Returns and interest rate: A nonlinear relationship in the Bogotá stock market

Transcription

1 Reurns and ineres rae: A nonlinear relaionship in he Bogoá sock marke Luis Eduardo Arango, Andrés González, and Carlos Eseban Posada * Banco de la República Summary This work presens some evidence of he nonlinear and inverse relaionship beween he share prices on he Bogoá sock marke and he ineres rae as measured by he inerbank loan ineres rae, which is o some exen affeced by moneary policy. The model capures he sylised fac on his marke of high dependence of reurns in shor periods of ime. These findings do no suppor any efficiency on he main sock marke in Colombia. Evidence of a non consan equiy premium is also found. The work uses daily daa from January 994 up o February JEL classificaion: C22, C52. Key words: nonlineariies, sock reurns, ineres rae, smooh ransiion regression, GARCH models. * The opinions expressed here are hose of he auhors and no of he Banco de la República, he Colombian Cenral Bank, nor of is Board. We hank Luis F. Melo for his commens and suggesions alhough any remaining errors are solely ours.

2 I. Inroducion Sock prices is one of he mos acive areas of economic research. Some models aimed o he sudy of asse prices have been developed. Thus, we have he exac presen-value formula, which considers he marke value of an asse as being deermined by he discoun presen value of he expeced dividend paymens. According o his view, he raional price of an asse will be less he higher he discoun rae. The Gordon growh model, which is perhaps he simples fundamenal-based mechanism o predicing sock prices, is an especial case where dividends are expeced o grow a a consan rae (see Heaon and Lucas, 999, for a recen applicaion of his model). The ineremporal equilibrium models of asse pricing build on he assumpion ha people rade asses wih he purpose of smoohing consumpion over ime. The resul of his is conained by he Euler equaion, where he marginal benefi of no consuming one more real dollar oday is equaed o he marginal benefi from invesing he dollar in any asse and selling i in order o consume he proceeds in he fuure (see Campbell, 999). Accordingly, his condiion describes he opimum. When he Euler equaion conains an sochasic discoun facor, he consumpion-based capial asse pricing model arises. These kind of ineremporal models can deal wih quesions abou he forces ha deermine he rae of reurn of he zero-bea asse (he riskless ineres rae) and he rewards ha invesors require for bearing risk (for a discussion a lengh of hese models see Campbell, 999). The Lucas asse-pricing model, as anoher example, considers consumpion as an exogenous variable and equal o oupu in equilibrium. In his case, he firs order condiions can be used o price asses so ha hese are funcion of consumpion (Lucas, 978). In his connecion, however, here are some poins of debae. Such poins focus on he driving forces or fundamenals ha deermine share prices, he channels hrough which asse prices affec economic aciviy, he informaion conained in hese prices abou fuure economic aciviy, he disincion beween large fundamenal and speculaive swings in asse prices and he response of policymakers o face hem. In his paper we explore he empirical link beween he asse prices on he Bogoá sock marke and he Colombian inerbank loan ineres rae (TIB). The obecive of his work is o describe he dynamics of he reurns and he way in which he TIB affecs he behaviour of he Bogoá sock index (IBB). Having a descripion of he dynamics is a good saring poin which could be easily improved o forecas reurns and elaborae a rading sraegy. The work is carried ou by using daily daa from January 994 up o February This frequency is usified given he velociy of boh he money and he sock markes o adus. The paper is organised as follows. Secion II shows some sylised facs and properies of IBB and TIB during he sample period. Secion III explains he empirical Some relaionship beween fundamenals and sock prices are associaed wih changes in corporae earnings growh, in consumer preferences, and in sock-marke paricipaion paerns (Heaon and Lucas, 999). As a source of non fundamenal variaions of sock prices are he acions of agens wih privileged informaion, poor regulaion mechanisms and imperfec raionaliy of invesors (Bernanke and Gerler, 999).

3 2 approach and shows he main findings and resuls. Finally, secion IV provides some conclusions. II. Properies of IBB Following Campbell e al. (997), he simple ne rae of reurn, beween daes - and is defined as: R, on he asse R = P P where P denoes he IBB a dae. Figure shows he IBB during he sample period. From his picure we observe ha he slope of any linear rend of he IBB, while posiive, would look raher small 2. Figure 2 presens he behaviour of he inerbank loan ineres rae (TIB) accompanied by he band of no inervenion of he cenral bank. When he TIB is on risk of crossing he band borders, cenral bank inervenes by couneracing he movemen dicaed by he marke, hus avoiding an inconvenien level of he ineres rae 3. This inervenion occurs hrough he REPO marke, in such a way ha TIB reflecs moneary policy somehow. Figure. Behaviour of he Bogoá sock index (IBB) //94 6/7/94 6//95 6/7/95 6//96 6/7/96 6//97 6/7/97 6//98 6/7/98 6//99 6/7/99 6//00 Source: Bogoá Sock Marke 2 I is clear ha a moving rend would be more sensible. 3 The higher levels of he TIB, however, occurred when he very cenral bank was acing in defence of he previous exchange rae band.

4 3 Figure 2. Inerbank loan ineres rae (TIB) and he inervenion band 80% 60% 40% 20% 0% 6/4/95 6/8/95 6/2/95 6/4/96 6/8/96 6/2/96 6/4/97 6/8/97 6/2/97 6/4/98 6/8/98 6/2/98 6/4/99 6/8/99 Source: Banco de la República Figure 3 shows he oin behaviour of reurns, R, and he firs difference of he TIB, TIB. The mean of he daily rae of reurn during he sample period is abou 0.029% while he sandard deviaion is.6%. Volailiy of reurns has been increasing, since beween January 994 and December 996 he deviaion was 0.99%, and.29% beween January 997 and February Figure 3. Reurns from he Bogoá sock index and he firs difference of he inerbank loan ineres rae (TIB) 35% 5% 25% 0% 5% 5% 5% -5% 0% -5% -5% -25% -0% -35% -5% 6//94 6//95 6//96 6//97 6//98 6//99 6//00 DTIB Reurns

5 4 Table shows he paricipaion of he upwards and downwards daily movemens of he IBB, when he variaion one, wo and hree days before had he same direcion. Mos of he index changes, alhough slighly, go downwards. However, on he mean-of-reurn basis, he upwards changes are higher han he downwards ones 4. A sriking propery is he auocorrelaion exhibied by he simple ne rae of reurn on he Bogoá sock marke, which may also be inerpreed as cuasiprobabiliies. For example, 67.7% (=3.2%+36.5%) of he movemens of he index repea he direcion of he change happened he day before. When we consider he changes of wo days before, he direcion of he change repeas 47.8% (=2.4%+25.9%) of he ime. Finally, when he changes of hree days before are considered, he direcion of he change repeas 33.6% of he ime 5. I follows ha financial asse reurns canno be considered independenly disribued over ime. Table. Percenage of he upwards and downwards variaions on he Bogoá sock index (IBB) having ino accoun he variaion of he one, wo, and hree days before Today s change Upwards Downwards No change or any Movemen oher combinaion of changes 47.2% 52.4% 0.4% Las day Upwards 3.2% Change Downwards 36.5% 32.3% Change of one and wo days Upwards 2.4% before Downwards 25.9% 52.7% Change of one, wo and hree Upwards 5.3% days before Downwards 8.3% 66.4% The saisical propery of posiive auocorrelaion of simple ne rae of reurn suggess ha here is a high proporion of invesors ha behave as chariss raher han fundamenaliss (for a recen applicaion of his see De Grauwe e al., 993), so ha he reurn process could be eiher a maringale or a fads process, somehing ha we explore nex. III. Empirical issues Under convenional procedures, we find evidence ha suggess ha boh IBB and TIB are nonsaionary processes. Given his resul, we nex move o check for any common 4 No especial paern is presen on a day-by-day basis. 5 As discussed by Fama (970) any es for marke efficiency involves a oin hypohesis abou he raionaliy of markes and he equilibrium expeced rae of reurn. The efficien marke view of he sock markes saes ha he reurn on socks is unforecasable and ha all informaion abou fuure prices is efficienly incorporaed ino he curren price (see MacDonald and Taylor, 99). Thus, if markes are efficien, he realised reurns are expeced o be serially uncorrelaed, a propery no exhibied by he reurns on he Bogoá sock marke.

6 5 rend beween he variables. However, by using he Johansen s procedure we do no fail o reec he hypohesis of a common rend beween hese wo variables 6. Wih hese preliminary resuls, apar from he nonlineariy beween share prices and ineres rae suggesed by he heory we also check he hypohesis of a negaive (inverse) relaionship beween hese wo variables (Appendix ses ou he heoreical model used o illusrae he relaionship beween he variables). On he oher hand, considering boh Figure 3 and he scaerplo in Figure 4, where he reurn of day appears agains he reurn of day, he Bogoá sock marke seems o conform he sylised fac ha relaively volaile periods characerised by large reurns alernae wih quie periods of small reurns. In oher words, in he Bogoá sock marke large price changes occur in clusers of ouliers which have been associaed o he nonlinear fashion in which volailiy series evolves over ime (Cao and Tsay, 992) 7. This gives us anoher argumen o look for nonlineariies in he series of reurns iself. Figure 4. Scaerplo of he reurn of he Bogoá sock index (IBB) on day agains he reurn on day 2,0 8,0 R() 4,0 0,0-4,0-8,0-8,0-4,0 0,0 4,0 8,0 2,0 R(-) 6 Only when a small number of lags is considered, he variables are coinegraed, bu he residuals are auocorrelaed and non normal, differen from wha is required by MLE. In addiion, he exclusion es suggess ha wih a consan in he coinegraing equaion TIB is no required. Given his resul, weak exogeneiy was checked by using he Granger-causaliy es, where we found some evidence on ha TIB does no Granger-cause R, while he reverse is differen. 7 Any evidence of nonlineariy in financial ime series would sugges ha some forecasing improving could be obained in he shor erm by changing linear sraegy for he nonlinear one. However, as noed by Brooks (996), evidence of nonlinear dependence in marke reurns could cas some doubs on he informaional efficiency of financial markes, since i may possible o obain a rading rule o generae posiive reurns wih a probabiliy higher he 0.5. Noneheless, we recall ha our work focuses on he reurn dynamics and he way in which ineres rae affecs sock reurns.

7 6 In his work we use he smooh ransiion regression (STR) approach o model he behaviour of he sock prices as he alernaive o he linear sraegy (Granger and Teräsvira, 993; Teräsvira, 998; Sarno, 999; Franses and van Dik, 999; van Dik, e al., 2000). A version of his ype of models can be wrien as: x = )) z + u β (4.) w + ( π + π 2F( si, ; α where x is he dependen variable, he reurn on he Bogoá sock marke in his case; w = w,..., w ) is a vecor of K regressors which ener linearly wih consan ( K parameer vecor β, z = z,..., z ) is a vecor of M regressors, s = s,..., s ) ( M ( L is a ( L ) vecor of regressors whose elemens may include hose of w and z, and 2 u is an iid error process, wih E ( u ) = 0, and Var ( u ) = σ. F is a ransiion funcion bounded by 0 and whose parameers are denoed by α. I is assumed ha E ( w u ) = 0, E ( z u ) = 0, E ( s u ) = 0. Some lagged elemens of x may be included in w and z alhough weak exogeneiy of he remaining elemens of hem wih respec o he parameers of ineres in (4.) is required. Weak saionariy of x, w, and z processes is assumed. Noice ha when he ransiion funcion F = 0, he STR model (4.) will be linear. However, he vecor of regressor coefficiens, π + π 2F( s i, ; α), will depend, in general, on he values of he ransiion variable s. The ransiion funcion can ake he logisic form, in which case we have a logisic STR (LSTR) model: F( s ; α) = ( + exp{ γ ( s c)}), γ > 0 (4.2) or as an exponenial funcion, where we have an exponenial STR (ESTR) model as: F( s 2 ; α) = exp{ γ ( s c) }, γ > 0 (4.3) where he slope parameer γ sands for he speed of he ransiion and c for he hreshold, ha is where he ransiion occurs. ( 0 : γ = 0 The null hypohesis o check is ha of lineariy H ). However, (4.) eiher wih (4.2) or (4.3) is only idenified under he alernaive ( H : γ > 0), a fac ha invalidaes he asympoic disribuion heory. This problem has been overcome hrough

8 7 he auxiliary regression (Granger and Teräsvira, 993; Teräsvira, 994; Teräsvira, 998): x = w + λ z + λ z s + λ z s + λ z s + v β (4.4) 0 where he null hypohesis of lineariy becomes H : λ = λ = λ 0), wih power ( = agains boh he LSTR and ESTR. The choice beween LSTR and ESTR can also be done using equaion (4.4), following he sequence proposed by Teräsvira (998). The approach requires firs he esimaion of a linear model (Table 2). By looking a he resuls, i is remarkable he significance of he coefficien of TIB as well as is sign. Thus, a his sage and given he way in which we have calculaed R and TIB we have evidence of a negaive and nonlinear relaionship beween P and ineres rae given by: P = P φ P TIB TIB ). ( 2 Table 2. Benchmark linear model Dependen variable: R Variable Coefficien Sandard - saisic p Value error Consan R R R TIB R Mean of dependen variable Sandard deviaion of.4997 Sandard error of regression.030 dependen variable N 482 The procedure coninues wih he selecion of he ransiion variable for which he lags of boh R and TIB were checked. According o he resuls, we do no fail o reec he null hypohesis of lineariy when he ransiion variable is R - since he marginal level of significance is he lowes. Given his resul, he selecion process allows us o choose an exponenial smooh ransiion regression model (ESTR). As noed by Lundbergh (999) and elsewhere, given ha we are dealing wih high-frequency financial series, he assumpion ha he error sequence generaed by he STR model for he condiional mean has a consan condiional variance is no realisic and is somehing o be esed (see also Engle, 982; Ballie and Bollerslev, 989). We firs es he null hypohesis of a consan condiional variance agains he u follows an ARCH(s) or a GARCH(p, q) process. For ha we use alernaive ha { } he ess suggesed by Teräsvira (998).

9 8 Since we do no fail o reec he null, we fi a ESTR-GARCH model o he daa, following a wo sep esimaion 8. The firs sep consiss of esimaing he condiional mean, while he second sep consiss of he esimaion of he condiional variance using he residuals of he ESTR model. On his basis, a GARCH(,) model for he innovaions was adused in such a way ha he gain of he model as a whole (ESTR- GARCH), measured in erms of he raio of he variances, is according o he esimaes included in Table 3. Table 3. Smooh ransiion regression (ESTR) GARCH model Dependen variable: R Variable Coefficien Sandard - saisic p- value error Linear par Consan R TIB Non linear par Gamma C Consan R Dependen variable: (Orhogonalised) Residual Consan ARCH () GARCH () Sandard deviaion of Mean of dependen variable he regression Durbin-Wason Sandard deviaion of dependen variable.0044 Some commens on hese resuls are in order. Firs, i is remarkable he presence of TIB - in he linear par of he model wih he righ sign which evidences a nonlinear and negaive relaionship beween IBB and TIB. Second, he model renders evidence of a ime-varian equiy premium, depending upon if he nonlineariy is acive or no. When i is no acive, and he model is in essence linear, he premium is negaive. However, when by virue of he value of he ransiion variable he nonlineariy is acive he equiy premium will be posiive (or less negaive). Third, he ESTR-GARCH model suggess ha he adusmens of he reurns owards he premium are symmeric, wihou concern on wheher he changes are downwards or upwards. Fourh, he auoregressive par (linear and nonlinear) of he model ESTR- GARCH could be associaed o he fads componen which allows for share reurns o deviae from he ones dicaed by he fundamenals. Finally, based on he esimaed value of he coefficien corresponding o R - and is sign, he value of R - helps o 8 This procedure generaes consisen esimaes, alhough a endency o yield over-parameerisaion. This is because some effecs of he non consan condiional variance may a firs be capured by he esimaed condiional mean (see Lundbergh, 999). A oin or simulaneous esimaion leads o more parsimonious models.

10 9 forecasing he conemporaneous reurn, which is in conformiy wih he facs of Table. Noneheless, he value of he ineres rae coefficien seems, numerically speaking, very low, regardless of being significan. In addiion, he lag of TIB,, seems oo long for R o receive he impac of a TIB, unless, assuming high ransacion coss, he agens migh wai a few days o make sure ha he new level of he TIB is emporary or permanen before modifying heir posiions. In Figure 5 we observe he ransiion funcion generaed by he ESTR model. A negaive variaion of.73% in R - (value of coefficien c in he regression of Table 3) a he end of he day before rading acivaes he nonlineariy of he process.,0 Figure 5. Transiion funcion 0,8 F(.) 0,5 0,3 0, DIBB(-) Under convenional ess applied o hese STR-ype of models 9, we do no find evidence of remaining nonlineariies, auocorrelaion, insabiliy of parameers, and addiional ARCH-GARH phenomena 0. Under he BDS es (see Table 4), we do no reec he null hypohesis of having iid residuals (see he Appendix 2 for some deail on BDS es). Table 4. Resuls of BDS es on he sandardised and orhogonalised residuals of he RTSE-GARCH model m-dimension ε No presened here bu available from he auhors on reques. 0 However, according o he Jarque-Bera saisic we do no fail o reec he null hypohesis of normaliy.

11 0 IV. Some conclusions This work was aimed o describe he dynamics of he reurns on he Bogoá sock marke and o find any empirical link, if a all, beween his variable and he shor erm ineres rae, as measured by inerbank loan ineres rae. The nonlinear economeric model for he reurns is suggesed by he sylised fac of relaively volaile periods, characerised by large reurns, alernaing wih quie periods of small reurns on he Bogoá sock marke. Differen from wha he basic heoreical model predics, he share prices index does no behave as a maringale. Raher, he model for reurns seems o have a componen (linear and nonlinear) of fads given is mean-revering propery. In oher words, we find ha, in he firs days of increase (decrease) of he IBB, an agen can hink ha if oday he index moved upwards (downwards), hen i could be expeced, wih a high probabiliy, ha omorrow i will also move upwards (downwards). This ype of predicabiliy of he movemen of he index or rading rule, raised by he posiive auocorrelaion, would no suppor any argumen in favour of he marke efficiency. The model gives some evidence abou he exisence of a non consan equiy premium, depending upon if he nonlineariy is acive or no. When i is no acive, and he model is in essence linear, he prime is negaive. However, when by virue of he value of he ransiion variable he nonlineariy is acive he premium will be posiive. A he same ime, we find evidence of he nonlinear and negaive effec ha, according o he heory, he ineres rae exers on he sock prices. However, ha impac is raher small and oo lagged somehow. This lag lengh could be explained if he agens wai some days before acing in order o disinguish emporary movemens of ineres rae from he permanen ones, because of supposed high ransacion coss. Appendix : The ineres rae, he sock prices, and he random walk view Le us assume ha he represenaive agen, of an economy populaed by a coninuum of hem, maximises he lifeime uiliy funcion (see Campbell e al., 997, chaper 8, and Sargen 987, chaper 3, for furher deails): E u( c ) = β 0 < β < where c is consumpion a ime. The consumer is consrained by : All variables are measured in unis of consumpion goods.

12 where ( p ( p + d ) [( p d + ) s + = + d ) s + y c p p is he price of a share of an enerprise a ime, ] d are nonnegaive random dividends paid o he owner of he sock a he beginning of ime, s is he number of shares held by he agen a he beginning of 2, and y is a random process associaed wih he labour income. E is he mahemaical expecaion condiional on he informaion available a he beginning of ime. Here, we consider ( p + d) + p as he gross yield on shares beween daes and +. The Euler equaion ha describes he opimal consumpion and porfolio plan is given by: = β E p + + d p + u ( c u ( c ) ) + If we have ino accoun ha he expeced value of a produc of sochasic variables is equal o he produc of he expeced value of each variable plus he covariance of he variables, we can rewrie he laer expression as: = β E p + + d p + E u ( c + u ( c ) + β cov ) assuming ha ( d ) p, u ( c ) u ( c )) p + + d p + u ( c, u ( c cov p =0, which is he case when u ( c ) is linear, and ha E u ( c + ) / u ( c ) is consan (equal o for convenience), he random walk model of sock prices obains, so ha he above Euler equaion reduces o: + ) ) β p = E + + ( p + d ) which is a firs order univariae process for he sock prices. The soluion for his expression is given by: 2 In his case, ( 0 d 0 ) s0 p + is given.

13 2 p = E = β d + ω (A.) β where ω is a maringale process. A maringale is a variable ha follows any random process such ha: E =, ω + ω ω ω 2,..., where ω could be defined o be uncorrelaed bu no necessarily saionary. The implicaion of his is ha dependence beween higher condiional momens (variances) could be presen (Mills, 993, pg. 9). An alernaive for he maringale is, for example, he AR() process pu forh by Shiller (98) as a model of sock markes fads. In general, he fads erm added o he random walk also allows for prices differen from he hose dicaed by he fundamenals. The fads erm is assumed o be mean-revering, so ha he price will have a endency o reurn o fundamenals. The presen-value involved in (A.) renders a nonlinear relaionship beween p and he ineres rae, r, even making he seady-sae assumpion ha β=/(+r). Appendix 2: The BDS es Brock, Decher and Scheinkman (987) developed he BDS es as a se of ess based on he correlaion inegral. The BDS disinguishes beween he null hypohesis of having random iid variables and he alernaive hypohesis of deerminisic chaos. The mehod builds on he concep of correlaion dimension of Grassberger and Procaccia (983). The inegral correlaion calculaed for a ime series is defined as: i > i m m ( ε x x ) C( T, m, ε ) = lim H i T 2 (A2.) T m where T=N-m+, N is he lengh of he series, he vecors of sequences { x } are he m-hisories build from he ime series iself, H is he Heaviside funcion which is zero when he equaliy akes place and one oherwise,. is he supreme norm and ε is a m m small bu posiive number. The wo poins x i and x will be spaially correlaed if he Euclidian disance is less han a give radiusε of an m-dimensional ball cenred a one m of he wo poins, i.e., x m i x < ε. The spaial correlaion beween all poins on he aracor for a given ε is deermined by couning he number of hese pairs locaed in a ball around every poin.

14 3 The correlaion dimension is defined as: D C ( m) lnc( T, ε, m) lim ε 0 lnε = (A2.2) The BDS es ransforms he correlaion inegral ino one ha, asympoically, is disribued as a normal variable under he null hypohesis of iid agains a non specified alernaive hypohesis. The BDS es observes he dispersion of he poins in a number of spaces which dimension goes from 2 up o m. The correlaion inegral of a whie noise process wih a embedding dimension m can be wrien as he m-power of he m correlaion dimension wih embedding dimension equal o : C ( T,, ε ). The BDS es can hen be wrien as: BDS = σ C( T, m, ε) C( T,, ε) m N 2 m [ C( T, m, ε ) C( T,, ε) ] (0,) (A2.3) The null hypohesis of whie noise (iid) is reeced when BDS excess in absolue value he seleced criical values. This will happen if he dispersion of he poins in he consecuive spaces is no in line wih he expeced dispersion for he case of whie noise.

15 4 References Ballie, R.T. and T. Bollerslev (989) The message in daily exchange raes: a condiional variance ale, Journal of Business and Economics Saisics, 7, 3, Bernanke, B. and M. Gerler (999) Moneary Policy and Asse Price Volailiy, Federal Reserve Bank of Kansas Ciy Economic Review, Fourh quarer, 7 5. Brock, W.A., W.D. Decher, and J.A. Scheinkman (987) A es for Independence based on he Correlaion Dimension, Mimeo. Universiy of Wisconsin. Brooks, C. (996) Tesing for non-lineariy in daily serling exchange raes, Applied Financial Economics, 6, Campbell, J.Y. (999) Asse prices, consumpion, and he business cycle, in J.B. Taylor and M. Woodford (eds.), Handbook of Macroeconomics, Elsevier, New York. Campbel, J.,Y., A.W. Lo, and A. C. MacKinlay (997) The Economerics of Financial Markes, Princeon Universiy Press, Princeon. Cao, C.Q. and R.S. Tsay, (992) Nonliear ime-series analysis of sock volailiies, Journal of Applied Economerics, 7, De Grauwe, P., H. Dewacher and M. Embrechs (993) Exchange Rae Theory. Chaoic Models of Foreign Exchange Markes, Blackwell, Oxford. Dik, D., T. Terasvira and P.H. Franses (2000) Smooh Transiion Auoregressive Models. A Survey of Recen Developmens, SSE/EFI Working Paper Series in Economics and Finance, No. 30. Engle, R.F. (982), Auoregressive condiional heeroskedasiciy wih esimaes of he variance of Unied Kingdom inflaion, Economerica, 50, 4, Fama, E.F. (970) Efficien capial markes: a review of heory and empirical work, Journal of Finance, 25, Franses P.H. and D. van Dik (999) Nonlinear ime series models in empirical finance, Manuscrip prepared for Cambridge Universiy Press. Granger, C. W. J. and T. Teräsvira (993) Modelling nonlinear economic relaionships, Oxford Universiy Press, Oxford. Grassberger, P. and I. Procaccia (983) Measuring he srangeness of srange aracors, Physica, 9D,

16 5 Heaon, J. and D. Lucas (999) Sock prices and fundamenals, in B.S. Bernanke and J.J. Roemberg (Eds.) NBER Macroeconomics Annual, The MIT Press, Cambridge Lucas, R. E., Jr. (978) Asse prices in an exchange economy, Economerica, 46, 6, Lundberch, S. (999) Modelling economic high-frequency ime series, Unpublished disseraion for he degree of Docor of Philosophy, Ph.D., Sockholm School of economics. MacDonald R., and M.P. Taylor (99) Financial markes analysis: An ouline, in M.P. Taylor (Ed.), Money and financial markes, Basil Blackwell, Cambridge. Mills, T.C. (993) The economeric modelling of financial ime series, Cambridge Universiy Press, New York. Sarno, L. (999) Adusmen coss and nonlinear dynamics in he demand for money: Ialy, 86-99, Inernaional Journal of Finance and Economics, 4, Sargen, T. (987) Dynamic Macroeconomic Theory, Harvard Universiy Press, London. Shiller, R. (98) Do sock prices move oo much o be usified by subsequen changes in dividends?, American Economic Review, 7, Teräsvira, T. (998) Modelling economic relaionships wih smooh ransiion regressions, in Handbook of Applied Economic Saisics, A. Ullah and D.E.A. Giles (Eds.), Marcel Dekker, New York.