peraps a value R> could perform beer. By lookng a Table s reasonably clear a =6 serves o oban acual confdence levels smlar o e nomnal ones. In Tables 3 and 4, e value of R was soug a makes e meod work well also for =6, wen e seres s nonseasonal. I was found a R=4 s preferable o R=3 n erms of avng less bas and more comparable resuls for e dfferen λ values. owever, n Tables, 3 and 4, e value of e esmaed auocorrelaon coeffcen was no consdered, because was no under our conrol. Te smulaons were carred ou w e sascal package S-Plus 000 (MaSof, Inc.). On e bass of ese smulaons, was concluded a e nomnal confdence level depends on e followng facors: () e acual confdence level, () e value of, and () e value of R. Tus, n order o calbrae e confdence nervals we esmaed e followng lnear regresson model (sandard errors n pareneses) w R = , σˆ = and sample sze=69 z R 0.46 nom = z ac (0.093) ( (0.009) (0.000) (0.036) Ts resul ndcaes a e Normal approxmaon derved prevously requres a sascally sgnfcan numercal correcon. W s equaon, e approprae z nom may be calculaed, gven e values of R and, as well as e desred z ac, correspondng o e acual confdence level. Suc a nomnal value can en be nroduced n expresson () o oban an approprae confdence nerval. Illusrave Applcaons Te Sales daase corresponds o e seasonal me seres provded by Cafeld and Proero (973). Te orgnal seres as N=77 observaons on sales of an engneerng frm. A me plo of e seres wou ransformaon appears n Fgure (a) and power-ransformed w λ=0.54 n Fgure (b). Ts ransformaon ndex was obaned as mnmzer of e coeffcen of varaon w =6 subseres and R= observaons per subseres (so a n=5 observaons were lef ou of e calculaons). In s case e auocorrelaon requred by e confdence nerval was esmaed as ρ ˆ = Te followng confdence nervals were obaned for e rue λ value. 99%: ( ,0.5646); 95%: (0.06,0.4846); and 90%: (0.066,0.4456). Fgure sows a grap of e coeffcen of varaon CV(λ) for ese daa, ogeer w a 95% confdence nerval for λ. Tus, w a confdence level of 95%, can be deermned a λ=0 s no suppored by e daa as e ndex of a varance sablzng power ransformaon. In oer words, e logarm s no a reasonable ransformaon o sablze e varance of s me seres. owever, values suc as λ=0.5 or λ=0.34, are reasonably adequae o represen e rue value of λ, even w 90% confdence. Ts resul s n agreemen w e basc concluson reaced by prevous auors (see Guerrero, 993). Now, for comparave purposes, assume a no auocorrelaon exss n e seres λˆ W ( λˆ ) = S / Z, for =,,, n suc a way a Vangel s (996) proposal can be used. In s suaon, e 00( - α) % confdence nerval s gven by / χ, α + σ : ( ˆ λ CV λ ) µ, ( ˆ χ α CV λ ) +

9 364 VARIANCE STABILIZING POWER TRANSFORMATION FOR TIME SERIES Table 4: Observaons below\above of a 00(-α)% nomnal confdence nerval for λ and acual α, w Model and R=3 (000 samples). λ Acual α α z nom average z ac \5 \ 7\ \ 7\54 8\ \3 7\55 \ \3 9\03 35\ \3 \05 37\ \0 0\7 3\ \0 0\9 4\ \5 \6 6\ \6 5\63 9\ \ 3\8 4\ \ 5\0 4\ \0 3\6 6\ \0 3\7 6\ \ \66 0\ \ \7 0\ \0 5\0 39\ \0 5\5 40\ \0 4\5 4\ \0 4\7 4\ \0 7\7 3\ \0 7\74 3\ \4 8\7 37\ \5 9\3 38\ Fgure. Sales daa. (a) Orgnal and (b) power-ransformed w λ=0.54.

10 GUERRERO & PERERA 365 Because λ ˆ = 0. 54, CV ( λ ˆ ) = , =6 and α=0.05, en χ 5,0.95 =. 5. Te confdence nerval ges defned by e λ values sasfyng e nequaly σ/µ 0.708, were sould be recalled a bo σ and µ are funcons of λ. ence, (see Fgure ) e 95% confdence nerval obaned s: ( ,0.577). Te correspondng nerval () on e assumpon ρ=0, sasfes e nequaly σ/µ 0.59 and becomes (-0.004,0.566). Bo nervals obaned on e no-auocorrelaon assumpon cover e value λ=0, bu Vangel s nerval s wder an ours. In s exercse, e auocorrelaon coeffcen canged from beng a negave value o zero, leavng everyng else consan. Ts cange produced a larger expandng facor of CV(λ), ence a wder nerval. Blowfly Daa Ncolson s blowfly daa ave been analyzed from several angles. Noably among ese s e one a employs a nonlnear model for ese daa, n place of a power ransformaon (see Young, 000). Nevereless, because we are manly concerned w e use of power ransformaons, we empasze e analyss presened n e paper by Cen and Lee (997). Tese auors used 8 observaons of e orgnal seres (from 8 o 99) for comparson w prevous works. Tey also menoned a oer auors used eer a logarmc or a square roo ransformaon (.e. λ=0 or λ=0.5). Ten, ey employed er meod, condonng on an auoregressve AR() model form, and made nferences on bo λ and e parameers of a model (mean, auoregressve coeffcen and error varance). Te pon esmae of e ransformaon ndex was obaned as e poseror mean of a dsrbuon obaned by Gbbs samplng w a unform pror on e se {0.30, 0.3,..., 0.50} Te esmaed value, λ ˆ = w sandard error 0.00, clearly dffers sgnfcanly from λ=0 and λ=0.5. owever, we beleve a Cen and Lee s meod s msleadng because condons on e model form, wle e oer meods agans wc ey compared er resuls are model-ndependen. Moreover, sould be recalled a e model form may cange dependng on e value of λ, as ndcaed by Goureroux and Jasak (00), us e AR() specfcaon mg be n doub. We appled our procedure o e daa employed by Cen and Lee, wou condonng on any gven model srucure. By so dong, λˆ =0.3997, w R=4 and =0; so a n= observaons were no used. Te pon esmae of e ransformaon ndex ook almos e same value as a obaned by Cen and Lee s meod. Te auocorrelaon became n s case ρˆ =0.05 and e confdence nervals were 99%: (-.0448,.67); 95%: ( ,.89) and 90%: (-0.338,.068). Tese nervals are nconclusve, because even w 90% confdence usng e daa n e orgnal scale, n a square roo scale or n logarms, produces essenally e same resuls (n erms of varance sablzaon). Ts resul would ave been expeced jus by lookng a e graps sown n Fgure 3, were no relevan canges are observed n e me seres beavor by cangng e scale. We calculaed agan e nerval proposed by Vangel (996) on e assumpon a ρ=0 (wc may be deemed reasonable snce ρˆ s ndeed close o zero) w λˆ =0.3997, CV(λ)= , =0 and α=0.05, so a χ 9,0.95 = 0.. Te correspondng 95% confdence nerval was defned by e λ values sasfyng e nequaly σ/µ , (see e grap of CV(λ) n Fgure 4) a s (-.7678,.393). Tus, e prevous concluson olds vald even f e assumpon ρ=0 were rue. Smlarly, e grap of CV(λ) sown n Fgure 4 sows wy e nervals are so wde: CV(λ) s exremely fla for e range of usual λ values employed n pracce. Ts s an example were e daa are bascally nsensve o e coce of a varance sablzng ransformaon. To es s dea, we esmaed e same AR() model for e daa w e followng coces of e ransformaon ndex: λ=,0.39,0.

11 366 VARIANCE STABILIZING POWER TRANSFORMATION FOR TIME SERIES Fgure 3. Blowfly daa. (a) Orgnal, (b) power-ransformed w λ=0.3997, and (c) log-ransformed. Fgure 4. Confdence nerval for λ w blowfly daa. Te Maxmum Lkelood esmaon resuls appear n Table 5, were may be observed a e esmaed AR coeffcens ( ˆφ ) are almos e same n e ree dfferen scales. Te Ljung-Box sascs Q(4-), wen compared agans a C-square dsrbuon w 3 degrees of freedom, sow no evdence of nadequacy. Te oer wo esmaed parameers (mean ˆφ 0 and resdual sandard error σˆ ) depend eavly on e scale of e analyss and do no allow a drec comparson. Te -sascs ndcae a e esmaed coeffcens are sgnfcanly dfferen from zero n e ree cases and e resdual graps (no sown) are also very smlar, sowng no evdence of nonconsan varance by vsual nspecon. Tus,

12 GUERRERO & PERERA 367 may be concluded a coosng one parcular power ransformaon, wn ose ndexed by λ=,0.39,0, depends on some creron dfferen from varance sablzaon. Peraps, e forecasng ably of e model sould be suded n e dfferen scales, as Cen and Lee (997) fnally dd, n order o selec e λ value, bu a ask was ousde e scope of s arcle. Table 5. Esmaon resuls of e AR() model for blowfly daa w dfferen coces of λ. λ ˆφ sa ˆφ sa σˆ Q(4 ) Concluson Ts arcle presens a procedure o calculae a confdence nerval for e rue ndex of a power ransformaon a bes sablzes e varance of a me seres. Ts s useful as enables a me seres analys o make sascal nferences abou e ransformaon ndex, wou relyng on a model-dependen meod. Te procedure was derved from a sudy of e approxmae mean and varance of e mnmum coeffcen of varaon employed for coosng e ransformaon. Ten, a small smulaon sudy allowed us o calbrae e confdence coeffcen. Ts calbraon was jusfed because our analycal resuls were derved from several approxmaons a may yeld naccurae resuls n praccal applcaons. Te coverage raes were found o be dependen on e nomnal sze of e confdence level, e subseres sze R and e number of subseres used. Te smulaons led o praccal conclusons. For nsance, e approprae subseres sze, wen ere s no seasonaly n e me seres, was found o be R = 4, wle e leng of e seasonal perod s adequae for a seasonal me seres (.e. R = for a monly me seres). A more exensve smulaon sudy would be requred o consder negave λ values as well as some oer me seres models, n order o ge more conclusve resuls. Te emprcal llusraons provded evdence on e use e meod may ave n praccal applcaons. Te frs example provded an emprcal confrmaon a our meod can be rused, because we obaned essenally e same resuls a were esablsed prevously by means of Maxmum Lkelood. owever, our meod was appled w less effor, and we dd no rely on knowledge of e model srucure of e me seres, as s requred by e Maxmum Lkelood meod. Te second llusraon esed e recommendaons derved from e smulaon sudy. In fac, was found a our meod led o sensble resuls and s relavely easy o apply. Fnally, s neresng o noe a e confdence nerval for e mnmum coeffcen of varaon can also be used o consruc confdence nervals for any coeffcen of varaon. Terefore, e resuls obaned ere may lead o furer researc n e area of nference for a coeffcen of varaon n general. References Box, G. E. P, Jenkns, G. M. & Rensel, G. C. (994). Tme seres analyss. Forecasng and conrol, 3rd ed. London: Prence-all. Cafeld C., & Proero, D. L. (973). Box-Jenkns seasonal forecasng: problems n a case-sudy. Journal of e Royal Sascal Socey, A-36, Cen, C. W. S., & Lee, J. C. (997). On selecng a power ransformaon n me-seres analyss. Journal of Forecasng, 6, de Brun, P., & Franses, P.. (999). Forecasng power-ransformed me seres daa. Journal of Appled Sascs, 6, Goureroux, C., & Jasak, J. (00). Nonlnear auocorrelograms: An applcaon o ner-rade duraons. Journal of Tme Seres Analyss, 3, Guerrero, V. M. (993). Tme-seres analyss suppored by power ransformaons, Journal of. Forecasng,,

13 368 VARIANCE STABILIZING POWER TRANSFORMATION FOR TIME SERIES Jennngs, D. E. (987). ow do we judge confdence-nerval adequacy? Te Amercan Sascan, 4, Suar, A., & Ord, J. K. (987). Kendall s advanced eory of sascs,, 5 ed. London: Carles Grffn & Co. Ld. Vangel, M. G. (996). Confdence nervals for a Normal coeffcen of varaon. Te Amercan Sascan, 5, -6. Young, P. C. (000). Socasc, dynamc modellng and sgnal processng: Tme varable and sae dependen parameer esmaon. In Nonsaonary and nonlnear sgnal processng, ed. W. J. Fzgerald, A. Walden, R. Sm, & P. C. Young, pp Cambrdge: Unversy Press. Appendx. Approxmae varances and covarances of funcons of random varables ) Teorem. Le X = ( X,..., X k )' be a k-dmensonal random vecor, g(x) be a real-valued funcon k defned on R and E ( X ) < for =,,k. Assume a e paral dervaves g' ( X ) = g( X) / X all g' E X denoe ' ( X ) evaluaed a E(X). Ten, e frs-order Taylor exs and le [ ( )] g k expanson g( X ) g[ E( X) ] + g' [ E( X) ][ X E( X )], so a E[ g( X) ] g[ E( X) ] and, f no all e g' [ E( X )] are zero, var[ g( X )] g' [ E( X) ] k k + g jj= = k = jj= k k { } var(x ) + g' [ E( X) ] g' [ E( X) ] Cov( X, X ) Smlarly, for wo funcons g ( X ) and ( ) [ E( X) ] g' E( X) [ ] Cov( X, ) ' j X j. j g X, cov[ g ( X),g ( X) ] g' [ E( X) ] g' [ E( X) ] var( ) Proof. Ts resul was esablsed by Suar and Ord (987, C. 0). j k X =. Expeced values, varances and covarance of m and se. I s known a E [ W ] = µ, var[ W ( λ )] = σ and corr[ W ( λ ), W ' ] = ρ f ' = ± and zero oerwse. Ten, ) E ( m) = µ, m = E W λ µ λ µ ' = ' = = E[ W µ ] + E{ [ W µ ][ W+ µ ]} = σ [ + ρ( ) / ] /, = = 3) [( ) ] [ ( ) ] ( ) E se = E E W λ µ m µ = = σ σ + ρ / / = ( ) σ ( ρ / ). ) var ( ) ( ) W ( ) [ [ ] ] Under Normal eory, w ρ = 0, e dsrbuon of ( ) se / σ s C-square w - degrees of freedom. Snce ρ canno be far away from zero, follows a ( )se / σ mus ave a dsrbuon close o a χ. Te varance of suc a dsrbuon s derved by assumng approxmaely vald e followng relaonsp a olds for a C-square dsrbuon: Varance = Mean, erefore

14 var [ ] ( se ) / σ ( )( ρ / ) GUERRERO & PERERA 369. From e Teorem n Appendx w k =, { [ ]} ( ) / g (X) = X, se) g' E( se ) var se ence, 4) E (se) E( se ) var( se) / σ 4 var( E (se ) ( / ) /( ) = σ { ρ / / [ ( ) ]}. ρ = σ /( ), Nex, e Teorem n Appendx appled w k=, X ( se m), ( X ) = 0 = g ( ) and g ( X ) = m, allows g X cov(se, m) g' [ E( se, m) ] g' [ E( se, m) ] cov(se, m) / E (se ) cov( se, m) Cov (se, m) = E =. Now, µ = {[ W µ ] ( m µ )} ( m ) = E W + W W X = se and =, g ( X ) = se, 3 ( λ ) µ ( λ ) µ ' ( λ ) µ E( m µ ) 3 = ' ' = = 3 E W µ = [ ] E( m µ ) ( a a θa a a + θ a a θa + θa a θ a a ) + E = en e normaly assumpon mples a e rd cenral momens of a, W (λ) and m are all zero. I follows a cov(se, m) = 0 and 5) cov( se, m) 0.