Copula-Based Factor Models for Multivariate Asset Returns

Transcription

1 economerics Aricle Copula-Based Facor Models for Mulivariae Asse Reurns Eugen Ivanov 1, *, Aleksey Min 2 and Franz Ramsauer 2 1 Deparmen of Economics, Universiy of Augsburg, Universiässr. 16, Augsburg, Germany 2 Deparmen of Mahemaics, Technical Universiy of Munich, Bolzmannsr. 3, Garching, Germany; min@um.de (A.M.); franz.ramsauer@um.de (F.R.) * Correspondence: eugen.ivanov@wiwi.uni-augsburg.de; Tel.: Academic Edior: Jean-David Fermanian Received: 29 Sepember 2016; Acceped: 3 May 2017; Published: 17 May 2017 Absrac: Recenly, several copula-based approaches have been proposed for modeling saionary mulivariae ime series. All of hem are based on vine copulas, and hey differ in he choice of he regular vine srucure. In his aricle, we consider a copula auoregressive (COPAR) approach o model he dependence of unobserved mulivariae facors resuling from wo dynamic facor models. However, he proposed mehodology is general and applicable o several facor models as well as o oher copula models for saionary mulivariae ime series. An empirical sudy illusraes he forecasing superioriy of our approach for consrucing an opimal porfolio of U.S. indusrial socks in he mean-variance framework. Keywords: COPAR model; dynamic facor model; mulivariae ime series; opimal mean-variance porfolio; vine copula JEL Classificaion: C58; C53; C10; G10 1. Inroducion I ook almos four decades before he saisical usefulness and araciveness of copulas was widely recognized afer he seminal papers by Frees and Valdez (1998), Li (2000), and Embrechs e al. (2002). Copulas are now a sandard ool for modeling a dependence srucure of mulivariae idenically and independenly disribued (iid) daa in applied science. The foundaion of he copula heory was laid by he famous Sklar s heorem (see Sklar 1959), which saes ha any mulivariae disribuion can be represened hrough is copula and marginal disribuions. If marginal disribuions are coninuous, hen he copula of a mulivariae disribuion is unique. This approach is paricularly flexible, since margins and he dependence srucure which is dicaed by he copula can be modeled independenly. To our knowledge, Darsow e al. (1992) iniiaed a heoreical applicaion of copulas o specify univariae Markov processes of firs order. Thus, condiional independence can be saed in erms of copulas, and his resuls in a copula counerpar of he Chapman Kolmogorov equaions for he ransiion probabiliies of a Markov process. Ibragimov (2009) generalized heir approach for univariae Markov processes of higher order as well as for non-markov processes. Furhermore, he inroduced new classes of copulas for modeling univariae ime series. Esimaion of copula-based saionary ime series models can sill be pursued in he classical framework as for iid daa. For example, Chen and Fan (2006) invesigaed heoreical aspecs of he wo-sep esimaion when marginal disribuions are fied non-paramerically in he firs sep and copula parameers are hen esimaed by maximum likelihood. The firs non-gaussian Vecor Auoregression (VAR) models were inroduced by Biller and Nelson (2003), where smarly chosen Gaussian VAR ime series were ransformed o achieve Economerics 2017, 5, 20; doi: /economerics

2 Economerics 2017, 5, 20 2 of 24 desired auocorrelaion srucure and marginal disribuions. Recenly, Brechmann and Czado (2015), Beare and Seo (2015), and Smih (2015) simulaneously developed copula-based models for saionary mulivariae ime series. Alhough hese models differ from each oher, heir generaliy consiss of an underlying R-vine pair-copula consrucion (see Aas e al. 2009) o describe he cross-secional and emporal dependence joinly. To capure he cross-secional dependence, Brechmann and Czado (2015) employ C-Vine, while Beare and Seo (2015) and Smih (2015) consider D-vine. Furher, Brechmann and Czado (2015) and Beare and Seo (2015) assume he exisence of a key variable, whose emporal dependence was explicily modeled, and his assumpion combined wih C- or D-vine for he cross-secional dependence resuls in a corresponding R-vine for a mulivariae ime series. In conras, Smih (2015) explicily modeled he emporal dependence beween mulivariae observaions and consruced a general D-vine for hem consising of D-vines for he cross-secional dependence. In he ime of big daa, facor models offer an elegan soluion o describe a high-dimensional panel daa wih a few unobservable (laen) variables, called facors. The idea behind facor models is ha observable variables are driven by wo orhogonal, hidden processes: one capures heir co-movemens and arises from a linear combinaion of he laen facors, whereas he oher covers heir individual naure in he form of idiosyncraic shocks. The dimension of he observed daa usually significanly exceeds he number of facors, and so a reducion in dimension akes place. In he seminal works of Sock and Wason (1999, 2002a, 2002b), facor models suppored he forecasing of univariae ime series. I was shown for large panel daa ha he unobserved facors can be consisenly esimaed, and his served for a consisen forecasing framework. In paricular, Sock and Wason (2002a) illusraed ha he forecasing of several macroeconomic variables based on facor models can ouperform hose obained from compeing models such as auoregression (AR), VAR, and leading indicaors. In he lieraure, facor models are classified as saic or dynamic wih respec o he sochasic dynamics of he unobserved facors. Saic facor models suppose iid normally disribued facors, while dynamic facor models assume ha he facor obeys a VAR model of order p 1. In his paper, we apply he copula auoregressive (COPAR) model of Brechmann and Czado (2015) o quanify he dependence srucure of esimaed unobserved facors in dynamic facor models. More precisely, we consider wo dynamic facor models and esimae hem separaely wih he maximum likelihood by employing he Kalman filer and smooher. The esimaed facors are hen combined wih a COPAR model, from which laen facors are simulaed for a forecasing purpose. Thus, our approach allows several esimaed dynamic facor models o be coupled wih a copula and admis a non-gaussian dependence srucure of simulaed laen facors. To gain informaion from he esimaed facors, a forecased variable of each marke is regressed on he corresponding simulaed facors and is previous value. I should be noed ha our modeling approach is differen from he facor copula modeling of Krupskii and Joe (2013) and Oh and Paon (2017). We firs esimae unobserved facors and hen fi copula for hem as for observable daa. Wih facor models, we reduce he daa dimension, and using auoregressive srucure of facors we decrease he number of copula parameers essenially. In conras, Krupskii and Joe (2013) and Oh and Paon (2017) rea iid daa and reduce he dimension of copula parameers by considering condiional independence wih respec o unobserved facors. Numerically inegraing he unobserved facors, a copula wih low dimensional parameers for observed iid daa is obained. For mulivariae ime series, Oh and Paon (2016) exended he facor copula models wih ime-varying parameric copulas. In he empirical applicaion, we consider monhly U.S. financial and macroeconomic panel daa o filer driving facors laer employed for a mean-variance porfolio opimizaion. Our main conribuion is a new mehod o improve porfolio performance using facor predicions sampled from he COPAR model. In conras o dynamic facor models, we explicily allow for non-gaussian cross-secional and emporal dependence beween facors. The forecased facors wih non-linear dependence srucure are used o assess he fuure variabiliy of mulivariae asse reurns. This allows us o consruc

3 Economerics 2017, 5, 20 3 of 24 an opimal porfolio in he mean-variance framework. For comparison, hree benchmark porfolios are consruced using dynamic facor models as well as empirical momens of observed asse reurns. Thus, our opimal porfolio ouperforms he benchmark porfolios according o several risk-adjused performance measures. The res of he paper is organized as follows. Secion 2 oulines dynamic facor models and heir esimaion in he maximum likelihood framework. Secion 3 briefly considers vine copulas. Secion 4 reviews COPAR models and discusses an algorihm o exend COPAR(1) model for anoher mulivariae observaion. Secion 5 presens our proposed mehodology for an opimal asse allocaion of 35 indusrial socks from S&P500 lised in Appendix A and compares i wih hree benchmark porfolios. Finally, we conclude and discuss furher research. Appendix B gives an overview on he considered monhly panel daa. Appendix C presens bivariae pair-copulas considered in a model selecion procedure for cross-secional dependence. Appendices D and E conain deailed numerical resuls for porfolio comparisons. Appendix F summarizes esing resuls of Granger causaliy beween esimaed facors. 2. Facor Models In our applicaion, we deal only wih dynamic facor models (DFMs). Furher, we resric our exposiion o he simples facor dynamics of order 1, since any VAR(p) can be wrien in VAR(1) form (see Lükepohl 2005, p. 15). Definiion 1. (Dynamic Facor Model) For any poin in ime, le X R m be a saionary vecor process wih zero mean. Le F R q, q m, denoe he mulivariae facor a ime, and le he vecor ε R m collec all idiosyncraic shocks. Then, a dynamic facor model of order 1 is given by X = ΛF + ε, F = AF 1 + u, wih consan marices Λ R m q and A R q q. The idiosyncraic shocks ε are iid Gaussian wih zero mean and covariance marix R R m m and he error vecors u R q are iid Gaussian wih zero mean and covariance marix Q R q q. Since X is a saionary process, Definiion 1 implicily assumes ha unobserved facor process F is also saionary. The saionariy of F can be ensured if he roos of he characerisic polynomial I q Az lie ouside of he complex uni circle. In his case, he moving average represenaion for F yields is saionary q dimensional zero-mean Gaussian disribuion (see Lükepohl 2005, pp ) given by ( N q (0, Σ F ) wih Σ F = A i Q A i). (1) i=1 For known or esimaed Q and A, he facors can be drawn from (1) by runcaing he infinie sum for a pre-specified error olerance of 10 5 for all enries of Σ F. Parameers Λ, A, R, and Q of he dynamic facor model in Definiion 1 can be esimaed in he maximum likelihood framework wih Expecaion-Maximizaion Algorihm (EM-Algorihm) of Dempser e al. (1977). This was firs done by Shumway and Soffer (1982) and Wason and Engle (1983), hough Shumway and Soffer (1982) assumed a known Λ, and Wason and Engle (1983) did no direcly maximize he log-likelihood of dynamic facor models. Recenly, Bork (2009) and Bańbura and Modugno (2014) derived he EM-Algorihm for he dynamic facor models in Definiion 1, on which we rely in our empirical applicaion. Noe ha he convergence properies of he EM-Algorihm has been heoreically shown for an exponenial family by Wu (1983). For he convenience of he reader, we ouline he esimaion procedure of Bork (2009) and Bańbura and Modugno (2014) and refer o he original works for furher deails. Ignoring he

4 Economerics 2017, 5, 20 4 of 24 unobservabiliy of he facors, he log-likelihood funcion of he model in Definiion 1 for a daa sample of lengh T can be derived by ieraive condiioning on observaions (e.g., in Bork 2009, p. 45 or Bańbura and Modugno 2014, p. 156). However, he facors F are unobservable, and herefore he log-likelihood is inegraed ou wih respec o he facor disribuion. This resuls in he expeced log-likelihood condiioned on he observed panel daa, which consiues he expecaion sep of he EM-Algorihm. In conras o he uncondiioned log-likelihood, here facors are replaced by heir corresponding condiional momens of firs and second order, which a single run of he Kalman filer and smooher given in (Bork 2009, p. 43) can provide. In he maximizaion sep of he EM-Algorihm, Bork (2009) and Bańbura and Modugno (2014) rea he condiional facor momens as consans, when he parial derivaives of he condiional expecaion of he log-likelihood wih respec o he model parameers are compued. Nex, hey search for he zeros of he arising sysem of linear equaions o deermine he maximum of he expeced log-likelihood funcion. The ieraive parameer updaes of he EM-Algorihm from Bork (2009) and Bańbura and Modugno (2014) is summarized in Theorem 1. Theorem 1. (EM-Algorihm as in Bork (2009) and Bańbura and Modugno (2014)) Assume he dynamic facor model in Definiion 1 and le he marix X = [X 1,..., X T ] R m T collec all panel daa. Le he index l 0 indicae he curren loop of he EM-Algorihm and le E [ ] X, θ (l) denoe he expecaion condiioned on he panel daa and he parameers esimaed in loop l. Then, he parameer updaes in loop (l + 1) are given as follows: Λ (l+1) = A (l+1) = R (l+1) = 1 T Q (l+1) = 1 T ( T X E [ ] ) ( F T X, θ (l) E [ F F =1 =1 ( T E [ ] ) ( F F 1 T X, θ (l) E [ F 1 F 1 =1 =1 ( T X X T Λ (l+1) E [ ] F X, θ(l) X =1 =1 ( T =1 E [ F F ] X, θ (l) ] ) 1 X, θ (l), ] ) 1 X, θ (l), ) A (l+1) T =1 E [ F 1 F, ] ) X, θ (l), where Z 1 sands for he inverse marix and condiional facor momens are compued using he Kalman filer and smooher for each loop l. The ieraive esimaion procedure in Theorem 1 requires a erminaion crierion. In our applicaion, we erminae he above EM-Algorihm as soon as he change in log-likelihood is smaller han Finally, noe ha he esimaed facors are unique up o roaion wih orhogonal marices. For forecasing purposes, one can ignore his fac, since esimaed parameers of facors in forecasing equaions will hen be ransformed correspondingly wih no effec on forecasing variable. We illusrae his poin laer in our applicaion. 3. Vine Copulas Since he saisical applicabiliy of vine copulas wih non-gaussian building bivariae copulas was recognized by Aas e al. (2009), vine copulas became a sandard ool o describe he dependence srucure of mulivariae daa (see Aas 2016 for a recen review). Moreover, Brechmann and Czado (2015), Beare and Seo (2015), and Smih (2015) have applied vine copulas o model emporal dependence of mulivariae ime series as well as he cross-secional dependence beween univariae ime series. In his secion, we review he COPAR model of Brechmann and Czado (2015), which is used o describe he sochasic dynamics and he dependence srucure of he esimaed facors from Secion 2. We sar

5 Economerics 2017, 5, 20 5 of 24 wih he concep of regular vines from Kurowicka and Cooke (2006). We do no consider illusraing examples on pair-copula consrucions, and refer insead o Aas (2016) or Czado (2010) for more inuiion on hem. Definiion 2. (Regular vine) A collecion of rees V = (T 1,..., T d 1 ) is a regular vine on d elemens if 1. T 1 is a ree wih nodes N 1 = {1,..., d} conneced by a se of non-looping edges E For i = 2,..., d 1, T i is a conneced ree wih edge se E i and node se N i = E i 1, where N i = d (i 1) and E i = d i are he number of edges and nodes, respecively. 3. For i = 2,..., d 1, e = {a, b} E i : a b = 1 (wo nodes a, b N i are conneced by an edge e in T i if he corresponding edges a and b in T i 1 share one node (proximiy condiion)). A ree T = (N, E) is an acyclic graph, where N is is se of nodes and E is is se of edges. Acyclic means ha here exis no pah such ha i cycles. In a conneced ree we can reach each node from all oher nodes on his ree. R-vine is simply a sequence of conneced rees such ha he edges of T i are he nodes of T i+1. A radiional example of hese srucures are canonical vines (C-vines) and drawable vines (D-vines) (see Czado (2010) and Aas e al. (2009)) in Figure 1. Every ree of a C-vine is defined by a roo node, which has d i incoming edges, in each ree T i, i {1,..., d 1}, whereas a D-vine is solely defined hrough is firs ree, where each node has a mos wo incoming edges. T 1 T 2 T T 1 T 2 T 3 T T Figure 1. C-vine (Lef) and D-vine (Righ) represenaion for d = 5. Regular vines are a powerful ool o sysemize all possible facorizaions of a d dimensional densiy as a produc of d univariae marginal densiies wih a produc of d(d 1)/2 condiional and uncondiional bivariae copulas (see Theorem 4.2 in Kurowicka and Cooke (2006)). Thus, he uncondiional and condiional bivariae copulas called pair-copulas of a given facorizaion can be uniquely mapped ono he edge se E of a paricular regular vine, and vice versa. Then, he condiional copulas and heir argumens depend on condiioned values. The dependence of condiional copulas on condiioned values is crucial, and allows for saisical applicaions only for a subclass of ellipical disribuions (see Söber e al. (2013)), since in his case, condiioned pair-copulas depend on condiioned values only hrough heir argumens. Aas e al. (2009) firs developed his observaion furher and wen beyond he ellipical world. Thus, hey considered regular vine facorizaions wih arbirary fixed condiional copulas and showed ha his resuls in valid mulivariae disribuions and copulas. In his paper, we also assume ha condiional copulas depend on condiioned values only hrough heir argumens, and so hey can be chosen from bivariae copula families. The number of possible vine srucures on d random variables can be immense. For d 4, only C- and D-vines are possible. For d > 4, here are d! 2 differen C- and D-vines. The oal amoun of regular

6 Economerics 2017, 5, 20 6 of 24 vine srucures has been compued by Morales-Nápoles e al. (2010), and is equal o d! 2 2(d 2 2 ). To selec condiional copulas on hese graphical srucures, we define he following ses as in Czado (2010). Definiion 3. (Condiioned and condiioning ses) For any edge e = {a, b} E i of a regular vine V, he complee union of e is he subse A e = { v N 1 : m = 1,..., i 1, e jm E m s.. v e j1 e ji 1 e }. The condiioning se associaed wih e is D e = A a A b. The condiioned ses associaed wih e are i(e) = A a \ D e j(e) = A b \ D e. The copula for his edge will be denoed by C e := C i(e),j(e) D(e). Given a regular vine, we specify a regular vine copula by assigning a (condiional) pair copula (wih parameers) o each edge of he regular vine. In doing so, we follow Czado e al. (2012). Definiion 4. (Regular vine copula). A regular vine copula C = (V, B (V), θ (B (V))) in d dimensions is a mulivariae disribuion funcion such ha for a random vecor U = (U 1,..., U d ) C wih uniform margins 1. V is a regular vine on d elemens, 2. B (V) = {C e e E m, m = 1,..., d 1} is a se of d (d 1) /2 copula families idenifying he uncondiional disribuions of U i(e),j(e) as well as he condiional disribuions of U i(e),j(e) U D(e), 3. θ (B (V)) = {θ e e E m, m = 1,..., d 1} is he se of parameer vecors corresponding o he copulas in B (V). To faciliae saisical inference, a marix represenaion of R-vines was proposed by Morales-Nápoles e al. (2010) and furher developed by Dissmann e al. (2013). To specify a d-dimensional R-vine in marix form, one needs several lower riangular d d marices: one ha sores he srucure of he R-vine, one wih copula families, and anoher wo wih he firs and second parameers. For a d-dimensional R-vine, he marix wih he srucure has he following form M = m 1,1.... m d 1,1 m d 1,d 1, m d,1 m d,d 1 m d,d where m i,j (1,..., d). The rules for reading from his marix are as follows. The condiioned se for an enry m i,j is he enry iself and he diagonal enry of he column m j,j, whereas he condiioning se is composed of variables under he enry; i.e., for m i,j, he condiioned se will be ( ) m i,j, m j,j, he condiioning se is (m i+1,j,..., m d,j ). Thus, m i,j denoes he node (m j,j, m i,j m i+1,j,..., m d,j ). We will assume ha he diagonal of M is sored in descending order, which can always be achieved by reordering he node labels, so ha we have m i,i = n i + 1. To illusrae he R-vine marix noaion, we consider he C-vine from Figure 1 and give below is R-vine marix represenaion.

7 Economerics 2017, 5, 20 7 of 24 The firs column encodes he following nodes Node is saved hrough m 1,1, m 2,1 given m 3,1, m 4,1 and m 5,1 ; 2. Node is saved hrough m 1,1, m 3,1 given m 4,1 and m 5,1 ; 3. Node 52 1 is saved hrough m 1,1, m 4,1 given m 5,1 ; 4. Node 51 is saved hrough m 1,1, m 5,1. In he sequel, we uilize C-vines o capure he cross-secional dependence of a mulivariae ime series a any ime poin. To capure he dependence beween mulivariae observaions, he firs ree of he C-vine for mulivariae observaion a ime poin is conneced o he firs rees of he C-vines for exising neighboring mulivariae observaions a ime poins 1 and + 1 wih one edge, correspondingly. This resuls in he firs ree of an R-vine for all mulivariae observaions reaed as one huge sample poin. Depending on he choice of C-vines and he connecion of he firs rees, he copula auoregressive model of Brechmann and Czado (2015) from he nex secion can be obained. Finally, noe ha he informaion on copula families and heir parameers is similarly sored in lower riangular d d marices. Each elemen of he R-vine marix below he diagonal specifies a condiional or uncondiional pair-copula depending on he diagonal enry above i. The family and parameers of his pair-copula are now enered a he same enry place of marices for copula families and parameers. Since he diagonal enries of he R-vine marix alone do no deermine any pair-copulas, no enries for copula family and parameers marices are needed on he diagonal. To avoid confusion wih space characer, we fill he main diagonal wih sign. 4. Copula Auoregressive Model R-vines have been mosly used o model conemporaneous dependence. We now presen a special R-vine srucure called copula auoregressive (COPAR) model from Brechmann and Czado (2015) which is designed o capure cross-secional, serial, and cross-serial dependence in mulivariae ime series, and allows a general Markovian srucure. Le {F, G } =1,...,T be an observable bivariae saionary ime series. To illusrae how he wo individual ime series are inerdependen, consider he mapping of dependencies for T = 4 in Figure 2. Verical solid lines represen he cross-secional dependence, horizonal solid lines and curved lines represen he serial dependence for each ime series, and doed and dashed lines represen he cross-serial dependence. F 1 F 2 F 3 F 4 G 1 G 2 G 3 G 4 Figure 2. Dependencies of a simulaneous bivariae ime series for T = 4. Verical solid lines: cross-secional dependence; horizonal solid lines and curved lines: serial dependence for each ime series; doed and dashed lines: cross-serial dependence. Tradiionally, R-vines were used o model only he cross-secional dependence (picured by he verical solid lines in Figure 2), bu under he assumpion ha oher dependencies are absen (i.e., daa

8 Economerics 2017, 5, 20 8 of 24 is iid). The COPAR model is designed o addiionally capure serial and cross-serial dependence. The following definiion of a COPAR model wihou Markovian srucure for wo ime series adoped o our noaion is aken from Brechmann and Czado (2015). The vecors (F s,..., F ) and (G s,..., G ) are denoed as F s: and G s:, respecively. Definiion 5. (COPAR model for he bivariae case) The COPAR model for saionary coninuous ime series {F } =1,...,T and {G } =1,...,T has he following componens. (i) (ii) Uncondiional marginal disribuions of each ime series are independen of ime. An R-vine for he serial and beween-series dependence of {F } =1,...,T and {G } =1,...,T, where he following pairs are seleced. 1. Serial dependence of {F } =1,...,T : The pairs of serial D-vine copula for F 1,..., F T ; ha is, F s, F F (s+1):( 1), 1 s < T. (2) 2. Beween-series dependence: F s, G F (s+1):, 1 s T, (3) and G s, F F 1:( 1), G (s+1):( 1), 1 s < T. (4) 3. Condiional serial dependence of {G } =1,...,T : The pairs of a serial D-vine copula for G 1,..., G T condiioned on all previous values of {F } =1,...,T ; ha is, G s, G F 1:, G (s+1):( 1), 1 s < T. (5) Pair copulas of he same lag lengh s, s, are idenical. We associae 1. copula C s F := C F s,f F (s+1):( 1) wih (2), 2. copulas C s FG F s,g F (s+1): and C s GF G s,f F 1:( 1),G (s+1):( 1) wih (3) and (4), respecively, and 3. copula C s G := C G s,g F 1:,G (s+1):( 1) wih (5). Le us explain he above definiion on an example of a bivariae ime series {F, G } =1,...,4 wih four observaions. Using a D-vine, Equaion (2) capures he dependence srucure of F 1, F 2, F 3, and F 4 expressed by he op lines connecing hem in Figure 2. For s = 1 and = 4, Equaion (3) describes he condiional dependence beween F 1 and G 4 condiioned on F 2, F 3, and F 4. Thus, Equaion (3) capures he condiional dependence beween F s and G for s < refleced by he dashed lines in Figure 2. For s =, Equaion (3) models he uncondiional dependence beween F and G expressed by he verical lines. Similarly, Equaion (4) describes he condiional serial dependence beween univariae ime series illusraed by he doed lines in Figure 2. However, Equaions (3) and (4) are no symmeric wih respec o condiioned ses. In paricular, he condiional disribuion of F s and G for s < is independen of G 1,..., G 1. This already indicaes ha he firs ime series {F } =1,...,T plays a key role in he sochasic dynamics of he ime series, since i is sufficien o describe he condiional dependence beween F s and G for s <. Therefore, he univariae ime series are no inerchangeable. Using a D-vine, Equaion (5) finally describes he dependence srucure of G 1, G 2,..., G T (connecing boom lines in Figure 2) condiioned on F 1, F 2,..., F T. Due o he propery ha copulas of same lag lengh are idenical, he COPAR model defines a saionary bivariae ime series. Now, we expand he COPAR model o an arbirary number of variables wih Markov srucure. We consider q univariae ime series observed a T ime poins; ha is, {F 1, } =1,...,T,..., { F q, }=1,...,T. We denoe a random vecor of q variables observed a imes = 1,..., T as F = ( F 1,,..., F q, ), ime series of individual variables for i = 1,..., q as F (i) = (F i,1,..., F i,t ), and also inroduce a vecor F l:q,s: = ( F l,s,..., F l,,..., F q,s,..., F q, ). Finally, we define a COPAR(k) model, which is COPAR model for a mulivariae ime series wih a Markovian srucure of he k-h order. Since unobservable facors

9 Economerics 2017, 5, 20 9 of 24 in Secion 2 are also denoed by F, our noaion above should no lead o a confusion. Our modeling approach uilizes esimaed facors even if hey are no observed. Therefore, for reader convenience, we denoe here mulivariae ime series wih F and presen COPAR models in erms of F. Definiion 6. (COPAR(k) model for q ime series of lengh T) The COPAR(k) model for a q-dimensional saionary ime series F R q 1, = 1,..., T has he following componens. (i) (ii) Uncondiional marginal disribuions of each ime series are independen of ime. An R-vine for he serial and beween-series dependence of F R q 1, = 1,..., T, where he following pairs are seleced. 1. Serial dependence of F (1) : The pairs of serial D-vine copula for F 1,1,..., F 1,T ; ha is, F 1,s, F 1, F 1,(s+1):( 1), 1 s < T. 2. Beween-series dependence of F (i) and F (j) for i < j, i, j = 1,..., q: F i,s, F j, F 1:(i 1),1:, F i,(s+1): 1 s T, and F j,s, F i, F 1:(i 1),1:, F i:(j 1),1:( 1), F j,(s+1):( 1) 1 s < T, 3. Condiional serial dependence of F (i) for 2 i q: F i,s, F i, F 1:(i 1),1:, F i,(s+1):( 1), 1 s < T, whereas (a) (b) copulas i and for lag lengh s > k are independen copulas, copulas for i, j = 1,..., q for he same lag lengh s, s are idenical. Definiion 6 inroduces he COPAR model, if a) is negleced. The dependence of F (i) is modeled condiioned on F (1),..., F (i 1), and consequenly, he order of variables canno be simply inerchanged. The number of pair copulas in COPAR models for ime series wih Markovian srucure does no depend on T, and is less han he number needed for a general R-vine. Wih respec o he number of pair-copulas, he following resul from Brechmann and Czado (2015) holds. Lemma 1. The number of copulas needed for a COPAR(k) model of q univariae ime series is q 2 k + q(q 1) 2. Noe, he number of parameers in a VAR(k) for he beween-series dependence of q ime series no couning parameers for marginal disribuions is equal o he number of pair-copulas in a COPAR(k) model. Therefore, he number of parameers in a COPAR(k) model is bounded by a muliple of he number of VAR parameers, depending on he number of parameers of he involved copula families. In conras, a general R-vine requires qt(qt 1) 2 pair-copulas, resuling in a huge number of parameers. Figure 3 illusraes he firs 4 rees of a COPAR model for hree univariae ime series F, G, and H wih T = 4 (i.e., for four observaions). Examining he firs ree, we observe ha i is a sequence of conneced C-vines, where he cenral nodes are he ime poins of he firs facor F. Thus, uncondiional conemporaneous dependence is modeled wih a C-vine.

10 Economerics 2017, 5, of 24 G 2 H 2 G 4 H 4 F 1 F 2 F 3 F 4 G 1 H 1 G 3 H 3 F 2,G 2 F 2,H 2 F 4,G 4 F 4,H 4 F 1,F 2 F 2,F 3 F 3,F 4 F 1,H 1 F 1,G 1 F 3,G 3 F 3,H 3 F 1,H 2 F 2 F 1,G 2 F 2 F 3,G 4 F 4 F 3,H 4 F 4 F 2,G 1 F 1 F 1,F 3 F 2 F 2,F 4 F 3 G 1,H 1 F 1 F 2,G 3 F 3 F 2,H 3 F 3 G 2,H 2 F 1:2 F 3,G 2 F 1:2 F 1,F 4 F 2:3 F 2,G 4 F 3:4 F 2,H 4 F 3:4 F 2,H 1 F 1,G 1 G 1,G 2 F 1:2 F 1,G 3 F 2:3 F 1,H 3 F 2:3 Figure 3. R-vine srucure of hree-dimensional ime series for T = 4. The marix represenaion of he R-vine srucure of his COPAR model is given by H 4 H 1 G 4 H 2 H 1 F 4 H 3 H 2 H 1 H 3 G 1 H 3 H 2 H 1 G 3 G 2 G 1 H 3 H 2 H 1 F 3 G 3 G 2 G 1 G 1 H 2 H 1 H 2 G 4 G 3 G 2 G 2 G 1 H 2 H 1 G 2 F 1 F 1 G 3 G 3 G 2 G 1 G 1 H 1 F 2 F 2 F 2 F 1 F 1 F 1 G 2 G 2 G 1 H 1 H 1 F 3 F 3 F 2 F 2 F 2 F 1 F 1 F 1 G 1 G 1 G 1 F 4 F 4 F 3 F 3 F 3 F 2 F 2 F 2 F 1 F 1 F 1 F 1 and he marix of copulas for hree-dimensional COPAR model is given by C3 H C2 H C3 HG C1 H C2 HG C HF C3 GH C1 HG C HF 3 2 C2 H C2 GH C3 G C1 HF C1 H C GH 2 C1 GH C2 G C3 GF C2 GH C1 GH C HF 2 C GH C1 G C2 GF C1 GH C2 G C1 HF C1 H C3 FH C3 FG C1 GF C GH C1 G C2 GF C1 GH C GH 1 C2 FH C2 FG C3 F C2 FH C2 FG C1 GF C GH C1 G C HF 1 C1 FH C1 FG C2 F C1 FH C1 FG C2 F C1 FH C1 FG C1 GF C GH C FH C FG C1 F C FH C FG C1 F C FH C FG C1 F C FH C FG. If a COPAR(k) model is esimaed based on he firs T mulivariae observaions, hen i can easily be exended o T + 1 observaions. This allows us o sample he (T + 1)-h observaion according o he esimaed dependence srucure beween he k h subsequen mulivariae observaions. Le us

11 Economerics 2017, 5, of 24 illusrae his poin for a COPAR(1) model and he above example wih hree univariae ime series. The marix represenaion of he R-vine srucure of his COPAR(1) model remains unchanged, and he marix of copulas for hree-dimensional COPAR(1) model is now simplified o he marix in (6), where 0 sands for independence copula C1 H C1 HG C1 HF C1 H 0 C1 GH C1 GH 0 C GH C1 G 0 C1 GH 0 C1 HF C1 H 0 0 C1 GF C GH C1 G 0 C1 GH C1 GH C1 GF C GH C1 G C1 HF C1 FH C1 FG 0 C1 FH C1 FG 0 C1 FH C1 FG C1 GF C GH C FH C FG C1 F C FH C FG C1 F C FH C FG C1 F C FH C FG. (6) Now, if we wan o expand his COPAR(1) model by adding a new ime poin (i.e., T T + 1 in our case, 4 5), he marix represenaion will be changed as follows 1. Add hree blank columns o he lef of he marix and add (H T+1, G T+1, F T+1 ) o he diagonal; 2. Under F T+1 comes (H 1,..., H T ), hen (G 1,..., G T ) and (F 1,..., F T ) ; 3. Under G T+1 comes (H 1,..., H T ), hen (G 1,..., G T ) and (F 1,..., F T+1 ) ; 4. Under H T+1 comes (H 1,..., H T ), hen (G 1,..., G T+1 ) and (F 1,..., F T+1 ). The expanded marix represenaion is as follows, where he new columns are marked in bold: H 5 H 1 G 5 H 2 H 1 F 5 H 3 H 2 H 1 H 4 H 4 H 3 H 2 H 1 G 4 G 1 H 4 H 3 H 2 H 1 F 4 G 2 G 1 H 4 H 3 H 2 H 1 H 3 G 3 G 2 G 1 G 1 H 3 H 2 H 1 G 3 G 4 G 3 G 2 G 2 G 1 H 3 H 2 H 1 F 3 G 5 G 4 G 3 G 3 G 2 G 1 G 1 H 2 H 1 H 2 F 1 F 1 G 4 G 4 G 3 G 2 G 2 G 1 H 2 H 1 G 2 F 2 F 2 F 1 F 1 F 1 G 3 G 3 G 2 G 1 G 1 H 1 F 2 F 3 F 3 F 2 F 2 F 2 F 1 F 1 F 1 G 2 G 2 G 1 H 1 H 1 F 4 F 4 F 3 F 3 F 3 F 2 F 2 F 2 F 1 F 1 F 1 G 1 G 1 G 1 F 5 F 5 F 4 F 4 F 4 F 3 F 3 F 3 F 2 F 2 F 2 F 1 F 1 F 1 F 1 We also expand he marix of copulas by adding hree columns o he lef of (6), as follows:.

12 Economerics 2017, 5, of C H C HG C HF 1 C1 H C1 HG 0 0 C GH C1 HF C1 H 0 C GH C G 1 0 C1 GH C1 GH C GF 1 C GH C1 G 0 C1 GH 0 C1 HF C1 H C1 GF C GH C1 G 0 C1 GH C1 GH C1 GF C GH C1 G C1 HF C FH 1 C FG 1 0 C1 FH C1 FG 0 C1 FH C1 FG 0 C1 FH C1 FG C1 GF C GH C FH C FG C F 1 C FH C FG C1 F C FH C FG C1 F C FH C FG C1 F C FH C FG. Thus, he hree new columns are jus replicaions of he previous hree columns by expanding he uninerruped sequences of zeros wih an addiional zero. 5. Empirical Applicaion The idea o model asse reurns wih facors arising from some observed daa and idiosyncraic componens is quie popular in modern finance heory. The mos prominen example is he capial asse-pricing model (CAPM) of Sharpe (1964), Liner (1965), Mossin (1966), and Treynor (2012), which is a one-facor model wih he marke reurn as he only common driver of asse prices. Anoher well-known approach is he arbirage pricing heory (APT) of Ross (1976). In his case, a muli-facor model describes he reurn of an asse as he sum of an asse-specific reurn, an exposure o sysemaic risk facors, and an error erm. A hird example is Sock and Wason (2002a), who exraced facors from a large number of predicors o forecas he log-reurns of he Federal Reserve Board s Index of Indusrial Producion. Ando and Bai (2014) provide furher empirical evidence ha sock reurns are relaed o macro- and microeconomic facors. In his secion, laen facors from U.S. macroeconomic daa are exraced and hen used for porfolio opimizaion. We consider 35 asses from S&P 500, which are classified as Indusrials according o Global Indusry Classificaion Sandard. We assume ha he esimaed facors have he mos predicion power for hese asses. The U.S. panel daa includes such economic indicaors as governmen bond yields along he curve, currency index, main commodiy prices, indicaors of money sock, inflaion, consumer consumpion, and indusrial producion gauges. Alogeher, we have 22 ime series lised in Appendix B. Each series conains monhly daa from 31 January 1986 o 30 November 2016, 371 daa poins in oal. Nex, we spli he panel daa ino financial and macroeconomic groups according o Tables A2 and A3 in Appendix B. In he sequel, each ime series is ransformed in order o eliminae rends and achieve is saionariy. Tables A2 and A3 in Appendix B also conain informaion on considered daa ransformaions. Furher, we consider hree facor models: separaely for he wo groups of he panel daa, and one join model for all monhly indicaors. Thereby, we aim o illusrae ha modeling nonlinear dependence beween esimaed facors of differen groups of panel daa wih COPAR may lead o a beer asse allocaion. We consider he following hree DFMs for i = f in, macro, all: X (i) = Λ (i) F (i) + ε (i) (7) F (i) = A (i) F (i) 1 + u(i), (8)

13 Economerics 2017, 5, of 24 where X (i) R mi 1 collecs observed macroeconomic daa and F (i) R qi 1 is he vecor of facors. The dimension m i of he panel daa for i = f in, macro and all is equal o 9, 13 and 22, respecively. In he firs sep, we esimae hree DFMs one for each group (macro and financial) and for he full panel daa. The saring ime frame is from January 1986 o December 2005, and expands successively by one monh unil November 2016 is reached. For he hree DFMs, we apply he EM-Algorihm from Secion 2 o he panel daa from each expanding ime window and obain a monhly sequence of esimaed facors for each monh of he considered ime period. For i = f in, macro, and all, he EM-Algorihm requires he facor dimension q i, which is no known. In wha follows, we perform he model selecion for he facor dimension. The dimension of facors is seleced using principal componens analysis (see Jolliffe 2002). We choose he number of principal componens (PCs) such ha we capure more han 95% of he empirical daa variance, based on he iniial ime frame from January 1986 o December Figure 4 illusraes he fracion of variance capured by eigh or fewer PCs for he considered ime window. Thus, wo facors are sufficien for he financial group o capure 95.4% of he corresponding daa variabiliy. For macroeconomic indicaors, we need hree facors (96.5% of variabiliy), while four facors are enough for he whole panel daa (96.2% of variabiliy). Percenage of explained variance group fin join macro Number of PCs Figure 4. Amoun of principal componens (PCs) vs. capured variance for hree dynamic facor models (DFMs) considered up o December In general, he laen facors of DFMs may follow a VAR model of order p. Iniiaed by a referee s commen, we also consider auoregressive order p = 2 for he join DFM wih all monhly indicaors and compare is forecasing performance for p = 1 and p = 2. Our decision crierion is based on he roo-mean-square error (RMSE) of poin predicions defined for univariae observaions x 1,..., x T as RMSE = T =1 (x ˆx ) 2 T, where ˆx 1,..., ˆx T are prediced values. Thus, we compue he RMSE for each ime series and ake heir average value. Thereby, our firs predicion is done for January 2006 and ends in November 2016, resuling in 131 poin forecass. The averaged values of he RMSE for p = 1 and for p = 2 are and , respecively. Therefore, our iniial choice of p = 1 is jusified. Figure 5 illusraes he correlaion coefficiens of he financial and macro facors filered from daa up o Ocober The esimaed facors show some moderae linear dependence wihin groups and a weak linear dependence beween groups. Noe ha dynamic facor models assume facors o be mulivariae normally disribued. Neverheless, he esimaed facors could exhibi non-gaussian dependence, which we aim o capure using he COPAR model.

14 Economerics 2017, 5, of Macro3 Macro2 Macro1 value Fin2 1.0 Fin1 Fin1 Fin2 Macro1 Macro2 Macro3 Figure 5. Correlaions heamap as of Ocober To improve he join forecasing of asse reurns a monh, for a subsequen monh, we link he wo facor models; namely, we capure he dependence srucure of he esimaed facors wih a COPAR model. Thereby, we combine he esimaed facors from wo groups as a single five-dimensional vecor for each daa ime window. We pu he financial facors firs, hen he macroeconomic. Using fied marginal normal disribuion funcions, he filered facors are ranslaed o copula daa. We consider a COPAR(1) model, since he ransiion equaion of he DFMs is supposed o be a VAR(1). Furher, Brechmann and Czado (2015) discuss a sophisicaed copula family selecion procedure for wo ime series. Here, we follow a more simple approach consising of hree seps. In he firs sep, he selecion of copulas for conemporaneous cross-secional dependence of he filered facors is done by neglecing serial dependence. Noe ha he cross-secional dependence is described by a C-vine wih fixed order of variables. Therefore, we only have o selec copula families of pair-copulas. Thus, we consider he esimaed five-dimensional facors as iid daa and perform sequenial selecion of copula families for each pair-copula using Akaike informaion crierion. Copula families for he pair-copulas associaed wih he firs ree of he C-vine are seleced firs, hen wih he second ree, and so on. For more deails on model selecion for C-vines, we refer o Czado e al. (2012). Equaion (9) presens he seleced copula families for cross-secional dependence in R-vine marix noaion unil December 2005 (for copula families in (9), see Appendix C): Families = (9) I is remarkable ha none of he seleced copula families in (9) is Gaussian. Moreover, only 0.5% of families seleced for cross-secional dependence for all expending ime windows are he Gaussian copula. In he second sep, we use he Gaussian copulas as condiional pair-copulas o model he emporal dependence of he filered facors. Noe ha his simple approach does no imply ha he join disribuion of he ime-shifed facors is Gaussian excep for he firs componen of F ( f in) ; i.e., F ( f in) 1,, F ( f in) 1 1, 2,..., F ( f in) 1,. Non-Gaussian pair-copulas for cross-secional dependence desroy he k Gaussianiy of he ime-shifed facors, and only he join disribuion of he firs componen of F ( f in) remains Gaussian. In he hird sep, he R-vine marix of he COPAR(1) model for he esimaed facors

15 Economerics 2017, 5, of 24 ( ) F ( f in), F (macro) is consruced, and he maximum likelihood esimaion is performed. Copula selecion and parameer esimaion of he COPAR model for he facors is done for every expanding window of daa; ha is, as soon as he DFMs for boh groups are esimaed and he laen facors are filered ou. Finally, noe ha we also es Granger causaliy beween componens of he mulivariae ( ) ime series F ( f in), F (macro) in he whole ime period, and can confirm i. For his, we regress ( ) ( ) univariae series of F ( f in), F (macro) on he lagged values of F ( f in) 1, F (macro) 1. Table A11 in { } Appendix F summarizes all five linear regressions. For example, he firs componen of F ( f in) =1,...,T Granger causes almos all oher univariae { ime } series a leas a 10% of significance level wih he excepion of he second componen of F (macro) =1,...,T. Nex, we presen a condiional mehod for forecasing using COPAR, which follows Brechmann and Czado (2015) wih a small modificaion. For a predicion ime poin, we simulae facors from he esimaed COPAR model, bu condiioned on he pas values of he facors. The sampled facors are furher used o forecas asses reurns and o consruc an opimal mean-variance porfolio. Since we assume an auoregressive order of one, i is enough o condiion only on he las value of facors; i.e., we condiion only on Fˆ ( f in) T and Fˆ (macro) T o make a forecas for ime poin T + 1. The full algorihm for boh markes is given in Algorihm 1, employing noaion F 1:0, =. Algorihm 1: Condiional mehod of forecasing using COPAR. 1. Esimae COPAR model based on he firs T observaions and ge facor esimaes ( ( f in) ˆF = ˆF (macro), ˆF ) ; 2. Se j = 1; 3. Repea he following seps (a) If j = q + 1 hen Sop; (b) Deermine he esimaed condiional densiy of F j,t+1 ˆF 1:q,T, F 1:(j 1),T+1 on he equidisan grid on (0, 1) wih sep = 10 4, i.e., for u =, n = 1,..., 9999; n c ( Φ 1 ( F 1,T+1 ),..., Φ j 1 ( F j 1,T+1 ), u, Φ1 ( ˆF 1,T ),..., Φq ( ˆF q,t )) f ( F 1,T+1,..., F j 1,T+1, ˆF 1,T,..., ˆF q,t ) j 1 k=1 φ k ( F k,t+1 ) φ j ( Φ 1 j ) (u) q k=1 φ k ( ˆF k,t ), where φ k and Φ k are he esimaed Gaussian densiy and disribuion funcion of laen facor F (i) k, ; (c) Deermine he esimaed condiional cumulaive disribuion funcion on he above grid; (d) Simulae 10,000 F j,t+1 from he esimaed condiional cumulaive disribuion funcion and se forecas F j,t+1 o heir empirical mean; (e) Se j = j + 1 and go o (a); To illusrae he idea behind his mehod, we consider a small-scale example of wo variables G and H. Le us assume ha we observe some values of hese wo random variables a ime poin ; i.e., G = Ĝ, H = Ĥ. We wan o find he disribuion of G +1 and H +1 condiioned on he observed values ha we have observed. For his purpose, we consider he decomposiion of he condiional disribuion of G +1, H +1 G = Ĝ, H = Ĥ given by

16 Economerics 2017, 5, of 24 F ( G +1, H +1 G = Ĝ, H = Ĥ ) = F ( G +1 G = Ĝ, H = Ĥ ) F ( H+1 G = Ĝ, H = Ĥ, G +1 = Ḡ +1 ), where Ḡ +1 is some forecas. One has a free choice for forecas mehods for G +1. As Brechmann and Czado (2015), we op for he condiional mean esimaed by he sample mean. Nex, we firs esimae he condiional densiy, and hen, he condiional disribuion funcion. For his simple case of wo variables G and H, he esimaed densiy ˆf of G +1 G = Ĝ, H = Ĥ on a grid from 0 o 1 wih sep is esimaed. Then, we search for he esimaed disribuion funcion ˆF. Wih ˆf and ˆF, one can esimae he mean. In he nex sep, we use he esimaed facors o model he asse reurns of 35 indusrial socks from S&P 500 and he sampled facors o consruc an opimal porfolio in he mean-variance framework for each expanding ime window saring from January 2006 up o November We consider and esimae he following model for each asse reurn j = 1,..., 35: r j, = α j,0 + α j,1 r j, 1 + Γ j ˆF + v j,, (10) where he consans α j,0, α j,1, and he vecor Γ j consiue unknown regression parameers and he errors v j, are assumed o be iid Gaussian wih respec o. The error erms v j, are assumed o be independen across j due o he dimensionaliy of he error covariance marix. Moreover, we rea all linear regression separaely for each asse j due o he dimensionaliy of regression parameers and couple hem wih he common esimaed facors. In Secion 2, we have poined ou ha he esimaed facors are unique up o a roaion wih an orhogonal marix R; i.e., RR = R R is an ideniy marix. Since Γ j ˆF = Γ j R R ˆF holds, he impac of he roaed and unroaed esimaed facors ogeher wih heir regression coefficiens on asse reurns remains he same. To consruc an opimal porfolio in he mean-variance framework, we deermine porfolio weighs w a monh by solving he following opimizaion problem wih respec o w: max w E [r +1 ] s.. 1 T w = 1, w 0, w Var [r +1 ] w σ 2 monhly, where r = (r 1,,..., r 35, ) is a vecor of asse reurns a, E [r +1 ] and Var [r +1 ] are he expecaion and covariance marix of r +1. Thus, our opimal porfolio does no allow shor-selling. Since E [r +1 ] and Var [r +1 ] are unknown a monh, we esimae hem wih four differen mehods, resuling in four porfolios. The firs porfolio is he COPAR porfolio, which is consruced using asse reurns modeled wih (10). In his case, E [r +1 ] and Var [r +1 ] are empirically esimaed a ime poin based on 10,000 sampled facors from he COPAR model and 10,000 sampled errors from he esimaed univariae Gaussian disribuion. The second porfolio is he DFM porfolio similarly consruced wih sampled facors from he esimaed saionary disribuion (1) for (7) (8) and i = all (i.e., he DFM for he full panel daa). The hird porfolio called independen DFM uses sampled facors from he esimaed saionary disribuion (1) for (7) (8) and i = f in, macro; i.e., ( ) asses reurns are driven by F ( f in), F (macro). The fourh porfolio is he hisorical porfolio, which employs he empirical mean and covariance marix based on he daa up o ime. We consider he hisorical porfolio as a benchmark. Finally noe ha he comparison of hese four porfolios enables he assessmen of he economic relevance of facors. For he above mean-variance opimizaion problem, we choose hree monhly volailiies σ monhly = 2.89%, 3.75%, 4.62%, which correspond o annual volailiies of σ = 10%, 13%, 16%. These choices of σ s are pracically reasonable, and he opimizer diversifies he porfolio for hem. For

17 Economerics 2017, 5, of 24 higher values of σ, he opimal porfolio consiss of only one sock, if no consrains on maximal weighs are imposed. We sar o deermine an opimal porfolio for he panel daa up o January Then, we sequenially expand he ime window by one monh and find opimal weighs for he considered four porfolios and chosen level of volailiies. The performance of he four porfolios wih iniial invesmen of 1 USD during he ou-of-sample ime period up o November 2016 is illusraed in Figure 6. 10% % % Porfolio COPAR DFM His Ind Figure 6. Performance of four porfolios wih iniial invesmen of 1 USD over he ou-of-sample ime period from January 2006 o November 2016 for σ = 10%, 13%, 16%. COPAR: copula auoregressive; His: hisorical; Ind: independen DFM.

18 Economerics 2017, 5, of 24 We observe ha he COPAR porfolio ouperforms he DFM, independen DFM, and hisorical porfolios. Furher, wo porfolios based on DFMs deliver a higher overall reurn han he hisorical porfolio, and he independen DFM (abbreviaed in Figures and Tables as ind) is preferred over he DFM porfolio. To compare he four porfolios, we consider several porfolio performance and risk measures summarized in Appendix D. Firs, noe ha observed sandard deviaions of porfolio reurns are higher han prespecified ones. This is naural due o he predicion error. According o he Sharpe and Omega raio, he COPAR porfolio is preferred over all remaining porfolios. For boh risk measures, he independen DFM porfolio ouperforms he DFM porfolio. We explain his finding wih a forunae spli of he panel daa. Furher, if we consider 95% Value a Risk (VaR) and 95% Condiional Value a Risk (CVaR), hen he hisorical porfolio ouperforms he ohers excep for one case. For σ = 16% and 95% VaR, he COPAR porfolio is slighly superior. To saisically assess he differences in he Sharpe raios, we perform he Jobson Korkie Tes from Jobson and Korkie (1981). The null hypohesis of his es is ha he Sharpe raios of he wo considered porfolios are equal. The normalized and cenered es saisics are asympoically Gaussian disribued wih mean 0 and variance 1 under he null hypohesis. If he null hypohesis is rejeced, hen here is significan saisical evidence for differen Sharpe raios. Appendix E presens resuls of he Jobson Korkie Tes for all pairs of porfolios and σ = 10%, 13%, 16%. The COPAR porfolio significanly ouperforms he independen DFM porfolio, and his advocaes our approach o model esimaed facors wih a COPAR model. For σ = 10% and σ = 13%, we do no see a saisically significan difference in Sharpe raios for hisorical and COPAR porfolios. We explain his finding wih lower sandard deviaions of he hisorical porfolio reurns. 6. Conclusions and Final Remarks This paper applies copulas o capure he dependence srucure of esimaed laen facors from dynamic facor models. The proposed modeling approach is especially convenien when several facor models under consideraion are esimaed separaely. In his conex, we combine he filered laen facors wih he COPAR model of Brechmann and Czado (2015), which resuls in a non-gaussian dependence beween he facors. The gained flexibiliy of he facor dependence is hen used for modeling asse reurns and building opimal mean-variance porfolios. In our empirical sudy, we consider U.S. panel daa consising of 9 financial and 13 macroeconomic monhly observable indicaors. The naure of indicaors suggess he consideraion of wo separae dynamic facor models. We also rea a join dynamic facor model for all indicaors. Esimaed facors from he considered DFMs are used o model reurns of 35 indusrial socks from S&P500. Then, facors predicions from differen models spanning almos 11 years are employed for porfolio opimizaion in he mean-variance framework. Our main conribuion is a performance improvemen of porfolios based on DFMs. For his, we propose o capure he dependence srucure of filered facors from DFMs wih a COPAR model. This allows us o sample facor forecass from he esimaed COPAR model condiionally on pas values of esimaed facors. The gained facor predicions are hen uilized o consruc an opimal mean-variance porfolio. Thus, we compare he COPAR-based porfolio wih wo porfolios derived from DFMs, as well as wih he classical mean-variance approach uilizing empirical means and covariance marices. For he considered panel daa and indusrial socks, we observe he ouperformance of he COPAR porfolio in erms of he oal reurn, he Sharpe and Omega raio. The superioriy of he COPAR-porfolio in erms of he Sharpe raio is even saisically significan for several porfolio comparisons. A possible improvemen of he proposed approach is is exension wih a model selecion crierion for a general R-vine, which bes capures he cross-secional and emporal dependence. Thus, one depars from he COPAR model and generalizes i as well as he copula based mulivariae ime series models of Beare and Seo (2015) and Smih (2015). In general, one can compleely revise our approach and alernaively develop dynamic versions of copula facor models as proposed by

19 Economerics 2017, 5, of 24 Krupskii and Joe (2013) and Oh and Paon (2017) for iid daa. The firs mehodology in his direcion is provided by Oh and Paon (2016), who capure cross-secional dependence wih ime-varying copulas. These poins are he subjec of furher research. Acknowledgmens: The auhors wan o hank he edior and he anonymous reviewers for heir very helpful suggesions, which conribued essenially o he improvemen of our manuscrip. We are also obliged o Eike Brechmann, who kindly provided R-code for COPAR model, which served as an inspiraion for compuer implemenaion of his paper. Franz Ramsauer graefully acknowledges he suppor of Pioneer Invesmens during his docoral phase. Eugen Ivanov graefully acknowledges he suppor of Deusche Forschungsgemeinschaf during his docoral phase. The auhors also express graiude o Leibniz-Rechenzenrum for providing compuaional resources and assisance. Auhor Conribuions: Eugen Ivanov, Aleksey Min and Franz Ramsauer analyzed he daa, consruced he model and designed he esimaion procedure. Asse allocaion procedure based on dynamical facor models is proposed by Franz Ramsauer. Eugen Ivanov and Franz Ramsauer performed he complee compuaional implemenaion. All hree auhors wroe he paper. Conflics of Ineres: The auhors declare no conflic of ineres. The sponsors had no role in he design of he sudy, in he collecion, analyses, or inerpreaion of daa, in he wriing of he manuscrip, and in he decision o publish he resuls. Appendix A. Asses for Porfolio Opimizaion Table A1. Asses used for porfolio consrucion. Ticker Name GICS Sub Indusry CMI Cummins Inc. Indusrial Machinery EMR Emerson Elecric Company Indusrial Conglomeraes CSX CSX Corp. Railroads MMM 3M Company Indusrial Conglomeraes BA Boeing Company Aerospace & Defense CHRW C. H. Robinson Worldwide Air Freigh & Logisics CAT Caerpillar Inc. Consrucion & Farm Machinery & Heavy Trucks CTAS Cinas Corporaion Diversified Suppor Services DHR Danaher Corp. Indusrial Conglomeraes DE Deere & Co. Consrucion & Farm Machinery & Heavy Trucks ETN Eaon Corporaion Indusrial Conglomeraes EFX Equifax Inc. Research & Consuling Services GD General Dynamics Aerospace & Defense GE General Elecric Indusrial Conglomeraes LLL L-3 Communicaions Holdings Indusrial Conglomeraes LEG Legge & Pla Indusrial Conglomeraes NSC Norfolk Souhern Corp. Railroads PBI Piney-Bowes Technology, Hardware, Sofware and Supplies RTN Rayheon Co. Aerospace & Defense RHI Rober Half Inernaional Human Resource & Employmen Services ROK Rockwell Auomaion Inc. Indusrial Conglomeraes COL Rockwell Collins Indusrial Conglomeraes UNP Union Pacific Railroads UPS Unied Parcel Service Air Freigh & Logisics UTX Unied Technologies Indusrial Conglomeraes WM Wase Managemen Inc. Environmenal Services TXT Texron Inc. Indusrial Conglomeraes FLR Fluor Corp. Diversified Commercial Services FDX FedEx Corporaion Air Freigh & Logisics PCAR PACCAR Inc. Consrucion & Farm Machinery & Heavy Trucks GWW Grainger (W.W.) Inc. Indusrial Maerials MAS Masco Corp. Building Producs R Ryder Sysem Indusrial Conglomeraes LMT Lockheed Marin Corp. Aerospace & Defense NOC Norhrop Grumman Corp. Aerospace & Defense

20 Economerics 2017, 5, of 24 Appendix B. Underlying Panel Daa In he following ables, FRED sands for Federal Reserve Bank of S. Louis. Table A2. U.S. financial indicaors. Indicaor Descripion Transformaion Source USA Treasury 3 m Yield of U.S. governmen bonds wih mauriy of 3 monhs Monhly change FRED USA Treasury 1y Yield of U.S. governmen bonds wih mauriy of 1 year Monhly change FRED USA Treasury 2y Yield of U.S. governmen bonds wih mauriy of 2 years Monhly change FRED USA Treasury 3y Yield of U.S. governmen bonds wih mauriy of 3 years Monhly change FRED USA Treasury 5y Yield of U.S. governmen bonds wih mauriy of 5 years Monhly change FRED USA Treasury 7y Yield of U.S. governmen bonds wih mauriy of 7 years Monhly change FRED USA Treasury 10y Yield of U.S. governmen bonds wih mauriy of 10 years Monhly change FRED Gold WTI Crude Gold Fixing Price 10:30 A.M. (London ime) in London Bullion Marke, based in U.S. Dollars. Crude Oil Prices: Wes Texas Inermediae (WTI) Cushing, Oklahoma. Reurn Reurn FRED FRED Table A3. U.S. macroeconomical indicaors. Indicaor Descripion Transformaion Source M1 U.S. M2 U.S. Narrow money (M1) includes currency, i.e., banknoes and coins, as well as balances which can immediaely be convered ino currency or used for cashless paymens, i.e., overnigh deposis. Inermediae money (M2) comprises narrow money (M1) and, in addiion, deposis wih a mauriy of up o wo years and deposis redeemable a a period of noice of up o hree monhs. Reurn Reurn FRED FRED Unemploymen Unemploymen rae Reurn FRED CPI PPI Consumer Price Index (CPI) measures changes in he price level of a marke baske of consumer goods and services purchased by households. Producer Price Index (PPI) measures he average changes in prices received by domesic producers for heir oupu Reurn Reurn FRED FRED PCE Measures goods and services purchased by U.S. residens Reurn FRED Personal saving rae Personal saving as a percenage of disposable personal income (DPI), frequenly referred o as he personal saving rae, is calculaed as he raio of personal saving o DPI. Reurn FRED Payrolls All Employees: Toal Nonfarm Payrolls. Reurn FRED Unemploymen Unemploymen Reurn FRED Iniial claims Iniial claims Reurn FRED Housing Sars Housing Sars Reurn FRED Capaciy uilizaion Capaciy uilizaion Reurn FRED Dollar Index Trade Weighed U.S. Dollar Index Reurn FRED

21 Economerics 2017, 5, of 24 Appendix C. Copula Families Table A4. Copula families as numbered in he R-package VineCopula. Number Family 0 Independence copula 1 Gaussian copula 2 Suden s copula 3 Clayon copula 4 Gumbel copula 5 Frank copula 6 Joe copula 13 Roaed Clayon copula (180 degrees; survival Clayon ) 14 Roaed Gumbel copula (180 degrees; survival Gumbel ) 16 Roaed Joe copula (180 degrees; survival Joe ) 23 Roaed Clayon copula (90 degrees) 24 Roaed Gumbel copula (90 degrees) 26 Roaed Joe copula (90 degrees) 33 Roaed Clayon copula (270 degrees) 34 Roaed Gumbel copula (270 degrees) 36 Roaed Joe copula (270 degrees) Appendix D. Porfolio Characerisics Table A5. Porfolio characerisics for annual volailiy σ = 10%. COPAR DFM ind DFM his Log-reurn (Toal) Log-reurn (Monhly) Sd. deviaion (Annual) Sharpe Raio (Monhly) Omega Raio (Monhly) % VaR % CVaR Table A6. Porfolio characerisics for annual volailiy σ = 13%. COPAR DFM ind DFM his Log-reurn (Toal) Log-reurn (Monhly) Sd. deviaion (Annual) Sharpe Raio (Monhly) Omega Raio (Monhly) % VaR % CVaR Table A7. Porfolio characerisics for annual volailiy σ = 16%. COPAR DFM ind DFM his Log-reurn (Toal) Log-reurn (Monhly) Sd. deviaion (Annual) Sharpe Raio (Monhly) Omega Raio (Monhly) % VaR % CVaR

22 Economerics 2017, 5, of 24 Appendix E. Significance Tess for Sharpe Raios Table A8. Tes saisics of he Jobson Korkie Tes for pairwise comparison of Sharpe raios for four porfolios from he mean-variance opimizaion problem wih σ = 10%. COPAR DFM ind DFM COPAR DFM 1.02 ind DFM 1.89 * 0.03 his Noe: * p-value < 0.1, ** p-value < 0.05, *** p-value < Table A9. Tes saisics of he Jobson Korkie Tes for pairwise comparison of Sharpe raios for four porfolios from he mean-variance opimizaion problem wih σ = 13%. COPAR DFM ind DFM COPAR DFM 1.62 ind DFM 2.20 ** 0.42 his Noe: * p-value < 0.1, ** p-value < 0.05, *** p-value < Table A10. Tes saisics of he Jobson Korkie Tes for pairwise comparison of Sharpe raios for four porfolios from he mean-variance opimizaion problem wih σ = 16%. COPAR DFM ind DFM COPAR DFM 2.03 ** ind DFM 2.27 ** 0.95 his 1.95 * Noe: * p-value < 0.1, ** p-value < 0.05, *** p-value < Appendix F. Tesing for Granger Causaliy Table A11. Summary of linear regressions for esing Granger causaliy. Columns conain esimae of regression coefficiens, and heir sandard errors are given in brackes. F f in 1, 1 F f in 2, 1 F macro 1, 1 F macro 2, 1 F macro 3, 1 Consan Dependen Variable: y F f in 1, F f in 2, F1, macro F2, macro F3, macro (1) (2) (3) (4) (5) *** *** *** * (0.054) (0.060) (0.053) (0.056) (0.055) ** *** (0.041) (0.045) (0.040) (0.042) (0.042) *** ** (0.047) (0.052) (0.046) (0.048) (0.048) * *** (0.046) (0.051) (0.045) (0.047) (0.047) *** *** (0.043) (0.047) (0.042) (0.044) (0.044) (0.052) (0.058) (0.051) (0.054) (0.053) Observaions R Adjused R Residual Sd. Error (df = 363) F Saisic (df = 5; 363) *** *** *** *** *** Noe: * p-value < 0.1; ** p-value < 0.05; *** p-value < 0.01.

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