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

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1 Volailiy Bounds, Size, and Real Aciviy Predicion Belén Nieo Universiy of Alicane, Spain Gonzalo Rubio * Universiy CEU Cardenal Herrera, Spain This version: Ocober 16, 2012 Absrac This paper shows how o exrac fuure real aciviy informaion from opimallycombined size-sored porfolios. In paricular, we analyze he capaciy of he size-based model-free Hansen Jagannahan volailiy bound o predic fuure economic growh. We find ha he volailiy bound is a powerful in-sample and ou-of-sample predicor of fuure indusrial producion growh. The asymmeric sensiiviies of small and large companies hrough he business cycle are behind our findings. Alernaive volailiy bounds esimaed wih soring procedures based on book-o-marke, momenum, or dividend yield do no eiher show hese asymmeric sensiiviies or forecasing capaciy of oupu growh. JEL classificaion: G10; G12; E44 Keywords: Financial uncerainy; volailiy bounds; real aciviy; soring socks; predicabiliy; size An earlier version of his paper was circulaed under he ile Volailiy Bounds, Soring Socks, and Real Aciviy Predicion. I was presened a he Luxembourg School of Finance, Universidad de Navarra, he workshop on Financial Sysem Perspecives Afer he Crisis a he Universidad de Girona, he 2nd Inernaional Conference on Securiies Markes a he Spanish Securiies Exchange Commission, and he XX Foro de Finanzas a he Universidad de Oviedo. We hank seminar paricipans and Joss van Bommel, Tibor Neugebauer, Sara Ferreira-Filipe, Anonio Moreno, Germán López, Carmen Aranda, Buron Hollifiled, Ricardo Laborda, Anonio Rubia, and an anonymous referee for helpful commens and suggesions ha subsanially improved he conens of he paper. The auhors acknowledge financial suppor from Miniserio de Educación y Ciencia and Miniserio de Economía y Compeiividad hrough grans ECO (Belén Nieo, belen.nieo@ua.es) and ECO (Gonzalo Rubio, gonzalo.rubio@uch.ceu.es) respecively. Gonzalo Rubio also acknowledges financial suppor from Generalia Valenciana gran PROMETEO 2008/106, and Copernicus4/2011. We assume full responsibiliy for any remaining errors. * 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 he marke capializaion of rading asses is a key issue for opimally deecing he impac of financial uncerainy in fuure real aciviy. Size has always been a key research opic in boh Economics and Finance. Firm size is presen when analyzing apparenly differen opics like seasonaliy of sock reurns, economies of scale, marke power, synergy exernaliies, collaerals, and so on. In his paper we repor addiional evidence regarding he imporance of size. Our main conribuion is o show how o opimally exrac he informaion conained in sizesored porfolios o generae powerful in-sample and ou-of-sample predicions of real aciviy. Our forecasing resuls are herefore a consequence of combining boh he relevance of size, and he paricular esimaor we propose o predic fuure economic growh. 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 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 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 from 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, Leau, Malkiel, and Xu (2001) find ha sock volailiy a a marke, indusry, and firm level helps o predic GDP growh during he pos-war period. Goyal and Sana-Clara (2003) repor a posiive relaion beween average sock variance, ha is largely idiosyncraic, and fuure marke reurns. However, hey do no analyze he relaion beween his measure of risk and fuure real aciviy. 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 volailiies of he marke, indusry porfolios, and he 10-year zero coupon Treasury bond reurns, helps in predicing economic aciviy. 3 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 3

4 Finally, Nieo and Rubio (2011), using a consumpion-based parameric approach for measuring he uncerainy embedded in financial prices, also predic real aciviy. Specifically, hey use he volailiy of alernaive consumpion-based sochasic discoun facor specificaions as a measure of uncerainy and find ha his measure is able o forecas economic growh a boh shor and long horizons. 4 This paper employs a much simpler approach o invesigae he predicabiliy of real aciviy. In paricular, we propose he Hansen Jagannahan (HJ hereafer, 1991) volailiy bound as he predicor. Given a se of porfolio reurns and he average riskfree rae for he corresponding sample, we compue he volailiy bound wih a rolling window of five years of pas daa. We show ha his model-free volailiy bound is a powerful predicor of fuure economic growh for boh in-sample and ou-of-sample conexs. From a pracical poin of view, i is imporan o noice ha his approach requires only financial marke reurns. This implies ha our forecasing measure can be used in real ime when employing an ou-of-sample forecasing framework. This srongly conrass wih he highly parameerized approach followed by Nieo and Rubio (2011), in which he sochasic discoun facor is he marginal rae of subsiuion of consumpion. However, i should also be poined ou ha boh approaches rely on a volailiy bound. In he previous paper, he esimaion of he preference parameers is carried ou by imposing he resricion ha he sochasic discoun facors are inside he HJ volailiy fronier. Addiionally, and given his resricion, he squared pricing errors of 10 size-sored porfolios are minimized obaining volailiy bounds ha are precisely 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. The 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). 4

5 on he fronier. In his paper, he approach is no only much more relevan from a pracical poin of view, bu i is a much simpler way of obaining a bound ha lies on he fronier. Of course, he paricular combinaion of he mean and variance of he corresponding sochasic discoun facor is always differen in all cases. The second conribuion of he curren paper is o explain why size is he key issue when forecasing real aciviy using he volailiy bound. Changes in economic condiions represened by beer/igher credi condiions generae srong asymmeric effecs on size-sored porfolios. In paricular, he asymmeric sensiiviies of reurns and volailiies of small and large companies o credi scenarios hrough he business cycle explain our findings. I seems ha hese asymmeries in boh ime-series and on he cross-secion of size-sored porfolio reurns are he responsible of he forecasing power of he resuling HJ volailiy bound. Oher alernaive soring procedures based on book-o-marke, momenum, or dividend yield do no show hese asymmeric sensiiviies relaive o differen credi scenarios. I urns ou ha heir lack of sensiiviy o credi condiions significanly limis he forecasing capaciy of oupu growh from combining hese porfolios in he paricular proporions suggesed by he HJ volailiy bound. Finally, i should be recognized ha 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. Hence, we also invesigae he source of forecasing abiliy by analyzing he predicing abiliy of he componens of he bound. We find ha predicabiliy crucially depends on he ineracion beween he numeraor and denominaor of he bound of size-sored porfolios, and no on any paricular componen. Once again, his ineracion effec is no observed for any oher soring procedure. 5

6 The remainder of he paper is organized as follows. Secion 2 describes he daa employed in he analysis. Secion 3 presens he main in-sample and ou-of-sample predicabiliy resuls using size-sored porfolios. Secion 4 compares he predicing abiliy of he HJ measure wih respec o sandard sae variable predicors, and compeing measures of financial uncerainy. Secion 5 discusses he forecasing evidence using alernaive soring procedures, and Secion 6 explains he reasons underlying he forecasing capaciy of he bound when using size-sored porfolios, and no alernaive soring procedures. Secion 7 concludes wih summary and final remarks. 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. Addiionally, we collec daa for he daily 100 size-booko-marke value-weighed porfolios from July 1963 o December 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, 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 governmen bond and he risk free rae, and a defaul premium (Defaul) ha is he difference beween Moody s yield on Baa Corporae Bonds and he 10-year governmen bond yields. All hese series are colleced from January 1959 o December m 6

7 Given he real aciviy forecasing evidence from aggregae illiquidiy repored by Naes, Skjelorp, and Arne-Odegaard (2011) a quarerly frequency, we also use a marke-wide illiquidiy indicaor (Illiq) based on he aggregae illiquidiy raio proposed by Amihud (2002). 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. 5 Daily daa on VIX is obained from January 1990 o December 2010 from CBOE. This series is augmened from January 1986 o December 1989 using VXO also from CBOE. 6 In boh cases, we employ he las day of he corresponding monh o creae a final monhly opion-implied volailiy series from January 1986 o December We collec hree alernaive measures of monhly macroeconomic growh. We obain nominal consumpion expendiures on nondurable goods and services from he Table of he Naional Income and Produc Accouns (NIPA) available a he Bureau of Economic Analysis. 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 (ΔC) from January 1959 o December Monhly daa of he indusrial producion index (IPI) are 5 We hank Yakov Amihud for kindly providing his daa from January 1965 o December We updae his measure from January 1997 o December 2008 using daily daa from CSRP on all individual socks wih enough daa wihin a given monh. A leas 15 observaions of he raio wihin he considered monh are required for asse j o be included in he sample. An excepion has been made for Sepember 2001 requiring a leas 12 observaions in his case. 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. 6 VIX is he volailiy index for he S&P 500 index, while VXO refers o he S&P 100 index. 7

8 downloaded from he Federal Reserve, wih series idenifier G17, IP Mayor Indusry Groups from January 1927 o December Lasly, he monhly growh rae of gross domesic produc (GDP) is obained from he Macroeconomic Advisers web page. 7 These daa are available from April 1992 o December Finally, we also collec he quarerly cross-secional dispersion measures for quarerly forecass for GDP from he Survey of Professional Forecasers web page which is available from he fourh quarer of In-Sample and Ou-of-Sample Predicabiliy of Real Aciviy wih he Volailiy of he HJ Bound and Size-Sored Porfolios 3.1. The HJ Volailiy Bound We 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 valueweighed size-sored equiy porfolios using, 1 2 N N 1 M 1 EM ERV 1 EM ER, (1) where M is he sochasic discoun facor saisfying he firs-order pricing equaions, where N 1 and R 1 E M 1R j 1 E, M 1 1 R f 1, 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 7 hp:// 8 hp:// 8

9 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. Table 1 conains he descripive saisics of he volailiy bound esimaed from 1927 o The average volailiy bound is wih a volailiy of 0.148, and posiive skewness and high excess kurosis. 9 The auocorrelaion of he volailiy bound is also high and equal o suggesing, as expeced, ha he bound is quie persisen over ime. I is useful o discuss he implicaions of his auocorrelaion for he empirical resuls we repor below. As discussed in he inroducion, he esimaed volailiy bound is our main predicor variable in ypical forecasing regressions of fuure oupu growh on he volailiy bound and (possibly) oher compeing predicors. The convenional inference in a predicive regression assumes ha he explanaory variable is saionary. In ha case, firs-order asympoics implies ha he -saisic for esing he forecasing abiliy of he predicor is approximaely sandard normal in large samples. However, he null disribuion of he -saisic can be dramaically differen when he predicor is nonsaionary, since he disribuion is disconinue a he poin ha auocorrelaion equals one, and he innovaions in he predicor and in he variable o be prediced are correlaed. There is ample simulaion evidence and analyical sudies on he poor approximaion of he large-sample heory o he acual finie sample showing large size disorions. 10 Two possible soluions have been adoped in he case of highly persisen predicors. One approach is based on he exac finie sample heory as in Sambaugh (1999), and many oher following papers like Lewellen (2004), Amihud and Hurvich (2004), and Amihud, Hurvich, and Wang (2009). The idea behind his approach is o 9 The Sharpe raio for he marke as a whole urns ou o be for he same sample period. 10 See Ellio and Sock (1994). 9

10 eliminae he noise produced by he correlaion among innovaions. Unforunaely, he problem in pracice is ha i is no possible o cerainly know wheher a ime series has or no a uni roo and, herefore, he rue disribuion of he ess is unknown. The second approach is based on he local-o-uniy asympoics, where he predicor is assumed o be auoregressive wih a roo near o uniy. 11 Specifically, he larges auoregressive roo is modeled as 1 c T wih c a fixed consan, and T he number of observaions. 12 Deviaions from he uni roo are measured by he parameer c, which is he responsible of inducing non-cenraliy in he limiing disribuion. This device allows he predicor o be saionary bu nearly inegraed when c < 0. The larger he parameer c, in absolue value, he less persisen is he predicor. If his uncerainy abou he deviaions from he uni roo is ignored, as in convenional ess, he asympoic size exceeds he nominal level. Moreover, he correlaion beween he innovaions of he predicor and he dependen variable acs as a power parameer in he limiing disribuion. In fac, if his correlaion is zero, he -saisic is asympoically normal disribued. Therefore, when he correlaion is sufficienly low and/or he uni roo deviaion parameer is sufficienly high, he disorion in es size is unappreciable. Campbell and Yogo (2006) derive a prees for deermining if he predicor is sufficienly saionary, for a given level of correlaion, such ha he convenional criical values can be applied and abulae he resuls. In paricular, hey abulae he values of c for which he size of he righ-ailed -es exceeds 7.5%, for seleced values of he correlaion beween he residuals. Their abulaed values can herefore be used o consruc a prees o decide wheher inference based on he convenional -es is sufficienly reliable. Specifically, hey indicae ha, independenly of he 11 See Sock (1994). 12 Torous, Valkanov, and Yan (2004) obain he null asympoic disribuion for he -saisic under his framework. 10

11 auocorrelaion of he predicor, he size of he es is less han 7.5% when he correlaion beween he innovaions is equal or less han in absolue value. 13 In order o know if sandard inference can be applied o our predicing exercise, we compue he serial correlaion of our proposed predicor, he HJ volailiy bound, and he correlaion beween he residual from he predicive regression and an AR(1) process for he predicor. Tha is, we esimae he following ordinary leas squares (OLS) regression model, M IPI,, (2) M u M, (3) where IPI, is he growh of indusrial producion a horizons of one, hree, six, 12, and 24 monhs calculaed as IPI lnipi IPI, and M, is he volailiy bound of he sochasic discoun facor available a monh ha is esimaed wih five years of monhly daa up o monh and 10 size-sored porfolios. We compue he correlaion beween he residuals from he wo equaions, Corr ˆ, û, which is displayed in he second column of Table 1 for he alernaive forecasing horizons analyzed in he paper. Despie he apparenly high level of persisence of 0.971, he correlaions beween he innovaions in IPI and he HJ volailiy bound are near o zero for all horizons beween 1 and 12 monhs, ranging from o respecively. Even a he 24-monh horizon he correlaion is lower han indicaing ha sandard asympoic disribuions can be applied for esing he significance of he predicor Campbell and Yogo (2006) also propose a new Bonferroni es of sock reurn predicabiliy, wihin he local-o-uniy asympoics, which is more efficien han he previous available es due o Cavanagh, Ellio, and Sock (1995). 14 In he case of he predicive regression a he longes horizon, i should be recognized ha he residuals are much higher (serially) correlaed due o he overlapping naure of he long horizon daa. Convenional inference employs he Newey-Wes (1987) auocorrelaion-robus sandard error. This is also he procedure followed in our empirical approach. Torous, Valkanov, and Yan (2004) derive he limiing 11

12 Figure 1 show 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. Our paper formalizes he evidence suggesed by Figure 1, and discusses he reasons behind he forecasing abiliy when employing size-sored porfolios In-Sample Predicabiliy wih he HJ Volailiy Bound We now proceed o analyze he predicing capaciy of he bound using he 10 sizesored porfolios. Panel A of Table 2 repors he resuls from he following in-sample predicive OLS auocorrelaion-robus sandard error regressions:, M IPI, (4) M IPI, 1IPI, 2, (5) where he firs equaion is he key univariae predicive regression we analyze in he paper, and he second equaion akes ino accoun ha serial correlaion in indusrial producion growh is expeced since we make muli-sep ahead predicions. This suggess ha he forecasing regressions should also include lagged values of he dependen variable. Each row of all panels of Table 2 corresponds o a paricular predicion horizon from one o 24 monhs. Alhough we employ indusrial producion growh as he relevan measure of real aciviy, Panel B repors similar evidence using GDP and disribuion for he robus -saisic in a local-o-uniy framework and hey show ha, as in he case of he one-period horizon, i depends on he correlaion beween innovaions of he variable o be prediced and he predicor. 12

13 consumpion growh insead of IPI growh. Given daa resricions on some of he sae variables used laer, we run hese predicive regressions beween January 1965 and July 2010, alhough Panel C conains evidence for alernaive sub-periods. 15 The op lef of Panel A repors he key resuls of he paper. There is a negaive and significan relaionship beween he HJ volailiy bound and fuure indusrial producion growh. Boh he magniude of he coefficiens (in absolue value) 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 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. 16 Therefore, our measure of uncerainy conveys informaion abou fuure economic growh. 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 above. 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. Our previous discussion on local-o-uniy framework suggess ha we may employ he convenional sandard aysmpoics when esing he significance of he volailiy bound as a predicor. In any case, we provide furher evidence using he biascorreced -saisic proposed by Amihud and Hurvich (2004), and Amihud, Hurvich, Wang (2009) in all previously esimaed forecasing regressions. The negaive and 15 The only excepion corresponds o he forecasing resuls using GDP growh where he sample period goes from April 1992 o December I should be recalled ha we use monhly daa in all panels. 16 -saisics are repored below he esimaed coefficiens. 13

14 significan relaion beween he volailiy bound and fuure real aciviy is mainained for all horizons. In paricular, he adjused -saisic ranges from o for he one and 24 monh horizons respecively in he univariae regression, and from o when we add he lagged IPI growh in he regressions. 17 Panel B of Table 2 displays he forecasing evidence using GDP and consumpion growh as he variables o be prediced. In boh cases, here is significan evidence of he HJ volailiy bound forecasing fuure macroeconomic aciviy. The resuls are paricularly sriking in he case of consumpion growh. The bound, as a measure of financial uncerainy, seems o conain informaion for fuure consumpion growh. The volailiy bound is srongly and negaively correlaed wih fuure aggregae consumpion. Panel C of Table 2 conains similar evidence for hree alernaive sub-periods using IPI growh as a measure of real economic aciviy. The previous empirical evidence is mainained for he full period from January 1931 o December 2010, bu i seems o be especially imporan for he sub-period beween January 1965 and December The higher volailiy of macroeconomic variables in he US marke before he grea moderaion years experienced beween he mid eighies and 2007 may explain he sronger predicive abiliy of he bound during he firs sub-period Ou-of-Sample Predicabiliy wih he HJ Volailiy Bound 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 ess designed o generae more closely acual real ime forecass. We employ wo 17 These auhors sugges a regression mehod for hypohesis esing in predicive regressions in which he independen variables are persisen and heir innovaions are correlaed wih he dependen variable. The auhors simulaions show ha heir adjusmen ouperforms oher bias correcion mehods, such as hose suggesed by Sambaugh (1999) and Lewellen (2004), and oher boosrapping mehods. The deailed resuls are available from he auhors upon reques. 14

15 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 predicing variable. On he oher hand, he unresriced model conains such a variable 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 X u, s 1,, -. (6a) s Rs The predicion under he resriced model is and he predicion error will be R R Ŷ ˆ ˆ X, (6b) R 0 1 û R Y Ŷ. (6c) R Similarly, he unresriced model ha includes he HJ volailiy bound, he nex period predicion and forecasing error are Y s U 0 U 1 U 2 M u, s 1,, X -, (7a) s s Us Ŷ U ˆ U ˆ U X ˆ U ( M ), (7b) û U Y Ŷ. (7c) U 15

16 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 û 2 R û 2 U, T,,T P, (8) MSE MSE R U T 1 P 1 T P 2 ûr, (9) T 1 P 1 T P 2 ûu. (10) 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 T 1 where d P 1 T 1 2 MSE P 1 d, (11) Ŝ P d and Ŝ d is a consisen esimaor of he variance of he 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 Hence Ŝ d k jk k k j T P 1 P j 1 d d d d. (12) T j The McCracken (2007) saisic is an F-es given by MSER MSEU MSE F P 1. (13) 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 U 16

17 of boh saisics is also unknown and ha he asympoic disribuion can only be obained under resricive assumpions ha include non-nesed models. 18 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: û U Y ˆ U 0 ˆ U 1 X ˆ M, 1,,T P U 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 1-1, , 1,,T P. 3. We simulae a sequence of independen and idenically disribued N(0,1) random variables denoed by and generae arificial residuals by using he esimaes of he MA process: ˆ ˆ ˆ 1 1ˆ -1, ˆ - -, 2,,T P * uu We simulae an arificial series of he dependen variable using he arificial residual and imposing he null hypohesis: Ŷ * ˆ R ˆ R * X u, 2,,T P. 0 1 U 18 See Wes (1996) and Clark and McCracken (2001) for a discussion. 17

18 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. Panel D of Table 2 conains he iniial ou-of-sample resuls where we es for he absolue ou-of-sample performance of he volailiy bound, and is relaive performance wih respec o he lagged IPI growh as he compeing predicor. In he firs case, he unresriced model is given by expression (4), while he resriced model is jus a forecasing regression of fuure oupu growh in a consan, IPI, (14) In he second case, he unresriced model is given by equaion (5), and he resriced model is he AR(1) base case, IPI, 1IPI,. (15) The ou-of-sample resuls are similar in boh cases. The empirical resuls from he lef box of Panel D show ha he HJ volailiy bound is a srong ou-of-sample predicor of fuure growh. The relaive mean squared error, RMSE MSE U MSE R, is less han one for all horizons excep for he longes horizon of 24 monhs, and he null hypohesis of equal forecasing accuracy is rejeced for all horizons from one o 12 monhs. Below each of he es saisics employed, we repor he corresponding p-value obained hrough he fixed regressor boosrap explained above. These resuls imply ha he inclusion of he bound improves he forecasing capaciy of a consan. Similarly, when we include he lagged IPI growh as he compeing predicor, he economic and saisical resuls are mainained, alhough he RMSE is slighly closer o one relaive o he firs case. The volailiy bound significanly improves he ou-of-sample forecasing abiliy of he lagged IPI growh as he compeing predicor. 18

19 4. Compeing In-Sample and Ou-of-Sample Predicors of Real Aciviy 4.1. The Compeing Predicors 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 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 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 a poenial sae variable for forecasing real aciviy. Finally, 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 19

20 informaion abou fuure economic growh, even afer conrolling for oher financial leading indicaors In-Sample Predicabiliy wih Compeing Predicors We nex employ all five variables discussed above and compare heir in-sample predicing abiliy wih ha of he HJ volailiy bound as esimaed wih 10 size-sored porfolios. We run he following predicive OLS auocorrelaion-robus sandard error regressions wih individual predicors and wih pairs of predicors ha always include he HJ bound: IPI, 1 M X, (16) 2 where X is eiher he marke reurn, he log of he price-dividend raio, he defaul spread, he erm spread, or he Amihud raio as a proxy for marke-wide illiquidiy. The in-sample resuls are repored in Panel A of Tables 3.a o 3.e where each case corresponds o a paricular forecasing horizon from one o 24 monhs. 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. I is especially relevan he sysemaic increase in he R 2 once we add he volailiy bound in regression (16). Hence, our forecasing relaion is sysemaically esimaed wih higher precision once we add he volailiy bound. 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 defaul spread, and marke-wide illiquidiy signal a higher degree of uncerainy, and we also know ha 19 The popular sock marke volailiy is analyzed in he secion dealing wih compeing measures of financial uncerainy. 20

21 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 coefficien associaed wih he price dividend raio is esimaed wih much more precision. On he oher hand, his 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. A he hree-monh horizon, all predicors seem o be individually significan and wih he correc sign. In he combined regressions, he higher R 2 saisics are obained when adding he price dividend raio, or 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 combinaion of he HJ volailiy bound wih eiher he sock marke reurn, he price dividend raio, or he erm spread seems o be he appropriae sraegy for 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 volailiy bound wih he erm spread have he highes R 2 saisics a he 12- and 24- monh horizons. A he longes horizon, 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 21

22 capaciy a longer horizons. In all cases, he HJ volailiy bound calculaed wih 10 sizesored porfolios remains a srong predicor of real aciviy Ou-of-Sample Predicabiliy wih Compeing Predicors The ou-of-sample resuls are repored in Panel B of Tables 3.a o 3.e where each case, as before, corresponds o a paricular forecasing horizon from one o 24 monhs. The firs row for each forecasing horizon shows he relaive mean squared error, RMSE. Recall 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. As in Table 2, below each of he es saisics employed, we repor he corresponding p-value obained hrough he fixed regressor boosrap of Clark and McCracken (2011). 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 independenly 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 20 The empirical resuls remain he same when we include he lagged IPI growh in he previous muliple forecasing regressions. In hese muliple-predicor regressions, we also employ biased-adjused -saisic proposed by Amihud, Hurvich, and Wang (2009) using a diagonal marix for he auoregressive esimaed coefficiens. As in he case of simple forecasing regressions, he adjused -saisic associaed wih he volailiy bound is -2.27, -3.28, -3.24, -1.72, and for one, hree, six, 12, and 24.monh horizons respecively when we add he sronger compeior for each paricular horizon. They are he defaul spread, dividend yield, and erm spread respecively depending upon he horizon analyzed in he regressions. All deailed resuls are available from he auhors upon reques. 22

23 predicing horizon. Noe ha his is consisen wih he in-sample resuls conained in Panel A of Table 3. We conclude ha he volailiy bound using 10 size-sored porfolios is a srong ou-of-sample predicor of fuure real aciviy relaive o well known compeing predicors. The uncerainy embedded in sock prices of size-sored porfolios is a powerful indicaor of fuure economic growh Oher Measures of Uncerainy as Predicors of Real Aciviy Anoher relevan issue is relaed o he comparison beween he HJ volailiy bound, as a measure of financial uncerainy, wih radiional compeiors like sock marke volailiy or VIX. We now provide evidence regarding he forecasing abiliy relaive o hese measures and ohers, less convenional measures, like idiosyncraic risk, he volailiy of he SMB Fama-French risk facor, and he cross-secional dispersion measures for quarerly forecass for GDP from he Survey of Professional Forecasers. We use wo measures of cross-secional dispersion. Dispersion measure D2 is he difference beween 75 h percenile and 25 h percenile of he forecass for GDP divided by GDP growh. Dispersion measure D3 is percen log-difference beween he 75 h percenile and he 25 h percenile of he forecass for GDP. As in he previous analysis wih compeing sae variables, we now run again he following in-sample predicive OLS auocorrelaion-robus sandard error regressions: IPI, 1 M X, (16) 2 where X is now one of he alernaive uncerainy measures menioned above, and he forecasing horizons are hree, six, 12 and 24 monhs. The shores horizon is now 3 monhs given ha he cross-secional dispersion measures are only available a he quarerly frequency. 23

24 The idiosyncraic risk refers o he average sock variance proposed by Goyal and Sana-Clara (2003), which is largely idiosyncraic. This measure avoids imposing a paricular facor asse pricing model and, in our case, i is esimaed eiher wih daily or monhly daa using he 100 size-book-o-marke porfolios displayed in he Kenneh Frnech s web sie. Similarly, he volailiy of he SMB facor is also esimaed wih eiher daily or monhly daa. In boh cases, we employ eiher a rolling window of one monh of daily daa, or 60 monhs of pas monhly observaions. VIX, as he key reference measure of financial uncerainy, refers o he volailiy given by CBOE a he las day of he corresponding monh during he sample period. The sock marke volailiy is esimaed a each monh as he sandard deviaion of monhly reurns using a rolling window of five years of pas observaions, o be consisen wih our measure of he HJ bound. As discussed in he inroducion, 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. 21 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: 12 1 Rm Rm1k, (17) 2 12 where 2 is a scaled facor relaed o he use of absolue values. We also compue his esimaor of sock marke volailiy o provide a poenially ineresing comparison wih he radiional sandard deviaion of reurns. k1 21 See he similar argumens of Chauve, Senyuz, and Yoldas (2011). 24

25 The resuls are conained in Table 4 where each panel corresponds o a given forecasing horizon. The resuls again confirm he forecasing abiliy of he volailiy bound on fuure oupu growh. The slope coefficien is, in all cases, negaive and significanly differen from zero, independenly of he forecasing horizon and he addiional uncerainy measure. I is especially imporan o noice he sysemaic increase of he R 2 when adding he volailiy bound o he forecasing regression. The combinaion of he bound and VIX generae he highes R 2 a 3 and 6-monh horizons, and i also remains high a he longes horizons. Idiosyncraic volailiy and he volailiy of he SMB facor are also relevan forecasers especially when we use he daily-based daa esimaors for relaively shor horizons. A he hree-monh horizon, he volailiy of he sock marke compued as he usual sandard deviaion does no presen significan forecasing capaciy by iself. In he combined regressions, he coefficien associaed wih volailiy of he sock marke becomes negaive and esimaed wih more precision han in he individual regressions. This evidence suggess ha he volailiy bound is capuring somehing else han marke volailiy. As we will discuss laer his is indeed he case. The volailiy of he bound is he maximum Sharpe raio. The predicing abiliy of he bound heavily depends on he ineracion beween he numeraor and denominaor raher han on he individual componens of he bound. On he oher hand, he marke volailiy esimaed as in equaion (17) obains relaively beer resuls han he radiional rolling window esimaor a he shores horizon. However, and conrary o he sandard deviaion, is marginal forecasing abiliy improvemen when combined wih he volailiy bound is lower han he one observed wih he regular measure. Finally, we also perform he ou-of-sample analysis using he Fornari-Melle measure of marke volailiy as he compeing predicor of he volailiy bound in he 25

26 resriced regression. The RMSE for all horizons is always lower han one, ranging from a he 3 monh horizon o a he longes horizon. In all cases, he es saisics show ha he inclusion of he HJ volailiy bound always significanly improves he predicing capaciy of he sock marke volailiy. We herefore conclude ha he HJ volailiy bound improves he in-sample forecasing abiliy of compeing uncerainy measures, and he ou-of-sample capaciy of he slowly changing measure of sock marke volailiy. 5. Alernaive Porfolio Formaion Crieria We now employ hree addiional alernaive measures of he HJ volailiy bound by using he reurns of 10 book-o-marke-, momenum-, and dividend yield-sored porfolios and a rolling window of five years of pas monhly reurns. Panel A of Table 5 conains he descripive saisics of he bound for hese hree soring procedures. As for he size-sored porfolios, all of hem presen posiive skewness and excess kurosis. The momenum soring has especially high momens wih a paricularly high posiive skewness relaive o he res of he porfolios. The correlaions among he bounds are posiive bu low excep for he correlaion coefficien beween he volailiy bound esimaed wih dividend yield and book-o-marke porfolios. These low correlaions sugges imporan differences beween he alernaive esimaed bounds. Noe ha he ineracion beween he numeraor and denominaor of he bound, volailiy dispersion and he complex dynamic correlaion behavior among he 10 porfolios in each of he four ses employed can generae a poenially differen ime series paern in he HJ bounds. We perform he forecasing regressions of equaion (2) using he HJ bound esimaed wih he 10 porfolios of each se. Panel B of Table 5 repors he resuls. 26

27 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 induced by he differen characerisics of he soring procedures may generae a differen forecasing abiliy of real aciviy. For example, i is ineresing o observe ha he annualized volailiy dispersion beween he exreme porfolios urns ou o be he highes for he size-sored porfolios. In paricular, he smalles porfolio have an 18.6 percen higher annualized volailiy han he porfolio of 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. In any case, i seems ha soring procedures, and he corresponding ime-varying diversificaion effecs and sensiiviies of reurns and risks o he business cycle are relevan issues for forecasing producion growh wih volailiy bounds. 22 The quesion is: why soring seems o be so imporan for predicing real aciviy? As poined ou before, he HJ volailiy bound is he maximum Sharpe raio. I is herefore he case ha he volailiy bound changes over ime because he maximum Sharpe raio varies over ime. Our evidence may be driven by he expeced reurn par of he Sharpe raio, by he inverse of he volailiy, by he ineracion of he wo, or by any of he wo componens of he volailiy of he angen porfolio. We invesigaes he alernaive componens of he maximum Sharpe raio, SR T, as poenial sources of predicabiliy by running forecasing regressions of fuure oupu growh on he percenage of each of he componens on he absolue value of he maximum Sharpe 22 Pasor and Veronesi (2009) show ha he volailiy of he sochasic discoun facor depends on he dynamic associaed wih echnological adopions. In paricular, hey show ha, once he new echnology has arrived, he volailiy of he sochasic discoun facor ends o be fla as long as he probabiliy of adopion is low, and i increases very rapidly as he probabiliy increases. As wih hese porfolio ses, he volailiy bound esimaed from alernaive indusry-sored porfolios does no presen any significan predicing abiliy of fuure oupu growh. 27

28 raio. Therefore, we run forecasing regressions wih four alernaive independen variables: E R T SR R T f, T SR T, Var T SR T, and Cov T SR T (18) where he wo las componens correspond o he firs and second elemens of he variance of he angen porfolio given by, 2 T N N N 2 2 j j jtkt jk j k (19) j 1 j 1k 1 k j Var T Cov T The resuls srongly indicae ha he main driver of predicabiliy is he ineracion beween he numeraor and he denominaor of he volailiy bound raher han any of is componens. The only marginally significan forecasing capaciy appears o be relaed o he componens of he variance of he angen porfolio. In paricular, and only for he shores horizons, boh Var Cov T SRT and T SRT presen some evidence of predicing abiliy wih he righ sign. 23 And, more imporanly, his is he case only for he se of size-sored porfolios. The componens of he bounds for alernaive soring procedures do no presen any evidence of forecasing fuure economic growh. I is herefore he ineracion beween he numeraor and denominaor of he bound for sizesored porfolio he main driver of he forecasing evidence repored above. 6. Why Size is so imporan for Predicing Real Aciviy? There is a consolidaed lieraure ha predics ha changing credi marke condiions affec very differenly small and large firms. 24 Under asymmeric informaion, crediors 23 As before, he deailed resuls are available from he auhors upon reques. 24 See he fundamenal argumens based on imperfec capial marke heory provided by Kiyoaki and Moore (1997), and he empirical evidence repored by Pérez-Quirós and Timmermann (2000). 28