Economic Fundamentals and the Predictability of Chinese Stock Market Returns: a Comparison of VECM and NARMAX Approaches. Abstract

Transcription

1 Submed for Inernaonal Journal of Forecasng Economc Fundamenals and he Predcably of Chnese Sock Marke Reurns: a Comparson of VECM and ARMAX Approaches Yanbng Zhang, Xupng Hua *, Lang Zhao 3, Sephen A. Bllngs 4 Absrac Ths paper ams o evaluae he predcve relaonshps of sock marke reurns and macroeconomc varables n Chna over he perod January 999 o June 00. We consder and compare ou-ofsample predcve ably of hree dfferen modellng approaches, namely a conegraed vecor auoregressve and error correcon (VECM) model, a onlnear Auo Regressve Movng Average wh exogenous npus (ARMAX) model wh lnear erms, and a ARMAX wh nonlnear erms. The forecasng exercse shows ha boh ARMAX models ouperform he VECM model and ARMAX wh lnear erms has he bes ou-of-sample predcve ably. Our resuls also sugges ha he movemens n sock marke reurns may manly be explaned by s own nnovaons. However, macroeconomc condons sll conan useful nformaon for forecasng fuure sock reurns and canno be negleced n accounng for he dynamcs of he reurns. Keywords: sock reurn predcably, macroeconomc varables, ou-of-sample forecass, VECM, ARMAX JEL Classfcaon: C53, E44, G7 School of Publc Polcy and Managemen, Unversy of Tsnghua, Bejng, Chna, Emal: zhangyanbng@snghua.edu.cn Correspondng auhor, Busness School, Unversy of ongham ngbo, Chna, 99 Takang Eas Road, Unversy Park, ngbo 3500, Chna. Tel: ; Fax: ; Emal: Xupng.Hua@nongham.edu.cn 3 Deparmen of Auomac Conrol and Sysem Engneerng, Unversy of Sheffeld, UK, Emal address: cop06lz@sheffeld.ac.uk. 4 Deparmen of Auomac Conrol and Sysem Engneerng, Unversy of Sheffeld, UK, Emal address: s.bllngs@sheffeld.ac.uk.

2 . Inroducon Over he pas several decades, a large quany of leraure has focused on he queson of wheher macroeconomc varables conan nformaon ha s useful for forecasng sock marke ndces or reurns, such as Fama and French (989), Rapach e al (005), Harmann e al (008), and Laopods (00). The basc assumpon lyng behnd hs sream of leraure s ha economc fundamenal nformaon publcly avalable n he pas has some predcve relaonshps o he fuure sock marke reurns of ndces. I s no surprsng because macroeconomc varables are lkely o exer mporan nfluences on frms expeced earnngs or dscoun raes and are herefore lkely o be relaed o sock marke reurns. Exensve emprcal sudes have examned he predcably of sock marke reurns usng macroeconomc varables, such as GP/GDP or ndusral producon, moneary growh, neres raes, nflaon, exchange raes, and so on. In regard o Chnese sock marke reurns, some oher macroeconomc varables, such as ho money or speculave capal flow (Guo and Huang, 00), have been nvesgaed as well. Overall, he mxed resuls n he exsng leraure show ha s dffcul o deermne wheher economc fundamenals or whch parcular macroeconomc varables (f any) are relable ndcaors of sock marke reurns. In hs paper, we use Chna s experence o re-examne he predcably of sock marke reurns based on macroeconomc varables. Among developng counres, Chna s parcularly suable as a case sudy for analyzng he relaonshp beween sock marke reurns and economc fundamenals. On he one hand, Chna s sock marke s relavely new bu has developed really rapdly snce s esablshmen. Is wo offcal sock exchanges, he Shangha Exchange and he Shenzhen Exchange, were esablshed n December 990 and July 99, respecvely. Accompanyng he fas growng Chnese economy, he wo exchanges have expanded a a breahakng speed n erms of sze (marke capalzaon), radng volumes and number of socks raded. Accordng o he sascs publshed by World Federaon of Exchanges (WFE), n January 008 and July 009, Chna had overaken Japan wce o become he world s second- bgges sock marke by capalzaon. By end of 009, he Shangha and Shenzhen sock exchanges lsed around,700 companes and had a marke capalzaon of US$3.57 rllon. However, he Chnese sock marke can be noorously opaque and speculave. A renowned economs n Chna, Professor Jnglan Wu, has wo very well-known commens wh regard o he Chnese sock marke. In January 00, he openly condemned Chna s sock marke for beng worse

3 han a casno, ha s, lke a casno whou rules 5. Very recenly, he descrbed Chna s sock marke as beng n he age of robber barons and poned o nsder radng by governmen offcals as especally common 6. Overall, varous domesc and nernaonal meda and research repors end o depc he Chnese sock markes as opaque, chaoc, neffcen, and raher rraonal (Eun and Huang, 007). In such an neffcen and speculave marke as Chna s, wll economc fundamenals sll conan useful nformaon n predcng sock reurns? Ths paper ams o shed some lgh on hs neresng queson usng hree dfferen mehodologes. They nclude he co-negraed vecor auoregressve and error correcon (VECM) model, he onlnear Auo Regressve Movng Average wh exogenous npus (ARMAX) model only wh lnear erms, and fnally, he ARMAX wh boh lnear and nonlnear erms. The VECM s one of mos popular economerc frameworks n dealng wh mulple me seres, whle he ARMAX s a generalzaon of he ARMAX model and has been proved o be successful n modelng numerous real world nonlnear sysems and s parcularly powerful n selecng he mos sgnfcan model erms. Boh ypes of models are wdely-used ools for forecasng me seres. Our sudy s based on a daa se of monhly macroeconomc me seres daa from Chna over January 999 o June 00. The daa s spl no wo and he frs half daa over he perod 999:-008: wll be used o esmae he hree sock reurn models. Then he esmaed model parameers from he frs half daa are used o predc sock reurns n he ou-of-sample perod 009:-00:6, and he one-sep-ahead predcon errors usng a rollng sample are also calculaed. Summarzng our resuls, alhough he movemens n sock marke reurns may manly be explaned by s own nnovaons, macroeconomc condons sll conan useful nformaon for forecasng fuure sock reurns and canno be negleced n accounng for he dynamcs of sock marke reurns. Among all he macroeconomc varables we consder, neres rae and real effecve exchange rae appear o be he mos common and relable n relaon o sock marke reurn predcably across all he hree models. Sgnfcan nonlneary n sock reurn dynamcs condonng on macroeconomc nformaon are found as well. Our research also demonsraes he accuracy of ARMAX models n predcng ou-of-sample sock marke reurns movemens over he popular VAR/VECM approach n economerc analyses. I provdes evdence ha he ARMAX mehod ogeher wh he algorhms of erm selecon and parameer esmaon s able o provde a powerful represenaon for economc or fnancal me seres analyss, modellng and predcon. Ths s one unque perspecve ha makes hs sudy dsnc from 5 Chna Daly, 3 March 00, Sock marke causes heaed debae. 6 Anderln J., 00. Chna sars crackdown on nsder radng, Fnancal Tmes webse, 5 ovember 00. 3

4 many exsng sudes. To our bes knowledge, hs s he frs applcaon of ARMAX o analyss of sock marke reurn dynamcs. Mos sudes n he leraure apply lnear esmaon mehods, such as smple regressons or VAR/VECM analyses. Such models have an mporan defcency n selecng explanaory varables, especally her forms and lagged erms (Gupa e al, 00), and hence may face an over-fng problem whch can resul n less accuracy n ou-of-sample predcon. ARMAX no only provdes a powerful and ransparen model of he predcve relaonshp of sock marke reurns and economc fundamenals, bu also enables a fas parameer esmaon process. The res of he paper s organzed as follows. Secon revews he relevan leraure. Secon 3 presens he daa and he descrpve sascs, and hen Secon 4 nroduces he economerc mehodology. Secon 5 presens he man modellng and forecasng resuls and Secon 6 concludes.. Leraure Revew Predcng sock marke ndces or reurns s an mporan research opc and numerous heorecal and emprcal sudes have nvesgaed hs problem. Many sudes examne he sources of reurn varaon and have shown ha hey are relaed o economc fundamenals, such as Chen e al (986), Cozer and Rahman (988), Fama (98, 990), Schewer (990), ajand and oronha (998), asseh and Srauses (000), Flannery and Proopapadaks (00), Rapach e al (005), Du (006), and Araujo (009). In such sudes, he represenaves of economc fundamenals whch may mpac sock marke reurns nclude aggregae oupu as measured by GP/GDP or ndusral producon, moneary growh, neres raes, nflaon, exchange raes, he crude ol prce, unemploymen rae, and so on. Theorecally, sandard sock valuaon models pos ha sock prces are manly affeced by (a) shocks o expeced cash flows, and (b) shocks o dscoun raes (Fama, 990) and boh shocks should be relaed o fuure economc acvy movemens or depend on he healh or weakness of he economy (Laopods, 00). Expeced dvdends should reflec real economc acvy as measured by GDP or ndusral producon. Dvdends are affeced by oher mporan macroeconomc varables such as neres raes (whch nfluence he frm s rsk premum and dscoun rae) and nflaon (whch drecly nfluences he marke neres rae). The lnk beween exchange raes and sock reurns may also be explaned by a smple and nuve fnancal heory: an apprecang currency s generally accompaned by ncreased money supply and a declne n neres raes, whch resuls n a declne n dscoun raes and hence a rse n local sock reurns (Abugr, 008). Emprcal evdence on he lnk beween sock marke movemens and economc fundamenals across dfferen counres, or whch parcular macroeconomc varables are relable ndcaors of sock 4

5 marke reurns, neverheless, s raher mxed. For one hng, ample emprcal evdence confrms he lnkage beween economc fundamenals and sock marke reurns. Fama (990) shows ha monhly, quarerly and annual sock marke reurns are hghly correlaed wh fuure producon growh raes for Schwer (990) proves ha Fama s fndngs are robus for a much longer perod and fuure producon growh raes are able o explan a large fracon of he varaon n sock marke reurns. Papers by Fama (98), Cozer and Rahman (988) and ajand and oronha (998) documen negave relaons beween real sock marke reurns and measures of expeced and unexpeced nflaon n he Uned Saes, Canada and Japan. By ulzng orwegan daa, Gjerde and Sæem (999) fnd ha real neres rae changes affec boh sock reurns and nflaon, and he sock marke responds accuraely o ol prce changes. In Blson e al (00), he exchange rae varable s clearly he mos nfluenal macroeconomc varable, wh he reurns for welve markes beng sgnfcanly relaed o hs varable. Du (006) ndcaes ha he posve relaonshp beween sock marke reurns and nflaon n he 930s s manly due o srongly pro-cyclcal moneary polcy, whle he srong negave relaonshp of sock marke reurns and nflaon durng he perod of s largely caused by supply shocks ha were relavely more mporan n ha perod. asseh and Srauss (000) fnd he exsence of a sgnfcan and long run relaonshp beween sock prces and domesc and nernaonal economc acvy, such as ndusral producon, busness surveys of manufacurng orders, shor- and long-erm neres raes, n sx European economes. Flannery and Proopapadaks (00) sudy he mpac of macro condons on equy reurns usng 7 macro varables and fnd sx of hem have more predcve ably han ohers. Usng daa from ndusralzed counres afer he 970s, Rapach e al (005) consder many macroeconomc varables and found ha neres raes appear o be he mos conssen and relable predcors of sock reurns across he counres. Araujo (009) sudes he economc sources underlyng he co-movemen of real sock reurns n Lan Amerca and fnds ha macroeconomc shocks canno be negleced n accounng for dynamcs of real sock marke reurns. Chen (009) suggess ha among he macroeconomc varables we have evaluaed, yeld curve spreads and nflaon raes are he mos useful predcors of recessons n he US sock marke. However, n recen years here has been an ncreasng concern ha sock marke movemens could no be explaned by economc fundamenals. Bnswanger (000, 00, and 004) sugges ha here s a breakdown n he radonally srong relaon beween sock reurns and real economc acvy n he US n he early 980s. Such a breakdown mples he emergence of new sources of varaon n sock reurns durng he sock marke boom over he 980s and 990s, whch are no explaned by he radonal dscouned cash flow valuaon model accordng o whch sock prces should lead measures of real acvy. Hooker (004) nvesgaes he predcve power of several canddae macroeconomc facors for emergng marke equy reurns usng he Bayesan model selecon 5

6 approach and provdes evdence agans he sgnfcance of macro varables, wh he excepon of he exchange rae. Harmann and Perdzoch (006) examne he lnk beween exchange rae movemens and sock reurns and found ha exchange rae movemens per se do no help o explan sock reurns. Laopods (006) examnes he dynamc lnkages among sock prces, neres raes, nflaon, and economc acvy for he Uned Saes snce he 970s, and also repors ha here was no conssen, dynamc relaonshp beween real acvy and sock prces across dfferen moneary regmes durng he las hry years. Laopods (00) addresses he dynamc lnkages among sock prces and several fundamenal varables for he perod from 990 o 009 for France, Germany, Ialy, he UK and he US, and suggess ha her sock markes appear o move ndependenly of changes n economc fundamenals n he long run. When comes o Chna s sock marke, he hgh level of speculaon, volaly and he rapd growh have araced numerous emprcal sudes, bu mos of hem have focused on marke operaon and effcency. The evaluaons range from exremely favourable assessmens ha emphasze GDP growh raes, fnancal deph or he rapd growh of prvasaon o exremely negave assessmens ha favour marke effcency or shor-erm profably (Xu and Oh, 0). Only a few papers have nvesgaed he lnk beween macro explanaory facors and Chnese sock marke reurns and he macro varables nclude growh raes of ndusral producon, dscoun raes, ho money and he nflaon rae, such as Chen (007), and Guo and Huang (00). Agan, he resuls are no conclusve and he emprcal queson as o wha exen he macroeconomc facors affec he behavour of sock prces awas more exploraon. 3. Daa and Descrpve Sascs In our analyss, we use monhly me seres from January 999 o June 00. We gaher a sock prce ndex (he Shangha Sock Exchange Compose Index, SP) as he dependen varable, an aggregae oupu proxy, namely value-added ndusral oupu (VAI), a broad moneary aggregae (M), a nomnal shor-erm neres rae measured by year depos rae (R), a produc prce ndex (CPI), and an exchange rae measured by he real effecve exchange rae ndex (REER) are colleced. Followng Guo and Huang (00), we also defne a new varable ho money (HM) whch s calculaed by (change n foregn exchange reserves) mnus (rade and servce balance) mnus (foregn drec nvesmen). All ogeher 7 monhly seres are exploed, and each seres conans 38 observaons. All varables are measured n year-on-year growh raes (percenage changes) excep ho money and neres rae. The daa were colleced from he Wnd and Bloomberg daabase. Table provdes some summary sascs for hese varables as well as her frs dfferences. The over he sample perod, mos of he seres exhb sgnfcan skewness and kuross. The Jarque Bera 6

7 es sasc suggess rejecon of he null hypohess of normal dsrbuon for mos varables excep REER. Jus as recorded n he leraure (Premarane and Bera, 005), he sock marke reurns n our sample perod exhb boh fa als and some asymmery. We have also found ha here are sgnfcan auocorrelaons n boh sock marke reurns and n her varances, whle he laer s an ndcaon of volaly cluserng, jus as he Ljung-Box Perce and ARCH effecs es sascs show. Much of he emprcal evdence has shown ha sock marke reurns are far from a random walk and may move n smooh and predcable paerns. Table Summary sascs for economc varables and her frs dfferences Varable Mean Sd. Dev. Skewness Kuross Mn Max Jarquebera Ljung-Box- Perce ARCH es SP * 66.5* 0.9* SP * 58.76*.759 VAI * 4.4* 9.83* VAI * 5.74* 7.38* M * 68.8* 6.64* M * 80.9* 0.0 R * 364.8* 8.60* R *.00* 8.8* CPI * * 5.9* CPI * 78.3* REER * 8.60* REER * HM * 74.84* 4.3** HM * *.05* oes:. * and ** denoe he rejecon of normal dsrbuon, no seral correlaon or ARCH effecs hypohess a % and 5% sgnfcance levels, respecvely. Before proceedng wh modellng, we need o deermne wheher each seres s saonary or no. Table presens he resuls of un roo ess for saonary of boh level values and frs dfferences of he egh seres. Boh Augmened Dckey-Fuller and Phllps-Perron resuls are repored. As seen, Augmened Dckey-Fuller resuls sugges ha 6 varables are negraed of order one, namely follow an I() process, and only one varable, namely ho money, are saonary a he % sgnfcan level, whle he Phllps-Perron resuls sugges ha 5 of 7 varables are I(). Therefore we may use a VECM model o nvesgae he exen o whch macroeconomc varables are responsble for flucuaons n Chnese sock marke reurns. 7

8 Table Tess for Un Roo Varable Augmened Dckey-Fuller Phllps-Perron Levels Dfferences Levels Dfferences SP * * VAI -3.95** -.9* -8.49* -37.9* M * * R * * CPI * * REER * * HM -8.07* -.30* -8.4* -3.3* oes:. The able presens he resuls of un roo ess for saonary of all ndvdual me seres.. * and ** denoe he rejecon of he un roo hypohess a he % level and 5% level of sgnfcance, respecvely. 4. Mehodology Ths secon nroduces he wo economerc mehods ha we use o assess he predcve relaonshps of sock marke reurns and macroeconomc varables. 4. Vecor-auoregressve and error correcon (VECM) models One of mos popular economerc frameworks n dealng wh mulple me seres s he vecorauoregressve (VAR) model. A vecor auoregresson wh conegraon and error correcon erms, or VAR/VECM, s a sysem of ordnary leas-squares regressons, n whch each of a se of varables s regressed on lagged values of boh self and he oher varables n he se. Varous forms of VARs have proved o be convenen mehods of summarzng he dynamc relaonshps beween sock marke reurns and macroeconomc varables and many auhors have researched hs opc, ncludng he muvarae VAR approach of Gjerde and Sæem (999), a conegraon and vecor error correcon (VECM) approach of asseh and Srauss (000) and a srucural VAR of Araujo (009). Snce our research s o compare predcve ables of dfferen approaches, here s no need o adop a srucural conemporary relaonshp n our modellng, hence followng asseh and Srauss (000), we also adop a VECM model raher han a srucural VAR approach. Accordng o Johansen (988, 99) and Johansen and Juselus (990, 99), a p-dmensonal vecor me seres up o k-lags of Z s consdered and modelled as an Unresrced Vecor Auoregresson (VAR) nvolvng Z : Z A Z A Z, ~ nd(0, ) (4.) k k 8

9 where Z s a p marx, A s a p p marx of parameers, and s a normal dsrbued..d. random varables. The above equaon can be reformulaed no a Vecor Error Correcon Model (VECM) as follows: k Z Z Z,,,T (4.) where s he frs dfference operaor, s a p p marx of he form Z s he se of I varables, s he drf parameer, and T, where and are p r marces of full rank, wh conanng he r conegrang vecors and nclude he correspondng adjusmen coeffcen n each of he r vecors. 4. The ARAMX model Jus as n many real world sysems, he mechansms behnd sock marke movemens are usually unknown and deermnng he srucure of he sock reurn models s he mos dffcul par durng denfcaon. To cope wh hese complexes, we propose o use a onlnear AuoRegressve Movng Average wh exogenous (ARMAX) mehod o nvesgae he exen o whch economc varables are responsble for sock reurn flucuaons. As a generalzaon of he ARMAX model, he ARMAX mehod has been proved o be successful n modelng numerous real world nonlnear sysems. I was proposed by Leonars and Bllngs (985). Based on he ARMAX model specfcaon, he Orhogonal Forward Regresson (OFR) algorhm and Error Reducon Rao (ERR) defnon were subsequenly nroduced by Bllngs e al. (988, 989) and Bllngs and Zhu (994) o provde an effcen way o deermne he mos sgnfcan erms among he canddae model erm se. The ARMAX model has successfully modelled many real world nonlnear sysems ncludng chaoc elecronc crcus, waer managemen sysems, urbocharged desel engnes, (Bllngs and Coca, 00) and more recenly he vsual sysem of fru fles and oher complex bologcal sysems. I s also exremely useful n he lnear modellng case when he number of canddae varables or erms s hgh. The srucure of he model can be formaed by selecng he erms wh an ERR value above a chosen cuoff value. The fnal seleced model should hen conss only of he mos sgnfcan erms. The ARMAX model proposed by Leonars and Bllngs (985) exends he ARMAX model o he nonlnear npu/oupu case and ARMAX usually akes he form of a se of nonlnear dfference equaons 9

10 y u e,...,,,...,,,..., y f y y n u d u n e e n e (4.3) where lag us he npu vecor, y s oupu vecor and ny and nu are maxmum oupu and npu lag, ne s he maxmum nose f s unknown nonlnear mappng. The nose varable e, whch accommodaes he effecs of measurng nose, modellng errors and unmeasured dsurbances s assumed o be bounded and uncorrelaed wh he npu, s compued depend of he esmaon process. Snce he f s unknown, he denfcaon of he ARMAX model nvolves no only he deermnng parameers of he models bu also he srucure of he model from he npu/oupu daa. The polynomal represenaon of f s one of he common mplemenaons and has receved grea aenon because of he good approxmaon properes and he smple model srucure. Oher choces ncludng wavele are avalable. The nonlnear mappng f here would be chosen as a polynomal represenaon wh a fne degree n all varables and he model srucure s assumed o be lnear-n-he-parameers. Accordngly, he general form of he polynomal ARMAX represenaon can be wren as M y p x (4.4) where x represens he lagged varables, s he dscree me varable, and x represens,...,,,...,,,..., he npu vecor, y y ny u d u nu n y s he sysem oupu. u s, where ny and nu are maxmum oupus and npu lag, n s he maxmum nose lag, p are model erms whch are a lnear or nonlnear combnaon of he varables, modellng error, M s he number of all he dsnc erms and are he unknown model parameers. s he Model (4.4) ncludes all possble combnaons of he varables and he parameers assocaed wh hose erms can be hundreds or more when he order of nonlneary s hgh. In pracce, many of he erms wll no be sgnfcan or redundan and can be removed. However, he mos commonly used parameer esmaon mehod Leas Square (LS) s no capable of deermnng he sgnfcance from all he possble erms of a ARMAX model and he nformaon crera whch are usually employed o selec parsmonous models based on he same daa se may have compuaon burdens because of he large number of canddaes varables and he prohbvely hgh compuaonal needs of sandard 0

11 mehod when appled o large nonlnear models. Accordngly, he Orhogonal Leas Square (OLS) algorhm was derved by Korenberg, Bllngs and Lu(988) o overcome hese dffcules. OLS allows he sgnfcance of he model erms o be deermned base on he value of he Error Reducon Rao (ERR) of each erm. However, he orgnal OLS algorhm has one major drawback whch s he choce of he nal orhogonalzed poson. The value of ERR s may change dependng on he order he erms are seleced no he model. The Orhogonal Forward Regresson (OFR) was hen proposed o solve hs problem by Bllngs e al. (988). To save space here, we ls he mahemacal deals of he Orhogonal Leas Squares algorhm and he Orhogonal Forward Regresson mehod n Appendx A and B, respecvely. Overall he ARMAX model ogeher wh he algorhms for erm selecon and parameer esmaon s one of he mos powerful lnear and nonlnear modellng mehodologes o dae. 5. Emprcal Resuls In hs paper, parameers are esmaed for all hree models for he frs half perod 999:-008: nally. Then for he ou-of-sample perod 009:-00:6, one-sep-ahead predcons usng he rollng samples, namely one-monh forecass, are generaed. The modellng and forecasng performance of he hree approaches are compared. 5. Modellng usng VECM echnques Snce many varables n our VARs are negraed wh order, we adop he Johansen conegraon procedure o denfy he conegraon vecors and dscuss he long-run relaonshps by seng up he VECM model. Afer he conegraon relaonshps are se up, he VECM was hen used o esmae he mpulse response and varance decomposon funcons. Ths framework s a useful seng for analyzng sock reurns and macroeconomc acvy; because ncorporaes dynamc co-movemens or smulaneous neracons, allowng us o sudy he channels hrough whch macroeconomc varables affec sock reurns as well as her relave mporance. Pror o he esmaon of VAR, he lag lenghs are chosen based on Akake Informaon Creron (AIC), whch suggess ha he opmum lag order s 3 for he VAR and for he VECM. Hence a welve-lag srucure s specfed for he VECM sysem. We hen conduc boh race and Maxegenvalue ess for conegraon. Boh race and Max-egenvalue ess ndcae 3 conegraon relaonshps a a % sgnfcance level. From an economc vew, s dffcul o nerpre he resuls of conegraon analyss when here s more han one conegraon relaonshp and conegraon equaons may no necessarly have an economc meanng (Maddala and Km, 998).

12 In our VECM sysem, we nclude lags of seven varables, as well as hree conegraon equaons, and all ogeher we have 87 erms n he model specfcaon for he VECM, whch may be wren as: SP a CE0 a CE0 a CE03 a SP a 8 M a 39 3 M 4 a 76 a HM 5 SP a 87 a HM 6 VAI C a 7 VAI where a, a are model parameers and C s he consan. 87 represens he frs dfferences operaon. The errors of he VECM are checked and found o conan no auocorrelaon usng a Q- es on he resduals, whch confrms he robusness of he model. (5.) The mpulse responses of sock reurns o generalzed one sandard devaon nnovaons n all he oher sx macroeconomc varables are shown n Fgure. As seen, Chnese sock reurn generally show posve responses o one S.D. nnovaon n ndusral oupu (VAI), broad money supply (M), nflaon (CPI), and ho money (HM), bu exhb negave response o shor-erm nomnal neres rae (R) and real effecve exchange raes (REER). The posve response o shocks n nflaon rae (CPI) s no conssen wh he man sream leraure whch records a negave relaonshp beween sock reurns and nflaon, whch deserves more research n he fuure. Fgure Impulse responses of sock reurns o generalzed one S.D. nnovaons n macroeconomc acvy Response of SP o VAI Response of SP o M Response of SP o ITEREST Response of SP o CPI Response of SP o REER Response of SP o HM

13 Whle mpulse response funcons race he effecs of a shock o one endogenous varable on o he oher varables n he VECM, he varance decomposon separaes he varaon n an endogenous varable no he componen shocks o he VECM. Thus, he varance decomposon provdes nformaon abou he relave mporance of each random nnovaon n affecng he varables n he VECM. Table 3 provdes Cholesky varance decomposons of sock reurns, whch show ha around 93% of he sock reurn varance may be explaned by s own varance n he nex monh, and only 7% may be explaned by macroeconomc varables. A he frs several lags, economc fundamenals such as neres raes, and real effecve exchange rae generally have some effecs on he fuure varance of sock reurns. Table 3 Varance decomposons of sock reurns Perod S.E. SP VAI M R CPI REER HM Modellng usng he ARMAX echnque wh lnear erms Under ARMAX mehodology specfcaon, a reference model s needed o selec he srucure of he sock prce model. We es all lnear combnaons of varous varables. Therefore, n he reference lnear model we nclude all he seven varables above and her lagged erms up o lag so ha he reference model s gven by: SP a SP asp a3vai a4vai a84hm C (5.) There are 84 erms n model (5.) and we hen apply he ARMAX erm selecon algorhm OFR on hs model. The fnally seleced erms wh bgges ERR values are lsed n Table 4, where he seleced erms are ordered accordng o her sgnfcance. Table 4 Seleced erms for lnear sock marke reurn model Rank Term ERR Parameer SP SP HM R REER

14 From Table 4, accordng o he ERR rankng, he wo lagged erms of sock reurn (percenage changes) are he mos sgnfcan erms o descrbe he sock reurns, because hese accoun for.% of he oal varance of he oupu. The oher four erms whch are also sgnfcan n affecng he changes n sock prce growh rae nclude growh raes n lagged ho money, neres raes, and real effecve exchange rae, and hey accoun for around.65% of sock reurn varance n ogeher. Ho money has posve effecs, whle neres rae and real effecve exchange rae have negave effecs. Oher macroeconomc varables, such as value-added ndusral oupu, broad money supply and nflaon have relavely less lnear effecs on sock reurns. The esmaon resuls n Table 4 may be wren o form he model SP (5.3) SP SP HM R REER 9 The auocorrelaon of he modellng resduals ndcaes ha he assumpon Equaon (5.3) has no been rejeced as here s no ouler ou he 95% confdence bounds, whch confrms he robusness of our seleced model. 5.3 Modellng usng he ARMAX echnque wh nonlnear erms The non-lnear effecs of varous macroeconomc varables on sock marke reurns are nex nvesgaed usng he ARMAX approach wh boh lnear and nonlnear erms of up o second degree. The canddae model erm se s gven by P C, SP, SP,, SP, SP, SP SP,, HM, REER HM (5.4) In general he second degree nonlnear erms of he nonlnear sock reurn reference model are produces over 0000 poenal model erms. Due o he vas number of erms, he reference model wll no be lsed here. The cu-off ERR value s chosen as 6 based on resdual auocorrelaon and normaly ess and he fnally seleced nonlnear sock marke reurn model s lsed n Table 5. The OFR algorhm and ERR have seleced jus he erms n Table 5 as he mos sgnfcan from he housands of poenal model erms. Ths shows he power of he ARMAX approach. Table 5 Seleced erms for non-lnear sock marke reurn model Rank Term ERR Parameer R R R VAI R7 REER 4 M SP 5 SP

15 The fnally seleced non-lnear model herefore s SP R 8 R 63.4R VAI 5 3.0M ( ) SP( ) 0.8 SR 777.8R 7 REER (5.5) From Equaon (5.5) and Table 5, produc of he 8-monh and -monh lagged neres raes explan he varances n sock reurn growh rae mos wh an ERR value as hgh as 4.30%. Oher nonlnear erms ncludng he produc of -monh lagged neres raes and 5-monh lagged growh rae of value added ndusral oupu, he produc of he 7-monh lagged neres rae and -monh lagged real effecve exchange raes, he produc of -monh lagged broad money growh and - monh lagged sock reurn, and -monh lagged sock reurn self so ha all ogeher hese erms accoun for 34.05% of he sock reurn varances. The resuls n he nonlnear modellng generally show ha neres raes self and he combnaon wh ndusral oupu or real effecve exchange rae have mos explanaory power and may accoun for around 43.86% of he varance of he sock reurns. Producs of broad money supply and pas sock reurns, and -monh lagged sock reurns have a sgnfcan mpac on sock reurns as well. The resuls show ha by akng nonlnear effecs no accoun, macroeconomc varables exhb much more explanaory power han hose suggesed by lnear modellng mehods, summng up o a hgh of 44% of he varances n sock reurns by he frs hree erms whch are purely combnaons of macroeconomc varables. 5.4 Ou-of-sample forecasng analyss Forecasng ably s a key creron for evaluang model performance. In hs par we assess he ouof-sample predcon ably of hree models usng he rollng samples for an 8-monh perod. Esmaed parameers n Equaon (5.), (5.3) and (5.5) are used o produce he one-sep-ahead predcon for he perod beween 009/0 o 00/6 and all sages re-occur on a monhly bass va a rollng wndow framework. Snce all hree models nvolve exernal varables he predcon of hese varables may mpac he back esng resuls. Therefore, he values for hese varables are assumed o be known and predcon s solely used o compare he sock reurn models. The predced sock reurns by boh lnear and nonlnear models and realzed sock reurns are drawn n Fgure. The squared errors of he predced lnear and nonlnear sock reurn models over he perod are hen drawn ogeher n Fgure 3. 5

16 Fgure A comparson of he predced and realzed sock reurns Fgure 3 Square errors of lnear and nonlnear predced sock reurns. As seen from Fgure 3 and 4, he squared errors n he predced ARMAX sock reurns are generally lower han hose n he VECM; whle predced sock reurns from he ARMAX wh lnear erms has smaller mean square errors han hose of he ARMAX wh nonlnear erms. The mean of he square errors for predcons from he VECM, ARMAX lnear and ARMAX nonlnear are , , and , respecvely. One of he reasons ha he ARMAX wh nonlnear erms underperforms n comparson wh ARMAX wh lnear erms may be ha here are rarely changes n shor-erm nomnal neres raes snce 009, namely he frs seleced model erm 8 R R s generally equal o zero mos of he me of he predcon perod. In he fuure 6

17 research alernave measures of neres raes shall be consdered o see wheher he predcon resuls are nfluenced by hs whch requres furher research. Generally, we conclude ha he ARMAX model wh lnear erms has he leas predcon errors and hence he bes ou-of-sample modellng ably. 6. Conclusons In hs paper we have examned he emprcal lnk beween sock reurns and economc fundamenals n Chna over he perod January 999 o June 00. The macroeconomc varables we consder nclude value added ndusral oupu, broad money supply, shor-erm nomnal neres raes, nflaon rae, real effecve exchange rae and ho money. An ou-of-he-sample es of predcably has been conduced. Overall, boh he VECM and ARMAX wh lnear erms esmaon resuls generally confrm ha he movemens n sock marke reurns may manly be explaned by s own nnovaons. However, macroeconomc condons sll conan useful nformaon for forecasng fuure sock reurns and canno be negleced n accounng for he dynamcs of sock marke reurns. Frsly, neres rae sands ou n erms of predcably across all hree models, and a negave relaonshp beween neres rae and sock marke reurns has been found n lnear models, whch s conssen wh man sream leraure, such as papers by Fama (98), Cozer and Rahman (988) and ajand and oronha (998). Secondly, real effecve exchange rae exhb sgnfcan negave effecs on sock reurns across lnear modellng approaches as well. I confrms he conclusons of Blson e al (00). Thrdly, mxed evdence for predcably of ho money s found: s he mos sgnfcan macroeconomc varable n he ARMAX modellng and exhb posve mpac on Chnese sock reurns n boh he VECM model and he ARMAX model wh lnear erms, bu has lle mpac n affecng he fuure varances of sock marke reurns based on he VECM approach. Fourhly, ambguous evdence on he nflaon s presened as well. In he VECM model, mpacs he sock reurns sgnfcanly bu wh a posve sgn, whch conradcs a negave relaonshp n Fama (98), bu s conssen wh some fndngs n Du (006). Fnally, alhough sock reurns exhb posve responses o shocks n broad money supply and ndusral oupu, hese wo macroeconomc varables have he leas explanaory power n predcng he fuure sock reurns. A modelng exercse usng ARMAX wh non-lnear erms does show ha here s sgnfcan nonlneary n sock reurn dynamcs condoned on macroeconomc nformaon. The resuls n he nonlnear modellng generally show ha he neres rae self and s combnaon wh ndusral 7

18 oupu or real effecve exchange rae have mos explanaory power and may accoun for around a half of he sock reurn varance. I has also been found ha ARMAX models wh boh lnear and non-lnear erms ouperform he VECM model n ou-of-sample predcve ably, whle ARMAX wh lnear erms s he bes predcng model. The VECM model has an mporan defcency n selecng explanaory varables and her lagged erms, and hence may face an over-fng problem and exhb he leas accuracy n ou-of-sample predcons. We also noe here are some neresng areas for fuure research. One advanage of he VECM model s may help us learn abou he dynamc srucure of sock reurns and macroeconomc varables, whle he ARMAX model s parcularly good a selecng he mos sgnfcan erms n modellng and hence avods he over-fng rap and provdes he much less forecasng errors n he medum erm. ARMAX also auomacally fnds he sgnfcan lnear and nonlnear relaonshp n he daa In he fuure, an emprcal modellng sraegy o combne co-negraon and error-correcon model, wh ARMAX models s wang o be worked ou. Appendx A: Orhogonal Leas Square Algorhm If he marx P n equaon (4) s assumed o be full rank and can be orhogonally decomposed as P WA (A.) where A s an MMun upper rangular marx and W s an M marx wh orhogonal T w w w as,,..., wh d w, w w w. columns,,..., M W W D dag dd d M The symbol.,. refers o he nner produc of wo vecors. Snce wj wk 0 where j 0,,..., where,,..., G g g g M square soluon, T g can be gven by k. Then he equaon (3.4) becomes w s orhogonal, Y PA A WG (A.) s an auxlary parameer vecor. Accordng o he orhogonal leas 8

19 gˆ Y, w (A.3) w, w Accordng o Korenberg, Bllngs and Lu (988), he OLS procedure for ARMAX polynomal model should be defned as 0 0 w p w p w,,..., M r r r0 p w r r r wr, 0 (A.4) and gˆ 0 y (A.5) gˆ y w w (A.6) Then he parameers where 0,,..., M can be calculaed as M ˆ gc ˆ (A.7) 0 where c m c c, m M m rm r r (A.8) Then he orhogonalzed form of model (4.4) can be wren as M (A.9) y g w Afer akng square of boh sdes and akng he me average, he model (A.9) can become 9

20 M y g w (A.0) The reducon n mean-square error can be acheved by ncludng a erm P x model (3.3) whch s equal o g w n he (A.) Therefore, he Error Reducon Error (ERR) can be defned from ERR g w y 00%,,,..., M (A.) Appendx B: Orhogonal Forward Regresson Algorhm The OFR (We, Bllngs and Lu, 004 ) s a modfed verson of he OLS and n he nal sage of OFR, all erms p where,,..., M Then a frs eraon of orhogonalzaon, ˆ,,..., M and he nal g and ERR can be obaned by gˆ w. are consdered as poenal canddaes for y w w w s assumed o equal o p, ERR g w y for all The erm wh maxmum ERR s hen seleced as he mos sgnfcan erm of he model. Then hs erm s removed from he canddae erms and n he second sage, all erms lef from sage one are consdered as he possble canddaes. Accordng o he equaon (B.) w p w, gˆ w w y, ERR g w y (B.) where 0

21 w p w (B.3) The second mos sgnfcan erm wll be chosen wh max ERR n he second sage and hs erm wll be removed from he canddae erms. The eraon connues unl he unexplanable varance of he sysem m ERR where m M. The fnal seleced orhogonalzed model s whch s equvalen o reaches he pre-se desred olerance m (B.4) y w g m (B.5) y p x T The parameers can be calculaed by equaon AG wh G g, g,..., g m and a a 0 a A 0 0 m m am, m (B.6) The srucure of he model can be obaned once he seleced model has passed he valdaon ess such as hgher order correlaon es nroduced by Bllngs and Voon (986). Reference: Abugr B. A., (008) Emprcal relaonshp beween macroeconomc volaly and sock reurns: evdence from Lan Amercan markes, Inernaonal Revew of Fnancal Analyss 7, Araujo E., (009). Macroeconomc shocks and he co-movemen of sock reurns n Lan Amerca, Emergng Marke Revew 0, Bllngs, S. A., and Coca, D., (00). Idenfcaon of ARMAX and Relaed Models. Reseach Repor. Deparmen of Auomac Conrol and Sysem Engneerng, Unversy of Sheffeld, 786. Bllngs, S. A., Korenberg, M. and Chen, S., (988). Idenfcaon of nonlnear oupu-affne sysems usng an orhogonal leas-squares algorhm. In. Journal of Sysems Scence 9, Bllngs, S. A., Chen, S., and Korenberg, M. J., (989). Indenfcaon of MIMO non-lnear sysems usng a forward regresson orghogonal esmaor. Inernaonal Journal of Conrol 49:

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