Dynamic Bayesian Inference in GARCH Processes with Skewed-t and Stable Conditional Distributions

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 OECONOMICA 192, 2005

M a t e u s z Pipień*

DYN AM IC BAYESIAN INFERENCE IN GARCII PR O C E SSE S W ITH SKEW ED-T A N D STABLE C O N D IT IO N A L D IST R IBU TIO N S**

Abstract. In AR(1)-GARCH(1, 1) framework for daily returns, proposed and adopted by Bauwens and Lubrano (1997), Bauwens et al. (1999), Osiewalski and Pipień (2003), we considered two types o f conditional distribution. In the first model ( M ,) we assumed conditionally skewed-i distribution (defined by Fernandez and Steel 1998) while the second GARCH specification ( M 2) is based on the conditional stable distribution. We present Bayesian updating technique in order to check sensitivity o f the posterior probabilities o f considered specifications with respect to new observations included into dataset. We also study differences between Bayesian inference about tails and asymmetry o f the conditional distribution o f daily returns and between one-day predictive densities o f growth rates obtained from both models. The results o f dynamic Bayesian estimation, prediction and comparison o f explanatory power of models M , and M 2 are based on very volatile daily growth rates o f the WIBOR one-month interest rates and daily returns on the PLN/USD exchange rate.

Keywonis: stable distributions, skewed-! distributions, Bayesian updating, univariate GARCH. JEL Classification: С И , C32, C52.

1. INTRODUCTION

C om m only used tool in forecasting the volatility o f the financial tim e series, nam ely G A R C H processes, were initially defined as a white noise stochastic processes with conditionally heteroscedastic n o rm al d istribu tion . A fter Bollerslev’s (1986) definition o f G A R C H scheme, m o re lep to k u rto tic co n d itio n al d istrib u tio n s (th an those o f norm al) have been also proposed

* Dr (Ph.D., Assistant Professor), Department of Econometrics, Cracow University of Economics.

and applied. F o r exam ple, Bollerslev (1987) presented estim ation o f the conditionally S tudent-t G A R C H (with unknow n degrees o f freedom p aram e­ ter). N elson (1991) considered G A R C H -ty p e process with generalised error distribution (G E D ). Rachev and M ittnik (2002) present results o f m odeling the volatility o f the daily returns using G A R C H processes w ith conditional W eibull, D o u b le W eibull, m ixture o f norm als an d L aplace distribu tio ns.

G A R C H processes w ith co n ditional stable d istrib u tio n s have been also considered (e.g. M cC ulloch 1985, Liu and B rorsen 1995, P an o rsk a et al. 1995, M ittnik et al. 2002 and Rachev and M ittnik 2002). T h e m ain advantage o f stable G A R C H processes is the fact th a t co n d itio n al n o rm ality can be tested in this fram ew ork. A dditionally, stable d istrib ution s are able in general to c a p tu re heavy tailedness an d possible skew ness o f th e co n d itio n al d istrib u tio n o f returns.

F ernandez and Steel (1998) proposed a generalization o f S tudent-г distribu­ tion, nam ely the skewed S tudent-t distribution, which allowed in a very simple way for heavy tails as well as for possible distributional asym m etry. Osiewalski and Pipień (2003) presented Bayesian estim ation and fo recasting in G A R C H m odels w ith co nditional Skewed-£ distribution . T he m ain p u rp o se o f Pipień (2004) was Bayesian com parison o f A R (1)-G A R C H (1, 1) m odels with skewed-ŕ and stable conditional distributions. Pipień (2004) presented posterior probabili­ ties o f m odels, posterior distributions o f com m on and m odel specific param e­ ters as well as discussed differences between predictive d istrib ution s generated from b o th specifications. T hese em pirical results were based on three tim e series, nam ely daily returns o f the P L N /U S D exchange rate, daily returns on the W arsaw Stock E xchange index (W IG ) and daily g row th rates o f the W IB O R one m o n th zloty interest rate.

T h e m ain goal o f this p aper is to apply Bayesian u p d atin g technique in o rd e r to check sensitivity o f the posterio r p robabilities o f skewed-£ and stable G A R C H m odels, w ith respect to new ob serv atio n s included into d atase t. T h e results o f dynam ic Bayesian co m p arison o f ex p lan ato ry pow er o f conditionally skew ed-г G A R C H (1, 1) m odel (M x) and conditionally stable G A R C H (1 , 1) m odel (M 2) are based on daily g row th rates o f the W IB O R o n e-m o n th in terest rates (dataset A ) and daily re tu rn s o n the P L N /U S D exchange ra te (d a ta set B). In b o th cases (A and B) startin g from tim e series consisting o f 1 0 0 observations, every tim e when we updated daily observation

into d ataset, we recalculated posterior distribution o f p aram eters and posterior p ro babilities o f m odels M x an d M 2. We also study differences between B ayesian inference a b o u t tails and asym m etry o f the co n d itio n al distribu tion o f daily retu rn s o btained from both m odels. As a result o f ap plication o f d yn am ic Bayesian inference, we present highest p o sterio r density intervals o f tail and asym m etry param eters fo r m odel M l and M2 and one-step

2. SKEWED STUDENT-T AND STABLE DISTRIBUTION

F ollow ing the definition in F ern an d ez and Steel (1998) let d en o te by z a ran d o m variable with skewed-t distribution with v > 0 degrees o f freedom ,

m o d al p aram eter ц, inverse precision hO and asym m etry p a ram eter h > 0 у > 0 (z ~ Skt((v, n , h , y ) . T he density function o f the d istrib u tio n o f z is given by the form ula:

w here / , ( x | v, ц, h) denotes the value o f the density fu nction o f the S tu d e n t-г d istrib u tio n w ith v > 0 degrees o f freedom , m odal p a ram eter /i and inverse

precision h > 0, calculated a t p o in t x:

T h e sh ap e p a ram eter v > 0 controls tail behav ior, m o d e ц an d inverse precision h are the location and dispersion m easures. P ara m e ter у ca p ­ tu res possible asym m etry. In general y 2 is the ra tio o f the prob ab ility m asses on the righ t and on the left side o f the m o de o f th e distribu tion o f z. H ence, if у = 1, then z follows sym m etric S tu d en t-ŕ d istribu tio n. U n d er sym m etry (y = 1) it is also clear, th a t, for v > 1, E(z) exists and is equal to ц.

T h e class o f stable distrib u tio n s is defined as a p aram etric family of continuous random variables closed with respect to the o peratio n o f summing. H ence, for any finite subset {wt , ..., w„} o f stable ra n d o m variables, the linear com bination w = + ... + anwn has also stable distribution (al 5 ..., a„

are real num bers). A nalytic expression for the ch aracteristic fu nction o f stable ra n d o m variable is given as follows:

I ((v + D/2)) r (v/2) s/nvh [1 + (/i v r 1. ( x - p ) T (v+1)/2 (2) cp{t) - exp 1 c o (|i|, a)

]}•

(e.g., Z o lo tare v 1961). In (2) the shape p aram eter a e (0 ,2 ] called index o f stability o r ch aracteristic exponent) defines th e “ fatness o f tails” o f density function (large a im plies thin tails), ц and h are th e lo catio n and scale p a ra m e te rs, ß e [ - l , 1] is the skew nes p a ra m e te r (th e sym m etric stable

d istrib u tio n corresp o n d s to ß = 0). F o r a e [ l , 2] an d ß = 0 the location ц is also equal to th e expected value o f ran d o m variable w. W e den o te by w ~ Sta(<x, n , h , ß ) , th a t w is stably distributed with index o f stability a, locatio n p aram eter \x, scale h and skewness ß. T here are th ree cases, where the closed form expressions for the density o f th e stable ra n d o m variable is know n. A norm al distribution is the case with a = 2, a C auchy distribution is the case w ith a = 1 and ß — 0, a Levy d istrib u tio n co rresp o n d s to the case w ith a = 0.5 and a. = 1.

P ractical ap p licatio n o f stable ran d o m variables in econ om etric m odeling requires deriving density function o f ran d o m v ariable w. It can obtained as th e integral o f (2):

Ci) 1 +„°° ,

f s . a( w\ ci ,n, h, ß) = = - Í e‘w‘<p(t)dt, — 00

and h ave to be ap p ro x im ated by num erical in teg ration (e.g. M ittn ik et al. 1999, R achev and M ittn ik 2002).

3. COMPETING GARCH SPECIFICATIONS

L et d en o te by Xj the value o f a currency (stock m a rk e t index, interest rate, exchange rate) a t tim e j. F ollow ing Bauwens and L u b ra n o (1997), Bauw ens et al. (1999), Osiewalski and Pipień (2003) let assum e an A R (2) process for \nXj with asym m etric G A R C H (1, 1) error. In term s o f logarithm ic g row th rates y} = 1 0 0 In (xj/x j- !) o u r basic m odel fram ew o rk is defined by the follow ing equation:

(4) y j - ô = p - i y j - i - ö + ö ^ n x j - i + E j , j = 1 , 2 , . . .

In the first m odel, M ls we assum e for the e rro r term Ej in (4), th at Ej = Zj(hj)o s , w here Zj are independent, skewed S tu d en t-t ra n d o m variables, with v > 0 degrees o f freedom param eter, m ode e ( — o c , + o o ), unit precision and asym m etry param eter y > 0; i.e. Z j ~ i i Skt(v,Ci,Ly)- D efining hj we follow

G lo sten et al. (1993) asym m etric G A R C H (1 , 1) specification:

w hich allow s to m odel asym m etric re actio n o f co n d itio n a l d ispersio n m easu re hj to positive an d negative sign o f sho ck e j - i . T h e o riginal G A R C H (1 , 1) form u latio n proposed by Bollerslev (1986) can be obtained from (5) by im posing restriction a j a f = 1. In (7) we also tre a t h0 as an ad d itio n al p aram eter. M odel M , assum es, th a t th e co n d itio n al d istrib u ­ tion (given the past o f the process, y / j - i , an d th e p aram eters) o f the e rro r term e; is the skew cd-i distribu tio n w ith v > 0 degrees o f freedom

p a ra m e te r, m o d e ^ e ( — oo, + oo), inverse p recision h} an d asym m etry p aram eter у > 0:

e j l V j - u M i * Sk t ( v , £ i , h j , y ) . j = 1, 2, ...

In m odel M 2, Ej = w^hj )0 5, where Wj are independent stable rand om variables w ith a e (0,2], location p aram eter i 2 e — ° o ,+ со), u n it scale an d skewness

p aram eter ß e [ - \ , 1]; i.e. Wj~iiSta(oi,C2, 1,/D- Ju st like in m odel M j we

assum e for hj asym m etric G A R C H (1 , 1) process, (5). In specification M2

Ej has conditional (with respect to ц/ j- 1 and the param eters) Stable distribution

w ith a e (0,2], locatio n C2e ( — oo, + oo), scale p aram eter h° 5 and skewness

№ 1,1]:

Ejlij/j-u M 2 ~ Sta(a,C2, h j ' 5, ß), j = 1, 2, ...

Let d en o te by 0 = ( ô , p , ô 1, a 0, a l , a { , b í , h 0) the vector o f all com m on p aram eters for b o th , M t and M 2, m odels. W e d en o te by r\x = ( C i,v ,y ) the vector o f m odel specific param eters in M x; r/2 = (C2,ct,ß) g ro up s ad dition al p aram eters fo r M 2. In m odel M , the co nditional d istrib u tio n o f is the skew ed-i d is trib u tio n w ith v > 0 degrees o f freedo m p a ra m e te r, m o de

/Ąl) = ô + р(у}- j — S) -(- ln x j - i + Cih°'5, inverse precision hj and asym m etry param eter y > 0:

^ P ( y j \ ' V j - i M l ,0,Til ) = f SkAy]\n'}í), h],y), j = 1, 2, . . .

In specification M 2 yj has conditional stable d istrib u tio n with a e (0,2], location pfjZ) = ö + p{yy . 1 - S ) + S l ln x 7_ ! + C2hj S, scale param eter (hj)0 5 and skewness ß e [ - \ , \ \ .

(7)

Viyj W ] - u M 2,0,ri2) = f s ta{y)\u.,tf\hO

j i ,ß),

In b o th m odels th e conditional distrib u tio n o f y t is heteroscedastic, where tim e varying dispersion m easure hj follows asym m etric G A R C H (1 ,1) equation (5). T h e degrees o f freedom p aram eter, v > 0 an d the ch aracteristic exponent a e ( 0 ,2 ] enable also fat tails o f p(yj \ii/j-l , M i, 0, ^ ;) (i = 1,2). T he possible

asym m etry o f conditio n al d istrib u tio n o f y} can be m odelled in M t by p aram eter у > 0 o r - in m odel M2 - by ß e [ - \ , 1]. H ence, b o th sam pling

m odels are able to ca p tu re tw o generally app eared featu res o f financial tim e series, i.e. heavy tails and asym m etry o f the co n d itio n al distrib utio n. F o r a discussion o f p oten tial differences in ex p lan ato ry pow er o f m odels JV/j and M2 caused by definitions o f stable an d skew ed-i families sec

F ern a n d ez and Steel (1998) and Pipień (2004).

4. COMPETING BAYESIAN MODELS AND DYNAMIC UPDATING

W e d en o te by y(i) = ( y lt ..., y t) th e vector o f observed up to d ay t (used in estim ation in day i) daily grow th rates and by y ' f — i y t n , y t+k) the vector o f forecasted observables at tim e t. T he follow ing density represents the i-th sam pling m odel (i = 1, 2) a t tim e t :

(8) P ( y m , 0, r / d = П р ( > - М - ь М „ 0, r/i),

]= i

i = 1 , 2 , t = T, T + 1, T + T '

In specification M , the sam pling m odel is based o n the p ro d u c t o f the a p p ro p ria te skew ed-i densities calculated a t d a ta p o in t, nam ely o n (6),

while in m odel M2 the density (8) is based on th e p ro d u c t o f stable

densities (7). C on stru cted a t tim e t Bayesian m odel M ;, i.e. the jo in t d istrib u tio n o f th e observables ( / ' \ y'p) and th e v ecto r o f p aram eters

(9) p(yw , y f , 0, VjlM;) = pi y w, y f \ 0 , tit, M ) p(0, >/i|Mi), i = 1, 2, t = T, T + 1, ..., T + T '

requires fo rm u latio n o f the p rio r d istrib u tio n p(0, 7,|М (), which is invariant

w ith respect to t. In both m odels we assum e p rio r independence between vectors o f co m m o n and m odel specific p aram eters. In each m odel we also assum e th e sam e p ro p e r p rio r stru ctu re for 0:

p(0, >/; IM,) = p(0) ■ pir/i I M ;) i = l , 2.

O u r p rio r in fo rm atio n a b o u t the com m on param eters is reflected by the follow ing density p(0):

(1 0) p(0) = p(ô)p(p)p(ôl )p(a0)p(a J p i a t )p(bx )p(h0) ,

discussed in details in Osiewalski and Pipień (2003). In m odel we assume:

p O íJ M j) = p ( C l , v , y ) = p(Ci)p(v)p(y),

w here p(Ct ) is stan d ard norm al, p(v) is exponential w ith m ean 1 0 and p(y)

is log standard norm al. T he prior distribution o f the m odel specific param eters in G A R C H (1 ,1 ) m odel, with stable co nditional density, (M 2) is defined as follows:

Р(Чг \M z) = Ж г .

a>P)

P(C2)p(ot)p(ß),

w here p(C2) is stan d ard norm al, p(a) is uniform over interval (0,2] and p(ß)

is uniform over [ -1,1].

T h e p rio r stru ctu re for com m on p aram eters as well as m odel specific p rio r assu m p tio n s for M , was presented in Osiew alski and Pipień (2003). Ilc rc we om it restrictions v> 2 and y e ( e x p (-2), exp(2)), im posed previously

to g u aran tee existence o f the second m o m en t o f p( yJ\ y / j - i , M 1, 0 , r ] l ). T he p rio r d istrib u tio n in m odel M2 was discussed in details in Pipień (2004).

5. EMPIRICAL RESULTS

In this p a rt we present an em pirical exam ple o f d y nam ic Bayesian co m p ariso n o f My and M 2. We considered T + T + 1 = 1398 observations o f daily grow th rates, y r o f the W IB O R one m o n th zloty in terest rate from 20.03.1997 till 5.09.2002 (dataset A) and T + T + 1 = 1657 observations o f daily returns on the P L N /U S D exchange rates from 5.02.1996 till 4.09.2002 (d a ta set B). S ta rtin g at t = T = 100 (which relates to 7.08.1997 fo r d ataset A an d to 25.06.1996 for d atase t B) we calculated p o sterio r probabilities o f m odels M L and M 2, and posterior distribution o f p aram eters based on d a ta se t yw, for each t = 100 up to t = T + T + 1. As a result o f daily up d atin g o bservations into y(i) we obtained 1299 (for d a ta se t A) and 1558 (for d atase t B) p o sterio r probabilities o f m odels and p o sterio r distrib u tio n s o f un k n o w n param eters. T h e m ain p urpose o f the follow ing p resen tation is to check sensitivity o f the po sterio r probabilities (as well as o f Bayesian inference ab o u t skewness and tails o f conditional d istrib u tio n o f retu rn s) w ith respect to new observations dynam ically included into d a ta se t y(,). We also study differences in the predictive d istrib u tio n s o f fu tu re g ro w th rates o b tain ed from b o th m odels.

60 50 40 30 20 10 0 1 101 201 301 401 501 601 701 801 901 1001 1101 1201 1301 1401 1501 1601

Fig. 1. Modeled time series with descriptive statistics

F igure 1 presents our both tim e series A and B. In F ig u res 1A and IB o n th e left axis we p lotted the vales o f daily grow th rates o f the W IB O R o n e-m o n th zloty interest rates and daily returns on the P L N /U S D exchange ra te (black line). In case o f d atase t A (Figure 1A) huge outliers in the plot o f yj, caused by changes in the m o n etary policy, to g eth er w ith the regions o f alm ost no variability, depicts very an om alous behavior o f daily changes o f th e P olish zloty m iddle term interest rate. T im e series o f daily grow th rates o f P L N /U S D exchange rate is characterized by the presence o f sparsely occured o u tliers with short-lived o u tb reak s o f volatility. O n the right axis

in F ig u re 1A and IB we plotted values o f th e sam ple k u rto sis o f y (,\ £ = 100, ..., T + T (grey line). In case o f d a ta se t A the fatness o f tails o f the em pirical d istrib u tio n o f y (,) dram atically change with respect to £. In both cases we observe considerable variability o f sam ple kurtosis, which - fo r d a ta se t A - reaches values even greater th an 130 an d n o t less than

18. T h e vertical d o tted lines in F igures 1A and IB locates í = 1 0 0 . It con stitu tes the sho rtest d a ta se t used here in Bayesian inference in M { and M 2. S tartin g a t this point, we recalculated posterio r characteristics o f m odels M v and M 2 every tim e the single observ ation o f daily g ro w th rates was included in to y(I).

F igure 2 presents po sterio r probabilities P ( M , |y (,)) (black line) and P ( M 2\y{,)) (grey line) obtained by assigning equal p rio r m odel probabilities ( P ( M t ) = P ( M 2) = 0.5). In the first colum n o f the first row o f F igure 2 we present the results for d atase t A, while the p lot in the second colum n o f the first row relates to the d atase t B. T he bottom plots o f daily grow th rates y} (j = 100 to 1398) m ay help in visual assessm ent o f the influence o f new d a ta included into y u) on changes o f the posterior probabilities. In case o f datset A, the first 500 o b servations yield decisive su p p o rt for G A R C H m odel with skewed-£ conditional distribution. A lm ost zero posterior probability P ( M 2\yif)) m akes stable G A R C H com pletely im probable in th e view o f the d a ta y(,), for £ = 100 till a b o u t 560. F o r d atase t A we also observe d ra m a tic fall o f the posterio r p ro b ab ility P ( M 1\ylt)) for £ greater th an 600. It seems to be caused by th e region o f alm ost no variability o f W IB O R o n e-m o n th interest rate, which lies roughly between £ = 500 and 650. Inclusion those observations into d atase t m akes y (t) (for £ = 6 5 0 , ..., 700) look like an alm ost n o n volatile series w ith huge negative outliers. Ever since, the d a ta clearly s u p p o rt G A R C H m odel with stable conditional distribution. We observe th a t, for £ > 1100, the posterior probability o f m odel M x again starts to lift, m aking this specification m ore likely a posteriori. R egular fluctuations o f y } for j = 1100, ..., 1398 su p p o rted G A R C H m odel with skewed-£ co nditio n al d istrib u tio n .

F o r d a ta se t В we observe successive grow th o f th e streng th o f the d a ta su p p o rt in fav o r o f m odel M v S tarting from t = 100 o bservatio ns, for £ = 100, ..., 250, skewed-£ G A R C H m odel quickly receives th e m ajo rity o f the posterior probability. F o r £ > 250 some occasional outliers - and especially stru c tu ra l b re ak a t £ = 385 - tem p o ra rily red u ce p o ste rio r p ro b a b ility P i M j y 0), m ak in g specification M 2 m ore p ro b a b le in view o f the data. A fte r including £ > 1 1 0 0 observations the p o sterio r p ro bab ilities o f b oth specifications becom e insensitive w ith respect to new o b serv ation s included into dataset. F o r £ > 1100 the dataset В decisively reject stable G A R C H model.

In F igu re 3 we present plots reflecting dynam ic changes in location and dispersion o f the m arginal po sterio r d istrib u tio n s o f tail and asym m etry p aram eters o f the conditional d istrib u tio n o f yj in m odels M L and M 2.

P ( M [ / V = 1 0 0 , 1657 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 ---/ W i l v ) ----p{a/2 Lv) 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 y „ t = 100,..., 1657 100 200 300 400 500 600 700 800 900 10001100 12001300140015001600 I N> ' О О

Fig. 2. Dynamic posterior probabilities of models M 1 and M 2 for datasets A and В

M ate us z P ip ie ń

Tail param eters Figure 3.1 Figure 3.2 Asymmetry parameters Figure 3.3 1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4

— lower bound А/, — upper bound My — lower bound M 2 — upper bound М2 — symmetry under M\ — symmetry under М2

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300

Figure 3.4

— lower bound M \ — upper bound А/, — lower bound A/2 — upper bound M 2 — symmetry under M \ — symmetry under М2

100 200 300 400 500 600 700 800 900 1000110012001300140015001600

F o r each d a ta se t y(I) (for A t = 100, ..., 1398, and fo r B i = 100, 1657) we calculated 95% highest posterior density ( I I P D ) intervals for tail param eters a (M 2) and v (A /,) and for asym m etry p aram eters ß (M 2) and у (M j). Presented H P D intervals can be in terpreted as the B ayesian 95% credible intervals for estim ated param eters.

T h e H P D intervals, plotted on Figures 3.1 and 3.3 indicate fundam ental differences in inference ab o u t the tails o f the co nd itio n al d istrib u tio n o f y} in case o f d a ta se t A. Based on tim e series y(1) both m odels, for t = 100, ..., 650, su p p o rt different type o f conditional distribution o f retu rn rates. Given model M j, for t = 100, ..., 650, there is no d o u b t, th a t the second m o m en t o f the co n d itio n al d istrib u tio n o f y } exists. A t the sam e tim e, given m odel M 2, the d a ta locate index a in the regions th at would preclude conditional norm ality o f yj (cf. F ig ure 3.1). F ro m the definition o f stable ran d o m variables it is equivalent w ith non-existence o f the second conditional m om ent. Sim ilarly as for posterior probabilities o f both m odels, inference ab o u t tails o f the conditio­ nal d istrib u tio n o f yj changes for t greater th a t 600 and ad dition ally becom e quite unanim ous. A fter updating ab o u t t = 700 observations th e hypothesis o f existence o f the second conditional m om ent is strongly rejected in both models. F o r t greater th a n 700 H P D intervals for a (in M 2) and v (in M 2) are both tightly located around the value 1.5 precluding existence o f the variance o f the co n d itio n al d istrib u tio n o f у}.

F ig u re 3.2 in T ab le 3 presents the H P D intervals o f tail param eters in My and M 2 o b tain ed in d ataset B. B oth m odels yield differen t inform ation a b o u t existence o f conditional m om ents o f y}. F ro m the definition o f stable fam ily, stable G A R C H specification precludes existence o f second m om ent o f p ( y j \ y S j - i , M 2, 0, tj2). As seen from F igure 3.2 the H P D intervals for param eter a are very tight and located very close to value a = 2. A dditionally, location as well as spread o f th e H P D intervals o f p a ram eter a rem ains insensitive to new observations u p d ated in d atase t B. In spite o f significant changes in dispersion o f the H P D intervals o f p a ram eter v in m odel My , th ere is n o d o u b t th a t p ( y j W j - \ , M 2, 0, rj2) posses variance (cf. F ig ure 3.2). T h e p lo t o f lower bound o f the H P D intervals o f v show s, th a t for t > 120 m o re th an 95% o f the po sterio r probability o f p (v |y (l), M y ) is concentrated on the left side o f the value v = 2 (see F ig u re 3.2).

H P D 95% intervals o f asym m etry p aram eters are presented in F ig ure 3.3 (d a ta set A ) and F ig u re 3.4 (dataset B). By grey h o rizo n tal lines we located sym m etric cases o f the co n ditional distrib u tio n s (for My it is th e case with у = 1 and for M2 it corresponds to ß = 0). In case o f d a ta se t A , ju st like

for tail param eters, b o th m odels yield different conclusions ab o u t asym m etry o f the co nditional d istrib u tio n o f yj for t = 100, ..., 600. U n d er m odel My, d a ta se t y(I) (for t = 100, ..., 450) build p o sterio r d istrib u tio n o f у w ith very volatile location and dispersion. It m akes uncertainty ab o u t possible skewness

o f the co n d itio n al d istrib u tio n o f y} (given M x) very »sensitive to new observ atio n s u p d ated in d ataset. H uge negative outliers, to g eth er w ith the region o f no variability (£ = 500, ..., 650) leads to very tig h t p osterior d istrib u tio n p (y |y (,)) fo r í = 400, 600, w here 95% o f th e p o ste rio r p ro b ab ility is located at the very small region o f p a ram eter space. D ataset y (t) (for £ = 400, ..., 600) leaves no d o u b t th a t co nd itio n al d istrib u tio n o f daily grow th rates (given m odel M y) is skewed to the left. F o r £ greater th a n 600 the H P D in tervals for p a ra m e te r у quickly s ta rt to w iden. C onsequently, given m odel M ,, for £ greater th an 600, the d a ta y(,) d o not preclude sym m etry o f the conditional d istrib u tio n , because the value у = 1

lies am o n g low er an upper bound o f the 95% H P D interval. In m odel M 2 the H P D interval o f the asym m etry param eter ß seems to be m o re dispersed and less sensitive to new observations th an the H P D interval for p aram eter у in m odel M v Except for £ = 100, 150 and t = 480, ..., 650, the d ataset

y( 0 d o n o t preclude sym m etry o f the (stable) cond itio n al d istrib u tio n o f y}.

In m o st cases o f £ the value ß = 0 lies either in the in terio r o f 9 5% H P D interval o r is very close to its u p p er boun d. In m odel M 2, th e d a ta always su p p o rt the hypothesis o f left asym m etry o f the co n dition al d istrib u tio n o f yj, ra th e r th a n right asym m etry. Except fo r a very few cases o f £, the m ajo rity o f the p ro b ab ility o f the p o sterio r d istrib u tio n o f ß lies below the value ß = 0 (see F igure 3.3).

Q uite regular fluctuations o f daily return s o f P L N / U SD exchange rate (d ataset B) m akes, in m odel M „ inference a b o u t possible skewness o f p(yJ\i//j-l , M 1,0,Tii ) consistent with m odel M 2. F ro m F ig u re 3.4 we see, th a t, for £ = 100, ..., 500, b o th m odels su p p o rt sym m etric case, leaving great uncertainty about possible right or left asym m etry of p{yj \[)/j-i , M i, 0, j/j). F o r £ = 100, ..., 500 the H P D intervals o f ß an d у are very dispersed and its location and spread is very sensitive with respect to the new observations. B ut, for £ > 500, b o th specifications su p p o rt hypothesis o f righ t asym m etry o f co n d itio n al d istrib u tio n o f y;-. As in m odel M 2, fo r £ > 500, th e H P D intervals o f asym m etry param eter ß lies on the right side o f th e value ß = 0,

stable G A R C H su p p o rts right asym m etry stron ger th a n m odel M v In case o f M y the H P D interval for asym m etry p aram eter у includes sym m etric case (y = 0), b u t the m ajority o f p o sterio r prob ab ility m ass o f p (y \y l,\ M x)

is co n cen trated o n the right side o f the value у = 0.

F igure 4 presents quantiles o f o rd e r 0.95 and 0.05 o f the one-step predictive densities at tim e £ (predictive distrib u tio n s o f yt + 1 given y(t))

o btained from b o th m odels in d atasets A and B. F igures 4.1 and 4.3 plots the quantiles o f p( yt + i \ M h y(i)) in case o f d atase t A , while F ig ures 4.2 and 4.4 relates to d a ta se t B. A s usual, in the third row we p u t o u r tim e series (A and В ) in o rd e r to asses sensitivity o f spread o f considered predictive d istrib u tio n s w ith respect to new observations у у T im e varying inverse

quantiles 0.95, 0.05 o f/>(уж|Л/|У0)

Figure 4.1 _{Figure 4.2}

quantiles 0.95, 0.05 o fp (y l+\M 2y )

Figure 4.3 _{Figure 4.4}

Fig. 4. Quantiles o f orders 0.95 and 0.05 o f the one-step predictive densities obtained from models M 1 (it = 100, 1398) and M 2 (t = 100, 1657) ai eu sz n p ie n I D y n a m ic B ay es ia n In fe re n

precision in M j and scale param eter in M 2, which arc b o th m odeled by a sy m m ctric-G A R C H (l, 1) eq u atio n , m ake one day ahead predictive densities very sensitive to new observations included in observed tim e series. F o r b o th d atasets, spread o f p(y,+ l M i, y(,)) (as m easured by q uan tiles o f order 0.05 and 0.95) instantly responds to changes in the volatility (dataset B) o r occasional huge outliers (d ataset A). A dditionally, eith er d atase t A o r В indicate, th a t stable G A R C H m odel generate one day predictive densities m o re dispersed th an those obtained from m odel M ,.

Visible difference in distance between qu antiles o f o rd e r 0.05 and 0.95 o f th e predictive distrib u tio n s p(yl + 1 |M „ / ° ) ( i = 1, 2) m ay be the crucial

p o in t in analyzing discrepancies o f d a ta su p p o rt o f skew ed-i and stable G A R C H m odels. In the co n stan t location and scale fram ew ork F ern an dez and Steel (1998) com pared sam pling distrib u tio n s ob tained from skewed-t o r stable assu m p tio n ab o u t the erro r term . T h e b enchm ark o f com parison was em pirical d istrib u tio n o f m odeled time series. A s a one o f the results, w hich was also obtained fo r m any tim e scries by R achev and M ittnik (2002), F ern a n d e z and Steel (1998) re p o rt alm ost im perceptible differences in d a ta fit o f skew ed-i and stable regression m odels. T h e plots o f sam pling densities o b tain ed from location and scale skewed-t and stable m odels were very sim ilar, and fitted well to em pirical density. As seen from F igure 4, tak in g into consid eratio n the p osterior uncertainty ab o u t p aram eters, m akes the predictive densities (obtained from M , and M 2) very different. It seems th a t b o th m odels reflect different posterior info rm ation a b o u t com m on and m odel specific param eters. C onsequently, and M 2 yield different ex-ante un certainty a b o u t fu tu re grow th rates.

6. CONCLUSIONS

In A R (1 )-G A R C H (1 ,1) fram ew ork for daily return s, proposed and ad o p ted by Bauwens and L u b ran o (1997), Bauwens et al. (1999) Osiewalski and Pipień (2003), there are considered in the p ap er tw o types o f co n ­ ditio n al d istrib u tio n . In the first m odel ( M t ) we assum ed conditionally skew ed-i d istrib u tio n (defined by F ernandez and Steel 1998) while the second G A R C H specification (M 2) is based on the co n d itio n al stable d istrib u tio n . W e presented Bayesian updating technique in o rd e r to check sensitivity o f the po sterio r probabilities o f considered specifications, with respect to new o bservations included into d ataset. W e also studied differen­ ces betw een Bayesian inference ab o u t tails and asym m etry o f the co n ­ d itio n a l d istrib u tio n o f daily re tu rn s and th e o n e-step pred ictiv e d ist­ rib u tio n s o btained from both m odels.

Based on very volatile daily grow th rates o f th e W IB O R on e-m onth in terest rates (d a ta set A, 1398 observations) as well as o n daily returns on the P L N /U S D exchange ra te (dataset B, 1657 ob servations), we ca­ lculated the po sterio r probabilities o f m odels M , and M 2, and the p o ­ ste rio r d istrib u tio n o f p aram eters using d a ta s e t y (,\ fo r each t = 1 0 0

up to t = T + T + 1 (which is equal to 1398 for d a ta se t A and 1657 for B). T h e m ain em pirical result o f this p ap e r is g re at sensitivity o f the p o ste rio r m o d el p ro b a b ilities w ith respect to new o b se rv a tio n s o f yj included in to d a ta se t. D aily re tu rn s o f d a ta se t A ch a rac te rized by very weak variability with unexpected huge negative outliers decisively su p p o rted G A R C H m odel with conditional stable d istrib u tio n . A fter in­ cluding m ore volatile observations into d a ta se t A , we observed th a t the p o sterio r p ro b a b ility o f m odel M , started to increase. F o r d atase t В we observe successive grow th o f the strength o f the d a ta su p p o rt in favor o f m o d el M ,. F o r t > 1100 o b se rv a tio n s o f d a ily re tu r n s o f th e P L N /U S D exchange rate, skew ed-i G A R C H m odel receives th e whole p o sterio r p ro b ab ility , m aking stable G A R C H com pletely rejected by the d a ta se t B.

W e also checked conform ity o f inference a b o u t tails an d asym m etry o f the co n d itio n al d istrib u tio n o f daily returns. In case A , fo r s h o rt tim e series b o th m odels yielded different inform ation a b o u t existence o f m o m en ts as well as possible skewnes o f p(yJ\ y / j ^l , M l, 0, rj^- H ow ever, for datasets, which

consisted m o re th an 700 observations o f daily grow th rates o f W IB O R lm , b o th m odels pointed to qualitatively sim ilar results o f th e p ro p erties o f the co n d itio n al d istrib u tio n o f y y F o r d ataset В stable G A R C H m odels was n o t able to m odel properly tails o f the conditio n al d istrib u tio n o f returns. H P D intervals o f the degrees o f freedom p aram eter in m odel M i (skewed-£ G A R C H ) decisively su p p o rted hypothesis, th a t the second and th ird co n ­ ditional m o m en t o f p ( ^ | ^ - i , M 1, 0, t}x) exist. H ow ever, tightly co ncentrated

aro u n d value v = 3 po sterio r distribution p ( v |/ ° ) , precludes conditional n orm ality . D a ta se t В supp o rted m ore flexible skew ed-г G A R C H m odels, m ak in g co n d itio n al stability im p ro b ab le a posteriori.

B oth m odels built one day predictive d istrib u tio n s very sensitive to new o b se rv a tio n s includ ed. W e observed in sta n t re a c tio n o f th e spread o f p(y, + i |M j, y w) (i = 1, 2) on occasionally ap p eared outliers o r unexpected

in ten sificatio n s o f volatility. F o r b o th d a ta s e ts p re d ic tiv e d istrib u tio n s o b tain ed from m odel M2 has greater dispersion th a n those obtain ed from

skew ed-i G A R C H m odel. It seems th a t, in building predictive distribu tion s, p o ste rio r u ncertain ty ab o u t com m on and m odel specific p aram eters o f specifications M x and M2 lead u p to different ex an te u n certain ty a b o u t

Mateusz Pipień

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Mateusz Pipień

DYNAMICZNE WNIOSKOWANIE BAYESOWSKIE W PROCESACH GARCH ZE SKOŚNYMI T-STUDENTA I STABILNYM ROZKŁADEM WARUNKOWYM

(Streszczenie)

W artykule przedstawiono modele A R(1)-G A RC H (1,1) dla dziennych stóp zmian (por. Bauwens i Lubrano 1997, Bauwens i in. 1999, Osiewalski i Pipień 2003) z różnymi typami rozkładu warunkowego. W pierwszym przypadku (model M x) rozważono warunkowy rozkład skośny t-studenta (zdefiniowany przez Fernández i Steela 1998), podczas gdy model M 2 to

proces GARCH o warunkowym rozkładzie a-stabilnym. Prezentujemy bayesowską aktualizację rozkładów a posteriori i predyktywnych (wraz z napływem nowych danych) w celu zbadania, czy typ rozkładu warunkowego zadany w procesie GARCH wpływa na wnioskowanie o naturze procesów opisujących zmienność finansowych szeregów czasowych o dużej częstotliwości. Rezultaty dynamicznej estymacji wykorzystującej podejście bayesowskie zilustrowano na przykładzie dwóch szeregów czasowych, tzn. dziennych stóp zmian kursu walutowego PLN/USD oraz oprocentowań jednomiesięcznych lokat międzybankowych (WIBORlm).