Minimum Mean Squared Error Estimation of the Noise in Unobserved Component Models

Transcription

1 Mnmm Mean Sqared Error Esmaon of he Nose n Unobserved Componen Models Agsín Maravall ( Jornal of Bsness and Economc Sascs, 5, pp )

2 Absrac In model-based esmaon of nobserved componens, he mnmm mean sqared error esmaor of he nose componen s dfferen from whe nose. In hs arcle, some of he dfferences are analysed. I s seen how he varance of he componen s always nderesmaed, and he smaller he nose varance, he larger he nderesmaon. Esmaors of small-varance nose componens wll also have large aocorrelaons. Fnally, n he conex of an applcaon, he sample aocorrelaon fncon of he esmaed nose s seen o perform well as a dagnosc ool, even when he varance s small and he seres s of relavely shor lengh. Keywords: Seasonal adjsmen; Sgnal exracon; Tme seres; ARIMA models.

3 1. MINIMUM MEAN SQUARED ERROR ESTIMATOR OF THE NOISE COMPONENT Le an observable seres be he sm of several orhogonal componens, one of hem whe nose, as n = +, (1.1) where denoes an nobservable componen and s normally dencally ndependenly dsrbed (nd) (0, σ ). Parclar cases of (1.1) are he rendseasonal-(whe nose) rreglar and he sgnal-pls-nose decomposons of a me seres. The model-based approach o nobserved componens esmaon nvolves he se of models for he componens of he ype = ψ ( B ) a, (1.) where ψ (B) represens a raonal fncon of he backward shf operaor B and he a s are orhogonal whe noses, each one wh varance σ. Le he overall model for he observed seres, conssen wh (1.1) and (1.), be gven by = ψ ( B ) a, (1.3) where ψ (B) s also a raonal fncon n B, whch can be expressed as ψ ( B ) = θ( B ) / φ( B ), where θ (B) and φ (B) are polynomals n B of degree q and p, respecvely. Un roos may be presen n he aoregressve polynomal φ (B). The prevos characeraon of he model-based decomposon of a me seres can be appled o he so-called redced form approach, n whch he overall model (1.3) s assmed known and, from hs, he models for he componens are derved (see, e.g., Brman 1980; Hllmer and Tao 198). I can also be appled o he srcral form approach, n whch he models for he componens [.e., Eq. (1.) for all ] are drecly specfed (see, e.g., Engle 1978; Harvey and Todd 1983). The mnmm mean sqared error (MMSE) esmaor of he h componen s gven by ẑ [ ψ ( B ) ψ ( F ) / ψ ( B ) ψ ( F ) ] = k, 3

4 1 where k = σ / σ a and F = B denoes he forward shf operaor (see Cleveland and Tao 1976; Bell 1984). Smlarly for, whch wll be referred o as he nose or rreglar componen, he esmaor becomes û [ 1 / ψ ( B ) ψ ( F ) ] = k, (1.4) where k = σ / σ a. [Noce ha (1.4) wll be vald for all admssble decomposons of no (1.1); he dfferences among hese decomposons wll smply mply dfferen vales of σ ]. As a conseqence, he esmaor û gven by (1.4), wll follow a model dfferen from he heorecal model for, whch s whe nose. Ths s a well-known resl (see, e.g., Bell and Hllmer 1984) and, alhogh case for some concern, has no been he sbjec of mch analyss. The concern orgnaes from he consderaon ha f he am s o remove whe nose varaon from a seres, nvely wold seem desrable o remove somehng ha s somewha close o whe nose. Eqaon (1.4), however, garanees ha he esmaor û wll no be whe nose. How mporan n pracce can hs deparre from whe nose behavor be expeced o be? Cleveland and Tao (1976) derved he heorecal aocorrelaon fncon (ACF) of û, for her model-based nerpreaon of X11. I was ceranly dfferen from ha of whe nose, ye all aocorrelaons were small, none of hem exceedng n absole vale.. In fac, for he example hey dscssed, alhogh he emprcal ACF of û esmaed wh X11 was smlar o he heorecal ACF of he X11 rreglar, was also close, consderng he sample se, o ha of a whe nose varable (see Cleveland and Tao 1976, fg. D). Therefore, for seres obeyng models reasonably close o he model verson of X11 and for sandard sample ses, he dfference beween he models for and û wll be small and he deparre from whe nose behavor n he esmaor wll be of lle praccal mporance. For seres wh a dfferen srcre, however, he deparre can be sbsanal. Usng (1.3) n (1.4), û can be expressed as a fncon of he nnovaons ( a ) n he observed seres û 1 = k ψ ( F ) a, or θ ( F ) û = k φ( F ) a. (1.5) 4

5 Noce ha f a seres follows he model (1.3), also follows he model = ψ ( F ) e, where e s he backward nnovaon e = E ( + 1, +,...), a whe nose varable ndependen of all fre s. Therefore, correspondng o he model for ha ses he forward shf operaor and he backward nnovaons, û cold be expressed alernavely n erms of he backward shf operaor and he backward nnovaons as 1 û = k ψ ( B ) e. From expresson (1.5) several resls are mmedaely avalable. Frs, snce û can be expressed as a lnear fncon of presen and fre nnovaons, follows ha E û + j = 0 for j > 0. Hence, alhogh aocorrelaed, û canno be forecas. Second, snce σ > 0 n (1.1) mples a posve mnmm n he psedospecrm of, he seres wll be nverble and he roos of θ ( B ) wll le osde he n crcle. Ths, consderng (1.5), û wll always be saonary, wh fne varance. Frhermore, seng (who loss of generaly) σ a = 1, (1.5) mples V ( û * ) / σ = σ V ( ) (1.6) * where V denoes varance and s he nverse process From Hllmer (1976, expresson 4..7) s fond ha * θ ( B ) = φ( B ) a. ω ω ω ω σ [ φ ( e ) φ ( e ) / θ ( e ) θ ( e ) ] 1 * for all ω n he range ( π, π ). Ths σ V ( ) 1 and hence V ( û ) σ, wh eqaly holdng only n he rval case =. Therefore, he varance of he esmaor û s smaller han he varance of he heorecal componen. In fac, from (1.6) can be seen ha as he seres ges closer o nonnverbly (.e., as he nose componen becomes smaller), he rao V ( û ) / σ ends o ero and û ges closer o nonsaonary. Hence large dfferences beween he heorecal nose componen and s MMSE esmaor (n erms of varances and aocorrelaons) wll characere seres for whch he nose componen s of lle mporance. Fnally, Eqaon (1.5) shows ha when he observed seres s nonsaonary (he case of appled neres), he esmaor of he nose wll be nonnverble. The eros n he specrm of û wll reflec he fac ha he eros of φ ( B ) are assocaed wh freqences for whch he rao of he nose varance o he sgnal varance n he observed seres s ero. Conseqenly, n erms of nose 5

6 esmaon, hese freqences wll provde no nformaon and herefore can be gnored. As an example, consder he sgnal-pls-nose decomposon of he negraed movng average (IMA) (1,1) model n Box, Hllmer, and Tao (1978). The canoncal decomposon (.e., he one wh maxmm nose varance) of = (1 θ B ), a where = 1 B, no = + 1 yelds whe nose wh k = (1+ θ ) / 4. From (1.5), û can be expressed as he saonary aoregressve movng average (ARMA) (1,1) process ( 1 θ F ) û = k ( 1 F ) a, (1.7) so ρ 1 = ( 1 θ ) /, ρ k = θ ρ k 1 ( k > 1), and V ( û ) / σ = (1+ θ ) /. (1.8) Ths rao always les n he range (0,1). Boh varances wll be eqal when he observed seres s pre nose. On he conrary, as θ approaches -1 and he nose varance becomes smaller, he aocorrelaons of û become larger n absole vale and he rao (1.8) wll end o ero. When θ = -.9, for example, ρ 1 =.95, and he varance of he one-sep-ahead forecas error of s 40 mes larger han he varance of he heorecal nose componen, whch, n rn, s 0 mes larger han ha of s esmaor. Noce also ha he movng average (MA) facor ( 1 F ) n (1.7) mples a ero n he specrm of û for he ero freqency, for whch he psedospecrm of dsplays an nfne peak. The conclson s ha when he nose componen s relavely small, lle of wll be capred by MMSE esmaon. Sll, as shall be seen n he nex secon, even for relavely shor seres he nformaon conaned n hs esmaor can be of consderable neres. 6

7 . A DIAGNOSTIC CHECK One of he advanages of a model-based approach s ha provdes he gronds for analyss of resls by comparng heorecal models wh he obaned esmaes. In pracce, he resdal s no esmaed drecly sng (1.4), b as û = ẑ, (.1) afer he oher componens have been removed from he seres. If eqaons (1.1) (1.3) are correc and he componens orhogonal, hen (.1) can be rewren as û = k ψ ψ ( B ) ψ ( B ) ψ ( F ) ( F ) = ψ 1 ( B ) ψ ( F ) ψ ( B ) ψ ( F ) k ψ ( B ) ψ ( F ). Snce he erm n brackes s eqal o σ, expresson (1.4) s fnally obaned; hence û, comped as he resdal, shold also sasfy (1.5). The comparson of he heorecal aocorrelaons of û, obaned from (1.5), wh he emprcal ones for he esmaed nose (as sggesed orgnally by Cleveland and Tao 1976) provdes a naral, easy-o-compe way o evalae resls n a parclar model-based decomposon. Dscrepances beween he wo aocorrelaons wold reveal nadeqaces n he procedre ha can be de o an ncorrec specfcaon of he model for he observed seres. Alhogh comparson of he heorecal and emprcal aocorrelaons may n heory provde a check on he resls, how can he comparson be expeced o perform n pracce? Ths sse wll be addressed n he conex of an example, whch consders seasonal adjsmen of he monhly seres of nsrance operaons (IO s) one of he componens (small, hogh no rval) of he Spansh moneary aggregaes. The seres sars n Janary 1979 and ends n Ocober 1985; consss herefore of 8 observaons. Ths, we are analyng a relavely shor monhly seres. 7

8 Fgre 1: Theorecal Aocorrelaons of he Irreglar Componen Esmaor for he Arlne Model 8

9 When adjsng he seres wh he program X11 ARIMA (defal opon), all ess for he presence of seasonaly were hghly sgnfcan. The resls faled o pass, however, one of he qaly assessmen ess, whch ndcaed ha here was oo mch aocorrelaon n he rreglar. The lag-1 aocorrelaon of he esmaed rreglar was, n fac, ρ ˆ 1 =. 4, qe a dsance away from ρ 1 =., he heorecal vale mplc n he MMSE model-based approxmaon o X11 developed by Cleveland and Tao (1976). The dscrepancy beween he wo vales clearly ndcaes ha X11 canno be gven Cleveland and Tao s MMSE model-based nerpreaon n hs case, b I am no neresed n comparng a non-model-based mehod wh a model-based mehod (n hs respec, see he conssency wh he daa check of Bell and Hllmer 1984). My am s nsead o jdge he adeqacy of a parclar model-based decomposon. Seasonal adjsmen ofen reqres, n pracce, reamen of a large nmber of seres, makng nfeasble o perform prevos nvarae analyss for each of hem. Ths here s a need for some sandard model ha approxmaes reasonably well a large nmber of seres and can herefore be appled ronely (he common cenral model of Sms 1985). An obvos canddae among ARIMA models s he arlne model of Box and Jenkns (1970), gven by 1 1 = (1 θ 1 B ) (1 θ 1 B ) a, (.) whch s known o approxmae many seres enconered n pracce. Modelbased decomposon of no rend, seasonal, and whe nose rreglar mples ha he MMSE esmaor of he rreglar,, follows he process F ) (1 θ 1 F ) û = k (1 F ) (1 F ) a ( 1 θ, (.3) from whch he heorecal aocorrelaons can be comped. Fgre 1 dsplays he lag-1 and lag-1 aocorrelaons of û for he dfferen vales of he θ parameers whn he admssble regon ( 1 < θ 1 < 1, 0 < θ 1 < 1; see Hllmer and Tao 198). The parameer θ 1 has praccally no effec on ρ 1; and θ 1, excep for large negave vales, does no affec ρ 1. In addon, ρ 1 s always negave and les n he regon (0, -1). As for ρ 1, alhogh he range goes from -.5 o.5, nless θ 1 s close o 1, s vale wll also be negave. In boh cases, he smaller he θ parameer, he larger he correspondng aocorrelaon of û wll be n absole vale. 9

10 To see how close one can expec o ge o he heorecal aocorrelaons n a parclar realaon, he vales seleced for θ 1 and θ 1 are hose of he acal arlne model, θ 1 =. 4, θ 1 =. 6. [For a seres followng hs model, Cleveland and Tao (1976) showed ha he censs (X11) procedre works reasonably well (p. 584).] The heorecal sandard devaon and lag-1 and lag-1 aocorrelaons of û are gven n he frs colmn of Table 1. Table 1. Arlne Model Theorecal Smlaed dsrbon Esmaed rreglar componen Mean Sandard devaon rreglar componen ρ 1 ρ Sandard devaon* * In ns of σ a Alhogh he dsrbons of he aocorrelaon esmaors are complcaed o derve, hey can be approxmaed by smlaon. Two hndred ffy ndependen seres of 84 observaons (eqvalen o seven years of daa) were generaed for he Arlne model (wh σ a = 1). For each seres, a heorecally whe nose rreglar was esmaed sng Brman s program (see Brman 1980), and for each rreglar componen seres, he sandard devaon and aocorrelaons were esmaed. In hs way, emprcal dsrbons are obaned for hese esmaors; her means and sandard devaons are gven n he second and hrd colmns of Table 1. The esmaor of ρ 1 appears o be nbased, alhogh ˆρ 1 has a small bas. When he lengh of he seres was ncreased o 11 and hen o 14 years, however, he mean of ˆρ 1 became -.3 and -., respecvely, approachng he heorecal vale. (The same smlaon was repeaed wce wh 50 seres, and hen for mes sng 150 seres. The dfferences n he resls were mnor and none of he vales repored n Table 1, for example, changed by more han.01.) One sse ha has no been addressed s ha of revsons. The esmaor û defned by (1.4), s gven by a cenered and symmerc fler appled o he observed seres. Ths mples ha, for perods relavely close o he presen, observaons needed o complee he fler wll no be avalable ye. Replacng hem wh forecass, a prelmnary esmaor can be comped and, wh he passng of me, as forecass are eher pdaed or replaced wh new 10

11 observaons, he esmaor of wll be revsed (see Perce 1980). As a conseqence, he end vales of he esmaed rreglar seres wll be conamnaed by revson error. Ths conamnaon wll ceranly have an effec on he esmaors of he momens of û, b Table 1 shows ha even for seres as shor as seven years, he effec on bas s small. As for he precson, he sandard devaon of ˆρ 1 and of ˆρ 1 s of he order of.10. (For oher lags, he sandard devaons of he aocorrelaon esmaors were larger.) Table. Sandard Devaon of he Aocorrelaon Esmaor T Lag 1 Lag 1 1 / T Fgre : Emprcal Dsrbons of Aocorrelaon Esmaes for he Irreglar Componen : (a) Arlne Model; (b) Model (.4) 11

12 Fgre 3: Specrm of he Irreglar Componen and Is Esmaor [ Model (.4) ] 1

13 The seres lengh was ncreased o 11 and hen o 14 years, and agan 50 realaons were generaed n each case. Table compares he sandard devaons of ˆρ 1 and ˆρ 1 for he dfferen sample ses. I s seen ha 1 T seems o be a roghly correc (slghly conservave) approxmaon o he sandard devaons obaned. Fnally, he emprcal dsrbons of ˆρ 1 and ˆρ 1 are ploed n Fgre a, n whch hey are compared wh he Normal approxmaon wh he same mean and varance. The skewness esmaes are, respecvely,.3 and -.06, and he esmaes of kross are.83 and.80. I seems reasonable o conclde ha for he model consdered, even for relavely shor seres, he check of wheher ˆρ 1 and ˆρ 1 fall whn he range ± T of her heorecal vale may provde a sefl dagnosc ool. For he log of he seres of IO s, he aocorrelaon esmaes are gven n he las colmn of Table 1. Alhogh ˆρ 1 s accepable (borderlne), ˆρ 1 ( =.40 ) s nqesonably osde he range ( -.10, -.50 ). (Noce ha he esmae s close o he correspondng one obaned for he X11 rreglar.) Hence he check agrees wh he X11 error message and clearly ndcaes ha model-based decomposon of he seres IO sng he arlne model s no approprae. The resl s confrmed by he esmaor of he sandard devaon of û (.), whch also falls osde he accepance range (.38,.46.) In fac, a more adeqae model han he arlne one for he log of he IO seres s 1 1 = (1.106 B.496 B ) (1.437 B ) a, (.4) wh σ. 034 (abo half he se of a = σ a for he arlne model). MMSE modelbased decomposon of (.4) yelds he followng esmaor of a whe nose rreglar: (1.106 F.496 F ) (1.437 F 1 ) û = k (1 F ) (1 F 1 ) a, (.5 ) wh k = σ / σ a =. 013 when σ aans s maxmm ( canoncal ) vale. Fgre 3 compares he specrm of he (whe nose) wh ha of s esmaor û, and he dfference beween hem s remarkable. The eros n he specrm of û correspond o he n roos n he aoregressve par of he model for. Comparng he areas nder boh specra, s seen how V ( û ) nderesmaes σ [n fac, V ( û ) = / 8 ]. Ths we are lookng a a seres wh a very small σ rreglar, whch s n rn, srongly nderesmaed (n accordance wh he resl n he prevos secon). 13

14 From (.5), he heorecal varances and aocorrelaons of û can be derved. They are gven n he frs colmn of Table 3. A smlaon smlar o he one for he arlne model was done, and he resls are presened n he second and hrd colmns of Table 3 and n Fgre b. As n he arlne model case, he esmaors of he sandard devaon and ˆρ 1 and ˆρ 1 appear o be reasonably nbased and precse. The forh colmn of Table 3 dsplays he sandard devaon and aocorrelaons of he rreglar componen esmaed wh Brman s program; n hs case, he hree esmaes are comforably n agreemen wh her heorecal vales. Table 3. Model (.4) Theorecal Smlaed dsrbon Esmaed rreglar componen Mean Sandard devaon rreglar componen ρ 1 ρ Sandard devaon* * In ns of σ a In smmary, a seres wh an rreglar (or nose) componen varance as small as 1% of he varance of he one-sep-ahead forecas error has been consdered. The dscsson sggess ha, even n ha case and for seres as shor as seven years of monhly daa, he comparson beween he heorecal and emprcal second momen of he rreglar esmaor performs well as a way of evalang resls n MMSE model-based decomposon of me seres. ACKNOWLEDGMENTS I hank wo anonymos referees for her helpfl commens. [Receved Jly Revsed Janary 1986.] 14

15 REFERENCES BELL, W.R. (1984), "Sgnal Exracon for Nonsaonary Tme Seres". Annals of Sascs 1, BELL, W.R. and HILLMER, S.C. (1984), "Isses nvolved wh he Seasonal Adjsmen of Economc Tme Seres". Jornal of Bsness and Economc Sascs, BOX, G.E.P., HILLMER, S.C. and TIAO, G.C. (1978), "Analyss and Modelng of Seasonal Tme Seres", n Zellner, A. (ed.), Seasonal Analyss of Economc Tme Seres, Washngon, D.C.: U.S. Dep. of Commerce-Brea of he Censs, BOX, G.E.P. and JENKINS, G.M. (1970), Tme Seres Analyss: Forecasng and Conrol, San Francsco: Holden-Day. BURMAN, J.P. (1980), "Seasonal Adjsmen by Sgnal Exracon", Jornal of he Royal Sascal Socey A, 143, CLEVELAND, W.P. and TIAO, G.C. (1976), "Decomposon of Seasonal Tme Seres: A Model for he X-11 Program", Jornal of he Amercan Sascal Assocaon 71, ENGLE, R.F. (1978), "Esmang Srcral Models of Seasonaly", n Zellner, A. (ed.), Seasonal Analyss of Economc Tme Seres, Washngon, D.C.: U.S. Dep. of Commerce-Brea of he Censs, HARVEY, A.C. and TODD, P.H.J. (1983), "Forecasng Economc Tme Seres wh Srcral and Box-Jenkns Models: A Case Sdy", Jornal of Bsness and Economc Sascs 1, HILLMER, S.C. (1976), "Tme Seres: Esmaon, Smoohng and Seasonal Adjsmen", Ph.D. dsseraon, Unversy of Wsconsn, Sascs Deparmen. HILLMER, S.C. and TIAO, G.C. (198), "An ARIMA-Model Based Approach o Seasonal Adjsmen", Jornal of he Amercan Sascal Assocaon 77, PIERCE, D.A. (1980), "Daa Revsons n Movng Average Seasonal Adjsmen Procedres", Jornal of Economercs 14, SIMS, C.A. (1985), "Commen on 'Isses Involved wh he Seasonal Adjsmen of Economc Tme Seres' by W.R. Bell and S.C. Hllmer", Jornal of Bsness and Economc Sascs 3, 1,