Tail Risk and the Macroeconomy

Transcription

1 Tail Risk and he Macroeconomy Daniele Massacci Einaudi Insiue for Economics and Finance Ocober 3, 204 Absrac We empirically invesigae how ail risk relaes o macroeconomic fundamenals and uncerainy. We inroduce a novel univariae ime series model o sudy he dynamics of ail risk in nancial markes: we apply resuls from exreme value heory and build a ime-varying peaks over hreshold model; and we de ne he laws of moion for he parameers hrough he score-based approach. The resuling speci caion is an unobserved componens model. We furher propose a connecedness measure for he comovemen in ail risk among porfolios. We apply he model o daily reurns from U.S. size-sored decile sock porfolios and show ha ail risk is counercyclical: i goes up when fundamenals deeriorae and macroeconomic uncerainy increases; and larger rms end o respond more han smaller ones o changes in he underlying macroeconomic condiions. We furher show ha he degree of ail connecedness among porfolios is highly counercyclical. Evidence from inernaional developed markes srenghens our empirical ndings and shows ha ail connecedness has experienced an upward sloping rend. JEL classi caion: C22, C32, G2, G5. Keywords: Time-Varying Tail Risk, Score-Based Model, Tail Connecedness, Macroeconomic Fundamenals, Uncerainy. This paper grealy bene s from a conversaion and several suggesions from Andrew Harvey. I am graeful o seminar paricipans a EIEF and o Domenico Giannone for helpful commens. Financial suppor from he Associazione Borsisi Marco Fanno and from UniCredi & Universiy Foundaion is acknowledged. Address correspondence o Daniele Massacci, EIEF, Via Sallusiana 62, 0087 Roma, Ialy. Tel.: dm355@canab.ne.

2 Inroducion Financial risk measuremen is an imporan eld of research boh in academia and amongs praciioners: i is of cenral imporance o invesors in he conex of porfolio risk minimizaion; and o policy makers responsible for monioring sabiliy in nancial markes. I is a challenging and fascinaing ask, which needs deep knowledge of nancial markes on heir own and in connecion wih he broader economic environmen. Accurae risk measuremen requires processing informaion from (a leas) wo sources: economeric modelling of nancial markes (see Andersen e al. (203)); and macroeconomic assessmen of risk drivers (see Andersen e al. (2005)). In his paper we consider boh aspecs and provide a conribuion o he general opic of nancial risk measuremen. There exiss a voluminous lieraure providing economeric ools o measure risk wih respec o he cener of he condiional disribuion of nancial reurns or wihin a su cienly small neighborhood abou i (see Andersen e al. (203)): hose ools provide reliable risk measures during normal or ranquil imes. The widh of he neighborhood depends on he shape of he ails of he condiional disribuion of reurns: under gaussianiy, he ails are exponenially declining, periods of urmoil are ruled ou almos surely and he widh coincides wih he real line. Financial reurns are however condiionally fa-ailed (see Bollerslev (987)) and risk measures valid during ranquil periods are no informaive in periods of urbulen markes. This naurally shifs he aenion from he cener o he ails of he condiional disribuion of reurns and o he concep of ail risk: his can be di cul o measure in pracice since he condiional disribuion of nancial reurns during periods of disress may no be appropriaely described under sandard parameric assumpions. A powerful ool o measure ail risk is exreme value heory (EVT) (see Embrechs e al. (997)): his provides an approximaion o he disribuion of random variables along he lower and upper ail of he disribuion iself. EVT has been widely used in empirical nance o measure ail risk as relaed o he uncondiional disribuion of exreme reurns. Diebold e al. (998) discuss he applicaion of he power law o esimae ail probabiliies for nancial risk managemen. Longin and Solnik (200) and Poon e al. (2004) apply he peaks over hreshold (POT) of Picklands (975) and Davison and Smih (990) o model he join disribuion of exreme reurns in inernaional equiy markes: he building block of he POT is he generalized Pareo disribuion (GPD). Esimaion of he ails of he uncondiional disribuion of reurns is valid under he mainained 2

3 assumpion of random sampling: he daa generaing processes of nancial reurns however exhibi srucural breaks and ime-varying dynamics, which would lead o model misspeci caion if negleced (see Kearns and Pagan (997)). The lieraure on EVT as relaed o he uncondiional disribuion of exreme reurns has hen been exended along wo complemenary direcions. Quinos e al. (200) build formal ess for he null hypohesis of sabiliy in he index ha shapes he disribuion of exreme nancial reurns. Oher conribuions have modelled ime-varying ail behavior wihin he POT framework. Chavez-Demoulin e al. (204) use Bayesian mehods o build a nonparameric POT model for he condiional disribuion of exreme reurns hrough regime swiches driven by a Poisson process. Massacci (204) follows a frequenis approach o parameerize he ail index of he GPD as an auoregressive process wih innovaion equal o he previous period s realized exreme reurn: he choice of he innovaion in his model is ad hoc and calls for more generaliy. In his paper we build on he dynamic POT of Massacci (204) and generalize he law of moion for he ail index: we sill assume an auoregressive process; and we apply he score-based approach recenly proposed in Creal e al. (203) and Harvey (203). Following he score principle, he innovaion in he law of moion of he ail index depends on he previous period s suiably weighed realized score. The new model conribues o he lieraure in wo ways. Firs, he innovaion no longer is ad hoc chosen, bu i is moivaed by saisical argumens (see Blasques e al. (204)). Second, he model urns ou o be an unobserved componens model (see Harvey (989; 203)): i hen is a useful ool o invesigae he connecions beween ail risk and he macroeconomic environmen. We apply he model o measure ail risk dynamics along he lower and upper ails of he condiional disribuion of sock reurns. Following he discussion in Gabaix e al. (2003), we analyze ail risk across di eren rm sizes and markes : we focus on daily reurns from U.S. size-sored decile porfolios and equiy indices from inernaional developed markes, respecively. We sudy ail risk in each individual porfolio as well as he comovemen in ail risk across porfolios: o achieve he second goal, we inroduce he concep of ail connecedness, which builds upon he cumulaive risk fracion de ned in Billio e al. (202). Our empirical ndings are as follows. Wihin he U.S. sock marke, ail risk is highly persisen across all rm sizes: his is analogous o wha found in relaion o reurns volailiy (see Andersen e al. (2003) and references herein); and he degree of persisence increases wih rm size. We hen See Gabaix e al. (2003), p

4 invesigae he connecions beween he dynamics of ail risk and he macroeconomic environmen. We proceed by considering hree classes of macroeconomic indicaors: sock marke volailiy (see Schwer (989)), business cycle coinciden indicaors (see Sock and Wason (204)) and he uncerainy measures consruced in Jurado e al. (204). We carry ou a non-causal analysis based on sample correlaions (see Andersen e al. (203)), which shows ha ail risk as measured by our model is counercyclical and increases wih macroeconomic uncerainy; and a causal analysis based on a vecor auoregression (see Bloom (2009), Kelly and Jiang (204) and Jurado e al. (204)), which is able o quanify he e ec of macroeconomic uncerainy on ail risk. Consisenly wih he persisency paern we documen, we show ha he macroeconomic environmen a ecs large rms more han smaller ones: his complemens he resuls of Cenesizoglu (20), who shows ha a daily frequency large rms reac more han small ones o macroeconomic news. Using our proposed measure of ail connecedness, we uncover he linkages beween he comovemen in ail risk across porfolios and he macroeconomy: a negaive shock o he fundamenals or a higher level of uncerainy (or boh) are associaed o a higher level of connecedness. Our ndings from inernaional developed markes con rm hose from he U.S.. A marke level, ail risk is counercyclical and increases wih macroeconomic uncerainy. Tail connecedness among markes exhibi a similar paern. Ineresingly, ail connecedness also displays an upward rend: his is in line wih he ndings in Chriso ersen e al. (202), who show ha pairwise correlaions in developed markes have increased hrough ime. The remainder of he paper is organized as follows. Secion 2 reviews he relaed lieraure. Secion 3 lays ou he economeric model for he univariae condiional disribuion of exreme reurns and inroduces he idea of ail connecedness. Secion 4 presens empirical resuls from porfolios of U.S. socks. Secion 5 sudies inernaional sock markes. Finally, Secion 6 concludes. 2 Relaed Lieraure This paper relaes o several srands of economic lieraure, which are no independen of each oher. Our economeric model for he condiional disribuion of exreme reurns builds upon resuls from EVT hrough he POT of Picklands (975) and Davison and Smih (990). Balkema and De Haan (974) and Picklands (975) provide he mahemaical foundaion of he POT in a saic framework. Diebold e al. (998) discuss he pifalls of saic EVT wih dependen daa and volailiy clusering and sugges 4

5 o esimae he ime-varying ail of he condiional disribuion of reurns; Kearns and Pagan (997) con rm hrough a Mone Carlo analysis he inadequacy of he saic framework when nancial daa are in use. Chavez-Demoulin e al. (204) work in a Bayesian conex and model he condiional disribuion of exreme reurns hrough regime swiches governed by an underlying Poisson process. We work in a frequenis environmen ha allows o develop an unobserved componens model suiable o link ail risk o he macroeconomy (see Harvey (989; 203)). We also relae o Engle and Manganelli (2004): raher han using resuls from EVT, hey rely on a quanile auoregression wih a priori speci ed observable innovaion; hanks o our approach based on he POT, we build a more exible daa-driven unobserved componens model. We describe he economeric model for he condiional disribuion of exreme reurns in Secion 3.. The laws of moion we choose for he ime-varying parameers link our work o a voluminous lieraure in economerics. Cox (98) disinguishes beween wo classes of models wih dynamic parameers: observaion-driven models and parameer-driven models. In he observaion-driven approach, he laws of moion for he parameers are speci ed in erms of funcions of observable variables: popular examples are he ARCH model by Engle (982), he GARCH by Bollerslev (986; 987), he auoregressive condiional densiy by Hansen (994), and he recen score-based auoregressive models by Creal e al. (203) and Harvey (203). In parameer-driven models, he parameers are seen as sochasic processes wih an error componen: examples are sochasic volailiy models (see Shephard, 2005). As in Massacci (204), we follow he observaion-driven approach: raher han a priori specifying an ad hoc updaing mechanism in he laws of moion of he ime-varying parameers, we apply he more general daa-driven score-based principle of Creal e al. (203) and Harvey (203). We describe in deails our conribuion o his sream of lieraure in Secion 3.2. In Secion 3.4 we inroduce he concep of ail connecedness: his is informaive abou he comovemen in he probabiliy of ail evens in nancial reurns. Our measure di ers from wha proposed in Billio e al. (202) and Diebold and Yilmaz (204), who focus on reurns and realized volailiies as risk drivers, respecively. Tail connecedness canno be measured from he dynamic power law of Kelly and Jiang (204) and Kelly (204), who very ineresingly esimae ail risk from he cross secion of reurns under suiable assumpions. Finally, we relae o a large empirical lieraure on modelling he disribuion of nancial reurns. 5

6 Sochasic modelling of reurns races back o Bachelier (900), whose work implies ha heir uncondiional disribuion is Gaussian and he ails are exponenially declining. In laer work, Mandelbro (963) and Fama (965) have empirically rejeced he Gaussian assumpion and argued ha he uncondiional disribuion of reurns exhibis heavy ails: his leads o he noion of ail risk and o he applicaion of EVT for risk measuremen as in Embrechs e al. (997), Longin and Solnik (200) and Poon e al. (2004). In order o allow for ime-varying condiional momens, Engle (982) and Bollerslev (986) follow he observaion-driven approach o build economeric models under he mainained assumpion of condiional Gaussian disribuion; and Bollerslev (987) allows for a more exible suden- condiional disribuion. We do no impose any disribuional assumpion and provide an approximaion for he lower and upper ails of he condiional disribuion of reurns. We furher sudy how ails and ail connecedness relae o he macroeconomic environmen following similar seps as in Andersen e al. (2005) and Andersen e al. (203): we focus on business cycle indicaors (see Sock and Wason (204)) and macroeconomic uncerainy (see Bloom (204)) and perform a comprehensive analysis ha, o he very bes of our knowledge, is novel o he lieraure. We repor resuls for he U.S. and inernaional equiy markes in Secions 4 and 5, respecively. 3 Modelling he Condiional Disribuion of Exreme Reurns In his secion we propose a new framework o sudy he condiional disribuion of exreme reurns. We discuss he componens of he model in Secions 3. and 3.2: he dynamic POT advanced in Massacci (204); and he score-based updaing mechanism of Creal e al. (203) and Harvey (203). In Secion 3.3 we address he issues of esimaion and inference. We inroduce ail connecedness in Secion Condiional Disribuion of Exreme Reurns Le fr g T = be he ime sequence of random reurns on a porfolio of risky asses, where T denoes he size of he available sample. Following common pracice in he lieraure, we de ne exreme reurns in erms of exceedances wih respec o a xed hreshold value (see Longin and Solnik (200) and Chavez-Demoulin e al. (204)): we hen refer o he evens R < and R > as o negaive and posiive exceedances, respecively. For exposiional purposes, we focus on posiive exceedances: analogous resuls hold for negaive exceedances by an argumen of symmery. Le F (r jf ) be he condiional cumulaive 6

7 disribuion funcion of R, where F denoes he informaion se available a period : we le F (r jf ) be ime-varying; we also assume i is absoluely coninuous and posiive everywhere on he real line for all, so ha an underlying probabiliy densiy funcion for R exiss a each poin in ime. For a given quanile q of he condiional disribuion of R, i follows ha Pr (R q jf ) = F (q jf ); he even R > is hen assigned a condiional probabiliy p = Pr (R > jf ) = F ( jf ). We focus on posiive exceedances and work under he assumpion > 0. Le he condiional cumulaive disribuion funcion of posiive exceedances over be F (r jf ) = Pr ( R r jr > ; F ) = F (r jf ) F ( jf ) ; 0 < r ; F ( jf ) which is unknown wihou disribuional assumpions on he sequence of reurns fr g T =, namely wihou assumpions abou he analyical expression for F (r jf ). An approximaion for F (r jf ) is available under he POT of Picklands (975) and Davison and Smih (990): following Balkema and De Haan (974) and Picklands (975), he GPD is he only nondegenerae disribuion ha approximaes ha of exceedances as he hreshold approaches he upper bound of he condiional disribuion of R. Formally, le he condiional cumulaive disribuion funcion of he GPD as applied o R be 8 >< G (r jf ) = >: /k r + k ; r ; > 0; k 6= 0 > 0; + r exp ; r ; > 0; k = 0 > 0; () where () + denoes he posiive par of he argumen beween brackes: we allow boh he shape parameer k and he scale parameer o be ime-varying, so ha G (r jf ) is ime-dependen. The following uniform convergence resul as applied o F (r jf ) holds (see Balkema and De Haan (974) and Picklands (975)): lim sup!+ r <+ jf (r jf ) G (r jf )j = 0: (2) Given > 0, he suppor of G (r jf ) depends on k : he suppor is r and r /k for k 0 and k < 0, respecively. The shape of G (r jf ) depends on and k (see Smih (985), Davison and Smih (990) and Ledford and Tawn (996) for echnical deails). The scale parameer > 0 depends on he hreshold. The parameer k deermines he shape of he upper ail of he condiional 7

8 disribuion of reurns and i is independen of : he case k < 0 corresponds o disribuions wih nie suppor (e.g., uniform disribuion); he case k = 0 o an exponenially declining ail (e.g., Gaussian disribuion); and he case k > 0 o a heavy-ailed disribuion obeying he power law (e.g., suden- disribuion). The condiional disribuion of nancial reurns exhibi heavy ails (see Bollerslev (987)) and we assume k > 0 for all : he higher k, he higher he value of p = Pr (R > jf ). In wha follows, we inerchangeably refer o k as o he shape parameer or he ail index. Since k plays a key role in measuring ail risk, we aim a characerizing is dynamic properies. We rea realizaions of R below as censored a (see Ledford and Tawn (996)). We de ne Y = max (R ; 0), wih corresponding condiional cumulaive disribuion funcion Pr (Y y jf ) = H (y jf ): for y = 0 i follows ha H (y jf ) = p ; and for y > 0 we obain he approximaion /k H (y jf ) = p + p G y (r jf ) = p + k ; k > 0; > 0: + Le I () denoe he indicaor funcion; he condiional disribuion funcion of posiive exceedances is H (y jf ) = I (y = 0) ( p ) + I (y > 0) " p + k y /k + # ; k > 0; > 0: (3) The funcion H (y jf ) in (3) requires informaion abou p = Pr (R > jf ). Under k > 0, he condiional disribuion of reurns belongs o he maximum domain of aracion of he Fréche disribuion (see Embrechs e al. (997)). Following Massacci (204), we approximae p as a power law muliplied by a ime-varying funcion L (q ) slowly varying a in niy: formally, Pr (R > q jf L ) = L (q ) q /k (cq ) ; lim q!+ L (q ) = ; q > 0; k > 0; c > 0: (4) We parameerize he funcion L (q ) as /k q L (q ) = ; q > 0; k > 0: (5) + q From (4) and (5) i follows ha p = Pr (R > jf ) = /k ; > 0; k > 0 : (6) + 8

9 he parameerizaion of L (q ) in (5) ensures ha p lies wihin he uni inerval and i is a probabiliy measure for all > 0; in addiion, p is monoonically decreasing in and increasing in k and sais es lim!0 p =, lim!+ p = 0, lim k!0 p = 0 and lim k!+ p =. From (3) and (6), we obain he following analyical expression for H (y jf ): " H (y jf ) = I (y = 0) " +I (y > 0) + # /k /k + + k y /k + # ; > 0; k > 0; > 0: (7) The analyical formulaion in (7) combines wo models from EVT: he POT for he condiional cumulaive disribuion funcion of exceedances; and he power law for he condiional probabiliy of an exceedance. The model is an example of a dynamic censored regression model, he dynamic Tobi is a exbook example of (see Hahn and Kuerseiner (200) and references herein): compared o he dynamic Tobi, we replace he disribuional assumpion imposed on he underlying coninuous dependen variable wih an approximaion for he condiional disribuion of posiive exceedances dicaed by EVT. 3.2 Time-Varying Parameers We now parameerize he laws of moion for he ime-varying parameers k and in H (y jf ) in (7). We follow he observaion-driven approach and apply he score-based mechanism of Creal e al. (203) and Harvey (203). We resor o Harvey (203) o build a GPD wih ime-varying parameers 2. We de ne = /k and = /k and reparameerize () as G (r jf ) = + (r & ) ; > 0; > 0; > 0; + where and are shape and scale parameers, respecively: G (r jf ) hen comes from he Burr disribuion by seing o uniy a suiable shape parameer 3. I follows ha G (r jf ) wih > 0 is heavy-ailed and he limiing case! leads o an exponenially declining ail. In he res of he paper, we inerchangeably refer o as o he shape parameer or he ail index: he dynamics of are 2 See Harvey (203), Secion 5:3:5. 3 On he relaionship beween he GPD and he Burr disribuion see Harvey (203), Secions 5:3:4 and 5:3:5. 9

10 he main focus of his paper. By reparameerizing (6), we have p = Pr (R > jf ) = & ; > 0; > 0; (8) + which sais es lim!0 p =, lim!+ p = 0, lim &!0 p = and lim &!+ p = 0: from (3) and (8), he cumulaive disribuion funcion H (y jf ) becomes H (y jf ) = I (y = 0) " +I (y > 0) & + & + y + & + # ; > 0; & > 0; > 0: (9) The dynamics of are cenral o our work: i is hen imporan o undersand how he model in (9) relaes o oher speci caions ha allow o esimae he sequence of shape parameers f g T =. We obain a less informaive se up by replacing (9) wih ~H (y jf ) = I (y = 0) & & + I (y > 0) ; > 0; > 0 : + + ~H (~y jf ) does no use he informaion abou he magniude of he exceedance coming from he POT and provides a less e cien esimaor for han H (y jf ). As in he Hill esimaor (see Hill (975)), one could use informaion semming from exceedances only and de ne _H (y jf ) = I (y > 0) " + & + y + & + # & ; > 0; > 0; > 0 : _H (y jf ) ignores informaion from he probabiliy mass a y = 0 and e ciency issues arise. We specify he laws of moion for and by following similar seps as in Massacci (204). We wrie > 0 in exponenial form as ln = 0 + ln + 2 u : (0) 0, and 2 are scalar parameers; ln inroduces an auoregressive componen; and he updae u requires 2 6= 0 for ideni caion purposes (unless is a priori known o be zero). The scale 0

11 parameer > 0 eners he condiional disribuion funcion in (9) only when a posiive exceedance occurs (i.e., when y > 0) and we do no observe i over he enire sample period = ; : : : ; T : we model in exponenial form as ln = ' 0 + ' ln + ' 2 u ; () where ' 0, ' and ' 2 are scalar parameers. In wriing () we assume ha he componens ha deermine he dynamics in also drive he law of moion for. The model in () is more general han wha suggesed in Chavez-Demoulin e al. (2005), who assume he realized exceedance is he only driver of he scale parameer 4 (i.e., ' = 0 and u = y ): Massacci (204) empirically shows ha he ail index provides valuable informaion for undersanding he ime-varying properies of he scale parameer. The key componen in he laws of moion for and in (0) and () is he common updaing mechanism u : his is known given F under he observaion-driven approach. A naural choice would be u = y, as in Chavez-Demoulin e al. (2005) and Massacci (204): his easily implemenable soluion is however an ad hoc and resricive choice. We hen op for he daa-driven score-based mechanism of Creal e al. (203) and Harvey (203): his innovaive mehodology relaes he innovaion u o he score of he underlying likelihood funcion, which is a known quaniy given F. Formally, from (9) le h (Y jf ) = I (Y = 0) +I (Y > 0) & + " + & + Y & + # ; > 0; & > 0; > 0 : (2) according o he score-based mechanism, u is known given F and de ned as u = ( E ( 2 ln [h (Y jf ln [h (y jf (ln ) 2 jf ln where (see Appendix A for deails) he realized score wih respec o ln is ( " ln [h (y jf )] = I (y > 0) + ln + ln + & + #) 2 I (y = 0) & + & 3 & 7 5 ln + 4 See Chavez-Demoulin e al. (2005), p. 23.

12 and he informaion quaniy is E ( 2 ln [h (Y jf (ln ) 2 jf = & 8> < + + >: & 2 9> = & ln : + >; + A rs sigh he dynamic score-based updaing mechanism may seem o be an ad hoc choice. Creal e al. (203) show ha i ness a variey of popular models such as he GARCH (see Bollerslev (986)) and he ACD (see Engle and Russell (998)). Blasques e al. (204) prove ha his soluion is informaion heoreic opimal, in he sense ha he parameer updaes always reduce he local Kullback-Leibler divergence beween he rue condiional densiy and he model implied condiional densiy. Harvey (203) relaes he dynamic score-based updae o he unobserved componens described in Harvey (989): his las inerpreaion makes our model paricularly suiable o sudy he connecions beween ail risk and he macroeconomic environmen. 3.3 Esimaion and Inference 3.3. Threshold Value As explained in Secion 3., we keep in (9) xed over he enire sample and se i equal o a prespeci ed quanile of he empirical disribuion of reurns. The choice of is a delicae issue. On he one hand, a low makes many observaions wih y > 0 available o esimae he parameers of H (y jf ); a he same ime, he GPD may provide a poor approximaion o he rue unknown underlying disribuion. On he oher hand, a high is consisen wih he heoreical limiing resul in (2); however, only few exceedances become available. The choice of creaes a rade-o beween model misspeci caion and esimaion noise, which leads o a rade-o beween bias and e ciency. The lieraure has proposed rigorous mehods o selec, as recenly surveyed in Scarro and MacDonald (202). In his paper we follow Chavez-Demoulin and Embrechs (2004) and Chavez-Demoulin e al. (204) and se such ha 0% of he realized reurns are classi ed as exceedances: Chavez-Demoulin and Embrechs (2004) show ha small variaions of lead o lile variaion on he esimaed values of he parameers. 2

13 3.3.2 Maximum Likelihood Esimaion Le = ( 0 ; ; 2 ; ' 0 ; ' ; ' 2 ) 0 denoe he vecor of parameers ha characerize he laws of moion for and in (0) and (), respecively: from he observaion-driven approach, he condiional disribuion of Y given he informaion se F is known up o. Score-based models can be esimaed by maximum likelihood (see Creal e al. (203) and Harvey (203)): we hen suiably exend he maximum likelihood esimaor discussed in Smih (985) and Davison and Smih (990). From (9), he likelihood conribuion from he realizaion of Y is h (y jf ) = I (y = 0) +I (y > 0) & + " + & + y & + # ; > 0; & > 0; > 0 : he maximum likelihood esimaor ^ for he rue vecor of parameers solves ^ = arg max L () ; where L () = Q T = h (y jf ) is he likelihood funcion. In he saic case, Smih (985) provides su cien condiions for he maximum likelihood esimaor of he GPD o be consisen, asympoically normally disribued and e cien. In he dynamic se up we consider, we proceed as in Creal e al. (203) and conjecure ha under appropriae regulariy condiions ^ is consisen and sais es T /2 ^ d! N (; ) ; = 2 log L 0 = : 3.4 Tail Connecedness Given a se of porfolios of risky asses, we idenify connecedness measures wih respec o he underlying meric and risk drivers: Billio e al. (202) apply principal componens analysis o nancial reurns; Diebold and Yilmaz (204) build several measures from variance decomposiions o sudy realized volailiies. Le be he N vecor of shape parameers from N porfolios of asses: we rack he sequence f g T = and quanify connecedness using he risk fracion as in Billio e al. (202). Le ^ = T TP ( &) ( &) 0 = 3

14 be he N N sample covariance marix of, where & = T P T = is he sample mean of. The risk fracion is he raio beween he maximum eigenvalue of ^ and he sum of all eigenvalues of ^: i lies wihin he uni inerval by consrucion; and he higher he risk fracion, he higher he degree of connecedness among he elemens of, as he rs principal componen is able o explain a higher porion of he variance of. 4 Resuls from he U.S. Sock Marke In his secion we uilize he heory presened in Secion 3 in relaion o reurns from U.S. sock porfolios: we build all empirical models as described in Secions 3. and 3.2; we classify realized reurns as exceedances according o he crierion discussed in Secion 3.3.; we run maximum likelihood esimaion in line wih he mehodology presened in Secion 3.3.2; and we measure ail connecedness as in Secion 3.4. We perform he analysis in Ox 7: (see Doornik (202)); and we implemen he maximizaion algorihm wih saring values & and equal o he esimaes from he saic model obained by seing = 2 = 0 and ' = ' 2 = 0 in (0) and (), respecively. We provide deails abou daa, esimaion resuls, connecedness and links o he macroeconomy in Secions 4., 4.2, 4.3 and 4.4, respecively. 4. Daa We use daily observaions from he Cener for Research in Securiy Prices (CRSP) and consider wo ses of daa: he value-weighed price index for NYSE, AMEX, and NASDAQ; and he price indices for sizesored decile porfolios. From each index value, we consruc he reurn r as he percenage coninuously compounded reurn. The sample period begins in January 954 and ends in December 202, a oal of 485 daily observaions: he long ime series of available daa allows o conduc inference on exreme reurns, which we de ne by seing he hreshold as discussed in Secion We use daily observaions from daases previously employed in imporan empirical conribuions on risk measuremen in equiy markes. Given a sample of quarerly observaions for he value-weighed price index for NYSE, AMEX, and NASDAQ, Ludvigson and Ng (2007) follow a facor approach o analyze he risk-reurn relaion. Perez-Quiros and Timmermann (2000) use monhly daa for size-sored decile porfolios and show ha rms of di eren size exhibi heerogeneous responses o business cycle ucuaions. We gain informaion abou marke wide ail risk from he reurns on he value-weighed porfolio. Decile-sored porfolios le 4

15 us sudy ail risk dynamics across rm size (see Gabaix e al. (2003)) and measure ail connecedness: Decile and Decile 0 correspond o he smalles (i.e., small caps) and he larges rms (i.e., large caps), respecively, and rm size monoonically increases from he former o he laer. We show descripive saisics and correlaion marix for daily porfolio reurns in Table. Table abou here As expeced, reurns from he value-weighed index and from he porfolio of larges rms have similar feaures. The porfolio mean decreases in marke size, and he rs four decile porfolios have lower sandard deviaion han he remaining six: he porfolios from he wo ses of smaller rms are opimal in a mean-variance sense. Unlike in sandard asse pricing models, mean and sandard deviaion do no decline as one moves from smalles rms o larges rms porfolios: he decline in mean and sandard deviaion as a funcion of rm size is resored wih monhly daa (see Perez-Quiros and Timmermann (2000) and he di erence is due o he sampling frequency. All porfolio reurns exhibi negaive skewness and excess kurosis: he null hypohesis of Gaussianiy is always rejeced a any convenional level. Finally, he correlaion beween decile porfolios is higher he closer hey are ranked in erms of degree of marke capializaion. 4.2 Esimaion Resuls We now esimae he model in (9), (0) and (). We collec resuls for negaive and posiive exceedances in Tables 2 and 3, respecively. Table 2 abou here Table 3 abou here The resuls for negaive exceedances (see Table 2) show ha and 2 in (0) are posiive in all empirical models: is very close o uniy and ail risk is highly persisen boh a marke level and across rms; is also increasing in rm size. Analogous resuls apply o posiive exceedances (see Table 3), where he persisence of he shape parameer is more pronounced han i is for negaive exceedances. We provide furher insighs on he uncondiional disribuion of esimaed daily ail indices in Tables 4 and 5, where we repor descripive saisics and correlaion marix for boh negaive and posiive 5

16 exceedances, respecively. The sample period is , a oal of 4600 observaions: we exclude esimaes from 954 o 955 o minimize he e ecs induced by he saring values chosen in he esimaion algorihm previously discussed. Table 4 abou here Table 5 abou here In he case of negaive exceedances (see Table 4), he descripive saisics (see Panel A) show ha sample mean, sandard deviaion and median of decrease wih rm size: on average and median erms, ail risk is higher for larger rms han i is for smaller ones; and he volailiy of ail risk decreases in rm size. The average value of he shape parameer falls approximaely beween 3:549 (Decile 0) and 4:447 (Decile 2): he former value relaes o he porfolio of larges rms and resembles he uncondiional poin esimae discussed in Gabaix e al. (2003). The empirical disribuion of shape parameers is negaively skewed across all porfolios and he degree of skewness diminishes wih rm size. The pairwise correlaions beween ail indices are seizable and end o be higher he closer porfolios are ranked in erms of rm size (see Panel B). Similar resuls hold in he case of posiive exceedances (see Table 5). We provide a graphical represenaion of he resuls colleced in Tables 4 and 5 by ploing he sequences of daily shape parameers esimaed over he sample period for negaive and posiive exceedances in Figure : o highligh he main feaures, we concenrae on value-weighed, small caps (i.e., Decile ) and large caps (i.e., Decile 0) porfolios. Figure abou here Consisenly wih he resuls in Table 2, he sequence of ail indices for negaive exceedances from large caps is more persisen han he one from small caps and i is similar o ha from he value-weighed porfolio; i also shows a more pronounced cyclical behavior associaed o sronger counercyclical ail risk. Analogous consideraions hold for posiive exceedances, where he series are more persisen han he counerpars from negaive exceedances. The graphical analysis suggess ha he business cycle may a ec large rms more han small ones: we provide sronger evidence of his resul in Secion

17 4.3 Connecedness Analysis We now measure connecedness among decile-sored CRSP porfolios using he sraegy deailed in Secion 3.4. We rea he esimaed sequences of shape parameers as he objec of ineres (see Andersen e al. (2003)): his allows us o overcome empirical and heoreical issues relaed o esimaion noise. We allow for ime-varying connecedness o accoun for dynamic e ecs induced by evens such as he business cycle or nancial crises. We esimae he covariance marix on a daily frequency using a rolling window of 00 daily observaions, which provides a good balance beween esimaion error and exibiliy: he sequences of risk fracions for negaive and posiive exceedances run over he period , a oal of 4600 observaions. If we assume an underlying linear facor speci caion, he risk fracion is a funcion of he parameers from he underlying model: allowing for ime-varying risk fracion is equivalen o allowing for ime-varying parameers in he facor model for he shape parameers; and linear models wih ime-varying parameers are general forms of nonlinear models, as emphasized in Whie s Theorem repored in Granger (2008). Allowing for ime-varying risk fracion hen means allowing for an underlying nonlinear facor model. We provide a measure of connecedness: unlike Billio e al. (202) and Diebold and Yilmaz (204), we do no aemp o uncover he underlying nework srucure. We show he sequences of risk fracions for negaive and posiive exceedances in Figure 2. Figure 2 abou here Boh sequences exhibi high degree of ime dependence and counercyclical paern; when connecedness peaks, he maximum eigenvalue capures almos he enire risk fracion. We collec resuls from analyical analysis of he series of maximum eigenvalues in Table 6, which repors descripive saisics, correlaion marix and auoregressive coe cien obained from ing an auoregressive process of order one. Table 6 abou here The resuls clearly show ha ail connecedness is generally higher and less volaile for negaive exceedances han i is for posiive ones (see Panel A); i is also highly persisen (see Panel C). 7

18 4.4 Macroeconomic Deerminans We now sudy he links beween ail risk and he macroeconomy. We proceed in wo seps: we rs run a "non-causal" analysis in Secion 4.4. o place our work wihin a macroeconomic framework (see Andersen e al. (203)); we hen perform a srucural invesigaion in Secion o sudy he e ecs of uncerainy shocks on ail risk Non-Causal Analysis We relae he sequences of shape parameers for negaive and posiive exceedances and of maximum eigenvalues o key macroeconomic indicaors racking business cycle dynamics and macroeconomic uncerainy: he business cycle is a key driver of marke risk (see Andersen e al. (203)) and asymmerically a ecs he disribuion of sock reurns across di eren rm sizes (see Perez-Quiros and Timmermann (2000)); macroeconomic uncerainy ineracs wih ail risk in producing e ecs on real aciviy (see Kelly and Jiang (204)); and macroeconomic uncerainy is counercyclical (see Bloom (204)). We employ macroeconomic indicaors available a monhly frequency and aggregae daily sequences ino monhly values by compuing monhly medians: Kelly (204) calculaes averages wihin he monh; however, unlike momens, quaniles of disribuions are robus o ouliers and he median is likely o provide more accurae informaion abou he cenral endency of he monhly disribuion of ail indices (see Kim and Whie (2004)); quaniles are also equivarian under monoone ransformaions (see Koenker (2005)) and we can map monhly medians of ail indices ino monhly medians of condiional probabiliies of exceedances. We consider several macroeconomic indicaors. The rs one is he recession dummy (RD), which akes uni value during recession periods de ned according o he NBER classi caion and i is oherwise equal o zero: he binary RD indicaor models he business cycle as a discree process, a view implicily embedded in Perez-Quiros and Timmermann (2000). We hen look a hree ses of coninuous macroeconomic indicaors. Reurns volailiy is counercyclical (see Schwer (989)) and for each individual index we consider wo measures of volailiy: monhly realized volailiy (RV), compued as he sum of squared daily reurns wihin he monh (see Schwer (989)); and he monhly long-run marke volailiy measure (LRV) proposed in Mele (2007). For each porfolio, we compue RV and LRV from he corresponding sequence of reurns. Following Sock and Wason (204) we consider ve coinciden indicaors: indus- 8

19 rial producion (IP), nonfarm employmen (EMP), real manufacuring and wholesale-reail rade sales (MT), real personal income less ransfers (PIX) and he index published monhly by The Conference Board (TCB) 5 ; we compue log-di erences for IP (IP), EMP (EMP), MT (MT), PIX (PIX) and TCB (TCB); oher condiions being equal, an increase in IP, EMP, MT, PIX or TCB signals an improvemen in he underlying macroeconomic condiions. Finally, we analyze he linkages wih macroeconomic uncerainy hrough he measures U(h) proposed in Jurado e al. (204) for horizons h = ; 3; 6; 2 6. The indicaors RD, RV, LRV, U(), U(3), U(6) and U(2) are counercyclical; IP, EMP, MT, PIX or TCB are cyclical. We perform a correlaion analysis as suggesed in Andersen e al. (203). We collec he resuls for negaive and posiive exceedances and for he risk fracions in Tables 7, 8 and 9, respecively. Table 7 abou here Table 8 abou here Table 9 abou here Each able repors descripive saisics, correlaion marix, leas squares esimaes for he auoregressive coe cien from ing an auoregressive process of order one, and correlaions wih he macroeconomic indicaors discussed above. Due o daa availabiliy, we compue correlaions wih macroeconomic indicaors using di eren sample periods: 955 : : 2 for RD, RV and LRV; 960 : : 06 for IP, EMP, MT, PIX and TCB; and 96 : 0 20 : for U(), U(3), U(6) and U(2). Saring from negaive exceedances (see Table 7), he persisence in he sequence of monhly medians increases in rm size (see Panel C): his is consisen wih he paern in shown in Table 3. Tail risk is clearly counercyclical (see Panel D). Negaive correlaions arise across all porfolios wih RD, he volailiy measures RV and LRV, and he uncerainy measures U(), U(3), U(6) and U(2): higher ail risk in equiy markes is associaed wih higher volailiy and higher macroeconomic uncerainy. On he oher hand, we observe posiive correlaions wih IP, EMP, MT, PIX and TCB: higher ail risk is associaed wih a deerioraion of macroeconomic fundamenals. Correlaions wih macroeconomic 5 We hank Mark Wason for making he daase available online a hp:// 6 We hank Sydney Ludvigson for making he daase available online a hp:// 9

20 indicaors clearly increase wih rm size. This resul creaes ension wih Perez-Quiros and Timmermann (2000), who show ha monhly reurns from small caps are more a eced by he business cycle han reurns from large caps: our ndings complemen hose in Cenesizoglu (20), who shows ha a daily frequency large rms reac more han small ones o macroeconomic news; and sugges ha furher insighs on ail risk may be gained by looking a he scaling properies of he disribuion of reurns (see Manegna and Sanley (995) and Timmermann (995)). Over all invesmen porfolios, ail risk is mos highly correlaed wih he volailiy measures RV and LRV; i is also correlaed wih he macroeconomic uncerainy measures proposed in Jurado e al. (204), which we deal wih in greaer deails in Secion Analogous qualiaive resuls hold for posiive exceedances (see Table 8). As for he sequences of risk fracions (see Table 9), ail connecedness is counercyclical over boh sides of he condiional disribuion of reurns and i increases wih macroeconomic uncerainy Causal Analysis: he Role of Macroeconomic Uncerainy We now run a causal analysis and invesigae he impac of macroeconomic uncerainy on ail risk by esimaing impulse response funcions from a vecor auoregressive (VAR) similar o hose consruced in Bloom (2009), Kelly and Jiang (204) and Jurado e al. (204). We op for he uncerainy measures consruced in Jurado e al. (204): hese are de ned as he common variaion in he unforecasable componen of a large number of economic indicaors; hey measure macroeconomic uncerainy, as opposed o microeconomic merics available in he lieraure (see Bloom (204)); unlike measures of dispersion such as marke volailiy, hey are no conaminaed by facors relaed o ime-varying risk aversion and herefore provide a more accurae measure of he rue economic uncerainy (see Bekaer e al. (203) and Jurado e al. (204)). As in Kelly and Jiang (204), in our VAR speci caion U(h) is rs, followed by he monhly median values of shape parameers (i.e., he measure of ail risk), he Federal Funds Rae, log average hourly earnings, log consumer price index, hours, log employmen and log indusrial producion 7 : uncerainy is counercyclical (see Bloom (204)) and he inclusion of coinciden indicaors allows o measure more accuraely he impac of uncerainy on ail risk. Unlike Kelly and Jiang (204), we focus on he nancial implicaions of macroeconomic uncerainy, raher han on he e ec of ail risk on macroeconomic aggregaes: in his respec, our work relaes o Andersen e al. (2005). We esimae 7 Average hourly earnings, hours and employmen are calculaed for he manufacuring secor as in Bloom (2009). 20

21 he VAR over he period 96 : : 2 for h = ; 3; 6; 2 and in each model we selec wo lags according o he BIC crierion. Based on he resuls presened in Secion 4.4., we esimae impulse responses for negaive and posiive exceedances, and for value-weighed, small caps (i.e., Decile ) and large caps (i.e., Decile 0) porfolios. In Figure 3 we plo percenage changes in he condiional probabiliies of exceedances induced by a onesandard deviaion shock o uncerainy: we obain he responses from hose of he shape parameers hrough he equivariance propery of quaniles. Figure 3 abou here In he case of negaive exceedances and depending on he uncerainy horizon, he value-weighed porfolio experiences an increase in ail risk beween 6 and 2 monhs afer he shock, wih magniude relaed o h and comprised beween 2% and 2:5%; he recovery is compleed afer 38 o 45 monhs, and a slow convergence o he equilibrium follows. More visible resuls hold for he Decile 0 porfolio. The response of small caps o macroeconomic uncerainy shocks is less pronounced and exhibi an earlier iming. As for posiive exceedances, all porfolios have impulse responses wih lower peaks and slower recovery han he homologous from negaive exceedances. The resuls from impulse response analysis show ha uncerainy shocks impac large rms more han small ones. 5 Evidence from Inernaional Sock Markes We now urn he aenion o inernaional equiy markes (see Gabaix e al. (2003)). We use daily observaions from he Morgan Sanley Capial Inernaional (MSCI) indices. We consider he following eleven developed markes: Ausralia, Canada, France, Germany, Ialy, Japan, he Neherlands, Sweden, Swizerland, he Unied Kingdom and he U.S.. We ake he perspecive of an unhedged U.S. invesor and express all indices in U.S. dollars. From each index value, we consruc he reurn r as he percenage coninuously compounded reurn. The sample period begins in January 975 and nishes in December 202. We accoun for holidays by deleing observaions from rading days for which a leas one marke has a reurn idenically equal o zero: we end up wih 9300 daily daa poins; and we de ne negaive and posiive exceedances as in Secion 4. Inernaional equiy markes have di eren opening imes 2

22 and a synchronizaion issue arises (see Marens and Poon (200)): we however do no run a causaliy analysis beween markes and synchronizaion is no of rs order imporance. Table 0 shows ha all inernaional porfolio reurns exhibi negaive skewness and excess kurosis, and he null hypohesis of Gaussianiy is always rejeced a any convenional level. Table 0 abou here We collec resuls from model esimaion for negaive and posiive exceedances in Tables and 2, respecively. Table abou here Table 2 abou here The resuls for negaive exceedances (see Table ) show ha and 2 in (0) are posiive in all markes: is very close o uniy and i falls beween 0:972 (Ausralia) and 0:99 (Sweden). Analogous resuls apply o posiive exceedances (see Table 2): ranges beween 0:987 (Swizerland) and 0:997 (Canada). Resuls from inernaional markes con rm ha ail risk is highly persisen over boh sides of he condiional disribuion of reurns; and ha persisency is more pronounced along he righ ail. We provide furher insighs abou he empirical disribuion of he esimaed sequences of shape parameers for negaive and posiive exceedances in Tables 3 and 4, respecively. Table 3 abou here Table 4 abou here The sample period is , a oal of 9077 observaions: we exclude esimaes from 975 o 976 o minimize he e ecs induced by he saring values in he esimaion algorihm discussed in Secion 4. In he case of negaive exceedances (see Table 3), he descripive saisics (see Panel A) show ha he sample mean falls beween 2:483 (Ialy) and 3:229 (he U.S.) and ha he empirical disribuion is negaively skewed across all inernaional porfolios; pairwise correlaions are sizeable (see Panel B), ranging from 0:37 (Japan and he U.K.) o 0:796 (Germany and he Neherlands). Analogous resuls 22

23 hold in he case of posiive exceedances (see Table 4): excepions are he empirical disribuions for Germany, Japan and he Unied Kingdom, which are posiively raher han negaively skewed (see Panel A). In Figure 4 we plo he sequences of daily shape parameers over he sample period for negaive and posiive exceedances for Germany, Japan and he Unied Kingdom: he series con rm ha ail-heaviness is highly ime-varying. Figure 4 abou here Table 5 collecs resuls from analyical analysis of he series of risk fracions (esimaed wih a rolling window of 00 observaions) by reporing descripive saisics, correlaion marix and auoregressive coe cien obained from ing an auoregressive process of order one: i is imporan o noice ha means and medians are subsanially lower han hose repored in Table 6 in relaion o he U.S. marke. Table 5 abou here We shed ligh on his resul in Figure 5, which shows he sequences of risk fracions for negaive and posiive exceedances: boh sequences exhibi a clear upward rend wihin he second half of he sample, wih a corresponding increase in connecedness 8. Figure 5 abou here We presen his nding from a di eren angle by compuing he mean values for he sequences of risk fracions over he four overlapping sub-samples , , and : he gures (in percenage erms) for negaive exceedances are 59:46, 6:97, 67:79 and 77:2, respecively; and hose for posiive exceedances are 58:55, 60:88, 65:89 and 74:6, respecively. These values con rm he upward rend in ail connecedness graphically shown in Figure 5. Chriso ersen e al. (202) documen an analogous behavior in reurns correlaions in developed counries: our resuls are complemenary as we are ineresed in he ail indices as risk drivers. We nally conduc a non-causal analysis. We consider several macroeconomic indicaors sampled a monhly frequency. The rs one is realized volailiy (RV) (see Schwer (989)): for each ail index, 8 We run a formal saisical analysis o srenghen he visual inspecion we make in he paper: he resuls are available upon reques. 23

24 we compue i as he sum of squared daily reurns wihin he monh from he corresponding counry porfolio; as for he risk fracion, we proceed as in Gourio e al. (203) and consruc i as he average counry-level realized volailiy, and i hen proxies global volailiy. As coinciden indicaors, we include log-di erences for indusrial producion a counry level (IP) and for he G7 counries (IP_G7), he laer being a proxy for global economic aciviy 9. Uncerainy measures such as hose consruced in Jurado e al. (204) are no available for inernaional economies. We de ne macroeconomic uncerainy in erms of volailiy of indusrial producion growh (see Segal e al. (204)): we measure counry-speci c and global macroeconomic uncerainy as Mele s (2007) monhly long-run volailiy of IP (LRV_IP) and IP_G7 (LRV_IP_G7), respecively. Finally, we sudy he poenial connecions o U.S. speci c macroeconomic uncerainy by including U(h), for h = ; 3; 6; 2. Due o daa availabiliy, we compue summary correlaions over he period and collec he resuls in Table 6. Table 6 abou here The resuls con rm ha counry-level ail risk and ail connecedness (as measured by he risk fracion) are counercyclical along boh ails of he condiional disribuion of reurns, as evidenced by he summary correlaions calculaed wih respec o RV, IP and IP_G7; hey also increase in he level of uncerainy as measured by LRV_IP, LRV_IP_G7 and U(h). 6 Concluding Remarks Undersanding he connecions beween ail risk and he macroeconomy is crucial for risk measuremen and managemen. We build a novel univariae economeric model o describe he condiional disribuion of exreme reurns: he model exends he peaks over hreshold from exreme value heory o allow for ime-varying parameers, which we parameerize according o he recenly proposed score-based approach. We furher inroduce he concep of ail connecedness o sudy he comovemen in he shape of he ails of he condiional disribuions of univariae exreme reurns. We analyze he dynamics of ail risk on a daily basis as a funcion of rm size and of he underlying marke. We provide empirical evidence ha ail risk in he U.S. sock marke is counercyclical across all 9 We colleced he series for indusrial producion from he OECD websie. Daa for Ausralia and Swizerland are no available and are hen omied. 24