Allowing the Data to Speak Freely: The Macroeconometrics of the Cointegrated Vector Autoregression

Transcription

1 Allowing he Daa o Speak Freely: The Macroeconomerics of he Coinegraed Vecor Auoregression 16 November 2007 Corresponding Auhor: Kevin D. Hoover Deparmen of Economics Duke Universiy Box Durham, NC U.S.A. Søren Johansen Deparmen of Economics Universiy of Copenhagen Sudiesræde København K Denmark Kaarina Juselius Deparmen of Economics Universiy of Copenhagen Sudiesræde Copenhagen K Denmark Tel. +(919) Fax +(919) E mail kd.hoover@duke.edu Tel Fax soren.johansen@econ.ku.dk Tel Fax E mail kaarina.juselius@econ.ku.dk Session Tile: Complexiy and Dynamics in Macroeconomics: Alernaives o he DSGE Models Session Chair: David Colander, Middlebury College Discussans: Sephen Turnovsky, Universiy of Washingon Duncan Foley, New School Universiy

2 Absrac of Allowing he Daa o Speak Freely: The Macroeconomerics of he Coinegraed Vecor Auoregression An explicaion of he key ideas behind he Coinegraed Vecor Auoregression Approach. The CVAR approach is relaed o Haavelmo s famous Probabiliy Approach in Economerics (1944). I insiss on careful sochasic specificaion as a necessary groundwork for economeric inference and he esing of economic heories. In ime series daa, he probabiliy approach requires careful specificaion of he inegraion and coinegraion properies of variables in sysems of equaions. The relaionship beween he CVAR approach and wider mehodological issues and beween i and relaed approaches (e.g., he LSE approach) are explored. The specific o general sraegy of widening he scope of economeric models o idenify sochasic rends and coinegraing relaions and o nes heoreical economic models is illusraed wih he example of purchasing power pariy. Keywords: coinegraed vecor auoregression, VAR, CVAR, coinegraion, Johansen, Juselius, Haavelmo, ime series models, Walrasian mehodology, Marshallian mehodology JEL Codes: B41, C30, C32, C50, C51, C52

3 All economiss agree ha realiy is complex and ha he ools wih which we confron i are far simpler. Theoriss someimes deal wih his gap by asking very lile of he daa. Sar wih he sylized facs and develop relaively simply heories o accoun for hem. Unforunaely, sylized facs are ofen oo sylized o discriminae among plausible candidae heories or o provide a basis for accurae quanificaion. Alernaive approaches sar from he oher end and ask much of he daa. One European radiion, which derives from Trygve Haavelmo s The Probabiliy Approach in Economerics (1944), focuses on obaining good characerizaions of daa before esing and on drawing ou he implicaions of daa ha ough o consrain economic heorizing. The applicaion of he coinegraed vecor auoregression (CVAR) recouned, for example, in Kaarina Juselius s (2006) exbook and faciliaed by he CATS in RATS economeric sofware (Jonahan G. Dennis e al., 2006) is a special macroeconomeric case of he Probabiliy Approach. The message of he Probabiliy Approach and he CVAR approach can be summarized in he slogan: facs, no sylized facs. 1. Beween Daa and Theory All economerics aims ulimaely o confron heory and daa. Differen approaches differ in how hey conceive he relaionship and he problems ha i poses. To sar, hink of an ideal case such as one migh find in a physics exbook. The law of graviy is applied o he dropping of a ball from a ower. The law, ogeher wih an iniial condiion (he heigh of he ower), deermines he disance he ball falls for each ime... in heory. Of course, no objec conforms perfecly o he graviy law. If one had a generous enough noion of approximaion, if he ball were seel and he iniial heigh were no oo high, hen igh bounds of approximaion would work; bu no if he ball were syrofoam. Now here are hree choices: A) declare ha heory is no good; B) modify he original 1

4 heory o accoun for he facors such as air resisance; or C) aemp o assess empirically he combined forces ha mus be used o adjus he graviy law o is applicaion in he paricular case. In economics, as in physics, he difficuly is ha our heory holds ceeris paribus. When oher hings are no equal, here is always some residual lef unexplained which, if large, may render he heory empirically irrelevan. As scieniss, we can eiher aemp o elaborae he heory in such a way ha fewer and fewer ceeris paribus condiions are invoked (B) or we can aemp o provide an adequae empirical characerizaion of he facors ha deermine he iniial gap beween heory and daa (C). Sraegy C can be seen as he passive analogue o a conrolled experimen (cf. Haavelmo, 1944, esp. chaps. 1 and 2). I has an advanage over B, in he sense of providing clues as o how he heory needs o be developed clues ha would be helpful in sraegy B, bu for which sraegy B iself offers no inernal resources. Wha is more, sraegy C gives some hope of acually isolaing he acion of he graviy law and, herefore, in fac esing wheher i is a conribuing facor o a successful accoun of he daa. I is only when we can conrol for enough of he complicaing facors ha he underlying quadraic relaionship of he graviy law can be deeced. The exreme limi of sraegy B is wha Milon Friedman has called he Walrasian mehodology, by which he means no general equilibrium heory per se, bu o he idea ha one mos have a complee, deailed heoreical accoun in order o say anyhing useful abou he economy a all (see Kevin D. Hoover, 2006). The exreme limi of sraegy C is a compleely aheoreical analysis of daa. Boh exremes are hopeless: sraegy B because we lack he cogniive capaciy o elaborae a complee heory from firs principles; sraegy C because wihou some prior concepual noion we would never find a saring place for any invesigaion. Sill, he Probabiliy Approach leans oward sraegy C: he weigh of he analysis is on characerizing he daa and on using he daa o criicize and guide heorizing. In Friedman s erms, he approach is Marshallian or, as Hoover (2006) pus 2

5 i, archaeological : we learn abou he economy a piece a a ime by removing he overlay of derius o uncover he underlying srucure, guided by our heoreical concepion of wha we are looking for, which is esed and enriched by each new discovery. 2. Models of Theory, Models of Daa The CVAR approach builds on Haavelmo s (1944) grea insigh ha he gap beween heory and daa need no be reaed as an unsrucured residual of approximaion bu could iself be modeled saisically using he heory of probabiliy. The cos is ha we now need anoher level of modeling in addiion o heory a saisical model consruced in such a way ha i) heory has implicaions inerpreable in is erms and ii) daa are described fully enough ha is only residuals are idenically independen random errors ha is, unsysemaic noise. The payoff is ha such a saisical model warrans he use of likelihood mehods and provides a firm basis for deducions abou he implicaions for heory. The CVAR approach sees he world as a highly complex dynamic sysem, he properies of which mus be inferred from daa reflecing a single (nonreplicable) realizaion of a mulivariae, pah dependen process. Naurally, his daa generaing process mus be approximaed by simpler relaionships, which characerize he daa accuraely enough for our paricular purposes. The saisical model ies economic heory o he daa when i ness boh he daa generaing process and he heoreical model. Then he parameers of he heoreical model can be read as asserions abou parameers of he saisical model, which can be esed agains he daa provided ha he saisical model characerizes i accuraely. While we can never know for cerain ha our saisical model capures he daa generaing process, we can ofen find compelling evidence when i does no. Søren Johansen (2006, pp ) provides an example, which sars wih he unobservable daa generaing process: 3

6 (1) x = 0. 9 x ε, = 1, 2,..., 100, x 0 = 10, where he ε are idenically independenly disribued (i.i.d) N(0,1). Noe ha E(x ) = 1/(1 0.9) = 10 and var(x ) = 1/( ). Consider an economic heory ha predics ha he mean value of x is µ = 10 (i happens ha our heory is exacly rue, bu ha will no generally be he case). To es he heory we need o provide a model of he probabiliy process. One model is (2) x = µ + ε, 2 where he ε are i.i.d. N ( 0, σ ). Omiing deails, he maximum likelihood esimaes of (2) yield a 95 percen asympoic confidence inerval of ˆ µ ± 1.96 ˆ σ / T = ± Since 10 does no lie wihin he confidence inerval, i migh appear, hen, ha we have good grounds o rejec he hypohesis ha µ = 10. Of course, ha claim is only as good as he probabiliy model in which i is based. Model (2) models he errors erms as i.i.d normal. Given he daa generaing process (1), his assumpion will be violaed; a simple saisical es would show ha he esimaed residuals are serially correlaed. We can conclude, hen, ha he daa generaing process is no nesed in our saisical model, ha he esimaes are no reliable, and ha he heory has no been esed adequaely. An alernaive saisical model is given by (3) x = ρ x + µ ( 1 ρ ) + ε 1, 2 where again he ε are i.i.d. N ( 0, σ ) and E ( x ) = µ, if ρ < 1. Now, as i happens, he daa generaing process (1) is precisely nesed in (3). Again omiing deails, he maximum likelihood esimae of (2) yields a 95 percen asympoic confidence inerval of ˆ µ ± 1.96 ˆ σ /[(1 ρ ) T ] = ± The previous esimae of µ was spuriously precise. On he curren probabiliy model (3), he esimae is less precise and we canno rejec µ = 10. 4

7 We canno have sufficien condiions for knowing ha he daa generaing process is nesed in any conjecured probabiliy model. One message of our cooked example is ha we should check he necessary condiions. No only did his lead us o rejec (2) as an adequae descripion of he probabiliy model, he serial correlaion of he residuals from esimaing (2) naurally sugges models in he auoregressive class such as (3). Anoher message is ha our ess of heory will go seriously wrong if we base hem on saisical models ha fail accuraely o characerize he daa in imporan ways. The upsho of hese messages is ha key elemens of he CVAR approach are, firs, o ge he probabiliy model righ, which is judged by ruhless applicaion of diagnosic esing, and second, o judge heories in relaion o he esable resricions ha hey imply for he probabiliy model. Accuraely characerizing he persisence of he daa is a vial aspec he CVAR approach. In equaion (3), we assumed ha ρ < 1, which implies ha he x is mean revering or saionary i.e., inegraed of order zero or I(0). Bu if insead, ρ = 1, hen x would be a nonsaionary, uni roo process (i.e., I(1)) in which shocks would accumulae, forming a sochasic rend (i.e., a permanen, nondeerminisic shif in he mean). Saisical inferences ha fail o accoun for nonsaionariy (deerminisic or sochasic) will be misleading in a manner analogous o inferences based on equaion (2). One way o accoun for a uni roo is o ransform he daa o saionariy by differencing. Bu differencing hrows away all he long run informaion in he daa. Forunaely, when daa share a sochasic rend, a paricular linear combinaion of he levels of he variables will also be saionary. Such variables are said o be coinegraed. Coinegraion was formalized by Rober F. Engle and Clive W. J. Granger (1987), alhough i is implici in he earlier work of he London School of Economics (LSE) approach (see Grayham E. Mizon, 1995). Since economic heories frequenly have clear, esable implicaions abou degrees of persisence and coinegraion, hese dynamic properies are cenral o he CVAR approach. For example, a saionary coinegraing 5

8 relaionship among nonsaionary variables can frequenly be inerpreed as defining a long run equilibrium oward which variables are adjusing. This is he famous error correcion mechanism of he LSE approach. 3. The Coinegraed Vecor Auoregression Model The CVAR provides a simple linear sysem ha can characerize he probabiliy disribuion of a se of variables. While he imporance of coinegraion is widely acceped, he CVAR approach can be disinguished from is close ally, he LSE approach, which has mos ofen focused on singleequaions, even hough i can be readily generalized o sysems (David F. Hendry, 1995). And he CVAR approach can be disinguished from many oher applicaions of coinegraion in sysems of equaions by is focus on well specified, congruen saisical models a hallmark of Haavelmo s probabiliy approach. Recogniion ha macroeconomic ime series daa are ypically nonsaionary and coinegraed moivaed, firs, he developmen of likelihood based inference for he CVAR model (Johansen, 1996), including ess for deermining he numbers of sochasic rends and coinegraing relaions and ess of hypoheses on heir srucure and an applied macroeconomeric mehodology (Juselius, 2006), a pracical mehod of asking wihin he broad conex of a heoreical model, wha do he daa say when hey are allowed o speak freely? The necessary ess and mehods are implemened in CATS in RATS (Dennis e al., 2006). Two facs argue for he CVAR approach: firs, saisical evidence indicaes ha nonsaionary daa are pervasive; second, economic heory is mainly abou he adjusmen of one variable o anoher in search of individually opimal, sysemically more coordinaed oucomes. By combining differenced and coinegraed daa, he CVAR model responds o boh facs. Economic daa are analyzed as shor run variaions around moving longer run equilibria. Longer run forces are hemselves divided ino he forces ha move he equilibria (pushing forces, which give rise o 6

9 sochasic rends) and forces ha correc deviaions from equilibrium (pulling forces, which give rise o coinegraing relaions). Inerpreed in his way, he CVAR has a good chance of nesing a mulivariae, pah dependen daa generaing process and relevan dynamic macroeconomic heories. Unlike approaches in which he daa are silenced by prior resricions, he CVAR model gives he daa a rich conex in which o speak freely. Especially wih respec o persisence properies, i is worh recalling ha he CVAR is no he underlying daa generaing process; raher i is a good enough approximaion for a paricular problem. For example, we can (and usually should) approximae highly persisen daa by an exac uni roo, since ess based on χ², F, and disribuions and he assumpion of saionariy will go badly wrong when he daa generaing process has a near uni roo, unless we have a very long sample of, say, more han 5000 observaions (Johansen, 2006). Much of he focus in he CVAR approach is on he long run: can we idenify he sochasic shocks? wha are he coinegraing relaions? Juselius and Massimo Franchi (2007; also Johansen, 2006) show how o ranslae he assumpions underlying a dynamic, sochasic generalequilibrium (DSGE) model (Peer N. Ireland, 2004) ino a se of esable assumpions on coinegraing relaionships and sochasic rends in a CVAR. Accouning for (near) uni roos in he model provides a powerful ool o robusify saisical and economic inference. Mos assumpions underlying he dynamic, sochasic general equilibrium model and, hence, he real business cycle model were rejeced when properly esed. Srucuring he daa in his way offers a number of facs, for example ha i was shocks o consumpion ha have generaed he long business cycles, ha a heory model should replicae in order o claim empirical relevance. Thus, he CVAR approach provides boh a criical framework and consrucive insighs. 7

10 Johansen e al. s (2007) invesigaion of purchasing power pariy provides a more deailed case sudy. Call he logarihm of German prices p 1, he logarihm of U.S. prices p 2, and he logarihm of he exchange rae s 12. Inflaion raes ( p 1, p 2 ) display a sochasic rend; and, since inflaion raes are I(1), price levels are I(2). The exchange rae is I(1). Since prices are a higher level of inegraion han exchange raes, purchasing power pariy requires ha prices be coinegraed ha is, relaive prices mus be I(1) (pp = p 1 p 2 ~ I(1)) and exchange raes and relaive prices mus share a common rend, so ha ppp = p 1 p 2 s 12 ~ I(0). Then, he hypohesis of purchasing power pariy in he dollar/deuschmark case amouns o a complex hypohesis: {p 1 ~ I(2), p 2 ~ I(2), pp = p 1 p 2 ~ I(1), s 12 ~ I(1), ppp = p 1 p 2 s 12 ~ I(0), p 1 & p 2 are pushing, s 12 is adjusing}. Here adjusing means ha he shocks o he nominal exchange rae do no conribue o any sochasic rend, even he ones ha drive s 12 iself. A careful CVAR analysis, confirms ha p 1 ~ I(2), p 2 ~ I(2), pp ~ I(1), and s 12 ~ I(1); bu, conrary o he hypohesis, pp and s 12 have differen sochasic rends, so ha hey are no coinegraing (i.e., ppp ~ I(1)), and s 12 is pushing. The failure of purchasing power pariy requires ha we dig deeper. General equilibrium implies ha a persisen deparure from purchasing power pariy mus generae a similar persisen movemen somewhere else in he economy. Since exchange raes are involved in capial movemens as well as rade in goods and services, a naural place o look is in he behavior of ineres raes. Adding he ineres raes on German and U.S. long erm bonds (b 1 and b 2 ) generalizes he more specific original model. In he more general model, he I(1) rend in ppp is he same as he rend in he relaive bond yield (b 1 b 2 ), so ha ppp and he real ineres rae differenial are coinegraing: (ppp ω[(b 1 p 1 ) (b 2 p 2 )] ~ I(0), where ω is a consan parameer. Such a specific ogeneral approach, which sars wih a small model and works o a larger one, is jusified because coinegraion is a propery ha is invarian o widening he daa se. 8

11 How should one undersand hese findings economically? While i is beyond our presen purpose o make a deailed case, he failure of purchasing power pariy appears o be relaed o he join deerminaion of nominal exchange raes in he goods and he foreign exchange marke, and how he laer influences deerminaion of ineres rae. These findings are consisen wih he applicaion o he foreign exchange marke of he heory of imperfec knowledge economics as developed by Roman Frydman and Michael D. Goldberg (2007). Such a heory is pos Walrasian in he sense ha i rejecs a cenral ene of modern Walrasian macroeconomics he raional expecaions hypohesis. Bu more fundamenally, i is pos Walrasian (ha is Marshallian in Friedman s sense or archaeological in Hoover s) in ha i rejecs he privileging of a priori economic heory over empirical evidence. In he language of he CVAR approach, empirical evidence is he pushing force and economic heory is adjusing. References Colander, David, ed. Pos Walrasian macroeconomics: Beyond he dynamic sochasic general equilibrium model. Cambridge: Cambridge Universiy Press, Dennis, Jonahan G.; Juselius, Kaarina; Johansen, Søren; and Hansen, Henrik. CATS in RATS: Coinegraion analysis of ime series, sofware. Evanson, IL: Esima, Frydman, Roman and Goldberg, Michael D. Imperfec knowledge economics: Exchange raes and risk. Princeon: Princeon Universiy Press, Engle, Rober F. and Granger, Clive W. J. Co inegraion and Error Correcion: Represenaion, Esimaion, and Tesing, Economerica, 1987, 55(2), pp Haavelmo, Trygve. The Probabiliy Approach in Economerics, Economerica, 1944, 12(supplemen). Hendry, David F. Dynamic Economerics. Oxford: Oxford Universiy Press,

12 Hoover, Kevin D. The Pas as Fuure: The Marshallian Approach o Pos Walrasian Economerics, in Colander (2006), pp Ireland, Peer N. A Mehod for Taking Models o he Daa. Journal of Economic Dynamics & Conrol, 2004, 28(6), pp Johansen, Søren. Likelihood based inference in coinegraed vecor auoregressive models. Oxford: Oxford Universiy Press, Johansen, Søren. Confroning he Economic Model wih he Daa, in Colander (2006), pp Johansen, Søren; Juselius, Kaarina; Frydman, Roman; and Goldberg, Michael D. Tesing Hypoheses in an I(2) Model wih Applicaions o he Persisen Long Swings in he Dmk/$ Rae, unpublished manuscrip, Juselius, Kaarina. The coinegraed VAR model. Oxford: Oxford Universiy Press, Juselius, Kaarina and Franchi, Massimo. Taking a DSGE Model o he Daa Meaningfully. Economics The Open Access, Open Assessmen E Journal 2007, No , hp:// ejournal.org/economics/journalaricles/ Juselius, Kaarina and Johansen, Søren. Exracing Informaion from he Daa: A European View on Empirical Macro, in Colander (2006), pp Mizon, Grayham E. Progressive Modeling of Macroeconomic Time Series: The LSE Mehodology, in Kevin D. Hoover, ed., Macroeconomerics: Developmens, ensions and prospecs. Boson: Kluwer, 1995, pp