Modelling the Trend: The Historical Origins of Some Modern Methods and Ideas

Transcription

1 gareh.jones Deparmen of Economcs Secon name Modellng he Trend: The Hsorcal Orgns of Some Modern Mehods and Ideas by Terence Mlls and Kerry Paerson Deparmen of Economcs Unversy of Readng Wheknghs Readng RG6 6AA Uned Kngdom Henley Busness School, Unversy of Readng 2013

2 Tle page Modellng he Trend: he Hsorcal Orgns of Some Modern Mehods and Ideas by T. C. Mlls, Loughborough Unversy and K. D. Paerson, Readng Unversy Trend-lnes are nended o descrbe some magnary sae of normalcy wherewh o compare he acual condons depced by he curve. (Karl Karsen, 1926, p. 31.) The sgnfcance (and perhaps he usefulness) of secular rend, consdered merely as a problem n curve fng, s unceran, no only because long-erm undulaons whch we know o have no real sgnfcance appear n random seres, bu because he rend s largely ndeermnae. (Vcor von Szelsk, 1929, p. 245.) Absrac The developmen of he mehods of correlaon and regresson analyss a he urn of he 20 h cenury, led o her use n aempng o denfy relaonshps beween economc varables. However, cauon was soon expressed ha correlang seres wh secular rends was lkely o be msleadng. Afer some dscusson of mehods, lnear derendng by leas squares esmaon became he defaul mehod. By he 1920s, however, some voces of dssen expressed he vew ha lnear derendng was lkely o be napproprae n some, even many, cases. Ths led o a number of nnovave mehodologcal developmens, ncludng rollng wndow esmaon, movng negraon, nonlnear rends, srucural breaks, sgmod-ype smooh adjusmen funcons, he begnnngs of sochasc rends and he consrucon of smoohers and flers. Alhough generally falng o have an mpac a he me subsanally predae her curren use n economercs. Ths arcle esablshes precedence for hese deas and recreaes some of he emprcal examples and early smulaons. Keywords: rend, derendng, decomposon, nonlnear rends, srucural breaks, sochasc rends, flerng. JEL: B1, B2, B4, C1 (Word coun: 9, fgures = 12,234)

3 I. Inroducon Followng he developmen of correlaon and regresson analyss n he lae 19 h cenury (see, especally, Edgeworh, 1892, 1893, 1894; Galon, 1888, 1990; Pearson, 1896; and Yule, 1897, 1907), here was consderable neres n applyng hese hen novel echnques o uncoverng economc relaonshps and, especally, o usng hem n he consrucon of chars or baromeers of economc acvy wh a vew o forecasng key varables, generally n he shor o medum erm. The neres n forecasng was boh academc and commercal, wh many aemps a consrucng ndces, baromeers and dals of economc acvy: see Knauh (1923) and Fredman (2009). There was also a compeve edge o producng forecass: for example, Smh (1929) noed ha, however he forecas seres (or ndex hereof) was obaned, he momen of comparson of he acual wh he forecas was always eagerly awaed: Afer he forecasng ndex has been consruced n one fashon or anoher, here comes ha breahless momen of suspense n he sascan's lfe when he hopefully compues he coeffcen of correlaon beween hs new-born forecasng ndex and he seres whch s he subjec of forecas. One hng s prey sure: he hgher he correlaon coeffcen, he faser he sascan's pulse. (bd, p. 94) In he early sages of calculang correlaon coeffcens and esmang regressons, here was a concern abou separang ou he apparenly secular movemens n me seres from flucuaons of hgher frequency. There was a commonly held vew ha regressng one rended seres on anoher mgh mslead as o he srengh of assocaon beween he wo varables; hs, of course, was a concern ha resurfaced much laer (mos noably by Granger and Newbold, 1974), and led o he developmen of conegraon (Engle and Granger, 1987). A ypcal pracce was o frs remove he secular movemens n some way, usually based on a deermnsc vew of he rend, and hen correlae he resulng derended seres, ofen referred o as he cycles. A less frequenly used alernave was o dfference each seres frs and hen correlae he dfferences, as n Hooker (1905) and Suden (1914). In essence, he underlyng problem o be solved was wha Yule (1921) descrbed as he me-correlaon problem : he essenal dffculy of he me correlaon problem s he dffculy of solang for sudy dfferen componens n he oal movemen of each varable. (bd., p. 501). 1

4 Ths sudy akes as s sarng pon he developmen of he dea of a secular rend, he mehods for dealng wh, and he assocaed developmens n concep and mehodology. The modellng of rends s one of he mos mporan ssues n modern me seres economercs (for example see Phllps, 2005, and Whe and Granger, 2011), bu such concerns have been around for a cenury or more. Whle some of he early effors a modellng rends are well known o modern researchers, ohers are only jus beng redscovered: wha hey ypcally have s a remarkable prescence n suggesng mehods and ponng ou phenomena ha are a he core of modern economercs. The prmary perod of focus for hs sudy s he decade or so dang from he early 1920s, whch was a very producve perod for conceps and mehodology relaed o he rend. These developmens ncluded some ha are now reaed as commonplace: rollng wndow esmaon; movng negraon; nonlnear rends and sgmod-ype smooh adjusmen o mulple srucural breaks; quadraure; an early undersandng of sochasc rends; and me seres flers. All hese remarkable developmens are, n one form or anoher, par of he concepual bass of modern economercs, bu are lle referenced, f a all. Ths paper s srucured as follows. Secon 2 oulnes he domnan vew prevalng afer he nal developmen and modellng of rends a he begnnng of he 19 h cenury, whch was amed a removng he secular elemen (genercally referred o as he rend); he pons wnner n hs developmen beng derendng by a smple polynomal, usually a lnear rend. Secon 3 connues by nong ha, afer hs nal developmen, here were several crcal suggesons along he lnes ha derendng n hs way may no be adequae for some, f no many, economc me seres. Ths led o some mehodologcal developmens, such as rollng wndow esmaon, he concep of negraon and he possbly of nonlnear rends, whch, whls no aken up a he me, are now an mporan par of he economs s oolk for modellng me seres. The quesonng of he adequacy of polynomal derendng also led o frs houghs abou sochasc rends, whch s he opc of secon 4. In Secon 5 we noe ha derendng s, n essence, jus a flerng of he orgnal daa, leadng o a seres ha s smooher n some sense han he orgnal seres; we oulne n hs secon a number of mporan developmens ha underle much of he curren vew on smoohng me seres (for example, he nfluenal Hodrck-Presco fler). Secon 6 concludes. II. Decomposon of a me seres 2

5 By he urn of he 20 h cenury, applcaons of he heory of correlaon were becomng popular usng economc and socal daa, bu here was an awareness of he poenal problems n applyng correlaon and regresson mehods o he raw me seres. For example, Hooker (1901), n examnng he correlaon beween he marrage rae and rade (aken o be he value of expors per capa) over he perod , rased an mporan dffculy wh correlaon analyss when appled o me seres daa. As he saw, f he movemens n he wo seres ha are beng correlaed are produced by a combnaon of slow secular movemens and more rapd, say year o year, changes, hen he laer may be hghly correlaed whle he former may be unrelaed, so ha he overall correlaon beween he wo seres may urn ou o be small. Ths s exacly wha appeared o be happenng wh he marrage rae and rade whch, over he perod, exhbed declnng and ncreasng secular movemens, respecvely, hus producng a calculaed correlaon of jus 0.18, wh a probable error of Arguably, wha was really of neres was he correlaon beween he mnor oscllaons he shor run movemens n he seres. Wha was necessary, so was argued, was o remove he secular componen from each seres and hen correlae he resduals. The praccal queson, a problem ofen seen as curve fng, was how o remove he secular movemens. (For an hsorcal perspecve on hs ssue and s relaon o spurous correlaon see Aldrch, 1995.) II.. Movng Average and Lnear Trends In conras o removng he sample mean from he seres o be correlaed, Hooker (1901) proposed a movng average (MA) rend: To correlae he oscllaons of wo curves, I propose ha all devaons should be reckoned, no from he average of he whole perod, bu from he nsananeous average a he momen. The curve or lne represenng he successve nsananeous averages I propose o call he rend., (bd., p. 486). Hooker proposed a movng average rend n whch, havng denfed a perod of, say, p years, he nsananeous average for year s he average over he perod of lengh p, where s he mddle year. If p s odd, hen s exacly he medan year of he perod. Lnear derendng based on a pror leas squares regresson was also popular and, ndeed, became he defaul pracce. The example n Fgure 1 s from Persons e al. (1922, Chars 16 and 17), who consruced an Index of Brsh Economc Condons usng he mehodology esablshed by Persons (1916) for he consrucon of a busness baromeer for he Uned Saes. The seres llusraed here s for he fregh 3

6 receps of sxeen Brsh ralways, where he lnearly derended seres s shown n he lower char. Fgure 1 here II.. Hooker: dfferencng Hooker (1905) reurned o he ssue of derendng, bu suggesed an alernave mehod for removng secular rends, ha of dfferencng, whch consss smply n calculang he correlaon coeffcens of he dfferences beween successve values of wo varables (bd., p. 697). Ths correlaon coeffcen was appled o daly changes of corn prces, fndng, unlke n he marrage and rade example, ha he absolue szes of he correlaon coeffcens so obaned were consderably smaller han (ofen less han half he sze of) he correspondng correlaon coeffcens calculaed from he levels. The concluson drawn by Hooker (1905) from hs analyss seems parcularly prescen when vewed from a modern perspecve: Correlaon of he devaons from an nsananeous average (or rend) may be adoped o es he smlary of more or less marked perodc nfluences. Correlaon of he dfference beween successve values wll probably prove mos useful n cases where he smlary of he shorer rapd changes (wh no apparen perodcy) are he subjec of nvesgaon, or where he normal level of one or boh seres does no reman consan. I may even, n ceran cases, be desrable o combne he wo mehods, and o correlae he devaons from he mean n he one seres wh he successve changes of he oher. (bd., p. 703, alcs n orgnal) Hooker was hus clearly aware of he dsncon beween wha are now called negraed processes and of he dffcules nheren n modellng he relaonshps beween seres of dfferen orders of negraon. III. Developmens of conceps and mehods The debae on derendng versus dfferencng s reasonably well known, wh sgnfcan conrbuons beng Suden (1914), Cave and Pearson (1914), Elderon and Pearson (1915), Persons (1917), Yule (1921) and Pearson and Elderon (1923). The pons wnner, n praccal erms, of hs debae was derendng, alhough, n an mporan and largely unnoced conrbuon, Smh (1926) suggesed a mehodology ha combned boh mehods (see Mlls, 2011). In conras o hs well-known debae, hs secon oulnes some varaons o fng lnear rends eher o he orgnal daa 4

7 or o her logarhms, hus ndcang he breadh of hnkng a he me, even hough n large par hese developmens were generally no aken up eher a all or unl much laer on. III.. Rollng wndow esmaon Hall (1925) suggesed wha we would now recognse as a rollng wndow mehod of esmaon raher han smple leas squares (LS). He movaed hs concerns as follows: he secular rend may be a connuous funcon he form of whch we do no know and whch can only be represened by a group of known funcons. (bd., p. 15) The dea s o choose a wndow lengh, whch Hall refers o as a perod, say p, whch s fxed; he LS esmae of he rend s hen calculaed for each wndow, whch successvely ncremen by one perod and, herefore, lose one observaon from he begnnng of he prevous wndow. Hall used p 4 o llusrae hs mehod, wh T 24 observaons; sarng a 1, here are hen 21 wndows of lengh 4: n general, here wll be M such wndows n a sample. In dscussng he choce of p, Hall suggess usng he perod of he cycle (and hus agrees wh Hooker, 1901, who uses a movng average); oo shor a perod relave o he cycle may confuse he rend wh he cycle, bu oo long a perod does no acheve he am of nducng some sensvy no he rend esmaes. Evdenly n such a wndow, whch adops an add one, drop one sraegy, he change n he LS esmaes depends on he dfference beween observaons ( m 1) p 1 and mp 1, m 1,, M ; f hese are equal, he rend esmae wll be unchanged. Rollng wndow esmaon s popular n a number of presen day conexs, for example n un roo ess (Banerjee e al., 1992), n esmaon n he presence of srucural breaks and n forecasng (for example, Pesaran and Tmmermann, 2007), and s avalable as an esmaon opon n some commercal sofware packages, for example STATA. However, Hall s (1925) conrbuon n hs developmen seems o have been overlooked. Hall llusraed hs approach usng Bradsree s Index of Prces, a quarerly seres avalable from 1900q1 o 1924q2, and we use hese daa o revs Hall s resuls and o demonsrae he dfferen rends and cycles ha resul 1. He noed a possble break n 1 All programmng heren uses MATLAB. 5

8 he prce seres effecve from 1915q1, so he esmaed one lnear rend for 1903q1 o 1914q4 and one for 1915q1 o 1921q4 (alhough he avalable daa exended o 1924q2, and curaled he sample for he smple rend as he fxed rend s no adapable o approxmaely curren daa (bd., p. 17)). The resulng rends and correspondng cycles are shown n Fgures 2 and 3; Hall s preference was o use he resduals relave o he rend, whch we follow here and below (as % of rend). Evdenly, he rollng wndow rend, referred o by Hall as he movng secular rend, racks he daa much more closely han a sandard lnear rend, even one wh a break. Fgure 2 here Fgure 3 here In a subsequen arcle, Hall (1926) suggesed a mehod ha was less compuaonally nensve han compung he LS esmaes for M wndows. Insead of compung M ndvdual LS rend regressons and akng he ordnaes a he sequenally fed values p, p 1,, T, Hall (bd.,) proposed esmang he lnear rends a four observaons apar (assumng quarerly daa for llusrave purposes), so ha he sequence of wndows ends a p, p 4,, T (for smplcy T s assumed o be an neger mulple of p). Then, o complee he mssng quarers, we nerpolae lnearly beween p and p 4, beween p 4 and p 8, and so on. If T s no an neger mulple of p (for example, suppose ha wo quarers are needed o complee he perod), hen a rollng regresson of perod p s fed n he usual way o emporarly oban he mssng rend ordnaes, hese beng adjused rerospecvely once he 3 rd and 4 h quarer observaons become avalable. III.. Movng Inegraon Hall (1925) also has some neresng commens on how o oban he (relave) cycle (and so mplcly he rend) by a procedure descrbed as movng negraon. Le he orgnal daa as a funcon of me be denoed f ; f he funcon s connuous, hen he defne negral over a perod of lengh a s f ( ) d and, usng dscree daa, k a hs can be approxmaed as k y ka (noaonally we use y, whereas Hall uses x and, mplcly, 1). Decomposng y no he hree componens of rend u, seasonal v, and cycle w, hen he perod of a cycle of lengh 12, hen: y k u v w. Illusrang wh quarerly daa over k y k11 k k = u v k k w 11 k11 k11. 6

9 If he seasonal effecs are (exacly) perodc (and hence cancel over he seasonal k k11 perod), hen v 0 and, f he perod of he cycle s ndeed 3 years, hen k k = k w 0 k 11, so ha y k11 u k 11, he equaly holdng over he perod of he cycle. Ths las equaly mgh be hough o be conenous, unlke he cancellng of he seasonal componens, embodyng as does he assumpon ha he up and down phases of he cycle cancel. However, replacng he equaly assumpon wh approxmaon does no damage Hall s general argumen. Thus, assumng ha hs equaly holds for any perod concdng wh he lengh of he cycle, we have he sequence: k Y k y k11, k 1 Y k1 y k10, k 2 Y y, whch Hall descrbes as movng negraon, n hs case hree k2, k9 year movng negrals of he secular rend, referred o here as MI(12). Nex, as he seasonal effecs are elmnaed over a year, k3 k 3 agan a movng negral can be formed, hs me over a year, k 1 k1 k2 k X y, X he secular rend plus cycle. 2 k2 y, k1 k y = k u + k w k3 k k3, and X k y ; hese are he one-year movng negrals of, To oban a relave cycle wo chan ndces are consruced: frs, he secular rend plus cycle based on one year of negraon: X k X k 1 ; second, he secular rend based on hree years of negraon: Y k Y k 1. The dfference beween he wo ndces s he relave cycle (noe here as resduals relave o he rend): C k X k X Y Y k1 k k1 Inuvely hs dfference s due o he cycle beng he dfference of he rend from he rend plus cycle. Equally of neres, alhough no emphassed drecly by Hall, s he relave rend Y k Y k 1. As wh he LS rends, he llusraed hs movng negraon mehod usng Bradsree s Index of Prces. The resulng MI(12) relave rend and chaned acuals, y k y k 1, are shown n Fgure 4; whereas Fgure 5 shows he relave MI cycle and, for comparson, he cycle derved from he movng secular rend. The correlaon coeffcen beween he wo cycles s 0.9, whch suppors Hall s vew ha he resuls from he movng secular rend and movng negraon mehods are n close agreemen. (bd., p. 22) Alhough nnovave, Hall s deas, boh on rollng wndow esmaon and (movng) negraon and summaon, were no pcked up unl very much laer and, as far as we can ell, whou acknowledgemen o hs work. 7

10 III.. The arcan rend Carmchael (1928) observed ha a number of me seres dd no exhb unformly smooh growh abou whch here was cyclcal movemen. Ths led hm o consder a nonlnear rend ha would be suable n hree crcumsances: () napproprae projecon of a negave lnear rend, leadng, for example, o negave or rdculously small posve values when comparavely large posve values only are possble (bd, p. 13); () approxmaely lnear growh ha s resumed afer nerrupon by an abrup change n level; () as () bu wh a frs nerrupon, for example a sharp drop, followed by anoher abrup change n level, before resumpon of he prevous growh. Fgure 4 here Fgure 5 here Carmchael addressed he queson of wha form could capure such movemens and noed ha a cenral creron for a rend s o emphasze he mporance of reasonable projecon no he fuure (bd, p. 13, fn 2). Hs suggesed form uses he arcangen, he properes of whch we brefly revew. Throughou Carmchael worked n degrees raher han radans. The angen of an angle, z, ha vares from 90 o 90, vares self from o +; n general, f x an(z) hen z an 1 ( x), where 1 an s he nverse funcon referred o as arcan(z ) ; z s n degrees and he converson o radans, whch are now used more frequenly, s y z 180. Thus, n radans, he lms of arcan(z ) are ( 2) and, herefore, by usng a suable scalng consan, say c, he lms can be adjused: for example, seng lms of c resuls n Carmchael s suggeson was o use erms of he form c arcan(x ) for he nonlnear rend. As we llusrae below, under an approprae choce of parameers, he arcan funcon s close o he logsc funcon ( 1 exp( x)), favoured a he me n populaon sudes (for example, Pearl and Reed, 1923). Carmchael s conrbuon, however, was more han o sugges an alernave nonlnear rend, for he showed ha, n combnaon wh a lnear rend, he arcan funcon can be used o model wha we would now regard as srucural breaks, an dea ha was aken up much laer n he work of Teräsvra usng smooh ranson funcons (see for example Teräsvra, 1994). 8

11 The arcan funcon s llusraed n he op panel of Fgure 6, along wh he cenred logsc funcon (1 exp( x)) , where, for comparably, he arcan funcon s n radans and he calbraon s 4c, 1 c. The dervaves of he arcan and logsc funcons are shown n he boom panel of Fgure 6; boh approach zero as x and are clearly smlar elsewhere. Carmchael suggesed he followng funconal relaonshps o capure hs hree prevously delneaed cases: () y a carcan(x ) () y a bx carcan(x ) () y a bx carcan( x) d arcan( x ) We would now convenonally add me subscrps o he varables y and x and a random erm, say. Fgure 6 here In conras, he sandard lnear rend model s y a bx. The varable y may eher be he orgnal daa or her logarhms (noe ha Carmchael used logs o he base 10, raher han o he base e). The varable x s a dsance measure relave o an orgn 0 0. Case () s a smple arcan rend, whch allows for a posve asympoe; case () allows for one nerrupon o he lnear rend; and case () allows wo nerrupons. In case (), he frs arcan componen allows wha Carmchael (bd, p. 14) descrbes as a rapd change n he level as x ncreases hrough zero and he second componen allows a second change hrough he me-shfed pon x. Wha Carmchael s effecvely descrbng, n curren ermnology, s a swchng model wh a sngle or double smooh ranson adjusmen funcon. We use one of Carmchael s examples o llusrae he use of he arcan rend. The me seres s of annual daa for Anaconda Copper common sock prces over he perod 1908 o Whls he lnear rend s fed values are nvaran o he orgn of he me varable (alhough he esmae of he nercep wll change), ha s no he case wh he arcan rend, where s necessary o choose he orgn x 0 for he me varable and a scale for yearly changes n x. In hs case, he problem addressed by Carmchael s ha of a farly rapd declne n he seres approachng 1924, he projecon of whch would lead o negave prces. Carmchael acknowledged ha hs mehod requred maers of judgemen as o he mng of he swch pon and he 9

12 speed of adjusmen, as represened n he choce of x. On hese he wroe: The necessy for arbrary choce of orgn and un on he x-axs s admedly objeconable. Could [he arbrary choces of scale and orgn] be obvaed whou loss of flexbly of he curve, he echnque would be vasly mproved. (Clarfcaon n [.] brackes.) These are relavely smple maers for curren economerc echnques o deermne. The esmaed lnear rend, akng x 0 for 1917, was y x ; he arcan rend, akng x 0 for 1900 and x 0. 9, was y arcan( x) (oher varaons were red by Carmchael). The seres and he esmaed rends are shown n Fgure 7; he rends are projeced hrough o 1935 o show he effec of he dfferen rend assumpons, n parcular he LS rend projecs negave sock prces, whereas here s a posve lm o he arcan projecons, wh a lmng value of The correspondng LS and arcan cycles are shown n Fgure 8. In deermnng and 0, based on a graphcal nspecon of he me seres, Carmchael red wo possble orgns and fve values of x, no all of whch led o an asympoe ha was posve. Whls he arcan model s nonlnear, once he dependence on x x and he swch pon are aken no accoun, s smple enough o exend he condonal leas squares search over a wo-dmensonal grd of possble values of x and 0, say S X T, whch wll resul n a conssen esmaon mehod provded ha he populaon values of IV. x and 0 are ncluded n S. The begnnng of sochasc rends Hall s (1926) dea of usng rollng wndow esmaon, n effec o approxmae an unknown and flexble form for he rend, resuled from he vew ha he secular rend may be a connuous funcon he form of whch we do no know and whch can only be represened by a group of known funcons (bd., p. 15). Whls hs was a developmen of he lnear rend assumpon, was sll framed n he conex of a (saonary) decomposon of a seres no s secular, seasonal and cyclcal componens (bd, p. 14). Hall s movng negraon mehod, alhough mplcly que revoluonary, was seen as par of he same developmen. However, f we add o hs Carmchael s (1928) developmen of smooh ranson adjusmen funcons, ogeher wh conrbuons o be oulned n hs secon by Karsen (1924, 1926) and von Szelsk (1929), we see n he 1920s he begnnng of he developmen of nonsaonary and sochasc rends as appled o economc me seres. Fgure 7 here 10

13 Fgure 8 here IV.. Quadraure Alhough receved wh some crcsm, Karsen publshed wo arcles, n 1924 and 1926, whch showed a dealed undersandng of how some economc me seres could be generaed as paral sums of ohers and ha here could, herefore, be a par of seres where one was he cumulaon of he oher. Underlyng Karsen s deas was he concep of quadraure, due o Charles Edge of he Quadraure Bureau. Two forces are sad by Mr. Edge o be n quadraure when hey race curves such ha he flucuaons of one of he curves correspond o he flucuaons of a curve of he negraon or cumulaon of he daa of he oher curve. When a cause-and-effec relaon exss beween such wo phenomena, he frs may hen be sad o be cumulavely affeced by he second. (Karsen, 1924, p. 14). Moreover on he deal: The quadraure mehod herefore apples cumulaon o oban a seres represenng he negral funcon, and dfferencng o oban a seres of daa represenng he dervave. (bd., p. 15) A par of seres (or curves), x and y, are sad o be n quadraure f hey are relaed such ha f x crosses he zero axs hen y has a urnng pon. 2 The smples (bu no he only) cases are hose ha have a sock-flow relaonshp, and Karsen movaes quadraure wh wo such examples, one of a bank accoun ha cumulaes he ne deposs (seres x) no a runnng balance (seres y) and he oher he ne flow no a reservor (seres x) and he level of he reservor (seres y). Karsen pu hus: he 2 Karsen (1924, p. 15) gves a foonoe of defnons of quadraure, showng ha he erm has several meanngs n use a he me, for example as a mehod of numercal negraon and he algnmen of planes; s use here concerns he phase dfference beween wo sgnals (seres), a quarer cycle (90 ) dfference leadng o he use of he erm quadraure. For example, f a me seres s generaed by a sne curve, cumulang he erms resuls n a cosne curve ha s dsplaced by a quarer of a perod relave o he sne curve. However, Karsen recognsed ha economc me seres are no as smple as sne curves: Tha economc forces usually flucuae or pulsae, so ha he curves of economc sascs are perodc or cyclc, s a fac wh whch everyone s famlar. Somemes he waves are very smooh, somemes hey seem cu up wh shor mnor waves, presenng a very choppy appearance. Somemes he prmary or long-me waves are farly regular, remndng one of sne curves; somemes hey are very rregular, large waves and small waves, shor and long, all followng close upon each oher. Bu whaever her form, hese wave-lke rregulares are presen n almos all economc daa. (bd., p. 15) 11

14 peak and valley of one curve appearng smulaneously wh he nersecon of he zero-lne of he oher (bd., p. 24). A conemporary example of wha Karsen mean by quadraure can be llusraed by he now famlar case of a (pure) random walk. Consder a whe nose sequence T { } 1 and he paral sum sequence T T { y} 1 { s 1 } 1. By consrucon hese pared seres are n quadraure: whenever 0 or changes sgn, hen here s a local peak or rough n y. The conex of Karsen s (1926) arcle are he hree curves, A, B and C, of he Harvard Busness Indexes: he A curve s he ndex of speculaon condons (on he New York Sock Exchange); he B curve ha of busness condons whch, hough varably defned, were n essence based on bank clearngs; and he C curve s he ndex of money and bankng condons, essenally an neres rae measure. (The ndexes are based on seasonally adjused and deermnscally derended seres, so her value of normalcy s zero.) Karsen suggesed ha he seres A and B were a par of seres n quadraure, ha s wh A beng he negral and B he negrand, so ha local peaks and roughs n sock marke prces were pared wh busness condons crossng he normal rend or zero lne. In Karsen s vew, he msake made by he Harvard Commee was whch of he wo really moves frs (bd., p. 405), snce hey vewed he A curve as precedng he B curve bu, n quadraure, seres A s he cumulave seres and canno precede he B seres. He commened: Perhaps escaped he aenon of he Harvard Commee ha he curve of he level of prces n he sock marke, Harvard Curve A, has a op or boom, a hgh or low-waer mark, whenever he curve of he flow of money no or ou of he sock marke from or no he busness world - represened by Harvard Curve B, he curve of busness condons - crosses s normal rend or zero lne. (bd., p. 404) To examne Karsen s asseron from a modern perspecve, we consder he me seres properes of he Harvard A and B seres. A necessary condon for Karsen o be correc s ha f B s an I(0) seres, hen A should be I(1). The daa are monhly for he perod 1919m1 o 1929m12, and he A, B and C seres are graphed n Fgure 9. Fgure 9 here 12

15 A vsual nspecon suggess ha seres A and B do seem o have dfferen characerscs. As Karsen only had nformaon hrough o he end of 1925 avalable o hm, he frs sample perod s aken o end n 1925m12. The Dckey-Fuller (DF) ype un roo es sascs are repored n Table 1 (he C seres s referred o below), and hese confrm Karsen s asseron: he null hypohess of a un roo s clearly no rejeced for seres A, whereas here s a rejecon n he case of seres B, whch s he case for boh sample perods. Table 1 ADF ess for Harvard Indexes A, B and C Seres A: I(1), ˆ Seres B: I(1), ˆ Seres C: I(1), ˆ ; I(2), ˆ 1919:4 1925: :4 1929: Noes: he mananed regresson for he es sasc ˆ ncludes an nercep and rend, ˆ ncludes an nercep; he lag lenghs were chosen by AIC. 5% crcal values are approxmaely 3.47 and 3.45 for ˆ and 2.88 and 2.90 for ˆ (smaller sample sze frs). The null of a second un roo for each seres was unformly rejeced. 13

16 Karsen also consders he relaonshp beween he C seres and, jonly, he A and B seres. He vews hese as beng n double quadraure : he C Curve s o a grea exen he cumulave of boh he A and B Curves. The C seres relaes o he prce of money, whch s largely deermned by he demand for money eher for busness purposes or for nvesmen purposes, ha s, eher n he busness world or n he secury markes (bd., p. 415). Thus, n Karsen s vew, he drvng force of he A and C curves s he B curve, whch s no self explaned: To explan Curve B we mus have recourse o oher busness sascs (bd., p. 417). Schemacally, Karsen s mplc model may be expressed as: C A s1 1 2 s1 B s 1 B s j j B s1 In me seres erms, f he B seres s I(0), hen he C seres s I(2). The ADF ess for seres C (see Table 1) pon o a frm rejecon of he null of wo un roos n favour of one un roo. However, Karsen s argumens show an mplc awareness of he operaons of sngle and double paral summaon ha underle he generaon of sochasc, as opposed o deermnsc, rends. IV.. The response o quadraure Karsen s (1926) crcsm of he nerpreaon of he Harvard ABC curves was no generally well receved, prompng responses from he Harvard group, Bullock, Persons and Crum (BPC, 1927), and Hansen (1927) 3 ; von Szelsk (1929) was crcal of he concepual bass of he mehod and (much) laer Sasuly (1947, p. 267) referred o quadraure as he correlaon hoax. 4 3 Hansen (1927) was no crcal of he dea ha one seres (he A curve) could be he cumulaon of anoher (he B curve), bu ha even n ha case one may have forecasng power for he oher: Mr. Karsen's own arcle gves an excellen llusraon of he grea value of he Harvard A Curve as a forecaser. He concludes ha because of he nerrelaon of he A Curve and he B Curve, he A Curve begns o urn down (or up, as he case may be) a he pon when he B Curve s crossng he normal lne. From hs he esablshes hs rend for he B Curve. He uses he A Curve o deermne he exac pon a whch he B Curve crosses he normal lne. Why? Because s a smple maer o deec he hgh and low pons on he A Curve, bu s no so smple o fnd he pons a whch B crosses he normal lne. I s jus for hs reason ha he A Curve s useful as a forecaser. (bd, p. 369) 4 The dea ha Karsen s resuls were a hoax relaed n par o he dffculy ha BPC had n deermnng he mehod of derendng ha was used by Karsen (see BPC, 1927, p. 88, for a summary of he correspondence on hs ssue). Referrng o Karsen s proof of quadraure for he Harvard A and B curves, BPC commened: Hs "sascal proof" consss n premsng he quadraure heory, 14

17 The Harvard reply, alhough crcal of Karsen s heory and mehods, conaned some mporan nsghs no how sochasc rends arse. BPC refer o he cumulaon mehod as comprsng wo seres: seres I, he generang (or orgnal) seres, and seres II, he cumulave seres (of he ems of seres I). Seres I may be a ransformed seres, for example, devaons from rend or normal values, so as o resul n a sequence comprsng boh posve and negave values. To llusrae wha BPC saw as he erroneous and arbrary naure of Karsen s cumulaon argumen, hey generaed random numbers for seres I and hen cumulaed hese no seres II. The daa for hs, one of he frs Mone Carlo expermens, was generaed as follows. BPC ook he followng senence: "The cumulaon process creaes cyclcal flucuaons ou of random sequences que common n numbers of all knds," assgned o each leer s numercal place n he alphabe, wh A assgned 0, and demeaned he resulng seres. They found ha he cumulave seres (seres II) s recognsable as havng he characerscs of a random walk (bd, p. Char A). BPC noed ha: I should be remarked especally ha he successve posve and negave ems of he generang seres need no be arranged by groups n a wave-lke form n order ha he resulng cumulave seres shall exhb wave-lke flucuaons. On he conrary s que possble for a very dsordered successon of posve and negave ems when used as a generang seres o yeld a cumulave seres n whch he oscllaory flucuaons parake of he wave-lke form encounered n hese economc movemens. Despe he lack of relevancy of he quoed asseron o he problem of securng an ndex of busness flucuaons and he wholly accdenal sequence of numbers used as a generang seres, s a noable fac ha he resulng cumulave seres s self a no unaccepable pcure of he flucuaons n general busness condons snce he war. (bd, p. 82) These observaons were no, however, regarded consrucvely. BPC observed ha he cumulaon of random npus led o he creaon of cycles, an exaggeraon of cyclcal flucuaons, pronounced undulang movemens, and wha we would erm as spurous correlaon. On he laer hey noed: delberaely manufacurng a hgh coeffcen of correlaon o suppor, and offerng he coeffcen hus secured as proof of he valdy of he quadraure heory. (bd, p. 88) 15

18 Consequenly, when he cumulaon process has been used, hgh values of coeffcens of correlaon beween any wo seres (or smlary beween he curves) are very doubful evdence ha he seres are relaed or even vary ogeher. (bd, p. 83) IV.. Furher expermenal evdence BPC had, wh her smple smulaons, hghlghed a number of he key characerscs of I(1) processes and he (poenally spurous) correlaon of wo I(1) seres. These mplcaons were no developed consrucvely, bu von Szelsk (1929) ook he BPC analyss furher a an expermenal level: The purpose of hs paper s o show ha correlaons beween me sequences are of lle sgnfcance n hemselves, and n parcular ha he process of cumulaon or quadraure descrbed by Karl G. Karsen may nroduce spurous correlaon and pseudo-cycles. (bd., p. 241) Szelsk underook a number of smulaons and, o llusrae hs mehod and argumens, we recreae hs Expermen 1, whch was o ake random draws from slps of paper numbered from 4 o +4, wh replacemen. There were n 110 such draws, resulng n he sequence { x } ha s 1 y x s ; he hen creaed he paral sum of he x, so. A rend was hen fed by leas squares o y, say y~, wh Szelsk s chars suggesng ha a 2 nd or 3 rd order polynomal was used dependng on he vsual properes of he cumulaed seres, and, fnally, he devaons from he fed rend, y ~ y z y ~ y., were cumulaed no s 1 s s The recreaed resuls for one such expermen are shown n Fgures 10 and 11. Szelsk s raonale for hs expermen was o show ha he paral sum has rend-lke behavour (see Fgure 10), whch we would now recognse as a sochasc rend, and hus, n he mehodology of he me, would frs be deermnscally derended usng a low order polynomal fed by leas squares. The resul s a seres ha shows cyclcal or pseudo-perodc behavour (see Fgure 11). Fgure 10 here Fgure 11 here 16

19 In von Szelsk s orgnal Expermen 1, he found he correlaon beween seres A and B (he frs wo n hs smulaon) was , whereas he correlaon beween he correspondng seres as cumulaed devaons from rend was We can confrm he apparenly spurous naure of he correlaon n he cumulaed seres by replcang he expermen 6 10 mes; he bvarae correlaons converge o 0.076, o for he cumulave seres and o for he devaons from rend seres. Von Szelsk (1929) carred ou a number of oher expermens o llusrae he consequences of cumulang quas-random seres, such as con ossng and dce rollng. As a resul, he drew fve conclusons (bd., p.245) 5 : 1. Seres comprsng cumulaed random ems exhb every characersc of economc me seres, excep seasonal varaon. 2. The sgnfcance (and perhaps he usefulness) of secular rend, consdered merely as a problem n curve fng, s unceran, no only because long-erm undulaons whch we know o have no real sgnfcance appear n random seres, bu because he rend s largely ndeermnae. 3. The process of cumulaon ncreases he amplude of he longer cycles. 4. The valdy (or usefulness) of he devaons-from-rend mehod of dscoverng relaonshps beween me seres s open o queson, because may show a correlaon where n realy none exss, or may ncorrecly measure a rue relaonshp. 5. The cumulaon of me seres ends o nroduce spurous correlaon. Ths has been abundanly llusraed above. I need occason no surprse. Alhough he connecons are no ye drawn ogeher, one commen s worh recallng: Even a Ph.D from a school of busness could no ell whch of wo graphs was, le us say, a wholesale prce ndex, and whch he cumulaon of successve hrows of a dce. (bd., p.) 5 Szelsk commened: Chance sequences were obaned from a successon of dce hrows. These were cumulaed, and he resuls graphed. They had a srkng resemblance o he pon chars of a sock rader. Char fend would doubless pronounce hem ypcal examples of marke acon, and pon ou he perods of accumulaon, ressance pons, double ops, and so on. The wrer could no ress he empaon of hese chars, and made que a lo of paper money playng he ndcaons as he dce hrows ran he prce up and down he scale. The possbles of he cumulave mehod are remarkable. (bd., p. 245). 17

20 In rerospec s he nverse queson ha s of neres: were rends and busness cycles generaed by he cumulaon of random causes? Kuznes (1929) was wary of makng he nverse nference ha acually observed cycles were ndeed caused by cumulang random shocks. can one nver he proposon and say ha, herefore, cyclcal oscllaons may be conceved prmarly as resuls of summaons of random causes, and ha he characerscs of some of hese cyclcal oscllaons can bes be grasped as a resul of he underlyng random evens or of he process of cumulaon? (bd., p. ) The properes of seres ha resuled from cumulang underlyng random sequences were becomng known n he lae 1920s and early 1930s: Slusky s famous paper on he generaon of spurous cycles was publshed n Russan n 1927 and n Englsh n Kuznes (1929) publshed an nsghful paper on he generaon of cyclcal movemens by cumulaon or consrucng a movng average, bu he ruled ou he possbly ha me seres could be generaed by a smple cumulaon of all pas shocks: IV.v. Workng (w)e shall om he sragh cumulaon alogeher, because s oo far removed from realy. In economc lfe, here s no perpeual nfluence of an even once occurred on all he subsequen evens. (bd., p. 269). The essenally saonary vew of he world expressed by BPC, Kuznes and ohers, was no shared by Workng (1934) 6, who was aware of a growng sense of he mporance of wha we now recognse as I(1) seres, even hough he sandard pracce a he me was very much whn a rend saonary framework: I has several mes been noed ha me seres commonly possess n many respecs he characerscs of seres of cumulaed random numbers. The separae ems n such me seres are by no means random n characer, bu he 6 Workng noes ha I has several mes been noed ha me seres commonly possess n many respecs he characerscs of seres of cumulaed random numbers. The separae ems n such me seres are by no means random n characer, bu he changes beween successve ems end o be largely random. Ths characersc has been noed conspcuously n sensve commody prces. On he bass of he dfferences beween chan and fxed-base ndex numbers Kng has concluded ha sock prces resemble cumulaons of purely random changes even more srongly han do commody prces. (bd., p. 11) 18

21 changes beween successve ems end o be largely random. Ths characersc has been noed conspcuously n sensve commody prces. The fac ha seres commonly used as ndexes of busness acvy closely resemble seres obanable by cumulang random numbers has gven suppor o he heory ha so called busness cycles resul n large degree from cumulave effecs of ndependen random nfluences bearng on he busness suaon some favourably, some unfavourably. (bd., p. 11) Workng (bd.,) noed ha some of he pracces n drawng random numbers o llusrae he characerscs of cumulaed seres were unlkely o produce a random seres; for example, assgnng numbers o he leers of he alphabe and hen akng parcular senences and consrucng he sequence of assocaed numbers. Workng drew hs random numbers from ables of Tppe s Random Samplng Numbers (Tppe, 1927), whch were convered o form draws from a normal dsrbuon wh a known sandard devaon. He hen provded a sequence of 2,399 random numbers obaned n hs way, whch were cumulaed no a paral sum sequence and, for presenaonal purposes hs seres was arranged as 46 years of weekly daa. Ths was an example of wha Workng called a random-dfference seres. The resulng seres showed a noceable negave rend, whch Workng observed mgh exce suspcon ha he orgnal seres had a negave mean whch was no, however, suppored by a es o ha effec. In fac, he seres so creaed was an example of a largely negave random walk. Ths arfcally creaed seres was hen used as a bass of comparson wh acual me seres and Workng (as prevous auhors had done) noed he smlary wh acual me seres. Bu, unlke BPC (1927) and von Szelsk (1929), Workng dd ake he vew ha paral sum (or random-dfference) seres were relevan o economc analyss and no an rrelevan consruc. He ook he vew ha acual seres, wh he characerscs of rend and cycle, could have been generaed by he cumulaon of shocks. Workng compared hs arfcal me seres o a seres on whea prces, whch had been a subjec of neres o hm. Whls he noed some smlar characerscs, some dffered, parcularly n he bmodal naure of changes n whea prces, and he offered wha ranspred o be a prescen commen: 19

22 I fnd ha o he mporan exen ha whea prces resemble a randomdfference seres, hey resemble mos closely one ha mgh be derved by cumulang random numbers drawn from a slghly skewed populaon of sandard devaon varyng raher sysemacally hrough me., (bd., p. 24). From a modern perspecve, Workng s clearly suggesng a marngale dfference sequence havng me varyng varances, whch may be nerpreed as measurng volaly. V. Tme seres smoohers and flers Alhough movng averages had been nroduced by Hooker (1901) for he purposes of smoohng a me seres and exracng a rend, more sophscaed applcaons of he mehod were developed n he 1920s. These had her geness n acuaral scence, where he procedure of graduaon, by whch crude moraly raes are ransformed o produce smooher esmaes of moraly, uses eraed movng averages, as orgnally suggesed by Spencer (1904, 1907) and oulned n Whaker and Robnson (1924, chaper XI). Suppose ha he prmary seres o be graduaed s nroduced he noaon u. Whaker and Robnson m 2 j m 1 u u m j o denoe he sum of 2m 1 u s cenered on u, and he followng generalzaon o weghed movng sums, m w m,,w 0,,wm u j m w u j j Spencer s 21-erm movng average can hen be wren as he eraed movng average ,9,4 v u 55 7 Ths may be expanded o oban v 18 5 {60u 57u 1 u 1 47u 2 u2 33u 3 u3 u 4 u4 6u 5 u5 2u 6 u6 5u 7 u7 u u 3u u u u } or 20

23 v 0.171u u 1 u u 2 u u 3 u3 u 4 u u 5 u u 6 u u 7 u7 u u 0.009u u 0.003u u whch s a symmerc movng average conanng en leads and en lags of he prmary seres u o oban he graduaon v. A propery of such a graduaon s ha wll reproduce a cubc polynomal n me whou dsoron. In a sequence of papers, Henderson (1916, 1924) and Whaker (1923, 1924) ndependenly consdered he problem of desgnng a smooher (laer more commonly known as a fler) ha, as well as reproducng a cubc polynomal rend whou dsoron, would also sasfy ceran smoohness condons. The prmary condon was ha he fler should mnmze he varance of he hrd dfferences of he 3 smoohed seres,.e., should mnmze Var average, for example, s hen gven by v. Henderson s 23-erm movng v 0.148u u 1 u u 2 u u 3 u3 u 4 u u 5 u u 6 u u 7 u7 u u 0.016u u 0.011u u 0.004u u These flers evenually found her way no use as smoohers of economc me seres, wh Henderson flers beng adoped for rend esmaon n he X-11 seasonal adjusmen procedure, replacng he Spencer movng averages used n earler versons of he U.S. Bureau of he Census seasonal adjusmen procedure. Almos fory years afer her nroducon, Leser (1961) revsed he Whaker- Henderson approach and exended he mehodology by dervng he weghs usng he prncple of penalzed leas squares, n whch a lnear combnaon of wo sums of squares s mnmzed. The frs sum of squares conans he devaons of he observaons u from he fler v, he second conans he second dfferences of 2 successve smoohed values v, wh he lnear combnaon of he wo beng defned by he weghs of uny and,.e., for he observed sequence u,u,,u he mnmand s T 1 2 T 1 2 u 2 v v T, The frs erm measures he goodness of f of he fler, he second penalzes he deparure from zero of he varance of he second dfferences of he fler, so ha s 21

24 a measure of smoohness: hence s referred o as he smoohness parameer. Successve paral dfferenaon wh respec o he sequence order condons v leads o he frs 2 v v 1 2 v u v so ha, gven T and, v s a lnear funcon of u wh me varyng weghs: v T j 1 w, j u j Leser hen developed an algebrac mehod of obanng he coeffcens w,, provdng a number of examples n whch he soluons were obaned n laborous and excrucang deal. However, s hsorcal mporance les n he fac ha he mehod developed by Leser and based upon he Henderson-Whaker approach was exacly ha proposed some wo decades laer by Hodrck and Presco (1997) and whch has enered no macroeconomc modellng as he H-P fler. j As an example, Whaker and Robnson (1924) consdered graduang, or smoohng, he seres generaed by: u f where he are drawn from a unform dsrbuon rangng from o Fgure 12 shows u self, Henderson s 23-erm movng average v, and he H-P fler usng a sandard seng of 100: he laer wo graduaons are vrually ndsngushable and have a correlaon coeffcen n excess of Fgure 12 here VI. Concludng Remarks There s no doub ha one gans from he early leraure on correlaon and regresson analyss a sense of grea excemen, arsng from he promse of usng hese new ools of sascal analyss o dscover relaonshps beween varables. However, s also clear ha even from early use of hese echnques here was an awareness ha a me seres may have several co-exsen characerscs, he key dsncon beng beween he slowly movng (low frequency) componen, usually denfed as he rend, and he more rapd movemens, usually referred o as he cycle. Ths s a dsncon ha s famlar oday. In wha may be seen as he frs phase of he developmen of me seres analyss, a farly sandard approach was o derend he 22

25 me seres of neres, usually adopng a smple lnear rend funcon, n wha we would now refer o as a saonary decomposon of a me seres along he lnes of an unobserved componens model. In broad erms, afer he nal phase of neres, here were some que fundamenal mplc crcsms of hs sandard pracce. Ths quesonng phase we mgh refer o as phase wo. I recognsed ha lnear derendng was a praccal (raher han concepual) soluon o he me-correlaon problem. The rend was n general lldefned and, anyway, unlkely o be lnear 7. Thus, for example, Hall (1925, p. 14) commened ha: I s mprobable ha he secular rend s srcly a sragh lne because s unlkely ha any economc phenomena wll be encounered whch s growng or ncreasng a a consan rae. Hall, herefore, proposed a more flexble way of modellng he rend 8. In a parallel leraure n acuaral scence, a key ssue was o defne he smoohness requred when flerng a me seres, of whch a lnear rend was a specal case. Ths dscusson necessarly nvolved a defnon of he creron of smoohness, whch had been naed by Sprague (1887) based on he second dfference of a me seres and was furher developed n he 1920s by Henderson and Whaker, and forms he bass of he well-known Hodrck-Presco fler. Also n he 1920s, he quesonng of he suably of he derendng approach, n fac he fundamenal queson of wheher was even sensble o concepualse he rend as deermnsc, had begun, f no by accden hen wh an elemen of unnended consequences n he dscusson beween Karsen (1926) and Bullock, Persons and Crum (1927). Ths dscusson led o he concluson ha he cumulaon of random shocks could lead o a me seres ha looked remarkably lke some real world seres. Whls hs mgh lead, as n BPC, o a dsmssal of he dea ha cumulaed or negraed random shocks could be he daa generang process, he dea was aken up furher by Szelsk (1929), who crcally apprased he domnan derendng mehodology; an mporan concluson beng ha he rend s largely ndeermnae, 7 The queson of wha consued a rend was sll a maer of dscusson n he 1930s. For example, Frckey (1934), n a paper enled The Problem of he Secular Trend, sared from he bass ha he secular rend has ofen been defned n some such language as he followng: he gradual and perssen movemen of he seres over a perod of me whch, conrased wh he shor run flucuaons of he seres, s long. (bd., p. 199) 8 The domnance of he lnear or loglnear rend was, however, self perssen (ndeed as s oday as he defaul choce). For example, Bernsen and Cowden (1937) proposed he use of a specal paper whch ploed he lnes correspondng o dfferen consan raes of growh over a parcular lengh of me (known as soropc lnes). They noed ha: The ype of paper we have developed for graphc presenaon of rend daa s desgned o elmnae auomacally regular geomerc growh. I s probable ha mos mporan economc seres n common use show hs ype of growh. (bd., p.446). 23

26 an observaon ha we now undersand more fully n he conex of negraed seres (n whch he rend s sochasc). Unforunaely, Szelsk s promsed heorecal analyss of hs resuls dd no maeralse (or was no publshed), bu we now recognse ha hs expermens generaed negraed seres. Phase wo led o he promse of a new nerpreaon of he generaon of economc me seres, parcularly nsofar as he modellng of he rend was concerned 9. I presaged a move away from he smple wo-sep procedure of fng he rend by a smple polynomal, possbly allowng for breaks n he rend, usng a movng average or a nonlnear rend funcon, hen savng he resduals and usng hese n he subsequen analyss. However, hs promse was no fulflled unl a number of years laer. 9 Despe he promse of he developmen of wha we now refer o as sochasc rends, he mehodologcal paradgm remaned broadly he same for a number of years. For example, Kendall (1941) commened: In recen years he mahemacs of rend fng have been brough o an advanced sage of developmen. There are varous mehods advocaed by dfferen wrers for dfferen purposes; bu, pung asde graphcal mehods, whch nvolve personal judgmen and canno clam serous aenon for heorecal work, hey all depend on one of wo processes: eher (a) a polynomal of chosen degree s fed o he whole seres, almos nvarably by leas squares; or (b) a' movng average of chosen exen and wh chosen weghs s used o deermne rend values a dfferen pons of he seres., (bd., p. 43). 24