Modeling Stock Returns in the South African Stock Exchange: a Nonlinear Approach

Transcription

1 Modeling Sock Reurns in he Souh African Sock Exchange: a Nonlinear Approach Lumengo Bonga-Bonga and Michael Makakabule Deparmen of Economics, Universiy of Johannesburg, Souh Africa lbonga@uj.ac.za Tel: ; Fax: Absrac This paper invesigaes he relaionship beween sock reurns and macroeconomic variables, aking ino accoun asymmeric adjusmen behaviour in he sock marke. The sudy applies he Smooh Transiion Regression (STR) model o accoun for smooh asymmeric response of sock reurns from economic variables. The resuls show ha changes in dividend yield is an imporan facor in deermining he asymmeric behaviour of sock reurns on he Souh African sock marke. Furhermore, he forecas performance of he STR model is compared wih Ordinary Leas Square (OLS) and Random Walk models. The STR, as a nonlinear model, ouperforms he OLS and Random Walk models in an ou-of-sample forecas. The findings of he paper violae he weak and semi-srong form es of he efficien marke hypohesis. Keywords: sock reurns, smooh ransiion regression, forecas 1

2 1. Inroducion The debae surrounding he validiy of he Efficien Marke Hypohesis (EMH) is raging on. Cenral o he debae is he issue of wheher sock marke reurns are predicable or no. Advocaes of he EMH heory conend ha sock prices (reurns) incorporae all publicly available informaion, so ha an average invesor canno earn abnormal reurns based on his rading sraegy. Therefore, according o he heory, i is impossible o consisenly ouperform he marke by using any informaion ha he marke already knows. For Fama (1970), he predicion of fuure sock reurns based on movemens in macroeconomic variables is a fruiless exercise, since profimaximising agens will ensure ha all relevan informaion peraining o changes in macroeconomic variables are fully impounded ino curren reurns. According o he auhor, he implicaion is ha echnical analysis (which is a sudy of pas prices, chars, ec.) and fundamenal analysis (analysis of macroeconomic variables) would yield no beer performance han an invesor who adops a buy-and-hold sraegy (i.e. passive invesmen). The noion ha sock reurns are no predicable, as implied by EMH, has been grealy challenged by many academics and finance praciioners. A number of sudies have documened a robus predicable relaionship beween sock reurns and macroeconomic variables or he so-called marke anomalies (see Malkiel, 2003 for a comprehensive lieraure review of such anomalies). Macroeconomic variables such as inflaion raes, erm and defaul spread on bonds, aggregae oupu, money supply, exchange raes and unemploymen raes are found o have significan influence in explaining sock reurns (Rapach, Wohar & Rangvid, 2005). Pesaran and Timmerman (1995) provide evidence of predicable componens in sock reurns using macroeconomic variables such as ineres raes, dividend yields, economic growh (indusrial producion) and inflaion. The auhors find he exisence of a relaionship beween sock reurns and macroeconomic variables, even afer accouning for ransacion coss. They ascribe he exisence of predicabiliy in sock reurns o incomplee learning and he presence of ime-varying premia. In an aemp o examine wheher predicabiliy of sock reurns is aribued o ime variaion in expeced reurns, Schwer (1990) finds evidence of a srong relaionship beween sock reurns and macroeconomic variables, afer conrolling for imevarying risk premium and shocks. Hsieh (1991) shows ha sock reurns are no independenly and idenically disribued as assumed by EMH, and hus here exiss a possibiliy of characerising a nonlinear relaionship beween sock reurns and macroeconomic fundamenals. According o Summers and Schleifer (1990), nonlineariies in sock reurns could arise due o noise rading, long memory in sock reurns due o ime variaion in expeced reurns, and inernaional feedback effecs. Recen empirical sudies have found evidence of nonlineariies associaed wih sae dependence (i.e. regime swiches) in he relaionship beween sock reurns and macroeconomic variables, and ha such relaionship resembles asymmeric behaviour (McQueen & Roley, 1993). This view was suppored by Chang (2009), who found ha predicabiliy of sock reurns changes over ime, such ha he relaionship is sronger in bad imes (recession) han in good imes (economic boom). 2

3 In an aemp o beer describe nonlineariies due o regime changes in economic variables, mos sudies adoped he Markov regime-swiching models assuming nonlinear saionary process (see Hamilon, 1989) and he Threshold Auoregressive Model (TAR) (see Tong, 1990). In suppor of evidence of nonlineariies associaed wih regime-swiching process, Moolman (2004) made a significan conribuion in his field of research from he emerging marke perspecive in general and Souh Africa in paricular by invesigaing he relaionship beween sock reurns and macroeconomic variables. Using he Markov regime-swiching model, he auhor found evidence ha sock reurns on he Johannesburg Securiies Exchange (JSE) depends on he sae of he business cycle. Noneheless, in challenging he applicaion of he Markov swiching and TAR models in modelling sock reurns, Saranis (2001) argues ha hese models may be successful in capuring nonlineariy beween variables, bu in he conex of modelling sock reurns hese models are oo resricive in ha hey assume a sharp regime swich. For Saranis (2001), he use of Smooh Transiion Regression Models (STR) is appropriae, as he changes in regime are smooh, raher han abrup, in he sock markes. In addiion, Aslanidis e al. (2002) conends ha he STR model is more appealing and in line wih economic heory, in he sense ha economic agens reac differenly o changes in economic variables. As a resul, he degree o which such agens adjus o differen regimes is gradual, raher han insananeous or abrup, such as claimed by he Markov swiching and TAR models. In he ligh of he abovemenioned sudies, i should be eviden ha sock reurns can be prediced from macroeconomic variables, if a nonlinear model specificaion is used o accoun for asymmeric behaviour presen in he sock exchange marke. The predicabiliy of sock reurns should presen a challenge o he EMH. The main objecive of his paper is o examine he relaionship beween sock reurns and macroeconomic variables in Souh Africa, wih emphasis on smooh ransiion o capure nonlineariy in he sock exchange marke. The resuls of his paper will inform on he degree of he speed wih which he Souh African sock marke changes from one regime (bull marke) o anoher (bear marke). Furhermore, he ou-ofsample forecasing performance of he STR model is compared wih he simple linear (OLS) and random walk models. In doing so, he implicaions for marke efficiency will hen be assessed. The paper is srucured as follows: Secion 2 briefly summarises some of he lieraure concerning he relaionship beween sock reurns and macroeconomic variables. Secion 3 oulines he mehodology o be used in he paper. Secion 4 presens he daa and empirical resuls. Secion 5 concludes he paper and provides areas for fuure research. 2. Lieraure review A number of sudies were conduced o assess he deerminans of sock reurns. Fama (1970) was he firs o inroduce he asse pricing model based on he EMH heory. The auhor examined he behaviour of daily changes over a seleced 30 socks of he Dow Jones Indusrial Average for he period 1957 o 1962, and found consisen 3

4 evidence of serial posiive and negaive dependence on he daily changes in sock reurns. An imporan implicaion of he EMH is ha sock prices should follow a random walk in ha fuure price changes are random, and hus unpredicable (Mishkin and Eakins, 1998). The random walk hypohesis is relaed o he weak form of he efficien marke hypohesis, in ha he curren sock price already incorporaes all he informaion of he pas sock prices. The consequence of he EMH is ha no srucural model for sock reurn deerminaion can ouperform he random walk model. Noneheless, a number of sudies have rejeced he principle of EMH, whereby curren sock prices fully reflec all securiy marke informaion, in favour of srucural models (Li & Lam, 1995). These sudies argue ha here exiss a relaionship beween sock reurns and macroeconomic variables, and ha his relaionship is ime varying and canno be capured by convenional or radiional linear frameworks. In oher words, hese sudies asser ha once a proper funcional mehod is used o accoun for he relaionship beween sock reurns and macroeconomic variables, srucural models can ouperform he random walk model, and hus well-informed invesors can realise excess reurns. The lieraure analysing he relaionship beween sock reurns and macroeconomic variables using a nonlinear framework, has been in exisence for some ime. For example, Bredin, Hyde and O Reilly (2008) esed he forecasing abiliy of he STR model, using sock marke indices of six developed economies, i.e. Unied Saes, Unied Kingdom, Germany, Canada, France and Japan. For each counry, he auhors used he world index, changes in ineres raes, dividend yield, inflaion, exchange rae, indusrial producion and changes in oil prices o explain sock reurns. The resuls of heir sudy show ha he STR model ouperforms he linear model in explaining sock reurns. In addiion o he mulivariae (STR) sudies, a number of sudies have used he univariae smooh ransiion auoregressive (STAR) model o prove a nonlinear adjusmen of sock reurns. For example, Bradley and Jansen (2004) found ha he univariae linear model ouperforms he nonlinear model (STAR) in modelling sock reurns. As far as he Souh African lieraure is concerned, he economeric analysis of he relaionship beween sock reurns and macroeconomic variables is very limied, and confined o linear models. For example, Van Rensburg (2000) examines he impac of macroeconomic variables on he JSE sock reurns using he Arbirage Pricing Theory (APT) over he period January 1980 o December Wih he aid of he vecor auoregressive (VAR) echnique, he auhor found ha sock reurns on he Johannesburg Sock Exchange (JSE) are driven mainly by resource and indusrial secors in Souh Africa. Jefferis and Okeahalam (2000) used he co-inegraion echnique o examine he relaionship beween sock reurns and macroeconomic variables in Souh Africa, Boswana and Zimbabwe for he period 1985 o The auhors found ha sock reurns in hese counries were driven by real exchange rae, long-erm ineres raes and GDP. 4

5 Moolman (2004) applies a Markov regime-swiching model o assess he relaionship beween sock reurns and macroeconomic variables in Souh Africa. The auhor finds ha he degree o which sock reurns depend on macroeconomic variables, depends on he sae of he business cycle in Souh Africa. 3. Economeric mehodology Smooh ransiion models are receiving much aenion in he finance lieraure. According o Teräsvira (2003), he smooh ransiion model is essenially an exension of swiching regression model and can eiher be univariae or mulivariae. The univariae version is referred o as Smooh Transiion Auoregressive Model (STAR) whereas he Smooh Transiion Regression (STR) involves a mulivariae analysis. Teräsvira (1994:209) defines an STR model as a combinaion of he hreshold auoregressive model and he exponenial auoregressive model and can be expressed by he following equaion: y ' z ' z G(, c, s ) u, (1) Where G (, c, s ) is he ransiion funcion, θ = (θ 0, θ 1, θ p ) and = ( 0, 1 p ) are parameer vecors, z is he vecor of explanaory variables, c denoe he hreshold variable, and γ is he slope of he ransiion funcion. Anderson and Teräsvira (1992) disinguish beween wo forms of STR ha allow for ime varying in auoregressive decay. One form of STR model is known as Logisic STR (LSTR), which can be expressed as follows: K 1 G (, c, s ) 1 exp{ ( s c )}, 0 (2) k 1 k In erms of he above equaion, he ransiion variable increases in andem wih he logisic funcion. Teräsvira, Van Dijk, and Franses (2002:4) demonsrae ha as γ zero or, he ransiion funcion becomes abrup, such ha he model becomes an AR; in oher words, he STR model becomes indisinguishable from he linear (AR) model. Anoher form of STR model is known as he Exponenial Smooh Transiion Model (ESTR), and can be defined as follows: 2 GE (, c, s ) 1 exp{ ( s c 1 ), 0 (3) ESTR is a non-monoonous ransiion funcion and is ideal in cases where he dynamic behaviour of a process is similar in boh upswing and downswing (Teräsvira, 2003:224). According o Saranis (2001:461), he ESTR model suggess ha while he behaviour of economic variables in he ransiion period can differ, he regimes will sill have similar characerisics, and, as a resul, boh ESTR and LSTR models have he capabiliies of explaining asymmery in sock prices. Unlike he TAR and Markov swiching models, The STR model does no require a prior assumpion of abrup swiching beween regimes, bu raher allows he daa o 5

6 dicae wheher he regime change is abrup or smooh (Tong, 1990). I is his characerisic ha makes he STR models more appealing in heir applicaion in sock markes, simply because such markes are characerised by a large number of paricipans, i.e. raders, speculaors, analyss ec., and such paricipans reac differenly o economic news or public informaion (Saranis, 2001:460). Teräsvira (1994:210) proposes procedures in building an STR model. These include lineariy es, esimaion and evaluaion of he model. A lineariy es is performed for he purpose of selecing an appropriae ransiion variable. In choosing he ransiion variable, he modeller should be guided by economic heory. Terasvira (2003:227) suggess a lineariy es for each candidae ransiion variable. In erms of his approach, he variable wih he lowes p-value (sronges rejecion of lineariy) is chosen as he ransiion variable 1. Luukkonen, Saikkonen and Teräsvira (1988:493) argue ha esing for lineariy is no a sraighforward exercise, due o he fac ha he model is only idenified under he alernaive hypohesis. As a resul of idenificaion problem, he normal es procedures such as he Likelihood raio, he Lagrange Muliple and he Wald Tes will produce undesirable esimaions of parameers. Insead, Luukonen e al. (1988: 493) sugges ha one should approximae he alernaive model by adoping a Taylor series expansion of he ransiion funcion as a means o circumven he idenificaion problem. The Taylor expansion funcion is mahemaically expressed as follows: 3 y z ~ j 0 j z s u, 1,... T (4) j 1 β 0 and β j are he dimension column vecors of parameers. The null hypohesis of lineariy is H 0 : β 1 = β 2 = β 3 = 0, i.e. boh parameers are joinly esed for zero agains he alernaive hypohesis. The hypohesis is carried ou using he LM es, and he F- es is used insead of χ 2 disribuion. The ransiion funcion is derived from he auxiliary regression as shown in (4). The following ess mus be performed o discriminae beween LSTR and ESTR, i.e. o choose an appropriae STR model. (i) Tes of he null hypohesis H04: β 3 = 0 (ii) Tes of he null hypohesis H03: β 2 = 0/ β 3 = 0 (iii) Tes of he null hypohesis H02: β 1 = 0/ β 2 = β 3 = 0 The above hypoheses are esed using he F-es. The decision rule is ha he LSTR is chosen if he p-value of H04 or H02 is highly significan. Conversely, he ESTR is seleced if he p-value of H03 is highly significan. Should i happen ha he es fails o provide a clear-cu choice beween he wo opions, i is recommended o fi boh models and decide on he appropriae one a an evaluaion sage (Teräsvira, 2003:227). The chosen model can hen be esimaed and evaluaed as oulined in Eirheim and Teräsvira (1996:60), i.e. es of no remaining nonlineariy, no auocorrelaion and parameer consancy. 1 JMuli sofware ( auomaically selec he ransiion variable; however one sill has o examine wheher he variable chosen is sensible or in line wih economic heory. 6

7 4. Daa and empirical resuls 4.1. Daa In order o assess he relaionship beween sock reurns and macroeconomic variables, his sudy makes use of he following variables exraced from I-Ne Bridge Daabase, and Bloomberg: he JSE ALL Share Index reurn (ALSI), he ALSI dividend yield (DY), Rand/Dollar R/$ exchange rae (RAND), he FTSE Index (FTSE) and S&P 500 Index (SP500). The use of he FTSE and S&P 500 indices aims a capuring he posiive relaionship ha exiss beween he domesic sock exchange and he foreign exchange marke, especially he London and New York sock exchanges (Samouilham, 2006). The daa consis of weekly observaions from May 1988 o December High frequency daa, such as weekly daa, are essenial o capure he nonlinear relaionship ha exiss in he daa (McCulloch & Tsay, 2001). The paper esimaes he relaionship beween he JSE sock reurns and macroeconomic variables, wih he use of he STR mehod o assess he degree of regime changes in he Souh African exchange marke. Furhermore, he forecasing performance of he STR mehod is compared wih he random walk and linear mehods. 4.2 Empirical Resuls Table 1 in he Appendix presens he resuls of he uni roo ess for he variables used in he paper. The resuls show ha he null hypohesis of uni roo is rejeced. This implies ha all variables are nonsaionary. Furhermore, he resuls in Table 1 sugges ha all he variables are saionary a firs difference, I (1), as he null hypohesis of uni roo is rejeced a 1% level for all he variables a firs difference. To es he number of coinegraion relaions, an iniial vecor auoregressive model (VAR) was se up, wih a lag lengh of he VAR process, p 4, seleced according o he Hannan-Quinn (HQ) informaion crieria. The LM-es, no repored here, indicaed ha here is no serial correlaion in he VAR residual when he lag lengh of 4 is seleced. The resuls of he race and Max-eigenvalue ess of coinegraion, repored in Table 2, indicae he presence of one coinegraing relaion or rank r 1. Table 3 presens he resuls of he OLS esimaion when ALSI is endogenised. Though he resuls indicae ha all coefficiens are saisically significan, he linear model fails a number of specificaion ess, such as he presence of serial correlaion, represened by he Durbin-Wason saisics, as well as he CUSUM es which denoes ha he coefficiens are no sable, as shown in Figure 1. As a linear model fails o model sock reurns adequaely from macroeconomic variables, he nex sep in he paper consiss in esing wheher a nonlinear model will be appropriae for his model. Table 4 displays he resuls of he lineariy es. All he variables were seleced as candidae ransiion variables. As repored in Table 4, he resuls of he F-es show ha he null hypohesis of lineariy is rejeced for all he variables a 1% level of significance. The rejecion of he null hypohesis is highly 7

8 significan for DY. Moreover, An LSTR(1) model is implied by he lineariy es, given ha he null hypohesis H02 is highly significan. The fac ha dividend yield is seleced as a ransiion variable, is heoreically appealing. Exensive research exiss which demonsraes he nonlinear relaionship beween sock reurns and dividend yields. For example, McMillan (2004) adoped he Exponenial Smooh Transiion Regression models (ESTR) o demonsrae he abiliy of dividend yields in explaining he asymmeric behaviour of UK sock reurns. Gombola and Liu (1993) found he exisence of a negaive relaionship beween sock reurns and dividend yields during a bull marke. Conversely, he relaionship beween sock reurns and dividend yields was found o be posiive in a bear marke. The auhors aribue such behaviour o he so-called differenial yield effec. The resuls repored in Table 5 show ha he coefficiens of he explanaory variables are saisically significan in he linear and nonlinear par of he LSTR(1) esimaion. The resuls repored in Table 5 are repored in equaion form as follows: Alsi DY 0.382FTSE 0.145Rand 0.29SP500 (5) DY 0.428FTSE 0.139Rand 0.267SP500 1 exp( DY ) 1 The resuls repored in Expression 5 show a high value of he slope parameer, This resul is confirmed in Figure 2, ha shows a rapid ransiion beween he wo exreme regimes. The resuls sugges ha reurns in he Souh African sock marke are characerised by asymmeric cycles wih a relaively high degree of ransiion beween regimes deermined by he dividend raio, DY. In as far as regime change is concerned, he resuls indicae ha if he ransiion funcion, G, c, DY, moves oward one, ha is lim, he magniude of he posiive effec of he DY devaluaion of he rand on sock reurn is lower, compared o he case where lim DY 0. I is imporan o noe ha lim DY signals a bear marke wih decreasing price of socks, and lim DY 0 should indicae a regime relaed o a bull marke, characerised by increasing sock prices. Figure 2 shows ha mos observaions of he dividend yields are siuaed in he bear marke, during he period of he analysis. On assessing he imporance of a nonlinear mehod in modelling sock reurns, forecasing performance of linear, nonlinear and Random Walk models are compared. Table 6 presens he resuls of he ou-of-sample roo mean square error (RMSE) as a crierion for forecasing performance of he hree models. I is eviden from hese resuls ha he LSTR (1) ouperforms all he ohers models in an ou-of-sample forecas, in erms of he RMSE crierion. This suggess he imporance of a nonlinear mehodology in modelling sock reurns in Souh Africa. 8

9 5. Conclusion This paper aimed a assessing he imporance of a nonlinear model in assessing he relaionship beween sock reurns and macroeconomic variables in Souh Africa. The paper shows he imporance of he nonlinear model ha focuses on smooh ransiion beween regimes in explaining sock reurns. The smooh ransiion regression model is used oward his end. The resuls of his paper show ha he STR model is appropriae for modelling sock reurns in Souh Africa. Furhermore, he resuls of he STR model show ha he magniudes of some macroeconomic variables, in explaining sock reurns, varies according o regimes. These regimes are deermined by he size of dividend yield. The superioriy of he STR model, as a nonlinear model, over he compeing models, i.e. OLS and random walk in an ou-of-sample forecas, is confirmed when crieria such as he RMSE is used for assessing forecas performance. REFERENCES Anderson H., and Teräsvira, T. (1992). Characerizing nonlineariies in business cycles using Smooh Transiion Auoregressive Models. Journal of Applied Economerics, 7: Aslanidis, N., Osborn, D. and Sensier, M. (2002). Smooh Transiion Regression Models in UK Sock Reurns. Royal Economic Sociaey Annual Conference, Paper No.11 Bradley, M. & Jansen, D. (2004). Forecasing wih a nonlinear dynamic model of sock reurns and indusrial producion. Inernaional Journal of Forecasing, 20: Bredin, D., Hyde, S. & O Reilly, G. (2008). Regime changes in he relaionship beween sock reurns and he macroeconomy. Briish Accouning Associaion Annual Conference, pp Chang, K. (2009). Do macroeconomic variables have regime-dependan effecs on sock reurn dynamics? Evidence from he Markov regime swiching model. Economic Modelling, 26:1-17. Eirheim, Ø. & Teräsvira, T. (1996). Tesing he adequacy of smooh ransiion auoregressive models. Journal of Economerics, 7: Fama, E. (1970). Efficien Capial Markes: A Review of Empirical Work. Journal of Finance, 25: Gombola, J. & Liu, F. (1993). Dividend yields and sock reurns: Evidence of ime variaion beween bull and bear markes. The Financial Review, 28(3):

10 Hamilon, J. (1989). A New Approach o he Economic Analysis of Nonsaionary Time Series and he Business Cycle. Economerica, 57(2): Hsieh, D. (1991). Chaos and Nonlinear Dynamics: Applicaion o Financial Markes. Journal of Finance, 46 (5): Jefferis, K. & Okeahalam, C. (2000). The impac of economic fundamenals on sock markes in Souhern Africa. Developmen Souhern Africa, 17(1): Li, K. & Lam, K. (1995). Modelling Asymmery in Sock Reurns by a Threshold Auoregressive Condiional Heeroscedasic Model. The Saisician, 44(3): Luukkonen, R., Saikkonen, P. & Teräsvira, T. (1988). Tesing Lineariy agains Smooh Transiion Auoregressive Models. Biomerika, 75(3): Malkiel, B.G. (2003). The Efficien Marke Hypohesis and Is Criics. Journal of Economic Perspecives, 17(1): McCulloch, R.E. & Tsay, R.S. (2001). Nonlineariy in High-Frequency Financial Daa and Hierarchical Models. Sudies in Nonlinear Dynamics & Economerics, 5(1):1-17. McMillan, D.G. (2004). Nonlinear predicabiliy of shor run deviaions in UK sock marke reurns. Economics Leers, 84(2): McQueen, G. & Roley, V. (1993). Sock prices, news and business condiions. Review of Financial Sudies, 6: Mishkin, F.S. & Eakins, S.G. (1998). Financial Markes and Insiuions. Addison- Wesley. Moolman, H. (2004). An asymmeric economeric model of he Souh African sock marke, Docoral hesis. Preoria: Universiy of Preoria. Pesaran, M. H., & Timmermann, A. (1995). Predicabiliy of Sock Reurns: Robusness and Economic Significance. Journal of Finance, 50: Rapach, D.E., Wohar, M.E., & Rangvid, J. (2005). Macro variables and inernaional sock reurn predicabiliy. Inernaional Journal of Forecasing, 21: Samouilham, N.L. (2006). The Relaionship beween Inernaional Equiy Marke Behaviour and he JSE. Souh African Journal of Economics, 74(2): Saranis, N. (2001). Nonlineariies, cyclical behaviour and predicabiliy in sock markes: inernaional evidence. Inernaional Journal of Forecasing, 17: Schwer, G.W. (1990). Sock reurns and real aciviy: A cenury of evidence. Journal of Finance, 45:

11 Summers, H. & Schleifer, A. (1990). The Noise Trader Approach o Finance. The Journal of Economic Perspecives, 4(2): Teräsvira, T. (1994). Specificaion, Esimaion and Evaluaion of Smooh Transiion Auoregressive Models. Journal of American Saisical Associaion, 89(425): Teräsvira, T. (2003). Smooh Transiion Regression Modeling, in Lukepohl, H. and Krazig, M. (2003). Applied Time Series Economerics, Cambridge Universiy Press, pp Teräsvira, T. Van Dijk, D. and Franses H. (2002). Smooh Transiion Auoregressive Models: A Survey of Recen Developmens. Economeric Reviews 21: Tong, H. (1990). Non-linear Time Series: A Dynamical Sysem Approach. Oxford: Oxford Universiy Press. (Oxford Saisical Science Series,6). Van Rensburg, P. (2000). Macroeconomic variables and he cross-secion of JSE reurns. Souh African Journal of Business Managemen, 31(1): APPENDICES Table 1 Uni Roo Tes (Augmened Dickey-Fuller Tes) Variable ADF Tes Saisic (Level) ADF Tes Saisic (Firs Differences) ALSI S&P FTSE RAND DY *The criical values for he augmened Dickey-Fuller es saisic are , and for 1%, 5% and 10% respecively. Table 2 Johansen Co-inegraion Tes A. Trace Tes Null Hypohesis (Trace Tes) Saisic Criical Value (5%) Prob None * A mos A mos A mos A mos

12 B. Maximum Eigenvalue Tes Null hypohesis Saisic Criical Value (5%) Prob None * A mos A mos A mos A mos *denoes rejecion of he hypohesis a he 5% level Table 3 Linear Model Esimaion of sock reurn (ALSI) Variable Coefficien C * DY -4.70* FTSE * RAND 21.57* SP * Durbin-Wason *Denoes saisically significan a 1% level of significance. Sandard errors are correced using he Newes-Wes Heeroscedasiciy and Auocorrelaion Consisen Variance (HAC) Figure 1 Tes for sabiliy of Coefficien: CUSUM es _ Table 4 P-Value of he lineariy es Hypohesis P-Value DY FTSE Rand S&P 500 F-saisic H H H The p-value of he es of H03 is much larger han he corresponding H02 and H04 for DY; herefore he null hypohesis of lineariy is rejeced and LSTR (1) model is chosen. 12

13 Table 5: Esimaed STR model variable esimae SD -sa p-value _ Linear Par CONST DY FTSE Rand SP Nonlinear Par CONST DY FTSE Rand SP Gamma C R 2 : adjused R 2 : variance of ransiion variable SD of ransiion variable variance of residuals: SD of residuals: Table 6: Ou-of-sample forecasing resuls Model RMSE STR Linear Random Walk

14 Figure 2 Transiion Funcion 14