COMBINATIONS OF HIGH AND LOW FREQUENCY DATA IN MACROECONOMETRIC MODELS

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1 1 CHAPTER 1 COMBINATIONS OF HIGH AND LOW FREQUENCY DATA IN MACROECONOMETRIC MODELS L.R. Klein and E. Sojo, Universiy of Pennsylvania 1. ECONOMETRIC PRACTICE Economericians who ry o follow and projec he overall economy as closely as possible ( Economy Wachers ) frequenly base heir main forecass on macroeconomeric models, supplemened by he frequen flow almos daily of indicaive informaion. As soon as repors are prepared abou some specific area of economic aciviy, hey are released o he public. A he exreme, we have insananeous marke repors, originaing wih he sar of he day a he Inernaional Dae Line and moving wih he sun o Tokyo, Hong Kong, Sidney/Melbourne, Singapore, Frankfur/Paris, London, New York, Chicago, Los Angeles/San Francisco. These daa cover boh commodiy and financial marke repors. On a monhly basis, here are naional saisics for price indexes, wage raes, employmen, unemploymen, orders, shipmens, invenories, consrucion, expors, impors, and many oher indicaors. Mos macroeconomeric models are based on social accouning saemens ha are prepared a eiher quarerly or annual inervals. I is possible o consruc social accouns more frequenly, bu i has no been done on more han a fragmenary or sporadic basis. In a pracical sense, we should assume ha quarerly models will be used for some ime o come. They will use sample daa from quarerly social accouns, many of whose enries will be ime aggregaions of monhly or higher frequency daa. The use of high frequency daa will be for he consrucion of price deflaors bu also for he direc esimaion of nominal (curren value) enries in he accouns. The deflaors of consumer spending will make use of L.R.Klein and Marquez (eds.), Economics in Theory and Pracice: An Eclecic Approach,3-16. C 1989 by Kluwer Academic Publishers.

2 2 monhly daa on consumer prices; oher deflaors will use monhly prices from differen indexes wholesale, producers, commodiy markes wih he excepion of governmen employee compensaion, which is obained implicily (nominal/real). Personal income series are acually produced monhly, and heir enries in he naional income accouns are obained from monhly ax repors, social insurance records, company dividends when issued, and monhly public secor budge repors. On he expendiure side of he social accouns, consumer oulays are compued from monhly (weekly) reail sales repors, invenory invesmen from monhly invenory repors and expor/impor oulays from monhly rade saisics, which need o be convered o a naional accoun basis. The preparaion of social accouns is a major saisical aciviy involving he combining of many diverse pieces of informaion. Many of hese pieces are produced a high frequency inervals, bu he oal ariculaed repor is quarerly, a bes, and for many counries only annually. In using macroeconomeric models for forecasing he naional economy, praciioners are ied o he ime schedule for preparing and releasing social accouns. In he Unied Saes, he main lag is approximaely hree weeks. Afer he end of a calendar quarer, he firs round esimaes of he compleed quarer are released on abou he 20 h of January, April, July and Ocober. These lack profi saisics ha become available one monh laer. The updaing of daa files and preparaion of he forecas ypically akes abou 7 o 10 days, alhough i could be speeded up, if necessary. During he second and hird monhs of a quarer, forecass are examined and revised, parly an he basis of revisions in he quarerly social accouns on abou he 20 h of each monh and parly on he basis of high frequency daa ha provide new informaion abou shor run economic movemens. Agencies preparing official daa use high frequency daa for he esimaion of quarerly oals and mus frequenly round ou a quarer s esimaes by projecing missing monhs wihin he quarer. The economeric forecaser using a quarerly model herefore sars he projecions wih inpu daa for he previous quarer (lags) and solves for he curren quarer. Some hings are known abou he curren quarer a he ime he soluion is iniiaed. The soluion ypically has a fuure horizon of some welve quarers, or hereabous. Should he economeric forecaser simply sand by he model resul and ignore wha is

3 3 already known abou he curren quarer, or should he economerician regard he high frequency daa as added informaion, ouside he sample? Since economeric samples are small and daa are scarce, i seems sensible o make use of he added informaion. In pracice, his is done by adjusing he consan erm of relevan equaions (or using a nonzero error erm) in order o make he model oucome agree wih he new informaion. In a single equaion his amouns o replacing an esimaed equaion value y = f (, θ ) + 0 (1) x where y = dependen variable, x = independen variable (inpu daa), θ = esimaed parameer value by ) y = f ( x, θ + a (2) where a = adjusmen value. The original equaion, before esimaion, is y = f ( x, θ ) + e (3) In equaion (1) he value of θ is replaced by is esimae, which is, in a sense, a represenaive value from a sampling disribuion. In he same sense 0 is he mean of he disribuion of e. If he funcion f is linear, his is obviously equivalen o changing he consan erm of he equaion by he amoun a and leaving he zero mean of he disribuion of e in place. This widely praciced procedure has been criicized as being informal and subjecive. A defense can be made for he pracical procedure bu an objecive procedure ha can be digially replicaed is available. I has been observed ha ime series analysis of high frequency daa provide good forecass of economic magniudes for he shor run, say up o six monhs. Some analyss may claim validiy for longer horizons, bu experience wih many series suggess good performance in shor horizons. Such performance is no generally superior o model performance, bu i is as good as model performance. I has also been observed and formally raionalized ha combinaions of forecass, by differen procedures, reduce he risk of forecas error, and may be preferred o forecass from one mehod alone. In his spiri of inquiry we propose he esimaion of curren quarer forecass by ime series mehods, using as much high frequency daa as

4 4 possible, and using hese forecass as benchmarks o which o adjus a quarerly model. The ime series adjused model is hen exrapolaed over a lenghier horizon. This approach has been independenly pursued by economericians a he Federal Reserve Board and he Universiy of Michigan. 1 The presen paper uses differen high frequency models and differen economeric models han hose in oher sudies. This is, herefore, an independen invesigaion along lines similar o hose followed elsewhere. The idea of combining daa of differen frequencies in model analysis and forecasing is no new. In quie differen conexs hese combinaions have been frequenly made. In Projec LINK, where macroeconomeric models from 79 counries or regions are simulaneously relaed o one anoher hrough rade flows, we have consisenly combined quarerly models of he main indusrial counries wih annual models of he developing and cenrally planned economies. Final resuls are reduced o he lowes common denominaor, namely, annual daa, bu in geing he resuls, we firs evaluae quarerly models, add or average quarerly values ino annual values and hen combine he ime aggregaed daa wih he resuls from he annual models. This process needs o be ieraed, and in re-enering he quarerly models on he i-h ieraion, we mus disaggregae he final annual resuls obained on he (i-1)s ieraion. In anoher applicaion of mixed frequencies, we combine, a Wharon Economerics, shor erm business cycle resuls from a quarerly model wih medium erm (10 years or more) resuls from an annual model. Careful forecasing of zigs and zags is applied o he quarerly models, and he quarerly resuls are aggregaed (averaged). The annual model is hen adjused, as described above, o reproduce approximaely he same yearly ime pah ha he aggregaed quarerly resuls follow. Afer he annual model raverses he shor run business cycle span of hree years, i akes a life of is own for he remainder of he simulaion horizon. 2. THREE APPROACHES TO THE CURRENT QUARTER MODEL To make use of monhly or weekly daa in model consrucion, and o have a capabiliy of generaing curren (and subsequen) quarer forecass of major economic magniudes, we approach he problem from hree sides:

5 5 (i) he expendiure side (ii) he income side (iii) hrough unsrucured empirical indicaor relaions. For case (i), we consider he main enries on he expendiure side of naional income and produc accouns (NIPA) and also he main daa sources used by he US Deparmen of Commerce in consrucing heir accouns. We esimae ime series and oher inerrelaionships among high-frequency variables and hen esablish empirical bridge equaions beween he enries in he naional income and produc accouns and he high frequency daa. Where possible, he bridge equaions are esimaed from monhly daa on boh he indicaors and he NIPA componens. All NIPA componens, however, are no repored monhly. In such cases, we mus build quarerly bridge equaions by aggregaing he monhly indicaors ino quarers and correlaing hose wih quarerly NIPA series. Our esimaed equaions are designed so as o be able o exrapolae he high frequency daa series o fill ou he curren quarer and exend beyond for one or wo more quarers. In case (ii) we do he same hing for he main enries on he income side of he naional income accouns ha we do (i) for he expendiure side. In mos of he prior work done a he Federal Reserve Board and elsewhere, he emphasis has been on he expendiure side. Of course, we have he problem of reconciling he wo cenral aggregaes produced by he mehods of case (i) and case (ii) in reaching a single main aggregae o represen eiher gross naional expendiure (produc) or naional income. Case (iii) has no paricular accouning srucure. We simply pu ogeher all he quick informaion ha we can assemble a high frequency early in he quarer. Much of i is inercorrelaed. We hen exrac he leading principal componens of he quarerly aggregaes (averages) of hese several indicaors and regress major quarerly magniudes on he quarerly aggregaes (averages) of he principal componens. Le us firs consider he deailed procedures followed in case (i), he consrucion of he curren quarer esimaes of he expendiure side. The Bureau of Economic Analysis of he Unied Saes Deparmen of Commerce makes and releases is firs esimaes of a quarer s end. This is a preliminary esimae and is based on monhly or weekly daa, much of which are incomplee for he enire quarer being esimaed.

6 6 Our procedure is o use he same high frequency daa ha he BEA uses in making he preliminary forecas. In fac, a rigorous es of a model used for forecasing is o demand ha he model be as close o he ulimae esimae as is he preliminary figure. Afer all, he esimae of he official naional income saisicians represens a figure ha is as close as we, who do no have access o all he deails, could conceivably be able o come. We firs esimae nominal GNP and is componens on he expendiure side. Some of hese require a physical volume indicaor and an associaed price indicaor. Nex we esimae price deflaors for caegories of nominal (curren value) expendiure iems of he GNP. The real expendiure side esimaes are obained by deflaing nominal expendiures by he esimaed price deflaor. The nominal GNP is buil up from he following able, which liss he Naional Income and Produc Accoun series on he lef and he indicaor series on he righ hand side. NIPA (quarerly) Personal Consumer Expendiures Durable goods Auos and pars Furniure & household equipmen Oher durable goods Nondurable goods Food and beverages Clohing and shoes Gasoline and oil Oher nondurable goods Services Gross privae domesic invesmen Nonresidenial srucures Producers durable equipmen Moor vehicles INDICATOR (monhly) Reail Sales of: Uni sales and CPI, auos Furniure, home furnishings & equipmen sores Durable goods less specific caegories Food sores Apparel and accessory sores Gasoline service saions Nondurable goods less specific caegories Employmen in services Value of new nonresidenial consrucion Pu in place Personal consumer expend., auos & Pars

7 7 Oher equipmen Residenial srucures Nonfarm invenory change Manufacuring Reail rade Merchandise wholesalers Nonmerch. Wholesalers Oher Farm invenory change Expors, machinery & ranspor. equip. Impors, machinery & ranspor. equip. Manufacurers shipmens of nondefense Capial goods Housing sars and value of new Residenial consrucion pu in place Book value of manufacuring invenories Book value of reail rade invenories Book value of merch. wholesale Invenories (auoregressive inegraed moving average) Expors Merchandise expors Service expors Impors Merchandise impors Service impors Governmen purchases Federal Federal srucures Fed. Employee compens. Oher federal Sae & local employee compensaion Sae & local srucures Oher sae and local Deflaors of he GNP Durable goods Auos and pars Furniure & household equipmen Oher durable goods Nondurable goods Food and beverages Clohing and shoes Gasoline and oil Oher nondurables Services Gross privae domesic invesmen Nonresidenial srucures Merchandise expors Expors of goods Merchandise impors Impors of goods Ne oulays, federal governmen Federal governmen new consrucion pu in place Federal governmen employmen Sae & local governmen employmen Sae & local gov. new consrucion pu in place Consumer price indexes of: Durable goods New cars Furniure & bedding and household appliances Nondurable goods Food and beverages Apparel Moor fuel, oil, and coolan Services PPI, inermed. maerials for consrucion

8 8 Producers durable equipmen Moor vehicles Oher equipmen Residenial srucures Nonfarm invenory change Manufacuring Reail rade Merchandise wholesalers Nonmerch. wholesalers Oher Farm invenory change Merchandise expors Service expors Merchandise impors Service impors Governmen purchases Federal employee compensaion Srucures Oher Sae & local employee compens. Srucures Oher Producer price of finished goods, capial equipmen PPI, inermed. maerials for consrucion Implici (nominal/real) Implici (nominal/real) Implici (nominal/real) Implici (nominal/real) Implici (nominal/real) Implici (nominal/real) Uni value of expors Merchandise expors Uni value of impors Merchandise impors Implici (nominal/real) Nonresidenial srucures Producer price index, capial equipmen Implici (nominal/real) Nonresidenial srucures Producer price index, capial equipmen The daa in his able are used in wo ways. Firs, bridge equaions are esimaed. These are regressions of NIPA quarerly series on quarerly aggregaes of he indicaor series. Where here is no appropriae indicaor available, he NIPA series is esimaed direcly from an ARMA process. For any NIPA ype variable, y i, we esimae a quarerly ARMA process as p q yi + α ij yi, j = βijei, j (4) j= 1 j= 0 differencing y i when he auocorrelaions show signs of nonsaionariy. The α ij and β ij are esimaed by ime series echniques, esing he random errors, e i, for randomness; i.e.; we assume hese random variables o be whie noise errors. In he second sage, afer we have deermined he regressions of he NIPA variables on indicaors, where possible, we esimae monhly ARMA equaions for each of he indicaors. In applicaions of his sysem, we firs exrapolae he indicaors so ha values for complee quarers (he curren plus one or wo ahead) can be esimaed. We hen pu hese esimaed indicaors on he righ hand side of he bridge equaions o obain values for he NIPA variables in he curren and projeced quarers. A ypical equaion is CENGD = ICENGD (5) (1.01) (23.84) R 2 = 0.71, ρ 1 = CENGD = Consumer expendiures on gasoline and oil in curren prices and in firs differences ICENGD = Reail sales of gasoline service saions in curren prices and in firs differences The correlograms of residual error is fla, near zero, and he firs order serial correlaion is 0.13.

9 9 The indicaors are generally highly correlaed wih he corresponding NIPA variable. All he equaions of he sysem are lised in an annex. The GNP and various sub--caegories are obained from he relevan sums (or differences) of he componens. The componens in consan prices are esimaed by dividing he esimaed nominal series, lised above, by he esimaed price deflaors. The deflaors hemselves are esimaed in he same way ha he curren value series in he GNP are esimaed. The design of calculaions on he income side proceeds in an analogous way. We begin wih a able of relaionships beween NIPA and indicaor elemens. NIPA (quarerly) Gross naional produc Less: capial consumpion allowances Equals: ne naion. produc Less: business ransfer paymens Less: saisical discrepancy Plus: subsidies less curren surplus Equals: naional income Less: corporae profis & adjusmens Less: conribuions for social insurance Personal Employer Less: wage accruals less disbursemen Plus: ransfer paymens Governmen Business Plus: ineres paid by governmen & consumers Ne ineres paid by consumers Ne ineres paid (fed) Less: ineres paid o foreign Ne ineres paid (sae) Plus: dividends Equals: personal income Personal income Wage & salary disbursemens Commodiy producing indusries Manufacuring INDICATORA (monhly) From ideniy Fixed invesmen (see expendiure side) From ideniy Ne oulays, federal governmen (see expendiure side) From ideniy Indusrial producion Reail sales Wage & salary disbursemens Wage & salary disbursemens or personal conribuions for social insurance Toal ransfer from monhly personal income Ousanding consumer insallmen credi federal governm. deb, ineres rae, prime comm. paper, 6 monhs Ousanding consumer insallmen, credi, ineres rae, prime comm. paper, 6 monhs Federal governm. deb, ineres rae, prime comm. paper, 6 monh Personal dividends from monhly personal income From ideniy From ideniy Employmen manufacuring avg. weekly, hours, manufacuring

10 10 Oher Disribuive indusries Services Governmen Oher labor income Proprieor s income Farm Business & professional Renal income of persons Personal dividends Personal ineres income Ineres paid by consumers Oher Transfer paymens o persons Less: personal ax & nonax payms. Equals: disposable personal income average hourly earnings Employmen, oher average weekly hours, manufacuring average hourly earnings Employmen, wholesale & reail rade avg. weekly hrs, wholesale & reail rade avg. hourly earnings Employmen, services average weekly hours, services average hourly earnings, services Employmen Prices paid by farmers Prices received by farmers Indusrial producion reail sales Dividend o price raio, sock prices Ousanding consumer insallmen credi ineres rae, prime comm. paper, 6 monhs Ineres rae, prime comm. paper, 6 monhs Toal unemploymen Wage and salary disbursemens From ideniy The income side is naurally esimaed in curren prices. The values could be deflaed by some combinaion of price indexes ha are simulaneously projeced from he expendiure side, and he deflaed values could be aken as represenaive of real GNP. I is, however, less speculaive o confine he esimaes from he income side o curren values and o compare such esimaes for reconciliaion purposes wih curren value esimaes obained from he expendiure side. In he hird case, we aim o use as much high frequency daa as are available in he curren quarer o esimae jus hree magniudes real GNP, nominal GNP, and he GNP deflaor. Since hese are relaed by he ideniy PDGNP GNP = GNP ($) only wo series should be compued direcly from indicaor daa; he hird should be obained from he above definiional equaion. Early readings on a number of variables are colleced monhly. These are mean o be revealing abou he overall economy. Enough values are esimaed for he curren quarer so ha we have hree monhly figures (observed and projeced) for each indicaor variable. Leading principal componens are compued from he observed se of quarerly aggregaes of monhly variables. The hree aggregaes (PDGNP, GNP, and GNP($)) are regressed on he leading principal componens. Projeced values of he indicaor variables for he curren quarer and also for one quarer ahead are subsiued ino he regression on principal componens in order o obain curren and fuure quarer forecass of he desired aggregae. The source daa used o form indicaors of PDGNP, GNP and GNP($) are lised below.

11 11 INDICATORS (MONTHLY) PDGNP, Price deflaor of GNP Consumer price index Producer price index, finished goods Producer price index, inermediae maerials Average hourly earnings Average weekly hours Uni value index of impors Prices received by farmers GNP($), Nominal gross naional produc Value of shipmens, manufacuring Value of new orders, manufacuring Value of unfilled orders, manufacuring Nominal personal income Money sock (M1) Reail sales Ineres rae on 6 monh CDs Index of ne business formaion GNP, Real gross naional produc Value of shipmens deflaed by PPI inermediae maerials Value of new orders deflaed by PPI inermediae maerials Value of unfilled orders deflaed by PPI inermediae maerials Nominal personal income deflaed by CPI Money sock (M1) deflaed by CPI Reail sales deflaed by CPI Real ineres rae (6 monh CD rae less CPI inflaion rae) Indusrial producion index Employmen Average weekly hours Each indicaor variable is projeced ahead by an ARMA process ha has been fied o hisorical daa on he monhly indicaor values. Each indicaor is firs fied o he semilog rend formula lni = a + bt I = quarerly aggregaes of monhly indicaor T = chronological ime a,b = esimaed regression coefficiens Derended values are compued from I/exp(a+bT) Principal componens are exraced from he derended variables afer sandardizaion. Call he componens (PC) 1, (PC) 2, ec. We fi he empirical regressions + PDGNP = c c i ( PC) 0 i + GNP ($) = d 0 d i ( PC) + GNP = e e i ( PC) 0 i i i i i

12 12 The aserisk denoes ha derended values are used for dependen variables in he hree regressions. The residuals are examined for serial correlaion and well known auoregressive correcions are used in order o isolae whie noise errors from he principal componen regressions. Derended esimaes of he main variables are obained and hen ransformed back o original unis wih rend. The mean absolue percenage errors of esimae over he sample period are real GNP 0.28 nominal GNP 0.39 price deflaor APPLICATIONS The use of he hree differen curren quarer models (expendiure side, income side, and empirical indicaor model) is self eviden. As daa become available during he firs or second monh of any quarer, esimaes of he enire quarer and he nex fuure quarer are made. This could be done a any ime, bu ypically during he early days of he second monh. The hree models could be averaged. The expendiure and income side models could be brough close ogeher by minimizing he one reconciling iem, he saisical discrepancy. These esimaes can be used for he early quarers of a forecas, bu hey can also be used for he purpose of adjusing, or calibraing, a model so ha i sars from realisic values. In he process of making adjusmens o he consan or error erms of equaions in a model, i has been cusomary o bring he model ino agreemen wih observaions for period T and T-1, he laes observed periods jus before he forecas projecion period, T+1,T+2,T+3, The values of a in equaion (2) above are he adjusmen values. They can be chosen so ha y, he variables ha are o be forecased, agree wih he values of he curren quarer model. In oher words, choose a such ha y = f ( x, θ + a where ) y = forecas of y from he curren quarer model ) f ( x, θ = value of y compued from he esimaed srucural equaion solved in a deerminisic mode. Since y are compued from objecive equaions ha can be replicaed, his is an objecive adjusmen procedure. Some forecasers have a endency o adjus many equaions of a model subsanially more han one half. Our view is ha adjusmen should be kep o a minimum. This means ha only he equaions ha are mainly responsible for leading variables (like GNP) should be adjused. The resuls should be scruinized o be sure ha no variable is erribly far ou of line. In principle, i would be possible o consruc a loss funcion so ha a seleced number of adjusmen values were chosen in order ha he loss value is minimized. ' ' ' ' 1,..., x, x 1,..., θ ') F ( y, y = a subjec o: ' ' ' ' 1,..., x, x 1,..., θ ') F ( y, y = a

13 13 where F is a column vecor of funcions, y is a column vecor of endogenous variables (equal in number o he coun of elemens of F), and x is a column vecor of exogenous variables. x is a column of parameer esimaes, and a is a column vecor of adjusmen values. The weighs w i in L, show he relaive imporance of he various elemens of y ha are evaluaed in he curren quarer model. These weighs should reflec he variables for which he curren quarer model shows greaer possibiliies of improving he economeric model and may vary wih he monhly observaions we have abou he curren quarer. The problem is o search for values of a, such ha he soluion of he enire sysem brings y as close as possible, in he square, o y. Many of he elemens of a could, of course, be zero. In some models, i will be possible o bring L o zero. This is where i is possible o bring y o heir arges y wihou violaing any equaions of he sysem. In general, however, if all he values of y are fixed exacly a y, i may no be possible o find a consisen soluion of he sysem. In his case, we seek a leas-squares soluion for deviaion of y. In any even, if he number of nonzero elemens of a is o be kep quie small, here will be fewer nonzero elemens of a han here are arge values y of y. In such cases, he objecive will be o bring y near o y, in he sense of minimum squared deviaion. I should also be possible o choose values of seleced elemens of x in order o minimize L, leaving all a = 0. In mos cases, we shall have a priori informaion abou x for he shor run up o six monhs and will no, for he presen problems, wan o search for opimal values of x. NOTES 1. Carol Corrado and Mark Greene (1988), Reducing Uncerainy in Shor Term Projecions: Linkage of Monhly and Quarerly Models, journal of Forecasing 7, pp Mark Greene, E. Philip Howrey, and Saul H. Hymans, The Use of Ouside Informaion in Economeric Forecasing, in D.A. Belsley and E. Kuh (eds.) (1986), Model Reliabiliy, Cambridge: MIT. The esimaion of curren quarer models from high frequency daa was iniiaed many years ago by Oo Ecksein a Daa Resources, Inc.