MEASURING INFLUENCE IN DYNAMIC REGRESSION MODELS. Daniel Peña*

Transcription

1 Working Papr Jun 1990 Dpartamnto d Economía Univrsidad Carlos III d Madrid Call Madrid, Gtaf Madrid) MEASURING INFLUENCE IN DYNAMIC REGRESSION MODELS Danil Pña* Abstract. _ This articl prsnts a mthodology to build masurs of influnc in rgrssion modls with tim sris data. W introduc statistics that masur th influnc of ach obsrvation on th paramtr stimats and on th forcasts. Ths statistics tak into account th autocorrlation of th sampl. Th f irst statistic can b dcomposd to masur th chang in th univariat ARIMA paramtrs, th transfr function paramtrs and th intraction btwn both. For indpndnt data thy rduc to th D statistics considrd by Cook in th standard rgrssion modlo Ths statistics can b asily computd using standard tim sris softwar. Thir prformanc is analyzd in an xampl in which thy sm to b usful to idntify important vnts, such as additiv outlirs and trnd shifts, in tim sris data. c: Ky words Missing obsrvations; outlirs, Intrvntion analysis; ARIMA modls; Invrs autocorrlation function. * Dpartamnto d Economia, Univrsidad Carlos III d Madrid and Laboratorio d Estadistica, ETSII, Univrsidad Politécnica d Madrid.

2 1 missing valu approach. W can us any suitabl masur of distan 1. INTRODUCTION Th distinction btwn additiv and innovational outlirs in tim sris was introducd by Fox 1972). Sinc thn, th study of outlirs has bn an activ ara of rsarch in tim sris. Abraham and Box 1979), Martin 1980), Chang, Tiao and Chn 1988), Tsay 1986, 1988) and Abraham and Chuang 1989) ar som of th rlvant rfrncs. Th study of influntial obsrvations in tim sris, howvr, has rcivd littl attntion in th statistical litratur. Pña 1987, 1990) showd that th study of influntial obsrvations can b carrid out for univariat tim sris using a c, such as th liklihood distanc or th chang in th prdictiv or postrior distribution, to masur th modl-chang whn it is assumd that on obsrvation, or a missing. subst of obsrvations, is In this papr, th missing valu approach is applid to build masurs of influnc for dynamic transfr function) rgrssion modls. Whn th rsiduals ar not autocorrlatd, and thrfor w ar in th particular cas of th standard rgrssion modl, th statistic suggstd rducs to Cook's D. This papr is organizd as follows: sction 2 analyzs th problm of building a masur of chang in th ARMA paramtrs; sction 3 prsnts a global masur for th chang of th whol modlo It is shown that this masur can b dcomposd to study th ffct on th transfr function paramtrs and th ARIMA paramtrs. Sction 4 applis ths statistics to a dynamic systm. 2. INFLUENCE ANALYSIS IN UNIVARIATE ARlMA MODELS 2.1 A masur of influnc in univariat ARIMA modls Suppos th ARIMA modl 2.1) ~-----_._._._._ _._-----

3 polynomial oprators in th backshift oprator B with roots outsid th unit circl, and v = 1 - B is th diffrnc oprator. It is also assumd hr that th stationary sris vdz t has zro man, and at is a whit nois squnc of iid N O, G a 2) variabls. Lt k = p+q b th numbr of paramtrs. Lt us call ~ and th maximum liklihood stimators of th paramtrs with th complt data st and ~i) and i) thos wh~n obsrvation zi is missing ths computations will b discussd in th nxt sction, 5). Thn, th paramtr stimats ft and fti) of th autorgrssiv rprsntation 2 2.2) can b computd from vd />B) = B) 1l'B), 2.3) and th chang in th modl structur can b masurd by a Mahalanobis typ distanc 1l' 2.4) whr I:1l' G 2 a is th varianc covarianc matrix of th maximum liklihood stimator ft. As shown in Pfla 1990), by building th masur of chang 2.4) using th 1l' paramtrs, w avoid th problms linkd to nar cancllation btwn AR and MA structurs. For larg sampls, approximatly whr Zt-1 is th matrix -1 I:1l' = Zt-1 Z t-1)

4 3 Now, lt Zt = Zt-1ft b th vctor of forcasts and Zti) = Zt-1 fti) b th vctor of forcasts using fti) instad of ft. Thn --1 Z zi»,z 1T - 1T i) )' 2.5) t t t :E 1T Thrfor, 2.4) can b writtn as Z - zi»,z zi) ) t t t t 2.6) Equation 2.6) shows that, as in th rgrssion modl, th Mahalanobis distanc that masurs th chang in th paramtrs can b xprssd as th Euclidan distanc of th chang in som forcast vctors. Howvr, in th rgrssion modl iti) is com putd by rpiacing th i th obsrvation by i ts man, whras hr Zti) is computd by rplacing th ith obsrvation with its conditional xpctation givn th rst of th data. Pña 1990) has shown that th statistic 2.6) can b writtn as a function of th Iiklihood-ratio tst to chck for additiv outlirs Computing univariat influnc C Th computation of 2.6) rquirs th stimation of th paramtrs whn on obsrvation is missing. Jons 1980), Harvy and Pirc 1984), and Kohn and Ansly 1986) hav shown how to solv this problm by stting up th modl in stat spac form and applying th KaIman filtr. An altrnativ mthod is th following. Th conditional liklihood function of th paramtrs whn obsrvation zi is missing can b writtn s Ljung 1982) as 2 n-1) 2 1 f 1 8 ) ) L8,a 1Zi'U)=- 2 In a In i- 2a 2 S,u'Zi)Zi/n' 2.7 whr 8 is th vctor of ~ and paramtrs, Zi) is th obsrvd

5 4 data st without zi, u is th vctor of starting valus, fi a 2 is th varianc of th sampl distribution of th stimator of th missing valu givn th rst of th data, ~i/n is th xpctd valu of zi givn Zi) and SB, u, Zi)' ~i/n) is th rsidual sum of squars givn th vctor of paramtrs and starting valus, B, u, in a data st in which obsrvation zi has bn substitutd by ~i/n This stimation can b carrid out asily using intrvntion analysis Box and Tiao 1975) as follows. Lt 2.8) b an intrvntion modl whr Iti) is an impuls variabl that taks th valu 1 at t = i. Thn, it can b shown Pña 1987) that th liklihood function for th paramtrs can b writtn as 1 LB,a 2 Z,u) = - ~ In a 2 - SB,u,Zi),zi/n). 2.9) Thrfor, if th sampl siz is larg, and th trm In fi can b disrgardd, th stimation of th paramtrs using 2.7) or 2.9) will b practically idntical. In summary, 1ti) can b asily computd with any computr packag that includs intrvntion analysis. Th computation of th vctor of forcasts in 2.6) dpnds on th starting valus, but if th sampl is larg its ffct will b ngligibl. In th masur 2.6) th vctor Zti) dos hav zi as on of its componnts, bcaus Zti) = Zt-11ti) and Zt-1 includs it. If w wr intrstd in th influnc of zi on th forcast gnratd from th modl, w could monitor th chang with a vctor of forcasts that dos not dpnd on th ith obsrvation at all. In th vctor of forcasts gnratd by th intrvntion modl 2.8), obsrvation zi is always rplacd by ~i/n' and thrfor, w will suggst as a masur of th chang in forcast th statistic:

6 ) whr Zt INT 2.8) is th vctor of forcasts from th intrvntion modl 3. INFLUENCE IN TRAN5FER FUNCTION MODEL tatistics of influnc. 5uppos now that w hav an xplanatory variabl Xt for th tim sris Yt. Th variabl Xt can b ithr dtrministic or stochastic, but w considr th standard cas in which th infrnc is don conditional on th givn valus of Xt. Thn, on can writ th modl as Y = ~ x + ~ a t ób) t pb) t' 3.1) whr mb) = mo + m1b + óab a ) hav roots outsid lag rgrssion quation + mmbm and th unit circl, ób) = or also 1 as - ó1b - th standard Yt = E n'y t. + E a,x t. J -J ~-~ + a t, 3.2) whr ab) = vb)nb). Now, modl 3.2) can b writtn as y= Z n +'T X v + U, 3.3) whr Y is th n x 1 vctor of obsrvations of th output sris, Z is an n x h matrix of past valus of th output, n is a vctor h x 1 of cofficints, T is a triangular matrix with cofficints

7 ._ ' 6 -'Tr 1 1 -'Tr - 'Tr T = 1 O O o 3.4) -'Tr P o -'Tr P x is th n x s matrix of currnt and past valus of th xplanatory variabl and s is th dimnsion of th v vctor. Lt ft and ~ b th maximum liklihood stimators in modl 3.3). Th forcast vctor is y = Z 'Tr + T X v 3.5) whr T is th stimatd T matrix using ft. Suppos now that obsrvation Yi is missing, and lt fti), ~i) b th maximum liklihood stimators of th paramtrs. W will discuss thir computation in th nxt sction.) Thn, th vctor of forcasts using th nw paramtrs is 3.6) Following th sam principis usd in th univariat cas, w dfin as th global masur of influnc 3.7) Equation 3.7) shows that to comput PiB) it is ncssary to obtain only th vctor of forcasts from th standard transfr function formulation 3.1). Thrfor, th ordr h dos not nd to b spcifid, and will b qual to p + q + m + a, th numbr of paramtrs usd to comput th vctor of forcasts

8 3.2 A Dcomposition of th statistic. 7 C An obsrvation can b influntial bcaus it changs th paramtrs in th nois modl, in th transfr function or in both. As i t may b usful to idntify ths cass sparatly, w can dcompos th statistic to show ths ffcts. A possibl dcomposition is to us 3.8) if w lt c1 = p+q)/c, c2 = m+a)/c b th proportion of paramtrs in th nois and transfr function structur, w may writ 3.9) whr th trm - -2 Pi7T) = 7T - 7T i) ) I zi Z 7T - 7T i) ) / a p+q), 3.10) is a masur of th chang in th univariat paramtrs. A-1 = T'T, th scond trm can b writtn Ltting P i vl7t) = v - vi» I X'A- X) v - vi»/oa m+a), 3.11) and is proportional to th Mahalanobis distanc btwn {J and Ji) w will s in th nxt sction that 0a- 2 XI A-1X is th invrs maximum of th varianc covarianc matrix for th gnralizd liklihood stimator J), and rprsnts th chang in th transfr function param~trs g!vn th 7T paramtrs. Finally, th third trm is 3.12 )

9 8 ', and rprsnts th intraction btwn th chang in th transfr function and th nois paramtrs. This intraction trm will in many cass b ngativ, bcaus a dcras in th transfr function paramtrs will b linkd to a chang in th opposit dirction of th nois modlo W will s an xampl of this situation in sction 4. Th brakdown of statistic 3.7) into i ts componnts shows clarly that: 1) if thr ar no xplanatory variabls in th modl, 3.7) rducs to th univariat statistic prviously dfind, and 2) if th nois is whit and thrfor w hav a rgrssion modl, 3.7) rducs to th D statistic introducd by Cook 1977) for th rgrssion modlo 3.3 Computing Diagnostics To comput statistic 3.7) w nd to stimat th paramtrs of th modl whn on obsrvation is missing. To dscrib th natur of this stimation, lt us writ th modl as y = X v + R, 3.13) whr R is a vctor of nois that follows a multivariat normal distribution with man zro and positiv dfinit covarianc matrix a 2 A, such that A-1 = T'T. Thn, th log liklihood corrsponding to 3.13) is 2 n In a - t In IAI - 1 Y - Xv) 'A- 1 y - Xv), 3.14 ) 2 2a2 and, conditional on A,,th gnralizd last squars stimator of v is 3.15 ) with covarianc matrix

10 9 I:v) = X'-~X)a ) As th n paramtrs, and thrfor th A matrix, ar usually unknown, maximum liklihood stimation rquirs an itrativ algorithm in which, givn an initial n valu, th matrix A is dtrmind and an initial stimator of v computd with 3.15). Thn, th rror procss R is gnratd using 3.13), and th usual univariat tim sris stimation mthod applid to R to produc a nw valu for n. Th procdur is itratd until convrgnc. Now, it is wll known that whn A = I, th idntity matrix, and only on obsrvation is missing, th stimation of th paramtrs can b obtaind by including a dummy variabl in th rgrssion modlo W hav shown in sction 2.2 that, for larg sampls, this procdur works with univariat tim sris data. It will also work for modl 3.13) if w considr th X valus as fixd. Thn, if w stimat th modl 3.17) whr Ii) has a on in th ith position and zros lswhr, vi) will provid th stimator of th v paramtrs whn Yi is missing. As th A matrix is, in gnral, unknown, th stimation of 3.17) will rquir an itrativ algorithm. To dscrib its structur, lt us first analyz th cas in which th paramtrs n, and thrfor th matrix A, ar givn. Thn, using th Cholsky factorization -1 A = T'T and ltting Y = TY, X = TX, Ii) = TIi)' u = TR, th modl is 3.18 ) whr now E[UO'] = a 2 I. Thn, it is shown in th Appndix that th last squars stimators of th paramtrs ar ----._._-_._ _.

11 ) and 3.20) é' \, whr 3.21) and 3.22) To undrstand th maning of ths stimators, lt us considr th simpl dynamic rgrssion modl Thn, th stimator Bi) for B givn ~ modl will b obtaind from th 3.23) and it can b shown that it is givn by 3.24) whr ' l

12 '" 11-2 S iln) = I:X - <1>, x t X t - 1 ) and t '! i rt Y t =, <1>,. Yiln = Yi+1 + Yi-1) t = i 1 + <1>2) and X t = [X t t '! i p X'I = X i X i - 1 ) t = i 1. n 1 + <1>2) Not that Yi/n and xi/n ar th optimal stimators in th minimum man squar rror sns) of th missing valus Yi and xi. Both ar computd using th invrs autocorrlation function of th output nois) modl s Pña 1990). Thus, 3.24) is th stimator of th paramtr 8 using ths intrpolators instad of th obsrvd valus. Bsids, w. = y. 3.25) is th diffrnc btwn th obsrvd valu Yi and its stimator using all th information containd in th sampl. Whn th n paramtrs ar unknown, th ML stimator of all th paramtrs of th modl whn Yi is assumd to b missing can b computd as follows: C 1) Assuming initial valus for th paramtrs 80) = 1t0), '0'0» comput th stimator 1ti) 1'0'0) for th univariat paramtrs of th nois, as shown in sction ) Using 1ti) 1'0'0) comput th ML stimator of th transfr function paramtrs 'O'i) using 3.19) and 3.20). 3) Itrat until convrgnc.

13 12 In vry stp 1) or 2), using th initial valu for th paramtrs, th minimum man squar rror intrpolator for th missing valu is computd using whr ~iln can b 9iln or ~iln and in both cass p1k is th invrs autocorrlation function of th output procss computd using th currnt stimat of th paramtrs. Thn, nw sris ar constructd using 9i In and ~i In instad of Yit xi), and th maximum liklihood stimats ar computd. Ths valus provid a nw stimator of pik' that is usd to build nw stimators for th valus 9iln' ~iln' Th itrations ar rpatd until convrgnc. Not that although th natur of th stimators has bn analyzd using th linarizd or structural form of th modl "\ 3.2), all th computation shouldb carrid out in th parsimonious rprsntation 3.1). Hnc, th ordrs h and s in 3.3) nd not b spcifid. Thus, in practic w only nd to stimat th modl = w ~ I ~ ) + mi) B ) Xt + i) B) 3.26) Yt Oi) B) t>i) B) and th vctor Yi) ndd to comput 3.7) can b obtaind from th vctor Yi)INT of forcasts gnratd from modl 3.26) using 3.27) whr Qi' ~i) B) and ~i)b) ~om from th stimation of 3.26). Th valus fti) and ~i) ndd in th computation of statistics 3.10) and 3.11) can b obtaind using th rlations and oi) B)vi) B) = mi) B)

14 f " 13 i) B) 11 i) B) = <Pi) B) Th statistic 3.7) is dsignd to masur th chang in th paramtrs of th modl, and is basd on th vctor of forcasts using th vctor of paramtrs fti)' ~i) with th sampl data. An altrnativ masur that taks into account th chang in th forcast is AINT), ya AINT) Y - Y Y i) i) 3.28) whr Yi)INT is th vctor of forcast gnratd by th intrvntion modl 3.26). This masur rducs to 2.10) if thr wr no xplanatory variabls in th modlo In summary, th computation of 3.7) and 3.28) can b carrid out with a program that computs ML stimators for th paramtrs of a transfr function modlo It is only ncssary to introduc a dummy variabl an intrvntion impuls variabl) at vry point, stimat th modl and comput th vctor of forcasts. Thn, to obtain th statistic 3.7) w nd th vctor of forcasts 3.27), whras to obtain th statistic 3.28) w us dirctly th vctor of forcasts from th intrvntion modlo 3.4 Finding Influntial Points W can say that a point is influntial at lvl a if th paramtrs stimatd using a modifid sampl in which this point is assumd to b missing ar notincludd in th 1 - a joint confidnc rgion for th paramtrs stimatd with th complt sampl. For a tim sris modl, th joint 1 - a confidnc intrval for th vctor of paramtrs 8 is givn by Pristly, 1981, p.369) S8) - P ~a2. S8) :S Fp, n-pi 1-a),.

15 14 whr 5B) is th rsidual sum of squars for th paramtr vctor B, B th ML stimator, p th numbr of paramtrs, G a 2 th rsidual varianc and Fp, n-pi 1-Q) th 1-Q prcntil of th F distribution with p and n-p dgrs of frdom. For larg sampls, statistics 2.6) and 3.7) can b compard with a X 2 distribution with k and dgrs of frdom. Thn, if, for xampl, Pi 71) quals th 0,25 valu of th corrsponding X 2 distribution, assuming that th i-th point is missing would mov th stimat of 71 for ft. w to th dg of th 0,25 joint confidnc rgion This rfrnc distribution is only an approximation. Howvr, bliv that th main usfulnss of ths statistics is as xploratory tools, and a plot of th valus PiB) ovr tim will indicat if thr ar influntial points and will suggst possibl hypothsis to b tstd. 4 AN EXAMPLE '-. To illustrat th prvious procdurs, w analyz a dynamic systm rprsntd by two sris. Th input sris is th gas fd rat in a gas furnac, and th output is th c02 concntration. Box and Jnkins 1976) includd 296 pairs of data points with a sampling intrval of 9 sconds. To simplify th computations, w hav slctd a sampl starting with th first obsrvation and assuming that th systm was sampld vry 27 sconds and so takn on obsrvation out of vry thr in th whol data sto From now on w will us this sampl of 99 obsrvations. Tabl 4.1 givs th univariat modls for ths input Xt) and output sris Yt). Th transfr function modlling procdur in Box and Jnkins 1976) lads to L 2 Yt = B B )x t + N t.16) 09) 09) B +.20B )N = a aa =.68. t t.10).12) 4.1)

16 15 MODEL O'a Q36) B +.48B2 -.24B3)Xt = at B +.82B2 -.36B 3 )Yt = at Tabl 4.1 Modls for th fd rat and sampling). Q36) is th Ljung-Box rsidual corrlation cofficints. C02 concntration statistic computd 27 sco with 36 Tabl 4.2 shows th largst valus of th global influnc masur 3.7) and its brakdown according to 3.9). Ths statistics hav bn computd using intrvntion analysis as follows. To comput Pica) th following modl was fittd: and Lt B'1.') = ci), wi)wi)~i)~i» b th paramtrs stimatd with this modl and Yi)INT th vctor of forcasts gnratd with it, w comput.

17 16 which is th vctor of forcasts using th paramtrs Bt i) in th transfr function modlo Thn, Pi B) is computd using 3.7) To comput th componnts of Pi B) w hav ignord th covarianc btwn th stimators that wr small, and hnc Man Intrq. Rang PiB) Pivl") Pi'!) Di Y) Tabl 4.2 { Distribution of th influnc statistics in modl 4.1). Th thr important points 90, 96, 91) ar shown, as wll as th man and Intrquartil rang of th distribution of ach influnc statistic for th whol data sto \ Tabl 4.2 shows th diagnostic statistics PiB) and DiY) that ar plottd in figurs 1 and 2. It can b sn from thm that Y90 is th most influntial point both on th paramtrs of th modl

18 17 ' and on th forcasts. Th Tabl also shows that in this cas PiB) ~ and DiY) pinpoint th sam obsrvations, and although thr is a diffrnc in th scal in both masurs, obsrvation 90 is roughly fiv tims th valu of nxt larg ons. Howvr, th small valu of P90B) compard with a X 2 distribution with 5 dgrs of frdom indicats that its ffct on th paramtr stimats is small, that is, i t movs th stimator slightly within th j oint confidnc rgion for th paramtrs. This is confirmd by th brakdown of th statistic in th tabl. structur of th modl is robust to th sampl. Ths rsults man that th global To illustrat this fact, Tabl 4.3 shows th paramtrs stimatd assuming that Y90 was missing, and it can b sn that thir chang is small. Th largst rsiduals from modl 4.1) ar displayd in Tabl 4.4) whr it can b sn that all th largr valus ar concntratd at th nd of th sampl and aftr Y90. Th plot of th rsiduals Figur 3) shows som vidnc of a chang aftr Y90, and if w apply th outlir dtction tchniqus dvlopd by Chang, Tiao and Chn 1988) and Tsay 1986), obsrvation 90 is idntifid as an additiv outlir. Th siz of th outlir is stimatd as 2.12 with a t valu of 4.13). Thrfor, w conclud that obsrvation 90 is an additiv outlir, that it is th most influntial point both on th paramtrs and on th forcast figurs 1 and 2), but also that its ffct on th stimatd paramtrs is small, as shown by tabls 4.2 and 4.3. c w1 4>2 a w2 4>1 whol sampl valu 90 missing Tabl 4.3 Paramtrs of th modl with th whol sampl and assuming that obsrvation 90 is missing. Th valu of Y90 is 54.5 and its optimal stimat is

19 18 t I at/ja Tabl 4.4 Rsiduals gratr than 2Ja in th transfr function modlo ' Th larg rsiduals aftr obsrvation 90 sm to indicat an important chang aftr this point. Two hypothss may b considrd: th f irst is a lvl shift aftr this point, th scond a trnd shift. Ths ffcts may b modlld by a stp function or a linar trnd function aftr t=90. Thrfor, w will stimat th modls C, VYt =.24 S90)_.06) t 1.29B.09) B 2 ) V X t +.09) 1 Va t -.65B +.22B 2 )'.10).12) 4.2) and Yt = 90) 2 a t St -1.31B+1.82B )Xt B B~2~)'.10).78).09).09).11).12) 4.3) whr St90) is a stpfuncticin that taks th valu on for t ~ 90 and zro bfor. Modl 4.2) includs a linar trnd aftr t=90, and modl 4.3) a stp at this point. Th rsidual standard rror of at in 4.2) is.629, and.638 in 4.3), and th rsiduals of modl 4.2) aftr t=90 show a bttr bhavior than thos from modl 4.3). Not again th robustnss of th paramtrs of th modl comparing 4.2) with th rsults of Tabl 4.3. As th,

20 19 statistics of influnc for modl 4.2) kp showing an influntial ffct at t = 90 but nothing aftrwards, w will add an impuls at this point in modl 4.2), and w finally stimat th modl. 03) 2 90) B)a t VYt= V )St -1.40B+1.66B )Vx ~t 4.4).07).47).07).08) 1-.76B+.25B 2 )'.10).12) that has varianc.588. This modl shows that an unusual vnt rprsntd by an impuls) occurrd at t = 90 and producd a big ffct at this point, and from thn on a linar trnd of.24 units vry priod was addd to th sris. Not again that th paramtrs for th nois and transfr function in modl 4.1), and 4.4) ar similar. Although th modl for th nois in 4.4) sms vry diffrnt from th on in 4.1), th " wights for ths modls ar vry similar. Thrfor, w can conclud that th rlationship dscribd for modl 4.1) is wll dfind, that an intrvntion happnd at tim t = 90 producing an incrasing trnd in th output and making th procss non-stationary, and that modl 4.4) rprsnts an adquat approximation to dscrib th ffct of th intrvntion. To study th ffct of an anomalous vnt in th middl of th sampl, lt us assum now that in th original sampl at points 40 and 41 an rror of masurmnt is mad, and instad of th valu 59.4, th sam at both points) w obsrvd Lt us call this sris Yt. Thn th stimatd modl is Yt = B B 2 )X B +.28B 2 )-1 a, 4.5).22).18).. 18) t t..10).10) and sorn rlvant statistics for diagnosis ar displayd in Tabl 4.5. It can b sn that th most influntial point, as far as affcting ithr th paramtrs or th forcasts, is t = 41. Howvr, although th ffct on both th paramtrs of th transfr function Pivl") = 8.25) and th nois Pi") = 3.51) is strong, compard with a X2 with 2 dgrs of frdom) th global chang is

21 ..-._ _. 20 not vry larg, bcaus of compnsation ffcts btwn both parts. Not that th brakdown of th statistic PiB) allows us to say that if th objctiv of th xprimntation is to stimat th transfr function paramtrs w should conclud that modl 4.5) is not robust, bcaus obsrvation 41 is abl, by itslf, to modify th transfr function paramtrs significantly r Pi B) Pivil 7Ti) Pi7T) ' DiY) atlaa Tabl 4.5 statistics of influnc for modl 4.5). ar small. All th othr valus 5. CONCLUDING REMABKS í Th idntification of influntial obsrvations complmnts th study of outlirs. As 1s wll known from th standard rgrssion st up, whn th modl includs xplanatory variabls it is possibl to hav highly influntial points that ar not idntifid as outlirs. Th importanc of this analysis dpnds on th objctivs of th study. Whn th modl is built to intrprt th paramtrs or to tst th gain of th transfr function, w may want to know whthr or not th conclusions w draw from th data

22 21. ar vry much affctd by sorn small numbr of obsrvations which may or not b outlirs. Whn a point is idntifid as influntial in a givn modl, w should first chck if this point is also an outlir. If it is, th usual procdur is to incorporat it into th modl using dummy variabls s Tsay 1986, 1988). If th point is not an outlir, w fac ssntially th sam problm that has bn studid in standard rgrssion with high lvrag points. Thr is not nough information in which to vrify or dny th givn point basd just on th data. Th a priori knowldg of th problm undr invstigation must b usd to choos th appropiat modlo A wis stratgy may b to kp both modls -th on that includs th suspicious data and th on that assums this point is missing- and to chck thm with nw data as a mans of finding a bttr modlo In this lattr cas th study of influntial obsrvations may indicat th shadowy rgions of prsnt knowldg and suggst possibl hypothss to b xplord. \.

23 22 ACKNOWLEDGMENTS This rsarch has bn supportd by th Grant PBS7-0S0S Dirccion Gnral d Invstigación Cint1fica y Técnica dl Ministrio d Educación y Cincia, Spain and by th Graduat School of Businss at th Univrsity of chicago. to an anonymous rfr for hlpful commnts. Th author is gratful ~ \,

24 23 APPENDIX Th stimation of modl 3.18 ) will b. w. ]. v i) = I i) I i) XI Ii) I i) XIX x -1 I I XI Y Y and using th xprssion for th invrs of a partitiond matrix w. ]. = b -6CIX)-lXIÍi)b -bii) XX I X)-l X X)-l A whr and aftr som ar obtaind. straightforward oprations rsults 3.19) and 3.20)

25 24 REFERENCES Abraham, B., and Box, G.E.P. 1979), "Baysian analysis of sorn outlir problms in tim sris", Biomtrika, 66, 2, Abraham, B., and Chung, A. 1989), "Outlir dtction and Tim Sris Modling", Tchnomtrics, 31, 2, Box, G.E.P., and Jnkins, G.M. 1976), Tim Sris Analysis Forcasting and Control, Holdn-Day. Box, G.E.P., and Tiao, G.C. 1975), "lntrvntion analysis with applications to conomic and nvironmntal problms", Journal of th Amrican statistical Association, 70, Chang, l., Tiao, G.C., and Chn Ch. 1988), "Estimation of Tim Sris Paramtrs in th prsnc of outlirs", Tchnomtrics, 30, Clvland, W.S. 1972), "Th invrs autocorrlation of a tim sris and thir applications", Tchnomtrics, 14, Cook, D.R. 1977), "Dtction of influntial obsrvations in linar rgrssion", Tchnomtrics, 19, Fox, A.J. 1972), "Outlirs in Tim Sris", Journal of th Royal Statistical Socity. B, 34, Harvy, A.C., and Pirc, R.G. 1984), "Estimating Missing obsrvations in Economic tim Sris", Journal of th Amrican Statistical Association, 76, Jons, R.H. 1980), "Maximum liklihood Fitting of ARMA modls to tim sris with missing obsrvations", Tchnomtrics, 22, Kohn, R., and Ansly, C.F. 1986), "Estimation, Prdiction and Intrpolation for ARlMA modls with missing data", Journal of th Amrican statistical Association, 81, 395, Ljung, G.M. 1982), "Th liklihood function for a stationary Gaussian autorgrssiv-moving'avragprocss with missing obsrvations", Biomtrika, 69, 1, Martin, R.O. 1980), "Robust Estimation of Autorgrssiv Modls", in Dirction in Tim Sris, ds. D.R. Brillingr and G.C. Tiao. Hayward, CA: lnstitut of Mathmatical statistics.

26 ~---~~----~----~ , Fcwcas11ng Inf lunc 6r , s 4 3 " '" 1 ~ '" \-..., 1\ \. 1\ ',.J\ '.; \ I \.,'"\ O -Gt"aph R Figur 2 Plot of th statistic D.Y) in modl 4.1) 1

27 Pña, D. 1987), "Masuring th importanc of outlirs in ARlMA 25 modls", in Nw Prspctivs in Thortical and Applid statistics, ds. M. Puri t al, John Wily, pp Pña, D. 1990), "Influntial obsrvations in tim sris", Journal of Businss and Economic Statistics, 8, 2, Pristly, M.B. Prss. Tsay, R.S. 1986), of outlirs", 31, 393, ), Spctral Analysis and Tim Sris, Acadmic "Tim sris modl spcifications in th prsnc Journal of th Amrican statistical Association, Tsay, R.S. 1988), "Outlirs, lvl shift and varianc changs in tim sris", Journal of Forcasting, 7, 1-20.

28 Paramlr lntlu~lc 1.00 r O '";'t::;.""'-' I 0.00,~. \ ~ -Graph A Figur 1 Plot of th statistic p.a) ~ in modl 4.1)

29 Rsiduals iranstr tuncilon modl ""Zr , '-' 2 1 Figur 3 Plot of th rsiduals in modl 4.1)