Modeling the Volatility of Shanghai Composite Index

Transcription

1 Modeling he Volailiy of Shanghai Composie Index wih GARCH Family Models Auhor: Yuchen Du Supervisor: Changli He Essay in Saisics, Advanced Level Dalarna Universiy Sweden

2 Modeling he volailiy of Shanghai Composie Index wih GARCH family models Yuchen Du, 0 Absrac Since GARCH family models have been inroduced o he world, people ofen use hem o analyse volailiy and hey usually fi well. The aim of his paper is o find a beer GARCH model o fi he Chinese Shanghai Composie Index daa. This paper shows some properies of he common daily sock daa, hen i demonsraes he esimaion resuls of he GARCH model and he asymmeric power ARCH (APARCH). A las, we compare he modeling resuls beween GARCH model and APARCH model and conclude ha APARCH model wih AR() process has beer performance. Keywords: sock daa, volailiy, ARMA,GARCH, APARCH

3 CONTENT.Inroducion.... Background.... Lieraure Review....3 Aim...3.Daa...3. Index reurns Plos of daa Summary Saisics ARCH-LM es Auocorrelaion funcion Mehodology ARMA, ARCH, GARCH APARCH AIC & BIC... 4.Applicaion resuls... 5.Conclusion... 4 Reference... 5

4 .Inroducion. Background Nowadays, wih he developmen of economics all around he world, more and more people are concerned abou he everyday change of he financial marke. Mos people prefer a higher rae of reurn han jus puing money ino banks o ge less ineres. To inves in sock marke is a very common choice for hose invesors. I is well-known ha higher benefi usually comes wih higher risk in he invesmen environmen. Then i is reasonable ha he changes and rends issues of he sock marke are concerned by invesors much more. People can ge informaion abou he siuaion of he sock marke in many ways. Among all he symbols of he sock marke siuaion, volailiy is an essenial label. "For a definiion, volailiy is a measure for variaion of he price of a financial insrumen over ime" (Lin C, 996). Therefore he fuure sock price uncerainy could be presened as volailiy. Modeling and forecas volailiy need new models oher han he radiional linear regression models. The well-known radiional linear ools have heir own limiaions in he applicaion since hey always ignore he heeroskedasiciy of he daily sock daa.. Lieraure Review The demand of allowing invesors o handle volailiy which is reaed as he varying rends of he financial marke urges people o creae his new kind of model for heeroskedasiciy. The high-frequency sock marke daily daa usually has some special properies such as fa-ailness, excess kurosis and skewness. Modelling volailiy of his kind daa is a very ough ask. As menioned before, he mos imporan propery of he daa, heeroskedasiciy

5 means ha he error erm variance of he daa is no a consan. This propery caused he limiaion of radiional linear financial insrumens. The firs model of condiional heeroskedasiciy, he auoregressive condiional heeroskedasiciy (ARCH) was brough up by Engle (98). Then he Generalized ARCH (GARCH) model which is proposed by Bollerslev (986) and Taylor (986) has replaced he ARCH model in mos applicaions. Afer ha, many GARCH family models came ou, like Asymmeric Power ARCH (Ding, e al, 993), Exponenial GARCH (Nelson,99), Glosen- Jagannahan-Runkle GARCH (GJRGARCH) model composed by Glosen, Jagannahan and Runkle (993). Among he worldwide inroduced GARCH family models, he overwhelmingly mos popular GARCH model in applicaions has been he GARCH(,) model (Teräsvira, 006). GARCH family models are regarded as he bes ools o analyse financial daa considering he heeroskedasiciy. Among hem, GARCH (,) is widely used for his kind of daa which has fa-ailness, excess kurosis and skewness, because of hose properies, we can also connec i o he Asymmeric Power ARCH model. For he Shanghai composie index, Zhang, Cheng and Wang(005) has ried o use GARCH(,) and EGARCH(,) o analyse i. The ime period of he daa hey used for he paper is very shor and hey focused on he characerisic of EGARCH model in he empirical research..3 Aim The aim of his paper is o find he model which fis he high-frequency sock marke daa beer beween he mosly used GARCH(,) and APARCH(,) models. We also compare he resuls for all he models when hey are wih or wihou ARMA(,0) process

6 .Daa This paper uses he common sock index as he original daase. In People's Republic Of China, wo differen sock exchanges are operaed separaely. One is Shanghai sock exchange, he oher one is Shenzhen sock exchange. The Shanghai sock exchange is esablished much earlier and has larger amoun of lised companies, lised socks and more capializaion value han Shenzhen sock exchange. In summary, based on he echnology and he geographical advanage of Shanghai, he Shanghai sock exchange becomes he mos imporan and he main exchange in China. The above condiions are he reasons why he daa from he Shanghai sock exchange will be used in our applicaion. The Shanghai Sock Exchange was esablished on December.9h.990 and he ime period of he daase in his aricle sars from he very beginning and ends in December. 8h. 0.. Index reurns Here is he definiion of index reurn: le { y } be he ime series of he daily price of some financial asse. The serially reurn on he h day is defined as r = log( y ) log( y ) Someimes hese ime series of he reurn price are muliplied by 00 which make sure ha hey can be inerpreed as percenage changes. The muliplicaion may reduce numerical errors since he original reurns could be very small numbers.. Plos of daa Below is he plo of he daily daa of he Shanghai Composie Index, including everyday high, low and closing price daa and closing prices were 3

7 used as he arge o analyse. The Figure shows his ime-varying ime series daa has an obviously variabiliy and apparenly huge changes from 006 o 008. Figure : Plo of original daily daa Figure : Plo of index reurn daa The Figure is he plo of index reurn daa. The volailiys would be analysed by using he index reurn daa insead of he original index daa 4

8 hrough he way ha using he 00-imes-log difference of each daily closing price which is menioned in secion...3 Summary Saisics The Table demonsraes some summary saisics of he daa. There are 537 observaions in oal in his daase. Afer he log-difference process, we have 536 observaions. We can see ha he mean and median of he daa are boh very near zero and he sandard deviaion is quie small. The kurosis and skewness are excess which are exacly he characerisics of financial ime series. Table : Summary Saisics of Shanghai Index Reurns Values Min Max Median 0.06 Mean Sd. Dev..53 Skewness Kurosis 4.66 Number of obs. 536 Noe: This able shows summary saisics of reurns of Shanghai Composie Index daily closing price by using 00 imes log-differences..4 ARCH-LM es The ARCH-LM es is a Lagrange muliplier (LM) es which is frequenly used o es for he lag lengh of ARCH errors, in oher words, he ARCH-LM es is abou esing wheher he series has ARCH effecs a all (Engle, 98). 5

9 For he ARCH-LM es, we run a regression here, ε = α0 + ( α ε q i= i q) In his regression, α α = = α 0 is he null hypohesis and he = q = + e alernaive hypohesis is H α 0 or α 0 or α 0 : which means here would be a leas one of he esimaed parameer αi significan for sure. Besides, he es saisics follows χ disribuion wih q degrees of freedom. The p-value of he es in his paper equals o which is a very small one. Therefore he null hypohesis ha he series has no ARCH effecs has been rejeced. The resul of he es allows us o fi GARCH family models o he daa..5 Auocorrelaion funcion Figure 3 shows he auocorrelaion funcion resuls. If a financial ime series is auocorrelaed, i means ha he fuure values should depend on curren and pas value which suggess ha i is predicable. The resuls prove ha he reurns, reurn square, especially he absolue value of reurns are frequenly auocorrelaed. 6

10 Figure 3: Auocorrelaion Funcion of Reurns, squared reurns and absolue value of reurns 3.Mehodology Engle proposed a new model in 98 which is called AuoRegressive Condiional Heeroskedasiciy (ARCH). This is he firs model of condiional heeroskedasiciy. Aferwards in 986, Bollerslev brough up Generalized ARCH (GARCH) model. I replaced ARCH model in mos applicaions evenually. In 993, Ding, Engle and Granger, hey carried ou an asymmeric model called Asymmeric Power ARCH (APARCH). Models of Auoregressive Condiional Heeroskedasiciy (ARCH) are he mos popular way of parameerizing his dependence(teräsvira, 006). 3. ARMA, ARCH, GARCH The reurn series of a financial asse, { r }, is ofen a serially sequence wih zero mean and exhibis volailiy clusering. This suggess ha he condiional variance of { r } given pas reurns is no consan (Cryer and Chan, 008). This is he reason why we canno apply a simple linear regression o his kind of daa since hey consider he series as homoskedasic 7

11 series indeed. Here is he sandard ime series model: r = E( r Ω ) + ε Following Bollerslev (986), r here denoe a real-valued discree-ime process wih condiional mean and variance boh varies wih Ω -, where Ω - is he informaion se of all informaion hrough ime. r is also regarded as reurn in his paper. Then we can ake an AuoRegressive MovingAverage (ARMA) (p,q) model as he mean equaion, E ( r Ω )= µ ( θ ) µ ( θ ) = ϕ + ϕ θ ε 0 r + + ϕpr p + θε + + q q In financial applicaions, i is he common pracice o apply ARMA model o he reurn series as mean equaion. Since i may be oo resricive o assume ha he observed process is a pure GARCH, adding up an ARMA par considerably exends he range of applicaions. Then an ARCH model (Engle,98) can be reaed as he variance funcion, V ar(r Ω ) = E( ε Ω ) = h ( θ ) The ARCH(q) process funcion form is like his: h ω + αε = α ε q q The ARCH process inroduced by Engle(98) clearly recognizes he difference beween he uncondiional and he condiional variance allowing he laer o change over ime as a funcion of pas errors (Bollerslev, 986). As we menioned above, in mos applicaions, he ARCH model has been replaced by he Generalized ARCH (GARCH) model ha Bollerslev (986) and Taylor (986) proposed independenly. 8

12 The GARCH (p,q) process is given as h r = E( r Ω ) + ε ε Ω ~ N(0, h ) = ω + α ε β + + α qε q + βh + + p h p Here, he condiional disribuion of ε is supposed o be normally disribued wih zero mean and he condiional variance equals o h, same as ARCH process. We call he erm βih p i= i GARCH erm. For p=0, he process is simply he ARCH(q) process; for p=q=0, ε is only whie noise. Usually he simples GARCH(,) model works very well: r ε = E( r Ω ) + ε Ω- ~ N(0,h ) h = ω + αε + βh where ω > 0, α > 0, β > 0 3. APARCH In paricular, Ding, e al (993) invesigaed he auocorrelaion srucure of δ r, where r is he daily sock marke reurns, and δ is a posiive number as he funcion power. They found r had significan posiive auocorrelaions for long lags which we also concluded in secion.5. Moivaed by his empirical resul hey proposed a new general class of ARCH models, which hey call he Asymmeric Power ARCH (APARCH). The variance equaion of APARCH(p,q) can be wrien as: ε = z h 9

13 p i= z ~ N(0,) δ i γ iε i) + h = ω + α ( ε β h ω > 0, δ > 0, α > 0 < γ i i i q j= j j <, i =,, p, β > 0, j =, q This model is quie ineresing since i couples he flexibiliy of a varying exponen wih he asymmery coefficien (o ake he leverage effec ino accoun). Moreover, he APARCH includes seven oher ARCH exensions as special cases, such as: ARCH when δ =, γ = ( 0 i =, p), β = 0( j =, q) GARCH when δ =, γ = ( 0 i =, p) i i Taylor and Schwer GARCH(TS-GARCH) when δ, γ = 0 j j = i Glosen, Jagannahan, and Runkle(GJR-GARCH) whenδ = T-GARCH when δ = N-GARCH when γ = ( 0 i =, p), β = 0( j =, q) Log-ARCH Model when δ 0 i j 3.3 AIC & BIC The Akaike informaion crierion (AIC) is o measure he relaive goodness of fiing saisical models. I was developed by Hirosugu Akaike, under he name of "an informaion crierion" (AIC), and was firs published by Akaike in 974. In he general case, he radiional AIC defined as AIC = k ln( L) where k is he number of parameers in he saisical model, and L is he maximized value of he likelihood funcion for he esimaed model. In his paper, AIC is calculaed like his, 0

14 AIC = ( ln( L) + k) / N N here sands for sample size. Among a group of candidae models for he daa, he preferred model is he one wih he minimum AIC value. Based on ha, we can also come o anoher conclusion ha we need a maximum value of he log-likelihood funcion because ha he larger log-likelihood we have, he smaller AIC value we will ge. Besides, he Bayesian informaion crierion (BIC) is also a crierion for model selecion. I is based on he likelihood funcion and i is quie closed o Akaike informaion crierion (AIC). The formula for he BIC is: - ln p ( x k) BIC = ln L + k ln( n) The assumpion ha he model errors or disurbances are independenly and idenically disribued according o a normal disribuion is precisely suiable for he applicaion case in his paper. Therefore he AIC & BIC values are wo benchmarks for choosing a beer fiing model here. 4.Applicaion resuls Boh of he esimaions are processed under he condiional disribuion normaliy and processing wih he 5000 observaions. Afer he esimaion, as menioned before, he comparison beween GARCH models is based on hose crieria.

15 Table : Esimaion resuls of parameers in GARCH models wihin sandard error in brackes µ GARCH(,) 0.03 (0.0) AR() ω α β 0.06 (0.00) 0.59 (0.09) (0.0) ARMA(,0) +GARCH(,) 0.03 (0.0) (0.06) 0.06 (0.00) 0.59 (0.09) (0.0) γ APARCH(,) (0.007) (0.007) 0.4 (0.03) 0.83 (0.00) (0.04) δ (0.008) ARMA(,0) +APARCH(,) (0.06) 0.05 (0.05) (0.006) 0.5 (0.03) 0.89 (0.00) (0.05) (0.009) Table shows he esimaion resuls of GARCH models. Since he sandard GARCH(,) and APARCH(,) do no have an AuoRegressive process, hen hey do no have he parameer ϕ in ha process. For anoher, APARCH process has wo unique parameers γ and δ which GARCH process does no. From Table, we can see ha all he parameers come up wih very small sandard error which shows heir significance. Following he mehodology, he firs measuring of he comparison is log-likelihood which needs a larger value and hen he AIC value and BIC value are opposie which require smaller value. From Table 3, we can see ha wo APARCH models have boh smaller AIC value and BIC value, meanwhile, a larger log-likelihood which all suggess ha APARCH models fied his daa beer han he well-known sandard GARCH models. Beween hese wo APARCH models, ARMA(,0) plus APARCH(,) model performs even beer which cerifies ha when we

16 allow for an ARMA par, i will obain a beer resul. This resul shows ha even hough he GARCH models are capable of mos he analysis of financial reurn ime series, someimes i sill depends on he daa iself, ARMA or oher linear ools may remain useful in pracical cases. Bu he ARMA(,0)+GARCH(,) model does no show any beer resul in sandard GARCH models which shows limiaion and uncerainy of ARMA process. Table 3: Comparison of GARCH family models GARCH(,) ARMA(,0) +GARCH(,) APARCH(,) ARMA(,0) +APARCH(,) Log-likelihood AIC value BIC value Conclusion Among all he esimaions and comparisons based on some cerain crieria, all he parameers are significan which shows ha he Shanghai Composie Index daa could be modeled by GARCH family models appropriaely. The comparison resul suggess ha APARCH models could fi he daa beer. The bes performance is from ARMA+APARCH model which suggess ha i is efficien o combine he mean and variance funcion ogeher. However, adding ARMA process could no confirm ha he goodness-of-fi would always show a beer resul for sure, such as he case of sandard GARCH models, alhough i is considered o exend he range of applicaions. All he resuls above shows ha here may exis some special properies in Chinese sock index daa. The ime period of he daa for his paper is quie long and here are very obvious up and down during he year 006 and 008, 3

17 Tha could be a break in his daa which is on he very advanced level modeling and forecasing and far beyond he range of his paper. Anoher assumpion in his paper is ha he disribuion of he error erms is supposed o be normally disribued. We could also consider i under he suden- disribuion since disribuion has some properies such as fa-ailness which is exacly he same as he financial ime series daa. Esimaions and comparisons by considering differen disribuions could be more beneficial for he applicaions. The sock index is influenced by a lo of exernal facors such as inflaion, igh or loose moneary policy and so on. I could hardly be prediced or modeled very accuraely. Especially for China, he research of he composie index is much less han he oher famous sock index around he world. Considering more differen disribuions, more differen GARCH family models could be our furher sudy. 4

18 Reference []Akaike, H. (973). Informaion heory and an exension of he maximum likelihood principle. In B. N. Perov and F. Csaki (Eds.), Second inernaional symposium on informaion heory, pp []Bollerslev T.,(986). Generalized auoregressive condiional heeroskedasiciy. Journal of Economerics, 3(3), pp [3] Cryer J.D, Chan(008), Times Series Analysis wih Applicaions in R, nd ed., Springer Science Business Media, LLC [4]Ding, Z., Engle, R.F., & Granger, C.W.J., (993). A long memory propery of sock marke reurns and a new model. Journal of Empirical Finance,, pp [5]Engle R.F. (Jul,98). Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion. Economerica.50(4), pp [6]Glosen. L. R. Jaganahan. R. and Runkle. D., 993 On he relaion beween he expeced value and he volailiy of he normal excess reurn on socks. Journal of Finance 48: [7]Lin. C., 996 Sochasic Mean and Sochasic Volailiy A Three-Facor Model of he Term Srucure of Ineres Raes and Is Applicaion o he Pricing of Ineres Rae Derivaives. Blackwell Publishers. [8]Nelson D.B., (99); Condiional Heeroscedasiciy in Asse Reurns: A New Approach, Economerica, 59,

19 [9]Taylor, S., (986), Modelling Financial Time Series, Wiley, Chicheser. [0]Teräsvira T., (006), An Inroducion o Univariae GARCH Models. SSE/EFI Working Papers in Economics and Finance, No. 646 []Zhang, Cheng and Wang, (005), An Empirical Research in he Sock Marke of Shanghai by GARCH Model. Operaion research and managemen science, 4-4 6