Macro-Finance Deerminans of he Long-Run Sock-Bond Correlaion: The DCC-MIDAS Specificaion * Hossein Asgharian, Lund Universiy + Charloe Chrisiansen, CREATES, Aarhus Universiy ++ and Ai Jun Hou, Sockholm Universiy +++ January 13, 2015 * The auhors hank Andrew Paon for helpful commens and suggesions. Asgharian hanks he Jan Wallanders and Tom Hedelius Foundaion for funding his research. Chrisiansen acknowledges financial suppor from CREATES (Cener for Research in Economeric Analysis of Time Series) funded by he Danish Naional Research Foundaion (DNRF78). + Hossein Asgharian, Deparmen of economics, Lund Universiy, Box 7082, 22007 Lund, Sweden. Hossein.Asgharian@nek.lu.se ++ Charloe Chrisiansen, CREATES, Deparmen of Economics and Business, School of Business and Social Sciences, Aarhus Universiy, Fuglesangs Allé 4, 8210 Aarhus V, Denmark. cchrisiansen@creaes.au.dk. +++ Ai Jun Hou, School of Business, Sockholm Universiy, Sweden. ajh@fek.su.se.
Macro-Finance Deerminans of he Long-Run Sock-Bond Correlaion: The DCC-MIDAS Specificaion Absrac: We invesigae he long-run sock-bond correlaion using a novel model ha combines he dynamic condiional correlaion model wih he mixed-daa sampling approach. The long-run correlaion is affeced by boh macro-finance variables (hisorical and forecass) and he lagged realized correlaion iself. Macro-finance variables and he lagged realized correlaion are simulaneously significan in forecasing he long-run sock-bond correlaion. The behavior of he long-run sock-bond correlaion is very differen when esimaed aking he macro-finance variables ino accoun. Supporing he fligh-o-qualiy phenomenon for he oal sock-bond correlaion, he long-run correlaion ends o be small/negaive when he economy is weak. Keywords: DCC-MIDAS model; Long-run correlaion; Macro-finance variables; Sock-bond correlaion JEL Classificaions: C32; C58; E32; E44; G11; G12 1
1. Inroducion Socks and bonds are he wo main asse classes. Thus, i is of imporance o invesigae furher he behavior of he sock-bond correlaion. Here, we inroduce an innovaion o he lieraure by decomposing he oal sock-bond correlaion ino is long-run and shor-run componens and by using financial and economic variables o predic he long-run componen. We use he Dynamic Condiional Correlaion (DCC) model coupled wih he Mixed-Daa Sampling (MIDAS) approach. The new DCC-MIDAS model allows he long-run correlaion o be affeced by boh macro-finance variables and he lagged realized correlaion iself. The MIDAS regression is inroduced by Anderou and Ghysels (2004) and Ghysels e al. (2006). I allows daa from differen frequencies o ener ino he same model. This approach makes i possible o combine high-frequency reurns wih macro-finance daa ha are only observed a lower frequencies (such as monhly and quarerly). Engle and Rangel (2008) apply his echnique o he GARCH framework o form he spline GARCH model. Combining he spline GARCH framework and he volailiy decomposing approach (see Ding and Granger, 1996; Engle and Lee, 1999; Bauwens and Sori, 2009; Amado and Teräsvira, 2013), Engle e al. (2012) inroduce he GARCH- MIDAS model. The model has he advanage ha i allows us o direcly incorporae informaion on he macroeconomic environmen ino he long-run componen. Conrad and Loch (2012) use he GARCH-MIDAS framework o decompose he sock reurns ino shor-run and long-run componens. They examine he long-run volailiy componen using economic facors. Baele e al. (2010) and Colacio e al. (2011) apply he MIDAS echnique o he DCC model of Engle (2002) o decompose he comovemen of socks and bonds ino shor-run and long-run componens. Finally Conrad e al. (2012) exend he DCC-MIDAS model by allowing macro-finance variables o ener he long-run componen of he correlaion of crude oil and sock reurns. The comovemen of sock and bond reurns may sem from several sources. Sock and bond reurns are expeced o be correlaed because heir fuure cash flows and he perinen discoun raes can be affeced by he same economic facors. Previous research invesigaes he predicive power of various macro-finance variables for he sock-bond comovemen. Viceira (2012) finds ha he yield spread and he shor rae are imporan deerminans of he sock-bond comovemen. Campbell and Ammer (1993) decompose he bond and sock reurns ino unexpeced componens of fuure cash flows and fuure discoun raes and employ a vecor auoregression model wih asse reurns and macro variables. They show ha sock and bond reurns are influenced by differen facors, which migh be he reason why sock and bond reurns are no srongly correlaed. Sock and bond reurns may also be correlaed since hey are alernaive invesmens. There are a number of empirical sudies addressing he effec of money ransfer beween he wo markes on he asses liquidiy, volailiy, and reurns. Agnew and Balduzzi (2006) find ha invesors rebalance porfolios as responses o changes in asse prices, and ha his resuls in a negaive correlaion beween ransfers in socks and bonds, which in urn leads o a negaive correlaion beween reurns in hese wo markes. Baele e al. (2010) show ha liquidiy relaed variables hold predicive power for he sock-bond comovemen, whereas macroeconomic variables hardly do. In general, sock and bond comovemen is expeced o be posiive excep in periods of fligh-o-qualiy. Fligh-oqualiy implies ha he ransfer of money from he high-risk sock marke o he low-risk bond marke a imes of high uncerainy increases he bond prices relaive o he sock prices, which makes he sock-bond correlaion weaker and perhaps even negaive. Fleming e al. (1998) find ha here are volailiy linkages beween he sock, bond, and money markes due o cross marke hedging. Connolly e al. (2005, 2007) invesigae how he sock marke uncerainy (measured by 2
he VXO volailiy index) influences he sock-bond comovemen and show ha he comovemen is posiive (negaive) following periods wih low (high) uncerainy. In his paper, we sudy he impac of a large group of macro-finance variables on he long-run componen of he sock and bond reurn volailiy and correlaion. We have seleced a wide range of variables suggesed by differen sudies on sock-bond co-movemen. The variables include sandard macro-finance variables (shor rae, inflaion), a liquidiy variable (volume of S&P 500 fuure conrac), he equiy uncerainy variable (VXO), variables reflecing he curren sae of he economy (he indusrial producion growh, he unemploymen rae, he defaul spread, he producer confidence index (PMI), he consumer confidence index (CC), and he Naional Aciviy Index (NAI)), as well as he Survey of Professional Forecaser daa (SPF). Furher, differen from mos of he previous sudies, we use he bond and sock reurns a he daily frequency and oher macro-variables a quarerly frequency wihin he same model using he MIDAS echnique. We firs decompose he sock and bond volailiy ino is shor-run and long-run componens by esimaing a univariae GARCH-MIDAS model for sock and bond reurns, where we allow for he direc impac of a macro-finance variable on he long-run componen of he volailiy. We hen sudy he macro-finance variable s impac on he long-run correlaion wihin he DCC-MIDAS framework. For his purpose we esimae he model wih a number of differen specificaions of he long-run correlaion equaion, i.e., a specificaion ha only includes lagged realized correlaions, a specificaion wih only a macro-finance variable, and a specificaion wih boh lagged realized correlaion and a macro-finance variable. Our resuls indicae ha cerain macro-finance variables including inflaion, indusrial producion, he shor rae, he defaul spread, he S&P volume, he producer confidence, and he consumer confidence affec he long-run sock-bond correlaion. However, in order for he model o perform well, i is imporan o ake he lagged realized correlaion ino accoun in he MIDAS modeling, in addiion o he macro-finance variables. Second, we find ha he long run sock-bond correlaion is negaive when he sae of economic is weak, indicaing he exisence of he fligh-o-qualiy phenomenon. We also find ha survey daa conain rich informaion for deermining he bond and sock correlaions, which sugges ha he perceived sance of he economy is an imporan deerminan of sock and bond correlaion. This paper conribues o he lieraure in several ways. This is he firs sudy based on he DCC- MIDAS model which includes macro-finance variables direcly in he equaion for he long-run componen of he sock-bond correlaion. We use a broader range of specificaions of he DCC- MIDAS model compared o he exising lieraure. We use a wide range of macro-finance variables, including boh hisorical daa and forecased daa. By invesigaing he long-run sock-bond correlaion and relaing i o he economic variables, we are able o provide new empirical evidence on he fligh-o-qualiy phenomenon. Finally, by using a wavele approach, we provide furher indicaions of he usefulness of smoohing echnics such as he DCC-MIDAS for predicing he long-run componen of he sock-bond correlaion. The remaining par of he paper is srucured as follows. Firs, in Secion 2, we lay ou he economeric framework, including our suggesed DCC-MIDAS model wih macro-finance variables. Then, we inroduce he daa in Secion 3. In Secion 4 we discuss some opening resuls ha lead up o our main empirical findings in Secion 5. We conclude in Secion 6. 3
2. DCC-MIDAS Sock-Bond Correlaion Model This secion oulines he economeric models used in his paper. Firs, we discuss he bivariae DCC-MIDAS model of Colacio e al. (2011). Second, we inroduce he new DCC-MIDAS-XC model in which he long-run sock-bond correlaion depends on a macro-finance variable (denoed by X ) as well as he lagged realized correlaion (denoed by C ). Third, we inroduce forecas daa (denoed by F ) ino he model using he DCC-MIDAS-XCF specificaion. 2.1 The DCC-MIDAS Model I is convenien o describe wo relaed economeric models before we ge o he DCC-MIDAS model iself, ha is, he GARCH-MIDAS model, and he Dynamic Condiional Correlaion (DCC) model. We begin wih he univariae GARCH-MIDAS framework of Engle e al. (2010). Consider a reurn series on day i in a period (e.g., monh, quarer, ec.) ha follows he process: r = µ + τ g ε, i = 1,..., N. (1) i, i, i, ε i, Φ i 1, ~ N (0,1) where N is he number of rading days in he period and Φ is he informaion se up o day (i-1) of period. Equaion (1) expresses he variance ino a shor-run componen defined by g i, and a long-run componen defined by τ which only changes every period. The oal condiional variance is defined as: 2 i g i, i 1, σ = τ. (2) The condiional variance dynamics of he componen g i, follows a GARCH (1, 1) process, ( r µ ) 2 i 1, g i, = ( 1 β ) + α + βgi 1, τ α (3) where α > 0 and β 0, α + β < 1 and τ is defined as smoohed realized volailiy in he MIDAS regression: K k = 1 ( w w ) log( τ ) = m + θ ϕ RV (4) k 1, N i= 1 2 k 2 RV = r i. (5), K is he number of lags over which we smooh he realized volailiy. Following Asgharian e al. (2013), we modify his equaion by including he economic variables along wih he lagged realized volailiy (RV) in order o sudy he impac of hese variables on he long-run reurn variance: K K = m + θ1 ϕ k 1 2 k + 2 k 1, 2 k = 1 k = 1 Q ( w, w ) RV θ ϕ ( w w ) X log( τ ) (6) k 4
Q where X k represens a macro-finance variable (measured a quarerly frequency). Noe ha we use a fixed window for he MIDAS, which means ha he componen τ used in our analysis does no change wihin a fixed period. The weighing scheme used in equaions (4) and (5) is described by a bea lag polynomial as follows: w1 1 ( k ) ( 1 k ) w2 1 ϕ k ( w) = K K, k = 1,... K. (7) K w 1 2 1 1 w j j 1 j= 1 K K For w 1 = 1, he weighing scheme guaranees a decaying paern, where he rae o decay is deermined by w 2. In he bivariae DCC model of Engle (2002), he reurn vecor follows he process: r ~ N ( µ, ) H and he condiional covariance marix is specified as H = D R D, where D is a diagonal marix wih sandard deviaions of reurns on he diagonal and R is he condiional correlaion marix of he sandardized reurn residuals. The condiional volailiies for asse S and B (q SS,+1 and q BB,+1 ) follow regular univariae GARCH models, e.g., he GARCH(1,1) specificaion. These are esimaed firs and seperaely. Then in a second esimaion sep, heir condiional covariance is esimaed. The 1 2 1 2 condiional correlaion is given as R = diag ( Q ) Q diag ( Q ) and Q (in elemenary form) is specified as q SB ρ (1 a b ) + a ( ξ ξ ) b ( q ) (8) = SB, S, 1 B, 1 + SB, 1 hereby giving us he condiional correlaion as q SB, ρ SB, = (9) qss, + 1qBB, where ξ S, and ξ B, are he sandaized residuals from he univariae models. ρ SB, is he uncondiional correlaion beween he sandardized residuals. The DCC-MIDAS model of Colacio e al. (2011) is a naural exension and combinaion of he DCC model and he GARCH-MIDAS model. The DCC-MIDAS model uses he sandardized residuals from he univariae GARCH-MIDAS model o esimae he condiional volailiies and he dynamic correlaion beween he asse reurns. The condiional covariance is now given as: q SB, = SB, (1 a b ) + a ξ S, 1ξ B, 1 + bq SB, 1 ρ (10) K SB, = ϕ k ( wk ) CSB, 1 k=1 ρ (11) C SB, = k= N k= N ξ ξ 2 S, k S, k ξ B, k k= N ξ 2 B, k (12) 5
where ξ S, k and ξ B, k are he sandardized residuals from he GARCH-MIDAS model of differen reurn series. The correlaions can hen be compued as in eq. (8). The q SB, is he shor-run correlaion beween asses S and B, whereas ρ SB, is a slowly moving long-run correlaion. 2.2 The DCC-MIDAS-XC Model We provide a compleely new exension of he DCC-MIDAS model o allow a macro-finance variable and he lagged realized correlaion o affec he long-run sock-bond correlaion. This is similar o he Asgharian e al. (2013) exension of he GARCH-MIDAS model. We updae he long-run correlaion in eq. (10) so ha we have he DCC-MIDAS-XC model: q SB, = SB, (1 a b ) + a ξ S, 1ξ B, 1 + bq SB, 1 ρ (13) ρ SB, = exp (2 z exp (2 z SB, τ SB, τ ) 1 ) + 1 (14) z K K Q SB m +, τ = SB θ RC ϕ k ( w1, w2 ) RCSB, k + θ X ϕ k ( w1, w ) X 2 k (15) k= 1 k = 1 RC N ξ ξ S, i B, i i= 1 SB, = (16) N N 2 2 ξ S, i ξ B, i i= 1 i= 1 where RC SB, is he realized correlaion (measued a he quarerly frequency). Q X is a macrofinance variable measued a he quarerly frequency. The usage of he Fisher ransformaion in eq. (14) follows Chrisodoulakis and Sachell (2002). By imposing he parameer resricion ha θ RC = 0, he DCC-MIDAS-X model of Cornad e.al. (2012) appears. By imposing he parameer resricion ha θ x = 0, anoher new model appears, he DCC-MIDAS-C model, in which only he lagged realized correlaion affecs he long-run sockbond correlaion. 2.3 The Two-Sided Exension: DCC-MIDAS-XCF Engle e al. (2012) sugges ha he performance of he GARCH-MIDAS model can be improved by including he fuure values of he macro variables (i.e. so called wo-sided filer) when anicipaing he long erm volailiy. We apply he wo-sided filer here. We make use of he DCC-MIDAS-XC model simulaneously using forecased and observed macro-finance variables, i.e., he wo-sided version of he model, he DCC-MIDAS-XCF model. Imposing θ RC o be zero and applying he wo-sided filer of Engle e al. (2012), eq. (15) can be modified as follows: z Klag 0 Q SPF SB, = m + θ X ϕk ( w1, w2 ) X k + θ X ϕ k ( w1, w2 ) X. τ k k= 1 k= Klead (17) 6
Noice ha he fuure unknown values are replaced wih forecased daa. Ideally, we would model he impac of he forecased variables on he long-run dynamic correlaions according o eq. (17), i.e., he same parameer θ should be shared by boh he hisorical and he forecased daa, and i would be esimaed wih a wo-sided filer. In his case he opimal weighing schemes for he variables do no decay monoonically bu are raher hump-shaped. However, he forecasers perform Q he predicion given he firs release daa and no he finally revised daa, while X k used in he equaion is he hisorical (finally revised) daa. Hence, i is difficul o inegrae and combine he hisorical daa and he forecased daa based on he firs release daa wih a wo-sided filer. 1 Therefore, we decide o model he impac of he forecased daa wih a modified wo-sided filer in which we rea he forecased daa as an individual variable. The specificaion is in he following:, z Klag 0 Q SPF SB, = m + θ X ϕ k ( w1, w2 ) X k + θ FX ϕ k ( w1, w2 ) X k k = 1 k = Klead τ. (18) Inuiively, for he weigh of he forecased daa, we would expec ha he highes weigh should be given o he mos recen variables. Consequenly, we should also give he highes weigh o he mos leaded lags. Therefore, we se w 1 =1 for he weighing scheme of he hisorical daa, esimae w 2, and se w 2 =1 for he weighing scheme of he forecased daa while esimaing w 1. 2.4 Esimaion Mehod N is se o be he number of he rading days wihin each quarer, he oal number of lags is K lag = 16 quarers (four years), and he oal number of leads is K lead = 3. Following Engle (2002) and Colacio e al. (2011), we esimae he model parameers using a wo-sep quasi-maximum likelihood mehod. The quasi-maximum likelihood funcion o be maximized is T T ' 2 ' 1 ' ( T log( 2 ) + 2 log D + ξ D ξ ) ( log R + ξ R ξ ξ ) L = π ξ (19) = 1 where he marix D is a diagonal marix wih sandard deviaions of reurns on he diagonal, and R is he condiional correlaion marix of he sandardized reurn residuals. The model involves a large number of parameers, and i does no always converge o a global opimum by he convenional opimizaion algorihms. Therefore, we use he simulaed annealing approach for he esimaion (cf. Goffe e al. 1994). This mehod is very robus and seldom fails, even for very complicaed problems. 3. Daa We use a combinaion of quarerly macro-finance variables and daily sock and bond reurns. We consider he sample period from he firs quarer of 1986 o he second quarer of 2013. The expecaion daa are obained from he Survey of Professional Forecasers (SPF) daabase a he Federal Reserve Bank of Philadelphia. The survey is conduced by he American Saisical Associaion and he Naional Bureau of Economic Research. The remaining daa are obained from DaaSream. = 1 1 Conrad and Lonch (2012) allow he model o be enirely based on SPF expecaion and replace he firs release daa wih he corresponding real-ime SPF expecaions. 7
3.1 Sock and Bond Daa The wo main variables of ineres are he sock and bond reurns. The Realized Volailiy is calculaed based on he daily reurns from he selemen prices of he S&P500 fuures conracs raded a he CME and he 10-year Treasury noe fuures conrac raded a he CBT. 3.2 Macro-Finance Variables We have seleced a wide range of variables suggesed by differen sudies on he sock and bond reurn co-movemen. Inflaion and shor raes: These wo are he sandard variables feaured in macroeconomic models. They are expeced o affec boh he cash flow and he discoun rae. However, heir effecs on bond and sock reurns may differ. Because bonds have fixed nominal cash flows, inflaion may generae differen exposures beween socks and bond reurns. The prominen role of inflaion for predicing fuure sock-bond correlaion is documened by Li (2002a). I is well known ha he level of he ineres rae drives he inflaion. Therefore we include he shor-erm rae. Viceira (2012) documens ha he shor rae and he erm spread are boh key deerminans of he sock-bond correlaion. Liquidiy variable: The lieraure on bond (Amihud & Mendelson 1991) and equiy pricing (Amihud 2002) has increasingly sressed he imporance of he liquidiy effec, which may also be conneced wih he fligh-o-qualiy phenomenon. Crisis periods may drive invesors and raders from less liquid socks ino highly liquid bonds, and he resuling price-pressure effecs may include negaive sock-bond correlaions. Therefore, as in Baele e al. (2010), we include he rading volume of S&P500 fuure conracs as he liquidiy-relaed variable in he paper. Sae of economy variables: Ilmanen (2003), Guidolin and Timmermann (2006), and Aslanidis and Chrisiansen (2013) show ha he general sae of he macro economy provides informaion abou he fuure sock-bond correlaion. Aslanidis and Chrisiansen (2012) show ha he shor rae, he erm spread, and he VXO volailiy index are he mos influenial ransiion variables for deermining he regime of he realized sock-bond correlaion. Here we le prominen variables such as he indusrial producion growh, he unemploymen rae, he defaul spread, he producer confidence index (PMI), he consumer confidence index (CC), and he Naional Aciviy Index (NAI) represen he sae of he macro economy. Sock marke uncerainy: Many papers (e.g., Connolly e al. 2005, 2007 and Bansal e al. 2010) have used he VIX-implied volailiy measure as a proxy for sock marke uncerainy and shown ha he sockbond co-movemens are negaively and significanly relaed o sock marke uncerainy. As he daa sar in 1986, we use he VXO index as a proxy for sock marke uncerainy. In summary, we use he following quarerly macro-finance variables: Inflaion, compued as he log-difference of he seasonally adjused CPI. Indusrial producion growh, compued as he log-difference of he quarerly values of he indusrial producion. Unemploymen rae, compued as he firs differences of he quarerly unemploymen raes. Term spread, compued as he firs differences of he yield spread beween 10-year Treasury bond and 3-monh Treasury bill. Shor rae, compued as he firs differences of yield on he 3-monh US Treasury bill. 8
Defaul spread, compued as he firs differences of he yield spread beween Moody s Baa and Aaa corporae bonds. S&P500 volume is he firs differences of he volume of he S&P500 fuures conrac. VXO, defined as he log-differences of he volailiy index. PMI, defined as he log-differences of producer confidence index. CC, defined as he log-differences of consumer confidence index. NAI is he value of he Naional Aciviy Index. 3.3 Forecased Macro-Finance Variables The Survey of Professional Forecasers is conduced afer he release of he advance repor of he Bureau of Economic Analysis, implying ha he paricipans know he daa for he previous quarer when hey make heir predicions. Due o daa availabiliy, we only include he forecased inflaion rae, unemploymen rae, erm spread, and shor rae. 2 We use median forecass for he firs hree coming quarers. The forecased daa are denoed by, 1,2, 3 X SPF + k k =. 4. Opening Resuls: Sock-Bond Correlaion and Smoohed Variables We sar by invesigaing if smoohing of macro-finance variables srenghens he correlaion beween macro-finance variables and he sock-bond correlaion. We use he wavele approach o smooh he macro-finance variables and hen look a he correlaion of he smoohed variables and he fuure realized sock-bond correlaions a differen leads. A discree wavele approach divides a ime-series, z, ino a se of componens of differen ime frequencies. The smooh (low-frequency) componens of a ime series are represened by J ( l) J A J, = 2 2 2 z ν J,l, l and he deailed (high-frequency) pars are represened by ν d (20) υ d, (21) j 1 lps B j, = j j z υ j,l, l s s j where s is he scale facor, p is he ranslaion facor, and s is he facor for normalizaion across he differen scales. The index j = 1, 2,, J, he scale where J is he maximum scale possible given he number of observaions for z, and l is he number of ranslaions of he wavele for any given scale. The noaions ν J,l, and υj,l, are he wavele funcions. The scaling funcions are orhogonal, and he original ime series can be reconsruced as a linear combinaion of hese funcions and he relaed coefficiens: 2 The forecased indusrial producion is also available. However, we exclude i as he forecased daa are quie differen from he hisorical daa obained from DaaSream. 9
J J z = A, + B,. (22) The scale B j, capures informaion wihin 2 j-1 and 2 j ime inervals. To consruc he smoohed series, we exclude all B j, up o he frequency of ineres. For example, wih quarerly daa, eliminaing all B j, for j 3 excludes all he variaions ha belong o frequencies higher han 2 3 quarers, i.e., wo years. 3 j = 1 j Inser Figure 1: Wavele Correlaion Figure 1 shows he wavele correlaion of he realized sock-bond correlaion wih he nonsmoohed and smoohed values of he macro-finance variables. We use up o forh order wavele smoohing. We use a random walk model (lagged realised correlaion) as he benchmark for he comparison. Wihou smoohing of he macro variable, he random walk model ouperforms he macro-finance variables and shows he sronges correlaion wih he fuure realised correlaion. Sill, he correlaion is reduced as we increase he number of leads. More specifically, he correlaion beween realised bond-sock correlaions a ime and +1 is around 0.8. Beween ime and +4 i is around 0.6. The maximum correlaion beween macro-finance variables and fuure sock-bond correlaion is around 0.4 when we use no smoohing, bu for almos all of he macrofinance variables he correlaion increases when we we use he wavele smoohed series. Wih four levels of wavele smoohing (smoohing up o 16 quarers), he S&P volume has a sronger correlaion han he lagged realized correlaion iself, especially for longer forecas horizons. The wavele findings moivae ha smoohing echnics such as he DCC-MIDAS model are useful in modeling he long-run componen of he sock-bond correlaion. An advanage of he DCC- MIDAS over alernaive smoohing echnics such as he wavele echnich is ha he opimal smoohing level is endogenousely deermined by he daa for he DCC-MIDAS model. 5. DCC-MIDAS-XC Resuls In his secion we describe he cenral empirical resuls. 4 Firs, we show he univariae GARCH- MIDAS-XC resuls. Second, we show he resuls of he DCC-MIDAS-XC model where he macrofinance variables influence he long-run componen of he sock-bond correlaion. Third, we show he resuls from using forecass for he macro-finance variables in DCC-MIDAS-XCF model o esimae he long-run componen of he sock-bond correlaion. 5.1. Macro-Finance Deerminans of Long-Run Volailiy Table 1 shows he resuls from esimaing he various GARCH-MIDAS-XC specificaions for sock volailiy (Panel A) and bond volailiy (Panel B). Inser Table 1: GARCH-MIDAS-XC For sock volailiy he bes model fi is obained for he specificaions ha allow for boh realized volailiy and a macro-finance variable (smalles AIC), followed by he models wih only realized volailiy which is again followed by he models ha only include macro-finance variables. Mos of he macro-finance variables are significan in explaining he long-run componen of he sock volailiy even when aking he realized volailiy ino accoun, he only excepions being he defaul 3 See Gencay e al. (2001) for a deailed discussion on he wavele mehod. 4 Throughou we use he 10% level of significance. 10
spread and he VXO volailiy index. The bes fi is observed in specificaions where boh he realized volailiy and he macro-finance variable are significan simulaneously. This is he case for he inflaion rae, he PMI, and he NAI. These hree macro-finance variables are all measures of real economic aciviy, i.e., hey are relaed o he business cycle. The sign of he effec is differen across macro-finance variables. There is a posiive effec from inflaion, such ha he larger he inflaion rae is, he larger he long-run sock volailiy is. For he PMI and he NAI he effec is negaive, so ha he smaller he PMI or NAI is, he larger is he long-run sock volailiy. The signs of he effecs from he macro-finance variables imply ha he long-run sock volailiy is smaller in imes of posiive overall economic condiions (low inflaion, high producer confidence, and high aciviy). Our resuls confirm he couner-cyclical behavior of sock marke volailiy firs observed by Schwer (1989). The resuls are also consisen wih Conrad and Loch (2012). They employ he GARCH-MIDAS framework on he US sock marke and find ha long-erm sock volailiy is negaively relaed o measures of economic aciviy. For he bond volailiy he ranking of he bes performing models is similar o sock volailiy. I is preferable o include boh realized volailiy and macro-finance variables when describing he longrun volailiy, followed by realized volailiy alone, and macro-finance variables alone. Ye, only few of he macro-finance variables are significan when addiionally accouning for he realized volailiy (GARCH-MIDAS-XC specificaion), namely only he erm spread, he defaul spread, and he VXO volailiy index. For hese variables boh he realized volailiy and he variables hemselves are simulaneously significan. So, for he bond volailiy, fixed income relaed variables are of imporance, which is very differen for he sock volailiy resuls. I is worh noing ha he signs of he coefficiens o he erm spread and he defaul rae are opposie he signs hey have in he sock volailiy. To some exen he defaul spread is relaed o he business cycle condiions. The VXO volailiy index also provides informaion abou he sae of he economy, in ha large VXO is conneced wih high uncerainy. The effec from he variables upon he long-run bond volailiy is posiive, so ha he larger he erm spread, he defaul spread, and he VXO volailiy index is, he larger is he long-run bond volailiy. As for socks, his implies ha long-run bond volailiy is large when he general economic condiions are weak (large erm spread, defaul spread rae, and large VXO volailiy). To our knowledge, here are no previous sudies of he effec of macro-finance variables upon he long-run bond volailiy for comparison of our resuls. Inser Figure 2: Long-Run Sock Volailiy Inser Figure 3: Long-Run Bond Volailiy Figures 2 and 3 show he long-run volailiy for socks and bonds for he various specificaions. The long-run componen is a lo smooher when i is esimaed based on (significan) macro-finance variables han when i is based on lagged realized volailiy. For he combinaion based on (significan) macro-finance variables and lagged realized correlaion, he long-run componen is sill fairly smooh, bu a lile less so han wih only macro-finance variables. Thus, in order o obain sable long-run sock and bond volailiy, i is of imporance o ake ino accoun he sae of he economy (as measured by various macro-finance variables). 5.2. Macro-Finance Deerminans of he Long-Run Correlaion 11
In Table 2 we show he resuls where boh he lagged realized correlaion and one macro-finance variable a a ime is included in he long-run sock-bond correlaion equaion (he DCC-MIDAS-XC model). In addiion, we show he resriced versions wih only he realized correlaion (DCC- MIDAS-C) and wih only he macro-finance variables (DCC-MIDAS-X). Inser Table 2: DCC-MIDAS-XC The resuls from he DCC-MIDAS-X model show ha he sign of he influence of he macrofinance variables is posiive and significan for inflaion, indusrial producion, S&P rade volume, and NAI, and i is negaive and significan for unemploymen. This clearly indicaes ha he longrun sock-bond correlaion ends o be small/negaive when he economy is weak, and i suppors he previous lieraure on he exisence of he fligh-o-qualiy phenomenon. However, we do no find such a clear paern for he coefficiens relaed o hese variables in he DCC-MIDAS-XC model. The reason ha he coefficien of he macro-finance variables in he DCC-MIDAS-XC canno fully reflec he relaionship beween he economic condiions and he long-erm correlaion is ha he realized correlaion iself o a large exen already capures his effec (he coefficien of his variable is posiive and highly significan in all he cases). Therefore, he coefficiens of he macro-finance variables in his model indicae he impac on he long-erm correlaion afer considering wha is already capured by he variable realized correlaion in he model. The bes model fi (based on AIC) is obained in he models wih boh realized correlaion and a macro-finance variable which is followed by models wih he realized correlaion only. Amacrofinance variable alone gives he wors fi. This is similar o he ranking of he univariae models for he sock and bond volailiy. However, he variables ha influence he long-run sock-bond correlaion differ from hose ha influence he long-run sock and bond volailiy. The inflaion rae, he indusrial producion, he shor rae, he defaul spread, he S&P volume, he PMI, and consumer confidence are all significan variables when considered joinly wih he lagged realized correlaion for explaining he long-run sock-bond correlaion. Only he inflaion rae, he defaul spread, and he PMI are recurring from he long-run volailiy for socks and bonds,. The oher imporan macro-finance variables for explaining he long-run sock volailiy (NAI) and bond volailiy (erm spread) and VXO are no significan for he long-run sock-bond correlaion. The forecasing abiliy of he inflaion is consisen wih Ilmanen (2003) who finds ha changes in discoun raes dominae he cash flow expecaions during periods of high inflaion, hereby inducing a posiive sock-bond correlaion. This is, however, in conras wih Campbell and Ammer (1993) who repor ha variaions in expeced inflaion promoe a negaive correlaion since an increase in inflaion is bad news for bonds and ambiguous news for socks. The auhors also find ha variaion in ineres raes promoes a posiive correlaion since he prices of boh socks and bonds are negaively relaed o he discoun rae. The S&P volume is a measure of liquidiy. The larger he S&P volume is, he larger he long-run sock-bond correlaion is. So, high liquidiy implies large/posiive sock-bond correlaion. The usefulness of liquidiy in forecasing he long-run sock-bond correlaion is in line wih he findings in Baele e al. (2010) who show ha liquidiy relaed variables hold predicive power for he sockbond comovemen. Inser Figure 4: DCC-MIDAS-C Long-Run Correlaion Figure 4 shows he long-run componen of he correlaion as well as he daily correlaion semming from he DCC-MIDAS-C model. The long-run componen is a lo less variable, i.e., smooher han he oal correlaion. 12
Inser Figure 5: DCC-MIDAS-X Daily Correlaion Inser Figure 6: DCC-MIDAS-XC Daily Correlaion Figures 5 and 6 show ha he differen specificaions, i.e., he DCC-MIDAS-X and he DCC- MIDAS-XC, provide very similar esimaions of he daily correlaion. So, in his regard he specific model choice is of lile relevance. Inser Figure 7: Long-Run Correlaion DCC-MIDAS-XC Figure 7 shows he long-run correlaions for he various specificaions wih only lagged realized correlaion, only a macro-finance variable, and he combinaion. Similar o he long-run sock and bond volailiy, he long-run sock-bond correlaion is smoohes when only using macro-finance variables and he leas smooh when using only lagged realized correlaion. The smoohness falls inbeween for he combinaion of macro-finance variables and lagged realized correlaion. The graphical presenaion of he esimaed long-run correlaions underscores ha we ge a lo of innovaive and useful informaion by he new model specificaion ha is no oherwise available. Inser Figure 8: Mean Absolue Errors Figure 8 shows he mean absolue error (MAE) for predicing he correlaion up o four periods ahead using various models. The MAE is generally increasing wih he forecas horizon. A he onequarer horizon he MAE is lowes when only considering he effec from he realized correlaion on he long-run correlaion, bu for longer horizons he MAE is improved by considering boh he realized correlaion and he macro-finance variables. Thus, he MAE resuls emphasize he usefulness of he new DCC-MIDAS-XC model specificaion. Among he macro-finance variables, S&P volume performs bes in forecasing fuure volailiy, boh alone and in combinaion wih he realized correlaion. 5.3 Effec of Forecased Macro-Finance Variables Table 3 shows he resuls from esimaing he wo-sided models ha rely on boh hisorical observaions and forecass of four macro-finance variables, he DCC-MIDAS-XCF model. Inser Table 3: DCC-MIDAS-XCF Adding he forecased macro-finance variables improves model performance (lower AIC) compared o ha of he models based only on observed macro-finance variables. No surprisingly, he specificaion including all hree ypes of informaion (he realized correlaion, he observed macrofinance variable, and he forecased macro-finance variable) provides he bes fi of all. The forecass of he inflaion rae are no significan in predicing he long-run correlaion wih he mos general model, while all hree ypes of informaion have explanaory power for he long-run correlaion when we use oher macroeconomic variables (unemploymen, shor rae, and erm spread). The effec from he forecased variable is posiive in all cases. Ye, he effec from he hisorical observed unemploymen rae urns negaive when used in combinaion wih he unemploymen forecass. Thus, in oal, he effec from he unemploymen rae observaions and forecass work owards cancelling each oher ou. The shor rae and erm spread have posiive effecs from boh hisorical observaions and forecass. Thus, for hese wo variables he effecs upon he long-run correlaion are made sronger by adding he forecass daa. Inser Figure 9: Long-Run Correlaion DCC-MIDAS-XCF 13
Figure 9 shows he long-run correlaion for he specificaions based only on lagged realized correlaion, only macro-finance variables (hisorical and forecass), and he combinaion. There are large differences in he esimaed long-run correlaions depending on he model specificaion. Thus, he new model specificaion provides addiional informaion ha could oherwise no have been obained. So, his once again sresses ha he new model specificaion is highly relevan. 6. Conclusion In his paper we scruinize he long-run sock bond correlaion. We make use of he dynamic condiional correlaion model (DCC) combined wih he mixed-daa sampling (MIDAS) mehodology. We provide an exension of he exising DCC-MIDAS models by which we allow he long-run correlaion o depend upon he lagged realized correlaion iself (C) as well as a macrofinance variable (X). In addiion, exend he DCC-MIDAS-XC model o allow he corresponding forecased macro-finance variable o influence he long-run sock-bond correlaion. The empirical findings in his paper convincingly documen he usefulness of he new DCC-MIDAS-XC models. The esimaed long-run sock-bond correlaion is very differen depending on which variables ha eners ino is esimaion. When only a macro-finance variable is used, he long-run sock bond correlaion is very smooh, while i is fairly volaile when only he lagged realized correlaion is used. When boh he lagged realized correlaion and a macro-finance variable is used, he esimaed long-run sock-bond correlaion falls in-beween he smooh and variable exremes. This underscores ha i is imporan o ake boh he lagged realized correlaion as well as he macrofinance variable ino accoun when forecasing long-run sock-bond correlaion. The inflaion rae, he indusrial producion, he shor rae, he defaul spread, he S&P volume, he producer confidence, and he consumer confidence are all significan in forecasing he long-run sock-bond correlaion. Moreover, forecass of some macro-finance variables are helpful in forecasing he long-run sock-bond correlaion. The effecs from he macro-finance variables upon he long-run sock-bond correlaion are such ha he long-run sock-bond correlaion ends o be large when he economy is srong. This effec suppors he conjecure of he fligh-o-qualiy effec on he long-run correlaion componen. 14
References Agnew, J. and P. Balduzzi (2006). Rebalancing Aciviy in 401 (k) Plans. Unpublished manuscrip. Andreou, E. and E. Ghysels (2004). The Impac of Sampling Frequency and Volailiy Esimaors on Change-Poin Tess. Journal of Financial Economerics, 2, 290-318. Amado, C. and T. Teräsvira (2013). Modelling Volailiy by Variance Decomposiion. Journal of Economerics, 175, 142-153. Asgharian, H., A.J. Hou, and F. Javed (2013). Imporance of he macroeconomic variables for variance predicion A GARCH-MIDAS approach. Journal of Forecasing, Forhcoming. Aslanidis, N. and C. Chrisiansen (2012). Smooh Transiion Paerns in he Realized Sock-Bond Correlaion. Journal Empirical Finance 19, 454-464. Aslanidis, N. and C. Chrisiansen (2013). Quaniles of he Realized Sock-Bond Correlaion. CREATES Working Paper. Baele, L., G. Bekaer, and K. Inghelbrech (2010). The Deerminans of Sock and Bond Reurn Comovemens. Review of Financial Sudies 23, 2374 2428. Bansal, N., R.A. Connolly, and C.T. Sivers (2010). Regime-Swiching in Sock and T-Bond Fuures Reurns and Measures of Sock Marke Sress. Journal of Fuures Markes 30, 753 779. Bauwens, L. and G. Sori (2009). A Componen GARCH Model wih Time Varying Weighs. Sudies in Nonlinear Dynamics & Economerics 13, 1-33. Campbell, J.Y. and J. Ammer (1993). Wha Moves he Sock and Bond Markes? A Variance Decomposiion for Long-Term Asse Reurns. Journal of Finance 48, 3-37. Chrisodoulakis, G.A. and S.E. Sachell (2002). Correlaed ARCH (CorrARCH): Modelling he Time-Varying Condiional Correlaion Beween Financial Asse Reurns. European Journal of Operaional Research 139, 351-370. Connolly, R., C. Sivers, and L. Sun (2005). Sock Marke Uncerainy and he Sock-Bond Reurn Relaion, Journal of Financial and Quaniaive Analysis, 40, 161-194. Connolly, R., C. Sivers, and L. Sun (2007). Commonaliy in he Time-Variaion of Sock-Sock and Sock-Bond Reurn Comovemens. Journal of Financial Markes 10, 192-218. Colacio, R., R.F. Engle, and E. Ghysels (2011). A Componen Model for Dynamic Correlaions. Journal of Economerics 164, 45-59. Conrad, C. and K. Loch (2012). Anicipaing Long-Term Sock Marke Volailiy. Working Papers 0535, Deparmen of Economics, Universiy of Heidelberg. Conrad, C., K. Loch, and D. Rier (2012). On he Macroeconomic Deerminans of he Long-Term Oil-Sock Correlaion. Working Papers 0525, Deparmen of Economics, Universiy of Heidelberg. Ding, Z. and C.W.J. Granger (1996). Varieies of Long Memory Models. Journal of Economerics 73, 61-77. Engle, R. (2002). Dynamic Condiional Correlaion - A Simple Class of Mulivariae GARCH Models. Journal of Business and Economic Saisics 20, 339-350. 15
Engle, R. and G. Lee (1999). A permanen and ransiory componen model of sock reurn volailiy. in ed. R.F. Engle and H. Whie, Coinegraion, Causaliy, and Forecasing: A Fesschrif in Honor of Clive W.J. Granger, (Oxford Universiy Press), 475 497. Engle R, E. Ghysels, and B. Sohn (2012). On he Economic Sources of Sock Marke Volailiy. Review of Economics and Saisics, Forhcoming. Engle, R. and J.G. Rangel (2008). The Spline GARCH Model for Low Frequency Volailiy and Is Global Macroeconomic Causes. Review of Financial Sudies 21, 1187-1222. Fleming, J., C. Kirby, and B. Osdiek (1998). Informaion and Volailiy Linkages in he Sock, Bond, and Money Markes. Journal of Financial Economics 49, 111-137. Fleming, J., C. Kirby, and B. Osdiek (2003). The economic value of volailiy iming using realized volailiies. Journal of Financial Economics 67, 473-509 Gencay, R., F. Selcuk, and B. Whicher (2001). An Inroducion o Waveles and Oher Filering Mehods in Finance and Economics. Academic Press, San Diego, CA. Ghysels, E., A. Sinko, and R. Valkanov (2006). MIDAS Regressions: Furher Resuls and New Direcions. Economeric Reviews 26, 53-90. Goffe W.L., G.D. Ferrier, and J. Rogers (1994). Global Opimizaion of Saisical Funcions wih Simulaed Annealing. Journal of Economerics 60, 65 99. Guidolin, M. and A. Timmermann (2006). An Economeric Model of Nonlinear Dynamics in he Join Disribuion of Sock and Bond Reurns. Journal of Applied Economerics 21, 1 22. Gulko, L. (2002). Decoupling. Journal of Porfolio Managemen 28, 59-66. Harmann, P., S. Sraemans, and C. Devries. (2001). Asse Marke Linkages in Crisis Periods. Working Paper 71, European Cenral Bank. Ilmanen A. (2003). Sock bond correlaions. Journal of Fixed Income 13, 55 66. Li, L. (2002a). Macroeconomic Facors and he Correlaion of Sock and Bond Reurns. Working paper. Yale Inernaional Cener for Finance. Li, L. (2002b). Correlaion of Sock and Bond Reurns. Working Paper, Yale Universiy. Schwer, G.W. (1989). Why Does Sock Marke Volailiy Change over Time?.Journal of Finance 44, 1115-1153. Viceira, L.M. (2012). Bond Risk, Bond Reurn Volailiy, and he Term Srucure of Ineres Raes. Inernaional Journal of Forecasing 28, 97 117. 16
Table 1. Esimaion of he ime varying variances by using univariae GARCH-MIDAS The able repors he resuls of he univariae GARCH-MIDAS model for esimaing he imevarying socks and bonds. Panel A shows he resuls for he reurn variance for he socks and Panel B gives he esimaion resuls of he bond reurns. The firs row of each panel gives he resul of he model ha only includes he realized volailiy (RV) in he MIDAS equaion, he second par of he panel repors he resuls of he model which only includes differen macro-finance variables in he MIDAS equaion, and he resuls of he model wih boh RV and he macro-finance variables are repored in he las par of each panel. µ is he inercep erm in he mean equaion for reurns, α and β are he parameers of he shor erm variance (equaion 3), W RV and W X are he esimaed weigh parameers of he realized volailiy and he macro-finance variables respecively, m is he inercep erm in he long-run variance equaion, and θ RV and θ X are he esimaed parameers of he realized volailiy and he macro-finance variables in he long-run variance (equaions 4 and 6), respecively. The esimaions are based on daily daa for reurns over he period from 1989 unil 2013, and quarerly daa for RV and he macro-finance variables from 1986 unil 2013 (we use 12 lags in he equaion for MIDAS). ***, ** and * indicae significance a he 1%, 5% and 10% levels, respecively. Panel A. socks reurns µ α β W RV W X m θ RV θ X AIC RV 8 *** 0.069 *** 0.918 *** 1.037 *** -3.931 *** 37.116 *** -17982 Inflaion 8 *** 0.066 *** 0.926 *** 2.069 * -3.499 *** 0.398 ** -17973 Indusrial Prod. 8 *** 0.068 *** 0.921 *** 3.203 ** -3.565 *** -0.293 *** -17974 Unemploymen 8 *** 0.068 *** 0.921 *** 4.436 * -3.575 *** 0.219 *** -17974 Term spread 8 *** 0.092 *** 0.898 *** 0 *** -3.317 *** -0.999 *** -17945 Shor rae 8 *** 0.067 *** 0.924 *** 1.426 *** -3.519 *** 0.612 *** -17976 Defaul rae 8 *** 0.066 *** 0.925 *** 3.915-3.506 *** -0.046-17969 Volume S&P 8 *** 0.068 *** 0.922 *** 1.625 *** -3.554 *** -0.632 *** -17978 VXO 8 *** 0.070 *** 0.917 *** 1.358 *** -3.630 *** 0.996 *** -17975 PMI 8 *** 0.069 *** 0.919 *** 0 *** -3.577 *** -0.852 *** -17978 CC 8 *** 0.068 *** 0.920 *** 0 *** -3.604 *** -0.941 *** -17977 NAI 8 *** 0.069 *** 0.919 *** 5.071 ** -3.604 *** -0.272 *** -17976 Inflaion 8 *** 0.069 *** 0.919 *** 1.146 *** 1.931 ** -4.072 *** 51.844 *** 0.483 *** -17994 Indusrial Prod. 8 *** 0.070 *** 0.918 *** 86.407 3.362 ** -3.645 *** 6.560-0.287 *** -17976 Unemploymen 8 *** 0.069 *** 0.919 *** 1 *** 6.413-3.613 *** 3.028 0.198 * -17975 Term spread 8 *** 0.063 *** 0.925 *** 1 *** 8.691 ** -3.664 *** 3.986 0.218 *** -17979 Shor rae 8 *** 0.068 *** 0.922 *** 100.871 1.478 *** -3.593 *** 5.675 0.590 *** -17978 Defaul rae 8 *** 0.069 *** 0.920 *** 0 *** 1.021-3.677 *** 10.908 0.327-17976 Volume S&P 8 *** 0.070 *** 0.917 *** 6.668 1.679 *** -3.735 *** 16.479-0.569 *** -17981 VXO 8 *** 0.069 *** 0.920 *** 0 *** 1.808-3.621 *** 3.863 0.591-17976 PMI 8 *** 0.072 *** 0.910 *** 1 *** 1.130 *** -4.032 *** 40.544 *** -1.027 *** -17999 CC 8 *** 0.069 *** 0.917 *** 0 *** 5 *** -3.785 *** 16.176-0.796 *** -17984 NAI 8 *** 0.073 *** 0.912 *** 0 *** 8.645 * -3.855 *** 26.860 *** -0.182 ** -17985 Table 1. Esimaion of he ime varying variances by using univariae GARCH-MIDAS (coninued) Panel B. Bond reurns 17
µ α β W RV W X m θ RV θ X AIC RV 1 * 0.042 *** 0.936 *** 7.063 ** -5.787 *** 349.905 *** -27684 Inflaion 1 8 *** 0.952 *** 2.089 * -5.260 *** -0.234 * -27678 Indusrial Prod. 1 8 *** 0.953 *** 1.189-5.256 *** -0.101-27675 Unemploymen 1 8 *** 0.953 *** 2 *** -5.263 *** 0.232 ** -27678 Term spread 1 7 *** 0.951 *** 1.113 *** -5.296 *** 0.633 *** -27687 Shor rae 1 8 *** 0.951 *** 1.187 ** -5.270 *** 0.274-27677 Defaul rae 1 * 0.047 *** 0.946 *** 2.563-5.048 *** -9-27677 Volume S&P 1 8 *** 0.953 *** 0 *** -5.244 *** 8-27676 VXO 1 * 9 *** 0.950 *** 1.171 *** -5.284 *** 0.581 ** -27678 PMI 1 8 *** 0.952 *** 1.172-5.252 *** 0.274-27676 CC 1 8 *** 0.953 *** 2.294-5.245 *** 8-27675 NAI 1 8 *** 0.952 *** 1.060-5.272 *** -0.183 * -27677 Inflaion 1 * 0.042 *** 0.936 *** 7.778 ** 1.724-5.740 *** 316.993 *** -0.093-27685 Indusrial Prod. 1 * 0.042 *** 0.936 *** 7.051 *** 6.598-5.816 *** 371.899 *** 0.040-27684 Unemploymen 1 * 0.040 *** 0.941 *** 5.485 ** 0 *** -5.712 *** 295.283 *** 0.070-27684 Term spread 1 * 0.040 *** 0.937 *** 11.427 * 1.291 *** -5.664 *** 246.991 *** 0.454 *** -27694 Shor rae 1 * 0.042 *** 0.935 *** 8.009 ** 1.256 * -5.740 *** 309.064 *** 0.191-27686 Defaul rae 1 * 0.042 *** 0.933 *** 5.183 *** 75.366-5.870 *** 403.037 *** 0.085 *** -27691 Volume S&P 1 * 0.042 *** 0.937 *** 5.791 141.405-5.814 *** 368.820 *** 0.043-27675 VXO 1 * 0.042 *** 0.936 *** 6.713 ** 1.126 ** -5.743 *** 304.206 *** 0.460 * -27687 PMI 1 * 0.042 *** 0.936 *** 7.751 ** 1.501-5.778 *** 343.434 *** 0.129-27685 CC 2 * 0.042 *** 0.936 *** 6.441 *** 6.802-5.808 *** 362.093 *** -0.062-27685 NAI 1 * 0.042 *** 0.936 *** 7.960 ** 1.027-5.742 *** 314.439 *** -0.057-27684 18
Table 2. Esimaion of he ime varying sock-bond correlaions by using DCC-MIDAS The able repors he resuls of he bivariae DCC-MIDAS model for esimaing he ime-varying correlaion beween sock and bond reurns. The firs row of he able gives he resul of he DCC- MIDAS-C model ha only includes he realized correlaion (RC) in he MIDAS equaion, he second par of he able repors he resuls of he DCC-MIDAS-X model which only includes differen macro-finance variables in he MIDAS equaion, and he las par of he able gives he resuls of he model wih boh RC and he macro-finance variables, i.e. DCC-MIDAS-XC model. a and b are he parameers of he shor erm correlaion (equaion 13), W RC and W X are he esimaed weigh parameers of he realized correlaion and he macro-finance variables respecively, m is he inercep erm in he long-run correlaion equaion, and θ RC and θ X are he esimaed parameers of he realized correlaion and he macro-finance variables in he long-run correlaion (equaion 15), respecively. The esimaions are based on daily sandardized residuals from 1993 unil 2013, and quarerly daa for RC and he macro-finance variables from 1989 unil 2013 (we use 16 lags in he equaion for MIDAS). ***, ** and * indicae significance a he 1%, 5% and 10% levels, respecively. a b W RC W X m θ RC θ X AIC RC 0.049 *** 0.929 *** 3.233 ** -3 1.071 *** 40632 Inflaion 7 *** 0.956 *** 1.057 *** 0.068 1.617 *** 40636 Indusrial Prod. 9 *** 0.956 *** 0 * -8 0.454 *** 40654 Unemploymen 9 *** 0.956 *** 0 ** -3-0.441 *** 40653 Term spread 8 *** 0.959 *** 160.463-3 0.184 40656 Shor rae 6 *** 0.960 *** 211.566 0.291 1.198 * 40648 Defaul rae 5 *** 0.962 *** 73.144-0 0.423 40653 Volume S&P 0.042 *** 0.947 *** 1.156 *** -0.068 1.501 *** 40634 VXO 6 *** 0.961 *** 21.170-2 0.428 40654 PMI 6 *** 0.961 *** 6.547 2-0.480 40655 CC 5 *** 0.963 *** 11.961 0.052-0.839 40652 NAI 9 *** 0.955 *** 1.216 ** -5 0.396 *** 40653 Inflaion 0.056 *** 0.917 *** 4.480 * 0 *** 3 0.855 *** 0.504 *** 40620 Indusrial Prod. 0.052 *** 0.920 *** 4.916 ** 32.512-5 1.116 *** -0.079 ** 40628 Unemploymen 0.049 *** 0.929 *** 3.124 ** 3.295-5 1.050 *** -7 40632 Term spread 0.049 *** 0.929 *** 3.607 ** 105.302-1 1.070 *** 0.058 40630 Shor rae 0.051 *** 0.922 *** 7.005 ** 14.947-2 1.047 *** 0.132 *** 40624 Defaul rae 0.052 *** 0.919 *** 7.368 ** 14.975-2 1.030 *** 0.111 ** 40626 Volume S&P 0.053 *** 0.914 *** 12.921 ** 1.317 *** -8 0.690 *** 0.638 *** 40620 VXO 0.053 *** 0.915 *** 10.380 ** 11.339-8 0.981 *** 0.152 40628 PMI 0.053 *** 0.917 *** 8.599 ** 5.359 ** -2 1 *** -0.173 * 40628 CC 0.051 *** 0.921 *** 8.717 ** 5.511 ** 3 1.066 *** -0.238 ** 40627 NAI 0.052 *** 0.922 *** 4.980 * 103.644-7 1.107 *** -0.061 40630 19