Reassessing he Prebisch-Singer Hypohesis: Long-Run Trends wih Possible Srucural Breaks a Unknown Daes John T. Cuddingon, Rodney Ludema, and Shamila A Jayasuriya Georgeown Universiy Ocober 2002 Absrac: This paper reconsiders he Prebisch-Singer hypohesis regarding long-run rends in commodiy prices by considering rend and difference saionary models wih up o wo possible break poins a unknown daes. We show ha raher han a downward rend, real primary prices over he las cenury have experienced one or more abrup shifs, or srucural breaks, downwards. The preponderance evidence poins o a single break in 92, wih no rend, posiive or negaive, before or since.
. INTRODUCTION Since he publicaion of he Grill-Yang (987) paper and associaed long-span (900-86) daase, here has been a resurgence in empirical work on long-erm rends in commodiy prices.,2 The search for a secular rend á la Prebisch and Singer has shifed from he issue of daa qualiy o economeric issues involved in esimaed growh raes or rends in nonsaionary ime series. Modern ime series economerics, however, has augh us ha i is poenially misleading o assess long-erm rends by inspecing ime plos or esimaing simple loglinear ime rend models. Alhough he GY series (see Fig. below) does no appear o be mean saionary, i is criical o deermine he source of nonsaionariy before aemping o make inferences abou he presence of any rend. Possible sources of nonsaionariy are: A deerminisic ime rend A uni roo process, wih or wihou drif One or more srucural breaks in he mean or rend of he univariae process General parameer insabiliy in he underlying univariae model. The key economeric issues are, in shor, he possible presence of uni roos and parameer insabiliy in he univariae models being esimaed. To faciliae a discussion of hese issues and o pu he exising lieraure ino conex, we firs specify a general loglinear ime rend model ha may or may no have a uni roo. Second, we describe hree The MUV-deflaed GY series has recenly been exended hrough 998 by IMF saff economiss. We hank Paul Cashin of he IMF Research Deparmen for proving he daa. 2 To consruc heir nominal commodiy price index, GY weighed he 24 nominal prices by heir respecive shares in 977-79 world commodiy rade. To ge a real index, GY divided heir nominal commodiy price index by he a manufacuring uni value index (MUV), which reflecs he uni values of manufacured goods expored from indusrial counries o developing counries. This is a naural choice of deflaors, given PS s concern abou he possibiliy of a secular deerioraion in he relaive price of primary commodiy expors from developing counries in erms of manufacuring goods from he indusrial world. (GY also considered a U.S. manufacuring price index as a deflaor and concluded ha heir resuls were no much affeced by he choice of deflaor.)
ypes of srucural breaks in his framework, where here are sudden shifs in model parameers. A more general ype of parameer insabiliy, where parameers are hypohesized o follow random walks, is also considered. Third, we presen a char or marix showing he relaionship among various univariae models ha have appeared in he lieraure. Also included are logical exensions of wha has already appeared. 2. TREND STATIONARY VS. DIFFERENCE STATIONARY MODELS Aemps o esimae he long-erm growh rae or rend in an economic ime series ypically begin wih a log-linear ime rend model: ln( y ) = α + β + ε () In he PS lieraure, y=p C /P M is he raio of he aggregae commodiy price index o he manufacuring goods uni value. The coefficien β on he ime index is he (exponenial) growh rae; i indicaes he rae of improvemen ( β > 0) or deerioraion ( β < 0) in he relaive commodiy price y. I is imporan o allow for possible serial correlaion in he error erm ε in (). Economerically, his improves he efficiency of he parameer esimaes; economically, i capures he ofen-pronounced cyclical flucuaions of commodiy prices around heir long-run rend. The error process in () is assumed o be a general auoregressive, moving average (ARMA) processes: ( ρ L ) A( L) ε = B( L) (2) u I will be convenien in wha follows o facor he auoregressive componen of he error process in a way ha isolaes he larges roo in he AR par of he error process; his roo is 2
denoed denoed ρ. The erms (-ρl)a(l) and B(L) are AR and MA lag polynomials, respecively. The innovaions u in (2) are assumed o be whie noise. A criical issue will be wheher ρ <, indicaing ha he error process is saionary, or wheher ρ=, indicaing nonsaionariy due o he presence of a uni over ime. In his case, ()-(2) is referred o a he rend saionary (TS) model, indicaing ha alhough y iself is nonsaionary (unless β =0), flucuaions of y around is deerminisic rend line are saionary. If, on he oher hand, y (or equivalenly he error process in (2)) conains a uni roo, esimaing he TS model wih or wihou allowance for (supposed) srucural breaks will produce spurious esimaes of he rend (as well as spurious cycles). An appropriae sraegy for esimaing he rend β in his case is o firs-difference he model ()-(2) o achieve saionariy. The resul is he so-called difference saionary (DS) model, a specificaion in erms of growh raes raher han log-levels of he y series: ( L )ln( y ) D ln( y ) = β + v (3) where L and D are he lag and difference operaors, respecively. The error erm in (3) follows an ARMA process: A ( L) v = B( L) (4) u In he DS model, a significan negaive esimae of he consan erm, β, suppors he PS hypohesis. I has long been recognized ha esimaed parameers in TS or DS models will be biased, or even meaningless, if he rue parameers do no remain consan over ime. Suppose, for example, ha he rue growh rae equaled -4.0% in he firs half of he 3
sample, bu +2.0% in he second half. An economerician who ignored he shif in parameers migh incorrecly conclude ha he growh rae was a uniform -2.0 percen over he enire sample. One way o assess he srucural sabiliy of he TS and DS models is o esimae each model, appropriaely modeling any serial correlaion in each case, hen o calculae recursive residuals and he 2-sandard error bands for he hypohesis ha he recursive residuals come from he same disribuion as he hose from he esimaed models. This is done for he TS model (wih an AR() error process) in Fig. 4 (change number). The recursive residuals in 92 and 974 are large, suggesing srucural breaks. Figure 4 also shows p-values for an N-sep forecas es for each possible forecas sample. To calculae he p-value for 920, for example, one would use daa from 900 hrough 920 o esimae a TS-AR() model. This model is hen used o forecas y() for he remaining N years of he sample: 92-998. A es saisic ha incorporaes he forecas errors, comparing he forecas wih he acual value, for he N-seps ahead can be consruced o es he null hypohesis ha such forecas errors could have been obained from he underlying TS-AR() model wih no srucural break. The p-value for he null hypohesis of no srucural break gives he probabiliy of finding an even larger es saisic if he null is, in fac, rue. If he p-value is smaller han he size of he es, ypically.0 or.05, hen one should rejec he null hypohesis of no srucural breaks. As seen in Fig. 4, he p-values very near 0.00 in he 90-20 period indicae ha he es saisic is so large ha he probabiliy of finding a larger one under he null is virually zero. Tha is, his graph clearly shows ha if he model is fied wih pre-92 daa and used o forecas ino he fuure, here is clear rejecion of parameer sabiliy. If insead 4
one uses daa up hrough he 940s, or 950s, or 960s, on he oher hand, parameer sabiliy is no rejeced. If one uses daa hrough he early 970s o forecas commodiy prices hrough he end of he 990s, here is again insabiliy albei somewha less severe judging from he p-values on he lef-hand scale in he graph. This evidence cerainly suggess ha he issue of srucural breaks or parameer insabiliy (perhaps due o a uni roo) mus be aken seriously if one chooses he TS model for analyzing he long-erm rends in primary commodiy prices. If one carries ou he same exercise for he DS specificaion, he recursive residual and N-sep ahead forecas analysis again suggess ha here is a srucural break in 92. See Fig. 6. Wih he DS model, however, his appears o be he only roublesome episode. Fig.6 Evidence of Parameer Insabiliy in DS Model.2..0.00.04 -. -.2 -.3.08.2 0 20 30 40 50 60 70 80 90 N-Sep Probabiliy Recursive Residuals Wha is clear up o his poin? Regardless of wheher he TS or DS specificaion is chosen, here is evidence ha one or wo breaks or parameer insabiliy may be a problem. 5
From he work of Perron (989) and ohers, i is clear ha uni roo ess, which help us choose beween he TS and DS specificaions, mus ake ino accoun he possible presence of srucural breaks. 3. STRUCTURAL BREAKS AND PARAMETER INSTABILITY To consider he possibiliy of a change in parameers (α, β ) in he TS model or β in he DS model, 3 one ypically consrucs a dummy variable: DUM TB = 0 for all < TB and DUM TB = for all TB where TB is he hypohesized break dae. Using his levelshif dummy, as well as is firs difference (a spike dummy) and a dummy-ime rend ineracion erm, yields he TS wih break model and he DS wih break model, respecively: TS wih Break Model ln( y ) = α + α DUM TB + β + β ( TB ) * DUM TB + ε (5) 2 2 DS wih Break Model D (ln( y )) = α D ( DUM TB ) + β + β * DUM TB + ν (6) 2 2 These specificaions are general enough o encompass he hree ypes of breaks described in Perron (989) classic paper on esing for uni roos in he presence of srucural breaks (which will be discussed below). His model A ( Crash model) involves only an abrup shif in he level of he series; i.e. α 2 0, β 2 =0. In model B (he breaking rend model), here is a change in he growh rae, bu no abrup level shif: α 2 =0, β 2 0. Finally, he Combined Model, model C, has change in boh he level and growh rae: α 2 0, β 2 0. 3 I is also possible o allow for shifs in he model parameers ha describe he error process, is serial correlaion and variance, bu we do no consider his exension here. 6
Suppose ha one knows a priori, or decides on he basis of uni roo esing, wheher he TS or DS specificaion is appropriae. Then, if he break dae, TB, is assumed o be known, i is sraighforward o es for he presence of srucural breaks by examining he -saisics on α 2 and/or β 2. A es for a break of ype C could be carried ou using an χ2 (2) es for he join hypohesis ha α 2 =0 and β 2 =0. The laer is equivalen o (one varian of) he well-known Chow es for a srucural break. More recen work on ess for parameer sabiliy warns agains arguing ha he break dae TB is known. Andrews (993), 4, Ploberger, Kramer, and Konrus (989), 5 and Hansen (992), 6 for example, develop mehods for esing for he presence of a possible srucural break a an unknown dae using algorihms ha searches over all possible break daes. 4 Andrews (993) considers ess for parameer insabiliy and srucural change wih unknown breakpoins in nonlinear parameric models. He ess he null of parameer sabiliy subjec o hree alernaive hypoheses: a one-ime srucural change eiher wih a known change poin, wih an unknown change poin in a known resriced inerval, and wih an unknown change poin where no informaion is available regarding he ime of he change. The daa in he esimaed model mus be saionary or drifless random walks; hey can no be series wih deerminisic or sochasic ime rends. He derives he asympoic disribuions of hree es saisics based on he Generalized Mehod of Momens (GMM) esimaors Wald, Lagrange Muliplier, and Likelihood Raio-like saisic under he null hypohesis of consan parameers and provides he respecive criical values for each. 5 Ploberger, Kramer, and Konrus (989) propose a flucuaions es for he null hypohesis of parameer consancy over ime in a linear regression model wih non-sochasic regressors. Their es is based on successive parameer esimaes and does no require he locaion of possible shifs o be known. They derive he asympoic disribuion of he flucuaion es saisic and deermine he rejecion probabiliy of his es saisic based on he magniude of flucuaions in he recursive coefficien esimaes. They also show how heir ess is relaed o earlier CUSUM and CUSUM squared ess. 6 Hansen (992) ess he null of parameer sabiliy in a framework of coinegraed regression models, agains he alernaive hypoheses ha a single srucural break exiss a eiher a given or an unknown ime. He considers a sandard muliple regression model conaining I() variables ha are assumed o be coinegraed; he model parameers are esimaed using OLS. His specificaion also allows for deerminisic and sochasic rends in he regressors. He proposes hree ess Supχ2, Mean χ2, and L C ha es he null hypohesis of parameer consancy and simulaes asympoic criical values for each es. The Supχ2 es has greaer power agains he alernaive hypohesis of a one-ime break a an unknown dae. The meanχ2 es has greaer power when he alernaive is random walk parameers. Ineresingly, he shows ha he special case of an unsable inercep in under alernaive hypohesis can be inerpreed as an absence of coinegraion among he I() variables in he model. Hence his es can be inerpreed as a coinegraion es 7
Recenly, here have been aemps in he macroeconomics lieraure o exend he unknown break dae lieraure o consider wo (possible) break poins a unknown daes. (See, e.g. Mehl (2000)). An obvious issue ha his exension raises is: why only wo breaks raher han, say, hree, or four? Auhors developing parameer sabiliy ess have also considered he alernaive hypohesis where he parameers are assumed o follow a random walk. In his case, he model parameers are generally unsable, in a way ha can no be capured a one-ime shif a any paricular dae. This es of general parameer sabiliy is a good diagnosic es when assessing he adequacy of a paricular model specificaion. Hansen (992, p. 32) provides an excellen overview of he issue and possible approaches o dealing wih i: One poenial problem wih ime series regression models is ha he esimaed parameers may change over ime. A form of model misspecificaion, parameer nonconsancy, may have severe consequences on inference if lef undeeced. In consequence, many applied economericians rouinely apply ess for parameer change. The mos common es is he sample spli or Chow es (Chow 960). This es is simple o apply, and he disribuion heory is well developed. The es is crippled, however, by he need o specify a priori he iming of he (one-ime) srucural change ha occurs under he alernaive. I is hard o see how any non-arbirary choice can be made independenly of he daa. In pracice, he selecion of he breakpoin is chosen eiher wih hisorical evens in mind or afer ime series plos have been examined. This implies ha he breakpoin is seleced condiional on he daa and herefore convenional criical values are invalid. One can only conclude ha inferences may be misleading. An alernaive esing procedure was proposed by Quand (960), who suggesed specifying he alernaive hypohesis as a single srucural break of unknown iming. The difficuly wih Quand s es is ha he disribuional heory was unknown unil quie recenly. A disribuional heory for his es saisic valid for weakly dependan regressors was presened independenly by Andrews (990), Chu (989), and Hansen (990). Chu considered as well he case of a simple linear ime rend. Anoher esing approach has developed in he saisics lieraure ha specifies he coefficiens under he alernaive hypohesis as random walks. Recen exposiions were given by Nabeya and Tanaka (988), Nyblom (989), and Hansen (990). The preceding works did no consider models wih inegraed regressors. [Hansen (992), from which his quoe is aken] makes such an exension. where he null hypohesis is he presence of coinegraion. (In conrac, in he Engel-Granger and Johansen coinegraion ess, he null hypohesis is he absence of coinegraion.) 8
In siuaions where one is emped o argue ha here are several srucural breaks, i probably makes sense o ask wheher he siuaion migh be beer described a one of general parameer insabiliy. A Marix of Possible Univariae Specificaions and Tess As oulined above, he key issues in esimaing he long-erm rend in real commodiy prices involve he presence or absence of uni roos and parameer sabiliy. In order o organize our discussion of he exising lieraure on uni roos and srucural breaks, and o poin o direcion for fuure research, consider he alernaive univariae specificaions in char in Fig. 7. The models in he lef column assume ha he ime series in quesion, here he real GY commodiy price index, does no have a uni roo. Raher i is saionary or rend saionary. Those on he righ presume he presence of a uni roo. Going across he rows, we consider parameer sabiliy/insabiliy of various kinds. The firs row assumes he model parameers are consan over ime. The second row assumes ha here is a mos single break or parameer shif in parameers a a known dae. The hird row assumes he possible single break occurs a an unknown dae. The fourh row considers he possibiliy of wo or more breaks deermined by eiher formal or informal mehods where he break daes are known or unknown. Finally, he fifh row considers he case where he model parameers follow a random walk and hence are unsable over ime. For convenience he models are numbered for fuure reference. 9
Fig. 7: Alernaive Specificaions ADF/PP uni roo ess, F sa Model - TS Model wih Consan Parameers KPSS es Model 3 - DS Model wih Consan Parameers, F sa Sup Wald, LM, LR Model 2 - TS Model wih Single Break a Known Dae Perron Model 4 - DS Model wih Single Break a Known Dae Sup Wald, LM, LR ZA/BLS Model 5 - TS Model wih Single Break a Unknown PV Model 6 - DS Model wih Single Break a Unknown Mehl/Zanias Mean F Model 7 - TS Model wih Two or More Breaks a Unknown Daes Model 8 - DS Model wih Two or More Breaks a Unknown Daes Me Model 9 - TS Model wih Unsable (Random Walk) parameers Model 0 - DS Model wih Unsable (Random Walk) parameers Empirical economiss have long employed he TS model for esimaing long-erm growh raes. A number of hese auhors also considered he possibiliy of model 2 a TS model wih a srucural break a a known/predeermined dae. To formally compare models and 2, Chow-ype srucural break ess were employed. These ess are represened by he arrow running from model o model 2. The arrow emerges from he model ha is assumed o hold under he null hypohesis in he es and poins oward he model under he alernaive hypohesis. 0
The uni roo revoluion in ime series economerics emerged slowly in he mid 970s and exploded in he 980s. I sressed ha seriously biased (indeed inconsisen) esimaes of long-erm rends could resul if one employed simple log-linear rend models when, in fac, he underlying series had uni roos. Uni roo ess, such as hose of Dickey and Fuller (979) and laer Phillips-Perron (988), were proposed as a mehod for choosing beween so-called rend saionary (TS) and difference saionary (DS) models when esimaing growh raes or rends in economic ime series. The null hypohesis under hese ess is he presence of a uni roo. These are, herefore, represened by he arrow running from model 3 o model. Subsequenly, Kwiakowski, Phillips, Schmid and Shin [KPSS] (992) developed a es ha mainained mean saionariy or rend saionariy under he null hypohesis. This es is, herefore, represened by he arrow running from model o model 2. The work of Perron (989) was seminal in ha i demonsraed ha he uni roo and srucural break issues are inerwined. Perron showed how he presence of a srucural break a a known break dae TB would bias sandard uni roo ess oward nonrejecion of he null hypohesis of a uni roo. Tha is, if one used ADF ess o es model 3 agains model, when he rue model was in fac model 2, one was very likely o falsely accep he null hypohesis of a uni roo. This has become known as he Perron phenomenon. Perron wen on o develop uni roo ess ha allowed for he (possible) presence of a srucural break under boh he null and alernaive hypoheses. The Perron-Dickey-Fuller uni roo es is represened by he arrow running from model 4 (he null) o model 2 (he alernaive). Acually, he developed separae ess for breaks of ypes A,B, and C,
respecively, as described in he accompanying box, Figure 8. 7 The appropriae specificaion in his various examples was primarily based on eyeballing he daa (albei wih some knowledge of pos World War I economic hisory), boh o deermine he mos plausible break dae, TB, and he ype of break (A,B,C). Fig.8: Perron s (992) Model Specificaion for Carrying Ou P-ADF Uni Roo Tess in Presence of Break a Time TB 8 Model A: y k A A A A A = ˆ µ + ˆ β + ˆ φ DUM TB dˆ D( DUM ˆ TB ) y cˆ, + + α + i y i + i= Model B: y k B B B B B = ˆ µ + ˆ β + ˆ φ DUM ˆ TB DT ˆ y cˆ, + γ + α + i y i + i= Model C: y k C C C C C C = ˆ µ + ˆ β + ˆ φ DUM ˆ TB DT dˆ D( DUM ˆ TB ) y cˆ, + γ + + α + i y i + i= where: = ime rend and TB refers o he ime of break. DUM TB, = if TB, and 0 oherwise (level-shif dummy) D(DUM TB ) = if = T B, and 0 oherwise DT = (-TB)*DUM TB, eˆ (spike dummy) eˆ (ime-ineracion dummy) eˆ 7 Perron s work on srucural breaks disinguishes beween he Addiive Oulier Model and he Innovaional Oulier Model. In he former, he break occurs suddenly a he break dae. In he laer, he break akes he form of a shif in he srucure of he underlying model ha akes effec gradually over ime in exacly he same way ha an innovaion is perpeuaed by he ARMA process of he esimaed model. See Perron and Vogelsang (992) for a discussion of he wo models. Throughlu his paper, we use he innovaional oulier model. 8 This is sligh reworking of Perron s original specificaion in ha he iming of he dummy here reflecs he firs period of he new regime and he ime ineracion erm is wrien he same way in models B and C. This shows more clearly ha models A and B are nesed in C. 2
Table 3 shows how imposing resricions on he es equaion for model C above causes i o collapse o TS or DS models wih various break ypes. Unresriced, he model ness all of hese as special cases. Table 3 Model α(adf sa) d*d(dum) φ*dum γ*dt DS-no break 0 0 0 0 TS-no break 0 0 0 0 DS-break A 0 0 0 0 TS-break A 0 0 0 0 TS-wih single 0 0 0 0 oulier DS-break B 0 0 0 0 TS-break B 0 0 0 DS-break C 0 0 0 0 TS-break C 0 0 0 0 Subsequen auhors, noably Chrisiano (992), Banerjee-Lumsdaine-Sock (BLS) (992) and Zivo and Andrews (ZA) (992) were highly criical of Perron s assumpion ha he dae of he (possible) break was eiher known a priori or was deermined by inspecing he daa wihou adjusing he criical values in subsequen saisical ess o reflec his informal search procedure. This, of course, echoed concerns in he lieraure developing formal ess for parameer sabiliy (discussed above; see Hansen (992) quoaion). As Fig. 2 illusraes, uni roo process ofen exhibi apparen breaks even when, in ruh, here is none. So i s risky o assess he presence of breaks by eyeballing he daa. 3
BLS and ZA proposed a generalizaion of Perron-Dickey-Fuller (P-ADF) es ha reaed he possible break dae as unknown; hey propose an algorihm for searching over all possible break daes wihin he (rimmed 9 ) sample. There are a couple of noeworhy aspecs of his es, which we dub he ZAP-ADF es. Firs, i allows he srucural break under he alernaive hypohesis bu no under he null hypohesis of a uni roo. This is refleced in he arrow represening he ZAP-ADF es, which runs from model 3 o model 5 in Fig. 7. Second, he ZAP-ADF es is a es of he null hypohesis of a uni roo, condiional on he possible presence of srucural break a an unknown dae. I is no a es for he presence of srucural break (hence our phrase a possible srucural break ). In spie of his, he ZAP-ADF and P-ADF ess have repeaedly been represened as ess of srucural change in boh he applied macroeconomeric and commodiy price lieraures. [See, e.g., Enders (995), Leon and Soo (997), and Zanias (undaed).] Finally, he ZAP- ADF es assumes ha he ype of break is known a priori. 0 Thus, he ZAP-ADF es has he raher inconsisen feaure of esing for a uni roo, condiional on he possible presence of a known ype of srucural break (A,B,C) a an unknown dae! In conras o he ZAP-ADF es, he specificaion in Perron (989) permied he break under boh he null and alernaive hypoheses -- albei a a known dae. Perron and Vogelsang (992) developed a uni roo es ha allowed for a break a an unknown dae under boh he null and alernaive. However, his was done in he conex of comparing a TS model wih zero rend (β=0) o a DS model (wih β=0 here, as well). This, in effec, 9 For echnical reasons, i is ofen necessary o rim he firs and las 0-5% of he sample, so ha only break daes in he middle 70-80% of he sample are considered. 0 Tha is, while ZA criicize Perron (989) for assuming he iming of he break is known, hey accep his visual characerizaion of he mos plausible ype of break for each macroeconomic variable considered as hey demonsrae how heir es differs from his. 4
limied he analysis ex ane o breaks of ype A. This es is denoed PV, running from model 6 o model 5 in he marix. More recenly, Leybourne, Mills, and Newbold (LMN)(998) have poined o a very compelling reason for preferring uni roo esing procedures ha allow for he presence of a break under boh he null and alernaive hypoheses. They consider siuaions where he rue model is a DS model wih eiher a ype A or B break. In eiher case, (LMN (998, p.9), our emphasis) demonsrae ha here is a converse Perron phenomenon. 2 Specifically, if he break occurs early in he series, rouine applicaion of sandard Dickey-Fuller ess can lead o a very serious problem of spurious rejecion of he uni roo null hypohesis. They go on o emphasize ha: Of course, his problem will no occur when he es procedures ha explicily permi a break under he null as well as under he alernaive are employed, as for example in Perron (989, 993, 994) and Perron and Vogelsang (992). This is he case wheher he break dae is reaed as exogenous or as endogenous, as in Zivo and Andrews (992) or Banerjee e al. (992). Indeed, our resuls imply a furher moivaion for employing such ess when a break is suspeced, in addiion o he well-known lack of power of sandard Dickey-Fuller ess in hese circumsances. (998, p.98) As menioned above, some auhors have enerained he possibiliy ha economic ime series migh have more han one srucural break. For he mos par hese muliple breaks were idenified by casual daa inspecion, alhough here are now formal uni roo ess in he (possible) presence of wo srucural breaks a unknown daes. See, e.g., Mehl (2000). Unforunaely, he uni roo ess in he laer paper shares wo undesirable feaures They noe ha in he case of heir ype A break in he simples ype of DS model a drifless random walk, his implies ha he firs-difference of he series is whie noise wih a single oulier a he break dae. 2 LMN (998, p.9): I is well known ha if a series is generaed by a process ha is saionary around a broken rend, convenional Dickey-Fuller ess can have very low power. [i.e., he Perron phenomenon. ] In his paper, he converse phenomenon is sudied and illusraed. Suppose ha he rue generaing process is inegraed of order one, bu wih a break 5
of earlier work: he breaks are assumed o be of a known ype and he break is allowed under he alernaive hypohesis, bu no under he null hypohesis of a uni roo. 4. A NEW LOOK AT GROWTH RATES, POSSIBLE BREAKS AND UNIT ROOT TESTS In esing he PS hypohesis, our primary ineres is in he growh rae β in he deflaed GY index. Has i been negaive as PS prediced? Has i been relaively sable over ime? Or has his parameer shifed or drifed over ime, or exhibied a sharp srucural break or breaks? In our paricular applicaion, we are less ineresed in he presence or absence of uni roos per se han was he applied macroeconomeric lieraure. Unforunaely i is difficul o esimae he growh rae β wihou firs making a decision on he presence or absence of a uni roo firs. Ideally, we would also like o formally es for he presence of srucural breaks wihou prejudging he case of wheher he series has a uni roo. This objecive, however, appears o be beyond our reach a his ime. Our sraegy is o proceed as follows. Firs esimae augmened ZAP-ADF-like regressions allowing for a mos wo srucural breaks a unknown daes. Having searched for he wo mos plausible break daes, we hen es wheher each break is saisically significan. If boh breaks are significan, we assume wo breaks in wha follows. If only one break is saisically significan, we re-esimae he ZAP-ADF equaion wih a single break a an unknown dae and es he o see wheher he remaining break is saisically significan. 6
The Possibiliy of A Mos Two Break Poins We firs consider he possibiliy ha he GY series is characerized by (up o) wo srucural breaks of unknown ype (A,B,C) and a unknown daes. Our search algorihm considers all possible pairs of break daes (TB, TB2) in he rimmed sample. 3 For he ZAP-ADF equaion, hree dummies he spike, level-shif, and rend ineracion dummies - - are included for each of he wo hypohesized break daes in order o allow for breaks of ype A,B, or C under boh he null hypohesis of a uni roo and he alernaive hypohesis of rend saionariy. Tha is, he esimaed ZAP-ADF equaion is: y = ˆ µ + ˆ β + ˆ αy + dˆ D( DUM 2 TB2 ) + dˆ D( DUM + ˆ φ DUM 2 TB2, TB ) + ˆ φ DUM 2 TB, + γ * * DUM + γ * * DUM k TB2, i= cˆ y i i + eˆ TB, + (7) In each regression as differen pairs of break daes (TB, TB2) are considered, he number of lags of he dependen variable, k, is chosen using Perron s general o specific mehod so as o be reasonably confiden ha he residuals are serially uncorrelaed a each sage as we proceed. Exending Hansen (992), albei less rigorously a his poin, o cover siuaions wih wo break daes, we calculae an supχ 2 saisic o make an inference abou he exisence of srucural change and a meanχ 2 saisic o deermine he exisence of general parameer insabiliy in he daa. In his conex of he ZAP-ADF equaion, he supχ 2 saisic is he maximum value over all (TB, TB2) pairs of he Wald es saisic for he null hypohesis ha all six dummies (level, spike, and ime ineracion dummies for TB 3 When searching for wo break poins wih he use of spike, level-shif and rend ineracion dummies, i is easy o show ha he break poins mus be separaed by a minimum of wo periods o avoid perfec mulicollineariy among he dummies. If one does no allow for breaks under he null hypohesis, only under he alernaive (as in Mehl (2000)), hen he wo spike dummies are omied and he wo break need only be separaed by a single period o avoid perfec mulicollineariy. 7
and TB2) are equal o zero. Hence we will call i a supχ 2 (6) saisic. The meanχ2(6) saisic is simply he average of he χ2(6) saisics. As explained in Hansen (992), a significanly high supχ2 wih a relaively low meanχ2 implies he exisence of a single srucural break (or here wo srucural breaks) and no/low parameer insabiliy. On he oher hand, a high meanχ2 is indicaive of general parameer insabiliy raher han an abrup srucural change (or wo). In addiion, we also compue he χ2 saisic for o es he join significance of he hree ypes of dummies associaed wih each candidae break poin. These are denoed χ2 sa(3)_tb and χ2 sa(3)_tb2, respecively. See Table 4 for resuls. Table 4. Grid Search Resuls for Two Srucural Breaks 4 Type of Model Type of Srucural Break Dummies ZAP-ADF Model Wih Level, Spike & Time Ineracion Dummies Chosen Break Poins TB & TB2 92 & 974 Supχ2(6) 58.53 Meanχ2(6) 0.79 ADF sa for uni roo es a -5.93 (TB=92,TB2=974) χ2sa(3)_tb (92) 4.76 χ2sa(3)_tb2 (974) 7.77 According o he grid search based on he ZAP-ADF equaion, he wo srucural breaks are mos likely o have occurred in 92 and 974. 5 The sup χ 2 (6) saisic of 58.53 is presumably saisically significan (given ha he % criical value from he χ 2 (6) 4 In a Penium III processor, he program runs for approximaely 30 minues for he ZAP-ADF model. The maximum number of lags considered in he lagged dependen variable polynominal is six. 5 We also searched for wo breaks in he ZAP-ADF model wihou he wo spike dummies (which precludes a level-shif break under he null hypohesis of a uni roo). In his specificaion he mos prominen breaks are in 92 and 985. This esimaion produced a supχ 2 of 34.43, a meanχ 2 of 8.07, and an ADF sa of 7.29. 8
disribuion is 6.8. The criical value for he sup saisic mus be deermined via simulaion mehods, bu we know i will be higher han 6.8.) The mean χ 2 (6) value of 0.79, on he oher hand, is probably no saisically significan. (We know ha for models wih one break he criical values from he mean χ 2 (6) disribuion will be slighly lower han hose from he sandard χ 2 (6) disribuion. See Hansen, 992.) Fig. 9 shows a 3-D graph of he χ2 (6) values corresponding o alernaive break dae pairs. The supχ 2 (6) of 58.53 corresponding o (TB,TB2)=(92, 974) is, by definiion, he global maximum bu here are several local maximums. There are, in fac, ohers χ2(6) saisics ha are close in value o he supχ2 aained in (92, 974). The second highes supχ2 value of 56.33 occurs wih candidae break dae pair (92, 973), and he hird highes of 55.79 occurred a (92, 984). Noe ha here is a clear L-shaped ridge of high supχ2(6) values where eiher TB or TB2 is 92. This suggess ha here is a raher decisive break in 92. Placing he oher possible break dae almos any oher dae afer 923 in he rimmed sample ofen produces a high χ 2 (6) saisic. This migh be indicaive of general parameer insabiliy, raher han a second decisive break poin. Alernaively, here may be only a single break a 92, wih he daing of a second possible break being raher inconsequenial in deermining he value of he sup χ 2 (6) saisic. Turning o he wo break poins, considered separaely, he χ 2 (3)_TB and χ 2 (3)_TB2 suggess ha he srucural change in 92 is more prominen han he one in 974. Noe ha χ 2 (3)_TB2=7.77, which is less han he sandard % criical value for χ 2 (3) of.34. The appropriae criical value, given ha he break daes are chosen from search process ha maximizes χ 2 (6), mus be higher. Thus, we can safely conclude ha TB2 is 9
insignifican. A deerminaion on TB would require a calculaion of he appropriae criical values. A complemenary approach is o esimae he ZAP-ADF equaion wih a single possible break poin a an unknown dae, which we ake up nex. Fig. 9. 3-D Graph of he χ2(6) Saisic for he ZAP-ADF Equaion (92, 984, 55.79) supf sa (92, 974, 58.53) 60 50 40 30 20 0 0 (92, 973, 56.33) 962 972 982 50-60 40-50 30-40 20-30 0-20 0-0 979 952 969 959 Year 949 939 929 99 942 932 922 92 Year2 909 To reierae, wihou conducing an exensive Mone Carlo simulaion analysis we don know wheher he supχ2 or meanχ2 saisics are saisically significan. Similarly, we don know wheher he ZAP-ADF sa of 5.93 in Table 4 above is large enough o rejec he null hypohesis of a uni roo, condiional on he possible presence of wo breaks of unknown ype (A,B,C) and unknown daes. 20
The ZAP-ADF Tess wih A Mos One Break The above exercise is repeaed assuming, now, ha here is a mos one break a an unknown dae as in ZA/BLS and Perron-Vogelsang. 6 The ZAP-ADF equaion is: y = k ˆ + ˆ β + ˆ αy dd ˆ ( DUM TB ) DUM TB * DUM TB cˆ + + φ, + γ, + i y i + i= µ eˆ (8) ZA/BLS and Perron-Vogelsang (992) choose he break dae ha minimizes he - saisic on α c he ADF saisic. We he use an alernaive search algorihm, alhough using our oupu i is easy o compare o he ZA/BLS resuls. The alernaive we consider is similar o Andrews (992) and Hansen (992). For each and every possible break dae TB in he [.5,.85]-rimmed sample, we calculae he Wald χ 2 saisic for he join hypohesis ha he coefficiens on all hree break dummies are joinly insignifican. Tha is, H 0 : ˆ θ = ˆ γ = dˆ = 0. 7 Under he null, here is no break of Type A, B, or C. We plo he sequence of χ 2 saisics, as in Hansen (992), o ge some indicaion of wheher here migh be one or more breaks. The maximum in he sequence of χ 2 saisics, denoed sup χ 2 is deermined. The mean χ 2 is also calculaed. 8 A high value for he sup χ 2 saisic signals a possible srucural break (of ype A, B, or C); a high value for he mean χ 2 saisic, on he oher hand, suggess he parameer esimaes (in he ZAP-ADF equaion 6 Perron and Vogelsang also consider an algorihm ha selecs TB so as o maximize he absolue value of he -sasiic on DUM. In heir conex which precludes breaks in he growh rae (as i is idenically zero), his amouns o using he supχ2 saisic ha we employ. 7 Given ha we impose linear resricions, he Wald es oupu produces boh an χ2 -saisic and a Chisquare saisic. However, he Chi-square saisic is more appropriae since lagged dependan variables appear as regressors in our equaion specificaion. 8 These wo saisics are analogous o he supχ2 and meanχ2 saisics discussed in Hansen (992), for his regressions ha did no involve lagged dependen variables. 2
in his case) are unsable. We also plo he -saisics on he dummies and he ADF- saisic for each possible break poin. When our single break selecion procedure is applied o he deflaed GY index, he Wald es saisics for he various possible break poins are hose shown in Fig. 0. The sup χ 2 of 32.4 occurs in 92 and is a clear oulier in erms of magniude; he mean χ 2 = 4.9. Given ha sup χ 2 lies well above he sandard % criical value for χ 2 (3) of 3.28, and mean χ 2 lies well below he criical value, i is reasonable o conclude ha he real GY series is well characerized by a single break in 92, raher han muliple breaks or general parameer insabiliy. Fig. 0: The Sequence of Wald χ 2 Tes Saisics for he Join Hypohesis H 0 : ˆ θ = ˆ γ = dˆ = 0. 40 30 Wald Saisic 20 0 0 00 0 20 30 40 50 60 70 80 90 Year To ge a beer undersanding of wha is producing he large sup χ 2 value in 92, one can examine he sequence of -saisics on he individual dummy coefficiens shown in Fig. below. 22
Fig. 4 2 - S ai si 0-2 Spike Dummy Level Dummy Tim e Ineracion Dummy -4-6 00 0 20 30 40 50 60 70 80 90 Year A visual inspecion confirms he exisence of a spike dummy in 92. Formally, he -saisics in 92 are 5.2047, -0.3987, -0.24, and 0.2029 for he spike, level, ineracion, and rend dummies respecively. Even hough we do no have he correc criical values o inerpre he spike dummy a his poin, a -saisic of 5.2047 is presumably above he appropriaely calculaed criical value, implying a rejecion of he hypohesis of a zero coefficien on he spike dummy. 9 9 The coefficien on he spike dummy in 92 is 0.284. This urns ou o be a clear oulier compared wih ha of he res of he period. 23
Fig. 2: The Perron-ADF -saisics a various break daes -2.4 -sa on lagged dependend variable -2.8-3.2-3.6-4.0-4.4-4.8-5.2 00 0 20 30 40 50 60 70 80 90 Year Turn now o Fig.2, which shows he ADF -saisic for all possible (single) break daes. Our supχ 2 saisic idenified 92 as he year of he break. On ha dae, he ADF saisic has a value of 3.03. Presumably (awaiing correc criical values), he null of uni roo canno be rejeced a a reasonable level of significance. Given ha Fig. shows only he -saisic on he spike dummy is large, and he ADF saisic is small, i suggess ha he GY series is probably well described as a DS-break A model. I is ineresing ha he minimum value of he Perron-ADF saisic in Fig. 2 is he 4.99 value in 972. The ZA/BLS mehod for selecing he break dae would, herefore, have chosen 972 no 92 as he break dae. Given he value of he es saisic, one would fail o rejec he uni roo wih break hypohesis a he one or five percen significance levels; he respecive criical values are 5.57 and 5.30 (assuming ha he ZA asympoic criical values sill apply when a spike dummy is included in he ZAP-ADF 24
equaion as we do here). From Fig.0 showing he sequence of Wald saisics, on he oher hand, i appears ha he argumen ha he break occurs in 972 raher han 92 is weak. Comparing our algorihm o he ZA/BLS algorihm suggess ha he laer gives very lile weigh o he significance of he spike dummy. In effec, his amoun o no aking seriously DS (uni roo) wih a ype A break model. We believe his biases he resuls agains he uni roo hypohesis. Our algorihm should dominae he ZA/BLS algorihm in he siuaions described by LMN (998). They emphasize he need o allow for he break under boh he null and alernaive hypoheses. We add o his poin by sressing he need o apply an appropriae search algorihm for deermining he break poin. The ZAP-ADF ess conduced here consider up o wo break daes in he GY series. We enaively conclude ha he series is well characerized as a uni roo process wih a single level-shif break (ype A) in 92. Unforunaely, uni roo ess have nooriously low power, so he common failure o rejec he uni roo hypohesis hardly provides a definiive deerminaion of he rue daa generaing process. An alernaive approach is o consider he KPSS ess, which ake saionariy or rend saionariy, raher han nonsaionariy, as he null hypohesis. These ess are found in he appendix. Two oher argumens can also be invoked in making a choice beween he TS and DS specificaions: Plosser and Schwer (978) discuss he pros and cons of esimaing economic ime series regression models, of which log-linear ime rend models are a special case, is levels or firs-differences. More precisely, hey consider he relaive coss of over-differencing and under-differencing. Which sraegy is riskier: firs-differencing ()-(2), so ha he DS model in (3)-(4) is esimaed, when in fac here is no uni roo in ()-(2), or esimaing ()-(2) when here is, in fac, a uni roo? They argue ha he problem of nonsaionary disurbances (possibly in he levels regressions) are far more serious han he problems caused by excessive differencing (in he second differences regression, for example). (978, p.657). 25
Parameer insabiliy in he TS model may, in fac, be an indicaion ha he error process, in fac, has a uni roo. Thus, we should look carefully for differences in he degree of parameer insabiliy across he TS and DS specificaions. Given he uncerainy surrounding he quesion of uni roos, i seems reasonable o esimae boh TS and DS models wih one or wo breaks. (The models wih no breaks have already been esimaed above.) Begin wih he more general wo-break specificaion. 5. ESTIMATED TS AND DS MODELS WITH TWO BREAKS Below we will consider he TS and DS model in urn, using our search algorihm o choose he daing of wo break poins. 20 As discussed above, we need o include only he level-shif and ime ineracion dummies o allow for breaks of ype A, B, and C in he TS model. Thus he crierion for choosing he break daes (TB,TB2) is he supχ 2 (4) saisic from he se of all χ 2 (4) saisics esing he join significance of he wo dummies associaed wih all possible pairs of break daes. Analogously, in he DS specificaion, we need o include only he spike and level-shif dummies. The crierion is again a supχ 2 (4) saisic. Once he wo mos plausible break poins have been idenified in he TS and DS specificaions respecively, here are hree subsamples of he GY index o consider. There is a necessary o esimae he growh raes for each segmen: pre-tb, TB hrough TB2, pos-tb2. Esimaes of he rend segmens for boh he TS and DS specificaions are shown in Table 5. Also repored is he Wald es of he hypohesis ha each rend 20 Each specificaion requires he inclusion of wo dummies for each break dae. I can be shown ha he break daes mus be separaed by a leas one period o avoid perfec mulicollineariy. 26
coefficien is equal o zero. A rejecion of he hypohesis indicaes he presence of a significan rend in he respecive sub-period. Table 5. Grid Search Resuls for Two Possible Breaks a Unknown Daes (TB, TB2) 2 Type of Model Type of Srucural Break Dummies TS Model Level & Time Ineracion DS Model Level & Spike Chosen Break Poins TB & TB2 92 & 985 92 & 974 Supχ 2 (4) 34.43 47.40 Meanχ 2 (4) 8.07 3.35 (Segmened) Trend. pre_tb 0.0032 (0.836) 0.0027 (0.6559) 2. TB hrough TB2-0.0006 (0.298) 0.000 (0.9690) 3. pos_tb2-0.002 (0.5874) -0.009 (0.0307) χ2 sa(2)_tb 3.25 9.32 χ2 sa(2)_tb2 4.36 4.77 Noe:. The p-value for he hypohesis ha he rend coefficien is equal o zero is given in parenhesis. P values ha are higher han your chosen es size (say.05) indicae failure o rejec he null hypohesis of a zero rend for he given segmen of he daa. These p values ignore he fac ha TB and TB2 were chosen so as o maximize supχ 2 (6). Thus he p-values on he rend segmens are possibly inaccurae. Examining he able, we find ha sup χ 2 (4) saisics for boh he TS and DS specificaions are large (relaive o he sandard % criical value for χ 2 (4) of 3.28. The mean χ 2 (4) saisic for he DS model is very small, suggesing no issue of general parameer insabiliy. The mean χ 2 (4) saisic for he TS model is close enough o he sandard criical vale ha i is impossible o guess he oucome of a formal parameer sabiliy ess based on simulaed criical values. The TS model esimaion places he wo breaks in 92 and 985. Moreover, he χ 2 (2)_TB and χ 2 (2)_TB2 sas for 92 and 985, respecively, are similar in magniude, wih 985 being slighly larger (4.36 vs. 3.25, whereas he.0% criical value for 2 In a Penium III processor, he program runs for approximaely 20 minues each for he TS and DS models. In boh cases, he maximum number of lags of he dependen variable considered (k) was six. 27
χ2(2)=9.2.) To calculae he segmen-specific growh raes in he TS model, he formulas in he accompanying box are used. TS Model: ln( y Calculaing Segmen-Specific Growh Raes in he TS and DS Models k 0 + β + β 2DUM TB + β 3 * DUM TB + β 4DUM TB2 + β 5 * DUM TB2 δ i ln( y i ) i= ) = β + DS Model: ln( y ) = β k 0 + βdum TB + β 2D( DUM TB) + β 3DUM TB2 + β 4D( DUM TB2 ) + δ i ln( y i ) i= Segmen-Specific Growh Raes in he TS Model: Pre_TB growh rae: TB hrough TB2 growh rae: Pos_TB2 growh rae: β ( δ +... + β + β 3 ( δ +... + 3 ( δ +... + δ k δ k β + β + β 5 δ k ) ) ) Segmen-Specific Growh Raes in he DS Model: Pre_TB growh rae: TB hrough TB2 growh rae: Pos_TB2 growh rae: β 0 ( δ +... + β + β 0 ( δ +... + δ k δ k β 0 + β + β 3 ( δ +... + δ k ) ) ) 28
The resuling calculaions for he TS model growh raes and heir χ2 saisics (convenional p values noed) indicae ha he rend in all hree sub-periods are no saisically differen from zero. In conclusion, herefore, if one rejecs he uni roo hypohesis and acceps he TS model, he GY series is bes characerized as a zero-growh series ha has experienced wo significan downward level shifs (ype A breaks), firs in 92 and hen again in 985. Dependen Variable: GY Mehod: Leas Squares Dae: 0/3/0 Time: :25 Sample(adjused): 902 998 Included observaions: 97 afer adjusing endpoins Variable Coefficien Sd. Error -Saisic Prob. GY(-) 0.6276 0.097600 6.370075 0.0000 GY(-2) -0.3444 0.095788-3.282405 0.005 C.489242 0.202636 7.349340 0.0000 @TREND 0.002224 0.00737.280643 0.2036 DUM92-0.068788 0.025894-2.656577 0.0094 DUM92*@TREND -0.00263 0.00782 -.47682 0.433 DUM985-0.02324 0.24439-0.050425 0.9599 DUM985*@TREND -0.00040 0.002699-0.38527 0.700 R-squared 0.88084 Mean dependen var 2.02670 Adjused R-squared 0.87440 S.D. dependen var 0.0272 S.E. of regression 0.039538 Akaike info crierion - 3.544222 Sum squared resid 0.393 Schwarz crierion - 3.33875 Log likelihood 79.8948 F-saisic 93.9648 Durbin-Wason sa.840466 Prob(F-saisic) 0.000000 29
Fig. 3 A Segmened Trend Saionary Model? 2.3 2.2 2..5.0.05.00 2.0.9.8.7 -.05 -.0 -.5 00 0 20 30 40 50 60 70 80 90 GY RESID_TS2_85 FIT_TS2_85 GY_F2_85 Figure 3 shows he acual logged GY series, he fied values and residuals from he bes fiing TS specificaion wih wo breaks, and he forecased values saring in 900 in order o show he long-run rend segmens more clearly. The ess summarized in Table 5 above indicae ha he rend is insignificanly differen from zero in each of he hree segmens of he TS model: pre-920, 92-984, and pos-984. In conras o he TS model, he DS model idenifies he wo break years as 92 and 974, raher han 985. Noe ha for he DS model, he supχ2(4) is very large while he meanχ2(4) saisic is quie small. (For comparison, he sandard χ2(4)=3.28.) Also, 30
he 92 break has a much higher χ2(2) sa han he 974 break. Togeher hese χ2 saisics sugges ha, if one uses he DS specificaion, he GY series is well characerized by one (92) or possibly wo (92, 974) srucural breaks raher han general parameer insabiliy. Examining he χ 2 (2)_TB (=9.32) and he χ 2 (2)_TB2 (=4.77) saisics, i is clear ha he 92 break is significan, while he 974 break is no saisically significan. 22 Thus he DS specificaion requires only a single break in 92. This is consisen wih our ZAP-ADF ess, which found a single break and were unable o rejec he null hypohesis of a uni roo. 6. ESTIMATED TS AND DS MODELS WITH A SINGLE BREAK We now esimae DS and TS Models wih single breaks a an unknown dae. We firs search for one endogenous break in he GY series using he TS model. As one may recall, we need o include only he level dummy and he ime ineracion dummy in his paricular seup. We obain he supχ 2 (2) saisic ha ess he hypohesis ha hese wo dummies are equal o zero and graph i below. The maximum supχ 2 (2) has a value of 7.93 and occurs in 946. The % criical value for he sandard χ 2 (2) disribuion, however, is 9.2. Thus he supχ 2 (2) and meanχ 2 (2) (=2.88) sugges ha a TS model wih zero breaks is adequae! Thus, raher curiously, he wo-break model suggesed ha here are wo (marginally?) significan breaks in 92 and 985, while he one break model finds no break a all! If one believes he wo break model, he GY series has wo downward level 22 Wha abou he calculaed growh raes for each segmen in he DS specificaion if we assume here are TWO breaks? Resuls for he DS model are slighly differen from hose obained from he TS model. In spie of a saisically insignifican rend in each of he firs wo sub-periods, he DS model idenifies he exisence of a possibly significan 22 negaive rend of.09% in he pos-974 period. 3
shifs, bu no ongoing secular rend. If one believes he TS model wih no breaks, here is a saisically significan negaive ime rend! Fig. 4 χ2(2) sas TS Model wih One Endogenous Break 8 7 6 5 4 3 2 0 00 0 20 30 40 50 60 70 80 90 Year We now search for a single break in he GY series using he DS model. In his case, we include only he level and spike dummies in he esimaion. We now use he supχ2(2) saisic o es he hypohesis ha hese wo dummies are zero. Fig. 5 graphs he χ2(2) for he DS model. 32
Fig. 5 The χ 2 (2) Sequence for DS Model wih One Break 35 30 25 20 5 0 5 0 00 0 20 30 40 50 60 70 80 90 Year Here, he maximum supχ2(2) has a value of 32.26 and occurs in 92. The second highes supχ2(2) has a value of 6.28 and occurs in 975. In addiion, he meanχ 2 (2) saisic is.58, a conrasingly low value compared o eiher he supχ 2 or he % criical value of 9.2 from he sandard χ 2 (2) disribuion. Therefore, wih he DS specificaion, a single downward level shif in 92 bu wih no ongoing (sochasic) rend fis he daa well. 7. CONCLUSIONS Despie 50 years of empirical esing of he Prebisch-Singer hypohesis, a long-run downward rend in real commodiy prices remains elusive. Previous sudies have generaed a range of conclusions, due in par o differences in daa bu mainly due o differences in specificaion, as o he saionariy of he error process and he number, iming, and naure of srucural breaks. In his paper, we have aemped o allow he daa o ell us he proper specificaion. In our mos general specificaion (model 8, in Fig. 7), which allows for a uni roo, and searches for wo srucural breaks of any kind, we find he 33