Sorting Stocks, Volatility Bounds, and Real Activity Prediction. Belén Nieto University of Alicante, Spain

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1 Soring Socks, Volailiy Bounds, and Real Aciviy Predicion Belén Nieo Universiy of Alicane, Spain Gonzalo Rubio * Universiy CEU Cardenal Herrera, Spain This version: November 2011 Absrac This paper analyzes he capaciy of he Hansen Jagannahan volailiy bound o predic fuure economic growh. Our resuls show ha he porfolio soring procedure employed o consruc he daa used o esimae he volailiy bound is he key issue in he bound being able o predic real aciviy. We find ha he volailiy bound esimaed wih 10 size-sored porfolios is a powerful in-sample and ou-of-sample predicor of fuure indusrial producion growh. JEL classificaion: G10; G12; E44 Keywords: Volailiy bound, real aciviy; soring socks; predicabiliy; size The auhors acknowledge financial suppor from Miniserio de Educación y Ciencia hrough grans ECO (Belén Nieo, belen.nieo@ua.es) and ECO /ECON (Gonzalo Rubio, gonzalo.rubio@uch.ceu.es). Gonzalo Rubio also acknowledges financial suppor from Generalia Valenciana gran PROMETEO 2008/106, and Copernicus4/2011. * Corresponding auhor. address gonzalo.rubio@uch.ceu.es. 1

2 1. Inroducion Does financial uncerainy predic fuure real aciviy? The answer o his quesion is paricularly relevan afer he recen urmoil experienced by indusrial economies over he world. This paper shows ha changes in he uncerainy embedded in sock prices are a powerful indicaor of fuure economic growh. 1 However, i is also he case ha he informaion conained in sock reurn co movemens is he key issue for opimally deecing he impac of financial uncerainy in fuure real aciviy. I has been recognized for a long ime ha he sock marke is a leading economic indicaor. The original papers by Fama (1981, 1990), and Schwer (1990) argue ha sock reurns a monhly, quarerly and annual frequencies are highly correlaed wih fuure oupu growh raes and his predicing abiliy increases wih he lengh of he horizon. Similarly, Sock and Wason (2003) provide a comprehensive analysis of he forecasing capaciy of differen variables relaed o financial markes in forecasing producion and inflaion. They find ha shor and long ineres raes, he erm spread and he sock marke index improve he forecas of real gross domesic produc (GDP) growh, alhough hey also poin ou non-rivial insabiliy problems inheren in he predicive relaions. Addiionally, direc measures of uncerainy in financial markes seem o have relevan informaion abou macroeconomic variables in he fuure. Schwer (1989) suggess ha marke volailiy reflecs uncerainy abou fuure cash flows and discoun raes. However, he does no find evidence supporing his argumen since during his sample period volailiy rises afer he beginning of recessions. Campbell e al. (2001) find ha sock volailiy a a marke, indusry, and firm level helps o predic GDP 1 Bloom (2009) argues ha uncerainy shocks, approximaed by sock marke volailiy, cause firms wih non-convex labor and capial adjusmen coss o delay hiring and invesmen since higher uncerainy increases he real opion value of waiing. Aggregae growh produciviy hen falls afer he uncerainy shock because he adverse effecs in employmen and invesmen slow down he reallocaion from low- o high-produciviy firms, which explains he real aciviy growh rae in he economy. 2

3 growh during he pos-war period. More recenly, Fornari and Mele (2011) show ha a slowly changing measure of sock marke volailiy ha capures he long run uncerainy in he financial marke explains fuure rends of economic aciviy. 2 Moreover, his measure of sock marke volailiy, ogeher wih he erm srucure spread, anicipae all Naional Bureau of Economic Research recession episodes, including he recen financial and credi crisis. In addiion, Chauve, Senyuz, and Yoldas (2011) repor ha he long-run componen of financial volailiy, in he sense of Adrian and Rosenberg (2008) bu exraced from he realized volailiy of marke, indusry, and 10-year zero coupon Treasury bond reurns, helps in predicing economic aciviy. 3 Finally, Nieo and Rubio (2011), using a consumpion-based parameric approach for measuring he uncerainy embedded in financial prices, also predic real aciviy. 4 Specifically, hey use he volailiy of alernaive consumpion-based sochasic discoun facor specificaions as a measure of uncerainy. Working wih conemporaneous and long-run recursive preferences, hey argue ha he significan predicabiliy of his volailiy relies mainly on he join effec of heir componens, ha is, he volailiy of consumpion growh, sock marke volailiy, and he covariance beween consumpion growh and marke reurns. 5 2 Fornari and Mele (2011) jusify heir findings following he heoreical framework of Mele (2007, 2008), who shows he counercyclical and asymmeric naure of volailiy in recessions and expansions. 3 In relaed lieraure, Andreou, Ghysels, and Kourellos (2010) employ implied volailiy as a predicor of economic aciviy and Backus, Chernov, and Marin (2011) employ equiy index opions o quanify he disribuion of consumpion growh disasers. These auhors show ha opions sugges smaller probabiliies of exreme oucomes han have been esimaed from macroeconomic daa. I is imporan o poin ou ha no only lagged marke reurns and volailiy have been employed as leading indicaors of economic aciviy. Naes, Sklelorp, and Arne-Odegaard (2011) repor a srong relaion beween sock marke liquidiy and he business cycle. 4 The auhors also show some power in predicing sock marke reurns a relaively long horizons. Alhough hey show some predicing capaciy a shor horizons, he predicabiliy of sock marke reurns is much weaker han a long horizons. Our curren paper does no address he issue of predicing sock reurns. For recen lieraure on predicing fuure sock marke excess reurns, see, among many ohers, Campbell and Yogo (2006), Cochrane (2008), Goyal and Welch (2008), Brennan and Taylor (2010), Ferreira and Sana-Clara (2011), and Cochrane (2011). 5 The auhors also find similar effecs using non-separable durable and nondurable preferences. 3

4 This paper employs a much simpler approach o invesigae he predicabiliy of real aciviy. In paricular, we use he Hansen Jagannahan (HJ hereafer, 1991) volailiy bound from a model-free perspecive raher han a marginal rae of subsiuion approach. Given a se of porfolio reurns and he average risk-free rae for he corresponding sample, we obain he volailiy bound using he expression proposed by HJ and a rolling window of five years of pas daa. We show how he model-free volailiy bound is a powerful predicor of fuure economic growh for boh in-sample and ou-of-sample conexs. In he end, he HJ bound is he maximum Sharpe raio; hus our measure includes no only excess marke reurns bu also informaion abou correlaion or exposure o common shocks and marke volailiy. However, he paper s main finding is ha he predicabiliy of he bound depends on he soring procedure used o consruc he equiy porfolios employed in he bound s esimaion. Hence, he dynamic ineracion effecs beween individual socks seem o be a key issue in exracing he informaion conained in he sock markes abou fuure real aciviy. This paper is organized as follows. Secion 2 describes our daa and Secion 3 presens he main in-sample predicabiliy resuls, using size-sored porfolios. Secion 4 discusses he forecasing evidence using alernaive soring procedures and Secion 5 compares he predicing abiliy of he HJ measure wih respec o sandard sae variable predicors. Secion 6 performs he ou-of-sample analysis and Secion 7 furher invesigaes he reasons underlying he forecasing capaciy hroughou he principal componens of he variance covariance marices of he alernaive equiy porfolios employed in he paper. Secion 8 concludes wih a summary and final remarks. 4

5 2. Daa Mos sock marke daa are from Kenneh French s websie. We obain monhly daa from January 1927 o December 2010 for he marke reurn ( R ), he risk-free rae ( R f ), he small-minus-big (SMB) and high-minus-low (HML) Fama and French (1993) risk facors, and 10 value-weighed size-, book-o-marke-, momenum-, and dividend yield-sored equiy porfolios. Table 1 conains descripive saisics on hese porfolios. We observe he well-known size and value premia. On an annualized basis, small firms earn, on average, 7.4% more han large firms, while value firms earn 6.3% more han growh firms. Similarly, high-momenum companies obain a 14.4% higher average reurn han low-momenum firms, while high dividend yield socks achieve a 1.9% higher reurn, on average, han low dividend paymen socks. As expeced, we observe more dispersion in average reurns in size-, book-o-marke-, and momenum-sored porfolios han in dividend yield-sored socks. A he same ime, small, growh, and low-momenum socks presen higher volailiy han large, value, and high-momenum firms. Exreme dividend yield socks are more volaile han inermediae dividend yield firms, bu he high and low dividend yield porfolio volailiies are very similar. Finally, he correlaions beween small and large companies, value and growh firms, high- and low-momenum socks, and high and low dividend yield asses are found o be he smalles wihin a given soring caegory: 0.698, 0.714, 0.594, and for size-, booko-marke-, momenum-, and dividend yield-sored porfolios, respecively. The price-dividend raio in logs (PD) is compued from he original series on Rober Shiller s websie. Addiionally, yields for he 10-year governmen bond, he onemonh T-bill, and Moody s Baa Corporae Bond series are obained from he Federal Reserve Saisical Release. We hen compue wo sae variables based on hese ineres raes: a erm srucure slope (Term), compued as he difference beween he 10-year m 5

6 governmen bond and one-monh T-bill yields, and a defaul premium (Defaul) ha is he difference beween Moody s yield on Baa Corporae Bonds and he 10-year governmen bond yields. Given he real aciviy forecasing evidence from aggregae illiquidiy repored by Naes, Skjelorp, and Arne-Odegaard (2011), we also use a marke-wide illiquidiy indicaor (Illiq) based on he aggregae illiquidiy raio proposed by Amihud (2002). 6 This is he raio of he absolue daily reurn over he dollar volume for a given sock, which is closely relaed o he noion of price impac. This measure is averaged monhly and across all available socks o obain he marke-wide illiquidiy measure for each monh in he sample. As in Naes, Skjelorp, and Arne-Odegaard (2011), we demean he series relaive o a wo-year moving average of he series. We also obain nominal consumpion expendiures on nondurable goods and services from he Table of he Naional Insiue of Pension Adminisraors (NIPA). Populaion daa are from NIPA s Table 2.6 and he price deflaor is compued using prices from NIPA s Table wih he year 2000 as is basis. All his informaion is used o consruc monhly seasonally adjused real per capia consumpion expendiures on nondurable goods and services ( C). Finally, monhly daa of he indusrial producion index (IPI) are downloaded from he Federal Reserve, wih series idenifier G17, IP Mayor Indusry Groups. 7 6 The main advanage of Amihud s illiquidiy raio is ha i can be easily compued using daily daa during long periods. Moreover, Hasbrouck (2009) shows ha, a leas for US daa, Amihud s raio beer approximaes Kyle s lambda relaive o compeing measures of illiquidiy. 7 Wih he excepion of marke-wide illiquidiy, monhly daa for all hese sae variables are available from January 1965 o July The illiquidiy variable is available from January 1965 o December

7 3. In-Sample Predicabiliy of Real Aciviy wih he Volailiy of he HJ Bound We firs esimae he monhly HJ volailiy bound of he model-free sochasic discoun facor wih overlapping sub-periods of five years of monhly daa from he 10 sizesored porfolios, using [ ] 1 2 N N 1 ( M ) ( 1 E( M ) E( R) ) V ( 1 E( M ) E( R) ) σ, (1) where M is he sochasic discoun facor saisfying he firs-order pricing equaions, where N 1 and ( R) [ M R ] 1 E + 1 j + 1 =, [ M 1] 1 R f 1 E + + =, E are he N-vecors of ones and average gross reurns, respecively; 1 V is he inverse of he variance covariance marix of reurns; and R f is he gross risk-free rae. The monhly esimaed volailiy corresponds o he average level of he risk-free ineres rae for each of he five-year sub-periods. Unlike he work by Nieo and Rubio (2011), his procedure does no depend on any paricular consumpion-based sochasic discoun facor specificaion, so he poenial predicive relaion does no depend on any given consumpion dynamics. Figure 1 shows his rolling-window HJ volailiy bound and he Naional Bureau of Economic Research s recession bars for he period from 1931 o I shows how he bound ends o increase before macroeconomic recessions, reaching is hisorical peak well before and during he recen financial urmoil. Alhough he peaks of he bound end o occur during he corresponding recession monhs, he volailiy of he sochasic discoun facor always increases before he sar of a recession. Panel A of Table 2 conains he resuls from he following predicive ordinary leas squares (OLS) auocorrelaion-robus sandard error regressions: ( M ) ε τ, + τ = α + β σ + + IPI, (2) 7

8 where IPI is he growh of indusrial producion a horizons of one, hree, six, 12,, + τ and 24 monhs calculaed as IPI, + = ln( IPI+ IPI ), and ( M ) τ τ σ is he volailiy of he sochasic discoun facor available a monh ha is esimaed wih five years of monhly daa up o monh. Given daa resricions on some of he sae variables used laer, we run hese predicive regressions beween January 1965 and July The regression in expression (2) is esimaed wih ( M ) σ from he use of 10 sizesored porfolios, as well as wih he five smalles and five larges porfolios. This separaion allows one o analyze wheher he forecasing relaion is especially srong when he uncerainy measure racks he higher degree of sensiiviy of small companies o economic shocks. The firs block of Panel A of Table 2 repors he key resuls of he paper. There is a negaive and significan relaionship beween he volailiy of he sochasic discoun facor and fuure indusrial producion growh. Boh he magniude of (he absolue value of) he coefficiens and he R 2 value increase considerably wih he ime horizon, wih R 2 as high as (approximaely) 20 percen a he 24-monh horizon. If we inerpre ( M ) σ as a measure of he financial uncerainy embedded in sock prices, hese resuls show ha higher uncerainy has a negaive and significan impac on fuure real aciviy. Therefore, our measure of uncerainy conveys informaion abou fuure economic growh. 8 The resuls using he smalles or larges se of size-sored porfolios separaely also end o show a negaive relaionship beween ( M ) σ and fuure real aciviy. Once 8 Because he HJ volailiy bound is very persisen, we also calculae he bias-correced esimaor and he corresponding bias-correced -saisic proposed by Amihud and Hurvich (2004). These auhors sugges a regression mehod for hypohesis esing in predicive regressions in which he independen variable is persisen and is innovaions are correlaed wih he dependen variable. This produces biased esimaes and biased -saisics. The auhors simulaions show ha heir adjusmen ouperforms oher bias correcion mehods, such as hose suggesed by Sambaugh (1999) and Lewellen (2004). Consequenly, we replicae he forecasing regressions wih heir procedure. The resuls are qualiaively he same as hose repored in Table 2, and he predicing capaciy of he bound remains saisically significan. The resuls are available upon reques. 8

9 again, he longer he horizon in he regression, he sronger he predicing resuls. However, for he one-, hree-, six-, and 12-monh horizons, boh he magniudes of he coefficiens and he R 2 are smaller for boh ses of five porfolios han for he original 10 size-sored porfolios. For he longes horizon, he R 2 value for he original se and he five larges porfolios are 19.6 and 18.1 percen, respecively. I is somehow surprising ha he R 2 value when ( M ) σ is calculaed for he five smalles porfolios is relaively lower and equal o 12.6 percen, alhough he magniude of he negaive slope coefficiens are almos he same in all hree cases. Generally speaking, we can conclude ha forecasing capaciy seems o be sronger using all asses in he sock marke, as represened by he 10 size-sored porfolios, raher han employing eiher he ses of larges or smalles socks. Therefore, hese iniial resuls do no allow us o associae he forecasing abiliy repored wih he poenially greaer or lesser sensiiviy of alernaive equiy porfolios o economic shocks. To furher invesigae his finding, Panel B of Table 2 repors he resuls of he following forecasing regressions: 10 IPI, + τ = α + β1 σ 10 IPI, + τ = α + β1 σ Small ( M ) + β σ ( M ) 2 + ε+ τ, Big ( M ) + β σ ( M ) + ε, 2 + τ (3) Small Big where σ ( M ) and ( M ) σ are he volailiy of he HJ bound esimaed by expression (1) for he five smalles and five larges porfolios, respecively, and 10 ( M ) σ is he bound for he 10 size-sored porfolios. The ime series of hese hree HJ Small Big bounds are displayed in Figure 2. Alhough he series of σ ( M ) and σ ( M ) cross each oher in several poins in ime, depending on he paricular sae of he 10 economy, he series of ( M ) he HJ bound. σ is pracically always above he oher wo esimaions of 9

10 The resuls provided in Panel B of Table 2 show ha he regression coefficiens 10 associaed wih ( M ) Big σ ( M ) and ( M ) Small σ and R 2 are pracically he same as in Panel A. The inclusion of σ does no add any significan explanaory power of fuure economic growh once we conrol for he behavior of he HJ bound under all 10 size- Big sored porfolios. The only excepion occurs when we also employ ( M ) Big longes horizon. Even in his case, he coefficien associaed wih ( M ) 10 wih much less precession han he coefficien relaed o σ ( M ) σ a he σ is esimaed, and he magniude of Big 10 he σ ( M ) coefficien is (in absolue value) approximaely half he ( M ) coefficien. σ slope We conclude ha he forecasing abiliy of he volailiy of he sochasic discoun facor as characerized by he HJ bound lies in he use of he 10 size-sored porfolios raher han a subse of he five smalles or five larges porfolios. I seems ha he inclusion of all asses when esimaing he HJ bound is imporan o capure fuure real aciviy. 4. In-Sample Predicabiliy of Real Aciviy: Oher Porfolio Formaion Crieria We now esimae hree addiional alernaive measures of he HJ volailiy bound by using he reurns of 10 book-o-marke-, momenum-, and dividend yield-sored porfolios. As before, we employ a rolling window of five years of pas monhly reurns. Figure 3 displays he HJ bounds for he full sample period. We observe imporan differences beween he alernaive esimaed bounds. Noe ha he volailiy dispersion and he complex dynamic correlaion behavior among he 10 porfolios in each of he four ses employed can generae poenially differen ime series of he HJ bounds. I seems paricularly imporan o noe ha he HJ bound for he momenum porfolios 10

11 increases before he recessions a he end of he 1980s and a he beginning of he new cenury. These peaks are probably associaed wih he uncerainy generaed in hese porfolios afer he crash of Ocober 1987 and during he do-com bubble. On he oher hand, he highes peak before he acual crisis is clearly from he HJ bound esimaed wih he 10 size-sored porfolios. We perform he forecasing regressions of equaion (2) using he HJ bound esimaed wih he 10 porfolios of each se as well as wih he wo subses of five porfolios for all hree soring crieria. Panels A o C of Table 3 repor he resuls for he book-o-marke-, momenum-, and dividend yield-sored porfolios, respecively. Surprisingly, independenly of he forecasing horizon, none of he esimaes of he HJ volailiy bound consruced from hese porfolio ses presen significan predicing resuls. I may be he case ha he dynamics of he volailiy dispersion and he correlaion beween socks included in he alernaive sored porfolios induce a differen forecasing abiliy of real aciviy. Alhough we reurn o his issue in Secion 7 below, we poin ou ha he annualized volailiy dispersion beween he exreme porfolios conained in he descripive saisics of Table 1 urns ou o be he highes for he size-sored porfolios. In paricular, he smalles porfolios have an 18.6 percen higher annualized volailiy han he larges socks, while he dispersion is only 12.7 percen, 11.4 percen, and 0.9 percen for he book-o-marke-, momenum-, and dividend yield-sored porfolios. Similarly, he dispersion beween he minimum and maximum correlaions beween he porfolios is 0.28, 0.24, 0.35, and 0.26 for he size-, book-o-marke-, momenum-, and dividend yield-sored porfolios. The dynamics of hese volailiies and correlaions seem o be a poenially key facor in explaining he differen predicing capaciies of he alernaive HJ bound esimaes. If so, soring 11

12 procedures and he corresponding ime-varying diversificaion effecs would be a relevan issue for forecasing producion growh wih volailiy bounds. 5. In-Sample Predicabiliy of Real Aciviy: Compeing Predicors Given he significan predicing abiliy of he HJ volailiy bound esimaed wih 10 size-sored porfolios, we now invesigae how robus our forecasing resuls are o compeing predicor variables of real aciviy. We consider predicors relaed o ineres raes, sock marke reurns, and illiquidiy. The jusificaion of he selecion of hese alernaive predicors is presened in Secion 5.1 and he forecasing resuls are discussed in Secion 5.2. In addiion, lagged values of he dependen variable are included in he forecasing regression o pick up poenial auoregressive dynamics in indusrial producion, since we consider growh raes for periods longer han one monh. Secion 5.3 conains he resuls of his analysis Compeing Predicors of Real Aciviy The erm spread, measured as he difference beween he ineres raes on long and shor mauriy governmen deb, is probably he mos common financial leading indicaor of real aciviy. Among many ohers, Esrella and Hardouvelis (1991), Esrella and Mishkin (1998), Sock and Wason (2003), Ang, Piazzesi, and Wei (2006), and Fornari and Mele (2011) show he significan predicive conen of he spread for producion growh, including is capaciy o forecas a recession indicaor in probi regressions. Addiionally, here is a growing body of lieraure exploring he ransmission of credi condiions ino he real economy. Among recen papers, Mueller (2009) and Gilchris, Yankov, and Zakrajsek (2009) show he forecasing power of he erm srucure of credi spreads for fuure oupu growh. These auhors argue ha here 12

13 is a pure credi componen orhogonal o macroeconomic condiions ha accouns for a large par of he predicing capaciy of credi spreads. Moreover, as long as sock prices equal he expeced discouned value of fuure earnings and dividends, sock reurns should also be useful in forecasing oupu growh. This is he insigh of Fama (1981, 1990). On op of ha, given he well-known evidence of he aggregae dividend yield being a powerful predicor of fuure marke excess reurns, as discussed recenly by Cochrane (2011), he price dividend raio becomes an appropriae sae variable o use for forecasing real aciviy. Two oher sock marke indicaors have become popular in predicing oupu growh. Naes, Skjelorp, and Arne- Odegaard (2011) argue ha sock marke liquidiy ends o dry up before a crisis in he real economy. In fac, hey show ha measures of sock marke liquidiy conain leading informaion abou fuure economic growh, even afer conrolling for oher financial leading indicaors. Finally, here has been considerable recen aenion o financial sock marke volailiy as a predicor of real aciviy. Fornari and Mele (2011) argue ha i is imporan o exrac he long-run componen of sock marke volailiy when using his variable as a predicor of fuure growh. 9 To isolae exreme financial episodes ha may no be necessarily informaive abou he economy s fuure scenario, he auhors propose a simple moving average of he pas 12 monhs of absolue reurns as he appropriae forecaser of real aciviy In-Sample Predicabiliy wih Compeing Predicors We nex employ all seven variables discussed above and compare heir in-sample predicing abiliy wih ha of he HJ volailiy bound as esimaed wih 10 size-sored 9 See he similar argumens of Chauve, Senyuz, and Yoldas (2011). 13

14 porfolios. We run he following regressions wih individual predicors and wih pairs of predicors ha always include he HJ bound: IPI, + τ = α + β + β Illiq 7 1 σ ( M ) + β R + β σ ( R ) + ε + τ, 2 m 3 m + β PD 4 + β Def 5 + β Term 6 (4) where σ ( R m ) is he marke reurn volailiy esimaed a each monh wih overlapping sub-periods of five years of monhly reurns, o be consisen wih our measure of he HJ bound. The resuls are repored in Table 4. Independenly of he alernaive sae variable employed and forecasing horizon, he HJ volailiy bound has always a negaive and highly significan relaion wih fuure IPI growh. Hence, our forecasing relaion is sysemaically esimaed wih high precision. A he one-monh horizon, all sae variables presen some evidence of predicabiliy, excep he sock marke reurn. All predicors presen he expeced signs. The erm spread coefficien is posiive, while he res of he sae variable esimaors have he heoreically correc negaive sign. Noe ha increases in he volailiy of he marke, he defaul spread, and marke-wide illiquidiy signal a higher degree of uncerainy, and we also know ha increases in he dividend yield forecas fuure posiive marke excess reurns, which implies ha increases in he price dividend raio should predic negaive marke reurns and a negaive impac on real aciviy. Once we combine on an individual basis he HJ volailiy bound wih he res of he predicors, i urns ou ha he coefficiens associaed wih he volailiy of he marke reurn, he price dividend raio, and he defaul spread are esimaed wih much more precision. This resul does no seem o hold for he erm and marke-wide illiquidiy variables. I is especially relevan he combined effecs of he HJ bound and he defaul spread; he R 2 value a jus he one-monh horizon is 9.38 percen. 14

15 I is imporan o poin ou ha we display he resuls using he volailiy of he sock marke esimaed a each ime wih he pas five years of monhly daa. We also repea he regressions using he esimae suggesed in Fornari and Mele (2011): 12 π 1 ( Rm ) = σ Rm+ 1 k, (5) 2 12 where π 2 is a scaled facor relaed o he use of absolue values. This measure provides slighly beer resuls han he previous measure of marke volailiy. In paricular, he coefficien is and i is also esimaed wih higher precision, so ha he -saisic is raher han However, i does no change he conclusion abou he forecasing power of he HJ bound. A he hree-monh horizon, all predicors seem o be individually significan and wih he correc sign. Ineresingly, he volailiy of he sock marke loses forecasing capaciy, alhough, as when we use he esimaor given by expression (5), he coefficien is esimaed wih more precision, and he -saisic becomes In he combined regressions, he higher R 2 saisics are obained when adding he volailiy of he sock marke, he price dividend raio, and he defaul spread o he HJ volailiy bound. The regression wih he HJ bound and he price dividend raio presens an R 2 of 15.5 percen. Finally, for all oher longer horizons, he resuls are similar, excep ha he erm spread becomes much more relevan in forecasing oupu growh and he defaul spread loses is significan predicing abiliy. Hence, he combinaions of he HJ volailiy bound wih he sock marke reurn, he volailiy of he marke, he price dividend raio, and he erm spread seem o be relevan facors in predicing fuure producion growh a long horizons. A he six-monh horizon he highes R 2 is observed when combining he HJ bound wih he price dividend raio, while he combinaions of he k= 1 15

16 volailiy bound wih he erm spread have he highes R 2 saisics a he 12- and 24- monh horizons. A he longes horizons, he HJ bound and erm spread explain 28.3 percen of he variabiliy of fuure producion growh. To conclude, he defaul spread conveys informaion abou fuure economic growh a relaively shor horizons, while he erm spread has predicing capaciy a longer horizons. In all cases, he HJ volailiy bound calculaed wih 10 size-sored porfolios remains a srong predicor of real aciviy Lagging he Dependen Variable Since we make muli-sep ahead predicions, serial correlaion in indusrial producion growh is expeced. This suggess ha he forecasing regressions should also include lagged values of he dependen variables. Therefore, we now run he regression ( M ) + β2 IPI, τ ε τ, + τ = α + β1 σ + + IPI. (6) The resuls are shown in Table 5. The auoregressive srucure of IPI growh is confirmed for horizons of one, hree, and six monhs. However, he coefficiens associaed wih he HJ volailiy bound remain negaive and saisically significan in all cases. In fac, hese coefficiens are very similar o hose repored in Table 2. Therefore, alhough he inclusion of he lagged dependen variable helps predic real aciviy, lagging he dependen variable does no seem o have any effec on our previous conclusions regarding he imporance of he HJ volailiy bound as an ex ane uncerainy predicor of economic cycles. 6. Ou-of-Sample Tess The predicing ools employed so far examine he abiliy of he predicors had we been able o use he coefficiens esimaed by he full-sample regressions. We now consider 16

17 ess designed o generae more closely acual real ime forecass. We employ wo alernaive saisics for esing he ou-of-sample accuracy of wo compeing models: he -es proposed by Diebold and Mariano (1995) and he F-saisic of McCracken (2007). In our case, he wo compared models are always nesed. The resriced model conains only one of he compeing predicors already used in our in-sample ess: eiher he sock marke reurn, he volailiy of he sock marke due o Fornari and Mele (2011), he price dividend raio, he defaul spread or he erm spreads. 10 On he oher hand, he unresriced model conains such a predicor and he HJ volailiy bound esimaed wih 10 size-sored porfolios. We now briefly describe his mehodology. The oal sample period conains T + P observaions, where he iniial in-sample esimaion period employs informaion from 1 o T, and he ou-of-sample forecasing period goes from T + τ o T + P, τ being he forecasing horizon. A each forecasing period = T + τ,..., T + P, we esimae he wo compeing nesed models using informaion up o he previous τ periods, generae he predicion, and compue he forecasing error. More formally, he resriced model is Y s R 0 R 1 = β + β X τ + u, s = τ + 1, K, -τ. (7a) s Rs The predicion under he resriced model is and he predicion error will be Ŷ ˆ ˆ, (7b) R R R = β0 + β1 X τ û R = Y Ŷ. (7c) R Similarly, he unresriced model ha includes he HJ volailiy bound, he nex period predicion and forecasing error are Y s U U U ( M ) + u, s = τ + 1, K, τ = β0 + β1 X s τ + β2 σ s τ Us -, (8a) 10 Since he marke-wide illiquidiy variable conains daa only unil he end of 2008, our ou-of-sample ess do no employ his sae variable. 17

18 Ŷ U = ˆ U ˆ U X ˆ U β + β β σ ( M ), (8b) 0 1 τ + 2 τ û U = Y Ŷ. (8c) U We nex compue he vecor of loss differenials, denoed d, ha compares he wo square errors a each monh and he mean squared forecasing error (MSE) for each model: d 2 2 R U + = û û, = T + τ, K,T P, (9) MSE MSE R U = = T ( ) + 1 P + 1 = T + P 2 ûr τ τ, (10) T ( ) + 1 P + 1 = T + P 2 ûu τ τ. (11) The wo saisics for esing equal forecasing accuracy have he null ha he loss differenials are zero, on average. The Diebold Mariano (1995) saisic is a -es expressed as d = T 1 where ( ) + = T MSE = ( P τ + 1) d, (12) Ŝ P P τ + 1 d and Ŝ d is a consisen esimaor of he variance of he τ loss differenial ha admis heeroskedasiciy and auocorrelaion. We employ he Newey Wes (1987) specificaion and, following Clark and McCracken (2011), a lag d lengh of k = 1. 5 τ. Hence Ŝ d = k j= k k k j T + P 1 ( P j + 1) ( d d )( d d ) τ. (13) = T + τ j The McCracken (2007) saisic is an F-es given by 18

19 MSER MSEU MSE F = ( P τ + 1). (14) MSE I mus be noed ha he loss differenials are measured wih an error ha is due o he fac ha he bea coefficiens are unknown. This implies ha he exac disribuion of boh saisics is also unknown and ha he asympoic disribuion can only be obained under resricive assumpions ha include non-nesed models. 11 As previously poined ou, his paper compares nesed models. For his case, Clark and McCracken (2011) sugges deriving he asympoic disribuion by a fixed regressor boosrap, and hey show ha he es saisics based on he proposed boosrap have good size properies and beer finie-sample power han alernaive boosraps. This mehod is based on he wild fixed regressor boosrap developed by Goncalves and Killian (2004) bu adaped o he muli-sep framework of ou-of-sample forecass. To implemen his mehod, we use he followings seps. 1. We esimae boh he resriced and unresriced models using he full sample period and we compue he residuals from he unresriced model: U ( M ), = 1 + τ, K,T P ˆ U ˆ U û Y X ˆ U U = β0 + β1 + β2 σ + τ τ. 2. We assume and esimae an MA (τ 1) process o capure he implici serial correlaion in he residuals from a τ-sep-ahead forecas, uu τ 1 +. = ε + θ1ε -1 + K, + θ -1ε -( τ - ), = 1 + τ, K,T P 3. We simulae a sequence of independen and idenically disribued N(0,1) random variables denoed by η and generae arificial residuals by using he esimaes of he MA process: = η ˆ ˆ ˆ ε + θ1η 1ˆ ε -1 + K, + θ -1η -( τ -1) ˆ ε -( τ - ), = 2τ, K,T P * uu τ See Wes (1996) and Clark and McCracken (2001) for a discussion. 19

20 4. We simulae an arificial series of he dependen variable using he arificial residual and imposing he null hypohesis: Ŷ ˆ R ˆ R * = β + β X τ + u, = 2τ, K,T P. * 0 1 U + 5. We compue boh he MSE -saisics and MSE F-saisics using hese arificial daa as if hey were he original daa. 6. Repea seps 3 5 5,000 imes and he p-value is he percenage of imes he simulaed saisic is greaer han he real saisic. The ou-of-sample resuls are repored in Table 6. The firs row for each forecasing horizon shows he relaive MSE given by he expression RMSE = MSE U MSE R. Noe ha when he RMSE is less han one, he inclusion of he HJ volailiy bound as an addiional predicor improves he forecasing capaciy wih respec o any of he compeing sandard predicors. Below each of he es saisics employed, we repor he corresponding p-value obained hrough he fixed regressor boosrap explained above. The empirical evidence is quie conclusive. Mos of he ime, we show ha he inclusion of he HJ bound significanly improves he predicing capaciy of he model. The RMSE is pracically always less han one, and he p-values end o be very low. I urns ou ha his is he case independen of he forecasing horizon. The only variable ha compees on a similar basis regarding is capaciy o predic real aciviy is he erm spread. For horizons of one, hree, and six monhs he null of no difference beween he forecasing errors of he wo models is no rejeced. For horizons of 12 and 24 monhs, he RMSE is greaer han one and he null is rejeced, indicaing ha he model including only he erm spread has beer ou-of-sample performance. Therefore, he erm spread becomes a beer forecaser he longer he predicing horizon. On he oher hand, he defaul spread presens wih precisely he opposie behavior. Noe ha his is consisen wih he in-sample resuls conained in 20

21 Table 4. Finally, we should menion ha he sock marke volailiy consisenly shows a higher MSE han he HJ volailiy bound. In fac, he es saisics show ha he inclusion of he HJ volailiy bound always significanly improves he predicing capaciy of he sock marke volailiy. 7. Principal Componen Predicabiliy The finding ha predicabiliy of real aciviy occurs when HJ volailiy bound is esimaed by using size-sored porfolios is boh ineresing and surprising. I seems ha he ime-varying behavior of correlaions and variance dispersion beween socks may be he reason behind our resuls. This secion provides furher empirical evidence analyzing he principal componens from he se of porfolio reurns of he alernaive soring procedures. Principal componen analysis allows us o decompose he behavior of he whole se of porfolio reurns, wihin a given soring procedure, ino orhogonal componens each corresponding o a differen se of informaion. By definiion, he firs principal componen is he (normalized) linear combinaion of porfolio reurns wih maximum variance. Table 7 shows ha he firs hree principal componens explain 98.6, 96.0, 96.0, and 94.0 percen of he oal variabiliy of reurns for he size-, book-o-marke-, momenum-, and dividend yield-sored porfolios, respecively. The firs principal componen of he size-sored porfolios explains a higher percenage han he firs principal componens of he alernaive soring sraegies. Addiionally, we observe ha he correlaion coefficiens beween he firs principal componens of he book-o-marke-, momenum-, and dividend yield-sored porfolios are 0.97 for he hree pairs, while he correlaion beween he firs principal componens of hese porfolios and he size-sored porfolios is slighly lower and equal o 0.95, 0.93, and 0.91, respecively. The second principal componen of he size-sored 21

22 socks has a correlaion coefficien of 0.62 wih he second componen of he book-omarke sored porfolios, and much lower correlaion wih he res of he second principal componens. We also find correlaions higher han 0.45 beween he second principal componens from he book-o-marke- and dividend yield-sored asses and from he momenum- and he dividend yield-sored porfolios. Finally, correlaions beween he hird principal componens from he differen ses of porfolios are relaively much lower han in all oher cases. To undersand he economic facors behind hese principal componens, we nex perform he following regressions for each of he hree principal componens and each porfolio se separaely: PC = α + βx + u, i = 1,2, 3, (15) i, where X is, alernaively, he sock marke reurn, he price dividend raio, he SMB or HML Fama French facors, he defaul spread, he erm spread, he real consumpion growh, and he marke-wide illiquidiy facor. The resuls are repored in Table 8. The variabiliy of he firs principal componen from he size-sored porfolios is clearly explained by he sock marke reurn. However, R 2 is 86.6 percen, which is relaively lower han he percenage explained of he firs principal componen by he marke reurn when using alernaive soring procedures. The R 2 values for he book-o-marke-, momenum-, and dividend yield-sored porfolios are 92.9, 93.2, and 92.6, respecively. 12 As expeced, when we run he regression of he firs principal componen of he size-sored porfolio reurns ino he SMB facor, we find ha his facor explains 38.1 percen of he variabiliy of he firs componen. Hence, he firs principal componen of he size-sored socks is explained 12 The firs principal componen of he alernaive porfolio classificaions is mosly explained by he sock marke reurn and he Fama French facors. The price dividend raio, he defaul and erm spread, and he illiquidiy facor do no seem o be relevan in capuring he variabiliy of he firs principal componen; however, consumpion growh explains 3.8, 2.8, 2.9, and 4.0 of is variabiliy. 22

23 no only by he aggregae marke facor bu also for he difference beween he reurns of small and large asses. We do no observe a similar resul for oher porfolio ses; any of he firs principal componens in hese cases are basically explained hrough he sock marke reurn. For example, he SMB and HML facors only explain 8.8 percen and 0.6 percen of he variabiliy of he firs principal componen of he book-o-marke sored porfolio reurns. The second principal componen of he size-sored porfolios is explained, as before, by he sock marke reurn and he SMB facor, while he hird principal componen is basically he SMB facor wih an R 2 of 44.3 percen. On he oher hand, he second principal componen of he book-o-marke asses is mainly associaed wih he marke reurn and he HML facor, and is hird principal componen is he HML risk facor wih an R 2 of 22.4 percen. Noe ha his represens half of he explanaory capaciy of he SMB facor for he hird principal componen of he size-sored porfolios. Regarding he momenum- and dividend yield-sored porfolios, i seem ha he SMB and HML are relevan facors for he hird principal componen of he momenum soring, wih more explanaory capaciy for SMB han for HML. Lasly, he HML facor explains as much as 53.0 percen of he variabiliy of he second principal componen of dividend yield-sored reurns. To conclude, he size facor appears o be relevan only when we use he sizesored porfolios. In all oher cases, eiher he sock marke reurn and/or he HML facor explains he behavior of he principal componens. 13 Therefore, size seems o be a key characerisic in explaining he forecasing capaciy of he HJ volailiy bound relaive o he bound s alernaive measures. 13 Only he hird principal componen of he momenum soring has a higher R 2 for SMB han for HML. 23

24 To suppor his conjecure, we run predicing regressions using he HJ volailiy bound esimaed from he se of principal componens insead of he se of porfolio reurns. Each individual regression employs he HJ bound esimaed wih one, wo, or hree principal componens for each porfolio-soring procedure. We can hen check which of hese alernaive bounds generaes a sronger forecasing abiliy of real aciviy. Table 9 conains he resuls from he following predicive OLS auocorrelaionrobus sandard error regressions: PC where ( M ) ( M ) ε τ PC, + τ = α + β σ + + IPI, (16) σ now refers o he HJ volailiy bound esimaed wih he firs, he firs wo, or he firs hree principal componens from each se of porfolio reurns. Panel A of Table 9 conains he evidence from he HJ bounds esimaed wih he principal componens of size-sored porfolio reurns. I shows ha he firs principal componen does no produce significan predicing power. We need o add he second principal componen o capure forecasing abiliy similar o ha shown in Table 2 for he six-, 12-, and 24-monh horizons. Moreover, we even need o add he hird principal componen if we wan o obain forecasing capaciy a he shores horizons. Given he relevance of he SMB facor in explaining he second and hird principal componens of he size-sored porfolios, his resul suggess ha he dynamic behavior of he difference beween he reurns of small and large porfolios may be he ulimae reason behind he forecasing abiliy of he HJ volailiy bound. I is no only he influence of he ineracion beween he numeraor and denominaor of he maximum Sharpe raio ha helps predic real aciviy, bu also, and even more imporanly, he ime-varying behavior of small firms relaive o large ones. Finally, confirming he evidence provided in Table 3, Panels B o D of Table 9 show no evidence of predicabiliy when he volailiy bound is esimaed using 24

25 principal componens from book-o-marke, momenum- or dividend yield-sored porfolios. 8. Conclusions The uncerainy embedded in equiy porfolio reurns helps predic fuure economic growh. This paper s main conribuion is o show a new measure of capuring changes in uncerainy incorporaed in sock reurns ha forecas real aciviy ha is based on he HJ volailiy bound. However, daa employed in he esimaion of he volailiy bound seem o be he key issue in properly incorporaing uncerainy shocks ha convey informaion abou fuure economic growh. Alernaive equiy porfolio soring formaions lead o very differen conclusions regarding he forecasing abiliy of he bound. I urns ou ha soring socks on he basis of size generaes a very powerful leading predicor. We show ha he HJ volailiy bound, when employing daa on 10 size-sored porfolios, generaes significan predicions of real aciviy boh in sample and ou of sample. This is he case independen of he forecasing horizon and he compeing sandard predicor included in he predicing regressions. The inclusion of he HJ bound consruced wih size-sored porfolios significanly improves he ou-ofsample forecasing abiliy of such well-known predicors as he sock marke volailiy, he erm spread, or he defaul spread. Moreover, when we es for forecasing using he HJ bound esimaed from he hree principal componens of equiy porfolio reurns based on size, book-o-marke, momenum, and dividend yield, he only relevan predicion comes from he principal componens of he size-sored porfolios. I urns ou we need o include boh he second and hird principal componens of hese size porfolios in he esimaion of he HJ bound o find significan forecasing capaciy of real aciviy. These second and hird principal componens are significanly associaed 25

26 wih he differences in reurns beween he small and large porfolios. Size makes he difference. The dynamics of he ime-varying second momens of reurns among he size-sored equiy porfolios are a reasonable explanaion of our findings. A comprehensive examinaion along hese lines is lef for fuure research. 26

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31 Table 1 Descripive saisics of monhly reurns for size-, book-o-marke, momenum-, and dividend yield-sored porfolios, January 1927 o December P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 Means Size BEME Momenum Div. Yield Sandard deviaions Size BEME Momenum Div. Yield Correlaions P2 P3 P4 P5 P6 P7 P8 P9 P10 Size-sored porfolios P P Max P P Min P P P P P Book-o-marke sored porfolios P P Max P P Min P P P P P Momenum-sored porfolios P P Max P P Min P P P P P Dividend yield-sored porfolios P P Max P P Min P P P P P