Importance of the macroeconomic variables for variance. prediction: A GARCH-MIDAS approach

Transcription

1 Imporance of he macroeconomic variables for variance predicion: A GARCH-MIDAS approach Hossein Asgharian * : Deparmen of Economics, Lund Universiy Ai Jun Hou: Deparmen of Business and Economics, Souhern Denmark Universiy Farrukh Javed: Deparmen of Saisics, Lund Universiy Work in progress Absrac This paper aims o examine he role of macroeconomic variables in forecasing he reurn volailiy of he US sock marke. We apply he GARCH-MIDAS (Mixed Daa Sampling) model o examine wheher informaion conained in macroeconomic variables can help o predic shorerm and long-erm componens of he reurn variance. We invesigae several alernaive models and use a large group of economic variables. A principal componen analysis is used o incorporae he informaion conained in differen variables. Our resuls show ha including lowfrequency macroeconomic informaion ino he GARCH-MIDAS model improves he predicion abiliy of he model, paricularly for he long-erm variance componen. Moreover, he GARCH- MIDAS model augmened wih he firs principal componen ouperforms all oher specificaions, indicaing ha he consruced principal componen can be considered as a good proxy of he business cycle. Keywords: GARCH-MIDAS, long-erm variance componen, macroeconomic variables, principal componen, variance predicion * Tel.: ; fax: ; address: Hossein.Asgharian@nek.lu.se; Deparmen of Economics, Lund Universiy, Box 708, S-007 Lund, Sweden. We are very graeful o Jan Wallanders och Tom Hedelius sifelse and Bankforskningsinsiu for funding his research. 1

2 1. Inroducion A correc assessmen of fuure volailiy is crucial for asse allocaion and risk managemen. Counless sudies have examined he ime-variaion in volailiy and he facors behind his ime variaion, and documened a clusering paern. Differen varians of he GARCH model have been pursued in differen direcions o deal wih hese phenomena. Simulaneously, a vas lieraure has invesigaed he linkages beween volailiy and macroeconomic and financial variables. Schwer (1989) relaes he changes of he reurns volailiy o he macroeconomic variables and addresses ha bond reurns, shor erm ineres rae, producer prices or indusrial producion growh rae have incremenal informaion for monhly marke volailiy. Glosen e al. (1993) find evidence ha shor erm ineres raes play an imporan role for he fuure marke variance. Whielaw (1994) finds saisical significance for a commercial paper spread and he one year reasury rae, while Brand & Kang (00) use he shor erm ineres rae, erm premium, and defaul premium and find a significan effec. Oher research including Hamilon & Lin (1996) and Perez & Timmermann (000) have found evidence ha he sae of he economy is an imporan deerminan in he volailiy of he reurns. Since he analyses of he ime-varying volailiy are mosly based on high frequency daa, he previous sudies are mosly limied o variables such as shor erm ineres raes, erm premiums, and defaul premiums, for which daily daa are available. Therefore, he impacs of variables such as unemploymen rae and inflaion on volailiy have no been sufficienly examined. Ghysels e al. (006) inroduce a regression scheme, namely MIDAS (Mixed Daa Sampling) which allows inclusion of daa from differen frequencies ino he same model. This makes i possible o combine he high-frequency reurn daa wih macroeconomic daa ha are only observed in lower frequencies such as monhly or quarerly. Engle e al. (009) propose he GARCH-MIDAS model wihin he MIDAS framework o analyze he ime-varying marke volailiy. Wihin his framework, he condiional variance is divided ino he long-erm and shor-erm componens. The low frequency variables affec he condiional variance via he longerm componen. This approach combines he componen model suggesed by Engle and Lee (1999) 1 wih he MIDAS framework of Ghysels e al. (006). The main advanage of he 1 For he componen model see also Ding and Granger, 1996; Chernov, e al. 003.

3 GARCH-MIDAS model is ha i allows us o link he daily observaions on sock reurns wih macroeconomic variables, sampled a lower frequencies, in order o examine direcly he macroeconomic variables impac on he sock volailiy. In his paper, we apply he recenly proposed mehodology, GARCH-MIDAS, o examine he effec of he macroeconomic variables on he sock marke volailiy. Deparing from Engle e al. (009), our invesigaion mainly focuses on variance predicabiliy and aims o analyze if adding economic variables can improve he forecasing abiliies of he radiional volailiy models. Using GARCH-MIDAS we decompose he reurn volailiy o is shor-erm and long-erm componens, where he laer is affeced by he smoohed realized volailiy and/or by macroeconomic variables. We examine a large group of macroeconomic variables which include unexpeced inflaion, erm premium, per capial labor income growh, defaul premium, unemploymen rae, shor erm ineres rae, per capial consumpion. We invesigae he abiliy of he GARCH-MIDAS models wih economic variables in predicing boh shor erm and long erm volailiies. The performances of hese models are hen compared wih he GARCH (1, 1) model as a benchmark. In order o capure he informaion conained in differen economic variables and invesigae heir combined effec, we perform a principal componen analysis. The advanage of his approach is o reduce he number of parameers and increase he compuaional efficiency. Our resuls show ha including low-frequency macroeconomic informaion ino he GARCH- MIDAS model improves he predicion abiliy of he model, paricularly for he long-erm variance componen. Moreover, he GARCH-MIDAS model augmened wih he firs principal componen ouperforms all oher specificaions. Among he individual macroeconomic variables, he shor erm ineres rae and he defaul rae perform beer han he oher variables, when included in he MIDAS equaion. To our knowledge his is he firs sudy ha invesigaes he ou-of-sample forecas performance of he GARCH-MIDAS model. The paper also conribues o exising lieraure by augmening he MIDAS equaion wih a number of he macroeconomic variables. 3

4 The res of he paper is organized as follows: Secion presens he empirical models, and he daa and he economeric mehods are described in Secion 3, while secion 4 conains he empirical resuls, and Secion 5 concludes.. GARCH-MIDAS In his paper, we use a new class of componen GARCH model based on he MIDAS (Mixed Daa Sampling) regression. MIDAS regression models are inroduced by Ghysels e al. (006). MIDAS offers a framework o incorporae macroeconomic variables sampled a differen frequency along wih he financial series. This new componen GARCH model is referred as MIDAS-GARCH, where macroeconomic variables ener direcly ino he specificaion of he long erm componen. This new class of GARCH model has gained much aenion in he recen years by Ghysles e al. (004), Ghysels e al. (006) and Andreaou e al. (010a). Chen and Ghysels (007) exend he MIDAS seing o a muli-horizon semi-parameric framework. Chen and Ghysels (009) provide a comprehensive sudy and a novel mehod o analyze he impac of news on forecasing volailiy. Ghysels e al. (009) discuss he Granger causaliy wih mixed frequency daa. Koze (007) uses he MIDAS regression wih high frequency daa on asse prices and low frequency inflaion forecass. In addiion, a number of papers use MIDAS regression for obaining quarerly forecass wih monhly and daily daa. For insance, Bai e al. (009) and Tay (007) use monhly daa o improve quarerly forecas. Alper e al. (008) compare he sock marke volailiy forecass across emerging markes using MIDAS regression. Clemens and Galavao (006) sudy he forecass of he U.S. oupu growh and inflaion in his conex. Forsberg and Ghysels (006) show, hrough simulaion, he relaive advanage of MIDAS over HAR-RV (Heerogeneous Auoregressive Realized Volailiy) model, proposed in Anderson e al. (007). The GARCH-MIDAS model can formally be described as below. Assume he reurn on day i in monh follows he following process: r = µ + τ g ε, i = 1,..., N. (1) i, i, i, ε i, Φi 1, ~ N(0,1) 4

5 where N is he number of rading days in monh and Φ is he informaion se up o ( i 1) i 1, h day of period. Equaion (1) expresses he variance ino a shor erm componen defined by g i, and a long erm componen defined by τ. The condiional variance dynamics of he componen g, is a (daily) GARCH(1,1) process, as: i ( r µ ) i 1, g i, = ( 1 β ) + α + βg i 1, τ α () and τ is defined as smoohed realized volailiy in he spiri of MIDAS regression: K k = 1 ( w w ) τ = m + θ ϕ k 1, RV k (3) N = RV r i i= 1, j where K is he number of periods over which we smooh he volailiy. We furher modify his equaion by involving he economic variables along wih he RV in order o sudy he impac of hese variables on he long-run reurn variance: K K K = m + θ1 ϕk 1 k k 1 + k 3 k 1, k= 1 k= 1 k= 1 l v ( w, w ) RV + θ ϕ ( w, w ) X θ ϕ ( w w ) X τ (4) k where l X k represens he level of a macroeconomic variable and v X k represens he variance of ha macroeconomic variable. The componen τ used in our analysis, does no change wihin a fixed ime span (e.g. a monh). Finally, he oal condiional variance can be defined as: σ = τ. (5) i g i, The weighing scheme used in equaion (3) and equaion (4) is described by bea lag polynomial, as: 5

6 ( w) w1 1 ( k ) ( 1 k ) w 1 ϕ = K K k (6) K w w j j 1 j= K 1 K 3. Daa and Esimaion Mehod 3.1. Daa We use he US daily price index o calculae sock reurns. In our condiional variance model we use a number of financial and macroeconomic facors which have been found by previous sudies o be imporan for reurn variance. The following variables are used: Shor-erm ineres rae is a yield on he hree monhs US Treasury bill. Slope of he yield curve is measured as he yield spread beween a en-year bond and a hree-monh Treasury bill. Defaul rae is measured as he spread beween Moody s Baa and Aaa corporae bond yields of he same mauriy. Exchange rae is he nominal major currencies dollar index from he Federal Reserves. Inflaion is measured as he monhly changes in he seasonally adjused consumer price index (CPI). Growh rae in he Indusrial Producion index. Unemploymen rae. Daa cover he period from January 1991 o June 008. All he iems excep he exchange rae are colleced from DaaSream. 3.. Esimaion Mehod 3..1 Various model specificaions 6

7 We use hree differen model specificaions. The models differ wih respec o he definiion of he long-erm variance componen, τ, while he equaion for he shor-erm variance, g i, remains he same in all he hree cases. The hree specificaions are: The RV model: In his specificaion, we solely use he monhly realized volailiy (RV) in he long-erm componen of he variance, defined by he MIDAS equaion, τ, in equaion (3). We have no economic variables in his model. The RV + X l + X v model: Here, we augmen he model by adding boh he level and he variance of an economic variable o he MIDAS equaion, τ. This modificaion is supposed o capure he informaion explained by boh he macroeconomic facor and he monhly RV. The X l + X v model: In his specificaion, we only sudy he effec of macroeconomic variables, boh level and variance, on he long-erm variance componen, i.e. equaion for τ. By analyzing hese hree alernaives, we can invesigae o wha exen he long-erm variance can be explained by he pas realized reurn volailiy and he macroeconomic variables. 3.. Esimaion sraegy Our esimaions are based on he daily observaions on reurns, while we use monhly frequency in he MIDAS equaion o capure he long-erm componen. The realized volailiy is our preferred measure of he monhly variance, bu since daily daa are no available for mos macroeconomic variables, i is no possible o use his measure. We selec he squared firs differences as he measure of he variance of he economic variables. We esimae he models described above using an esimaion window and hen use he esimaed parameers o make ou-of-sample variance predicion. 3 We use a en-year esimaion window and keep he parameers over he subsequen year. The firs esimaion window sars in January We have also esimaed he model wih only he level or he variance of he economic variables in he MIDAS equaion. In order o save space, hese resuls are no repored bu are available upon reques. 3 We use several alernaive ime spans for he esimaion window, i.e. five, eigh and hen years. Our resuls show ha he esimaion accuracy reduces as we decrease he lengh of he esimaion window. We herefore selec o only presen he resuls wih a 10-year esimaion window. The resuls for oher esimaion windows are available upon reques. 7

8 1994 and ends in December 003. However, we also need hree years lagged daa before each ime period o compue he hisorical realized volailiy, which means ha he realized volailiy for January 1994 is esimaed wih daa from January 1991 o December The esimaion window is hen moved forward by one year unil December 007. Our ou-of-sample forecas covers he period January 004 unil June 008. We chose no o use daa afer he sar of he financial crisis 008, since he exreme ouliers of he period of he financial crisis make i impossible o make any reliable and accurae ou-of-sample comparisons of he models. One may address his issue by including jumps in he shor-erm componen of he GARCH-MIDAS srucure. However, i will significanly complicae he esimaion procedure. Furher, since we could only be able o analyze he jump effecs in he shor-erm movemens, i does no improve he predicion of he long-erm movemens, which is one of he essences of he GARCH-MIDAS srucure. We use he esimaed τ from he MIDAS equaion as he predicion of he long-erm variance (see equaions (3) and (4)). Since he values of τ are on a daily basis, we muliply his value wih he number of rading days wihin each monh. The esimaed daily oal variance ( σ ) is used as he predicion of shor-erm variance. The forecasing abiliy of he GARCH-MIDAS model is compared wih a simple GARCH (1.1) model, r η = σ z, z ~ N(0,1), (7) = µ + η, σ = ω + αη + 1 βσ 1 We predic he long erm volailiy wih he monhly observaions and for he shor-erm forecas we use he daily observaions. We compare he ou-of-sample predicions of he monhly variances from he GARCH MIDAS and he GARCH models wih he monhly realized volailiy measured as he sum of daily squared reurns in monh. To assess he shor-erm predicion abiliy of he models we compare he esimaed daily oal variance of he GARCH-MIDAS and he GARCH model wih he realized daily volailiy, measured as he squared reurns. 8

9 We employ a number of measures o evaluae he variance predicion of a specific model by comparing he model prediced variance wih he realized monhly volailiy, esimaed as he sum of he squared daily log reurns wihin each monh. We use wo loss funcions, he Mean Square Error (MSE) and he Mean Absolue Error (MAE), defined as 1 MSE = T 1 MAE = T T ( + 1 E ( σ + 1 ) = 1 T = 1 σ (8) + 1 E ( σ ) σ (9) MSE is a quadraic loss funcion and gives a larger weigh o large predicion errors comparing o he MAE measure, and is herefore proper when large errors are more serious han small errors (see Brooks and Persand (003)). We use he es suggesed by Diebold and Mariano (1995), DM-es, o compare he predicion accuracy of wo compeing models, ( d ) ( d ) + 1 E DM = ~ N( 0,1) (10) var d = e e A, B, where e A, and e B, are predicion error of wo rival models A and B, respecively, and E(d ) and var(d ) are mean and he variance of he ime-series of d, respecively. In addiion o hese measures we run he following regression of he realized variance on he prediced variance (see e.g., Andersen and Bollerslev (1998) and Hansen (005)). + 1 = + be σ + ( 1) u σ a + (11) If he prediced variance has some informaion abou he fuure realized volailiy, hen he parameer b should be significanly differen from zero. Furhermore, for an unbiased predicion we expec he parameer a o be zero and he parameer b o be equal o one. We also look a he R-square of his regression. The maximum likelihood mehod is used o esimae he model parameers. The likelihood funcion of he GARCH-MIDAS model involves a large number of parameers, which does no 9

10 always converge o a global opimum by he convenional opimizaion algorihms. We, herefore, use he simulaed annealing approach (see Goffe e al. (1994)) for esimaion. This mehod is very robus and seldom fails, even for very complicaed problems Weighs and number of lags in he MIDAS equaion During he esimaion, we have chosen several sraegies o simplify he esimaion and o make he model work more efficienly. Firs, we have o choose he weighs (w 1 and w ) in he bea funcions specified in equaion (6). We have hree alernaives: i) Taking boh w 1 and w as free parameers and esimaing hem wihin he model. ii) Fixing w 1 a priori and leing w be esimaed wihin he model. iii) Fixing a priori boh w 1 and w. Figure 1 illusraes he plo of he weighing funcion for wo choices of w 1 (1 and ) and wo choices of w (4 and 8). I shows ha he weigh funcion is monoonically decreasing as long as w 1 is equal o one. Given w 1 equal o one, increasing w will give a larger weigh o he mos recen observaions. A w 1 larger han one gives a lower weigh o he mos recen observaions. Alernaive (i) someimes resuls in very counerinuiive weighing paerns, e.g. a lower weigh for more recen observaions (w 1 larger han one). We, herefore, follow Engel e al. (009) and fix he weigh w 1 o one, which makes he weighs monoonically decreasing over he lags. Since here are no a priori preferences for he choice of w, we le he model defines w (alernaive (ii)) when esimaing he RV model. However, we keep he esimaed weigh from his model for he remainder of he specificaions. Second, we have o decide how many lags we should use in he MIDAS equaion (K in he equaions 3, 4 and 6). The oal lags are deermined by he number of years, or so called MIDAS years, and by he ime span ha will be used o calculae τ in equaions (3) and (4). This ime span can be a monh, a quarer, or a half year. Regarding he lengh of he ime-period used in our sudy and in order o have a sufficien number of ou-of-sample predicion, we decide o use a monhly ime span. In he lower graph of Figure 1, we plo he maximum values of he likelihood funcion using differen lags in he MIDAS equaion. I can be seen ha he opimum 10

11 value of he likelihood funcion increases wih he number of lags and i converges o is highes level a around 36 lag. We herefore limi he number of lags in he MIDAS equaion o 36 which resuls in hree MIDAS years Principal componens GARCH-MIDAS is compuaionally complex and he inclusion of several macroeconomic variables in one model will resul in idenificaion and/or convergence problems. Therefore we use one variable a a ime in he MIDAS equaion. In order o incorporae he informaion conained in differen variables in he same equaion, we also consruc principal componens based on hese variables. Since he macroeconomic variables have differen scales, we use he correlaion marix o consruc he principal componens. 4. Resuls and Analyses 4.1. Descripive analysis Table 1 shows he correlaion beween monhly observaions on he macroeconomic variables and he realized monhly volailiy of he US sock reurn (RV). Ineres rae, as expeced, has a high negaive correlaion wih slope (-0.70). Furher, he slope is higher when he unemploymen rae is high. Unemploymen and inflaion are also highly correlaed during he seleced ime span. Table shows he correlaions beween he principal componens and he macroeconomic variables. The firs principal componen (PC 1 ) has a high correlaion wih mos of he variables, paricularly wih ineres rae, slope, defaul rae, and unemploymen (average correlaion is 0.48). Since mos of hese variables are commonly used as a measure for business cycle we may consider he variable PC 1 as a proper proxy for he cycle. Similarly, we observe a relaively large correlaion beween some variables i.e., inflaion and ineres rae wih PC. Oher principal componens have eiher low correlaions wih he macroeconomic variables or only relaed o one specific variable (such as PC 3 and indusrial producion). We choose herefore only o include PC 1 and PC in he MIDAS equaion. Figure plos he monhly realized volailiy of he reurn, he macroeconomic variables, as well as he firs wo principal componens consruced based on he macroeconomic variables. A drasic flucuaion is observed in realized volailiy 11

12 beween he period 1997 ill mid of 00. This may indicae he effec of Asian crisis in 1998, he burs of he do-com bubble in 000 and he Sepember 11 incidence in 001. The las volaile period near indicaes he sar of he recen financial crisis. We can find a similar paern in he movemens of he PC 1 series. I shows a declining rend in he beginning, followed by a sharp increase in he values afer he financial urmoil in 001, which remains unil 003. An increasing rend around he period of signals he sar of he recen financial crisis. From he plo of PC, we can observe a coninuously increasing rend hroughou he sample period. The ineres rae paern is reversed of ha for PC 1 confirming he high negaive correlaion beween hem (-0.78). Similarly, he defaul rae is high during financial crisis of 1998, 001 and 007 compared o oher ime periods. The growh rae in indusrial producion is smooh besides some peak poins near The exchange rae changes slighly around 001, oherwise i seems sable hroughou he sample period. The inflaion has an opposie behavior o ha of PC, supporing heir highly negaive correlaion (-0.83). Similarly, he unemploymen rae increases afer he crisis of 001 and remains high for he nex couple of years. We can observe an increasing rend in he unemploymen rae afer he recen financial crisis of In-sample esimaions In Table 3, we presen he esimaed parameers of he in-sample fi for he firs esimaion period, saring on January 01, 1991 and ending on December 31, 003. The models are esimaed wih he firs wo principal componens and wih all he individual economic variables in he MIDAS equaion. In order o save space we only repor he resuls for PC 1 and PC. Mos of he parameers in he equaions for reurns and he shor-erm variance componen (g i ) are significan a he 5% level, indicaing a clusering paern in he shor-erm reurn variance. Turning o he long-erm componen, we can see he RV is significan a he 5% level in all he hree models, while he weigh w is only significan a he 10% level. In order o have he same degree of smoohness for all he variables we use w esimaed from he model wih only RV, when we augmen he model wih macroeconomic variables. The resuls show ha he level of PC 1 is significan along wih RV bu no is variance. However, if we exclude RV from he equaion of he long-erm componen, boh he levels and he variance of PC 1 are significan. I 1

13 shows ha RV capures he effec of he variance of PC 1. RV is sill significan a he 5% level when we use PC as a macroeconomic variable. The parameer for he variance of PC is also significan bu a he 10% level. However, only he level of PC is found significan if we exclude RV from he model. We may conclude ha he join effecs of he economic variables, capured by PC 1 and PC, conain some informaion abou he driving force of sock marke reurn variance. In Figure 3 we compare he esimaed shor-erm, long-erm and oal variance from he GARCH-MIDAS model where we only use he realized volailiy in he MIDAS equaion (RV model). In he firs par of he esimaion window, despie some large peaks in he shor-erm variance (possibly due o he Asian crises) he long-erm variance is quie low. Afer 000 we observe a subsanial increase in he long-erm variance componen, while he shor-erm componen is below he long-erm componen mos of he ime. Figure 4 illusraes he esimaed long-erm componen of he reurn variance given by he MIDAS equaion, for he firs in-sample period. We compare he resuls from he RV model wih wo alernaive specificaions, he RV model augmened wih a macroeconomic variable and a model which only includes he macroeconomic variable. In he firs graph he macroeconomic variables are represened by PC 1, while in he second graph we presen he esimaed variances wih PC. I shows ha he esimaed variance from he model RV+PC 1 follows mosly ha from he RV model, while he PC 1 model moves quie differenly. Comparing all he hree models, i seems ha he RV+PC 1 model combines he wo oher models, where RV deermines he variaions and PC 1 affecs mosly he level of he esimaed variance. All he hree models give a relaively similar paern, mos of he ime, when we use PC as he macroeconomic variable Ou-of-sample predicion In his secion, we analyze he abiliy of he GARCH-MIDAS model in forecasing he long-erm monhly variances, see equaions (3) and (4), and he oal daily variances, see equaion (5). The parameers are obained using a rolling 10-year esimaion window and are held consan during he subsequen year. Our ou-of-sample forecas covers he period from January 004 o June 008. We use hree alernaive MIDAS specificaions: he RV model ha only includes he 13

14 realized volailiy of sock reurns, he RV+X l +X v model ha includes he realized reurn volailiy as well as he level and he variance of he economic variables, and finally he X l +X v model wih only he level and he variance of he economic variables. As our primary choice of he macroeconomic variables in he GARCH-MIDAS model, we use he wo firs principal componens, PC 1 and PC. We use a en-year esimaion window and keep he parameers over he subsequen year. The firs esimaion window sars in January 1994 and ends in December 003. Table 4 repors he predicion performance of all he models using MSE and he DM es. As a benchmark we esimae he GARCH (1,1) model, where we use monhly observaions for comparison wih he GARCH-MIDAS long-erm variance componen and daily observaions when we compare i wih he GARCH-MIDAS oal variance. The esimaed MSE is based on he deviaion beween he variance forecased and he realized variance, where he realized monhly variances are esimaed as he sum of daily squared reurns in each monh, and he realized daily variances are he squared daily reurns. The lef panel of Table 4 shows he resuls for he long-erm variance componen. The GARCH- MIDAS model wih RV+PC1 has lowes MSE values for monhly predicions. This resul is confirmed by he DM-es (In order o save space, we only repor he DM-es when using he radiional GARCH and GARCH-MIDAS as he benchmark models). The model RV+ PC 1 significanly ouperforms boh he GARCH model and he RV model in he long-erm variance predicion. The GARCH-MIDAS model wihou any economic variable performs beer han GARCH bu he difference beween he models forecas is no saisically significan. The models wih PC 1 and PC alone, as a long-erm variance driving facor, perform very poorly and are significanly worse han boh GARCH and RV model. In he righ panel of he able, we display he findings from daily variance predicions. The RV+PC 1 model sill performs beer han he oher models, bu he differences are very small and saisically insignifican. In fac all he models perform beer han he GARCH model. In figure 5, we plo he resuls of he regression of he realized volailiy on he prediced variance. In general, if he prediced variance has some informaion abou he fuure realized volailiy, hen he slope parameer should be significanly differen from zero. Furhermore, for an unbiased predicion we expec he inercep parameer o be zero and he slope parameer o be 14

15 equal o one. The firs graph shows he -saisics for he inercep for boh daily and monhly variance predicions, and he slope parameers for daily and monhly variance predicions are presened in he second and hird diagrams, respecively. In accordance o he resuls above, he RV+PC 1 model shows a very srong abiliy in forecasing boh long-erm (monhly) and oal (daily) variances; i has a very close o zero inercep and a close o one slope esimaions in boh predicions. None of he oher models share hese properies for boh predicions, for example he RV model performs well a he daily predicion bu is slope is no significanly differen from zero in he monhly predicion. All in all, our ou-of-sample analysis shows ha adding proper macroeconomic informaion, measured by PC 1, o he long-erm variance componen of he GARCH-MIDAS model significanly enhances he predicion abiliy of he model. Now, i is ineresing o analyze he forecasing abiliy of he differen macroeconomic variables, separaely. Figure 6 plos he DMes resul of he RV+X l +X v model, using individual macroeconomic variables and he wo principal componens, and ha of he RV model. The GARCH (1, 1) model is used as he benchmark o compue he es saisics. According o he figure, all he saisics are negaive, which implies ha all he models give a lower forecas error han he GARCH model, in boh monhly and daily predicions. However, he es is only significan for monhly predicions and for hree cases, i.e. he specificaions wih PC 1, ineres rae, and defaul. Since he boh ineres rae and defaul are highly correlaed wih PC 1, he srong ou-of-sample performance of he model wih PC 1, can o a large exen be relaed o hese wo variables. 5. Conclusion In his paper, we have used he GARCH-MIDAS approach o forecas fuure variances. To esimae he long-erm componen of he variance, in addiion o he smoohed realized volailiy we use informaion from macroeconomic variables. A principal componen approach is employed o combine he informaion from a large number of variables, which include ineres rae, unemploymen rae, erm premium, inflaion rae, exchange rae, defaul rae, indusrial producion growh rae. We use a rolling window o esimae he parameers of he model and o make forecas for ou-of-sample variances. We compare he forecasing abiliy of GARCH- MIDAS models wih he radiional GARCH model. 15

16 Our findings show ha he GARCH-MIDAS model consiues a beer forecas han he radiional GARCH model. We show ha including he low-frequency (monhly) macroeconomic informaion no only significanly enhances he forecasing abiliy of he model for he long-erm (monhly) variance, i also improves he predicion abiliy of he model for high-frequency (daily) variances. However, he laer resul is no saisically significan based on he DM-es. The GARCH-MIDAS model ha includes he firs principal componen ouperforms all oher specificaions. The srong performance of he firs principal componen may be moivaed by is close connecion o he variables shor erm ineres rae and he defaul rae, which makes he firs principal componen a good proxy of he business cycle. The paper conribues o exising lieraure by (1) augmening he long-erm componen (MIDAS equaion) wih macroeconomic variables and () invesigaing he forecasing abiliy of he GARCH-MIDAS model. 16

17 References Alper, C. E., S. Fendoglu, and B. Saloglu (008). Forecasing Sock Marke Volailiies Using MIDAS Regression: An Applicaion o he Emerging Markes. MPRA Paper No Andersen, T. and Bollerslev, T. (1998). Answering he Skepics: Yes, Sandard Volailiy Models Do Provide Accurae Forecass, Inernaional Economic Review, 39, Anderson, T., T. Bollerslev, and F. Diebold (007). Roughing I Up: Including Jump Componen in he Measuremen, Modeling and Forecasing of Reurn Volailiy. The Review of Economics and Saisics, 89, Andreaou, E., E. Ghysels, and A. Kourellos (010a). Regression Models wih Mixed Daa Sampling Frequencies. Journal of Economerics, in-press. Bai, J., E. Ghysels, and J. Wrigh (009). Sae space models and MIDAS regression. Working Paper, NY Fed, UNC and John Hopkins. Brand, M. W. and Kang, Q. (00). On he relaionship beween he condiional mean and volailiy of sock reurns: A laen VAR approach. The Wharon School. Brooks, C. and G. Persand (003), The Effec of Asymmeries on Sock Index Reurn Value a Risk Esimaes, Journal of Risk Finance, 4, 9-4. Chen, X., and E. Ghysels (007). News Good or Bad- and Is Impac on Muliple Horizons. Working Paper, NC-Chapel Hill. Chen, X., and E. Ghysels (009). News good or bad and is impac on predicing fuure volailiy. Review of Financial Sudies (forhcoming). Chernov, M., Gallan, R., Ghysels, E. and Tauchen, G. (003), Alernaive models for sock price dynamics, Journal of Economerics, 116, Clemens, M. P., and Galavao, A. B. (006) Macroeconomic Forecasing wih mixed Frequency Daa: Forecasing US oupu growh and inflaion. Warwick Economic Research Paper No Universiy of Warwick. Diebold, F. and Mariano, S. (1995). Comparing Predicive Accuracy, Journal of Business & Economic Saisics, 13,

18 Ding, Z. and Granger, C. (1996), Modeling volailiy persisence of speculaive reurns: A new approach. Journal of Economerics 73, Engle, R., and Lee, G. (1999), A permanen and ransiory componen model of sock reurn volailiy. In ed. R.F. and H. Whie, Coinegraion, Causaliy, and Forecasing: A Fesschrif in Honor of Clive W.J. Granger, Oxford Universiy press, Engle, R., E. Ghysels, and B. Sohn. (009). Sock Marke Volailiy and Macroeconomic Fundamenals, Working Paper. Forsberg, L., and E. Ghysels (006). Why do absolue reurns predic volailiy so well? Journal of Financial Economerics, 6, Ghysels, E., P. Sana-Clara, and R. Valkanov (004). The MIDAS ouch: Mixed Daa Sampling Regression. Discussion Paper UNC and UCLA. Ghysels, E., A. Sinko, and R. Valkanov (006). MIDAS regression: Furher resuls and new direcions. Economeric Reviews, 6, Ghysels, E., A. Sinko, and R. Valkanov (009). Granger Causaliy Tess wih Mixed Daa Frequencies. Discussion Paper, UNC. Glosen, L. R., Jagannahan, R. and Runkle, D. E. (1993). On he relaionship beween he expeced value and he volailiy of he nominal excess reurn on socks, Journal of Finance 48, Goffe, W.L., Ferrier, G.D., Rogers, J. (1994). Global opimizaion of saisical funcions wih simulaed annealing, Journal of Economerics, 60, Hamilon, J. D., and G, Lin. (1996). Sock Marke Volailiy and he Business Cycle, Journal of Applied Economerics, 11, Hansen, P.R. (005). A es for superior predicive abiliy, Journal of Business and Economic Saisics, 3, Koze, G. L. (007). Forecasing Inflaion wih High Frequency Asse Price Daa. Working Paper. Universiy of Sellenbosch. 18

19 Perez-Quiros, G. and Timmermann, A. (000), Firm size and cyclical variaions in sock reurns, Journal of Finance, 55, Schwer, G. W., (1989). Why Does Sock Marke Volailiy Change over Time?, Journal of Finance, 44, Tay, A. S. (007). Mixed Frequencies: Sock Reurns as a Predicor of real Oupu Growh. Discussion Paper, SMU. Whielaw, R. (1994), Time variaions and covariaions in he expecaion and volailiy of sock reurns, Journal of Finance 49,

20 Table 1. Correlaion beween variables The able shows he correlaion beween monhly observaions on he macroeconomic variables and he realized monhly volailiy of he US sock reurn (RV). The macroeconomic variables are he yield on a hree monhs US Treasury bill (In. rae), he yield spread beween a en-year bond and a hree-monh Treasury bill (Slope), he unemploymen rae (Unemp), he growh rae in he indusrial producion (Ind. Prod), he monhly changes in he consumer price index (Inflaion), he monhly changes in he exchange rae (Exch) and he spread beween Moody s Baa and Aaa corporae bond yields (Defaul). Daa cover he period from January 1991 o June 008. RV In. rae Slope Unemp Ind. prod Inflaion Exch Defaul RV 1.00 In. rae Slope Unemp Ind. Prod Inflaion Exch Defaul

21 Table. The correlaion of principal componens wih he macroeconomic variables The able shows he correlaion beween he macroeconomic variables wih he principal componens (PC) consruced based on hese variables. The macro economic variables are he yield on a hree monhs US Treasury bill (In. rae), he yield spread beween a en-year bond and a hree-monh Treasury bill (Slope), he spread beween Moody s Baa and Aaa corporae bond yields (Defaul), he monhly changes in he exchange rae (Exch), he monhly changes in he consumer price index (Inflaion), he growh rae in he indusrial producion (Ind. Prod) and he unemploymen rae (Unemp). Daa cover he period from January 1991 o June 008. In. rae Slope Unemp Ind. prod Inflaion Exch Defaul Pc Pc Pc Pc Pc Pc Pc

22 Table 3. Esimaed parameers of he GARCH-MIDAS model The able shows he esimaed parameers of he GARCH-MIDAS model wih differen specificaions of he MIDAS equaion. The firs row of he able presens he resuls of he model wih only he realized volailiy (RV) of reurns in he MIDAS equaion, while he res rows of he able presen he esimaed parameers when we also include he level and he variance of he economic variables, X l and X v respecively, in he MIDAS equaion. We only presen he resuls obained for he firs and he second principal componens consruced based on seven macroeconomic variables. Daa cover he firs esimaion period saring in January 1991 and ending in December 003. mu alpha bea m RV level var w RV 0.07 ** ** ** ** **.677 * PC 1 RV+ X l +X v ** ** ** ** ** ** X l +X v 0.07 ** ** 0.94 ** ** ** PC RV+ X l +X v ** ** ** ** ** * X l +X v 0.07 ** 0.08 ** ** **.677

23 Table 4. Comparisons of he ou-of-sample predicion errors The able shows he resuls of he esimaed mean square error (MSE) and DM-es for he ouof-sample performance of he differen models in predicing daily and monhly variances. We use hree alernaive specificaions in he MIDAS equaion, a model ha includes only he realized volailiy of sock reurns (RV model), a model ha includes he realized reurn volailiy as well as he level and he variance of he economic variables (RV+X l +X v ), and finally a model wih only he level and he variance of he economic variables (X l +X v ). The lef panel shows he resuls for he long-erm variance componen, τ in equaions (3) and (4), while righ panel shows he resuls for he condiional daily oal variance (see equaion (5)). The resuls of he GARCH- MIDAS are compared wih corresponding GARCH esimaions. As he macro variables we use he wo firs principal componens, PC 1 and PC, in he MIDAS equaion. We use a en-year esimaion window and keep he parameers over he subsequen year. The firs esimaion window sars in January 1994 and ends in December 003. The realized monhly variances are esimaed as he sum of daily squared reurns in each monh, while for he realized daily variances we use he squared daily reurns. Ou-of-sample forecass cover he period from January 004 o June 008. The minus (plus) sign in each cell indicaes ha he model given in he row performs beer (worse) han he model given in he column. An aserisk implies a significan difference in he performance. Long erm variance Toal variance MSE GARCH RV model MSE GARCH RV model GARCH RV model RV+PC * -* RV+PC PC * +* PC * +*

24 Figure 1. The weighs and he number of lags in GARCH-MIDAS The upper graph shows he behavior of weighs as he funcion of he number of lags using differen values for w 1 and w. We selec wo alernaive values for w 1 (1 and ) and wo values for w (4 and 8). In he lower graph, we plo he maximized value of log likelihood funcion of he GARCH-MIDAS model wih differen lag values. The long erm componen (MIDAS equaion) includes only he realized reurn volailiy. Weighs and Lags Weighs ,4 1,8,4, Lags Lags and Likelihhod funcions Likelihood funcions Likelihhod Fs Lags 4

25 Figure. Plo of he realized volailiy and he economic variables The figure illusraes he monhly realized volailiy of he reurn and movemens of he seleced macroeconomic variables, as well as he firs principal componen consruced based on he macroeconomic variables. The daa ranges from January 1991 o June RV PC PC Ineres rae Defual rae Indusrial producion Exchange rae Inflaion Unemploymen rae Slope 5

26 Figure 3. Comparison of he long-erm, shor-erm and oal variance The figure illusraes he long-erm, shor-erm and oal variances esimaed by he GARCH- MIDAS model. The MIDAS equaion only includes he realized volailiy of sock reurns (RV model). The esimaion period covers he period from January 1991 o December 003, while a sample of 36 monhly observaions have been used o esimae he exponenially moving average of he realized volailiy in he MIDAS equaion RV Model Toal Shor Long 6

27 Figure 4. Esimaed long-erm variance The figure illusraes he esimaed long-erm variance, τ, based on hree alernaive specificaions of he MIDAS equaion, a model ha includes only he realized volailiy of sock reurns (RV), a model ha includes he realized reurn volailiy as well as he level and he variance of he economic variables (RV+X l +X v ), and finally a model wih only he level and he variance of he economic variables (X l +X v ). We illusrae he resuls for he firs wo principal componens consruced based on seven macroeconomic variables. The esimaion period covers he period from January 1991 o December 003, while a sample of 36 monhly observaions have been used o esimae he exponenially moving average of he included variables in he MIDAS equaion PC1 RV model RV+PC1 PC PC RV model RV+PC PC

28 Figure 5. Regression of he realized volailiies on he prediced variances The figure plos he resuls of he esimaed parameers from he regression of he realized volailiy on he prediced variance. The firs figure plos he -saisics for he inercep and he second and hird figures give he slope parameers for monhly and daily variance predicion, respecively, and he relaed 95% confidence inervals. We use hree alernaive MIDAS specificaions: RV includes only he realized volailiy of sock reurns, RV+X l +X v includes he realized reurn volailiy and he level and he variance of he economic variables, X l +X v conains only he level and he variance of he economic variables. As economic variables, we use wo firs principal componens, PC1 and PC, in he MIDAS equaion. The resuls of he GARCH- MIDAS are compared wih corresponding GARCH esimaions. The realized monhly volailiy is esimaed as he sum of daily squared reurns in each monh, while for he realized daily volailiy is compued as he squared daily reurn values of he esimaed inercep Monhly Daily Garch RV RV+PC1 RV+PC PC1 PC 95% confidence inerval for slope coefficien Monhly variance Garch RV RV+PC1 RV+PC PC1 PC 95% confidence inerval for slope coefficien Daily variance Garch RV RV+PC1 RV+PC PC1 PC 8

29 Figure 6. DM-es of he individual macrovariables The figure shows -values of he DM es for he ou-of-sample performance of he differen models in predicing daily and monhly variances. I indicaes he conribuion of each macroeconomic variable, PC 1 and PC in order o improve he predicion of long-erm variance. We use wo alernaive specificaions in MIDAS equaion, a model ha includes only he realized volailiy of sock reurns (RV model), a model ha includes he realized reurn volailiy as well as he level and he variance of he economic variables (RV+X l +X v ). -values of he DM-es Monhly Daily -3 9