RUHR. Gold Price Forecasts in a Dynamic Model Averaging Framework Have the Determinants Changed Over Time? ECONOMIC PAPERS #506

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1 RUHR ECONOMIC PAPERS Dirk G. Baur Joscha Beckmann Rober Czudaj Gold Price Forecass in a Dynamic Model Averaging Framework Have he Deerminans Changed Over Time? #506

2 Imprin Ruhr Economic Papers Published by Ruhr-Universiä Bochum (RUB), Deparmen of Economics Universiässr. 150, Bochum, Germany Technische Universiä Dormund, Deparmen of Economic and Social Sciences Vogelpohsweg 87, Dormund, Germany Universiä Duisburg-Essen, Deparmen of Economics Universiässr. 12, Essen, Germany Rheinisch-Wesfälisches Insiu für Wirschafsforschung (RWI) Hohenzollernsr. 1-3, Essen, Germany Ediors Prof. Dr. Thomas K. Bauer RUB, Deparmen of Economics, Empirical Economics Phone: +49 (0) 234/ , Prof. Dr. Wolfgang Leininger Technische Universiä Dormund, Deparmen of Economic and Social Sciences Economics Microeconomics Phone: +49 (0) 231/ , Prof. Dr. Volker Clausen Universiy of Duisburg-Essen, Deparmen of Economics Inernaional Economics Phone: +49 (0) 201/ , Prof. Dr. Roland Döhrn, Prof. Dr. Manuel Frondel, Prof. Dr. Jochen Kluve RWI, Phone: +49 (0) 201/ , Ediorial Office Sabine Weiler RWI, Phone: +49 (0) 201/ , Ruhr Economic Papers #506 Responsible Edior: Volker Clausen All righs reserved. Bochum, Dormund, Duisburg, Essen, Germany, 2014 ISSN (online) ISBN The working papers published in he Series consiue work in progress circulaed o simulae discussion and criical commens. Views expressed represen exclusively he auhors own opinions and do no necessarily reflec hose of he ediors.

3 Ruhr Economic Papers #506 Dirk G. Baur, Joscha Beckmann, and Rober Czudaj Gold Price Forecass in a Dynamic Model Averaging Framework Have he Deerminans Changed Over Time?

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5 Dirk G. Baur, Joscha Beckmann, and Rober Czudaj 1 Gold Price Forecass in a Dynamic Model Averaging Framework Have he Deerminans Changed Over Time? Absrac The price of gold is influenced by a wide range of local and global facors such as commodiy prices, ineres raes, inflaion expecaions, exchange rae changes and sock marke volailiy among ohers. Hence, forecasing he price of gold is a nooriously difficul ask and he main problem a researcher faces is o selec he relevan regressors a each poin in ime. This combinaion of model and parameer uncerainy is explicily accouned for by Dynamic Model Averaging which allows boh he forecasing model and he coefficiens o change over ime. Based on his framework, we sysemaically evaluae a large se of possible gold price deerminans and use boh he predicive likelihood and he mean squared error as a measure of he forecasing performance. We carefully assess which predicors are relevan for forecasing a differen poins in ime hrough he poserior probabiliy. Our findings show ha (1) DMA improves forecass compared o oher frameworks and (2) provides clear evidence for he imevariaion of gold price predicors. JEL Classificaion: C32, G10, G15, F37 Keywords: Bayesian economerics; dynamic model averaging; forecasing; gold Sepember Dirk G. Baur, Kuehne Logisics Universiy, Hamburg; Joscha Beckmann, UDE and Kiel Insiue for he World Economy; Rober Czudaj, UDE. Thanks for valuable commens are due o he paricipans of he 12h INFINITI Conference on Inernaional Finance, Prao/Ialy, he 10h BMRC-DEMS Conference on Macro and Financial Economics/Economerics, London/UK, and he roaing lecure of he Ruhr Graduae School in Economics, Essen/ Germany. All correspondence o: Rober Czudaj, Universiy of Duisburg-Essen, Deparmen of Economics, Chair for Economerics, Essen, Germany, rober.czudaj@uni-due.de

6 1 Inroducion There is now a large and growing academic lieraure on he financial economics of gold. Mos sudies focus on one of he key properies of gold, e.g. he inflaion hedge propery of gold (e.g. Blose, 2010, Jasram, 2009, and Beckmann and Czudaj, 2013), he currency hedge propery of gold (e.g. Capie e al., 2005 and Sjaasad and Scacciavillani, 1996) or he safe haven propery of gold (e.g. Baur and Lucey, 2010) bu do no enerain gold price forecass. 1 Ineresingly, here is no esablished forecasing model for gold prices and no framework ha simulaneously considers and evaluaes all poenial facors and heir dynamics. This paper aims o fill his significan gap in he lieraure by adoping he Dynamic Model Averaging approach which is well-suied for forecasing under condiions of uncerainy. The quesion wheher poenial deerminans of he gold price are also useful in a ime-varying forecasing framework is of grea ineres for a broad audience. A significan in-sample, i.e. conemporaneous and hus non-predicive, relaionship beween he gold prices and paricular deerminans over a specific period does no provide a guaranee for success in erms of ou-of-sample predicabiliy. Hence, evaluaing he evoluion of gold price forecass based on our framework offers an ineresing perspecive for researchers and marke paricipans. Overall, a researcher who aims a forecasing gold prices faces a variey of issues: Firsly, he deerminans of gold poenially change over ime resuling in model uncerainy, i.e. he se of variables ha deermine he price of gold varies. This can be illusraed by sudying he dynamics of he gold price over he las decades: Having mosly flucuaed around a consan value in he 1980s and 1990s, he rapid increase observed in he 2000s migh be aribued o variables which were no considered as imporan deerminans up o ha poin in ime. The ask is furher complicaed by he fac ha deerminans migh be imporan hroughou he sample bu o a differen degree, resuling in parameer uncerainy. I is, for example, well esablished, ha he impac of sock marke movemens on gold varies over ime and is generally higher during imes of urbulence. Inflaion or inflaion 1 Baker and Van Tassel (1985) and Pierdzioch e al. (2014) are noable excepions. The laer also ry o accoun for model and parameer uncerainy, however hey only consider six differen predicors and apply model selecion and aggregaion mehods ha are based on recursive and rolling-window OLS esimaes. The selecion of he opimal forecas model and he weighing of several forecas models solely depend on he in-sample fi based on informaion crieria. The aspec which makes our forecass superior compared o simple selecion and aggregaion mehods is ha our model weighs are based on he forecasing performance of he models in previous periods. 4

7 expecaions may also be relevan only a cerain periods of ime, e.g. when inflaion expecaions exceed a specific hreshold. In summary, a researcher faces boh uncerainy abou he variables o include in a forecasing model - model uncerainy - and uncerainy abou he ime-variaion of he parameers - parameer uncerainy. Rafery (1995) describes a common approach o accoun for model uncerainy by firs fiing he full model, screen he -saisics for he parameers, remove he variables for which hese are small and hen re-esimae he resuling, reduced, model. Rafery (1995) argues ha researchers ofen proceed using he seleced model as if i were he only model ha had ever been considered. By choosing among a large number of models one increases he likelihood of finding significan variables by chance alone. 2 An inuiive soluion o avoid his model uncerainy is he combinaion of differen models. An ad-hoc approach is o use a simple average of differen forecass. More sophisicaed frameworks rely on Bayesian echniques, explicily accouning for a large degree of uncerainy by assigning probabiliies o each model condiional on previous forecasing performance. However, due o he need of deermining a ransiion marix Bayesian inference can suffer from a compuaional burden if many regressors are included. This becomes obvious if m regressors are considered resuling in 2 m possible model combinaions. In oher words, a forecasing model wih m = 14 variables resuls in 16, 384 model combinaions. If he forecasing model is allowed o change a each poin in ime he number of combinaions of models which mus be esimaed is even much higher. Agains his background, we adop he Dynamic Model Averaging (DMA) approach inroduced by Rafery e al. (2010) and applied by Koop and Korobilis (2012) for forecasing inflaion. Such a framework has he advanage ha boh he forecasing model and he coefficiens in each model are allowed o change over ime. Anoher advanage is ha a direc comparison wih forecass based on he bes model a a cerain poin in ime according o Dynamic Model Selecion (DMS) can be drawn. From an economic poin of view, he imporance of each predicor over ime can be analyzed by disenangling he dynamics of all underlying models simulaneously. A key ingredien is ha a model will have more weigh a ime if i has forecased well in he period prior o. In oher words, sequenial learning is a key par of he forecasing procedure. Our main conribuion is manifold. We analyze wheher forecass based on Dynamic Model Averaging 2 Among ohers, Deckers and Hanck (2014) have called aenion o his mulipliciy problem. 5

8 are able o sysemaically ouperform he random walk for forecasing he price of gold. We also address he imporan quesion wheher more parsimonious models should be preferred for forecasing. Finally, we sysemaically analyze he imporance of differen gold price predicors over ime. In doing so, we examine how he weighs aached o regressors evolve over ime and wheher model averaging or model selecion provides superior resuls in erms of he forecasing performance. We consider Bayesian and classical crieria o assess he adequacy of he obained forecas. This paper is srucured as follows: Secion 2 presens he Dynamic Model Averaging framework. Secion 3 inroduces he daa used in he esimaion. Secion 5 describes and discusses he esimaion resuls and Secion 6 summarizes he main findings and provides concluding remarks. 2 Mehodology In our forecasing seup we consider K differen models, M 1,...,M K, which are given by differen subses of possible predicors x =(x j : j =1,...,m), 3 and we allow for he uncerainy which model is he bes a each poin in ime. We apply a sae space model which comprises an observaion and a sae equaion given by y = x (k) θ (k) θ (k) + ε (k), (1) = θ (k) 1 + δ(k), (2) where y denoes he endogenous variable (i.e. he price of gold) and θ (k) parameers for k =1,...,K. The measuremen noise ε (k) noise δ (k) is a vecor of regression is disribued as N(0,V (k) ) and he process as N(0,W (k) ). Boh he sae vecor θ (k) as well as he predicor vecor x (k) are differen for each model and he quaniies belonging o M k are represened by he superscrip (k). For k =1 Eq. (1) and (2) can be referred o as a sandard sae space model, which can be esimaed adapively wih sandard Kalman filering. 4 If he process is governed by model M k a ime, henl = k. The evoluion of he model changes can be deermined by a K K ransiion probabiliy marix P,wherep k,l = P [L = l L 1 = k] denoes 3 x includes he ime h informaion se and also includes an inercep. This means ha x conains he predicors observed a 1, when using a forecas horizon of h =1. 4 See Rafery e al. (2010) for deails. 6

9 is elemens. Since K =2 m, he dimension of his marix depends on he number of predicors m and unless m is very small, P will be an enormous marix. Hence, inference will be imprecise and he compuaion will ake much ime (Koop and Korobilis, 2012). Therefore, we avoid hese problems by he use of an approximaion proposed by Rafery e al. (2010) ha includes a forgeing facor. This will be illusraed furher below. In our model seup he underlying sae is characerized by he pair (Θ,L ), where Θ =(θ (k) : k = 1,...,K), and he probabiliy disribuion of (Θ,L )is p(θ,l )= K k=1 which will be updaed each ime as new daa becomes available. p(θ (k) L = k)p(l = k), (3) The esimaion consiss of a predicion and an updaing sep. Suppose a firs ha we know he condiional disribuion of he sae a ime 1 given he ime 1 informaion se Y 1 = y 1,...,y 1 given by p(θ 1,L 1 Y 1 )= K k=1 p(θ (k) 1 L 1 = k, Y 1 )p(l 1 = k Y 1 ). (4) The condiional disribuion of θ (k) 1 in Eq. (4) can be approximaed as follows θ (k) 1 L 1 = k, Y 1 N(ˆθ (k) 1, Σ(k) 1 ). (5) Then he predicion sep is a wo-sep echnique: firs, we use he model predicion equaion o predic he model indicaor L and second, we use he parameer predicion equaion for a condiional predicion of he parameer θ (k) given L = k from he firs sep. The model predicion equaion is π 1,k P [L = k Y 1 ]= K π 1 1,l p k,l. (6) As menioned above, insead of specifying he ransiion probabiliy marix P, we use an approximaion ha replaces Eq. (6) by π 1,k = l=1 π 1 1,k α K. (7) l=1 πα 1 1,l α is he forgeing facor menioned above wih 0 <α 1, his will ypically be slighly less han 7

10 uniy. The advanage of his approximaion applying a forgeing facor is ha i does no require an Markov Chain Mone Carlo (MCMC) algorihm o draw he ransiion beween he models, since we only need informaion abou π 1 1,k and π 1,k, bu no abou P. 5 Then he parameer predicion equaion is θ (k) L = k, Y 1 N(ˆθ (k) 1,R(k) ), (8) where R (k) =Σ (k) 1 + W (k). (9) However, same as for P, o specify each m m marix W (k) is compuaionally infeasible. Therefore, we again use an approximaion proposed by Rafery e al. (2010) ha includes a facor of forgeing. Hence, we replace Eq. (9) by R (k) = λ 1 Σ (k) 1, (10) where λ denoes he forgeing facor wih 0 <λ 1 ha is also ypically slighly below uniy. Therefore, W (k) =(λ 1 1)Σ (k) 1. The forgeing facor ensures ha daa which is lagged by i ime periods, ges he weigh λ i. If we se λ =0.99 and use monhly daa, his means ha he observaions hree years ago receive around 70% as much weigh as he observaion of he las period. Hence, he idea is similar o applying a rolling window regression wih a window size of 1 1 λ.6 The predicion sep is followed by he updaing sep, which also consiss of wo seps: model updaing 5 The erm forgeing facor suggess ha more recen informaion is more relevan han informaion more disan in he pas and ha his is independen of good or bad forecass. However, he forgeing facor α implies ha a model will receive more weigh a ime, if i has produced a good forecas in he recen pas, where he forecasing performance is measured by he predicive densiy given in Eq. (11). 6 The forgeing facors are of crucial imporance: In a sense, he key quesion is wheher an approach based on previous esimaion errors is superior o simpler frameworks ha are based on consan coefficiens. The aracive feaure of forgeing facors in his conex is ha hey allow conrolling he degree of insabiliy in he coefficiens. This is imporan since i is unclear wheher rapidly changing coefficiens are useful even if a high uncerainy regarding he relevan regressors or he corresponding elasiciy exiss. The reason is ha fas changing coefficiens migh inflae esimaion errors. Comparing resuls based on differen forgeing facors offer a possible assessmen of his issue. 8

11 and parameer updaing. 7 Firs, he model updaing equaion is π,k = π 1,k f k (y Y 1 ) K l=1 π 1,lf l (y Y 1 ), (11) where f l (y Y 1 ) denoes he predicive densiy of model l evaluaed a y given by N(x (l) ˆθ (l) 1,V(l) + x (l) R (l) x (l) ). In order o esimae he observaion innovaion variance V (l) for each model, we apply a rolling window version of he recursive mehod of momens esimaor suggesed by Rafery e al. (2010): ˆV (l) = 1 j= +1 [ ( y x (l) ˆθ (l) 1 ) 2 x (l) R (l) x (l) ], (12) where denoes he window size and has been se o 12 (i.e. one year) in our applicaion. Using a rolling window is moivaed by he fac ha he error variance is subjec o change over ime and a rolling window allows a beer approximaion of his changing paern han a simple recursion. Second, parameer updaing is done via θ (k) L = k, Y N(ˆθ (k), Σ (k) ), (13) where wih e (k) ˆθ (k) = (k) ˆθ 1 + R(k) x (k) (V (k) + x (k) R (k) x (k) = y x (k) ˆθ (k) 1 as he one-sep-ahead predicion error of model k, and ) 1 e (k) (14) Σ (k) = R (k) R (k) x (k) (V (k) + x (k) R (k) x (k) ) 1 x (k) R (k). (15) Noe ha R (k) x (k) (V (k) +x (k) R (k) x (k) ) 1 is he Kalman gain or he blending facor which minimizes he poserior error covariance. More precisely, if he observaion innovaion variance V (k) hen he acual observaion y is rused more han is predicion x (k) is zero, ˆθ (k) 1. On he oher hand, if he prior esimae for he error covariance R approaches zero, hen he acual observaion y is rused less and less while is predicion x (k) ˆθ (k) 1 is rused more and more. 7 Such a sequenial learning procedure offers a greaer degree of flexibiliy han adoping simple averages. As will be oulined below, he case of equal weighs across all models is embedded in our approach and serves as a prior assumpion for all models. 9

12 The process given by Eq. (11) and (13) is ieraed each ime new daa becomes available and i is iniialized by using he noninformaive prior π 0 0,k =1/K for k =1,...,K, and a relaively diffuse prior on he parameers: θ (k) 0 N(0, 100). Then he one-sep-ahead predicion of y (i.e. he price of gold) is given by ŷ DMA = K k=1 π 1,k ŷ (k) = K k=1 π 1,k x (k) ˆθ (k) 1. (16) Hence, he gold price predicion is a weighed average of he model-specific predicions ŷ (k),where he weighs are given by he poserior predicive model probabiliies π 1,k a each poin in ime. 8 These probabiliies can be used o evaluae he imporance of a model and herefore of a predicor ha is included in his model over ime. Using his framework we are able o analyze if and how he gold price deerminans have changed over ime. 3 Daa There is no esablished forecasing model for he price of gold. However, here are several facors ha are viewed o significanly influence he price of gold. These facors can be derived from he properies gold is generally associaed wih, i.e. he inflaion hedge and sore of value propery, he safe haven propery, he currency hedge propery and he porfolio diversifier propery. The variables ha are commonly used o empirically es hese properies are consumer price indices, sock prices, exchange raes and foreign currency reserves. Whils exchange raes represen he currency hedge propery, he combinaion of exchange raes and foreign currency reserves represens he role of gold as a diversifier for cenral bank foreign currency reserves. We also include ineres raes o analyze he cos of carry (e.g. Blose, 2010) and commodiy price indices and silver as poenial sysemaic facors of gold. 9 I is worh menioning ha some of he well-esablished properies of gold, such as he inflaion hedging funcion, depend on he sample period under invesigaion (e.g. Baen e al., 2014). This provides furher evidence for he imporance of parameer insabiliy. Overall, our framework enables us o analyze wheher some of he well-esablished (non-predicive) in-sample gold price deerminans such as he currency hedge are useful for forecass of he gold price. Theoreically, 8 For differen forecas horizons h Eq. (16) can be generalized o ŷ DMA = K k=1 π h+1 h,kx (k) ˆθ (k) h. 9 Escribano and Granger (1998) analyzed he co-movemen of gold and silver in a co-inegraion framework. 10

13 i is well possible ha even a perfec in-sample fi implies no ou-of-sample predicabiliy (see also Rossi, 2005). We use (i) he MSCI world sock price index and (ii) he S&P500 composie price index as represenaive sock price indices, (iii) a GARCH(1,1) process of he reurns of he MSCI world index as a measure of sock price volailiy, 10 (iv) he S&P GSCI commodiy price index, (v) he CRB commodiy price index, (vi) he price of silver, (vii) he US consumer price index, (viii) a global composie price index, (ix) he US dollar rade-weighed index, (x) he euro rade-weighed index, (xi) US 3-monh Treasury bill and (xii) US 10 year Treasury bond yields, (xiii) he Barclays Capial US aggregae bond index, and (xiv) a global foreign currency reserves index. The price of gold is he 3pm London fixing price denominaed in US dollars. All oher variables are also denominaed in US dollars if applicable. We assume ha changes in he price of gold y depend on a combinaion of he above facors as follows where x (k) y = x (k) θ (k) + ε (k), (17) is a vecor of variables observed a h ha defines model k. The daa is obained from Thomson Reuers Daasream for he 40-year sample period from 1975 o 2014 a a monhly frequency. Due o saionariy consideraions each series is aken as log-firs-difference. Ineres raes are he only excepion; hese are aken in firs differences. Figure I illusraes he evoluion of he price of gold and gold reurns over he sample period. The ime-series plos demonsrae significan variaions of he price of gold in he lae 1970s and a clear bull marke from The srong flucuaions during he mid-sevenies can be aribued o he he breakdown of Breon Woods and he firs major oil price shock. *** Inser Figure I abou here *** 10 We have also considered he CBOE volailiy index (VIX), however he corresponding ime series sars in 1986 and would herefore resric our sample period. Thus, we make use of a GARCH(1,1) process of he reurns of he MSCI world index as a proxy for he VIX, since he correlaion beween boh series is fairly high. 11

14 Figure II presens he ime-series of gold reurns exceeding 5% and 10% hresholds for boh negaive and posiive reurns. The graphs demonsrae ha volailiy of gold reurns clusers and ha he volailiy is relaively high around 1980 and The 10% graph provides paricularly srong evidence for he clusering wih almos no large absolue reurns presen in he 20-year period beween 1985 and *** Inser Figure II abou here *** The srong variaion in gold reurns and gold prices suggess ha he facors influencing gold prices also vary over ime. 4 Forecas Performance The usefulness of gold price forecass can be assessed based on differen crieria: he simples benchmark includes a naive random walk forecas. Regardless of heir simpliciy, random walk forecass have urned ou o be a ough benchmark o bea when i comes o forecasing financial variables such as exchange raes (Meese and Rogoff, 1983) or sock prices where even sophisicaed models and mehods frequenly fail o provide superior forecas. Table I repors some forecas error saisics a differen forecas horizons (h =1, 3, 12) for our DMA and DMS model in comparison o several alernaives including he simple random walk and oher model averaging echniques, where eiher one or boh forgeing facors are fixed. Anoher simple bu useful benchmark is he hi raio which corresponds o he percenage of correc direcion forecass. Such a measure is useful wihin porfolio decisions where he direcion raher han he magniude of he forecas maers. In addiion, we do no only include classical forecas error crieria such as he mean absolue forecas error (MAFE) or he roo mean squared forecas error (RMSFE), bu we also consider a Bayesian crieria: he sum of he log predicive likelihood (log(pl)). This has he advanage ha we do no only evaluae he poin forecass bu also consider he enire predicive disribuion f l (y Y 1 ). 12

15 *** Inser Table I abou here *** Overall, i becomes eviden ha a forecaser is beer off using our flexible approach, which accouns for boh model and parameer uncerainy. More precisely, Table I reveals ha DMA or DMS wih forgeing facors of a leas 0.95 provides more accurae forecass compared o less flexible alernaives such as BMA or DMA wih a forgeing facor fixed o uniy, i.e. no forgeing. In addiion, for a forecas horizon of one monh he sign of he gold reurn is prediced correcly in a leas 50% of he cases while less flexible alernaives perform slighly worse. In addiion, DMS seems o be superior compared o DMA for a slower forgeing of he pas (e.g. forgeing facor of 0.99) while DMA appears o be beer for a faser rae of forgeing (e.g. values of 0.95 or 0.90). Our overall evidence, ha our approach is (1) clearly superior o a simple random walk and (2) flexible enough o ouperform model averaging echniques ha do no imply any forgeing (BMA) or do no allow for model and parameer uncerainy a he same ime, also holds for higher forecas horizons. This is suppored by boh classical and Bayesian crieria. In he following secion we discuss he poserior inclusion probabiliies achieved by our approach wih a relaively slow rae of forgeing (α = λ =0.99) in order o show ha he imporance of several gold price deerminans has changed over ime. 5 Time-Varying Imporance of Gold Price Deerminans Figure III shows he ime-varying number of predicors (including he consan) included in he model aached o he highes inclusion probabiliy. The graph provides clear evidence ha shrinkage is needed o provide accurae gold forecass. In mos cases, wo or hree regressors urn ou o be he opimal choice. Once again, he high-volailiy periods a he beginning and he end of he sample urn ou o be excepions. While frequen swiches are observed a he beginning of he sample unil he sar of he eighies, he model also briefly includes six or seven regressors hree imes afer I is quie remarkable ha he algorihm drops down o hree regressors almos immediaely a each poin in ime. This demonsraes he flexibiliy and he learning of he DMA approach: More regressors are included o accoun for unexpeced changes bu a more parsimonious model is seleced quickly aferwards. 13

16 *** Inser Figure III abou here *** Figures IV-VI provide he probabiliy ha a predicor is useful for forecasing a ime based on he weigh aached by DMA o models which include he corresponding regressor. An inspecion of he differen graphs suggess ha he inclusion probabiliy always exceeds 0.5 during a leas one poin in ime, suggesing ha each variable is imporan a some sage over he sample period. Ineresingly, he imporance of some variables such as he US consumer price index significanly decreases afer he firs years while oher variables such as silver prices become increasingly imporan as ime evolves. The overall high volailiy of he probabiliies during he beginning of he sample can be raced back o he economic urbulence afer he firs oil price shock. 11 Many probabiliies also show significan changes around he Millennium. Taking Figure II ino accoun, i becomes eviden ha periods of large absolue gold reurns coincide wih hose changes in he inclusion probabiliies. This demonsraes he flexibiliy of our forecasing approach since changing paerns of he gold price are refleced in changes of he weighs aached o he differen regressors. Furhermore, some regressors are favored once volailiy of he gold price increases bu he probabiliies decrease again shorly aferwards, illusraing he feaure of fas changing weighs. We have already shown in he previous secion ha his flexibiliy pays off in erms of forecas accuracy compared o oher averaging echniques. Figure IV presens he poserior inclusion probabiliies for he MSCI world sock marke index, he S&P500 composie sock marke index, he GARCH(1,1) of he MSCI sock reurns, he S&P GSCI commodiy price index, and silver prices. The sock marke indices boh exhibi inclusion probabiliies clearly below 0.5 on average wih he S&P500 index displaying larger bu more volaile inclusion probabiliies han he MSCI world index afer The inclusion probabiliies provide weak evidence for he imporance of sock markes o forecas gold prices in general. There is also no significan evidence for an increased inclusion probabiliy during financial crisis or urmoil consisen wih gold s safe haven saus. More specifically, posiive changes of he inclusion probabiliies around he Asian financial crisis and he Russian crisis in 1997 and 1998 and he urmoil following he Lehman 11 However, i is worh noing ha he probabiliies a he very beginning of he sample period should be inerpreed wih cauion, since hese could depend on he choice of he iniial condiions, i.e. noninformaive prior on he model probabiliies. 14

17 Brohers bankrupcy in 2008 are small and insignifican. The picure is differen for he esimaed GARCH volailiy of he MSCI sock index reurns. The inclusion probabiliy is significanly higher compared o he reurn series and increases rapidly beween 2000 and The GSCI commodiy price index exhibis inclusion probabiliies close o one around 2006 bu say below 0.5 for mos pars of he sample. The resuls for he CRB display he same paern bu are more volaile a he beginning of he sample and are close o zero afer. The inclusion probabiliies for silver prices are mosly higher and more volaile wih relaively high values beween 1995 and 2005 bu values below 0.5 a he end of he sample period. The resuls for sock and commodiy prices confirm ha sysemaic facors are only parly imporan o forecas gold prices. The overall paern for sock prices suggess ha in-sample (nonpredicive) safe haven or hedge funcions of gold agains socks which have been frequenly esablished over he period under invesigaion are no necessarily relaed o ou-of-sample predicive power. In addiion, he safe haven effec is ofen more pronounced on a shor-erm basis and no significan for monhly daa. These periods may hus be oo shor o gain weigh in he inclusion probabiliies of our monhly daase. *** Inser Figure IV abou here *** *** Inser Figure V abou here *** Figure V presens he poserior inclusion probabiliies for he US consumer price index (CPI), a global (world) consumer price index as well as US long-erm and shor-erm ineres raes. The inclusion probabiliies of he US CPI indicae ha consumer prices have played a big role during he high inflaion regimes in he 1970s and an insignifican role during he grea moderaion period beween 1980 and mid-2000, in which inflaion raes have been relaively low. The inclusion probabiliies have increased from values of zero in 2000 o values close o 0.4 in This increase lends weigh o he view ha quaniaive easing afer he financial crisis in 2008 implies higher inflaion expecaions. In conras, he global price index exhibis relaively large inclusion probabiliies in he 1980s and 15

18 reduced and more volaile probabiliies in he 1990s consisen wih he lower inflaion raes and expecaions during he grea moderaion. Remarkably, he inclusion probabiliies increase around 2005 and are slighly above 0.5 for he pos-2008 crisis period. These findings suppor he inflaion hedge propery of gold a a global level and demonsrae ha gold is raher influenced by world price levels han by US price levels. 12 We analyze price levels and ineres raes in one graph as hese series generally co-move as prediced by he Fisher effec. We focus on US ineres raes as a global benchmark. The inclusion probabiliies for 10-year US governmen bond yields display an inclusion probabiliy above 0.5 in he 1970s and increases afer 1985 and The shorer mauriy 3-monh T-bill provides weaker evidence, i.e. lower and more volaile inclusion probabiliies which decrease unil 2000 bu increase significanly aferwards and surpass he 10-year bonds a he end of he sample. A possible explanaion for his paern is ha moneary policy currenly faces a zero lower bound environmen. The radiional role of long-erm governmen bonds for anchoring inflaion is less imporan since marke paricipans pay more aenion o shor-run changes and announcemens made by moneary auhoriies. *** Inser Figure VI abou here *** Figure VI conains plos for he US dollar rade-weighed index, he euro rade-weighed index, he world FX reserves of cenral banks, and he Barclays capial aggregae bond index. All series yield relaively low inclusion probabiliies for mos of he sample period beween 1980 and unil The low inclusion probabiliies beween 1985 and 2000 migh be surprising a firs bu less so if he high volailiy of he US dollar relaive o he gold price over his period is aken ino accoun. Of all indices, world FX reserves display he highes probabiliy a he end of he sample. This probably mirrors he fac ha cenral banks, mos noably, China and India, have only recenly sared o diversify heir FX reserves and purchased gold. This migh have resuled in expecaions of ongoing gold purchases by cenral banks and is refleced in he low inclusion probabiliies on average and he 12 The causaliy paern beween gold reurns and inflaion is ambiguous. Considering ha gold prices incorporae inflaion expecaions, one migh raher expec ha gold forecass inflaion bu no ha inflaion forecass gold. However, a possible explanaion for our findings is ha changes in curren inflaion also inroduce changes ino expeced inflaion. An increase in expeced inflaion migh hen generae addiional demand for gold as marke paricipans use gold o hedge agains higher inflaion. 16

19 increase of he inclusion probabiliies o values above 0.5 in he lae 2000s. Our resuls provide srong suppor for he ime-variaion of facors which drive he gold price. The mos obvious paern is ha mos regressors display inclusion probabiliies close o zero beween 1985 and 2000 when gold reurns were low and he price of gold relaively consan. On he oher hand, mos inclusion probabiliies increase significanly afer 1995 when he gold price becomes more volaile. The large variaion in he probabiliies suggess ha simple averaging over all models is an inadequae sraegy since his would resul in equal and consan weighs for all models (as mirrored by he prior) over he full sample period. The resuls also emphasize ha a model ha focuses on gold price forecass provides differen resuls compared o models ha focus on he facors ha influence he price of gold conemporaneously and hus in-sample. For example, exchange raes exhibi a srong conemporaneous, in-sample effec bu essenially no predicive, ou-of-sample effec. 6 Summary and Concluding Remarks This paper proposes Dynamic Model Averaging o forecas he price of gold and conribues o he gold economics lieraure wih a novel economeric mehodology o forecas gold prices. The paper sresses ha accouning for boh model and parameer uncerainy is paricularly imporan for an asse for which no pricing model exiss and ha is influenced by a variey of local and global facors ha poenially change over ime. The esimaion resuls show ha parsimonious forecasing models wih only hree variables clearly dominae richer models wih up o foureen variables. However, i becomes also clear ha he inclusion of more predicors is beneficial due o he fac ha our averaging approach is more flexible o capure abrup changes. In addiion, he facors ha influence fuure gold prices vary significanly hrough ime and can be clearly disinguished from resuls ha focus on he in-sample and non-predicive relaionships wihin a classical regression model. More precisely, we find ha in-sample funcions of gold such as he safe haven propery, he currency hedge propery or porfolio diversifier propery do no auomaically ranslae ino ou-of-sample predicabiliy. The resuls also show ha he Dynamic Model Averaging approach ouperforms alernaives such as he random walk and Bayesian Model Averaging. An appealing feaure of DMA is is abiliy o learn and is flexibiliy in erms of he 17

20 weighs aached o regressors: as an example, periods of large gold reurns coincide wih rapid changes of he inclusion probabiliies. An ineresing avenue for fuure research could be an analysis of he usefulness of gold prices for forecasing socks, bonds, exchange raes or inflaion raes wihin a Dynamic Model Averaging framework. 18

21 References Baker, S.A. and R.C. Van Tassel (1985), Forecasing he Price of Gold: A Fundamenalis Approach, Alanic Economic Journal, 13(4), Baen, J.A., C. Ciner, and B.M. Lucey (2014), On he Economic Deerminans of he Gold-Inflaion Relaion, Resources Policy, 41, Baur, D.G. and B.M. Lucey (2010), Is Gold a Hedge or a Safe Haven? An Analysis of Socks, Bonds and Gold, The Financial Review, 45, Beckmann, J. and R. Czudaj (2013), Gold as an Inflaion Hedge in a Time-Varying Coefficien Framework, The Norh American Journal of Economics and Finance, 24, Blose, L.E. (2010), Gold Prices, Cos of Carry, and Expeced Inflaion, Journal of Economics and Business, 62(1), Capie, F., T.C. Mills, and G. Wood (2005), Gold as a Hedge agains he Dollar, Journal of Inernaional Financial Markes, Insiuions and Money 15(4), Deckers, T. and C. Hanck (2014), Variable Selecion in Cross-Secion Regressions: Comparisons and Exensions, Oxford Bullein of Economics and Saisics, forhcoming. Escribano, A. and C. Granger (1998), Invesigaing he Relaionship beween Gold and Silver Prices, Journal of Forecasing, 17, Jasram, R.W. (2009), The Golden Consan: The English and American Experience , (wih updaed maerial by Jill Leyland), Edward Elgar. Koop, G. and D. Korobilis (2012), Forecasing Inflaion using Dynamic Model Averaging, Inernaional Economic Review, 53, Meese, R.A. and K. Rogoff (1983), Empirical Exchange Rae Models of he Sevenies: Do hey fi ou of Sample?, Journal of Inernaional Economics, 14(1-2),

22 Pierdzioch, C., M. Risse, and S. Rohloff (2014), On he Efficiency of he Gold Marke: Resuls of a Real-Time Forecasing Approach, Inernaional Review of Financial Analysis, 32, Rafery, A.E. (1995), Bayesian Model Selecion in Social Research, Sociological Mehodology, 25, Rafery, A.E., Karny, M. and P. Eler (2010), Online Predicion under Model Uncerainy via Dynamic Model Averaging: Applicaion o a Cold Rolling Mill, Technomerics, 52(1), Rossi, B. (2005), Tesing Long-Horizon Predicive Abiliy wih High Persisence, and he Meese- Rogoff Puzzle, Inernaional Economic Review, 46(1), Sherman, E. (1982). Gold: A Conservaive, Pruden Diversifier, Journal of Porfolio Managemen, 8(3), Sjaasad, L. A. and F. Scacciavillani (1996). The Price of Gold and he Exchange Raes, Journal of Inernaional Money and Finance 15,

23 Figure I Gold prices and gold reurns This Figure presens he evoluion of monhly gold prices (op panel) and monhly gold reurns (boom panel). The plos illusrae significan variaion of he gold price hrough ime. gold prices gold reurns

24 Figure II Large absolue gold reurns The plos show he larges posiive and negaive gold reurns if absolue reurns exceed 5% (op plo) and 10% hresholds (boom plo). The graphs clearly indicae volailiy clusering in paricular in he 1970s and in he pos-2008 period. gold reurns > 5% gold reurns > 10%

25 Figure III Opimal model size The graph shows he ime-varying number of predicors (including he consan) included in he model aached o he highes inclusion probabiliy

26 Figure IV DMA poserior inclusion probabiliies for sock prices and commodiy prices The plos show he poserior inclusion probabiliies for he MSCI world sock marke index, he S&P500 composie sock marke index, he GARCH(1,1) process of he reurns of he MSCI world index (proxy for he VIX), he S&P GSCI commodiy price index, he CRB commodiy price index, and silver prices. The graphs provide he probabiliy ha a predicor is useful for forecasing a ime based on he weigh aached by DMA o models which include he corresponding regressor. MSCI S&P π π GARCH(1,1) 0.8 S&P GSCI 0.7 π 0.7 π CRB 1 Silver 0.6 π 0.9 π

27 Figure V DMA poserior inclusion probabiliies for consumer prices and ineres raes The plos show he poserior inclusion probabiliies for he US consumer price index, a global (World) consumer price index and US long-erm and shor-erm ineres raes. US CPI W CPI π 0.9 π long-erm 0.8 shor-erm 0.6 π 0.7 π

28 Figure VI DMA poserior inclusion probabiliies for effecive dollar FX rae and world FX reserves The plos show he poserior inclusion probabiliies for he US dollar rade-weighed index, he world FX reserves of cenral banks, he euro rade-weighed index, and he Barclays Capial US aggregae bond index. US dollar FX reserves π 0.9 π Euro 0.5 US Bonds 0.6 π 0.45 π

29 Table I Comparison of forecas models h =1 h =3 h =12 MAFE RMSFE Hi raio log(pl) MAFE RMSFE Hi raio log(pl) MAFE RMSFE Hi raio log(pl) DMA (α = λ =0.99) DMA (α = λ =0.95) DMA (α = λ =0.90) DMS (α = λ =0.99) DMS (α = λ =0.95) DMS (α = λ =0.90) BMA (α = λ = 1) DMA (α =0.99, λ = 1) DMA (α =1,λ =0.99) Random walk Noe: The able repors he mean absolue forecas error (MAFE), he roo mean squared forecas error (RMSFE), he hi raio, and he sum of he log predicive likelihood (log(pl)) for each model alernaive. 27