GARCH Models of Time Series on DAM

A C T A U N I V E R S I T A T I S L O D Z I E N S I S

FOLIA O EC O N O M IC A 206, 2007

Alicja Ganczarek*

GA RC H M O D ELS O F TIM E SERIES O N DAM**

Abstract. In this paper an analysis o f the time series on the D ay Ahead M arket (D A M ) o f the Polish Power Exchange is presented. In this analysis Generalized Autoregressive C onditional H eteroscedasticity (G A R C H ) models are used to describe the time series o f rates o f return o f price o f electric energy on D A M . This analysis is based on the data from July 2002 to June 2004.

Key words: Polish Power Exchange, D ay Ahead M arket, Balance M arket, Autoregressive C onditional H eteroscedasticity, Generalized Autoregressive Conditional H eteroscedasticity, M axim um L ikelihood M ethod, Akaike's inform ation criterion, Schwarz’s consistent criterion, H an n an -Q u in n ’s consistent criterion, Rissanen’s stochastic com plexity criteria.

1. IN T R O D U C T IO N

The D ay Ahead M arket (DAM ) was the first m arket, which was es­ tablished on the Polish Power Exchange. This whole-day m arket consists o f the tw enty-four separate, independent m arkets where participants can freely buy and sell electricity. The breakthrough in the development of the Polish Power Exchange was m ade 1st July 2000, when the first transaction was completed on the D A M . Advantage o f the Exchange is that all the participants o f m arket can buy and sell electric energy, independently whether there are producers or receivers o f electric energy.

Since 1st July 2002 Balance M arket (BM) - technical m arket, which looks after balance on Polish energy m arket, has introduced additional price: Price Accounting Deviations of sale PADs and I rice Accounting

* M .S c., D epartm ent o f Statistics, University o f E conom ics in Katowice. ** Research supported by Polish scientific grant K B N 1 H 02B 024 27.

Deviations of purchase PA Dp. These prices should helpin expectation future dem and for the electric energy on whole-day and futures m arket.

A lot o f the empirical results show that the time series of rates of return aren ’t dependent only on the firs m om ent o f the data:

- the volatility o f rates o f return is characterizes with volatility clustering, it is cause o f heteroscedasticity and the growing o f variance of terror term,

- the rates o f return have the leptokurtic distribution and the fat-tailed, the distribution o f the returns d a ta have the substantially heavier tails than a norm al distribution,

- the volatility of rates o f return is inverse correlation with the volatility o f their variance - leverage effects,

- the long m em ory processes in the series o f variance, the squares returns d ata are characterizes with the significant autocorrelation coefficients.

R. F. E n g l e (1982) introduced the Autoregressive Conditional H etero­ scedasticity (A R C H ) m odel, which incorporated into variance equation some o f the stylized characteristics comm on to the second of m om ent o f financial basset price inform ation.

T he A R C H (q) m odel is defined as 2. M E T H O D O L O G Y Z t = p + y/'ht et

(

)

^ - m ean of rates of return, noise e, ~ N (0,1),

N (0 A ),

c, - coefficient, c0, cq > 0, c, > 0 (i = 1,..., q — 1),

if £ ct < 1, then the time series Z, is strict stationary,

A m ore generalized version o f A R C H , the Generalized Autoregressive C onditional Heteroscedasticity GA RC H , was formulated by Engle’s graduate student T. B o l l e r s l e v (1986). In comparison to the A R C II model, the G A R C H m odel allows a potentially more complete representation o f the dynam ic nature o f the process by which the conditional variance in financial m arket d ata m ay evolve.

The G A R C H (p, q) model is defined as

Z, = /л + \lh,E, (3)

where c0, cq, bp > 0 and otherwise coefficients are nonnegativees,

if £ c ,+ í]f y < l> then the time serieS

Z'

ÍS StrÍCt stationary‘

i ■« 0 i - 1

The process G A R C H is characterizes with return to mean. I he mean long-term variance o f this process is defined as

V = ---— ---- :— _{' q P} (5)

1 - S c , - I Л

A n effective m ethod used to estimate the coefficients in A R C H (q) and GARCHQ?, q) m odels is maximum likelihood m ethod (ML). 'I he coefficients are the results o f maximum o f a function

1„L " t a 2 ^ X > * , 4 £ f ®

2 2. , = , í ™ t t

where Z „ ..., Z N arc empirical rates of return.

A lot o f different model selection criteria are proposed in selecting an optim al A R C H model. The m ost o f the standard class o f these model selection procedures involve minimizing some loss function.

One o f the m ost popular models is H. A k a i k e ’ s (1973) information criterion, which takes the form

A l С = -21n L + 2k (?)

where к is num ber o f the coefficients.

G. S c h w a r z (1978) developed a consistent criterion based on Bayesan argum ents

where N is the sample size.

E. J. H a n n a n and B. G. Q u i n n (1979) proposed the consistent criterion for the order o f an autoregressive Rusing the law o f the iterated logarithm

J. J. R i s s a n e n (1987) developed a m odel selection criterion, which is a sample approxim ation to a m easure of stochastic complexity

The A kaike’s (7) and Schwarz’s (8) criterions are m ost popular and very often used. H. M i t c h e l l and M. M c K e n z i e (2003) resumed and com pared a lot o f the used criteria. The results o f their work, based on simulated d a ta suggest, that HQ and R C L provide a superior level of perform ance for A R C H and G A R C H process com pared to the m ore com ­ m only used criteria.

In this part of paper the results of estim ation o f A R C H and G A R C H m odels are presented. T o analysis the hourly logarithm ic rates o f return of price o f electric energy on DAM were noted from 01.07.2002 to 30.06.2004 are used. The program s such as: EX CEL, G R E T L and STA TISTICA are used to calculate. The volatility of rates o f return on D A M is characteristics with volatility clustering (Fig. 1).

B1C = -21n L + 2 k \ n N (8)

HQ = - ln L 4- 2/c ln (ln N))

(9)

(10)

o 100 200 300 400 500 600 700 Time in hours from 1 V I I 2002 lo 30 V I 2004 (I)

Fig. 1. Tim e series p lot o f rates o f return o f price o f electric energy 1.07.02-30.06.04. S o u r c e : author’s own com putations.

The rates of return have the leptokurtic distribution (Fig. 2) and fat-tailed (Fig. 3).

- 0 . 3 0 - 0 . 2 5 - 0 .2 1 - 0 . 1 6 - 0 . 1 2 - 0 . 0 / - 0 . 0 2 0 . 0 2 0 . 0 7 0 .1 1 0 . 1 6 0 . 2 0 0 . 2 5 0 . 2 9 Value of rales of retun of price of electric energy (Z,)

Fig. 2. H istogram o f logarithmic rates o f return o f price o f electric energy S o u r c e : author’s ow n com putations.

n o r m a l d i s t r i b u t i o n z l ( p r i c e ) = 2 . 7 5 1 5 E - 6 + 0 . 0 7 4 9 X

Theoretical quantile

Fig. 3. Q uantile-quantile p lot o f logarithm ic rates o f return o f price o f electric energy S o u r c e : author’s ow n com putations.

F igure 4 shows autocorrelation for 168 lagged variables and their square. T he price of electric energy is characterize with daily, weekly and yearly seasonal. T he significant autocorrelation coefficients m ean also, th at the logarithm ic rates and square o f logarithm ic rates o f return of price of electric energy have the long mem ory processes in the series of variance.

In Tab. 1-3 the results o f estimate the A R CH (q) and G A R C H (p, q) models, by G R E T L program , are presented. F o r q > 1 and p > 2 these m odels can obtain convergence.

T a b l e 1

A R C H (1) m odel results for 17 276 observations

Coefficient Std. error (-statistic p-value

-0.001 709 0.000 656 -2 .6 0 5 600 0.009 179

Co 0.005 092 0.000 193 26.409 400 < 0 .0 0 0 01

Th e au to c o rr e la ti o n ra nk (( - p, p = 1 ... ... 1 6 8 ) 1 + . 0 4 1 . « 3 7 6 1 3 + . 0 1 0 . 0 0 7 6 2 5 + . 1 2 9 . 0 0 7 6 3 7 + . 0 0 9 . 0 0 7 6 4 9 + . 1 1 5 . 0 0 7 6 6 1 + . 0 1 1 . 0 0 7 6 7 3 + . 1 1 7 . 0 0 7 6 8 5 + . 0 1 9 . 0 0 7 6 9 7 + . 1 0 3 . 0 0 7 6 1 0 9 + . 0 2 2 . 0 0 7 6 1 2 1 + . 1 1 4 . 0 0 7 6 1 3 3 + . 0 1 9 . 0 0 7 6 1 4 5 + . 1 1 6 . 0 0 7 6 1 5 7 + . 0 1 1 . 0 0 7 6 -1.0 -0.5 0.0 0.5 V a l u e o f a u t o c o r r e l a t i o n o f r a t e s o f r e t u r n o f p r i c e o f e l e c t r i c e n e r g y ( r ( Z f , Z f .p ) ) 2 9 . 0 8 . 0 0 0 0 1 0 2 5 . . 0 0 0 0 7 0 1 9 . . 0 0 0 0 7 7 6 8 . . 0 0 0 0 1 2 1 E 2 . 0 0 0 0 1 2 7 E 2 . 0 0 0 0 1 6 7 E 2 . 0 0 0 0 1 7 4 E 2 . 0 0 0 0 2 H E 2 . 0 0 0 0 2 2 0 E 2 . 0 0 0 0 2 5 8 E 2 . 0 0 0 0 2 6 5 E 2 . 0 0 0 0 3 0 9 E 2 . 0 0 0 0 3 1 6 E 2 . 0 0 0 0 0 1.0 1 + . 1 8 5 . 0 0 7 6 - - - '- - - s ’ — ’- - - 5 9 3 . 7 . 0 0 0 0 1 3 + . 1 0 8 . 0 0 7 6 _{W 2 9 2 2 . . 0 0 0 0} 2 5 + . 1 5 5 . 0 0 7 6 ŕ - - - 8 1 5 5 . . 0 0 0 0 3 7 + . 0 9 5 . 0 0 7 6 в _{9 6 3 3 . . 0 0 0 0} 4 9 + . 1 5 2 . 0 0 7 6 в — 1 3 2 Е 2 . 0 0 0 0 6 1 + . 0 9 5 . 0 0 7 6 ff e _{1 4 6 Е 2 . 0 0 0 0} 7 3 + . 1 5 7 . 0 0 7 6 w * — 1 7 9 Е 2 . 0 0 0 0 8 5 + . 0 7 5 . 0 0 7 6 w 1 9 0 Е 2 . 0 0 0 0 9 7 + . 1 8 3 . 0 0 7 6 f f " - 2 2 5 Е 2 . 0 0 0 0 1 0 9 + . 1 0 2 . 0 0 7 6 a _{2 4 0 Е 2 . 0 0 0 0} 1 2 1 + . 1 6 3 . 0 0 7 6 _{2 7 3 Е 2 . 0 0 0 0} 1 3 3 + . 0 9 6 . 0 0 7 6 Ц _{2 8 7 Е 2 . 0 0 0 0} 1 4 5 + . 1 5 5 . 0 0 7 6 Ь 3 2 7 Е 2 . 0 0 0 0 1 5 7 + . 1 1 1 . 0 0 7 6 . 0 0 0 0 3 4 5 Е 2 0 . 0 0 . 0 0 0 0 1.0 -0.5 0.0 0.5 1.0 T h e a u t o c o r r e l a t i o n o f s q u a r e o f r a t e s o f r e t u r n o f p r i c e o f e l e c t r i c e n e r g y (r[Z}, Z tí))

Fig. 4. A utocorrelation p lot o f logarithm ic rates and square o f logarithm ic rates o f return o f price o f electric energy S o u r c e : author’s ow n com putations.

to

T a b l e 2 G A R C H ( 1, 1) m odel results for 17 276 observations

Coefficient Std. error (-statistic p-value

0.000 441 0.000 472 0.933 200 0.350 734

c o 0.000 005 0.000 003 1.623 200 0.104 570

C1 0.015 065 0.004 710 3.198 100 0.001 386

ft, 0.984 006 0.005 129 191.843 500 < 0 .0 0 0 01

S o u r c e : author’s own com putations.

T a b l e 3 G A R C H (2 ,1 ) model results for 17 276 observations

Coefficient Std. error t-statistic p-value

A 0.000 302 5 0.000 461 6 0.655 300 0 0.512 266 0

c 0 0.000 010 9 0.000 006 9 1.573 100 0 0.115 714 0

Cl 0.031 981 8 0.009 984 6 3.203 100 0 0.001 362 0

ft, 0.194 275 0 0.026 399 3 7.359 100 0 < 0 .0 0 0 01 ft 2 0.771 826 0 0.031 379 6 24.596 500 0 < 0 .0 0 0 01 S o u r c e : author’s own com putations.

T a b l e 4 M odel selection results for 17 276 observations

A R C H ( l ) G A R C H (1, 1) G A R C H (2, 1) Sum o f coefficients 0.317 5 0.999 5 0.998 4 Log-likelihood 19 323.58 22 192.37 22 244.39 A lC -3 8 643.17 -4 4 378.75 - 4 4 480.77 B1C -3 8 608.14 -4 4 326.21 - 4 4 410.72 HQ -1 9 314.47 -2 2 178.71 - 2 2 226.16 R C L -1 9 311.05 - 2 2 173.71 - 2 2 219.50 V 0.007 4 0.005 5 0.005 7 S v 0.086 2 0.074 0 0.075 4

All param eters in A R C H (l) model are significance. In G A R C H models significance are only these coefficients, which are responsible for the lagged variables o f volatility.

In Tab. 4 we com pare these three models based on criterions, which were presentedin second p art of this paper. In all models the sums of the coefficients are less then one, so all models are strict stationary. The G A R C H (2, 1) m odel has the smallest loss function. We can write the G A R C H (2, 1) m odel based on results from Tab. 3:

Ż, = 0.000 302 5 + V V t,

ht = 0.000 010 9 + 0.031 918 8 Z t2_i + 0.194 575 0/jt_ , + 0.771 826 0 h t_ 2.

The m ean long-term variance o f this process equals 0.0057, so the hourly residuals standard deviation of rates of return for this data set equals 7.54%.

In the next step the rests o f GARCH(2, 1) model are analyzed. On the Fig. 5 the residuals plot against time is presented.

Time in hours from 1 VII 2002 to 30 V I 2004 (Í)

Fig. 5. Residuals plot against time the G A R C H (2, 1) model S o u r c e : author’s own com putations.

The empirical rates of return are described well by the generalized autoregressive conditional heteroscedasticity, if time series

N (0 .1 ) (11)

Л

where ht, fi - are the characteristics, which are estimated on base the Z, process.

The G A R C H models describe well the real process Z t, if the time series o f residuals (11) have normal distribution.

U nfortunately the time series o f residuals of G A R C H (2, 1) model have the leptokurtic distribution (Fig. 6) and fat-tailed (Fig. 7).

3 5 0 0 § 3 0 0 0 § f 2 5 0 0 О CĽ о 2 000 0 1 1 5 0 0 "О '8 i 1000 _о ф е 5 0 0 =3 Z о -3.4685 -2.5511 -1.6338 -0.7164 0 .2010 1.1184 2 .0 3 5 8 2.9531 Value of the residuals of G A R C H (2 ,1 ) (£,)

Fig. 6. Histogram o f time series S o u r c e : author’s own com putations.

8 6 4 сг

I 0

The param eters o f residuals series (Tab. 5) dem onstrate the difference between norm al distribution and the distribution of empirical residuals. The residuals have the right asymmetry and leptokurtic distribution. Standard deviation is close to one but m ean o f residuals equals -0.0067.

T a b l e 5 Parameters o f distribution o f time series £,

Parameters Values M ean -0 .0 0 6 7 M edian -0 .0 0 6 2 M ode -Standard deviation 1.000 3 K urtosis 4.556 9 Skewness 0.540 0

S o u r c e : author’s own com putations.

4. C O N C L U SIO N

This empirical exercise shows, that the hourly rates or return o f price of electric energy depend on the lagged variables of volatility. A lthough that, the classical G A R C H models aren’t well described the rates of return, they are better than models, which establish the const variance at time. The sensible difference between empirical and theoretical distribution means that the Generalized Autoregressive C onditional Heteroscedasticity models based on norm al distribution shouldn’t be used to describe the rates o f return of prices o f electric energy. In an attem pt to capture the leptokurtosis common to financial returns data, the A R C H family o f models m ay be extended to assume some other density. Typically m odification to the standard class of m odel G A R C H involves replacing the standard norm al density with some other assumed distribution for example r-density or the G E D density.

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Alicja Ganczarek

M O D E L E GARC H SZ E R E G Ó W C Z A SO W Y C H N A R D N

W pracy została przedstawiona analiza szeregów czasow ych stóp zwrotu cen energii elektrycznej notow anych na rynku dnia następnego (R D N ) Towarowej G iełdy Energii SA od lipca 2002 d o czerwca 2004 r. za pom ocą modeli G A R C H . Celem pracy jest odpow iedź na pytanie, czy m odele G A R C H efektywnie opisują kształtowanie się cen energii elektrycznej na parkiecie polskiej giełdy energii i czy m ożna je w ykorzystyw ać d o m odelow ania szeregów czasow ych stóp zwrotu cen energii elektrycznej.