The Economic and Social Review, Vol. 29, No. 4, October, 1998, pp. 369-382
Testing Distributional Models for the I r i s h
E q u i t y Market
J O H N C O T T E R *
University of Ulster
Abstract: T h i s s t u d y o u t l i n e s d i s t r i b u t i o n a l properties a n d tests t h e a p p l i c a b i l i t y of different models for t h e I S E Q i n d e x a n d i t s m a j o r constituents. A S t a b i l i t y u n d e r A d d i t i o n s procedure is a p p l i e d to raw, f i l t e r e d a n d rescaled r e t u r n s . The f i l t e r i n g process uses G A R C H d , 1) a n d G A R C H - M ( 1 , 1) specifications, r e m o v i n g t h e influence o f t e m p o r a l anomalies a n d non-synchronous t r a d i n g effects. Rescaling involves the s t a n d a r d i s i n g of r e t u r n s using the t i m e v a r y i n g models estimates of conditional variance. Results s u p p o r t t h e o r d i n a r y stable a n d m i x t u r e s o f stables d i s t r i b u t i o n s . T h e lack o f n o r m a l i t y is affected by f i r s t a n d second m o m e n t dependence.
I I N T R O D U C T I O N
T
he I r i s h equity m a r k e t has attracted greater investor a t t e n t i o n i n recent years w i t h a n almost sevenfold increase i n t u r n o v e r since 1990. M u c h of the focus is due to the p o p u l a r i t y of equity m a r k e t s i n general, b u t is also a response to improvements i n t r a d i n g conditions, and the general s t r u c t u r e of t h e D u b l i n Stock E x c h a n g e . O f p a r a m o u n t i m p o r t a n c e to i n v e s t o r s a n d researchers is t h a t they correctly assess the r i s k r e t u r n r e l a t i o n s h i p t h a t the m a r k e t offers. M o s t of the research e x a m i n i n g t h i s relationship applies p r i c i n g models a n d statistical tests assuming n o r m a l i t y , and t h u s there is a constant variance. However, i f n o r m a l characteristics are lacking, i t may lead to e m p i r i c a l problems. For instance, findings for any procedure t h a t requires a constantP a p e r p r e s e n t e d a t t h e T w e l f t h A n n u a l Conference o f t h e I r i s h Economic Association.
variance, for example, regression analysis, can be questioned i f a t i m e v a r y i n g second moment is documented.
M a n y a l t e r n a t i v e d i s t r i b u t i o n a l models have been suggested for t h e i r a p p l i c a b i l i t y i n correctly describing characteristics of equity r e t u r n s and these can be classified under t w o headings. F i r s t the stable paretian f a m i l y i n c l u d i n g the m i x t u r e s of stables (Ghose and Kroner, 1995), and second the A R C H f a m i l y i n c l u d i n g the G A R C H - M (Poon and Taylor, 1992) specifications. These models have both recognised and have s i m i l a r a b i l i t y i n r e m o v i n g the causes of non n o r m a l i t y such as conditional heteroskedasticity ( M i l l s , 1996). I n particular, w h i l e t h e k u r t o s i s findings show an i m p r o v e m e n t for t h e adjusted data i n contrast to r a w r e t u r n s , i t s t i l l persists (Taylor, 1986).
L i t e r a t u r e on changing characteristics has indicated two p a r t i c u l a r d i s t r i b u t i o n a l types: m i x t u r e s of stables and m i x t u r e s of normals. The m i x t u r e s of stable p a r e t i a n model involves either a changing of the scale or skewness parameters. I n contrast, the m i x t u r e s of normals model involves price series w i t h the same mean and changing variance. Causes of the changing moments have focused on m a r k e t anomalies due to t r a d i n g and temporal conditions, as w e l l as nonsynchronous data. These features have been documented for the I r i s h equity m a r k e t w i t h t i m e periods i n v o l v i n g t h i s paper's r e a l m of analysis, w i t h evidence of seasonalities and t h i n t r a d i n g (Cotter, 1997; and M u r r a y , 1995).
Models incorporating conditional dependence have been developed and applied to equity m a r k e t prices w i t h v a r y i n g success. Conditional heteroscedastic based approaches have received voluminous empirical attention i n finance, and a com prehensive review can be found i n Pagan (1996). The basis of these approaches is t h a t the source of leptokurtosis is due to heteroscedasticity i n the variance of a d i s t r i b u t i o n . As m a n y of the A R C H related models may be applicable to equity r e t u r n s , i t is necessary to have a j u s t i f i e d rationale for e x a m i n i n g any p a r t i c u l a r specification. G a l l a n t et al. (1992) find t h a t temporal and t r a d i n g factors have an influence on both the conditional mean and variance of US data. Incorporating b o t h t h e first and second moment influences together suggests a r i s k r e t u r n r e l a t i o n s h i p .1 G i v e n t h i s r e l a t i o n s h i p , a G A R C H - M m o d e l i n t r o d u c e d by M c C u r d y and M o r g a n (1988) can be applied. The application of t h i s model to speculative data has been extended to deal w i t h autoregressive effects.
Previous analysis of t i m e series properties of the ISEQ include m e a s u r i n g autocorrelation and d e t e r m i n i n g day of the week effects (Lucey, 1994). T h i s paper readdresses t h i s issue by testing the applicability of different d i s t r i b u t i o n a l models for t h e I S E Q and F T S E indexes u s i n g a S t a b i l i t y u n d e r A d d i t i o n s procedure. As the I r i s h m a r k e t is heavily influenced by a number of r e l a t i v e l y
large stocks, an analysis of different sized equities provides a more detailed p i c t u r e of t h e market's operation. The data is examined under three forms: raw, f i l t e r e d a n d rescaled r e t u r n s . A f i l t e r i n g process removes t r a d i n g and t e m p o r a l anomalies influencing the r a w r e t u r n s t h r o u g h G A R C H ( 1 , 1) a n d G A R C H - M ( 1 , 1) models. Rescaling involves standardising the filtered r e t u r n s
by u s i n g the estimates of the conditional variance, obtained from the t i m e v a r y i n g
frameworks. These approaches determine how i n f l u e n t i a l the anomalies are i n d e t e r m i n i n g d i s t r i b u t i o n a l conclusions. Causes of non-normal properties and the robustness of our G A R C H models are also detailed.
The o u t l i n e of the paper is as follows: i n i t i a l l y , a description of the data is given, s h o w i n g s u m m a r y statistics. T h i s is followed by a methodology section where a description of the f i l t e r i n g and rescaling procedures, as w e l l as the s t a b i l i t y under additions test, is given. Next, results are presented and discussed i n t h e e m p i r i c a l findings section. Finally, conclusions are detailed.
I I D A T A
There has been considerable progress i n the s t r u c t u r e of Dublin's e q u i t y m a r k e t w h i c h bears w e l l for i t s future. Whereas the I r i s h stock exchange is s m a l l by i n t e r n a t i o n a l standards, there has been a substantial i m p r o v e m e n t i n i t s c a p i t a l i s a t i o n v a l u e i n recent t i m e s . Two i m p o r t a n t s t r u c t u r a l factors positively influencing the size of the index have been the introduction of relatively large companies from the State owned sector, for example Greencore and I r i s h Life, a n d food co-operatives w h i c h w e n t public, such as Waterfood Foods. I n an i n t e r n a t i o n a l context, there is considerable room for more increases i n size, as FTSE's value represents over 100 per cent of U K G D R I n comparison, I S E Q represents u n d e r 50 per cent of I r i s h G D R
W i t h very l i t t l e interest i n the first t w o markets, these were replaced i n 1996 w i t h the Developing Companies M a r k e t .
S u m m a r y statistics for daily r e t u r n s between J a n u a r y 1987 and December 1997 are shown i n Table 1, where details of the first four moments plus the Jarque-Bera n o r m a l i t y test are presented. Each of the moments is calculated a s s u m i n g t h e other variables are constant. The more commonly used com pounded returns are generated using the first difference of the n a t u r a l logarithms of a l l closing daily prices, k n o w n hereafter as raw returns:
l o g ( Pt) - l o g ( Pt_ , ) (1)
[image:4.476.74.433.330.454.2]The F T S E 1 0 0 index is chosen as a m a t t e r of comparison, w h i c h is also the case for major constituents of the I r i s h m a r k e t , namely, A l l i e d I r i s h B a n k , B a n k of I r e l a n d and Jefferson Smurfit. The other equities included cover a representative size r a n g e i n m a r k e t c a p i t a l i s a t i o n t e r m s , S i l v e r m i n e s , ( w i t h a m a r k e t capitalisation at the end of 1997) of under IR£75m, and Grafton of between IR£75 and IR£500m.
Table 1: Summary Statistics of Equity Returns
Series Mean Max Min Standard Skewness" Kurtosisa Jarque-Bera
(%) (%) (%) Deviation Test (%)
I S E Q 4 . 4 1 x l 02 8.54 - 1 2 . 7 7 9.72x10-' - 1 . 5 8 * 23.61* 67923.9* F T S E 1 0 0 3 . 7 3 x l 02 7.59 - 1 3 . 0 2 9.27x10"' - 1 . 6 7 * 25.89* 80273.4* A l l i e d I r i s h B a n k 5 . 9 7 x l 0 '2 13.11 - 1 5 . 6 8 1.60 -0.58* 11.59* 16274.1* B a n k o f I r e l a n d 7 . 9 6 x l 0 "2 12.10 - 1 3 . 7 0 1.56 - 0 . 0 8 8.78* 9256.2* Jefferson S m u r f i t 2 . 8 2 x l 0 '2 24.06 -24.95 1.97 - 0 . 6 7 * 25.52* 78103.5* G r a f t o n l . l l x l O "1 34.83 -29.48 2.16 - 0 . 2 8 * 71.61* 613742.0* S i l v e r m i n e s - 3 . 3 7 x l 0 "3 35.67 - 3 8 . 5 7 2.77 - 0 . 2 0 * 3 8 . 3 1 * 172866.0*
* represents s i g n i f i c a n t at 5 per cent level.
(a) A n o r m a l d i s t r i b u t i o n has a skewness a n d k u r t o s i s coefficient of zero a n d three respectfully.
r e t u r n s , w i t h a h i g h p r o p o r t i o n of companies of s i m i l a r size to G r a f t o n and Silvermines h a v i n g r e t u r n s following d i s t i n c t courses. T h i s represents a good choice for investors seeking holdings w i t h a wide range of r e t u r n possibilities.
A second i m p o r t a n t performance i n d i c a t o r for i n v e s t o r s is e q u i t y r i s k , conventionally measured by the s t a n d a r d deviation of r e t u r n s . V o l a t i l i t y for t h e I S E Q is less t h a n 1 per cent per day, and very s i m i l a r to FTSE's r i s k levels. There is also a h i g h degree of s i m i l a r i t y i n the r i s k p a t t e r n of both indices t h r o u g h t i m e w i t h , for example, 1996 e x h i b i t i n g low v o l a t i l i t y a n d 1987 h i g h v o l a t i l i t y . Investors a t t r a c t e d to f i r m specific, r a t h e r t h a n m a r k e t r i s k , have a diverse range to choose from on the D u b l i n m a r k e t as the v o l a t i l i t y of the representative equities differ significantly from the m a r k e t index. Also, f i r m specific r i s k varies inversely w i t h company size. Investors l o o k i n g for h i g h l y r i s k y options should note t h a t as f i r m size decreased (for instance A l l i e d I r i s h B a n k to Silvermines), t h e corresponding v o l a t i l i t y measure increased.
T h e issue of n o r m a l i t y has often been addressed by e x a m i n i n g b o t h t h e skewness a n d kurtosis coefficients ( B a d r i n a t h and Chatterjee, 1991). W h i l e there is no clear consensus r e g a r d i n g the sign of t h e skewness coefficient, generally t h e m a g n i t u d e is s m a l l a n d close to zero for speculative prices, thereby suggest i n g a s y m m e t r i c a l d i s t r i b u t i o n . The skewness coefficient, p, examines t h e concentration of r e t u r n s to t h e r i g h t or left of the mean value. O u r findings contradict t h e hypothesis t h a t the data is symmetric, based on the test statistic (t = p/(SE P)). Negative skewness is p r o m i n e n t for our data i n Table 1, w i t h b o t h i n d e x e s e x h i b i t i n g s i m i l a r values. E x t r e m e n e g a t i v e factors such as t h e w o r l d w i d e effects of t h e A s i a n m a r k e t crises manifest themselves as outliers, l e a d i n g to a conclusion of asymmetry. Skewness findings are incorporated i n t h e s t a b i l i t y u n d e r additions procedure w h e n t e s t i n g different d i s t r i b u t i o n a l hypotheses.
1 4 0 0
1 2 0 0
1 0 0 0
8 0 0
u
82 6 0 0
4 0 0
2 0 0
0
'<? % % *<b °o °J °e
ISEQ
Figure 1: Histogram of ISEQ with Normal Curve
I I I M E T H O D O L O G I C A L I S S U E S
A s t a b i l i t y u n d e r additions test is applied to examine the d i s t r i b u t i o n a l properties of r a w equity r e t u r n s . T h i s focuses on the kurtosis coefficient and allows for interdependency w i t h other distributional moments. Two other distinct sets of ISEQ's r e t u r n s are tested u s i n g the s t a b i l i t y under additions test. F i r s t , r e s i d u a l s f r o m a f i l t e r i n g t e c h n i q u e based on G A R C H specifications are examined. Second, the filtered r e t u r n s are rescaled u s i n g the t i m e v a r y i n g models' estimates of conditional variance. T h i s section discusses the s t a b i l i t y u n d e r additions test, the f i l t e r i n g , and rescaling processes.
3.1 Stability Under Additions Test
The s t a b i l i t y under additions test relies on the stable p a r e t i a n model defined by i t s characteristic function:
log<|>(t) = i 8 t - Y l t l a ( l + P ( t / l t l ) t a n (not/2))
for 0 < a < 2 (2)
where t is any real number, 8 is the location parameter, y i s the scale parameter,
P is the skewness parameter, and a is the kurtosis parameter a n d is also the
The characteristic exponent describes the shape of the areas around the mean a n d t h e t a i l s of the d i s t r i b u t i o n . Specific types of stable d i s t r i b u t i o n s realise different values. The special case of the n o r m a l d i s t r i b u t i o n is described w h e n a = 2. Speculative data from spot markets have a kurtosis value between 1 a n d 2 ( H a l l et al., 1989). I n t h i s case, the d i s t r i b u t i o n has a finite mean, b u t t h e v a r i a n c e is i n f i n i t e . I n practice t h i s i n f i n i t e variance i m p l i e s t h a t e r r a t i c behaviour is shown by r e t u r n s for large samples and t h i s is indicated by larger s t a n d a r d deviations t h a n for Gaussian processes. T h i s conclusion falls n e a t l y i n t o t h e v o l a t i l i t y c l u s t e r i n g arguments t h a t support t i m e v a r y i n g models such as G A R C H w h e r e periods o f h i g h v o l a t i l i t y c a n occur consecutively, as demonstrated by fat tails. The second d i s t r i b u t i o n a l moment generally aligned w i t h n o n - n o r m a l i t y is the skewness coefficient, for example, see Table 1. The stable d i s t r i b u t i o n recognises skewness by allowing the parameter take on values i n t h e i n t e r v a l - 1 < P < 1.
D i s t r i b u t i o n a l characteristics are tested for using the stability under additions test, developed by McCulloch (1986). T h i s approach uses a fractal approach s i m i l a r to F a m a and Roll's (1971) test, b u t has the additional property of allowing the skewness parameter to take on a range of values ( - 1 < P < 1). As n o n - t r i v i a l skewness is found for ISEQ's r e t u r n s , i t is best to apply a procedure t h a t incorporates v a r y i n g skewness i n the calculation of the kurtosis coefficient. Also, t h i s newer method has a n advantage of r e m o v i n g a s m a l l asymptotic bias i n F a m a a n d Roll's estimates of a. Five sample quantiles, and linear interpolations of t a b u l a t e d index numbers are the basis of t h i s technique. Indexes for b o t h a a n d P coefficients are measured u s i n g the following:
V = V'9 5 ~X ,°5
a Y Y
a n d
v _ X-95 + X .0 5 - 2 X .5
A- 7 5 A- 2 5
(4)
where Va and Vp are sample estimates of corresponding population values. These indexes are dependent on each other. I n contrast, s u m m a r y statistics assume t h a t each coefficient is c a l c u l a t e d w i t h other p a r a m e t e r s b e i n g constant. Interdependent values for a and P are t a b u l a t e d by McCulloch (1986).
observations, t h a t is = a2 = ... = an. However, w h e n a p p r o x i m a t i n g sample values, a estimates can be affected by s a m p l i n g error, and the extent of t h i s could be compounded w i t h the s t a b i l i t y under additions test w h e n y o u move to larger sums of observations, as the sample size decreases. To control for possible s a m p l i n g error, the n u m b e r of o v e r l a p p i n g sums of observations is l i m i t e d to 20.
3.2 Filtering Technique
The purpose of the f i l t e r i n g technique is to remove documented anomalies from the data, thereby generating smooth r e t u r n s series'. B y using (5) as the r e l a t i o n s h i p between the sequences (Yt) and |Zt}, the e s t i m a t i o n of the variance parameter relies on a l i n e a r G A R C H (p, q).
Yt= b Zt +et (5)
E x t e n d i n g (5) to i n c l u d e an e s t i m a t e of t h e c o n d i t i o n a l v a r i a n c e gives a G A R C H - M model, outlined i n (6).
Yt = b Zt + c ht + et (6)
I n t h i s case, the t i m e v a r y i n g model involves a j o i n t estimation of t h e mean i n v o l v i n g the disturbance t e r m plus the hetraskedastic process t h a t estimates the second moments.
The specific G A R C H - M filtering process adopted here is similar i n its objective to t h a t of Engle and N g (1993). I t involves a j o i n t estimation of our mean equation i n c o r p o r a t i n g the conditional variance, as w e l l as the heteroscedastic process t h a t estimates the second moments and is given i n (7). The model removes t e m p o r a l anomalies, such as the day of the week effects (see G a l l a n t et al., 1992 for a discussion); autocorrelation effects w h i c h r e s u l t i n nonsynchronous data (for e x a m p l e , t h i n t r a d i n g ) ; a n d t h e influence of c o n d i t i o n a l v a r i a n c e . A G A R C H ( 1 , 1 ) specification s i m i l a r to (7) w i t h o u t the measure of the conditional variance is included, due to mixed evidence for a G A R C H - M application w i t h the FTSE100 (Poon and Taylor, 1992).
Yt = a0 + I ABXt_8 + A1 1DM+ A1 2DT + A1 3Dw
(7) + A1 4DT h + A1 5DF + A1 6DH + A1 7ht + et
F r i d a y a n d DH = 1 i f a holiday. O t h e r w i s e they are 0. T h e measure o f t h e conditional variance is denoted by ht. T h e residuals, et, are the filtered r e t u r n s series.
T h e risk measure is allowed to v a r y over t i m e a n d is assumed to follow a G A R C H (1,1) process given by E q u a t i o n (8) where adjustments are made for day of the week a n d holiday effects, a n d a t r e n d .
ht = b0a e2_ i + P ht- i + bxDM + b2DT + b3Dw + b4DT h + b5DF +
b6DH - b7Tt+ b8T2 E J V ^ ~ N ( 0 , ht) (8)
L i n e a r a n d quadratic t r e n d variables, Tt and T2 are included i n the variance specification, b u t n o t directly i n the mean equation ( G a l l a n t et al., 1992).
3.3 Reseated Returns
A r e l a t e d set o f r e t u r n s t o those generated by the f i l t e r i n g process are the
rescaled r e t u r n s , a n d these are obtained as follows:
where ht is a n estimate of the actual conditional variance.
I t i s common t o focus i n on a simplified v a r i a t i o n o f (9), where t h e filtered r e t u r n s series is divided by t h e conditional standard deviation, as the results are similar. B o t h Taylor (1986) a n d Bollerslev (1988) suggest t h a t m u c h of the serial dependence exhibited i n t h e second moments o f a n asset's r e t u r n can be removed i f one examines t h e standardised r e t u r n s after a p p l y i n g an A R C H r e l a t e d approach.
I V E M P I R I C A L F I N D I N G S
The e m p i r i c a l findings for the applicability of various d i s t r i b u t i o n a l models for the I r i s h e q u i t y m a r k e t are now discussed. The s t a b i l i t y under additions test is applied to r e t u r n s obtained from different f i l t e r i n g procedures based on r e l a t e d t i m e v a r y i n g A R C H models. These models are estimated by m a x i m u m l i k e l i h o o d methods u s i n g the a l g o r i t h m developed by B e r n d t et al. (1974).2 Possible causes of t h e d i s t r i b u t i o n a l findings are i d e n t i f i e d by m e a s u r i n g a u t o c o r r e l a t i o n values for the e q u i t y data. Also, the a b i l i t y of the f i l t e r i n g technique to remove A R C H effects is analysed using the autocorrelation findings.
The values for the characteristic exponent for the r a w compounded r e t u r n s are s h o w n i n Table 2. The classifications are 1, 2, 5, 10, 15 and 20 loosely corresponding to daily, bidaily, weekly, biweekly, t r i w e e k l y and m o n t h l y con t i n u o u s l y compounded r e t u r n s . A n u l l hypothesis of the I S E Q index belonging t o a n o r m a l d i s t r i b u t i o n is rejected conclusively u s i n g t h e s t a b i l i t y u n d e r
[image:10.474.84.448.431.517.2]a d d i t i o n s t e s t as none of the k u r t o s i s coefficients have values e q u a l t o 2. The FTSE100 and major constituents of the I r i s h m a r k e t are also not n o r m a l l y d i s t r i b u t e d . The characteristic exponent findings suggest t h a t the u n d e r l y i n g d i s t r i b u t i o n characterising the I S E Q is a m i x t u r e s of stables d i s t r i b u t i o n as the a coefficient values increase across the sums of observations, and reach 2 for m o n t h l y r e t u r n s . I n d i v i d u a l I r i s h equities have s i m i l a r kurtosis coefficients to t h e I S E Q for d a i l y r e t u r n s as one w o u l d expect from assets w h i c h are so p r o m i n e n t i n the makeup of the m a r k e t portfolio. I n contrast, the characteristic exponent for the FTSE100 is r e l a t i v e l y higher. There is no clear evidence to support the o r d i n a r y stable d i s t r i b u t i o n for the data analysed.
Table 2: Estimates of the Characteristic Exponent for Raw Equity Returns
Series n = 1 n = 2 n - 5 n = 10 n = 15 n = 20
ISEQ 1.48 1.49 1.56 1.59 1.81 2.00
FTSE100 1.87 1.72 1.99 2.00 2.00 2.00
Allied Irish Bank 1.49 1.54 1.77 2.00 1.70 1.76
Bank of Ireland 1.39 1.56 1.62 1.72 1.70 1.76
Jefferson Smurfit 1.43 1.48 1.57 1.64 1.66 1.54
n is the different sums of observations.
I n order to d e t e r m i n e w h e t h e r t h e stable p a r e t i a n family, a n d i n par ticular, the m i x t u r e s of stables is not only the most applicable classification, b u t also, a n d of more importance, whether I r i s h equity r e t u r n s actually follow t h i s
d i s t r i b u t i o n , t h e issue is investigated further. The s t a b i l i t y under additions is r e a p p l i e d t o t h e e q u i t y d a t a after f i l t e r i n g u s i n g G A R C H a n d G A R C H - M specifications. T r a d i n g a n d seasonality effects ( a n d c o n d i t i o n a l v o l a t i l i t y influences i n t h e case o f a G A R C H - M ) are thereby removed from t h e r a w compounded returns. These filtered returns are t h e n rescaled using the measures of conditional variance from the G A R C H models, and the findings are shown i n Table 3 .3 I n general, b o t h G A R C H filters generate s i m i l a r conclusions, b u t the k u r t o s i s values for the I S E Q index are higher for the G A R C H ( 1 , 1 ) specification. N o r m a l i t y is again rejected for the equity r e t u r n s .
Table 3: Estimates of the Characteristic Exponent for Rescaled Returns
Series n = l n =2 n = 5 n = 10 n = 15 n =20
GARCHU, 1)
ISEQ 1.51 1.56 1.65 1.45 1.71 1.64
FTSE100 1.92 1.73 1.82 1.97 1.86 1.87
Allied Irish Bank 1.56 1.63 1.70 1.87 1.68 1.63
Bank of Ireland 1.45 1.59 1.60 1.55 1.55 1.48
Jefferson Smurfit 1.43 1.48 1.56 1.57 1.70 1.59
GARCH-M(1, 1)
ISEQ 1.48 1.47 1.54 1.44 1.31 1.38
FTSE100 1.85 1.78 1.79 1.77 1.70 1.74
Allied Irish Bank 1.48 1.54 1.59 1.55 1.59 1.64
Bank of Ireland 1.44 1.59 1.77 1.77 1.58 1.83
Jefferson Smurfit 1.41 1.47 1.57 1.63 1.56 1.41
n is the different sums of observations.
K u r t o s i s values for daily r e t u r n s r e m a i n r e l a t i v e l y unchanged from r a w to rescaled r e t u r n s , b u t the patterns across the sums of observations changes. As t h e c h a r a c t e r i s t i c exponent values for t h e I S E Q i n d e x r e m a i n reasonably constant, t h i s supports the hypothesis t h a t the data is from an o r d i n a r y stable d i s t r i b u t i o n , i n contrast to the m i x t u r e s of stables hypothesis. T h u s the I S E Q index indicates a n i n f i n i t e variance, as demonstrated by the characteristic of v o l a t i l i t y clustering. S i m i l a r conclusions can be made for the i n d i v i d u a l equities a n d t h e F T S E 1 0 0 index. O b t a i n i n g these differing conclusions from r a w to rescaled r e t u r n s indicates a benefit from applying the G A R C H filters as i t shows t h a t t h e i n f l u e n c e o f t r a d i n g anomalies is i m p o r t a n t i n d e t e r m i n i n g t h e d i s t r i b u t i o n a l characteristics of equity data.
A s u m m a r y of the extent to w h i c h first and second moment correlation occurs i n t h e I r i s h equity m a r k e t is shown i n Table 4. The ISEQ index is not s t r i c t w h i t e noise as significant dependence exists i n the levels for 11 of 30 lags. S i m i l a r conclusions can be made for the other series analysed. Strong A R C H effects are i n d i c a t e d for t h e I S E Q i n d e x a n d these are stronger t h a n the f i r s t order dependence, corresponding to a general finding for equity markets (Taylor, 1986). A s t h e other series' i n d i c a t e s i m i l a r f i n d i n g s , i t is no s u r p r i s e to see t h e
application i n finance of a vast number of models t h a t are i n h e r e n t l y setup to deal w i t h such a characteristic.
Table 4 : Summary of Autocorrelation Values for Equity Returns
Nos. of Significant Lags
Level Returns (Rtl GARCHd.l) GARCH-M(l.l)
ISEQ 11 (196.0) 2 (31.2) 0 (7.4)
FTSE100 5 (61.6) 1 (23.1) 1 (20.6)
Allied Irish Bank 4 (51.7) 2 (28.8) 1 (90.7)
Bank of Ireland 3 (60.8) 1 (33.8) 2 (30.6)
Jefferson Smurfit 3 (47.6) 0 (21.3) 2 (31.7)
Nos. of Significant Lags
Squared Returns (Rtl G A R C H d . l ) GARCH-M(l.l)
ISEQ 21 (1.63x103) 17 (662.0) 0 (1.87xl0-2)
FTSE100 12 (1.55x103) 1 (36.5) 6 (552.0)
Allied Irish Bank 14 (498.0) 2 (39.6) 0 (2.5)
Bank of Ireland 15 (375.0) 1 (29.2) 11 (210.0)
Jefferson Smurfit 6 (176.0) 6 (95.4) 0 (0.6)
Notes: (a) Ljung Box statistics are in parentheses.
(b) The series (Rt) refers to the unfiltered returns whereas the others are the respective G A R C H d . l ) and GARCH-M(l.l) filtered returns.
G A R C H - M model m a y not be fully appropriate i n describing the behaviour of these r e t u r n s (Poon a n d Taylor, 1992).
V C O N C L U S I O N S
T h i s study examines d i s t i n c t d i s t r i b u t i o n a l hypotheses for the I r i s h equity m a r k e t , by focusing on the I S E Q index. The major constituents, as w e l l as the F T S E 1 0 0 are included for comparative purposes. The paper uses a s t a b i l i t y u n d e r additions test to determine kurtosis values, a l l o w i n g the moments of a d i s t r i b u t i o n to be interdependent on each other. The key to the procedure is c a l c u l a t i n g the characteristic exponent of a stable model, w h i c h can t a k e on d i s t i n c t sets of values, thereby d e t e r m i n i n g the u n d e r l y i n g d i s t r i b u t i o n of the data analysed. ISEQ's r e t u r n s are tested i n three forms after a p p l y i n g a f i l t e r i n g process to remove data anomalies, as w e l l as g i v i n g estimates of the conditional variance. These are filtered, rescaled (obtained from standardising the filtered r e t u r n s ) a n d r a w c o m p o u n d e d r e t u r n s . Two filtering processes, n a m e l y , G A R C H Q , 1) and G A R C H - M ( 1 , 1 ) specifications are u t i l i s e d . The d i s t r i b u t i o n a l findings are analysed as to determine the influence of dependency i n b o t h the m e a n and variance parameters.
Excess kurotosis and skewness are found i n I r i s h e q u i t y r e t u r n s . Support is offered for the I S E Q belonging to a n o r d i n a r y stable d i s t r i b u t i o n after removal of t e m p o r a l and nonsynchronous t r a d i n g effects. T h i s d i s t r i b u t i o n a l finding allows for v a r y i n g parameters such as scale and skewness, and more importantly, the existence of v o l a t i l i t y clustering. As ISEQ's raw compounded r e t u r n s indicate m i x t u r e s of stables d i s t r i b u t i o n , i t demonstrates the benefits of filtering. B o t h first and second m o m e n t dependence are documented for the r a w r e t u r n s . The filters remove m u c h of the dependence, w i t h the G A R C H - M ( 1 , 1) d o m i n a t i n g t h e G A R C H d , 1) model for the I S E Q index.
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