Dynamc Regressons wh Varables Observed a Dfferen Frequences Tlak Abeysnghe and Anhony S. Tay Dearmen of Economcs Naonal Unversy of Sngaore Ken Rdge Crescen Sngaore 96 January Absrac: We consder he roblem of formulang and esmang dynamc regresson models wh varables observed a dfferen frequences. The sraegy adoed s o defne he dynamcs of he model n erms of he hghes avalable frequency, and o aly ceran lag olynomals o ransform he dynamcs so ha he model s exressed solely n erms of observed varables. A general soluon s rovded for models wh monhly and quarerly observaons. We also show how he mehods can be exended o models wh quarerly and annual observaons, and models combnng monhly and annual observaons. Key Words: Varables of dfferen frequences, dynamc regressons, emoral aggregaon, sysemac samlng, lag olynomals, seral correlaon. JEL Classfcaon: C ------------------------ Corresondence o: Tlak Abeysnghe, Dearmen of Economcs, Naonal Unversy of Sngaore, Ken Rdge Crescen, Sngaore 96. Emal: TlakAbey@nus.edu.sg, Ph. (65) 87 66, Fax. (65) 775 66. * The auhors would lke o hank he NUS Economercs Readng Grou for her valuable commens.
. Inroducon Economc daa are avalable n a varey of frequences. Economerc models, on he oher hand, are ycally consruced for use wh daa observed a he same frequences. Daases for use n any one economerc alcaon are hus assembled a he frequency of he lowes frequency varable, wh he daa seres avalable a hgher frequences convered o he lower frequency hrough emoral aggregaon or sysemac samlng, deendng on wheher he corresondng varables are flow or sock varables resecvely. A researcher may, for nsance, be neresed n modelng he relaonsh beween ouu and emloymen: f ouu s observed quarerly and emloymen monhly, a model ncororang hese wo varables would have o be secfed a a quarerly frequency, wh quarerly emloymen fgures sysemacally samled from he monhly fgures. Ths aer develos a modelng sraegy ha avods he need for all daa seres whn an economerc alcaon o be samled a he same me nervals. Dynamc regresson models are formulaed whch nclude varables observed a dfferen frequences. There are clear advanages o such a modelng aroach. Consder he case where he deenden varable s avalable quarerly whle he ndeenden varable s observed monhly. By allowng he ndeenden varable o be ncluded n he model a he hgher frequency, monhly mullers would be avalable ha would oherwse be los had he monhly daa been convered no quarerly observaons. The model would erm udang of quarerly forecass as monhly daa becomes avalable. Includng monhly dynamcs may also mrove one-quarer ahead forecass. A long hsory of aers has dscussed he effecs of sysemac samlng and emoral aggregaon on model srucure, arameer esmaes, forecasng and causal relaonshs (Zellner 966, Brewer 97, We 98, Wess 98, among ohers), bu hese works focus on
suaons where all he varables n he model are avalable a one frequency whereas he heorecal model of neres s defned a a hgher frequency. Our am s o develo a way of ncludng varables a her hghes frequences avalable, even f hese frequences are no he same across all varables. The sraegy adoed n hs aer s ha of Abeysnghe (998, 999), whch s o defne an auoregressve dsrbued lag model wh he dynamcs of he model defned n erms of he hghes frequency avalable among he varables. The roblem hen s one of mssng observaons, and our soluon s o aly ceran lag olynomals o ransform he dynamcs so ha he model s exressed solely n erms of he observed varables. Abeysnghe (998, 999) consdered a smle model wh an AR() srucure, wh he deenden varable samled less frequenly han he ndeenden varable. Our conrbuon n hs aer s o rovde a soluon for he general AR() case for models combnng monhly and quarerly, quarerly and annual, and monhly and annual observaons. We also ndcae how hese resuls can be exended o oher combnaons of frequences. We begn by nroducng he dynamc models ha we consder n hs aer. Focusng on he case where he model conans monhly and quarerly daa, we show how a sraghforward alcaon of lag olynomals can ransform he dynamc model so ha only observed frequences aear. The coeffcens of hese lag olynomals are smle funcons of he auoregressve arameers n he orgnal model. Esmaon and esng ssues are dscussed. Secon exends he mehod o quarerly-annual and monhly-annual combnaons, and we conclude n secon.
. The Basc Model The basc auoregressve dsrbued lag model ha we consder s ( y = α + β( x + ε, ε ~ d (, σ ) () where ( = +... L + L + + L, β... ( = β + β L + β L + + β L r r. We refer o hs as ARX(,r) model. The varables x and y are assumed o be avalable a dfferen frequences, and he me subscr s defned n erms of he hghes frequency. For examle, f x s monhly and y s quarerly hen =,,,T would reresen monhs. The model can nclude more han one regressor hough for exosonal uroses we wll say wh jus one regressor. Our aroach can also be exended o he ARMAX class of models, bu we leave ou he MA srucures o kee he exoson clear. In all our examles we wll assume ha s he deenden varable ha s observed wh he lower frequency, hough our resuls can easly be adaed for he reverse case. If he lower frequency varable y reresens a sock varable, and x s observed a m mes he frequency of y, hen only every mh observaon of y s avalable, and he observed daa se would comrse x, x,..., x } and y, y,..., y } where we have assumed for { T { m m T noaonal smlcy ha he frs avalable observaon of y s a = m and ha T s a mulle of m. In he quarerly-monhly case, m =. If, on he oher hand, y reresens a flow varable, hen wha s observed of y a every mh erod s an aggregaon of m flows recorded a he hgher frequency. The ARX(,r) can be modfed o handle he case of a low-frequency flow varable by emorally aggregang he varables o oban ( Y = α + β( X + ( + L + L +... + L m ) ε, ε ~ d (, σ ) ()
where Y ) m m = ( + L + L +... + L y and X ( + L + L +... + L ) x =, and he lag olynomals ( and β ( are as revously defned. Agan, under our assumons, wha s observed of Y are he values a m, m,, T whereas X s avalable a all lags. As he mehods we roose are smlar for boh he sock as well as he flow varable cases, we wll focus on he case where he low frequency varable s a sock varable, and refer o he flow varable case only when dfferences arse. Noe ha n he usual way of dealng wh msmached frequences, he hgher frequency daa s sysemacally samled, or emorally aggregaed deendng on wheher he varable s a sock or a flow. In our framework, wheher or no he hgher frequency (ndeenden) varable s aggregaed deends on wheher he low frequency (deenden) varable s a flow or a sock. The naure of he hgher frequency daa s nconsequenal.. Monhly-Quarerly Daa Consder frs he smle case wh an ARX(,r) srucure ( + y = α + β( x + ε () where x (=,,,T) s observed a monhly nervals whereas y s observed only quarerly, so only every hrd observaon of y s avalable,.e., he observed values of y comrse { 6 T y, y,..., y }. The sraegy adoed n Abeysnghe (998) s o ransform he model so ha only he observed frequences aear. Ths nvolves mullyng boh sdes of () by a lag olynomal ( = ( + L + L ) = ( L + ) whch wll conver he model o L ( + L ) y = ( + ) α + β( ( L + L ) x + ν () Noe ha Abeysnghe (998) adoed a fraconal me subscr whch we do no follow here. 5
o be esmaed over = τ, τ =,,..., T /. We wll refer o he lag olynomal ( as he ransformaon olynomal, and he lower frequency as he observed frequency. In hs case, he ransformed error erm v = ( ε sll manans he d roery a he observed frequency, and () can be esmaed by a non-lnear LS echnque. One of he advanages of hs aroach s ha alhough y s quarerly, he monhly mullers or mulse resonses can easly be worked ou from () usng ( β( once he arameers have been esmaed. In he general ARX(,r) case he necessary ransformaon olynomal wll be a lag olynomal of order, ( = ( + L + L +... + L ). Alyng hs ransformaon o () gves ( L ) ( y = () α + ( β( x + v (5) where v = ( ε. Noe ha he olynomal π ( = ( ( s of order. Seng he coeffcens of he unobserved lags of hs olynomal o zero,.e., π = π, j =,,, j j =, wll rovde relaonshs from whch we can solve for he coeffcens of ( n erms of he s. For llusraon, consder he ARX(,r) case where ( = ( + L + ). Mullyng L hs olynomal wh he ransformaon olynomal ( of order wll gve us he followng lag olynomal of order 6: + ( + ( + ) L + ( + + ) L + ( + + ) L + ( + 5 + ) L + + ) L L 6 Seng he coeffcens of lags,, and 5 o zero and solvng for he s wll yeld he followng soluon 6
= = =., =,, Thus he ARX(,r) model ( + ) L + L ) y = α + β( L x + ε can be exressed n observed frequences as 6 ( + ( ) L + L ) y = () α + ( β( x + ( ε (6) where ( = + L + L + L + L wh he s as defned above. The followng heorem rovdes he general soluon o he roblem of fndng he coeffcens of he lag ransformaon olynomal ( = ( + L + L +... + L ) for he ARX(,r) case. Theorem : Le = and + = + =... = =. If = c, j j= j j, =,,,,, where = j f rem = oherwse c, j, hen ( + L + L +... + L )( + L + L +... + L ) = ( + π L + π L +... + π L ) where π k = f k = j or j for some j =,,,. Proof: See Aendx A. 7
The erm j rem refers o he remander of quoen j,.e., we have c =, j f he dfference beween he subscrs of j and j s dvsble by, and oherwse. For convenence, he coeffcens of ( for he AR() hrough o he AR(5) case are abulaed n Aendx A. The case where y conans a un roo (a he hgher frequency) can easly be handled. A rocess wh a un roo a he hgher frequency wll dslay a un roo a he lower frequency afer alcaon of he ransformaon olynomals. In he quarerly-monhly ARX(,r) case, hs can be verfed by smly subsung = no he AR olynomal n (6) and seng L =. The un roo ARX(,r) case can be handled by facorng L ou of he -order AR olynomal n (), and alyng he ransformaon for he ARX( ) case followed by he ransformaon for ARX() wh =. We llusrae hs rocedure n he quarerly-monhly ARX(,r) case wh a un roo. Le ( = ( + L + L )( ). Mullyng hs olynomal L wh he ransformaon olynomal ( of order as n he ARX(,r) case wll gve us he followng lag olynomal: 6 ( + ( ) L + L )( y = () α + ( β( x + ( ε (7) Mullyng (7) by ( L ) = ( + L + L ) gves ( + ( ) L 6 + L )( L ) y = () () α + ( β( ( x + ( ( ε (8) where "( L ) x = X s a movng sum of x. The formulaon n (8) s suable for he suaon where x s a saonary varable. For examle, ( L )y may be he quarerly nflaon rae and x he monhly unemloymen rae. If 8
x s also a un roo rocess bu no conegraed wh y, he ( oeraor mus be aled hroughou equaon (7) and as a resul "( L )( x reduces o ( L )x, and ( ε becomes he whe nose rocess. In hs case modelng s done usng he quarerly dfferences of boh y and x. If y and x are I() rocesses and conegraed, hen he model revers back o he orgnal form (6) and can be esmaed n level form whou mosng he conegrang resrcon. Beng a dynamc model, sandard ess aly (Sms e al., 99).. Esmaon and he Auocorrelaon Problem We have noed n he quarerly-monhly ARX(,r) sock varable case ha he ransformed error rocess v = ( ε s no serally correlaed a he observed lags. Esmaon of he model arameers can herefore be carred ou usng a non-lnear leas squares mehod. However, he ransformed errors wll be auocorrelaed n he general quarerly-monhly ARX(,r) flow varable case for as well as he quarerly-monhly ARX(,r) sock varable case for. In he sock varable case, ( s of order and herefore v = ( ε sysemacally samled a every rd observaon wll be an MA(q) rocess where q n[ / ] where n[.] s he neger oeraor (Brewer, 97). For he flow varable case, v = ( ( + L + L ) ε and so wll follow an MA(q) rocess wh q n[ ( + ) / ]. To ge a feel for he sze of he auocorrelaons nvolved we exlore some smle cases below. For a general MA( q ~ ) rocess v q~ = ( + θ L +... + θ~ q L ) ε sysemacally samled a every m erods, he jh auocorrelaon a he observed frequency, ρ mj, can be comued as ~ q γ mj ρ mj = where γ = ( ε ε ) = σ θ θ, θ γ = mj mj E mj + mj, j =,,,, n[ q ~ /m]. In he = 9
quarerly-monhly ARX(,r) flow varable case, he ransformed errors v = ( ( + L + L ) ε wll follow an MA() rocess a he observed frequency. Afer subsung for he orgnal AR arameers, he observed frequency-frs order auocorrelaon of v s ρ m ( ) = + 5 + Fgure los hs auocorrelaon for saonary values of. The auocorrelaon roblem aears o be small; for values of (,), whch s he more lkely regon for economc daa (recall ha our AR coeffcens have sgns ha are he reverse of he convenonal secfcaon), ρ m s less han.. Unforunaely, here s no reason o exec he auocorrelaon roblem o be small for he oher cases. Fgure los he frs auocorrelaon of v = ( ε for he quarerly-monhly ARX(,r) sock varable case, whch also follows an MA() rocess when sysemacally samled a he observed frequency. The auocorrelaon s seen o le beween.5 and.5 for values of and n he saonary range. A lo of ρ m n he ARX(,r) flow varable case shows hs auocorrelaon o range from abou. 6 o. 6. The major obsacle osed by he auocorrelaon roblem s he nconssency of he nonlnear LS esmaor of he ransformed model. Snce he auocorrelaons, and herefore he MA arameers, deend on he AR arameers a smle alernave o leas squares s o use a nonlnear IV esmaor. Afer comung he auocorrelaons from he esmaed s, he MA arameers can be derved by solvng he se of non-lnear equaons gven n Box e al. (99,., eq. 6..). The same rocedure can be used o esmae σ = var( ) and he sandard errors ε of he IV esmaor can be recomued by relacng σ v by σ (noe ha σ σ v ). One has o go hrough he rouble of dervng he MA arameers only f he model s desgned for
forecasng. If he objecve s o derve he mulse resonses, hen he MA arameers do no ener he calculaons and can be gnored. The success of he IV esmaor deends on he qualy of he nsrumen used. One ossbly s o use lagged deenden varables y -(+j), j=,,.. as nsrumens, alhough hs may no work well f s large. Mone Carlo sudes carred ou n relaon o a flow ARX(,) model shows ha n small samles he LS and IV bas could be smlar and may be neglgble f he auocorrelaon s small (Abeysnghe, 999). Anoher raccal roblem s he choce of he lag orders and r. As observed n Abeysnghe (998) f s known he choce of r s no dffcul. Sarng wh a large value for r one can es downward o choose an arorae value for r. Comlcaons arse n he choce of because he form of he ransformaon olynomal ( deends on. One ossbly s o rea (5) as a reduced form and esmae as a lnear model. The number of sgnfcan lags would ndcae he arorae order of he lag olynomal (. If, for nsance, he coeffcen on 6 y s sgnfcanly dfferen from zero whle hose of y 9, y, are no, hs would mly =. If r* lags of x are sgnfcan, hs would sugges r = r* (ncluson of y 6 would necessarly mly he ncluson of a leas four lags of x ). The dsadvanage of hs aroach s ha some reduced form arameers mgh be very small, even f he orgnal srucural arameers are no, and n small samles hese arameer esmaes may urn ou o be sascally nsgnfcan. In summary, he raccal mlemenaon of our modelng aroach mgh ake he followng form: f un roo varables are nvolved, es for conegraon by converng all hgh
frequency varables o he low frequency avalable. If conegraon canno be rejeced, use he level varables for modelng, oherwse use dfferenced daa. Esmang (5) as a reduced form, as descrbed n he revous aragrahs, would sugges suable values of and r, afer whch (5) can be esmaed usng a non-lnear IV echnque. We sugges overfng o see f he chosen and r are suffcen. Noe ha he sandard es s alcable here. If he resduals aear o be emrcally whe nose, gnorng he MA srucure of he ransformed model would robably be nconsequenal, and he esmaed model may be u o use. In hs case a non-lnear LS esmaon of he model mgh be beer as he LS esmaor s more effcen han he IV esmaor; f he resduals reman whe nose under he LS mehod, he LS esmaes would be referable for nference. If resdual auocorrelaon s resen, he MA arameers can be derved as descrbed earler n hs secon. An alernave s o denfy an ARMA model for he error erm and esmae hem ogeher wh he model arameers as n he Box-Jenkns ransfer funcon nose model aroach,.e., generalze he ARX model o an ARMAX srucure.. Exensons o Quarerly-Annual and Monhly-Annual Cases Anoher emrcally moran case s where he deenden varable s observed annually and he ndeenden varable s observed quarerly. The general sraegy n hs case wll be o aly he ransformaon gven n he followng heorem wce. The frs ransformaon wll conver he quarerly lag srucure no bannual erms, and he second ransformaon wll conver he bannual srucure no an annual srucure. Inegraon and conegraon are nvaran o emoral aggregaon and sysemac samlng (Marcellno, 999).
Theorem : Le = ( ), =,,,, hen ( + L + L +... + L )( + L + L +... + L ) = ( + π L + π L +... + π L ) where π k = f k = j for some j =,,,. Proof: See aendx A. For examle, consder he AR() case ( + ) L + L ) y = α + β( L x + ε. We have o conver hs model o a form n whch he lag srucure on y only conans he lags n mulles of. Theorem suggess alyng he ransformaon ( = ( L + ) once o oban a lag srucure n mulles of for y o oban: L ( + ( ) L + L ) y = ( + ) α + β( ( L + L ) x + ( L + L ) ε. Alyng a second ransformaon ( ( ) L + ) gves us L ( + ( = ( + ( ( ) L + )( ( + ) L 8 L ) y + ) + ) ω + β( L + L )( L + L ) ε L )( ( ) L + L ) x A smlar dea can be aled o he monhly-annual case: frs ransform he lag srucure on y o he bmonhly form (usng Theorem ), followed by a ransformaon o he bannual form (usng Theorem ) and fnally o he annual form (agan usng Theorem ). As n he monhly-quarerly case, hese ransformaons creae a roblem of auocorrelaon of he ransformed error erm; he ransformed error erm follows an MA rocess a he observed frequences n all cases. For each, he fnal ransformaon marx wll be of
order, and he ransformed error wll follow, a he observed lags, an MA(q) rocess where q n[ / ] for he sock varable case and q n[ ( + ) / ] for he flow varable case. Fnally, we noe ha he above ransformaons can easly be adaed o he case where he ndeenden varable s observed less frequenly han he deenden varable. Now he ransformaon olynomal ( has o be worked ou n relaon o β( n (). To aly he revous resuls β( can be wren as β ( L ) = β * * r ( + β L +... + β ) where * β = β / β, =,,..., r. r L. Concludng Remarks Ths aer has rovded a modelng aroach whch allows varables observed a dfferen frequences o be framed whn a sngle model whou converng he hgher frequency varable no a lower frequency va sysemac samlng or emoral aggregaon. Ths aroach enals a number of advanages. Frsly, we can recover he mulse resonses or mullers a he hgh frequency me uns. Ths nformaon s oally los f one were o use he sandard sysemac samlng or emoral aggregaon aroach. Secondly, hs aroach s lkely o rovde beer forecass comared o hose based on he sandard aroach. Thrdly, forecas udang can easly be done as and when he hgh frequency daa become avalable. The cases ha we cover are mosly suable for macroeconomc analyss, where daa are usually avalable n monhly, quarerly or annual frequences. An exenson o oher combnaons of frequences may be fruful, esecally for areas lke fnance. Oher ossble avenues for fuure research nclude he exenson of our mehods o vecor auoregresson models and for causaly esng. For an llusrave alcaon see Abeysnghe (998).
References Abeysnghe, T., 998, Forecasng Sngaore s Quarerly GDP Growh wh Monhly Exernal Trade, Inernaonal Journal of Forecasng, 55-5. Abeysnghe, T., 999, Modelng Varables of Dfferen Frequences, Inernaonal Journal of Forecasng, forhcomng. Box, G.E.P., G.M. Jenkns, and G.C. Rensel, 99, Tme Seres Analyss: Forecasng and Conrol, rd ed., (Prence-Hall, Inc., New Jersey). Brewer, K.R.W., 97, Some Consequences of Temoral Aggregaon and Sysemac Samlng for ARMA and ARMAX Models, Journal of Economercs, -5. Marcellno, M., 999, Some consequences of emoral aggregaon n emrcal analyss, Journal of Busness and Economc Sascs 7, 9-6. Sms, C.A., J.H. Sock, and M.W. Wason, 99 Inference n Lnear Tme Seres wh Some Un Roos, Economerca 58, -. We, W.W.S., 98, Effec of Sysemac Samlng on ARIMA models, Communcaons n Sascs: Theory & Mehods, 89-98. Wess, A.A., 98, Sysemac Samlng and Temoral Aggregaon n Tme Seres Models, Journal of Economercs 6, 7-8. Zellner, A., 966, On he Analyss of Frs Order Auoregressve Models wh Incomlee Daa, Inernaonal Economc Revew 7, 7-76. 5
Aendx A Proof of Theorem By mullyng ( + L + L +... + L )( + L + L +... + L ), and subsung he exressons for from he heorem, we see ha π k akes he form π k = k = = k k = j= c, j j j k where = j f rem = oherwse c, j. Noe ha he subscrs of j, j and k add u o m. Noe also ha for k s mus sum o zero, e.g., all erms of, mus sum o zero, lkewse all Consder any one erm n he summaon n 5 erms mus sum o zero, and so on. π k conanng, a, b and k a b (where a, b and k a b are no necessarly dsnc). a, b and may aear because j = a, k a b j = b and k = k a b. There are sx ossbles, wh he corresondng values for and j, as follows 6
j j k j a b k a b a + b b a a k a b b k - b k a b b a k - a b a + b a b b k a b a k a k a b k a b a b k b a + b k k a b b a k a b + a k We now show ha π k = for each of hese 6 cases, when k akes he form j or j for any osve neger value j. Ths amouns o showng ha c n each case. k = j=, j = For hese 6 cases, we have o dvde he roblem no 8 sub-cases, 9 each for he cases where k akes he form j and j, and deendng on wheher a akes he form m, m or m, and wheher b akes he form n, n or n, where m and n are arbrary neger values. The label he egheen sub-cases as follows 7
case k a b case k a b n n m n m n n n n n 5 j m n j m n 6 n 5 n 7 n 6 n 8 m n 7 m n 9 n 8 n The followng able shows k c, j for each of he 8 x 6 cases, and comues c j = j=, for each case : j case 5 6 7 8 9 5 6 7 8 b a - - - - - - k a b - - - - - - a b - - - - - - k a b - - - - - - a + b k - - - - - - b + a k - - - - - - m n c n j n= j=, 8
k In all cases c, hence =. A smlar exercse for k of he form j wll show ha = j=, j = π k n ha case π k n general. Q.E.D. Proof of Theorem By mullyng ( + L + L +... + L )( + L + L +... + L ), we see ha π k akes he form, π k = k = = k = k ( ) k for k =,,...,. If k of he form j hen here s an even number of erms n he summaon, wh he ak a erms cancelng ou he k aa erms, herefore π k = f k s odd. Q.E.D. 9
Aendx A The followng able rovdes he coeffcens of he ransformaon olynomal ( + L + L +... + L ) for he ARX(,.) case where he deenden varable s observed quarerly and he ndeenden varable s observed monhly., j refers o he coeffcen n he jh cell ndcaed by row and column ARX(j,.). ARX(,.) ARX(,.) ARX(,.) ARX(,.) ARX(5,.),,,,,,,,, +,,,,, 5 5, + 5, 5 6 6, 6, 5 7 7, + 5 8 8, 5 9 5 5
Fgure Auocorrelaon n he Quarerly-Monhly Flow Deenden Varable ARX(,r) Case.5..5 ρ..5 -.5 -. - -.5.5
Fgure Auocorrelaon n he Quarerly-Monhly Sock Deenden Varable ARX(,r) Case.5 =.9 =.5 ρ = -.5 =. -.5 - -.5 - -.5.5.5 Noes: The dashed orons of he grahs show values of ρ n he non-saonary range of and.