Non-renewable resource prices. A robust evaluation from the stationarity perspective

Transcription

1 MPRA Munich Personal RePEc Archive Non-renewable resource prices. A robus evaluaion from he saionariy perspecive María José Presno and Manuel Landajo and Paula Fernández Universiy of Oviedo (Spain) 9 November Online a hps://mpra.ub.uni-muenchen.de/453/ MPRA Paper No. 453, posed November 7:38 UTC

2 NON-RENEWABLE RESOURCE PRICES. A ROBUST EVALUATION FROM THE STATIONARITY PERSPECTIVE María José Presno, Manuel Landajo & Paula Fernández Universiy of Oviedo November 9, ABSTRACT The bulk of he lieraure invesigaing persisence properies of non-renewable resource prices series has focused on applicaion of uni roo ess. This paper conribues o he debae, applying a mehodology which allows () robus deecion of he presence and (if so) he number of changes, () inference on saionariy of he series, and (3) esimaion of change locaions. In conras o previous papers, he analysis is carried ou from he perspecive of saionariy esing, incorporaing quadraic rends and he possibiliy of smooh changes. For a classical daabase, we find significan evidence of rend saionariy in mos of he series, suggesing ha shocks are mosly of a ransiory naure. Excepions are silver and naural gas, wih saionariy being rejeced for all he specificaions considered in he paper. Finally, he knowledge of he sochasic characerisics of he series allows robus deecion of change poins which appear o be relaed o economic evens. Keywords: non-renewable resource prices, srucural changes, saionariy es, sequenial procedure.. INTRODUCTION There is a large body of lieraure which invesigaes he ime series properies of nonrenewable resource prices. The reasons for his ineres include boh heoreical and economeric issues. From a heoreical poin of view, if prices are rend saionary, shocks dissipae and policy effors o resore price following a shock are unwarraned. On he conrary, in he case of a sochasic rend, policy inervenion is sensible in order o overcome he permanen effec of a shock. The auhors are wih he Deparamen of Applied Economics, Universiy of Oviedo, Avenida del Criso s/nº, 339 Oviedo, Asurias (Spain). mpresno@uniovi.es (M.J. Presno); landajo@uniovi.es (M. Landajo); pfgonzal@uniovi.es (P. Fernández).

3 Researching he poenial saionariy of real non-renewable resource prices can be vial in an assessmen of heories validiy. One of he models more debaed is he Hoelling model. In a world of cerainy, he Hoelling model predics ha non-renewable resource prices are rend saionary; while in an uncerain conex, according o Slade (988), prices may be difference saionary. A number of papers have examined his issue in an effor o undersand which heory or heories bes describe he observed behaviour. In his respec, Slade (988) represens one of he firs aemps o analyze ime series properies of naural resource prices in order o evaluae he Hoelling model; applying uni roo ess and considering a linear rend model, Slade finds evidence of sochasic rends. Afer Slade s (98) assessmen ha he progress effec can lead o a U-shaped price pah, Agbeyegbe (993) and Berck and Robers (996) exended he uni roo analysis incorporaing a quadraic rend, and Ahrens and Sharma (997) and Lee e al. (6) considered breaks and found furher evidence agains he uni roo hypohesis. Using a differen mehodology - Kalman filer mehods- Pindyck (999) esimaed a model where prices rever o a quadraic rend ha shifs over ime. In he field of energy, ime series properies of prices have been analysed in order o invesigae he efficien marke hypohesis. Lee and Lee (9) found evidence in favour of he broken saionariy hypohesis, implying ha energy prices are no characerized by an efficien marke. Maslyuk and Smyh (8) focused on crude oil, and highlighed reasons o analyse he sochasic properies of is price: if oil prices are rend revering, his is consisen wih crude oil being sold in a compeiive marke where prices rever o long-run marginal cos, which changes only slowly. Also, some sudies 3 have linked shocks o crude oil prices o oupu and inflaion, he naural rae of unemploymen, movemens in sock marke indices, and flucuaions in business cycles. Finally, here are links beween oil price shocks and invesmen decisions (e.g. Dixi and Pindyck, 994; Finn, ), and he price of oil occupies a cenral sage in heories in he area of environmenal and resource economics. For insane, Sinn (8) poins ou ha oil is one of he main sources of carbon emissions and exends he Hoelling model in order o consider he global warming, and Holland (8) shows ha oil prices, raher han producion, are a beer indicaor of impending resource scarciy. Knowledge of he sochasic properies of ime series is also imporan in economeric esimaion and subsequen applicaion of he models o forecasing and decision making. Kraukraemer (998) includes a survey of his lieraure. 3 See Maslyuk and Smyh (8) for a review.

4 Berck and Robers (996) focused on ARMA and ARIMA models o forecas naural resource prices, bu in heir esimaion hey did no consider srucural breaks. Lee e al. (6) overcame his limiaion, and sressed ha pre-esing for uni roos and including srucural breaks can improve he accuracy of forecasing non-renewable resource prices. Pindyck (999) applied Kalman filer echniques o obain forecass of energy prices and sressed he relevance of knowing he sochasic properies of prices in erms of long-run forecasing and for firms making invesmen decisions. In he field of energy, Felder (995) highlighed he imporance of long-erm fuel price forecass in asceraining generaion planners o make correc choices in deermining heir fuel mix and evaluaing fuel diversificaion sraegies. Papers which invesigae persisence properies of non-renewable resource prices have focused on uni roo ess resuls, allowing in some cases for srucural breaks in he rend funcion. In his paper we conribue o he debae by incorporaing resuls from new ime series mehodologies which provide more robus conclusions abou he naure of he resource price ime pahs. For his we examine a classical daabase of naural resource real prices series from 87 o 99. These daa were analysed previously using ohers mehodologies, hus making comparabiliy of conclusions easier. Firs, we apply saionariy ess. The above papers analysed he null hypohesis of a uni roo agains he alernaive of rend saionariy -in some cases wih a break-, bu we consider he esing problem in he reverse direcion: he null of rend saionariy (around linear and quadraic rends wih breaks or smooh ransiions) agains he alernaive of a uni roo. So, saionariy ess complemen uni roo ess. Furhermore, as uni roo ess are known o have low power under saionary bu highly persisen processes, saionariy ess provide a useful means o confirm resuls from uni roo ess. In his paper we apply saionariy ess allowing for breaks and smooh ransiions in he rend funcion, bu we use a new mehodology for heir reamen. Some researches have shown ha here is a circular esing problem beween ess on he parameers of he rend funcion and uni roo/saionariy ess, so Perron and Yabu (9), Harvey e al. () and Kejriwal and Perron () proposed approaches ha are robus o uni roo and saionary errors in order o es for sabiliy of he rend funcion and o obain a consisen esimae of he rue number of breaks. Wih his basis, our analysis begins by deecing he presence, and if so, he number of breaks in he series. For his we apply he above mehodologies. In heir mainsream versions hese echniques consider linear rends, bu he pecualiriies of non- 3

5 renewable resource prices ime series have led us o exend he basic mehodology in order o allow for nonlinear (quadraic) rends. Once we have checked he number of changes, we apply wo alernaive ypes of saionariy ess in order o disinguish he sochasic properies of he daa: (i) allowing for breaks - which occur insananeously- and (ii) allowing for smooh ransiions -whose effecs are gradual, he ransiion beween wo regimes being smooh-. The laer approach is aracive since o assume ha he change akes place insananeously may be unrealisic in many economic applicaions, and also allows us o eliminae he very subsanial size disorions which appear when we apply saionariy ess wih a break when he series really exhibis smooh changes (see Landajo and Presno, ). Finally, knowing he sochasic characerisics of ime series is useful in order o ackle consisen esimaion of he break/midpoin smooh ransiion daes. Concreely, wih breaks in he level and/or he slope, consisen esimaes of he break daes are obained from a level or firs-differenced specificaion according o wheher a saionary process is presen or no. To summarize, his paper conribues o he debae abou emporal properies of a classical daabase of naural resource real price series, bu applying a powerful mehodology ha enables (i) robus deecion of presence, and if so, he number of breaks in he level and/or he slope of he rend funcion, (ii) o carry ou inferences on saionariy of he series, condiional on he presence of breaks and smooh ransiions, and (iii) esimaion of breaks/midpoin smooh ransiion locaions. The res of he paper is srucured as follows: secion inroduces he mehodology, including Mone Carlo experimens o obain criical values adaped o he characerisics of he ime series. Secion 3 repors he empirical resuls and a discussion. The paper closes wih a summary of conclusions.. METHODOLOGY In his secion we inroduce he mehodology for he analysis. Firs, we esimae he number of changes in ime series, applying some approaches which are robus o saionary/inegraed errors. Upon his basis, we examine he sochasic properies of he daa via he analysis of he null of saionariy around linear and quadraic rends, wih he number of changes deeced in he previous sage. For his, we implemen he Landajo and Presno () proposal, applying wo ypes of saionariy ess: allowing for breaks and allowing for smooh ransiions. Finally, we esimae change locaions. 4

6 .. Mehods o esimae he number of changes Informaion abou he absence or presence (and in his case, he number) of changes is vial o devise uni roo and saionariy ess wih good properies. Tradiionally, boh kinds of ess were applied considering a priori he presence of one or more changes; however, i seems more suiable o research firs he poenial presence of such breaks, since he inclusion of dummy variables o cope wih nonexisen breaks leads o reducion in he power of uni roo/saionariy ess. On he oher hand, knowing he naure of persisence in he noise componen is necessary in order o es for srucural breaks. Inference based on a srucural change es from firsdifferenced daa -which conveys o assume a uni roo- leads o ess wih poor properies when he series conains a saionary componen. On he oher hand, applicaion of he es on he level of he daa enails differen limiing disribuions -so, differen asympoic criical values- depending on a uni roo is presen or no. So, here is a circular problem beween ess on he parameers of he rend funcion and uni roo/saionariy ess. Perron and Yabu (9) solved his circular problem. They proposed an approach o assess he presence of a srucural change in he linear rend funcion of a univariae ime series wihou any prior knowledge as o wheher he noise componen is saionary or conains an auoregressive uni roo. Kejriwal and Perron () exended he mehodology, and developed a sequenial procedure ha allows one o obain a consisen esimae of he number of breaks in he linear rend funcion. Their proposal provides a procedure o es he rend funcion for breaks in slope or simulaneous breaks in slope and level, bu no breaks solely in level. Harvey e al. () filled his gap and derived ess which allow for muliple level shifs. Nex we inroduce hese ess. All hese proposals consider a linear rend, bu we exend hem in order o incorporae a quadraic rend o cope wih nonlineariies. The Perron and Yabu (9) es Perron and Yabu (9) proposed an approach o esing he sabiliy of he rend funcion based on a Feasible Quasi Generalized Leas Squares procedure ha uses a super-efficien esimae of he sum of he auoregressive parameers α when α=. They consider he daa generaing process: 5

7 y u = x Ψ + u ' = α u ν = d( L) e + ν () for =,,T, where x is a (r x ) vecor of deerminisic componens, Ψ is a (r x ) vecor of = i unknown parameers, d( L) d L, i d <, d( ) i= i i, i. i. d. (, σ ) i= e, and u is some consan. I is assumed ha -<α. If α=, u is an inegraed process of orden (u ~I()), so y is a difference saionary process wih a possibly broken rend; when -<α<, u ~I(), and he series is rend saionary wih a possibly broken rend. The null hypohesis o be esed is RΨ=a, where R is a (q x r) full rank marix and a is a (q x ) vecor, wih q being he number of resricions. Perron and Yabu (9) considered hree linear Models including a shif: Models I and II only allow for a shif in inercep and in slope, respecively, and Model III allows for boh a shif in inercep and slope a T b =[λt] for some λ (,), where [.] denoes he larges ineger ha is less han or equal o he argumen. In our analysis, we jus consider 4 Model III, where x =(,,DU,DT ), DU =(>T ), DT =(>T )(-T ), wih (.) being he indicaor funcion, and Ψ=(β,β,δ,η ). For he general case, he hypohesis of ineres is δ =η =. We also consider an unresriced case 5 which allows esing for sabiliy of he slope parameer while he inercep may vary across regimes. For ha, we only es wheher a shif in slope is presen (η wih δ unresriced, so he hypohesis of ineres is η =). In his case, criical values corresponding o Model II are used. Since we also analyse he case of quadraic rends, we exend he procedure, o include a new model: Model III (quadraic), where x =(,,,DU,DT ), wih Ψ=(β,β,β,δ,η ). The hypoheses of ineres are δ =η = and η = for he general and unresriced cases, respecively. Appendix A.. describes he es procedure. Perron and Yabu (9) provided asympoic criical values for his es, alhough our simulaions 6 (see Table below) indicaed ha in finie samples of moderae size, criical values may differ considerably from heir asympoic counerpars; so, finie sample criical 4 We skip Model I because he sequenial procedure by Kejriwal and Perron () does no allow for shifs jus in inercep, so, in order o analyse his kind of change, we apply he Harvey e al. () procedure. 5 As shown in Kejriwal and Lopez (), he join es for Model III has power agains processes which are characerized by shifs only in he level and i is likely o rejec he null of sabiliy even when here is no change in he slope of he rend funcion. So, in order o disinguish beween changes in level or slope, hey recommend he unresriced proposal. 6 Malab codes for he procedures used in his paper are avalaible from he auhors upon reques. 6

8 values are obained. More precisely, we generaed criical values for samples wih T= and T=5 in Models III and III (quadraic), in he general and unresriced versions. Based on simulaions we found ha he rimming parameer ε=.5 and δ=.5 in (3a) see Appendix A..- led o good resuls in finie samples, so we considered hese values boh o obain criical values and in applicaions. Table repors criical values. As a reference, he las wo columns conain he asympoic criical values (Perron and Yabu, 9). As expeced, as T increases, finie sample criical values approach he asympoic ones, bu for T= here are remarkable differences beween boh values. Concreely, for T= criical values are longer han asympoic ones, and applicaion of he laer in finie samples would lead o an increase in rejecions of he null hypohesis of sabiliy and o size disorions in he es, so, in his paper we use he finie sample criical values. [Inser Table ] The Kejriwal and Perron () sequenial es Kejriwal and Perron () exended he Perron and Yabu (9) es and proposed a sequenial procedure ha allows one o obain a consisen esimae of he number of breaks while being agnosic o wheher a uni roo is presen. The procedure proceeds by esing he null hypohesis of l changes agains he alernaive hypohesis of l+ changes. In our analysis, given he number of sample observaions, we allow a maximum of wo breaks 7. The firs sep of he procedure conducs he Perron and Yabu es for no break versus one break. Condiional on rejecion, he esimaed break dae is obained by a global minimizaion of he sum of squared residuals. The sraegy proceeds (by using he mehodology by Perron and Yabu, 9) o es for he presence of an addiional break in each of he segmens obained from he esimaed pariion. The es saisic for he null of one versus wo breaks can be expressed as: where ExpW ( ) { ( i max ExpW ) } = () i (i) ExpW is he one-break es in segmen i. The null hypohesis of a single change agains he alernaive of wo is rejeced if ( ) ExpW is sufficienly large. 7 This is in accordance wih Kejriwal and Perron (), who recommended ha he maximum number of breaks should be decided wih regard o he available sample size. Oherwise, he sequenial es will be based on a small number of observaions in each subsample, leading o low power and/or size disorions in he ess. 7

9 As in Perron and Yabu (9), we considered Model III (for he general and unresriced proposals), derived he quadraic case (Model III, quadraic, general and unresriced), and generaed finie sample criical values. Simulaions showed good resuls for he rimming parameer ε=. and for δ=.. Table repors criical values and shows ha small sample criical values are again longer han he asympoic ones. [Inser Table ] The Harvey e al. () es for breaks in level The sequenial procedure by Kejriwal and Perron () does no allow esing for muliple breaks solely in level. Harvey e al. () filled his gap and proposed robus ess for deecing muliple breaks in level condiional on a sable underlying slope. The model is: y u = β + β + = ρu + ε, n i= δ DU i =,..., T ([ λ T ]) i + u, (3) =,..., T λ i Λ, where Λ = [ λ L, λ U ], wih λ L and λ U being rimming parameers which saisfy λ < λ <. < L U The null hypohesis is δ = for i=,, n, and he alernaive is ha here is a leas one break in level. The es is based on he quaniies: where i M = max M S S = = Λ,[ mt ] / ( ˆ ωυ ) T / ( ˆ ω ) T M u ˆ m β T M (4) M,[ mt ] = m T i= y + i m T i= m T y i+ m is he window widh, and mus saisfy he consrain λu λl n + = n m max, which provides an upper bound for he maximum number of breaks assumed o be presen. ˆβ 8

10 denoes he OLS esimaor of he rend coefficien, and ωˆ υ and esimaes appropriae for he case of I() and I() shocks, respecively. The proposed es is: U cv max S, cv S ξ = ξ ωˆ u are he long-run variance (5) where cv ξ and cv ξ denoe he asympoic criical values of S and S, under I() and I() errors respecively, a significance level ξ. The decision rule rejecs he null if U> κ cv, where κ ξ is a posiive scaling consan. A rejecion informs us ha a leas one level break is presen. Harvey e al. () also proposed a sequenial procedure for deermining he number of level breaks, n U. The procedure, adaped o a maximum of wo breaks, is in Appendix A.. We exended he above procedures in order o include a quadraic rend. The model is: y u = β + β + β = ρu n i= + ε, =,..., T + δ DU i ([ λ T ]) i + u, =,..., T ξ (6) ξ In his case, M in (4) is replaced by m M = max M Λ m [ ] ˆ T ˆ, mt β β ( + ) T (7) where ˆβ is he OLS esimaor for he quadraic rend coefficien. This modificaion is also included in he sequenial procedure o deermine he number of breaks. Table 3 repors finie sample criical values for linear and quadraic cases 8. [Inser Table 3].. Saionariy analysis. The Landajo and Presno () es Once he number of changes is analysed, we apply saionariy ess allowing for breaks and smooh ransiions. 8 Following Harvey e al. () recommendaion, we chose m=. and m=.5 for he window widh, and considered λ.5, λ =. 85 and λ., λ =. 9 respecively. Our choices imply ha he maximum L = U L = U number of breaks allowed in he model is n max =8 and 6, respecively. Harvey e al. () does no consider his las case, alhough we included i in order o analyse he possibiliy of changes a boh ends of he sample, as in Perron and Yabu (9) and Kejriwal and Perron (). 9

11 Landajo and Presno () exended previous resuls on saionariy esing o nonlinear models which may include several endogenously deermined changes. Their approach may be applied o a wide range of models, including smooh deerminisic componens, and some non-smooh cases (e.g., breaks) may be seen as limiing cases of he considered srucures. The following error-componens model is analysed: where ( /T,θ ) y ( / T, ) ε, = µ + f θ +, T µ = µ + u ; =,..., T; T =,,... f is a smooh funcion of ime (i.e. a rend) wih θ being a vecor of free parameers, { ε } and { } E ( ) σ > and ( ) ε = ε u u are independen zero-mean error processes wih variances E σ ; { } = u µ sars wih (8) µ, which is assumed o be zero. As for he linear smooh ransiion models we consider logisic sigmoidal changes of he forms: Model I: ( / T, ) = β + β / T + δ [ + exp{ γ ( T λ) }] f θ (9) / [ { }] Model III: ( / T, ) = β + β / T + ( δ + η / T ) + exp γ ( T λ) f θ () / λ [,], γ>. λ deermines he relaive posiion of he iming of he ransiion midpoin T b ino he sample and γ conrols he speed of ransiion (gradual for small γ, and converging o a break as γ increases). So, smooh ransiion models are a very flexible class. The above specificaions allow he analysis of series affeced by a smooh change in level (Model I), and boh in level and slope (Model III). Lagrange Muliplier (LM) saionariy esing relies on he following seing: H σ u q =, H : q > () : σ ε The LM saisic o es () has he expression: T T = ˆ T E = Sˆ σ () where E = e i i= denoes he forward parial sum of he residuals of nonlinear leas squares (NLS) fiing and ˆ σ is a suiable esimaor for he long-run variance of ε }. {

12 .3. Esimaion of change locaions Finally, informaion on saionariy of he ime series can be exploied o faciliae more accurae esimaion of he break daes. Resuls by Perron and Zhu (5) show ha, in he presence of a break in slope, he esimaes of he break daes from he level specificaion are consisen irrespecive of he noise componen is saionary or has a uni roo; however, Kejriwal and Lopez () concluded via Mone Carlo simulaions ha more accurae esimaes of he break daes can be obained by esimaing a specificaion in firs differences when a uni roo is presen, and lower mean squared errors are observed when esimaing a level model in he I() case. In models wih pure level shifs, consisen esimaes of he break daes may be obained using he procedure suggesed by Harvey e al. () in he uni roo case, and by minimizing he sum of squared residuals from he level specificaion in he saionary case. 3. EMPIRICAL RESULTS This secion includes an empirical analysis of he ime series properies of non-renewable resource prices. To ensure comparabiliy beween our conclusions and previous papers we seleced series similar o hose analysed by Slade (98) and Berck and Robers (996), and idenical o he daa used by Ahrens and Sharma (997) and Lee e al. (6). Daa are annual prices for he period beween 87 and 99, deflaed by he producer price index (967=), and include aluminium, biuminous coal, copper, iron, lead, naural gas, nickel, peroleum, silver, in and zinc. In his long period of ime, non-renewable resource prices have suffered changes due, among ohers, o macroeconomic facors -such us changes in ineres raes and exchange raes-, business cycle phases -recessions and expansion periods-, or poliical evens -such us wars or hreas-, so our analysis begins wih he esimaion of he number of changes. 3.. Esimaion of he number of changes In he firs sage we applied ess o ascerain if breaks are presen. These ess are usually applied o evaluae he join significance of he inercep and slope dummies. However, in order o disinguish beween changes in level or slope, we followed a sraegy similar o he Kejriwal and Lopez () proposal. The firs sep ess for one srucural break using he Perron and Yabu (9) procedure and considering he more general Model III. A rejecion by his es can be caused by a change in level and/or slope. So, in he second sage he unresriced es (designed o deec a break in slope while allowing he inercep o shif) is

13 applied. A rejecion by his es can be inerpreed as a change in he growh rae regardless of wheher he level has changed. Given evidence in favour of a break we hen proceed o es for one versus wo breaks using he Kejriwal and Perron () es. According o he number of observaions in our analysis we allow for a maximum of wo breaks. Bai and Perron (998) and Prodan (8) poin ou ha a poenial problem associaed wih his sequenial procedure is ha single break ess may suffer from low power in finie samples in he presence of muliple breaks, especially if hey show opposie signs. So, we repor he resuls of he one versus wo breaks es independenly of he resuls from he single break es. Condiional on a sable slope in he firs sep, we focus on changes in he level of he series, applying he Harvey e al. () es in order o esimae he number of level breaks. Figure illusraes he sequence. Rejec changes Rejec Tes ( vs. breaks) Kejriwal and Perron () es Rejec Tes ( vs. break) (unresriced model) change Fail o rejec Tes ( vs. break) (general model). Perron and Yabu (9) es Fail o rejec Tes for changes in he level Harvey e al. () es Fail o rejec changes ( low power?) Figure. Sequenial applicaion of change ess. The analysis was carried ou for he linear and quadraic models. Table 4 repors resuls. Columns ExpW and ExpW(/) show figures from Perron and Yabu (9) and Kejriwal and Perron () ess respecively. Column U includes resuls from Harvey e al. () es for m=.5 (firs row) and m=. (second row), which led o idenical conclusions in all cases. Column n U repors he number of level breaks deeced from he sequenial es.

14 Conclusions abou he number of changes for he linear and quadraic models are similar, and he inclusion of a quadraic rend only allows modificaion of resuls manifesly for he coal and iron series. Also, we find one pure level shif jus in he coal series (lineal model). Aluminium, gas, silver and zinc series show clearly wo changes, boh for he linear and quadraic models, while peroleum and in have jus one. For copper, lead and nickel (for he las wo ones, only in he quadraic model) we find conradicory resuls from ExpW and ExpW(/) ess, which can be explained by low power of he ess in finie samples in he presence of muliple breaks. In hese cases of doub we will show furher resuls abou saionariy for boh models ( and breaks). 3. Saionariy analysis Once he kind and number of changes were found, we applied he saionariy ess for he linear and quadraic specificaions. For boh models, we considered wo ypes of changes: breaks and smooh ransiions. The las specificaion is of ineres since i incorporaes he possibiliy of gradual, insead of insananeous, changes and allows he nonlinear naure of he series o be capured. Tables 5 and 6 display resuls and criical values a he %, 5% and % significance levels (columns c.v.) for he saionariy ess, under he break and smooh case, respecively. 9 Column Model indicaes he specificaion considered and beween parenheses is he number of changes deeced in he previous sage. In case of doub on he specificaion (change in level and/or growh rae), we chose he more general one in order o avoid disorions in he size of he es, since he inclusion of irrelevan componens jus leads o sligh reducion in power. In order o ake residual auocorrelaion ino accoun, we used he Barle window, wih bandwidh l T seleced by using he daa-driven device proposed by Kurozumi (), wih he pre-specified values k =.5,.8,. 9 (in Kurozumi's noaion, l T = l A k ). Similar conclusions are obained for each k value. Table 6 also includes he fied λ and γ values. In conras o Lee e al. (6), who concluded ha including a quadraic rend implies few significan differences in heir uni roo analysis wih breaks, we observed ha he null of saionariy is generally favoured. In paricular, for he break case, aluminium and peroleum 9 In order o compue criical values for he saionariy es, he Mone-Carlo-based boosrap proposed by Landajo and Presno () was used (see Appendix A.3). NLS fiing of he smooh ransiion models was implemened by using Levenberg-Marquard algorihm, wih he consrains λ [,] and <γ<5 imposed on each smooh change. A preliminary grid search over, (λ, γ) pairs was carried ou before he gradien descen algorihm was iniiaed. 3

15 become saionary; and he same occurs for aluminium, coal and copper in he smooh analysis, alhough he conrary effec akes place for nickel. These changes in conclusions could be due o a decrease in size disorions as a consequence of a more suiable specificaion, alhough a facor o keep in mind is also he loss of power associaed wih he inroducion of a new deerminis componen. Resuls in Table 6 show ha esimaed speeds of ransiion are clearly low for he gas series. In his case, he specificaion of a smooh change leads o he rejecion of he null hypohesis of saionariy a lower significance levels. A similar behaviour happens for silver, copper, coal and nickel series, which are characerized by a relaively smooh/medium change esimae. For he in series he fied speed of ransiion is medium. However, in his case he saionariy es moves from rejecion o non rejecion of he null hypohesis when he more flexible smooh change specificaion is considered. A similar behaviour is observed for he peroleum series in he linear case. One explanaion for his fac may be found in Landajo and Presno (), who poin ou ha if he change is smooh and i is misspecified as a break, he saionariy es suffers size disorions which lead o incorrecly rejecing he null; on he conrary, he inroducion of a smooh ransiion in series wih a break jus leads o sligh reducions in he power of he saionariy es. In order o carry ou a confirmaory analysis, we compared our resuls wih Lee e al. (6) uni roo es conclusions for he same specificaions (linear or quadraic model and number of breaks). Their research concludes ha series are saionary in more cases han ours, which looks somewha unexpeced, since saionariy ess end o favour he null of saionariy. In general erms, conclusions from boh ess coincide excep for coal (jus for he linear model), aluminium, gas, silver and in (for boh, linear and quadraic, models) series, for which opposie conclusions are found. Cheung and Chinn (996) summarized he resuls of heir confirmaory analysis, and concluded ha conradicions due o non rejecion of saionariy ess and uni roo ess can be impued o he low power of he ess. This could be he case of he aluminium series for he quadraic model and coal for he linear one. The reverse, ha is, a rejecion for boh ess can be explained by he exisence of more complex daa generaing processes. I could be he case of aluminium (for he linear model), gas, silver and in (for boh models). We noe ha mos of hese cases are series where he esimaion of he smooh ransiion model displays a leas one low/medium value of he speed of ransiion parameer. This seems o sugges he suiabiliy of considering hese smooher and more flexible models. 4

16 [Inser Table 4] [Inser Table 5] [Inser Table 6] Due o he variey of models considered and ha some conclusions vary according o model specificaion, we oped for applying model selecion crieria for linear/quadraic and break/smooh models, considering he number of changes deeced in he firs sage. Concreely, we compued he Schwarz informaion crierion (SIC), he Akaike informaion crierion (AIC) and he adjused R-Squared. Table 7 repors resuls. Wih he excepion of zinc and iron, in all cases he quadraic model is seleced. Also, he smooh ransiion model is seleced for coal, copper, iron, gas, peroleum and in series. Exceping peroleum, all hese series have a leas one change wih low/medium speed of ransiion, and some of hem are marked as problemaic in he break-based confirmaory analysis. This appears o suppor he presence of nonlinear paerns in he relaive prices of naural resources. For he seleced models, he null of saionariy is no rejeced a 5% significance, excep for gas and silver; a % significance he null is also rejeced for lead, nickel and in. Prevalence of nonlinear feaures in relaive prices of primary commodiies is documened in he lieraure. Balagas and Hol (9) sudied he daase compiled by Pfaffenzeller e al. (7), which includes 4 relaive primary commodiy price series, among hem some non-renewable resourses: aluminium, copper, lead, silver, in and zinc. Balagas and Hol (9) conduced ess of he linear uni roo model agains models belonging o he family of smooh ransiion auoregressions (STAR). In heir analysis hey found ha he null is rejeced for mos prices of primary commodiies, among hem, all he series corresponding o non-renewable resourses. They posulae ha nonlineariy is due o impossibiliy of negaive sorage. Harvey e al. () analysed he inegraion properies of he same daase wih a differen mehodology. They generalized he Ellio e al. (996) uni roo es which considers a linear rend along he direcion of allowing for a local quadraic rend erm, and suggesed a es procedure based on a conservaive union of rejecions decision rule. In heir analysis i is confirmed he rejecion of he uni roo hypohesis wih boh ess for aluminium and zinc, and he non rejecion for silver and in; however, heir resuls are mixed for copper and lead. In hese cases, he rejecion rule leads o he rejecion of he null of uni roo, alhough jus a he marginal % for copper. Our conclusions are largely in accordance o he above references. See Balagas and Hol (9) for a review. Daa differ from ours, since hese are indices of primary commodiy prices relaive o he price of manufacures, observed annually over he period 9-3 and measured in logarihms. 5

17 [Inser Table 7] By way of summary, we find ha he series are saionary, exceping silver and gas. In he case of in, lead and nickel he rejecion is a he marginal % level significance. For he gas series, he conclusion of no saionariy is robus o all rend specificaions and confirms Pindyck (999) asserion: sae variables for coal and naural gas are esimaed o be random walks or somehing very close o a random walk. For he coal series, our resuls agree wih his conclusion for he linear case, bu no for he finally seleced quadraic specificaion. Resuls for anoher energy series peroleum- lead o non rejecion of he null of saionariy for he seleced quadraic model. Previous papers found evidence for quadraic rends in peroleum series (e.g. Slade, 98 and Lee e al., 6), and concluded ha he inclusion of breaks allows rejecing he null of uni roo (e. g. Posali and Picchei, 6 or Lee e al., 6). Conclusions for he silver series are also robus o he model specificaions considered in his research, and for all of hem we rejec he null of saionariy, confirming he efficien marke hypohesis. Xu and Fung (5) remarked ha precious meal commodiies are characerized by sandard qualiy and sorage characerisics ha enable arbirage in crossmarke fuures rading, so, undersanding wheher shocks o hese prices are persisen or ransiory has direc relevance o arbirageurs and speculaors in he commodiy rading marke. In our sudy i is confirmed ha shocks are persisen, so prices canno be prediced using hisorical daa and i is no possible for invesors o make profis using echnical analysis. A conclusion abou non-saionariy of silver series could be expeced since is price, as ha of gold, is deermined in clearly speculaive markes. Also, informaion crieria selec he break model, which is in accordance o Mainardi s (998) asserion ha precious meal price peaks are quickly followed by downward pressures brough abou by facors such as inernaional ineres raes hikes or slack world economic aciviy. In he case of in, conclusions depend on he model. For he seleced one -a quadraic rend model wih one smooh change- he rejecion of he null of saionariy is jus a he % significance level; however, he rejecion is a lower levels when a break is considered. In he case of non linear models, he uni roo es by Balagas and Hol (9) rejecs he null a he marginal % and he procedure by Harvey e al. () leads o no rejec. So, conclusions are mixed, bu a he habiual 5% significance level our resuls seem o indicae saionariy of he in series. 6

18 3.3. Esimaion of change locaions and discussion of resuls Upon he above informaion abou saionariy of he series more accurae esimaes of he change daes can be obained. According o Kejriwal and Lopez (), in he case of uni roo he change daes are esimaed from a specificaion in firs differences, while for saionariy series a model in levels is considered. Table 8 repors change poins for all he models considered in he sudy. In he case of smooh ransiion models we repor esimaions of λ and γ. As expeced, when a smooh model is seleced, a leas one of he parameers of speed of ransiion is low/medium valued. For he specificaion seleced according o informaion crieria we repor parameer esimaes of he complee model. Mos of he series show negaive slopes and he quadraic model is seleced in many cases. Kraukraemer (5) poins ou ha for mos of he wenieh cenury, naural resource commodiy price rends have been generally fla or decreasing, especially for minerals series. Since hese are non-renewable resources, one migh expec hey would be more subjec o increasing scarciy and herefore increasing prices; however mineral prices generally declined, wih he excepion of he period from 945 unil he early 98s, when many non-renewable prices (copper, iron, nickel, silver, in, coal, gas) showed an upward rend, paricularly afer he 973 embargo. Kraukraemer (5) poins ou ha his seems o mach he U-shaped price curve ha would occur as depleion exered enough upward pressure on price o overcome he downward force of echnological progress. However, he economy responds o price increases in a variey of ways: subsiuions, research and developmen, new reserves are discovered, new mehods for recovering resources or reducing he cos of using lower-qualiy reserves are found As a resul, mos mineral prices declined since he early 98s. [Inser Table 8] Nex we commen specific resuls for each resource and discuss hisorical and economic evens which ook place. Aluminium. Linear models deec a change dae which maches wih he oubreak of World War I in Europe in 94. Then, shorages of aluminium meal began o appear, and prices rose because of he increased demand for aluminium in war maerials. In March 98, he presiden of he U.S. imposed price conrols on aluminium meal, and is use for miliary equipmen and essenial civilian needs was placed under Governmen regulaion. This fac could mark he oher urning poin, deeced around he end of World War I. However, he quadraic rend models deec changes before, a he end of he 9 h cenury and in he early 9 s. The firs dae maches wih he use of innovaions, such as he Hall- 7

19 Heroul process, which led o he mass commercial producion of aluminium; as producion levels coninued o increase, producers kep he price low o encourage is use by consumers. This way, in he early 9 s, producers held aluminium meal prices a a low seady level in order o compee agains copper in he elecrical indusry and oher appliances (Plunker, 999). Copper. In his case a quadraic rend is seleced. Our resuls agree wih Slade (98), who compares linear and quadraic rends, and finds he laer provides he bes fi for copper prices. The rend is quadraic, falling for a ime and hen rising. For he seleced model, he change midpoins are deeced a 898 and 98. The firs change is quie sharp and may be relaed o he period of depression by he end of he 9 h cenury. However, in 98 he change has a medium speed. Edelsein (999) poins ou ha when he recession began in 98, world mine producion was reaching peak levels, and he resuling oversupply depressed copper prices (no sharply, according o our resuls) for 5 years. For he oher specificaions, a change in 98 is deeced, corresponding o World War I. During war ime copper was mos needed, so some sudies deec srucural breaks in he Firs, and even in he Second World War. Iron. A linear model wih wo smooh changes (wih midpoins around 874 and 956) is seleced. The firs change is quie sharp and coincides wih he Long Depression which began wih he Panic of 873. Concreely, some researches show ha he iron indusry as a whole fel he effecs of he depression beween 875 and 886. A he ime, railroadbuilding indusry and iron were closely relaed: he firs one peaked in 87 and he second one reached is highes price in 87. So, he fall in railroad consrucion in 875 enailed ha boh consumpion and prices of iron declined. Following Sürmer (), he Long Depression exhibied a negaive aggregae demand shock, which had effecs on he iron prices. The change in 956 could be relaed o he period afer he Korean War. Lead. Informaion crieria selec a quadraic model wih breaks in 947 and 98, which coincide wih previous sudies, such us Kellard and Wohar (6) or Yang e al. (). The firs change could be relaed o he period afer World War II, when oal demand for lead acceleraed wih elecronic developmens (e.g. primary elevision and video display ubes) and demand for leaded gasoline. Smih (999) poins ou ha wih he near phaseou of lead in gasoline, pains, solders, and waer sysems, and he imposiion of expensive environmenal producion conrols, he indusry experienced hard imes beween 98 and 8

20 986. Also, lead consumpion declined subsanially a he beginning of he eighies due o recession. Nickel. Models show robus conclusions abou change poins, alhough a quadraic model wih wo breaks (in 973 and 988) is finally seleced. Sainless seel producion and nickel prices are closely relaed. In fac, in he lae 99 s, sainless seel producion accouned for more han 6% of world nickel consumpion and was he primary facor in nickel pricing (Kuck, 999). So, nickel prices, reflecing consumpion, rose slighly from 97 unil 975, when he cumulaive effec of opening several new producion faciliies began o be fel. In 975, U.S. demand for nickel weakened, parly because of he erminaion of miliary operaions in Vienam and he crisis of he early 97 s. In he eighies, price peaked in 988 and declined aferwards. Kuck (999) poins ou hree facors which were responsible for his increase: he subsanial and unforeseen increase in demand for sainless seel, reducion in world producion capaciy because of low meal prices during he early and mid-98 s and he decreased availabiliy of sainless seel scrap. Silver. All he models deec changes a he beginning of he 8 s. I coincides wih he excepional speculaive movemen occurring in , when he Hun brohers aemped o corner he silver marke. Parameer esimaes show a sharp increase followed by a negaive slope. This fall in prices afer he break poin appears as a reacion o he high levels obained beforehand. Tin. A smooh change wih he midpoin in 976 is deeced 3. Carlin (999) remarks ha during he 9-year run of he in agreemens ( ), he Inernaional Tin Council suppored he price of in by buying and selling in from is buffer sockpile; however, he buffer sockpile was no sufficienly large, mainly o defend he arificial ceiling prices, and in prices rose, especially from 973 hrough 98 when rampan inflaion plagued he American and many foreign economies. 976 is precisely he midpoin of his period of increase in prices. This dae is roughly in accordance o Lee e al. (6) -who deec a break in 974 which hey relae o he energy crisis of he early 97s-, and Kellard and Wohar (6) -in As Kellard and Wohar, we observe ha afer an iniial rachering up of prices, he rend slope is negaive. As in he case of silver, his fall in prices seems o be a reacion o he previous high levels. Sürmer () ascribes he increase of prices o a 3 As remarked previously, resuls from saionariy esing in in series are mixed, so we also esimaed he change poin under non-saionariy for he seleced model, finding a change midpoin in 977, which is very similar o he one found in he saionary case. 9

21 specific demand shock ha migh be explained by susained purchases of in by he inernaional buffer sock unil is collapse in 985. Zinc. Changes are deeced around World War I, and are relaed o he imporance of zinc in he war indusry. Wih he onse of World War I in 94, he demand for zinc ammuniion producs -bronze and brass shell casings- ripled he value of zinc. However, as he war ended so did he boom. For energy series (peroleum, coal and naural gas), informaion crieria selec he quadraic model, which is in accordance o Pindyck (999) specificaion. He poins ou ha a quadraic U-shaped rend line is consisen wih models of exhausible resource producion ha incorporae exploraion and accumulaion of proved reserves over ime, as well as echnological change. Coal. The seleced model esimaes a smooh change a he beginning of he 96 s and a fas one in 974. The Sudy of Coal Prices by he U.S. governmen found ha he second change was due o he OPEC oil embargo sared in December 973, which raised prices for subsiues (coal and naural gas). Oher facors were he anicipaion of he Unied Mine Workers' srike during he second half of 974 and he coninuing increase in labor coss, a rend which had begun in 97. Following Ellerman (994), he rends in he price of coal roughly coincide wih he dominan realiies of he indusry: loss of marke share o oil and naural gas during he 95s and 96s, rising crude oil prices in he 97s, and over-capaciy in he 98s. He poins ou hese condiions probably conribued o he observed changes in price, bu changing produciviy is he major explanaory facor for he rend in coal prices in U.S. since World War II. Gas. Differen changes are found depending on he specific srucures (breaks/smooh ransiions) considered. The break model capures he firs oil shock and he period of he full liberaion of he U.S. gas marke. From 954 o 978, he price of naural gas ranspored hrough he inersae pipeline sysem was regulaed by he Federal Power Commission, and prices changed very lile from year o year. A parial deregulaion of wellhead prices occurred wih he Naural Gas Policy Ac in 978. Bu in he meanime, prices began o rise in he mid-97s, a period of urmoil in inernaional energy markes as a resul of he firs oil shock. So, when full de-regulaion finally became effecive in 985, gas prices had risen, creaing a long-erm marke surplus, he gas bubble. Prices suqsequenly rereaed. Saring

22 in 984, he Federal Energy Regulaory Commission iniiaed a series of orders 4 inended o resrucure he buy-sell relaionship among producion, ransmission and disribuion companies. Lee and Lee (9) also deec srucural breaks in many counries around 985, which hey relae o he 985 crash in oil prices. Peroleum. Informaion crieria selec a quadraic model. This is in accordance wih Ahrens and Sharma (997), Lee e al. (6) and Li and Thompson (), who emphasized he imporance of allowing for a nonlinear specificaion when examining he ime series properies of oil price. All he models deec a change in 98-98, coinciding wih he Iranian Revoluion and Iran-Iraq War, which caused oil prices o peak owards he end of 98 as well as imporan effecs on he world economy 5. Kilian (9) poins ou ha he increase in he real price of oil afer 979 appears o be driven mainly by he superimposiion of a sharp increase in precauionary demand in 979 (due o he poliical uncerainy in he Middle Eas: Khomeini s arrival in Iran, he Iranian hosage crisis and he Sovie invasion of Afghanisan) on a slower-moving srong increase in real economic aciviy ha sared wo years earlier, wih only minor conribuions from oil supply shocks. Subsequenly world peroleum consumpion declined in he early 98s due mainly o he developmen of new echnologies, oil subsiuion by oher energies (especially in power generaion) and more efficien energy use. Alhough Saudi Arabia shu down producion in , he nominal and he real price of oil declined significanly; in 986, he Saudis abandoned hose effors, causing he price of oil o collapse. Parameer esimaes in Table 8 agree wih hese evens, and show ha afer a large increase, oil prices experienced a negaive rend, he adjusmen being relaively fas. 4. CONCLUSIONS This paper employs a powerful ime series mehodology in order o invesigae he sochasic properies and he change poins of a daabase of nonrewable resource real prices. For his we have considered four model specificaions which combine linear and quadraic rends wih poenial breaks and smooh ransiions beween regimes. 4 For insance, order no. 38 released uiliy buyers (such as he local disribuion companies) from he commimen o purchase he ransporaion capaciy hey reserve; as Davous (8) shows, his led o a deep fall in average wellhead prices. 5 In fac, he Naional Bureau of Economic Research characerizes he economic difficulies a his ime as being wo separae economic recessions, wih he firs one, due o he Iranian Revoluion ending in July 98, bu followed very quickly by a new shock beginning in July 98 wih he Iran-Iraq War.

23 Our resuls indicae ha mos of he series are saionary, bu here are wo clear excepions: naural gas and silver. This fac means ha real price shocks on hese resources are mosly permanen in naure, following ha heir markes are efficien in he weak sense, so prices canno be prediced using hisorical daa and i is no possible for invesors o make profis using echnical analysis. Also, boh resources share he characerisic ha heir prices were regulaed and afer he deregulaion suffered a bubble : in 98, when he Hun brohers ried o corner he silver marke and in he mid-eighies, when full de-regulaion in gas marke finally became effecive. In conras o ohers papers, we find ha oil prices are saionary around a quadraic rend. As Pindyck (999) noed, his is consisen wih crude oil being sold in a compeiive marke where prices rever o long-run marginal cos, which changes only slowly. In his case, echnical analysis is useful in order o predic prices and making profis. Demarcaion beween saionary and non-saionary resource real prices series also has implicaions in erms of sabilizaion policies. Reinhar and Wickham (4) argue ha design and feasibiliy of sabilizaion and hegding sraegies depend very much on he naure of shocks. Boh are useful in dealing only wih emporary and, preferably, shor-lived shocks, while permanen shocks require adjusmen and, possibly, he implemenaion of srucural policies. Also, if a shock is emporary, bu is effecs are widespread and persis for many years or he price series has a varying rend, price sabilizaion may be equally cosly and difficul o implemen. From our analysis, silver and gas are difference saionary and so sabilizaion policies would be ineffecive, and for he res of series, hese policies may be difficul o implemen. This fac would explain, for insance, he gradually selling off of he in buffer sock (see Ghoshray, ). Informaion abou sochasic properies of he series also allowed more accurae esimaes of he change daes in prices, which are relaed o economic and hisorical evens. Following Kilian (9), some of he changes could be relaed o supply, aggregae demand and specific demand shocks. An example of he firs one is he negaive shock o supply which increased he real price of oil a he beginning of he eighies. Aggregae demand shocks are characerized for being relaively similar for all he resources. Insances include he Long Depression, World Wars I and II and he subsequen reconsrucion, he indusrial expansion of Souh Korea and Japan in he 96s or he recessions in 974 and 98, and hey seem o explain changes in copper, iron, lead or zinc prices. Specific demand shocks evolve differenly across markes, and include precauionary demand, demand shocks due o he spread of echnological innovaion or non-lineariies in he inensiy of use. Sürmer ()