Does he ime horizon of he reurn predicive effec of invesor senimen vary wih sock characerisics? A Granger causaliy analysis in he domain Yong Jiang, Zhongbao Zhou chool of Business Adminisraion, Hunan Universiy, Changsha 41008, China Absrac Behavioral heories posi ha invesor senimen exhibis predicive power for sock reurns, whereas here is lile sudy have invesigaed he relaionship beween he ime horizon of he predicive effec of invesor senimen and he firm characerisics. To his end, by using a Granger causaliy analysis in he domain proposed by Lemmens e al. (008), his paper examine wheher he ime horizon of he predicive effec of invesor senimen on he U.. reurns of socks vary wih differen firm characerisics (e.g., firm size (ize), book-o-marke equiy (B/M) rae, operaing profiabiliy (OP) and invesmen (Inv)). The empirical resuls indicae ha invesor senimen has a long-erm (more han 1 monhs) or shor-erm (less han 1 monhs) predicive effec on sock reurns wih differen firm characerisics. pecifically, he invesor senimen has srong predicabiliy in he sock reurns for smaller ize socks, lower B/M socks and lower OP socks, boh in he shor erm and long erm, bu only has a shor-erm predicabiliy for higher quanile ones. The invesor senimen merely has predicabiliy for he reurns of smaller Inv socks in he shor erm, bu has a srong shor-erm and long-erm predicabiliy for larger Inv socks. These resuls have imporan implicaions for he invesors for he planning of he shor and he long run sock invesmen sraegy. Keywords: Invesor senimen; ock reurns; Granger causaliy; Frequency domain approach; Time horizon Corresponding auhor, Email: jiangziya.ok@163.com.. 1
1. Inroducion Behavioral heories posi ha invesor senimen exhibis predicive power for sock reurns, which has been exensively invesigaed in recen years (Baker and Wurgler, 006, 007; ambaugh e al., 01; Chung e al., 01; Huang e al.,015; Aloui e al., 016; You e al., 017). For example, Baker and Wurgler (007) documen ha invesor senimen has a predicive power wih respec o equiy reurns. chmeling (009) shows ha he invesors senimen acs as a significan predicor for sock reurns for 18 indusrialized counries. ambaugh e al. (01) find ha invesor senimen is a significan negaive predicor for he shor legs of long-shor invesmen sraegies. Dergiades (01) argues ha he invesor senimen has a non-linear causaliy o sock reurns by employing a nonlinear Granger causaliy model. Li e al. (017) use a quanile Granger non-causaliy es model find a nonlinear causal relaionship beween invesor senimen and U.. sock reurns. The common deficiency of he above-menioned sudies is ha he causaliy assumpion is limied o one paricular daa, whereas hey canno analyze hese componens separaely, ha is, he long-erm componens and he shor-erm componens (see Breiung and Candelon, 006; Lemmens e al., 008). Consequenly, hey canno idenify wheher he ime horizon of predicive effec of invesor senimen on he reurns of sock is in he shor erm or in he long erm. Up o now, no lieraure examine wheher he ime horizon of he sock reurn predicive effec of invesor senimen varies for differen firms. In his paper, we address his issue by decomposing he Granger causaliy (GC) in he domain, using domain causaliy approach developed by Lemmens e al. (008) based on specral approach. The key idea of his approach is ha a saionary process can be described as a weighed sum of sinusoidal componens wih a cerain. Insead of compuing a single GC measure for he enire relaionship, he GC is calculaed for each componen separaely. This analysis makes i possible o deermine wheher he predicive effec of invesor senimen is concenraed on shor-erm or long-erm. To he bes of our knowledge, he analysis of GC from invesor senimen o he reurns of socks wih differen levels of firm characerisics has no ye been explored in he domain. By doing his, we provide evidence ha he invesor senimen has differen long-erm and shor-erm predicive effecs on sock reurns by considering differen levels of firm characerisics. (e.g., firm size, book-o-marke equiy rae, operaing profiabiliy
and invesmen). The paper proceeds as follows: ecion presens he daa and mehodology, ecion 3 provides he empirical findings and ecion 4 concludes.. Daa and mehodology.1 Daa The daa are a a monhly, spanning he period beween July 1965 and epember 015. The U monhly invesor senimen index is aken from Baker and Wurgler (007) 1. The equally weighed porfolio reurns formed on firm characerisics: firm size (ize), book-o-marke equiy rae (B/M), operaing profiabiliy (OP) and invesmen (Inv), which are colleced from he websie of Kenneh R. French.. Frequency domain Granger causaliy es In his paper, we follow he bivariae GC es over he specrum of Lemmens e X al. (008). Le and be wo saionary ime series of lengh T. The goal is o es wheher Granger-causes Y a a given. Pierce's (1979) X Y measure for GC in he domain is performed on he univariae innovaions series, and, derived from filering he X and Y as univariae ARMA processes, which are whie-noise processes wih zero means, are possibly correlaed wih each oher a differen leads and lags. Le () and () be he specral densiy funcions, or specra, of and a a 0,, defined by 1 ik ( ) ( k) e (1) k 1 ik ( ) ( k) e, () k where k) Cov(, ) and k) Cov(, ) represen he auocovariances of ( k and a a lag ( k k. The idea of he specral represenaion is ha each ime series may be decomposed ino a sum of uncorrelaed componens, each relaed o a paricular. The specrum can be inerpreed as a decomposiion of he series variance by. The porion of he variance of he 1 hp://pages.sern.nyu.edu/~jwurgler/main.hm. hp://mba.uck.darmouh.edu/pages/faculy/ken.french/index.hml. 3
series occurring beween any wo frequencies is given by he area under he specrum beween hose wo frequencies. In oher words, he area under () and () beween any wo frequencies and and (, d). d, gives he porion of he variance of respecively, due o cyclical componens in he band The cross specrum represens he cross covariogram of wo series in he domain. I allows deermining he relaionship beween wo-ime series as a funcion of. Le () be he cross-specrum beween and series, which defined as i k e C ( ) ( ) ( k) (3) k ( ) iq where C ( ), called cospecrum and ( ), called quadraure specrum are respecively, he real and imaginary pars of he cross-specrum and Q 1 i 1.Here (k) Cov(, k ) represens he cross-covariance of and a a lag k. The specrum Q () beween he wo series and a a can be inerpreed as he covariance beween he wo series and ha is aribuable o cycles wih. The cross-specrum can be esimaed non-paramerically by 1 i k ˆ ( ) w k ˆ ( k) e (4) M M wih he empirical cross-covariances ˆ (k) Cˆ ov(, ) and window weighs w, k k for k M,, M. Eq. (4) is called he weighed covariance esimaor, and he weighs w k are seleced as he Barle weighing scheme i.e. 1 k / M. The consan M deermines he maximum lag order considered. The specra of Eqs. (1) and () are esimaed in a similar way. This cross-specrum allows us o compue he coefficien of coherence h ( ) defined as ( ) h ( ) (5) ( ) ( ) Coherence can be inerpreed as he absolue value of a specific correlaion coefficien, which akes values beween 0 and 1. Lemmens e al. (008) 4
have shown ha under he null hypohesis ha h ( ) 0, he esimaed squared coefficien of coherence a he., wih 0 when appropriaely rescaled, converges o a chi-squared disribuion wih degrees of freedom, denoed by. d ( n 1) hˆ ( ) (6) where d sands for convergence in disribuion, n T /( M k M w ). The null k hypohesis h ( ) 0 versus h ( ) 0 is hen rejeced if hˆ,1 ( ) (7) ( n 1) wih being he 1 quanile of he chi-squared disribuion wih degrees,1 of freedom. The coefficien of coherence in Eq. (5) gives a measure of he srengh of he linear associaion beween he wo-ime series, by, bu does no provide any informaion on he direcion of he relaionship beween he wo processes. Lemmens e al. (008) have decomposed he cross-specrum (Eq. (1)) ino hree pars: (i), he insananeous relaionship beween and ; (ii), he direcional relaionship beween and lagged values of ; and (iii), he direcional relaionship beween and lagged values of, i.e., 1 1 ik ik (0) ( k) e ( k) e ( ) (8) k k1 The proposed specral measure of GC is based on he key propery ha no Granger-cause does if and only if ( k) 0 for all k 0. The goal is o es he predicive conen of i.e., relaive o which is given by he second par of Eq. (8), The Granger coefficien of coherence is hen given by 1 1 ik ( ) ( k) e (9) k ( ) h ( ) (10) ( ) ( ) Therefore, in he absence of GC, h ( ) 0 for every in 0,. The 5
Granger coefficien of coherence akes values beween zero and one (Pierce, 1979). Granger coefficien of coherence a is esimaed by ˆ ( ) hˆ ( ) (11) ˆ ( ) ˆ ( ) wih ˆ ( ) as in Eq. (4), bu wih all weighs w 0, for k 0. The disribuion of he esimaor of he Granger coefficien of coherence is derived from he disribuion of he coefficien of coherence (Eq. (6)). Under he null hypohesis ˆ h ( ) 0, he disribuion of he squared esimaed Granger coefficien of k coherence a, wih 0 is given by d ( n 1) hˆ ( ) (1) where 1 k M k n is now replaced by n T /( w ). ince he weighs, wih a posiive index k, are se equal o zero when compuing ( ), in effec only he w wih negaive indices are aken ino accoun. The null hypohesis ( ) 0 k ˆ w k hˆ versus h ˆ ( ) 0 is hen rejeced if hˆ,1 ( ) (13) ( n 1) Aferward, we compue Granger coefficien of coherence given by Eq. (11) and es he significance of causaliy by making use of Eq. (13). 3. Empirical resuls This secion repors he resuls of causaliy ess in he domain for wo bivariae sysems: invesor senimen and each sock reurns of he porfolio for U. The variables have been filered using ARMA models o obain he innovaion series. Boh Augmened Dickey-Fuller (ADF) es (Dickey and Fuller, 1981) and Philip's Peron (PP) es (Phillips and Perron 1988) rejec he null hypohesis of a uni roo in all-ime series a he 5% significance level. 6
Table 1 Uni roo and saionary es ADF No rend Trend No rend Trend enimen -3.447(4)*** -3.41(4)*** -3.47(1)** -3.367(1)** ize-small -18.90(0)*** -18.905(0)*** -18.819(3)*** -18.80(3)*** ize-middle -1.08(0)*** -1.01(0)*** -0.888(4)*** -0.87(4)*** ize-big -.81(0)*** -.63(0)*** -.37(5)*** -.19(5)*** B/M-low -.733(0)*** -.716(0)*** -.76(5)*** -.698(4)*** B/M-middle -.886(0)*** -.867(0)*** -.874(9)*** -.855(9)*** B/M-high -1.854(0)*** -1.86(0)*** -1.806(3)*** -1.813(3)*** OP-low -.093(0)*** -.078(0)*** -.084(4)*** -.070(4)*** OP-middle -3.06(0)*** -3.043(0)*** -3.053(7)*** -3.034(7)*** OP-high -.999(0)*** -.980(0)*** -.996(6)*** -.977(6)*** Inv-small -.865(0)*** -.847(0)*** -.843(6)*** -.85(6)*** Inv-middle -3.355(0)*** -3.337(0)*** -3.378(8)*** -3.360(8)*** Inv-big -.371(0)*** -.35(0)*** -.354(3)*** -.335(3)*** Noes: **and ***indicae significance a he 5% and 1% level, respecively. The numbers in parenheses are he opimal lag order in he ADF and PP es based on he chwarz Info crierion and Newey-wes bandwidh. In Fig.1, he esimaed Granger coefficiens of coherence are ploed versus he ( 0, ). This coefficien assesses wheher, and o wha exen, he invesor senimen is Granger causing he sock reurns of he porfolio a ha. The higher he esimaed Granger coefficien of coherence, he higher he Granger causaliy a ha paricular. If he coefficien is higher han he 5% criical value, he invesor senimen is said o significanly Granger cause he sock reurns a he. Noe ha he lag lengh ranslaed ino a cycle or periodiciy of T monhs by M PP T, he can be T /, where T is he period. In his paper, we disinguish beween he long-erm componens (low frequencies) and he shor-erm componens (high frequencies) of a ime series. We define he long-erm componens o have a cycle larger or equal o 1 monhs, which corresponds o he 0. 5. The shor-erm componens have a cycle smaller han 1 monhs, which corresponds o a 0. 5. As shown in Fig.1, for small ize socks (boom 30% quaniles), he null hypohesis of no causaliy is rejeced for all frequencies a he 5% significance level, i indicaes ha he sock invesor senimen has highly significan predicive power for he sock reurns in he shor erm and long erm. However, for he big ize socks (op 30% quaniles), he Granger coefficiens of coherence corresponding o he low 7
frequencies hardly reach saisical significance. I suggess ha he sock invesor senimen have no predicabiliy o he sock reurns in long-erm. Meanwhile, he null hypohesis should be rejeced when (.5, 3) (cycles of.1.5 monhs), which reveals ha invesor senimen Granger-causes sock reurns of big ize socks in he shor erm. This finding is in line wih Lemmon and Porniaguina (006) who find ha invesor senimen can significan predic he reurns of small size socks. Fig.1 shows ha he invesor senimen is a poor predicor for he reurns of smaller B/M socks in long erm bu i has highly significan predicabiliy in shor erm. And for bigger B/M socks, he invesor senimen has no only shor-erm bu also long-erm predicabiliy for reurns. More specifically, we find ha for smaller B/M socks, when he 0. 5, here exiss a larger Granger coefficiens of coherence in he range (.5, 3) which can rejec he null hypohesis of no causaliy. I reveals ha in he shor erm, he invesor senimen is a rich predicor of reurns of smaller B/M socks. However, when he 0. 5, he null hypohesis canno be rejeced a he 5% significance level, which indicaes ha he invesor senimen doesn' have long-erm predicabiliy for he sock reurns. Wih regard o bigger B/M socks, i is found ha he null hypohesis always can be rejeced a he 5% significance level in low and high. I confirms ha he invesor senimen no only has shor-erm bu also long-erm predicabiliy for he sock reurns. Fig.1 displays ha he invesor senimen has predicabiliy for he reurns of lower OP socks boh in shor erm and in long erm. However, he invesor senimen jus has a predicabiliy for he reurns of larger OP socks only in shor erm. In paricular, for lower OP socks, he Granger coefficiens of coherence mosly are larger han he 5% criical value beween he frequencies of 0 and, which means ha he invesor senimen has persisen predicabiliy for he reurns of lower OP socks. However, for he higher OP socks, i can rejec he null hypohesis of no causaliy only when is in he range (.4,.5) (cycles of.1.6 monhs), i reveals ha invesor senimen only Granger-causes sock reurns of higher OP socks in he shor erm. Nex, we find ha i has a differen predicabiliy from invesor senimen o he reurns of socks wih low and high invesmen (see boom panel of Fig.1). pecifically, i indicaes ha he invesor senimen is a poor predicor of reurns of 8
smaller Inv socks in he long erm bu i has significan predicabiliy in he shor erm. And for bigger Inv socks, he invesor senimen no only has a shor-erm bu also a long-erm predicabiliy for he sock reurns. ize-small ize-middle ize-big B/M-low B/M-middle B/M-high OP-low OP-middle OP-high Inv-small Inv-middle Inv-big Fig.1. Granger coefficiens of coherence for sock reurns formed on ize, B/M, OP, Inv a 3 Deciles. The dashed line represens he criical value, a he 5% level, for he es for no Granger causaliy. The paper considers he robusness of he resuls wih respec o he sock reurns of he porfolio a 10 Deciles (see Figs.-5). I proves he robusness of our main resuls as follows. Firsly, as shown in Fig., for he sock reurns of porfolio formed on ize from Low10 o 8-Dec, i is proven ha he Granger coefficiens of coherence mosly are larger han he 5% criical value in all range ( 0, ). However, for he sock reurns of porfolio formed on ize in 9-Dec and Hi-10 deciles (bigger ize socks), he null hypohesis of no causaliy can be rejeced only when is in he range 9
(.5, 3). I is in line wih our foregoing finding ha he invesor senimen has srong predicabiliy for he sock reurns of smaller ize socks boh in he shor erm and long erm, bu for big socks, only has a shor erm predicabiliy. ize-lo 10 ize--dec ize-3-dec ize-4-dec ize-5-dec ize-6-dec ize-7-dec ize-8-dec ize-9-dec ize-hi 10 Fig.. Granger coefficiens of coherence for sock reurns formed on ize a 10 Deciles. The dashed line represens he criical value, a he 5% level, for he es for no Granger causaliy. econdly, as shown in Fig.3, for he sock reurns of porfolio formed on B/M from Low10 o 6-Dec deciles, i is found ha he Granger coefficiens of coherence value larger han he 5% criical value only when 0. 5. For he higher B/M socks (7-Dec,8-Dec, 9-Dec and Hi-10 deciles), we find ha i always has a larger Granger coefficien of coherence o rejec he null hypohesis of no causaliy in he whole range ( 0, ). I proved ha he invesor senimen has significan predicabiliy for he reurns of lower B/M socks boh in he shor erm and long erm, whereas only has a predicabiliy for he reurns of larger B/M socks in he shor 10
erm. B/M-Lo 10 B/M--Dec B/M-3-Dec B/M-4-Dec B/M-5-Dec B/M-6-Dec B/M-7-Dec B/M-8-Dec B/M-9-Dec B/M-Hi 10 Fig.3. Granger coefficiens of coherence for sock reurns formed on B/M a 10 Deciles. The dashed line represens he criical value, a he 5% level, for he es for no Granger causaliy. Thirdly, as shown in Fig.4, for he sock reurns of porfolio formed on OP a Low10 and -Dec deciles (smaller OP socks), i indicaes ha he null hypohesis of no causaliy mosly can be rejeced when he 0. 5 or 0. 5. For he higher OP socks (3-Dec,4-Dec,5-Dec,6-Dec,7-Dec,8-Dec, 9-Dec, and Hi-10), i can rejec he null hypohesis of no causaliy jus when he 0. 5. These findings assure ha he invesor senimen has significan predicabiliy for he reurns of lower OP socks boh in he shor erm and long erm, whereas only has a predicabiliy for he reurns of larger OP socks in he shor erm. 11
OP-Lo 10 OP--Dec OP-3-Dec OP-4-Dec OP-5-Dec OP-6-Dec OP-7-Dec OP-8-Dec OP-9-Dec OP-Hi 10 Fig.4. Granger coefficiens of coherence for sock reurns formed on OP a 10 Deciles. The dashed line represens he criical value, a he 5% level, for he es for no Granger causaliy. Finally, as shown in Fig.5, for smaller Inv socks (e.g. from 3-Dec o 8-Dec), he null hypohesis of no causaliy can be rejeced only when he is in he range (, 3) (cycles of.1-3.1 monhs). For bigger Inv socks (9-Dec and Hi-10), i always can find higher Granger coefficiens of coherence han he 5% criical value when he 0. 5 or 0. 5. I confirms again ha he invesor senimen only has predicabiliy in he sock reurns of smaller Inv socks in he shor erm, bu has a srong shor erm and long erm predicabiliy for larger Inv socks. 1
Inv-Lo 10 Inv--Dec Inv-3-Dec Inv-4-Dec Inv-5-Dec Inv-6-Dec Inv-7-Dec Inv-8-Dec Inv-9-Dec Inv-Hi 10 Fig.5. Granger coefficiens of coherence for sock reurns formed on Inv a 10 Deciles. The dashed line represens he criical value, a he 5% level, for he es for no Granger causaliy. 4. Concluding remarks This paper aims o invesigae wheher he ime horizon of reurn predicive effec of invesor senimen varies wih sock characerisics (e.g., firm size, book-o-marke equiy rae, operaing profiabiliy and invesmen) by employing a Granger causaliy es in he domain proposed by Lemmens e al. (008). The evidence shows ha invesor senimen has differen predicive effecs whose ime horizon can be long-erm or shor-erm on sock reurns wih differen firm characerisics. More specifically, we find ha 1) he invesor senimen has srong predicabiliy in he sock reurns for smaller ize socks boh in he shor erm and long erm, bu only has a shor-erm predicabiliy for bigger ize socks. ) he invesor senimen has a significan predicive effec on he reurns for lower B/M socks boh in he shor erm and long erm, whereas only has a predicabiliy for he 13
reurns of larger B/M socks in he shor erm. 3) he invesor senimen has significan predicabiliy for he reurns of lower OP socks boh in he shor erm and long erm, whereas only has a predicabiliy for he reurns of larger OP socks in he shor erm. 4) he invesor senimen merely has predicabiliy for he reurns of smaller Inv socks in he shor erm, bu has a srong shor-erm and long-erm predicabiliy for larger Inv socks. These resuls have imporan implicaions for he invesors for he planning of he shor and he long run sock invesmen sraegy. Acknowledgmens We graefully acknowledge he financial suppor from he Naional Naural cience Foundaion of China (Nos. 7177108, 71431008) and Hunan Provincial Naural cience Foundaion of China (No. 017JJ101). References Aloui, C., Hkiri, B., Lau, C. K. M., & Yarovaya, L. 016. Invesors senimen and U Islamic and convenional indexes nexus: A ime- analysis. Finance Research Leers, 19, 54-59. Baker, M., & Wurgler, J. 006. Invesor senimen and he cross-secion of sock reurns. The Journal of Finance, 61(4): 1645-1680. Baker, M., & Wurgler, J. 007. Invesor senimen in he sock marke. The Journal of Economic Perspecives, 1(): 19-151. Chung,. L., Hung, C. H., & Yeh, C. Y. 01. When does invesor senimen predic sock reurns?. Journal of Empirical Finance, 19(), 17-40. Dergiades, T. 01. Do invesors senimen dynamics affec sock reurns? Evidence from he U Economy Economics Leers, 116(3), 404-407. Dickey, D.A., Fuller, W.A., 1979. Disribuion of he esimaors for auoregressive ime series wih a uni roo. Journal of he American aisical Associaion 74 (366), 47 431 Huang, D., Jiang, F., Tu, J., & Zhou, G. 015. Invesor senimen aligned: A powerful predicor of sock reurns. The Review of Financial udies, 8(3), 791-837. Phillips, P. C., & Perron, P. 1988. Tesing for a uni roo in ime series regression. Biomerika, 75(), 335-346. Pierce, D. A. 1979. R-squared measures for ime series. Journal of he American aisical Associaion, 74, 901 910 Lemmens, A., Croux, C., & Dekimpe, M. G. 008. Measuring and esing Granger causaliy over he specrum: An applicaion o European producion expecaion surveys. Inernaional Journal of Forecasing, 4(3), 414-431. Lemmon, M., & Porniaguina, E. 006. Consumer confidence and asse prices: ome empirical evidence. The Review of Financial udies, 19(4), 1499-159. Li, H., Guo, Y., & Park,. Y. 017. Asymmeric Relaionship beween Invesors' enimen and 14
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