Modelling Beta Risk for New Zealand Industry Portfolios

Transcription

1 Modellng Beta Rsk for New Zealand Industry Portfolos Xao-Mng L Department of Commerce, Massey Unversty (Albany), New Zealand Abstract In modellng the beta rsk of the New Zealand ndustry portfolos, we extend the prevous analyses of three major technques to nclude the stochastc volatlty model and the Schwert and Segun approach based on the stochastc volatlty model, and perform the modfed Debold and Marano test adjusted for ARCH that has not yet been appled n the prevous, smlar studes for forecast evaluaton. It s found that, n the case of n-sample forecastng, the stochastc volatlty model s the optmal technque, whle the model may be favoured for out-of-sample forecastng, unlke pror works on the market betas of other countres ndustry portfolos that suggest that the Kalman flter approach s preferred. JEL classfcaton: G1; C1 Key words: Tme-varyng beta; Stochastc volatlty; ; Kalman flter; New Zealand Address for correspondence: Dr Xao-Mng L, Department of Commerce, Massey Unversty (Albany), Prvate Bag , North Shore MSC, Auckland, New Zealand. Tel: ext Fax: Emal: x.n.l@massey.ac.nz

2 1. Introducton Snce the emprcal study of Fama and French (1992), possble tme varaton n systematc rsk (.e., market betas) for fnancal assets has been subject to extensve nvestgaton n the fnance lterature. Consderable emprcal evdence generated seems to challenge the theory that mantans a tmenvarant lnear structure n the captal asset prcng model (CAPM). Such CAPMs are uncondtonal, the underlyng assumpton beng that market rsk or beta s constant. On the other hand, condtonal CAPMs allow beta to be condtoned on rsk factors and hence to be tme-varyng. Applyng the condtonal CAPM, many researchers are able to argue that the beta stablty assumpton does not hold, and to provde the estmates of the tme seres of beta. The recent such works on tme-varyng beta nclude the followng: McKenze, Brooks, Faff and Ho (2000) for US banks; Le, Brooks and Faff (2000) for Australan fnancal sector companes; Faff, Hller and Hller (2000) for UK ndustry portfolos; Brooks, Faff and McKenze (1998) and Faff and Brooks (1998) for Australan ndustry portfolos; and Brooks, Faff and Arff (1998) for the Sngapore stock market. Some earler works that only performed tests of stablty found evdence of beta nstablty for Korea, Fnland, Hong Kong, Sweden, the US and so on. See, for example, Bos and Fetherston (1992); Bos, Fetherston, Martkanen and Perttunen (1995); Cheng (1997); Wells (1994); Bos and Fetherston (1995); and Brooks, Faff and Ho (1997). Whle there s now a consderable body of research n ths area for many natons, smlar work on the New Zealand market betas s regrettably lackng. The New Zealand stock market s relatvely small, compared to those studed n the aforementoned artcles. As at 30 Aprl 2003, there were 224 companes wth lsted securtes on the NZSE Market and 213 securtes quoted. These securtes have a total market captalsaton of NZ$42.3 bllon. In the four months ended 30 Aprl, 2003, 2,494 mllon shares wth a value of NZ$6,131 mllon were processed on the market. Despte ts relatvely small sze, though, New Zealand has been among the least-regulated economes and has had the freest stock market n the world snce radcal deregulatory reforms n For example, an mportant 1

3 feature of the Lstng Rules s the provson of an ndependent market survellance system, under whch the Market Survellance Panel s an ndependent body charged wth the admnstraton and enforcement of ts lstng rules and free of statutory controls by the government. As a result, the New Zealand captal market s found to have become more globally ntegrated than before (Chay and Eleswarapu, 2001). In such an envronment, New Zealand companes and ndustral sectors are facng greatly ncreased competton from ther counterparts n other economes for the rasng of captal on the one hand, but may also be beneftng from an enhanced level of captal nflows on the other. Therefore, the relatonshps between rsk factors and returns for New Zealand ndustry portfolos as emboded n market betas, as well as ther nstablty, should be of nterest to both domestc and foregn nvestors. The fndngs about to what degree and n what form beta rsk n New Zealand ndustry portfolos s unstable wll help portfolo managers and securty analysts n constantly updatng and re-estmatng such relatonshps. Investgatons nto beta nstablty have generally adopted two approaches: testng for constancy of beta and modellng of changng beta. As consderable evdence aganst the assumpton of constant beta has been generated n the lterature, we decded to manly adopt the second approach whch has been wdely utlsed n prevous studes to determne tme-varyng betas for varous markets and/or sectors. Tme-seres modellng s capable of provdng estmates of the beta seres and the forms of beta varaton. Three hghly popular technques of ths approach nclude the -type model (Bollerslev, 1990), the Schwert-Segun (SS) model (Schwert and Segun, 1990), and the Kalman flter model. The -type models are used to calculate the tme-varyng condtonal covarance of portfolo and market returns and the tme-varyng condtonal varance of market returns before the tme seres of beta s computed. The SS model s a sngle factor model of return heteroscedastcty, n that only the condtonal varance of market returns s obtaned from a process and then used to generate the condtonal beta seres. The Kalman flter approach does not look at the behavour of 2

4 return volatlty; nstead, t estmates recursvely the parameters ncludng beta n the smple market model. Brooks, Faff and McKenze (1998) (examnng the Australa ndustry portfolos), Faff, Hller and Hller (2000) (examnng the UK ndustry portfolos) and Le, Brooks and Faff (2000) (examnng the Australa fnancal sector stocks) all fnd that the Kalman flter approach s superor to the type and SS models. As the second contrbuton of ths paper, however, we consder not only these three approaches, but also the stochastc volatlty (SV) model. The SV model dffers from the -type models n that the former nvolves two nose processes whle the later only one. Introducng an addtonal nnovaton s supposed to ncrease substantally the flexblty of the model n descrbng the evoluton of condtonal varance. Yu (2002), for example, fnds that the SV model outperforms eght unvarate models ncludng the -type models n terms of forecastng performance. Despte these, the use of the SV model for econometrc evaluaton of beta rsk has not become common n the emprcal fnance lterature, probably because of the dffculty n parameter estmaton. In ths study, we apply a quas-lkelhood method (Ruz, 1994) n estmatng the SV model. We want to see f the SV model s superor n predctng beta rsk n the New Zealand context. Another dstncton of our study s that we conduct a new verson of the Modfed Debold and Marano (DM) test n forecast evaluaton. The test s developed by Harvey, Leybourne and Newbold (1999), and ts statstcs are adjusted for the presence of ARCH n forecast errors. Many pervous studes on modellng tme-varyng beta rsk faled to perform such forecast evaluaton tests (except one by Faff, Hlluer and Hller (2000) whch, nevertheless, used the verson of the Modfed DM test not adjusted for ARCH), and they reled on vsual nspectons on the sze of some accuracy measures (such as mean square errors). However, a vsual nspecton often leads to erroneous conclusons after comparng competng forecasts. Therefore, to enable such comparsons to be relably made, we resort to formal testng procedures. 3

5 The remander of ths paper s structured as follows. The next secton brefly outlnes sx modellng technques employed n ths study. Secton 3 reports and comments on the results of descrptve statstcal tests, model estmaton, comparng dfferent condtonal betas, and assessng nsample and out-of-sample forecastng performances of the sx models. Conclusons are presented n secton Competng Approaches Sx approaches to estmatng constant/tme-varyng betas are adopted and ther results are compared n ths paper. In ths secton, we outlne these competng approaches. () Ordnary Least Square (OLS) To begn wth, beta rsk s frstly treated as a constant. The results wll be used as a benchmark wth whch comparsons of tme-varyng betas' results are made later. The smple market model by whch the estmates of constant betas are obtaned va the OLS method s R =? + R + e (1) OLS t mt t where R t denotes returns on the portfolo of an ndustry, R mt returns on the market as a whole,? and OLS a portfolo's mean returns and systematc rsk respectvely. If the error term e t s assumed to be ndependently and dentcally dstrbuted (..d), and the covarance of R t and R mt (Cov(R t, R mt )) and the varance of R mt (Var(R mt )) are constant, the OLS procedure can be appled to obtan the pont estmates of? and OLS. 4

6 () -type models It s well documented n the emprcal fnance lterature that the error term e t s not an d, and the varance-covarance matrx of portfolo and market returns are not constant over tme. In ths case, beta, defned as Cov( R, R )? (2) Var( R ) t mt t mt n the market model R =? + R + e, now becomes non-constant, and clearly the OLS method t mt t s rrelevant here. One approach to estmatng ths condtonal beta t s to estmate condtonal covarance Cov(R t, R mt ) and condtonal market varance Var(R mt ). However, by makng some smplfyng assumptons, the problem can be reduced to estmatng the followng two smple (1,1) models: 2 Var( Rt)? ht? a? b? t,? 1? ch t,? 1, a>0, b >0, c? 0, and b? c? 1 (3) for the condtonal varance of portfolo returns R t, and 2 Var( Rmt)? hmt? am? bm? mt,? 1? ch m mt,? 1, a m>0, b m>0, cm? 0, and bm? cm? 1 (4) for the condtonal varance of market returns R mt. The error terms n equatons (3) and (4) are the condtonal means of, respectvely, ndvdual portfolo returns and market returns; namely R t =? t and R mt =? mt. Then the condtonal covarance of portfolo and market returns may be computed accordng to Cov( R, R )?? hh (5) t mt m t mt where? m, assumed to be constant, s the correlaton coeffcent between R t and R mt. For more detals about how some smplfyng assumptons lead to the above three equatons, see Bollerslev (1990) and Le, Brooks and Faff (2000), for example. Substtutng equatons (3), (4) and (5) nto (2) wll generate the full tme seres of t. Equatons (3) and (4) can be replaced by other -type models, 5

7 such as the exponental model and the threshold ARCH model, to take account of asymmetrc effects between postve and negatve returns. But ths s beyond the scope of the present paper. Another parameter n R =? + R + e to be estmated s the condtonal ntercept?. t mt t The followng equaton has been proposed to serve ths purpose:?? R?? R (6) t mt where R t s the mean ndustry return, the mean condtonal beta, and R mt the mean market return. () The Schwert and Segune (SS) model Usng the SS model, the tme-varyng condtonal beta s calculated as SS t 2? 1? (7) hmt where the estmates of? 1 and? 2 come from the regresson model R R???? R??? e (8) mt t 1 mt 2 t hmt and the condtonal market varance h mt s obtaned by estmatng equaton (4). As for?, t may be estmated usng equaton (6) but wth replaced by SS, the mean value of the condtonal SS beta. (v) The Kalman flter algorthm Instead of obtanng the estmates of condtonal varances frst and then computng tme seres of beta, the Kalman flter approach drectly estmates a tme-varyng beta on a state-space model. Wth ths approach, equaton (1) s modfed to become a measurement equaton below R =? + R + e KAL t t t mt t 2 et N (0,? ) (9) 6

8 wth the transton equatons specfed as? = T? +? t a a,t-1 at? N (10) at 2 (0, qa) = T +? KAL KAL t b,t-1 bt? N (11) bt 2 (0, qb) We do not arbtrarly set T a and T b equal to 1, but let them be estmated freely from the data. In other words, we let the data determne whether? t and KF t follow an AR(1) or a random walk process. (v) The stochastc volatlty (SV) model The SV model s expressed as: R t???? t? t=? + exp( 0.5ht )? t,? t ~ NID(0,1);? 2? ht???? ht,? 1? u t, ut ~ NID(0,? u ). where h t = 2 ln( t )?. Movng? to the left-hand sde and transformng R t -? by takng logarthms of ts squares, one obtans a lnear state-space model: ln? ( R?) t??? E[ln(? t )]? h+v t t?? 1.27? ht? v, t vt ~ NID(0,? v /2);? 2? ht???? ht,? 1? u, t? < 1, ut ~ NID(0,? u ) (12) where v t = ln( 2 2? t ) - E[ln(? t )]. The above equaton mples that, f? t ~ NID(0,1), the mean and varance of ln(? ) are equal to and 2 t 2 s v/2 respectvely (Ruz, 1994). The error terms v t and u t are assumed to be uncorrelated. The ntroducton of u t nto the condtonal varance equaton makes current volatlty h t stochastc. u t represents the shocks to the ntensty of the flow of new nformaton as measured by h t, whle v t represents the contents of the news. The SV model (12) can be used to ft the portfolo returns and, f the subscrpt s replaced by m, the market return. A dfference between the (1,1) model and the SV model s that the former typcally cannot capture the excess kurtoss observed n the return seres completely, whle the latter can (Franses and van Djk, 2000, ). But lke the approach, the purpose of estmatng the 7

9 SV model s to obtan the tme seres of condtonal varances of the portfolo return and the market return, h t and h mt, before generatng the beta seres based on:? SV m ht? t? (13) h mt Equaton (13) can then be substtuted nto equaton (1) for OLS t to obtan the market model R t =? + SV R mt + e t, and equaton (6) used to estmate? wth replaced by SV, the mean value of the condtonal SV beta. To estmate an SV model, we need a quas-maxmum lkelhood (QML) method va Kalman flterng (Ruz, 1994), or a Monte Carlo Markov Chan (MCMC) method (Jacquer, Polson and Ross, 1994). Jacquer, Polson and Ross (1994) compare the relatve performance of the QML and MCMC estmators, and show that for the hgh volatlty cells (var(h t )/E[h t ] 2 = 10), the QML domnates the MCMC estmator. Such coeffcents of varaton centre around 10 n the cases of the NZ ndustry portfolos. In addton, "[T]he real advantage of the QML approach s ts speed and adaptblty to many stuatons" (Jacquer, Polson and Ross, 1994). Thus, we decde to employ the QML method n ths study. For a more detaled descrpton of QML estmaton va Kalman flterng, we drect the reader to Jacquer, Polson and Ross (1994) and Yu (2002). (v) The SS model n conjuncton wth the SV model (SSSV). The sxth approach adopted n ths paper s to obtan the condtonal varance of the market return from estmatng the SV model and then use t to calculate the SS tme-varyng beta. The equatons to be used for ths purpose are smlar to (7) and (8), namely: SSSV t 2? 1? (14) hmt R R??? R?? e (15) mt t 1 mt 2 t hmt 8

10 but wth h mt estmated from the SV model (12) rather than from the (1,1) model (4). Agan? s estmated usng equaton (6), provded that condtonal SSSV beta. s replaced by SSSV, the mean value of the 3. Data, Estmaton and Fndngs () Descrptve statstcs The data employed n ths study are daly stock prce ndexes of sx group sectors and thrteen sngle sectors, and NZSE TOTAL as the market ndex. The sx group sectors are as follows. (1) PRIMARY, ncludng Agrculture & Fshng (Agrculture hereafter), Mnng, Forestry & Forest Products (Forestry hereafter), and Buldng Materals & Constructon (Buldng hereafter). (2) ENERGY. (3) GOODS, ncludng Food & Beverages (Food hereafter), Textles & Apparel (Textles hereafter), and Intermedate & Durables (Intermedate hereafter). (4) PROPERTY. (5) SERVICES, ncludng Transport, Ports, Lesure & Toursm (Lesure hereafter), Consumer, Meda & Telecommuncatons (Meda hereafter), and Fnance & Other Servces (Fnance hereafter). (6) INVESTMENT. All twenty ndexes used are Gross,.e., adjusted for dvdends assumng that dvdends are renvested, and they are value-weghted. Snce January 3, 1997 s the earlest date from whch the ndexes of the sx group sectors and the thrteen sngle sectors started to be calculated by the NZSE, the data sample perod covered n ths study s from January 3, 1997 to August 28, 2002, wth 1,424 observatons. We obtaned these data from the Datastream database and Datex. Twenty tme seres of contnuously compounded percentage return are calculated from the prce data (.e., R t = (lnp t - lnp t-1 )?100%), and ther summary statstcs are set out n Table 1. The sngle sectors Consumer, Food, Fnance and Textles and the group sector GOODS rank the top fve n terms of ther mean returns (all greater than %) over the perod. On the other hand, nvestors who nvested n the followng ndustry portfolos seem to have suffered a loss: INVESTMENT, Transport, 9

11 Forestry and PRIMARY. In partcular, the INVESTMENT sector generated the lowest, negatve mean return ( %). The fact that some of the NZ ndustry portfolos exhbt negatve mean returns s n contrast wth the Australan case where all ndustry portfolos enjoyed postve mean returns (See Brooks, Faff and McKenze, 1998). Wth regard to the rsk of each portfolo, the fve portfolos wth negatve mean returns are also rsker than others wth postve mean returns. Although the PROPERTY sector dd not yeld the hghest long-run average returns, ts rsk was nevertheless the lowest (SD = ); and ths seems to concde wth the Australan case (See Brooks, Faff and McKenze, 1998). The portfolo returns of all ndustres but Agrculture demonstrate more or less negatve skewness, and all exhbt excess kurtoss. The market return also suffers the problems of negatve skewness and severe excess kurtoss. Thus, t s not surprsng that the Jarque-Bera normalty tests fal to reject non-normalty n all cases. In addton, the ARCH test statstcs show that the twenty return seres have ther varances changed over tme, mplyng that ther rsks are not constant throughout the sample perod. The non-constancy of the varance of the market return already suggests that the systematc rsk of each portfolo could be tme-varyng. As a prelmnary analyss, Table 1 also reports the OLS estmates of beta n equaton (1) and the assocated CuSumSq test statstcs. Although all betas are sgnfcant at a hgher than 1% level, the null hypothess that the regresson coeffcents n equaton (1) are stable must be rejected at a hgher than 1% level. These CuSumSq test results further ndcate the possble non-constancy of beta for all ndustry portfolo returns wth no excepton. () Model estmaton Gven ts wdespread popularty n the lterature, the parsmonous (1,1) model (3) s ftted to the return data of each of the nneteen portfolos, and (4) to the market ndex NZSE TOTAL. We use the general (1,1) model rather than other -type models manly for the sake of makng our results comparable wth those for the US, the UK and Australa that were based also on the 10

12 (1,1) model (See, for example, McKenze, Brooks, Faff and Ho, 2000; Faff, Hller and Hller, 2000; Le, Brooks and Faff, 2000; and Brooks, Faff and McKenze, 1998). The estmaton results are presented n Table 2. One can see that all the ARCH and terms have ther coeffcents b and c postve, sgnfcant statstcally at a hgher than 1% level, and summed to less than unty. The Forestry sector s the most persstent ( b? c? ) whle the Food sector has the least degree of persstence ( b? c =0.6681). Thus, the uncondtonal varance of each return seres s postve and fnte. Snce these nneteen ndustry portfolos and the market portfolo satsfy the necessary condtons of the model, they can be ncluded n the remander of the analyss. Table 2 also gves the correlaton coeffcent? m between each portfolo return seres and the market return seres. The Mnng sector has the lowest correlaton coeffcent wth the market, whle the SERVICES sector has the hghest Ths range of correlaton coeffcent n New Zealand s larger than n Australa (0.609~ See Brooks, Faff and McKenze, 1998) and the UK (0.47 ~ See Faff, Hller and Hller, 2000). Moreover, about half of the nneteen portfolos exhbt a? m lower than 0.5, ndcatng that, n general, the relatonshp between the market return and that of each sector s less sgnfcant n New Zealand than n Australa and the UK. Wth the seres of the condtonal varance for the market return generated by the model, all data needed to estmate the SS model (8) become avalable. Table 3 shows the estmated coeffcent values and some statstcs of ths regresson equaton. Coeffcent? 1 s sgnfcant at the 1% level n all cases, whle coeffcent? 2 s sgnfcant at the 1% level n fourteen cases, at the 5% level n two cases, and nsgnfcant n three cases. Regardng explanatory power, the ncluson of the term R- m/h mt nto the market model ncreases R 2 only margnally (comparng R 2 n Table 3 wth R 2 for? OLS n Table 7), even for those sectors whose? 2 are hghly sgnfcant. It s not surprsng to observe the hghest R 2 value (0.8531) for the SERVICES sector and the lowest R 2 value (0.0543) for the Mnng ndustry, for the reasons as gven above on the correlaton coeffcent. Agan, New Zealand ndustres 11

13 llustrate a larger varablty n R 2, and overall a lower explanatory power of the market return and changng beta rsk for varatons n ndustry portfolo returns, than the Australan and the UK counterparts as reported n Brooks, Faff and McKenze (1998) and Faff, Hller and Hller (2000). Ths s especally true n the case of the NZ Mnng sector where almost no explanatory power of the regresson equaton (8) s observed. The tme-varyng charactersatons of condtonal mean (? t ) and condtonal beta ( KF t ) as n equatons (10) and (11) can be estmated usng the Kalman flter technque. When estmatng the system, startng values of KF 2 2?,, T, T, q and q were provded, but ther fnal estmates were t t a b a b obtaned by maxmsng the log-lkelhood functon at convergence. As Table 4 shows, every ndustry sector s able to reach convergence when applyng the Kalman flter algorthm. It appears that the movements n the condtonal mean and beta are charactersed by a random walk process. Ths s because the null hypotheses T a =1 and T b =1 cannot be rejected at the 10% level (not reported) whle the null hypotheses T a =0 and T b =0 can be decsvely rejected at a hgher than 1% level (reported n Table 4), for all sectors. The estmated standard devatons q a and q b are also provded n Table 4, but snce constrants q,q? 0 were mposed to prevent them from takng negatve values, they follow a b nonstandard dstrbutons and t s meanngless to ndcate ther levels of statstcal sgnfcance based on the t-ratos. Ths paper also pursues the stochastc volatlty model whch many prevous studes on tmevaryng beta have excluded from consderaton. We provde the estmates of the four parameters t,?, s u and pof equaton (12) n Table 5. Based on the t-values (not shown),? and? are dfferent from zero at a hgher than 1% sgnfcance level for all of the twenty return seres. Moreover, all estmates of? are also dfferent from unty at a hgher than 1 % level. Ths reveals that all h t follow a statonary AR(1) process; that s, the condton 0 <? < 1 s satsfed for all sectors. Even f h t could take a negatve value due to the presence of u t n equaton (12), ths does not affect the fact that the 12

14 2 2 condtonal varance of returns ( s? Var( R )=exp(0.5 h )) s always postve. We do not ndcate the t t t level of statstcal sgnfcance for s u and s v, the standard devatons of u t and v t, for the same reason as n the Kalman flter model that the non-negatvty constrants were mposed durng the estmaton process. The reported estmates of s u and s v may gve us an dea of how these sectors dffer n terms of the varablty of the shocks to the ntensty of the new nformaton flow, and the varablty of the contents (large/small, postve/negatve) of the news. So far the Schwert and Segun approach employed n the lterature has nvolved -type models only, and t s certanly of nterest to see how ths approach would perform f based on the SV model nstead. For ths purpose, we estmated equaton (15) agan and the results are reported n Table 6. Comparng ths table wth Table 3, t turns out that the SSSV model s somewhat worse than the SS model. The goodness of ft as measured by R 2 drops n all cases, and 2 becomes nsgnfcant for eghteen return seres, ffteen more than the SS model. Ths suggests that the SV model n conjuncton wth the SS approach mght be ncapable of capturng the tme-varyng feature of systematc rsk, as we wll see later. () Dfferent condtonal betas and ther varatons Based on the model estmaton results presented n the prevous subsecton, nneteen condtonal beta seres were obtaned from each of sx approaches. We dscuss and compare them n ths subsecton. Usng the procedures as outlned n secton 2 for the sx approaches, sx dfferent beta seres were generated for each of the nneteen portfolos, wth the OLS beta beng constant and the remanng fve betas changng over tme. Table 7 sets out the mean values of the fve tme-varyng betas along wth ther ranges of values (low/hgh), and the OLS beta (as well as the assocated coeffcent of determnaton R 2 ). Table 8 then provdes the correlaton coeffcents between mean betas n varous pars. Three messages emerge from the two tables and mert some dscusson. Frst, on average, the 13

15 mean values of the SSSV beta are the closest to, whle the mean values of the SV beta are the farthest from, the OLS pont estmates of beta, and the SV betas take the hghest values n all cases. Second, n terms of the correlaton wth the OLS beta, the mean beta ranks number 1 ( ), followed by the mean Kalman flter beta ( ), the mean SS beta ( ), the mean SV beta ( ) and the mean SSSV beta ( ) (See Table 8). Such correlaton reflects how smlar a parametersaton of rsk the correspondng approach provdes to the OLS approach. The hghest correlaton occurs between the mean Kalman flter beta and the mean SS beta ( ), and the lowest between the mean SV beta and the mean SSSV beta ( ), accordng to Table 8. Ths s surprsng as t suggests that nvolvng a common model (the SV model) does not necessarly ensure a smlar parametersaton of rsk, and nvolvng totally dfferent models need not lead to vastly dfferent parametersatons of rsk. The thrd message pertans to the second moment. The SV betas exhbt the wdest range of varaton and the SSSV betas the second. For example, the dfference between a low value and a hgh value of the SV beta could be as large as 132.2, n the case of the Agrculture sector. But no negatve values are observed wth ths approach (as well as the modellng technque). Negatve values are generated n some cases by the SS model, the Kalman flter algorthm and the SSSV model, although the ranges of varaton under the frst two approaches are moderate. The occurrence of negatve betas s not unque to our study here: Brooks, Faff and McKenze (1998) and Le, Brooks and Faff (2000) also report such a fndng whch nvolves the use of daly data and the Kalman flter technque. So what we are most concerned about s the extraordnarly large ranges of beta varaton assocated wth the SV and SSSV models. Brooks, Faff and McKenze (1998) argue that the Kalman flter approach often generates, n the ntal stages of estmaton, very large outlers (n the absolute term), and thus exclude some start-up observatons from the analyss of the Kalman flter betas as a remedy. Snce the SV model also nvolves the use of the Kalman flter technque, we tred the same soluton but the problem remaned. An examnaton of the condtonal varance of each return seres 14

16 generated by the SV model reveals that large outlers actually occur throughout the entre data sample perod, and not just concentrate n the ntal stages of estmaton. Thus droppng the frst few tens or even hundreds of observatons does not help. On the other hand, there s a queston about such a treatment. Is a large range of varaton n the values of tme-varyng beta really a problem f aberrant observatons do somehow belong to the tme seres of condtonal varances or f temporary perods of hgh or low volatlty are ndeed part of the underlyng data-generatng process? It s well known that outlers yeld large values of skewness and kurtoss, but the nature of the SV model s such that t can capture the excess kurtoss completely as mentoned earler. Ths probably explans why the SV betas and the SSSV betas vary so wdely, gven that ther values depend crucally on the condtonal varances generated by the SV model. Arbtrarly removng outlers may well lead to, rather than avod, an unfar bas. Therefore, a reasonable range of the observed values of beta need not mply that the underlyng approach s superor to others. (v) Forecastng performance To formally assess the sx modellng technques, we follow prevous studes n the lterature by comparng ther forecastng performances wth each other. Two most popular measures to evaluate forecast accuracy are the mean absolute error (MAE) and the mean square error (MSE). They are defned by 1 N MAE?? ˆRt? Rt (16) t? 1 N t? 1 N? ˆR? 2 t Rt 1 N MSE??? (17) where N denotes the number of observatons on ˆR t, ˆR t represent the forecasts of a return seres, and R t are the actual observatons of the return seres. We use these two measures n ths study as crtera: the smaller they are, the better the underlyng approach. To make sure that an MAE or MSE that looks 15

17 smaller than the other n a par-wse comparson s ndeed so, we performed tests for the equalty of predcton mean absolute errors and mean square errors proposed by Harvey, Leybourne and Newbold (1999). Let us frst compare the n-sample forecastng performance of each modellng technque. Usng the estmates of? and? for each modellng technque and the data on market returns R mt to predct the n-sample values of ndustry returns ˆR t, we calculated the mean absolute error (MAE) and the mean square error (MSE) for each ndustry portfolo. Table 9 reports all the MAE and MSE measures. Table 10 presents the Modfed DM test statstcs LS * for the followng two null hypotheses: (1) The two forecasts have equal absolute error (e.g., H 0 : E(d t ) = 0 wth d t = e 2 tsv, -e 2, tkf ); and (2) the two forecasts have equal mean square error (e.g., H 0 : E(d t ) = 0 wth d t = e tsv, - e, tkf ). The crtcal value wth whch these statstcs are compared s that of the t dstrbuton at the 10% sgnfcance level. Based on the LS * statstcs, we hghlght wth bold fgures the smallest MAE and MSE n Table 9 that are sgnfcantly dfferent from others for each ndustry portfolo. In some cases we have more than one smallest MAE or MSE measures, snce they are found to be nsgnfcantly dfferent from each other accordng to Table 10. For example, the Agrculture sector has the smallest MAE assocated wth the SV model and the second smallest MAE assocated wth the KF technque. However, the LS * statstcs n Table 10 reveal that the two MAEs are all sgnfcantly smaller than other MAEs at a hgher than 1% level, but are not dfferent from each other gven LS * =-0.7. Thus, both are consdered to be the smallest. Table 9 shows that, n terms of the MAE measure, the SV approach yelds the smallest forecast error n eghteen of nneteen nstances, among whch four number 1 postons are shared wth the Kalman flter approach. The remanng number 1 poston s taken solely by the Kalman flter approach for the Textles sector. That s, none of the other four approaches could ever have the chance to gan a gold medal (.e., the smallest forecast error). It s nterestng to observe from Table 10 that, for every ndustry portfolo, the Kalman flter MAE s smaller than all other MAEs but the SV one at a hgher 16

18 than 1% sgnfcance level. Ths suggests that f the SV model dd not enter the n-sample forecastng competton, all gold medals t receves would go to the Kalman flter approach makng t a champon. There are, however, some dfferences between the MAE and MSE measures. Usng the MSE crteron, the number of gold medals receved by the SV model falls to ffteen whle that by the Kalman flter model rses to ffteen; and n twelve cases the gold medals are shared between the two. Despte such dfferences, the proporton of the MAE/MSE measures put together that are the smallest for each approach can be used to rank the sx modellng technques: the SV model s the best (86.842%), the Kalman flter model s the second best (52.632%), and the remanng are equally the worst (0%), n terms of n-sample forecastng performance. It appears, therefore, that f the stochastc model was not consdered, we would end up wth the same concluson as reached n Brooks, Faff and McKenze (1998), Le, Brooks and Faff (2000), and Faff, Hller and Hller (2000) that the Kalman flter approach domnates all other methods. Next, let us consder out-of-sample forecastng. To conduct out-of-sample forecast tests, we dvded the sample perod nto two sub-perods, wth sub-perod 1 spannng from January 3, 1997 to December 31, 2001 and sub-perod 2 from January 1, 2002 to August 28, Sub-perod 1 was retaned for estmatng the sx models' parameters, whle sub-perod 2 was used for computng 1-day ahead forecasts by pluggng the sub-perod-1 parameter estmates nto the correspondng model. In dong so, we assumed that the market return R mt n the out-of-sample perod s known, and that ndvdual portfolo returns are known up to the date on whch ther 1-day-ahead forecasts are made especally where they are needed for predctng the 1-day-ahead values of volatlty (and hence 1-dayahead values of beta) n the, SS, SV and SSSV models. The resultng MAE and MSE measures and the correspondng modfed DM test results are gven n Table 11 and Table 12. Unlke the fndng by Brooks, Faff and McKenze (1998) for the Australan ndustres that the out-of-sample evdence does not alter the concluson about the superorty of the Kalman flter approach as supported by the n-sample evdence, we have found that the results obtaned from n- 17

19 sample forecastng no longer apply to the out-of-sample cases for the New Zealand ndustres. Note that the authors dd not perform the modfed DM tests for the null hypothess of no dfference n MAE and MSE for each condtonal beta estmaton model: had they done so, ther conclusons would probably have been altered qualtatvely. A notceable observaton emerges from Table 11: the boundary between dfferent modellng technques becomes blurred n terms of out-of-sample forecastng performance. The modfed DM test results reported n Table 12 ndcate that the null hypothess of the equalty of forecastng errors cannot be rejected at the 10% level n many cases. Thus, for the Buldng and Mnng sectors, the MAE measures demonstrate no statstcal dfference across the sx modellng approaches. The same story apples to the MSE measures for the Agrculture, INVESTMENT and Mnng sectors. Nevertheless, we can stll rank the sx approaches as follows (usng the method descrbed above): the approach (92.105%), the Kalman flter approach (86.842%), the SS approach (76.316%), the OLS and SSSV approaches (57.895%), and the SV approach (35.842%). It s nterestng to see that the model, once placed n the last poston n the n-sample forecastng competton, replaces the SV model as number 1 n the out-of-sample forecastng competton, whle the SV model now becomes such a loser that t ranks number 6. One possble reason may be that the ablty to capture contemporaneous shocks (u t ) to volatlty (ln(h t )) s the man advantage of the SV model over the (1,1) model (and perhaps over the Kalman flter model as well). However, n the out-of-sample perod, contemporaneous shocks are no longer ncorporated n the estmaton or updatng of the parameters, such as?,?,? v and? u, n the SV model. Wthout the advantage of beng able to utlse nformaton on contemporaneous shocks to volatlty, the SV model s unsurprsngly eclpsed by the (1,1) model and other models n the out-of-sample forecastng competton. To sum up, when comparng n-sample forecast errors, the SV model overwhelmngly domnates the other fve technques n generatng more accurate measure of beta rsk; but when usng 18

20 out-of-sample forecasts to assess the worth of a model, the approach outperforms the other fve ones, albet supported by less clear-cut evdence. In addton, Table 11 seems to suggest that that, for all sectors but transport, the forecastng accuracy of each of the sx models s generally mproved n the out-of-sample cases as compared to the n-sample results n Table 9, because the out-of-sample MAEs and MSEs become mush smaller than ther n-sample counterparts. Brooks, Faff and McKenze (1998) obtaned smlar results for the Australan ndustres, but t s not clear whether ther comparsons were made over the same perod for both n-sample and out-of-sample forecasts. When we compared n-sample and out-of-sample MAEs and MSEs over the same sample perod (.e., sub-perod 2) for each sector and each modellng approach based on the Modfed DM tests, such results are bascally reversed. For most sector/approach grds, n-sample MAEs and MSEs are sgnfcantly dfferent from, and smaller than, out-of-sample MAEs and MSEs. Ths s true especally n the cases of the Kalman flter, SV, SSSV, SS and OLS models; the only excepton s the approach where a number of n-sample MAEs and MSEs are found to be greater than out-of-sample MAEs and MSEs at a hgher than 10% level. Moreover, the basc concluson that the SV model s superor to all other approaches n terms of n-sample forecastng performance s not altered whether sub-perod 2 or the entre data sample perod s consdered. To preserve space, however, we do not present our results n the form of tables here, but they are avalable from the author upon request. 4. Concluson Modellng tme varaton of systematc rsk has been a central theme of several studes that look at some well-developed, large captal markets lke the US, the UK and Australa. Ths paper adds to the lterature a study on the New Zealand equty market by modellng the beta rsk of the New Zealand ndustry portfolos over the perod from January 3, 1997 to August 28, In dong so, however, we extended the prevous analyses of three major modellng technques to nclude the stochastc volatlty 19

21 model and the Schwert and Segun approach based on the stochastc volatlty model, and performed the Modfed DM test adjusted for ARCH that has not yet been appled n the prevous, smlar studes for forecast evaluaton. Evdence generated from ths study ndcates that the betas of all the NZ ndustry portfolos are also unstable. But whch modellng technque s more capable of capturng movements n systematc rsk? Manly by the n-sample MAE and MSE crtera, pror works for the UK and Australa have provded overwhelmng evdence n favour of the Kalman flter approach. Ths would also be true n the New Zealand case nvestgated here f the stochastc volatlty model was not consdered. By applyng the stochastc modellng technque n generatng beta observatons, we have found overwhelmng evdence that ths approach replaces the Kalman flter approach to become the optmal technque under the n-sample MAE and MSE crtera. However, the superorty of the stochastc volatlty approach s not mantaned n out-of-sample testng: the model s most favoured here, although the supportng evdence s less clear-cut than that for the stochastc volatlty model n the case of n-sample forecastng. One message from ths result s that, even wthout consderng the stochastc volatlty model, the Kalman flter technque s not the best to forecast returns out-of-sample for the New Zealand ndustry portfolos, unlke ther Australan counterparts. However, ths result s not to suggest that the superorty of the model s unversal for any out-of-sample forecastng perods. Indeed, the ablty of the stochastc model to capture contemporaneous shocks to volatlty suggests the use of ths model for short-run out-of-sample forecastng by rollng over the sample and re-estmatng the model. Further emprcal research may be pursued n ths drecton. 20

22 References Bollerslev, T. (1990). Modellng the coherence n short-run nomnal exchange rates: a multvarate generalzed ARCH model. Revew of Economcs and Statstcs 72, Bos, T. & Fetherston, T. A. (1992). Market model nonstatonarty n the Korean stock market, n Rhee, S G and Change, R P (eds): Pacfc Basn Captal Market Research Volume 3. Amsterdam: Elsever Scence Publshers. Bos, T. & Fetherston, T. A. (1995). Nonstatonarty of the market model, outlers, and choce of the market rate of return. Advances n Pacfc-Basn Fnance Management 1, Bos, T., Fetherston, T. A., Martkanen, T., & Perttunen, J. (1995). The nternatonal comovements of Fnnsh stocks. The European Journal of Fnance 1, Brooks, R. D., Faff, R. W., & Arff, M. (1998). An nvestgaton nto the extent of beta nstablty n the Sngapore stock market. Pacfc-Basn Fnance Journal 6, Brooks, R. D., Faff, R. W., & Ho, Y. K. (1997). A new test of the relatonshp between regulatory change n fnancal markets and the stablty of beta rsk of depostory nsttutons. Journal of Bankng and Fnance 21, Brooks, R. D., Faff, R. W., & McKenze, M. D. (1998). Tme-varyng beta rsk of Australan ndustry portfolos: a comparson of modellng technques. Australan Journal of Management 23, Chay, J. B. & Eleswarapu, V. R. (2001). Deregulaton and captal market ntegraton: A study of the New Zealand stock market. Pacfc-Basn Fnance Journal 9, Cheng, J. W. (1997). A swtchng regresson approach to the statonarty of systematc and nonsystematc rsks: the Hong Kong experence. Appled Fnancal Economcs 7, Faff, R. W. & Brooks, R. W. (1998). Tme-varyng beta rsk for Australan ndustry portfolos: an exploratory analyss. Journal of Busness Fnance and Accountng 25, Faff, R. W., Hller, D., & Hller, J. (2000). Tme-varyng beta rsk: an analyss of alternatve modellng technques. Journal of Busness Fnance and Accountng 27, Fama, E. F. & French, K. R. (1992). The cross-secton of expected stock returns. Journal of Fnance 47, Franses, P. H. & van Djk, D. (2000). Non-lnear tme seres models n emprcal fnance. Cambrdge: Cambrdge Unversty Press. Harvey, D. I., Leybourne, S. J., & Newbold, P. (1999). Forecast evaluaton tests n the presence of ARCH. Journal of Forecastng 18, Jacquer, E., Polson, N. G., & Ross, P. E. (1994). Bayesan analyss of stochastc volatlty models. Journal of Busness and Economc Statstcs 12,

23 Le, F., Brooks, R., & Faff, R. (2000). Modellng the equty beta rsk of Australan fnancal sector companes. Australan Economc Papers 39, McKenze, M. D., Brooks, R. D., Faff, R. W., & Ho, Y. K. (2000). Explorng the economc ratonale of extremes n generated betas: the case of US banks. The Quarterly Revew of Economcs and Fnance 40, Ruz, E. (1994). Quas-maxmum lkelhood estmaton of stochastc volatlty models. Journal of Econometrcs 63, Schwert, W. G. & Segun, P. J. (1990). Heteroskedastcty n stock returns. Journal of Fnance 45, Wells, C. (1994). Varable betas on the Stockholm exchange Appled Fnancal Economcs 4, Yu, J. (2002). Forecastng volatlty n the New Zealand stock market. Appled Fnancal Economcs 12,

24 Table 1 Summary Statstcs and Beta Pont Estmates for NZ Industry Portfolos Industry Sector Mean Skewness Kurtoss SD JB ARCH? OLS CuSumSq Agrculture *** *** *** *** Buldng *** *** *** *** Consumer *** *** *** **** ENERGY *** *** *** *** Fnance *** *** *** *** Food *** *** *** *** Forestry *** *** *** *** GOODS *** *** *** *** Intermedate *** *** *** *** INVESTMENT *** *** *** *** Lesure *** *** *** *** Meda *** *** *** *** Mnng *** *** *** *** Ports *** *** *** ** PRIMARY *** *** *** *** PROPERTY *** *** *** *** SERVICES *** *** *** *** Textles *** *** *** *** Transport *** *** *** *** Market ndex *** ***?? Note: Ths table presents summary statstcs for daly returns on the ndexes of New Zealand s sx group sectors and thrteen sngle sectors and on the NZSE TOTAL ndex coverng the perod from January 3, 1997 to August 22, The beta pont estmates generated by the OLS estmaton of equaton (1) are presented n the eghth column. The fnal column gves the CuSumSq test statstc for the stablty of each beta pont estmate. Whte's heteroscedastcty-adjusted t-ratos are used for beta pont estmates. ** Sgnfcantly dfferent from zero at the 5% level. *** Sgnfcantly dfferent from zero at the 1% level. 23

25 Table 2 The Estmaton Results of the (1,1) Model Sector a b c b? c? m Agrculture * *** *** Buldng *** *** Consumer * ** *** ENERGY *** *** Fnance *** *** *** Food *** *** *** Forestry ** *** *** GOODS ** *** *** Intermedate * *** *** INVESTMENT *** *** *** Lesure ** *** *** Meda * *** *** Mnng * *** *** Ports * *** *** PRIMARY ** *** ** PROPERTY *** *** *** SERVICES * *** *** Textles *** *** *** Transport ** ** *** Market ndex ** *** *** Note: Ths table presents the estmaton results of the (1,1) model n equaton (3) ftted to each ndustry portfolo return and n equaton (4) to the market return. The fnal column gves the correlaton coeffcent between each ndustry portfolo return and the market return. * Sgnfcantly dfferent from zero at the 10% level. ** Sgnfcantly dfferent from zero at the 5% level. *** Sgnfcantly dfferent from zero at the 1% level. 24

26 Table 3 The Estmaton Results of the Schwert and Segun Model Sector? 1 2 R 2 Agrculture *** *** Buldng *** *** Consumer *** *** ENERGY * *** ** Fnance *** *** *** Food *** *** *** Forestry *** *** GOODS ** *** Intermedate *** INVESTMENT *** *** *** Lesure *** *** Meda *** *** Mnng *** *** Ports ** *** *** PRIMARY *** *** PROPERTY *** *** SERVICES *** *** Textles * *** *** Transport *** ** Note: Ths table presents the estmaton results of the Schwert and Segun model n equaton (8) R t =? +? 1 R mt +? 2 R mt /h mt + e t where R t s the daly return for ndustry portfolo, R mt s the daly return for the market ndex, and h mt s the condtonal volatlty of the market return estmated usng the (1,1) model n equaton (4). The fnal column gves the coeffcent of determnaton. * Sgnfcantly dfferent from zero at the 10% level. ** Sgnfcantly dfferent from zero at the 5% level. *** Sgnfcantly dfferent from zero at the 1% level. 25

27 Table 4 The Estmaton Results of the Kalman Flter Model Sector T T a b q a qb Agrculture *** *** Buldng *** *** Consumer *** *** ENERGY *** *** Fnance *** *** Food *** *** Forestry *** *** GOODS *** *** Intermedate *** *** INVESTMENT *** *** Lesure *** *** Meda *** *** Mnng *** *** Ports *** *** PRIMARY *** *** PROPERTY *** *** SERVICES *** *** Textles *** *** Transport *** *** Note: Ths table presents the estmaton results of the Kalman flter model n equatons (10) and (11) for each ndustry portfolo return seres. *** Sgnfcantly dfferent from zero at the 1% level. The t-tests on T a =1 and on T b =1 (not reported here) show that T a and T b are not sgnfcantly dfferent from unty at the 10% level for all ndustry portfolos. 26

28 Table 5 The Estmaton Results of the Stochastc Volatlty Model Sector? t? s u s v Agrculture *** *** ***a Buldng *** *** ***a Consumer *** *** ***a ENERGY *** *** ***a Fnance *** *** ***a Food *** *** ***a Forestry *** *** ***a GOODS *** *** ***a Intermedate *** *** ***a INVESTMENT *** *** ***a Lesure *** *** ***a Meda *** *** ***a Mnng *** *** ***a Ports *** *** ***a PRIMARY *** *** ***a PROPERTY *** *** ***a SERVICES *** *** ***a Textles *** *** ***a Transport *** *** ***a Market ndex *** *** ***a Note: Ths table presents the estmaton results of the stochastc volatlty model n equaton (12) ftted to each ndustry portfolo return and to the market return. *** Sgnfcantly dfferent from zero at the 1% level. a Sgnfcantly dfferent from one at the 1% level. 27