Stavros Degiannakis

Alexandra Livada

Epaminondas Panas
email: panas@aueb.gr

Keywords: ARCH model, GED distribution, Leverage effect, Levy-stable distribution, Rolling sample ARCH model, GED distribution, Leverage effect, Levy-stable distribution, Rolling sample, Spill over, Value at risk C32, C52, C53, G15

In this paper an asymmetric autoregressive conditional heteroskedasticity (ARCH) model is applied to some well-known financial indices (DAX30, FTSE20, FTSE100 and SP500), using a rolling sample of constant size, in order to investigate whether the values of the estimated parameters of the model change over time. Although, there are changes in the estimated parameters reflecting that structural properties and trading behaviour alter over time, the ARCH model adequately forecasts the one-day-ahead volatility. A simulation study is run to investigate whether the time variant attitude holds in the case of a generated ARCH data process revealing that either in that case the rolling-sampled parameters are time-varying. The rolling analysis is also applied to estimate the parameters of a Levy-stable distribution. The empirical findings support that the stable parameters are also time-variant.

= = = = t t t t t t t t t d i i t t t t g z V z E f z z (1)
where is a vector of unknown parameters, ( ) . f is the density function of t z , ( ) . g is a linear or non-linear functional form and t is a vector of predetermined variables included in information set I at time t . By definition, ( ) t is serially uncorrelated with mean zero, but with a time varying conditional variance equal to ( )

2 t , or ( ) ( ) ( ) 2 1 , 0 | t t t f I
. [START_REF] Engle | Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation[END_REF] in his seminal study assumed that t z are normally distributed, whereas [START_REF] Bollerslev | A Conditional Heteroskedastic Time Series Model for Speculative Prices and Rates of Return[END_REF] and Nelson (1991) introduced the student t and the generalized error distributions, respectively, in order to model the leptokurtosis of the conditionally distributed ( ) t . In the case of modeling a leptokurtotic and asymmetric conditional distribution of ( ) t , the generalized t (Bollerslev et al. 1994), the skewed generalized t [START_REF] Theodossiou | Financial Data and the Skewed Generalized t Distribution[END_REF], the skewed student t [START_REF] Lambert | Modeling Skewness Dynamics in Series of Financial Data[END_REF] and the skewed generalized error [START_REF] Bali | Modeling the Dynamics of Interest Rate Volatility with Skewed Fat-Tailed Distributions[END_REF]Theodossiou 2002) distributions were utilized. Since very few financial time series have a constant conditional mean of zero, an ARCH model can be presented in a regression form by letting t be the unpredictable component of the conditional mean

( ) v ve z f v v z t v t 1 2 ) ( 1 2 1 = + , (4)
for

< < z , < v 0
, where ( )

. denotes the gamma function and

( ) ( )

2 1 2 3 1 2 1 v v v .
(5)

The conditional variance specification has the form of the exponential GARCH, or EGARCH model, which is suggested by Nelson (1991). The EGARCH model captures the asymmetric effect exhibited in financial markets, as the conditional variance, 2 t , depends on both the magnitude and the sign of lagged innovations. Assuming GED distributed innovations with EGARCH specification for the conditional variance we take into account that i) the unconditional distribution of innovations is symmetric but with excess kurtosis and ii) their conditional distribution is asymmetric and leptokurtotic.

Parameter allows for the leverage effect. The leverage effect, first noted by Black (1976), refers to the tendency of changes in stock returns to be negatively correlated with changes in returns volatility, i.e. volatility tends to rise in response to 'bad news' and to fall in response to 'good news'. If 0 = then a positive surprise, ( ) 0 > t , has the same effect on volatility as a negative surprise, ( ) actually reduces volatility while a negative surprise increases volatility. Moreover, the logarithmic transformation ensures that the forecasts of the variance are non-negative.

0 < t . If
Parameter 0 allows us to explore the contribution of non-trading days to volatility.

According to Fama (1965) and French and Roll (1986) information that accumulates when financial markets are closed is reflected in prices after the markets reopen. The conditional mean is modeled such as to capture the relationship between investors' expected return and risk2 ( 1 µ ), the non-synchronous trading effect3 ( 2 µ ), and the inverse relation between volatility and serial correlation4 ( 3 µ ).

Model ( 3) is expanded in order to take into account the phenomenon of volatility spill over from one market to the other5 . For t y denoting the daily return of the FTSE20 index, the conditional variance is modeled in the following form ( where the parameters 1 and 2 account for the volatility spill over from U.S. and German stock markets to the ASE market, respectively. The ASE in co-operation with the London-based Financial Times Stock Exchange has introduced the FTSE20 index, which includes the 20 largest companies ranked by their capitalization and trading volume. The daily conditional volatilities of the SP500 and DAX30 index returns at time . These are regarded as exogenous variables that have been estimated according to framework (3).

The data set consists of the FTSE20 index daily returns in the period from January 3 rd , 1996 to July 5 th , 2002 and the conditional variance of the DAX30 and SP500 returns from January 2 nd , 1996 to July 5 th , 2002. In order to estimate the conditional variance of the DAX30 and SP500 indices, their daily returns are used for the periods of January 14 th , 1992 to July 5 th , 2002, and January 7 th , 1992 to July 5 th , 2002, respectively. Figure 1a plots the FTSE20 daily returns.

INSERT FIGURE 1 ABOUT HERE

Maximum likelihood estimates of the parameters are obtained by numerical maximization of the log-likelihood function using the Marquardt (1963) algorithm that is computed as

( ) ( ) ( ) ( ) ( ) ( ) ( ) = = + + = T t i t T t i t i t i i l l l 1 1 1 1 ! , (7)
where is the log likelihood contribution for each observation t , is the identity matrix and ! is a positive number chosen by the algorithm. The process is repeated until the maximum of the percentage changes in the coefficients is smaller than 0.001%.

( ) v a , , , , , , , , , , , 2 ,

INSERT TABLE 1 ABOUT HERE

Table 1 presents the estimated parameters of model (3). The estimated risk premium is positively, though weakly related with the conditional variance (coefficient 1 µ ). The coefficient 2 µ , which allows for first order autocorrelation, is insignificant. Daily serial correlation is inversely related to the conditional volatility of the FTSE20 index, which is consistent with the results of LeBaron (1992) (coefficient 3 µ ). The estimated value of coefficient is -0.064 and statistically insignificant, which implies that surprises of same magnitude but opposite signs have the same effect on volatility. The estimated parameters 1 and 2 are insignificant, indicating that there is no evidence of volatility spillover from Frankfurt and Chicago indices to ASE market. The estimated coefficient v is 1.335 with a standard error of 0.043, so the distribution of the standardized innovations is significantly thicker tailed than the normal distribution. The estimated value of 0 is about 0.187 and statistically significant. Thus, a non-trading day contributes less than a fifth as much to volatility as a trading day.

. R o l l i n g -s a m p l e d p a r a m e t e r s o f t h e a s y m m e t r i c A R C H m o d e l

Our purpose is to examine if the estimated parameters of the asymmetric ARCH model change over time and whether there is any impact of time-varying estimated parameters on volatility forecasting accuracy. The ARCH process is estimated, at each of a sequence of points in time, using a rolling sample of constant size equal to 1000 trading days, a sample size that is preferred 6 by the majority of applied studies.

We produce one-day-ahead conditional volatility predictions for the trading days 2 presents the percentage of rolling-sampled estimations, which are outside of the 95% confidence interval. That is, the rolling estimations of coefficient 1 are outside the 95% confidence interval of the full-sampled 1 estimation in the 54.40% of the cases. An important characteristic, which is extracted from the 6 Engle et al. (1993), Engle et al. (1997), Noh et al. (1994), [START_REF] Angelidis | The Use of GARCH Models in VaR Estimation[END_REF] note that the size of the rolling sample turns out to be rather important while Frey and Michaud (1997), Hoppe (1998) and Degiannakis and Xekalaki (2006) comment that the use of short sample sizes generates more accurate volatility forecasts, since it incorporates changes in trading behaviour more efficiently.

INSERT FIGURE 2 ABOUT HERE INSERT TABLE 2 ABOUT HERE

The percentage of estimated rolling parameters that are statistically different from the parameter values estimated using the full data sample, as presented in Table 3, is also indicative for the changes of the estimated values across time. There are four parameters, whose rolling-sampled estimators differ statistically significant from their full-sampled estimators in more than 10% of the trading days.

INSERT TABLE 3 ABOUT HERE

The values of the rolling parameters indicate that the characteristics of the ASE market change during the examined period. Table 4 presents the percentage of the trading days for which the estimated rolling parameters are statistically insignificant. The coefficient 0 , which accounts for the contribution of non-trading days to volatility, is not statistically different to zero for the 57.07% of the rolling cases, although it is significant in the full sample, at 1% level of significance. Therefore, the statistical inference based on the estimated values of parameter 0 would lead to an insignificant contribution of non-trading days to volatility in about the half number of trading days. Moreover, in the full sample, FTSE20 index is characterized by an inverse relation between volatility and serial correlation. On the contrary, the values of rolling 3 µ are not different to zero in most of the cases. However, there are parameters, such as 1 , 1 and v , whose estimations are statistically significant in both full and rolling sample. On the other hand, the values of the coefficients 2 µ , and 1 are statistically insignificant in both rolling

INSERT TABLE 4 ABOUT HERE

In Figure 3.a, the 95% in-sample confidence interval of the FTSE20 index of daily returns is plotted from 11 th January 2000 to 5 th July 2002. There are 31 (4.99%) violations of the confidence interval, which reflect a correctly specified model that fits data satisfactory. However, a model that uses a large number of parameters may exhibit an excellent in-sample fit but a poor out-of-sample performance. Studies such as Heynen and Kat (1994), Hol and Koopman (2000) and Pagan and Schwert (1990) examined a variety of volatility prediction models with in-sample and out-of-sample data sets. We investigate the possibility that model over-fitting can be occurred and evaluate the performance of the estimated ARCH model by computing the out-of-sample forecasts. In the sequel, the one-day-ahead 95% prediction intervals are constructed.

Let us compute the one-day-ahead conditional mean,

+ + + = + + + 4 2 | 1 3 2 µ µ µ µ , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), ln ln 1 1 1 ln ln 2 | , 30 2 2 | , 500 1 | | | | 1 1 0 1 0 2 | 1 t t
+ + + + + + = + + (8) where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t t t t t t t t t t t t t v a , , , , , , , , , , , 2 1 1 1 0 0 3 2 1 0 µ µ µ µ
is the parameter vector that is estimated using the sample data set which is available at time t , ( ) 3.b plots the one-day-ahead 95% prediction interval, which is constructed as the one-day-ahead conditional mean ± 2.06 times the conditional standard deviation, both measurable to t I information set, or

( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 / 1 1 3 1 2 = t t t t t v v v E . Figure
( ) ( ) t t t t t v GED y | 1 | 1 025 . 0 , ; 1 , 0 + + ± , where ( ) ( ) a v GED t , ; 1 , 0 is the ( ) a 1 100
quantile of the GED distribution. Hence, each trading day, ( t ), the next trading day's, ( 1 + t

), prediction intervals are constructed, using only information available at current trading day, t . There are 29 observations (4.67%) outside the 95% prediction intervals.

INSERT FIGURE 3 ABOUT HERE

For a more formal method of evaluating forecasting adequacy, we apply two hypotheses tests that measure the forecasting accuracy in a VaR framework. One-dayahead VaR at a given probability level, a , is the next trading day's predicted amount of financial loss of a portfolio, or Kupiec (1995) introduced a likelihood ratio statistic for testing the null hypothesis that the proportion of confidence interval violations is not larger than the VaR forecast. The test statistic, which is asymptotically

( ) ( ) ( ) t t t t a v GED a VaR | 1 1 | 1 , ; 1 , 0 1 + + = .
2 1 X distributed, is computed as = K LR ) ) 1 ( ) [ln(( 2 n N n N n N n ] ) 1 ( ln( n N n p p
, where n ( )

( ) = + + < N i t t t a VaR y d 1 | 1 1 2 / ( ) ( ) 2 / 1 | 1 1 a VaR y d t t t > + + + , for ( ) ( ) 1 2 / | 1 1 = < + + a VaR y d t t t if t t t VaR y | 1 1 + + < and ( ) ( ) 0 2 / | 1 1 = < + + a VaR y d t t t
otherwise, is the number of trading days over the out-of-sample period N that a violation has occurred, and p is the expected frequency of violations. Christoffersen (1998) developed a likelihood ratio statistic that jointly investigates whether i) the proportion of violations is not larger than the VaR forecast and ii) the violations are independently distributed. The statistic is computed as Despite the fact that the values of the estimated coefficients change over time, the model adequately forecasts the one-day-ahead volatility. Thus, at least in the case of ASE market, changes in the values of the estimated parameters do not indicate inadequacy of the model in describing the data. On the contrary, model's parameters should be re-estimated on a daily base in order to reflect any changes that have been occurred in the stock market and have been incorporated in the prices of assets.

= C LR ) ) 1 ( 2ln( - n n N p p )) ) 1 ( ) 1 ( 2ln(

. T h e a s y m m e t r i c A R C H m o d e l f o r o t h e r s t o c k m a r k e t s

In this section we investigate whether the values of the estimated parameters change over time in other stock markets as well. The asymmetric ARCH model in framework ( 6) is applied for the DAX30, FTSE100 and SP500 stock indices:

t t A t A t A y e y t A µ µ µ µ µ + + + + = 1 , 3 2 2 , 1 0 , 4 2 , , ( ) ( ) ( )
( ) ( ).

ln ln 1 1 1 ln ln 2 1 , 2 2 1 , 1 , , , 1 1 0 0 2 , + + + + + + = t C t B t A t t A t t A t t t A L E L L N a (9)
In order to account for the volatility spill over effect from one market to the others, when (9) is estimated for stock market A (for instance SP500), the daily conditional volatilities of stock markets B and C (that is FTSE100 and DAX30 respectively) are regarded as exogenous variables. Table 5 presents the estimated parameters of the above model (9) for each market separately. The data set used is from January 3 rd , 1996 to July 5 th , 2002. Briefly discussing the values of the parameters, we note that i) the relation of the conditional variance with the risk premium, although positive, is statistically insignificant (coefficient 1 µ ), ii) the non-synchronous trading effect is not present in the estimated models (coefficient 2 µ ) and iii) concerning the case of the SP500 stock index, the daily serial correlation is inversely related to its conditional volatility (coefficient 3 µ ). Moreover, the leverage effect is not present in the German stock market. On the contrary, for the SP500 and FTSE100 stock indices, the estimated value of parameter is statistically significant at 1% level of significance. The volatility spill over effect is statistically significant for the U.K. stock market. Regarding the SP500 index daily returns, there is evidence that volatility spillovers from Frankfurt to Chicago stock market. Finally, for the

INSERT TABLE 5 ABOUT HERE

Following the approach presented in section 3, the rolling parameters of the ARCH models are estimated using a rolling sample of 1000 observations 7 . As in the case of the ASE market, the values of the rolling parameters differ from their fullsampled estimations.

. E x t e n s i o n s

In order to investigate whether the phenomenon of time-variant values of estimated parameters, in the four stock markets considered, is related to a specific structural characteristic of the model specification, we estimate another ARCH specification. Degiannakis (2004) and Giot and Laurent (2003) used an ARCH model with the APARCH volatility specification of Ding et al. (1993) and the skewed student-t distribution for the standardized innovations. We estimated such a model for our datasets and found similar qualitative results. The estimated parameters are time varying.

We have also re-estimated model (9) using alternatively i) larger sample sizes of rolling parameters, ii) the BHHH algorithm (Berndt et al. 1974) instead of the Marquardt algorithm in estimating the maximum likelihood parameters and iii) the same starting values at each point in time, instead of the estimates at time 1 t as starting values for the likelihood algorithm at time t . Despite the slight changes occurred in each case, the rolling parameters are time-variant for all cases.

. R o l l i n g -s a m p l e d p a r a m e t e r s f r o m s i m u l a t e d p r o c e s s e s

A simulation study could shed light in rolling-sampled estimated parameters behaviour.

A series of simulations is run in order to investigate if the time-variant attitude holds even in the case of an ARCH data generating process. We generate a series of 32000 values from the standard normal distribution, ( ) 9, about 58% of the 30000 conditional variance rolling-sampled parameters are outside the 95% confidence interval of the parameters estimated using the whole sample set of the 30000 simulated data. The procedure is repeated for an

1 , 0 ~. . . N z d i i t . Then an AR(1)GARCH(1,
AR(1)EGARCH(1,1) conditional variance form, ( ) ( ) 2 1 1 1 1 1 1 1 0 2 ln ln + + = t t t t t t a a

&

, but the results are robust to the choice of the conditional variance specification.

A series of 32000 values from the first order autoregressive process are also produced. The AR(1) process is created as

t t t z y y + + = 1 12 . 0 0001 . 0 , for ( ) 1 , 0 ~. . . N z d i i t .
Dropping out the first 1001 data, 30000 maximum likelihood rolling-sampled estimates of the parameters are also obtained. As far as the case of the AR(1) process is concerned, we infer that the rolling estimated parameters are time-invariant, as on average the 5% of the estimated rolling parameters are outside the 95% confidence levels.

Both the AR(1)GARCH(1,1) and the AR(1) processes were simulated for various sets of parameters, but there are no qualitative differences to the fore mentioned conclusions. Moreover, a series of simulations were repeated i) for ARCH volatility forms without any conditional mean specification, ii) based on estimation procedures of the most well known packages, EVIEWS ® 4.1 and OX-G@ARCH ® 3.4, iii) for larger rolling samples of 5000 values, iv) for non-overlapping data samples, but there are no qualitative differences in any of these cases 9 .

So, the simulation study provides evidence that the time-variant attitude of rolling-sampled parameters estimations characterizes not only the examined data sets but the ARCH data generating process itself as well.

INSERT TABLE 9 ABOUT HERE

. R o l l i n g -s a m p l e d p a r a m e t e r s f r o m a L e v y -s t a b l e d i s t r i b u t i o n

The study of the shape of stock returns can be dated back to 1900 where Louis Bachelier in his Ph.D. Thesis, 'The Theory of Speculation', first presented it -see [START_REF] Bachelier | Theorie de la Speculation[END_REF]. Mandelbrot (1963) and Fama (1965) made the first re-examination of the unconditional distribution of stock returns. Mandelbrot (1963) concluded that price changes can be characterized by a stable Paretian distribution with a characteristic exponent, a , less than two, thus exhibiting fat tails and infinite variance. Fama (1965) examined the distribution of thirty stocks of the Dow Jones Industrial Average; his results were consistent with Mandelbrot's. Thereafter, it has been accepted that the stock returns distributions are fat-tailed and peaked. In an attempt to model the unconditional distribution of stock returns several researchers have considered alternative approaches. See for example, Blattberg and Gonedes (1974), Bradley and Taqqu (2002), Clark (1973), Kon (1984), McDonald (1996), Mittnik and Rachev (1993), Rachev and Mittnik (2000). De Vries (1991), Ghose and Kroner (1995) and Groenendijk et al. (1995) demonstrate that ARCH models share many of the properties of Levystable distribution but the true data generating process for an examined set of financial data is more likely ARCH than Levy-stable. A number of studies, such as Liu and Brorsen (1995), Mittnik et al. (1999), Panorska et al. (1995), [START_REF] Tsionas | Likelihood-Based Comparison of Stable Paretian and Competing Models: Evidence from Daily Exchange Rates[END_REF], examined the properties of ARCH models with Levy-stable distributed innovations.

9 All the simulation studies are available to the readers upon request. The probability density function of a stable distribution cannot be described in a closed mathematical form. By definition, a univariate distribution function is stable if and only if its characteristic function has the form

( ) ( ) ' ( ' ) * ' + ' , - = a t t t i t t i t a , 1 exp . & / , ( 10
)
where In estimating the parameters of the stable distribution of index returns, we adopt the estimation procedure suggested by McCulloch (1986)

1 = i , R t 0 with ( ) ' ' + ' ' , - = 1 = . 1 , log 2 1 , 2 tan , a t a a t # #2 .
( ) ( ) ( ) ( ) ( ) 25 . 0 75 . 0 ˆ05 . 0 95 . 0 ˆx x x x = 2 3 (11) ( ) ( ) ( ) ( ) ( ) ( ) 05 . 0 95 . 0 ˆ50 . 0 2 05 . 0 95 . 0 ˆx x x x x + = & 3 . ( 12
)
Since ( )

a 3 is monotonic in a and ( ) & 3
is monotonic in & , we are able to find a and & by inverting ( )

a 3 and ( ) & 3 : ( ) ( ) ( ) & 3 3 , 1 a g a = , ( 13
) ( ) ( ) ( ) & 3 3 & , 2 a g = . ( 14
)
McCulloch tabulated 1 g and 2 g for various values of ( )

a 3 and ( ) & 3
. A similar procedure is also applied for the scale and location parameters. An alternative procedure to estimate the parameters of the stable distribution is the regression method proposed by Koutrouvelis (1980).

Following a procedure similar to that of ARCH modelling, the parameters of the stable distribution are estimated, at each of a sequence of points in time, using a rolling sample of constant size equal to 1000 trading days. Thus, the rolling-sampled parameters are estimated for the trading days of January 1996 to July 2002 for the DAX30, FTSE100 and SP500 indices and from January 2000 to July 2002 for the FTSE20 index.

The empirical findings, for the case of the Greek stock market, are graphically summarized in Figure 4, which plots the rolling-sampled estimates of parameters along with the 95% confidence interval of the parameters estimated using the full data sample. 4 shows that the estimates of a are less than two. The case of FTSE20 reveals that 92% of the a 's rolling-sampled estimates are between 1.44 and

Inspection of Figure

INSERT FIGURE 4 ABOUT HERE

In Table 10, we present the estimates of the parameters of stable distribution based on all data available as well as the standard deviation of the rolling-sampled estimated parameters. The estimates of a do not approach two in any of the examined indices. However, there are estimated rolling parameters that are statistically different from the parameter values estimated using the full data sample. For example, the rolling-sampled estimates of the tail index ( a ) are statistically different to the full sample estimated parameter in the 51.46% of the trading days for the case of the SP500 index.

In 9.59% and 9.42% of the trading days the rolling estimates of parameter & are statistically different to the relevant full-sampled values for the DAX30 and FTSE100 indices, respectively, whereas the location ( ) parameters are time-variant in none of the cases. Another important parameter of the stable distribution, from the point of view of portfolio theory, is the scale parameter, . As far as the FTSE20 index is concerned, the rolling-sampled estimates of the scale parameter differ statistically from its fullsampled value in the 56.48% of the trading days. Hence, the parameter estimates, using the full data sample are statistically different from the parameter values estimated using the rolling samples of constant size for one parameter in each index.

INSERT TABLE 10 ABOUT HERE

. D i s c u s s i o n

We estimated an asymmetric ARCH model using daily returns of the FTSE20, DAX30, FTSE100 and SP500 indices and concluded that although the estimated parameters of the model change over time, the model does not lose its ability to forecast the one-dayahead volatility accurately. Gallant et al. (1991), Stock (1988) and Lastrapes (1990) have noted, the explanation of the ARCH process must lie either in the arrival process of news or in market dynamics in response to the news. Based on some earlier work by Clark (1973) and Tauchen and Pitts (1983), Gallant et al. (1991) provided a theoretical interpretation of the ARCH effect. They assumed that the asset returns are defined by a stochastic number of intra-period price revisions and information flows into the market in an unknown rate. As the daily information does not come to the stock market in a constant and known rate, the estimation of the ARCH stochastic process that explains the dynamics of the stock market could be revised at regular time intervals. In our case the ARCH process is estimated using daily returns.

Thus, the parameters of the model may be revised on a daily base, because of the observed phenomenon of changes in the estimated parameters. If we used data of higher frequency, i.e. ten-minutes intra-daily returns, the estimated model may be revised more frequent than on a daily base. The change of the estimated values of the parameters has to be further examined in intra-daily high-frequency data sets, on a future research.

Furthermore, the rolling parameter analysis was applied to the unconditional distribution of returns. The empirical results indicate that in all cases -DAX30, FTSE20, FTSE100 and SP500 -the parameters of the asymmetric stable distribution of stock returns change across time. We observed the phenomenon of parameter changing across time for both the conditional (ARCH process) and the unconditional (Levy-stable) distribution of returns.

Altering the method of model estimation, the rolling-sampled parameters remain time-variant. Even in the case of a simulated ARCH process, the property of time varying rolling-sampled parameters holds. To the best of authors' knowledge, this is the first study that investigates the phenomenon of time varying estimated parameters either i) in real-world financial data or ii) in a simulated data generating process. However, the theoretical interpretation of this phenomenon has to be further investigated. Notes: The 95% confidence interval is constructed as ( )

F

1000 1621 025 . 0 , 335 . 1 ; 1 , 0 ˆ S GED ±
, where denotes the parameter vector estimated using the full data sample, S ˆ is the standard deviation of ˆ and ( )

a v GED , ; 1 , 0 is the ( ) a 1
percentile of the GED distribution, with v denoting the tail thickness parameter.

. I n t r o d u c t i o n

In the recent literature, regarding the description of the characteristics of financial markets, one can find a vast number of specifications of both ARCH and Stochastic Volatility (SV)

processes that have been considered for. However, the SV models1 are not as popular as the ARCH processes in applied studies. The purpose of the present study is to apply an asymmetric ARCH model to some well known financial indices, using a rolling sample of constant size, in order to observe the changes over time in the values of the estimated parameters. A thorough investigation is conducted by comparing the parameters of the fullsampled estimated model to the parameters of the rolling sub-sample estimated models. We conclude that the values of the estimated parameters change over time, indicating a data set that alters across time reflecting the information that financial markets reveal. The analysis is extended to simulated time series indicating that the time-varying estimated coefficients characterize the ARCH data generating process itself.

In ARCH modelling, the distribution of stock returns has fat tails with finite or infinite unconditional variance and time dependent conditional variance. Estimation of stable distributions is an alternative approach in modelling the unconditional distribution of returns.

Thus, we adopt the estimation procedure of McCulloch (1986) and the parameters of the Levy-stable distribution are estimated at each of a sequence of points in time, using a rolling sample of constant size. The empirical findings suggest that the parameters of the unconditional distribution are also not constant over time.

Reviewing the relevant literature we notice absence of studies showing that although the parameters of a well-specified model vary significantly over time, their time varying attitude does not influence model's forecasting ability. The main object of our study is to provide evidence that model's parameters should be re-estimated on a frequent base in order to reflect any changes that have been occurred in the stock market and have been incorporated

in the prices of assets. The paper is divided in six sections. Section 2 lays out the asymmetric ARCH model that is applied in the FTSE20, DAX30, FTSE100 and SP500 stock indices. In section 3, the estimated rolling-sampled parameters of the asymmetric ARCH model are discussed. In section 4, a simulation study examines whether the parameters are time-varying in the case of a generated ARCH process. In section 5, the unconditional distribution of returns is estimated and the phenomenon of time-variant parameters is investigated in the Levy-stable distribution.

Finally, in section 6 we summarize the main conclusions.

. A n a s y m m e t r i c A R C H m o d e l

A wide range of proposed ARCH models is covered in surveys such as [START_REF] Andersen | DM-Dollar Volatility: Intraday Activity Patterns, Macroeconomic Announcements, and Longer-Run Dependencies[END_REF], [START_REF] Bera | ARCH Models: Properties, Estimation and Testing[END_REF], Bollerslev et al. (1992), Bollerslev et al. (1994), Degiannakis and Xekalaki (2004) and Poon and Granger (2003). The Nobel price award to R.F. Engle for ARCH volatility modeling is the uncontested proof of the contribution of ARCH models in time series and econometric modelling (Diebold 2003). A plethora of studies applied ARCH models to predict future volatility by updating the available information set at each of a sequence of points in time. Among others, Balaban and Bayar (2005) tested in 14 countries the relationship between stock market returns and their forecast volatility, [START_REF] Blair | Forecasting S&P 100 volatility: the incremental information content of implied volatilities and high frequency index returns[END_REF] compared the information content of implied volatilities and intraday returns in the context of forecasting S&P100 volatility, [START_REF] Wei | Forecasting stock market volatility with non-linear GARCH models: a case for China[END_REF] forecast China's weekly stock market volatility and [START_REF] Yu | Forecasting volatility in the New Zealand stock market[END_REF] predicted stock price volatility using daily New Zealand data. [START_REF] Angelidis | The Use of GARCH Models in VaR Estimation[END_REF], Degiannakis (2004), Brooks and Persand (2003) and Giot and Laurent (2003) predicted

Value-at-Risk (VaR) measures, while Degiannakis and Xekalaki (2001), Engle et al. (1997) and Noh et al. (1994) used rolling ARCH models to forecast volatility of options.

An ARCH process, ( ) θ ε t , can be presented as

( ) ( ) ( ) ( ) ( ) ( ) ( ), ,... , ,...; , ,...; , 1 , 0 ~2 1 2 1 2 1 2 . . . - - - - - - = = = = t t t t t t t t t d i i t t t t g z V z E f z z υ υ ε ε σ σ θ σ θ σ θ ε (1)
where θ is a vector of unknown parameters, ( )

ε μ μ σ μ μ μ σ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + = - - 1 , 3 2 2 , 1 0 , 4 2 , , t A t t z , σ ε = , ( ) v GED z d i i t ; 1 , 0 ~. . . , ( ) ( ) ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - Ψ Δ - + + + = t A t t A t t A t t t A L E L L N a , , , 1 1 0 0 2 , 1 1 1 ln ln σ ε γ σ ε σ ε δ σ , ( 3
)
where denotes the generalized error distribution (GED), v is the tail thickness parameter of the GED,

( v GED ; 1 , 0 )
L is the lag operator and is the number of non-trading days preceding the day. The density function of a GED random variable is given by

t N th t ( ) v ve z f v v z t v t 1 2 ) ( 1 2 1 Γ = + -- λ λ , (4) for , ∞ < < ∞ - z ∞ ≤ < v 0
, where ( ) . Γ denotes the gamma function and

( ) ( ) 2 1 2 3 1 2 1 ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ Γ Γ ≡ - - v v v λ .
(5)

The conditional variance specification has the form of the exponential GARCH, or EGARCH model, which is suggested by Nelson (1991). The EGARCH model captures the asymmetric effect exhibited in financial markets, as the conditional variance, , depends on both the magnitude and the sign of lagged innovations. Assuming GED distributed innovations with EGARCH specification for the conditional variance we take into account that i) the unconditional distribution of innovations is symmetric but with excess kurtosis and ii) their conditional distribution is asymmetric and leptokurtotic. Parameter γ allows for the leverage effect. The leverage effect, first noted by Black (1976), refers to the tendency of changes in stock returns to be negatively correlated with changes in returns volatility, i.e. volatility tends to rise in response to 'bad news' and to fall in response to 'good news'. Moreover, the logarithmic transformation ensures that the forecasts of the variance are non-negative. Parameter 0 δ allows us to explore the contribution of non-trading days to volatility. According to Fama (1965) and French and Roll (1986) information that accumulates when financial markets are closed is reflected in prices after the markets reopen. The conditional mean is modeled such as to capture the relationship between investors' expected return and risk 2 ( 1 μ ), the non- synchronous trading effect 3 ( 2 μ ), and the inverse relation between volatility and serial

correlation 4 ( 3 μ ).
Model ( 3) is expanded in order to take into account the phenomenon of volatility spill over from one market to the other 5 :

( )

( ) ( ) ( ) ( . ln ln 1 1 1 ln ln 2 1 , 2 2 1 , 1 , , , 1 1 0 0 2 , - -+ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - Ψ Δ - + + + = t C t B t A t t A t t A t t t A L E L L N a σ δ σ δ σ ε γ σ ε σ ε δ σ ) (6)
2 The relationship between investors' expected return and risk was presented in an ARCH framework, by Engle et al. (1987). They introduced the ARCH in mean model where the conditional mean is an explicit function of the conditional variance. 3 According to Campbell et al. (1997), 'The non-synchronous trading or non-trading effect arises when time series, usually asset prices, are taken to be recorded at time intervals of one length when in fact they are recorded at time intervals of other, possible irregular lengths.' 4 LeBaron (1992) found a strong inverse relation between volatility and serial correlation for SP500, CRSP and Dow Jones returns. As LeBaron stated, it is difficult to estimate 4 μ in conjunction with when using a gradient type of algorithm. So, 3 μ 4 μ is set to the sample variance of the series. 5 Engle et al. (1990) evaluated the role of the information arrival process in the determination of volatility in a multivariate framework providing a test of two hypotheses: heat waves and meteor showers. Using meteorological analogies, they supposed that information follows a process like a heat wave so that a hot day in New York is likely to be followed by another hot day in New York but not typically by a hot day in Tokyo. On the other hand, a meteor shower in New York, which rains down on the earth as it turns, will almost surely be followed by one in Tokyo. Thus, the heat wave hypothesis is that the volatility has only country specific autocorrelation, while the meteor shower hypothesis states that volatility in one market spills over to the next. See also Kanas (1998). δ account for the volatility spill over from B and C stock markets to the A stock market, respectively. In order to account for the volatility spill over effect from one market to the others, when ( 6) is estimated for stock market A, the daily conditional volatilities of stock markets B and C are regarded as exogenous variables that have been estimated according to framework (3)6 .

The data set used in this paper consists of the Financial Times Stock Exchange 20 (FTSE20) index for Greece, the Deutscher Aktien Index 30 (DAX30) for Germany, the Financial Times Stock Exchange 100 (FTSE100) index for U.K. and the Standard & Poor's 500 (SP500) index for U.S.A. The period covered for the FTSE20, DAX30, FTSE100 and SP500 is from January 3 rd 1996, January 14 th 1992, January 9 th 1992 and January 7 th 1992 to July 5 th 2002, respectively. A thorough investigation is conducted by comparing the parameters of the full-sampled estimated model to the parameters of the rolling sub-sample estimated models. Maximum likelihood estimates of the parameters are obtained by numerical maximization of the log-likelihood function using the Marquardt (1963) algorithm.

INSERT TABLE 1 ABOUT HERE

Table 1 presents the estimated parameters of model ( 6) for each market separately.

The standardized residuals, , and their squared values, , from all models obey the standard assumptions of autocorrelation and heteroskedasticity absence. Indicatively, we present the Ljung-Box Q-statistic for the null hypothesis that there is not autocorrelation up to 20 1 ,

- t A t σ ε 2 , 2 - t A t σ ε
th order computed on and . Briefly discussing the values of the parameters, we note that i) the relation of the conditional variance with the risk premium, although positive, is

statistically insignificant (coefficient 1 , - t A t σ ε 2 , 2 - t A t σ ε 1 μ ), ii
) the non-synchronous trading effect is not present in the estimated models (coefficient 2 μ ) and iii) concerning the cases of the FTSE20 and SP500 stock indices, the daily serial correlation is inversely related to its conditional volatility (coefficient 3 μ ). Moreover, the leverage effect is not present in the Greek and German stock markets. On the contrary, for the SP500 and FTSE100 stock indices, the estimated value of parameter γ is statistically significant at 1% level of significance. The volatility spill over effect is statistically significant for the U.K. stock market. Regarding the SP500 index daily returns, there is evidence that volatility spillovers from Frankfurt to Chicago stock market. Finally, for the FTSE20, DAX30 and SP500 cases, parameter v is statistically different to the value of 2 at any level of significance, justifying the use of a thick-tailed distribution. The estimated value of 0 δ is about 0.187 and statistically significant only in the case of the Greek market indicating that a non-trading day contributes less than a fifth as much to volatility as a trading day.

. R o l l i n g -s a m p l e d p a r a m e t e r s o f t h e a s y m m e t r i c A R C H m o d e l

Our purpose is to examine if the estimated parameters of the asymmetric ARCH model change over time and whether there is any impact of time-varying estimated parameters on volatility forecasting accuracy. The ARCH process is estimated, at each of a sequence of points in time, using a rolling sample of constant size equal to 1000 trading days, a sample size that is preferred 7 by the majority of applied studies.

We produce one-day-ahead conditional volatility predictions for the trading days of 11 th Δ for FTSE20 and SP500. However, there are rolling parameters which do not change significantly 7 Engle et al. (1993), Engle et al. (1997), Noh et al. (1994), [START_REF] Angelidis | The Use of GARCH Models in VaR Estimation[END_REF] note that the size of the rolling sample turns out to be rather important while Frey and Michaud (1997), Hoppe (1998) and Degiannakis and Xekalaki (2006) comment that the use of short sample sizes generates more accurate volatility forecasts, since it incorporates changes in trading behaviour more efficiently. 8 Figures of the estimated rolling parameters for the DAX30, FTSE100 and SP500 indices, similar to Figure 1, are available upon request.

INSERT FIGURE 1 ABOUT HERE INSERT TABLE 2 ABOUT HERE

The percentage of estimated rolling parameters that are statistically different from the parameter values estimated using the full data sample, as presented in Table 3, is also indicative for the changes of the estimated values across time. There are four parameters, in the case of the Greek market, whose rolling-sampled estimators differ statistically significant from their full-sampled estimators in more than 10% of the trading days. Although, in the case of the FTSE100 index, only the rolling estimators of 1 Δ parameter differ statistically from their full data sample estimator, in the case of the SP500 index there are four parameters, which show a statistically significant difference from their full-sampled estimators in more than 20% of the trading days.

INSERT TABLE 3 ABOUT HERE

The values of the rolling parameters indicate that the characteristics of the markets change during the examined period. According to Table 4, which presents the percentage of trading days that the rolling parameters are statistically insignificant, there are parameters whose rolling-sampled estimations are statistically insignificant while their full-sampled estimations are significant. For example, parameters 3 μ and 1 δ for the SP500 index, as well as parameter γ for FTSE100 index, although they appear to be significant in the full sample, almost all their rolling-sampled estimations are insignificant at 5% level of significance.

Therefore, in the full sample, an inverse relation between volatility and serial correlation

INSERT TABLE 4 ABOUT HERE

However, although the estimated parameters are time varying, the in-sample and outof-sample forecasting ability of the model is accurate. There are 31, 19, 17 and 29 cases, or 4.99%, 2.99%, 2.66% and 4.57%, observed returns outside the 95% confidence intervals for the FTSE20, DAX30, FTSE100 and SP500 indices, respectively. In Figure 2.a, the 95% insample confidence interval of the FTSE20 index of daily returns is plotted from 11 th January 2000 to 5 th July 2002. However, a model that uses a large number of parameters may exhibit an excellent in-sample fit but a poor out-of-sample performance. Studies such as Heynen and Kat (1994), Hol and Koopman (2000) and Pagan and Schwert (1990) examined a variety of volatility prediction models with in-sample and out-of-sample data sets. We investigate the possibility that model over-fitting can be occurred and evaluate the performance of the estimated ARCH model by computing the out-of-sample forecasts. In the sequel, the one-dayahead 95% prediction intervals are constructed. Let us compute the one-day-ahead conditional mean,

( ) ( ) ( ) t t t t t I y E y | 1 | 1 θ + + ≡
, and conditional variance,

( ) ( ) ( ) t t t t t I E | 2 1 2 | 1 θ ε σ + + ≡ , using the following formulas: ( ) ( ) ( ) ( ) ( ) t A t t t t A t t t t A y e y t t t A , 3 2 2 | 1 , 1 0 | 1 , 4 2 | 1 , ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + = + - + + μ σ μ μ σ μ μ , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), ln ln 1 1 1 ln ln 2 | , 2 2 | , 1 | , | , | , | 1 1 0 1 0 2 | 1 , t t C t t t B t t t A t t t t A t t t A t t t t t t t t t A E L N a σ δ σ δ σ ε γ σ ε σ ε δ σ + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - Ψ Δ - + + + = + + (7) F o r P e e r R e v i e w where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t t t t t t t t t t t t t v a , , , , , , , , , , , 2 1
1 1 0 0 3 2 1 0 δ δ γ δ μ μ μ μ θ Δ Ψ ≡
is the parameter vector that is estimated using the sample data set which is available at time , t ( )

t t t t I E | | ε ε ≡
denotes the prediction error conditional on the information set that is available at time t , and ( )

t t t t A I E | 2 | , ε σ ≡
is the conditional standard deviation which is computed by the ARCH model, in equation ( 6), using the information set available at time t . Note that for , the expected value of its absolute price is equal to

( v GED z t ; 1 , 0 ~)( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 / 1 1 , F o r P e e r R e v i e w
the out-of-sample period that a violation has occurred, for

N ( ) ( ) 1 2 / | 1 1 = < + + a VaR y d t t t if and t t t VaR y | 1 1 + + < ( ) ( ) 0 2 / | 1 1 = < + + a VaR y d t t t
otherwise, and p is the expected frequency of violations. Christoffersen (1998) developed a likelihood ratio statistic that jointly investigates whether i) the proportion of violations is not larger than the VaR forecast and ii) the violations are independently distributed. The statistic is computed as , where proportion of violations in all the cases, except for the 95%-VaR of the FTSE100 index. In the case of Kupiec's test the p-values are 70.28%, 6.08%, 3.45% and 96.37% for 95%-VaR and 8,15%, 13.63%, 56.56% and 52.70% for 99%-VaR, for the FTSE20, DAX30, FTSE100 and SP500 indices, respectively. Testing the null hypothesis of whether the violations are equal to the expected ones as well as if they are independent, we observe that the relative p-values are 40.03%, 16.42%, 0.15% and 95.19% in the 95%-VaR case and 17.98%, 32.51%, 7.10% and 73.92% in the 99%-VaR case, for the FTSE20, DAX30, FTSE100 and SP500 indices, respectively.

= C LR ) ) 1 ( 2ln( - n n N p p - - )) ) 1 ( ) 1 ( 2ln(

C LR

Despite the fact that the values of the estimated coefficients change over time, the model adequately forecasts the one-day-ahead volatility. Thus, changes in the values of the estimated parameters do not indicate inadequacy of the model in describing the data. On the contrary, model's parameters should be re-estimated on a daily base in order to reflect any changes that have been occurred in the stock market and have been incorporated in the prices of assets 10 . 10 In order to investigate whether the phenomenon of time-variant values of estimated parameters is related to a specific structural characteristic of the model specification, we estimate another ARCH specification. Degiannakis Dropping out the first 1001 data, 30000 maximum likelihood rolling-sampled estimates of the parameters are also obtained. As far as the case of the AR(1) process is concerned, we infer that the rolling estimated parameters are time-invariant, as on average 5% of the estimated rolling parameters are outside the 95% confidence levels.

( ) (2004) and Giot and Laurent (2003) used an ARCH model with the APARCH volatility specification of Ding et al. (1993) and the skewed student-t distribution for the standardized innovations. We estimated such a model for our datasets and found similar qualitative results. The estimated parameters are time varying. We have also reestimated model (6) using alternatively i) larger sample sizes of rolling parameters, ii) the BHHH algorithm (Berndt et al. 1974) instead of the Marquardt algorithm in estimating the maximum likelihood parameters and iii) the same starting values at each point in time, instead of the estimates at time t as starting values for the likelihood algorithm at time t . Despite the slight changes occurred in each case, the rolling parameters are timevariant for all cases. Both the AR(1)GARCH(1,1) and the AR(1) processes were simulated for various sets of parameters, but there are no qualitative differences to the fore mentioned conclusions.

1 , 0 ~. . . N z d i i t 1 -
Moreover, a series of simulations were repeated i) for ARCH volatility forms without any conditional mean specification, ii) based on estimation procedures of the most well known packages, EVIEWS ® 4.1 and OX-G@ARCH ® 3.4, iii) for larger rolling samples of 5000 values, iv) for non-overlapping data samples, but there were no qualitative differences in any of these cases11 .

So, the simulation study provides evidence that the time-variant attitude of rollingsampled parameters estimations characterizes not only the examined data sets but the ARCH data generating process itself as well.

INSERT TABLE 5 ABOUT HERE

. R o l l i n g -s a m p l e d p a r a m e t e r s f r o m a L e v y -s t a b l e d i s t r i b u t i o n

In this section, we investigate whether the phenomenon of parameter changing across time is related with the unconditional distribution of returns also. Mandelbrot (1963) and Fama (1965) made the first re-examination of the unconditional distribution of stock returns. Mandelbrot (1963) concluded that price changes can be characterized by a stable Paretian distribution with a characteristic exponent, , less than two, thus exhibiting fat tails and infinite variance. Fama (1965) examined the distribution of thirty stocks of the Dow Jones Industrial Average; his results were consistent with Mandelbrot's. Thereafter, it has been accepted that the stock returns distributions are fat-tailed and peaked. In an attempt to model the unconditional distribution of stock returns several researchers have considered alternative approaches. See for example, Blattberg and Gonedes (1974), Bradley and Taqqu (2002), Clark (1973), Kon (1984), McDonald (1996), Mittnik and Rachev (1993), [START_REF] Panas | Estimating Fractal Dimension Using Stable Distributions and Exploring Long Memory Through ARFIMA Models in Athens Stock Exchange[END_REF], Rachev and Mittnik (2000). The probability density function of a stable distribution cannot be described in a closed mathematical form. By definition, a univariate distribution function is stable if and only if its characteristic function has the form

( ) ( ) ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - - = a t t t i t t i t a , 1 exp ω β γ δ ϕ , ( 8
)
where In estimating the parameters of the stable distribution of index returns, we adopt the estimation procedure suggested by McCulloch (1986). The estimation procedure is a quantile method and works for distribution is the regression method proposed by Koutrouvelis (1980).

1 - = i , R t ∈ , ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 2 tan , πα ω a t if 1 ≠ a and ( ) t a t log 2 , π ω - = if 1 = a .
Following a procedure similar to that of ARCH modelling, the parameters of the stable distribution are estimated, at each of a sequence of points in time, using a rolling sample of constant size equal to 1000 trading days. The empirical findings, for the case of the Greek stock market, are graphically summarized in Figure 3, which plots the rolling-sampled estimates of parameters along with the 95% confidence interval of the parameters estimated using the full data sample. Inspection of Figure 3 shows that the estimates of are less than two. The case of FTSE20 reveals that 92% of the a 's rolling-sampled estimates are between 1.44 and 1.55. The parameter a β is greater than zero, which implies skewness to the right.

The rolling values of β are positive and range from 0.003 to 0.22 but there are not outside the 95% confidence interval for any case13 .

INSERT FIGURE 3 ABOUT HERE

In Table 6, we present the estimates of the parameters of stable distribution based on all data available as well as the standard deviation of the rolling-sampled estimated parameters. The estimates of do not approach the value of two in any of the examined indices. However, there are estimated rolling parameters that are statistically different from the parameter values estimated using the full data sample. For example, the rolling-sampled of the trading days for the DAX30 and FTSE100 indices, respectively, whereas the location (δ ) parameters are time-variant in none of the cases. Another important parameter of the stable distribution, from the point of view of portfolio theory, is the scale parameter, γ . As far as the FTSE20 index is concerned, the rolling-sampled estimates of the scale parameter differ statistically from its full-sampled value in the 56.48% of the trading days. Hence, the parameter estimates, using the full data sample are statistically different from the parameter values estimated using the rolling samples of constant size for one parameter in each index.

INSERT TABLE 6 ABOUT HERE

. D i s c u s s i o n

We estimated an asymmetric ARCH model using daily returns of the FTSE20, DAX30, FTSE100 and SP500 indices and concluded that although the estimated parameters of the model change over time, the model does not lose its ability to forecast the one-day-ahead volatility accurately. Furthermore, the rolling parameter analysis was applied to the unconditional distribution of returns. We observed the phenomenon of parameter changing across time for both the conditional (ARCH process) and the unconditional (Levy-stable) distribution of returns. Even in the case of a simulated ARCH process, the property of time varying rolling-sampled parameters holds. One possible reason for parameter instability might be that the behaviour of the market participants has undergone fundamental changes.

Parameters instability indicates a change in market behavior but we can not determine the source of that change. The term 'a data set that alters', could incorporate a wide range of possible sources, i.e. financial legislation, market microstructure, market participants' perspective, technological revolution or even macroeconomic policy. Gallant et al. (1991), Stock (1988), Lamoureux and Lastrapes (1990) and Schwert (1989) among others have aimed at explaining the economic interpretation of the ARCH process. As Engle et al. (1990) and Lamoureux and Lastrapes (1990) have noted, the explanation of the ARCH process must lie either in the arrival process of news or in market dynamics in response to the news. Based on some earlier work by Clark (1973) and Tauchen and Pitts (1983), Gallant et al. (1991) provided a theoretical interpretation of the ARCH effect.

They assumed that the asset returns are defined by a stochastic number of intra-period price revisions and information flows into the market in an unknown rate. As the daily information does not come to the stock market in a constant and known rate, the estimation of the ARCH stochastic process that explains the dynamics of the stock market could be revised at regular time intervals. In our case the ARCH process is estimated using daily returns. Thus, the parameters of the model may be revised on a daily base, because of the observed phenomenon of changes in the estimated parameters. If we used data of higher frequency, i.e.

ten-minutes intra-daily returns, the estimated model might be revised more frequent than on a daily base.

To the best of authors' knowledge, this is the first study that investigates the phenomenon of time varying estimated parameters either i) in real-world financial data or ii) in a simulated data generating process. A natural extension of this study would be to analyse the change and the relative economic interpretation of the estimated values of the parameters in intra-daily high-frequency data sets.

A c k n o w l e d g e m e n t

We would like to thank the editor and an anonymous referee for their constructive comments that helped to improve significantly an earlier version of the paper. , where denotes the parameter vector estimated using the full data sample, is the standard deviation of and

R e f e r e n c e s

  surprise increases volatility less than a negative surprise. If 1 < , a positive surprise

  of 11 th January 2000 to 5 th July 2002. The daily conditional volatility of the SP500 and DAX30 returns, were estimated by framework (3), using, also, a rolling sample of constant size equal to 1000. Since the ARCH model is estimated at each point in time, we use the maximum likelihood estimates at time 1 t as starting values for the iterative maximization algorithm at time t . Figure 2 depicts the rolling-sampled estimated parameters of the model as well as the 06 . 2 ± times the conditional standard deviation confidence interval of the parameters estimated using the full data sample. From visual inspection, the estimated rolling parameters are, clearly, out of the confidence interval bounds in many cases. Table

  estimated parameters, is the fact that the estimated values do not fluctuate in a mean reverting form but they change gradually. Sudden changes of the values of the rolling estimated parameters, which are characterized by a mean reverting form, should indicate an improperly maximum likelihood estimation procedure. On the other hand, gradual changes of the estimated coefficients indicate a data set that alters from time to time, forcing us to believe that the values of the estimated parameters reflect the information that financial markets reveal.

  estimations. Hence, we may infer that the values of the estimated parameters change across time, reflecting the individual features of particular periods that characterize financial markets.

  DAX30 and SP500 cases, parameter v is statistically different to the value of 2 at any level of significance, justifying the use of a thick-tailed distribution. The continuously compounded returns of the underlying indices are plotted in Figures 1.b to 1.d.

  Dropping out the first 1001 data, maximum likelihood rollingsampled estimates of the parameters are obtained by numerical maximization of the loglikelihood function, using a rolling sample of constant size equal to 1000. According to Table

  The particular distribution represented by its characteristic function is determined by the values of four parameters: a , & , and . The parameter a , 2 0 < 2 , is called the characteristic exponent. It measures the thickness of the tails of a stable distribution.The smaller the value a , the higher the probability in the distribution tails. If 2 < a then we have thicker tails than the tails of normal distribution. Thus, stable distributions have thick tails and consequently increase the likelihood of the occurrence of large shocks.The skewness parameter & , is skewed to the left. As & approaches one, the degree of skewness increases. The scale parameter , 0 > , is a measure of the spread of the distribution. It is similar to the variance of the normal distribution, Cauchy distribution.

  . The parameter & is greater than zero, which implies skewness to the right. The rolling values of & are positive and range from 0.003 to 0.22 but there are not outside the 95% confidence interval for any case 10 .

  Figure 1. FTSE20, DAX30, FTSE100 and SP500 continuously compounded daily returns from January 3 rd , 1996 to July 5 th , 2002.

Figure 3 Figure 4 .

 34 Figure 3.a. In-sample 95% confidence interval of the FTSE20 index daily returns for the ARCH model (11 th January 2000 to 5 th July 2002).

  January 2000 to 5 th July 2002. Since the ARCH model is estimated at each point in time, we use the maximum likelihood estimates at time 1 t as starting values for the iterative maximization algorithm at time . Figure 1 depicts the rolling-sampled estimated parameters for the FTSE20 index as well as the t 06 . 2 ± times the conditional standard deviation confidence interval of the parameters estimated using the full data sample 8 . From visual inspection, the estimated rolling parameters are, clearly, out of the confidence interval bounds in many cases. Table 2 presents the percentage of rolling-sampled estimations, which are outside of the 95% confidence interval of the full-sampled parameters. Characteristic examples of the change in the parameter values are 1 Ψ and for DAX30 as well as v 1

  such as γ (leverage effect), and 0 δ (contribution of non-trading days to volatility). An important characteristic, which is extracted from the rolling-sampled estimated parameters, is the fact that the estimated values do not fluctuate in a mean reverting form but they change gradually. Sudden changes of the values of the rolling estimated parameters, which are characterized by a mean reverting form, should indicate an improperly maximum likelihood estimation procedure. On the other hand, gradual changes of the estimated coefficients indicate a data set that alters from time to time, forcing us to believe that the values of the estimated parameters reflect the information that financial markets reveal.

  , but the values of rolling 3 μ are not different to zero in most of the cases. Of course, there are parameters whose estimations are statistically different to zero in both the full sample and the rolling samples (i.e. the parameter 1 Δ for the FTSE20, DAX30 and SP500 indices). Hence, we may infer that the values of the estimated parameters change across time, reflecting the individual features of particular periods that characterize financial markets.

  . Under the null hypothesis, the is asymptotically chi-squared distributed with two degrees of freedom. The main advantage of Christoffersen's test is that it can reject a VaR model that generates either too many or too few clustered violations. Both tests do not reject the null hypothesis of correct

  l i n g -s a m p l e d p a r a m e t e r s f r o m s i m u l a t e d p r o c e s s e s A simulation study could shed light in rolling-sampled estimated parameters' behaviour. A series of simulations is run in order to investigate if the time-variant attitude holds even in the case of an ARCH data generating process. We generate a series of 32000 values from the standard normal distribution,. Then an AR(1)GARCH(1,1) process is created, Dropping out the first 1001 data, maximum likelihood rolling-sampled estimates of the parameters are obtained by numerical maximization of the log-likelihood function, using a rolling sample of constant size equal to 1000. According to Table5, about 58% of the 30000 conditional variance rollingsampled parameters are outside the 95% confidence interval of the parameters estimated using the whole sample set of the 30000 simulated data. The procedure is repeated for an AR(1)EGARCH(1,1) conditional variance form, are robust to the choice of the conditional variance specification.A series of 32000 values from the first order autoregressive process are also produced. The AR(1) process is created as

  The particular distribution represented by its characteristic function is determined by the values of four parameters: , a β , γ and δ . The parameter , measures the thickness of the tails of a stable distribution. The smaller the value , the higher the probability in the distribution tails. If a 2 < a then we have thicker tails than the tails of normal distribution. Thus, stable distributions have thick tails and consequently increase the likelihood of the occurrence of large shocks. The skewness parameter β , measure of the asymmetry of the distribution. The distribution is symmetric, if 0 = β. As β approaches one, the degree of skewness increases. The scale parameter γ , 0 > γ , is a measure of the spread of the distribution. It is similar to the variance of the normal distribution, the scale parameter γ is finite for all stable distributions, despite the fact that the variance is infinite for all 2 < a . The location parameter δ ,

  of the other parameters. Essentially, McCulloch suggests that if we have a random variable x , which follows a stable distribution and denotes the quantile of this distribution by th p ( ) p x , then the population quantile can be estimated by the sample quantile ( ) p x ˆ. McCulloch's estimator uses five quantiles to estimate

Page

  estimates of the tail index ( ) are statistically different to the full sample estimated parameter in the 51.46% of the trading days for the case of the SP500 index. The rolling estimates of parameter a a β are statistically different to the relevant full-sampled values in 9.59% and 9.42%

Figure 1 .

 1 Figure 1. The rolling-sampled estimated parameters of the ARCH model and the 95% confidence interval of the parameters estimated using the full data sample.

Figure 2 Figure 3 .

 23 Figure 2.a. In-sample 95% confidence interval of the FTSE20 index daily returns for the ARCH model (11 th January 2000 to 5 th July 2002).

  ,

	ln	2 t	=	a	0	+	ln	1	+	N	t	0	+	1	1	1	L	1	L			t t		E		t t	+	L	t t	(6)
															+	1	ln	2 SP	500	,	t	1	+	2	ln	2 DAX	30	,	t	1

Table 6

 6

		presents the percentage of rolling parameters, which are
	outside the 95% confidence interval of the full-sampled parameters. Characteristic
	examples of the change in the parameter values are 1 and v for DAX30 as well as 1
	for SP500. However, there are rolling parameters which do not change significantly
	across time, such as	(leverage effect), and 0 (contribution of non-trading days to
	volatility).
		INSERT TABLE 6 ABOUT HERE
	7 Figures of the estimated rolling parameters for the DAX30, FTSE100 and SP500 indices, similar to
	Figure 2, are available upon request.

  . The estimation procedure is a McCulloch suggests that if we have a random variable x , which follows a stable distribution and denote the th p quantile of this distribution by ( ) McCulloch's estimator uses five quantiles to estimate a and & as follows:

	quantile method and works for	0	.	6	a	2	and any value of the other parameters.
	Essentially, p x , then the
	population quantile can be estimated by the sample quantile ( ) p x ˆ.
	F o r
	P
	e
	e r
				R e v
							i e
							w

Table 1 .

 1 Parameter estimates for the FTSE20 index returns using data from January 3 rd , 1996 to July 5 th , 2002.

	o r
	P
	e
	e r
	R e v
	i e
	w

Notes: With v =1.335, the 97.5% point of the generalized error distribution is 2.06. With v =1.335, the 99.5% point of the generalized error distribution is 2.94.

Table

2

. Percentage of rolling-sampled estimated parameters that are outside the 95% confidence interval.

Table 3 .

 3 Percentage of rolling-sampled estimated parameters that are statistically different from the parameter values estimated using the full data

	sample.
	Parameter 0 µ	5% sign. Level 21.86%	1% sign. Level 1.29%	Parameter 1	5% sign. Level 7.40%	1% sign. Level 0.00%
			1 2 µ µ 3 µ	0.96% 0.00% 0.00%	0.00% 0.00% 0.00%	1 1	18.97% 0.00% 0.00%	10.13% 0.00% 0.00%
			a	0	17.20%	3.86%	2	12.54%	0.16%
				0	0.00%	0.00%	v	1.29%	0.00%
	Table 4. Percentage of rolling-sampled estimated parameters that are statistically
	insignificant at 5% and 1% levels of significance.
	Parameter	5% sign. Level	1% sign. Level	Parameter	5% sign. Level	1% sign. Level
	µ	0		30.06%	76.21%	1	0.00%	0.00%
	1 2 µ µ 3 µ		32.80% 99.84% 65.11%	97.11% 100% 87.78%	1 1	0.00% 100% 100%	0.00% 100% 100%
	a	0		0.00%	0.48%	2	89.55%	99.84%
			0		27.65%	57.07%	v	0.00%	0.00%

Table 5 .

 5 Parameter estimates for the DAX30, FTSE100 and SP500 index daily returns (January 3 rd , 1996 to July 5 th , 2002). =1.735, v =1.858, v =1.689, the 97.5% point of the generalized error distribution are 2.00, 1.98 and 2.00, respectively. With v =1.735, v =1.858, v =1.689, the 99.5% point of the generalized error distribution are 2.70, 2.65 and 2.72, respectively. For the DAX30 index, parameters 1 and 2 present the volatility spillover from the FTSE100 and SP500 indices, respectively. For the FTSE100 index, parameters 1 and 2 present the volatility spillover from the DAX30 and SP500 indices, respectively.

	Parameter		Coefficient			Standard error			Coefficient /
									Standard error
	µ	0	DAX30 FTSE100 SP500 DAX30 FTSE100 SP500 DAX30 FTSE100 SP500 0.000 -0.000 -0.001 0.001 0.000 0.001 0.596 -0.067 -1.027
	µ µ 2 1 3 µ	1.995 0.024 -0.075	1.251 0.005 0.144	4.297 -0.100 0.333	2.709 0.061 0.138	3.924 0.060 0.126	4.013 0.062 0.121	0.736 0.398 -0.544	0.319 0.078 1.140	1.071 -1.620 2.745
	a	0	-9.858	-1.326	-4.059	0.919	0.522	0.685 -10.727	-2.538	-5.929
		0	0.095	0.012	0.039	0.050	0.036	0.040	1.880	0.342	0.956
		1	0.190	0.056	0.060	0.007	0.062	0.044	27.847	0.892	1.378
		1	0.973	-0.001	0.785	0.013	0.185	0.028	73.455	-0.003	28.040
			-0.068	-0.108	-0.236	0.079	0.036	0.079	-0.856	-2.969	-2.975
		1	-0.008	0.694	0.081	0.012	0.144	0.035	-0.688	4.822	2.295
		2	0.004	0.201	0.041	0.012	0.095	0.031	0.386	2.116	1.314
	v	1.735	1.858	1.689	0.029	0.095	0.038	-9.137	-1.495	-8.184
	Notes: With v For the SP500 index, parameters 1 and 2 present the volatility spillover from the DAX30 and
	FTSE100 indices, respectively.

Table 6 .

 6 Percentage of rolling-sampled estimated parameters that are outside the 95% confidence interval. (Values in parenthesis present the lower and upper bounds of the 95% confidence interval).

				DAX30		FTSE100		SP500
	µ	0	(-0.001 0.002) 33.18% (-0.001 0.001) 24.11% (-0.002 0.001)	20.66%
	1 µ	(-4.989 8.978)	0.00% (-8.762 11.263) 0.80%	(-5.978 14.572)	16.48%
	µ		2	(-0.133 0.182) 0.00%	(-0.148 0.157)	1.28% (-0.258 0.058)	0.00%
	µ	3	(-0.431 0.281) 0.00%	(-0.178 0.465)	12.32 %	(0.022 0.644)	0.48%
	a	0	(-12.227 -7.489)	3.20% (-2.659 -0.007) 16.64% (-5.812 -2.306) 24.00%
			0	(-0.035 0.224) 0.00%	(-0.080 0.105)	0.00% (-0.065 0.142)	0.00%
			1	(0.172 0.207) 62.24% (-0.104 0.215)	0.00% (-0.052 0.173)	20.96%
			1	(0.939 1.007) 22.08% (-0.472 0.471)	1.12% (0.713 0.857) 60.48%
				(-0.271 0.136) 0.00%	(-0.201 -0.015)	1.12% (-0.439 -0.033)	0.48%
		1	(-0.038 0.022)	3.04% (0.327 1.062)	0.48% (-0.009 0.171)	0.00%
			2	(-0.025 0.034)	1.60% (-0.041 0.444)	0.00% (-0.039 0.121)	35.36%
	v	(1.660 1.811) 46.72% (1.616 2.100)	0.48% (1.591 1.787)	9.44%

Table 7 .

 7 Percentage of rolling-sampled estimated parameters that are statistically different from the parameter values estimated using the full data sample.

			DAX30	FTSE100	SP500
	Parameter	5% sign. Level	1% sign. Level	5% sign. Level	1% sign. Level	5% sign. Level	1% sign. Level
	0 1 2 µ µ µ 3 µ	13.67% 0.00% 0.00% 0.00%	0.80% 0.00% 0.00% 0.00%	4.02% 0.16% 1.13% 3.22%	0.00% 0.00% 0.00% 0.64%	14.15% 8.52% 0.00% 0.00%	4.34% 0.64% 0.00% 0.00%
	a	0	16.72%	7.40%	0.48%	0.00%	24.28%	6.59%
		0	0.00%	0.00%	0.00%	0.00%	2.73%	0.00%
		1	0.00%	0.00%	0.00%	0.00%	7.56%	0.00%
		1	2.57%	0.00%	14.47%	5.79%	31.67%	3.54%
			5.14%	0.00%	4.50%	0.00%	36.17%	10.13%
		1	0.00%	0.00%	0.80%	0.32%	0.00%	0.00%
		2	0.00%	0.00%	0.16%	0.00%	24.92%	0.00%
	v	16.72%	0.32%	0.00%	0.00%	0.00%	0.00%

Table 8 .

 8 Percentage of the rolling-sampled estimated parameters that are statistically insignificant at 5% and 1% levels of significance.

				DAX30									FTSE100	SP500
	Parameter	5% sign. Level	1% sign. Level				5% sign. Level	1% sign. Level	5% sign. Level	1% sign. Level
	µ	0	88.36%	99.37%				94.69%	100.00%	66.35%	84.28%
	µ µ 2 1 3 µ	93.87% 100.00% 100.00% 100.00% 100.00% 100.00%				99.22% 99.22% 79.69%	100.00% 100.00% 100.00% 100.00% 57.08% 87.26% 96.56% 92.77% 100.00%
	a	0	0.00%	0.00%					17.81%	40.78%	1.57%	18.08%
			0	81.45%	100.00% 100.00% 100.00% 100.00% 100.00%
			1	0.00%	0.00%					100.00% 100.00% 100.00% 100.00%
			1	0.00%	0.00%					31.25%	38.91%	0.00%	0.00%
				100.00% 100.00% 100.00% 100.00%	0.00%	44.18%
			1	100.00% 100.00%					0.00%	0.16%	94.03% 100.00%
			2	100.00% 100.00%				67.97%	96.56%	59.91%	91.19%
	v	0.00%	0.00%						0.00%	0.00%	0.00%	0.00%
	Table 9. AR(1)GARCH(1,1) simulated process. Percentage of rolling-
	sampled estimated parameters that are outside the 95% confidence
	interval.
						y	t	=	µ	0	+	µ	1	y	t	1	+	t
					t	=	z	t			2 t	,	z	t	i	. i ~. . d	N	( ) 1 , 0
						2 t	=	a	0	+	1 a	2 t	1	+	a	2	2 t	1
									µ	0			µ	1	a	0	1 a	a	2
	Simulated Values			0.005		0.150	0.040	0.0500	0.900
	Estimated Values (Full Data Sample)			-0.003 0.158	0.037	0.0138	0.895
	Rolling parameters outside the 95% c.i.		11.70% 3.32% 73.17% 30.88% 72.17%

Table 10 .

 10 Stable parameter estimates, using the full data sample, of the FTSE20, DAX30, FTSE100 and SP500 index daily returns, their standard errors and the percentage of rolling-sampled estimated parameters that are statistically different from the parameter values estimated using the full data sample at 5% level of significance. The standard error of parameter a is computed as the standard deviation of the rolling-sampled estimated parameters,( )

		Tail index a	Skewness &	Location	Scale
		FTSE20
	Coefficient	1.48303	0.07799 -0.00033 0.01005
	Standard error	0.05606	0.07965	0.00143 0.00081
	5% sign. Level	0.32%	0.00%	0.00%	56.48%
		DAX30
	Coefficient	1.58306	-0.14798 0.00101 0.00754
	Standard error	0.15725	0.18828	0.00069 0.00217
	5% sign. Level	1.53%	9.59%	0.12%	0.00%
		FTSE100
	Coefficient	1.68238	-0.06489 0.00046 0.00591
	Standard error	0.10944	0.25581	0.00039 0.00165
	5% sign. Level	2.13%	9.42%	0.49%	0.00%
		SP500
	Coefficient	1.49172	-0.11841	0.0005 0.00525
	Standard error	0.07160	0.09609	0.00052 0.00218
	5% sign. Level	51.46%	5.00%	0.00%	0.00%
	Notes:

  Figure 2. The rolling-sampled estimated parameters of the ARCH model and the 95% confidence interval of the parameters estimated using the full data sample.

	0.00					10.000				0.20
	0.00					8.000				0.15
	0.00 0.00 0.00 0.00				2% 4% 6% 8% 10%	-2.000 0.000 2.000 4.000 6.000		0% 2% 4% 6% 8%		-0.05 0.00 0.05 0.10
	-0.01 10/1/00			10/1/01	-4% 0% -2% 10/1/02	-4.000 10/1/00	10/1/01	-2% -4%	10/1/02	-0.10 10/1/00	10/1/01	10/1/02
		µ0			-6% 95% Conf. Interval	µ1		-6% 95% Conf. Interval	µ2	95% Conf. Interval
					-8%			-8%
	0.70 0.50 0.60 0.30 0.40 0.00 0.10 0.20 10/1/00	µ3		10/1/01	-10% 2/1/1996 -6% -4% -2% 0% 2% 4% 6% F o c. FTSE100 2/1/1999 -3.00 -5.00 -7.00 -9.00 -13.00 -11.00 10/1/02 10/1/00 2/1/2002 r 95% Conf. Interval 0	10/1/01	-10% 2/1/1996 -6% -4% -2% 0% 2% 4% 6% 10/1/02 95% Conf. Interval 2/1/1998 d. SP500 2/1/2000 0.40 0.35 0.30 0.25 0.15 0.20 0.00 0.05 0.10 10/1/00 2/1/2002 0	10/1/01	10/1/02 95% Conf. Interval
	0.50 0.45 0.35 0.40 0.30 0.25 10/1/00 0.10 0.05 0.00 -0.05 -0.15 -0.10 10/1/00	2	1	10/1/01 10/1/01	-10% -8% 2/1/1996 10/1/02 95% Conf. Interval 2/1/1998 P 2/1/2000 1.00 0.95 e 2/1/2002 0.85 0.90 e 0.80 r 0.75 10/1/00 10/1/01 1 R 0.10 0.15 e -10% -8% 2/1/1996 10/1/02 95% Conf. Interval 0.05 v 0.00 -0.05 i -0.15 -0.10 e 10/1/02 -0.25 -0.20 10/1/00 10/1/01 10/1/02 w 2/1/1999 95% Conf. Interval 95% Conf. Interval	2/1/2002 10/1/00 0.10 0.12 0.06 0.08 0.04 -0.02 0.00 0.02 -0.06 -0.04 -0.08 1 1.55 1.60 1.50 1.45 1.40 1.30 1.35 1.20 1.25 10/1/00 v	10/1/01 10/1/01	10/1/02 95% Conf. Interval 10/1/02 95% Conf. Interval

  information set I at time t . Since very few financial time series have a constant conditional mean of zero, an ARCH model can be presented in a regression form by letting t

																			ε be the
	unpredictable component of the conditional mean
																y	A	t ,	=	E	(	y	A	t ,	|	I	t	-1	) t ε +	,	(2)
	where	y	A	,	t	=	ln	( P A	,	t	A P	, t	1 -	)	denotes the continuously compound rate of return from time	t	-	1
	to , and t		t P , A	is the asset price A at time . In order to investigate the characteristics of stock t
	market A, we apply an ARCH model of the following form:
																			A	t
								r o F t A y	A	t		e	y	A	t	t
															P e
																	e
																			r
																			R
																			e
																			v
																			i e w
																			. f is the density function of , t z	g	( ) .	is a linear
	or non-linear functional form and		t υ is a vector of predetermined variables included in

Table 1 .

 1 Parameter estimates for the FTSE20, DAX30, FTSE100 and SP500 index daily returns (January 3 rd , 1996 to July 5 th , 2002).

	Angelidis, T., Benos, A. and Degiannakis, S. (2004). The Use of GARCH Models in VaR
	Estimation. Statistical Methodology, 1(2), 105-128.
	Balaban, E. and Bayar, A. (2005). Stock returns and volatility: empirical evidence from
	fourteen countries. Applied Economics Letters, 12, 603-611.
	Barndorff-Nielsen, O.E., Nicolato, E. and Shephard, N. (2002). Some Recent Developments in
	Stochastic Volatility Modelling. Quantitative Finance, 2, 11-23.
	Journal of Economic Surveys, 7, 305-366.

[START_REF] Andersen | DM-Dollar Volatility: Intraday Activity Patterns, Macroeconomic Announcements, and Longer-Run Dependencies[END_REF]

. ARCH and GARCH Models, Encyclopedia of Statistical Sciences Vol.II, (eds.) Samuel Kotz, Campbell B. Read and David L. Banks, New York, John Wiley and Sons Inc.

[START_REF] Bera | ARCH Models: Properties, Estimation and Testing[END_REF]

. ARCH Models: Properties, Estimation and Testing.

a Indicates that the coefficient is statistically significant at 1% level of significance. b Indicates that the coefficient is statistically significant at 5% level of significance.

Table 2 .

 2 Percentage of rolling-sampled estimated parameters that are outside the 95% confidence interval. (Values in parenthesis present the lower and upper bounds of the 95% confidence interval).

	Page 63 of 68		Submitted Manuscript
	μ	0	FTSE20 (-0.002 0.000)	56.48%	DAX30 (-0.001 0.002)	33.18%
	μ 2 1 μ	(-1.780 7.485) (-0.075 0.181)	7.04% 0.00%	(-4.989 8.978) (-0.133 0.182)	0.00% 0.00%

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 3 .

 3 Percentage of rolling-sampled estimated parameters that are statistically different from the parameter values estimated using the full data sample.

			FTSE20	DAX30	FTSE100	SP500
			5%	1%	5%	1%	5%	1%	5%	1%
	Parameter	sign.	sign.	sign.	sign.	sign.	sign.	sign.	sign.
	0 1 2 μ μ μ 3 μ	Level 21.86% 1.29% 13.67% 0.80% Level Level Level 0.96% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%	Level 4.02% 0.16% 1.13% 3.22%	Level 0.00% 14.15% Level 0.00% 8.52% 0.00% 0.00% 0.64% 0.00%	Level 4.34% 0.64% 0.00% 0.00%
	a δ	0 0	17.20% 3.86% 16.72% 7.40% 0.00% 0.00% 0.00% 0.00%	0.48% 0.00%	0.00% 24.28% 0.00% 2.73%	6.59% 0.00%
	1 Ψ	7.40%	0.00%	0.00%	0.00%	0.00%	0.00%	7.56%	0.00%
	1 1 Δ γ δ 2 δ	18.97% 10.13% 2.57% 0.00% 0.00% 5.14% 0.00% 0.00% 0.00% 12.54% 0.16% 0.00%	0.00% 14.47% 5.79% 31.67% 0.00% 4.50% 0.00% 36.17% 10.13% 3.54% 0.00% 0.80% 0.32% 0.00% 0.00% 0.00% 0.16% 0.00% 24.92% 0.00%
	v	1.29%	0.00% 16.72% 0.32%	0.00%	0.00%	0.00%	0.00%

Table 4 .

 4 Percentage of the rolling-sampled estimated parameters that are statistically insignificant at 5%

	and 1% levels of significance.
			FTSE20	DAX30	FTSE100	SP500
	Parameter 0 μ	5% sign. Level 30.06%	1% sign. Level 76.21%	5% sign. Level 88.36%	1% sign. Level 99.37%	5% sign. Level 94.69%	1% sign. Level 100%	5% sign. Level 66.35%	1% sign. Level 84.28%
	μ μ 2 1 3 μ	32.80% 99.84% 65.11%	97.11% 100% 87.78%	93.87% 100% 100%	100% 100% 100%	99.22% 99.22% 79.69%	100% 100% 96.56%	57.08% 100% 92.77%	87.26% 100% 100%
	a	0	0.00%	0.48%	0.00%	0.00%	17.81%	40.78%	1.57%	18.08%
	δ	0	27.65%	57.07%	81.45%	100%	100%	100%	100%	100%
	1 Ψ	0.00%	0.00%	0.00%	0.00%	100%	100%	100	100
	1 1 Δ γ δ 2 δ	0.00% 100% 100% 89.55%	0.00% 100% 100% 99.84%	0.00% 100% 100% 100%	0.00% 100% 100% 100%	31.25% 100% 0.00% 67.97%	38.91% 100% 0.16% 96.56%	0.00% 0.00% 94.03% 59.91%	0.00% 44.18% 100% 91.19%
	v	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%	0.00%

Table 5 .

 5 AR(1)GARCH(1,1) simulated process. Percentage of rollingsampled estimated parameters that are outside the 95% confidence interval.

		y	t		=	μ	0		+	μ	1	y	t	-1	+	ε	t
	t ε =	t z σ	2 t	,	z	t	i	. i ~. . d	N	( ) 1 , 0
	σ	2 t	=	a	0	+	1 a	ε	2 -+ 1 t	a	2	σ	2 -1 t
					μ	0								μ	1	a	0	1 a	a	2
	Simulated Values			0.005			0.150	0.040	0.0500	0.900
	Estimated Values (Full Data Sample)		-0.003 0.158	0.037	0.0138	0.895
	Rolling parameters outside the 95% c.i.	11.70% 3.32% 73.17% 30.88% 72.17%

Table 6 .

 6 Stable parameter estimates, using the full data sample, of the FTSE20, DAX30, FTSE100 and SP500 index daily returns, their standard errors and the percentage of rolling-sampled estimated parameters that are statistically different from the parameter values estimated using the full data sample at 5% level of significance.

		Tail index a	Skewness β	Location δ	Scale γ
		FTSE20
	Coefficient	1.48303	0.07799 -0.00033 0.01005
	Standard error	0.05606	0.07965	0.00143 0.00081
	5% sign. Level	0.32%	0.00%	0.00%	56.48%
		DAX30
	Coefficient	1.58306	-0.14798 0.00101 0.00754
	Standard error	0.15725	0.18828	0.00069 0.00217
	5% sign. Level	1.53%	9.59%	0.12%	0.00%
		FTSE100
	Coefficient	1.68238	-0.06489 0.00046 0.00591
	Standard error	0.10944	0.25581	0.00039 0.00165
	5% sign. Level	2.13%	9.42%	0.49%	0.00%
		SP500
	Coefficient	1.49172	-0.11841	0.0005 0.00525
	Standard error	0.07160	0.09609	0.00052 0.00218
	5% sign. Level	51.46%	5.00%	0.00%	0.00%
	Notes: The standard error of parameter of the rolling-sampled estimated parameters, is computed as the standard deviation a ( ) t a ˆ, for T t ,..., 1 = trading days, i.e.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

The relationship between investors' expected return and risk was presented in an ARCH framework, byEngle et al. (1987). They introduced the ARCH in mean model where the conditional mean is an explicit function of the conditional variance.

According toCampbell et al. (1997), 'The non-synchronous trading or non-trading effect arises when time series, usually asset prices, are taken to be recorded at time intervals of one length when in fact they are recorded at time intervals of other, possible irregular lengths.'

LeBaron (1992) found a strong inverse relation between volatility and serial correlation for SP500, CRSP and Dow Jones returns. As LeBaron stated, it is difficult to estimate 4

Engle et al. (1990) evaluated the role of the information arrival process in the determination of volatility in a multivariate framework providing a test of two hypotheses: heat waves and meteor showers. Using meteorological analogies, they supposed that information follows a process like a heat wave so that a hot day in New York is likely to be followed by another hot day in New York but not typically by a hot day in Tokyo. On the other hand, a meteor shower in New York, which rains down on the earth as it turns, will almost surely be followed by one in Tokyo. Thus, the heat wave hypothesis is that the volatility has only country specific autocorrelation, while the meteor shower hypothesis states that volatility in one market spills over to the next. See alsoKanas (1998).

| 1 1 0 | 1 µ

Figures depicting the rolling-sampled estimates of the parameters for the DAX30, FTSE100 and SP500 indices are available upon request.

The reader who is interested in SV models is referred to[START_REF] Barndorff-Nielsen | Some Recent Developments in Stochastic Volatility Modelling[END_REF],Chib et al. (1998),Ghysels et al. (1996),Jacquier et al. (1999),Shephard (2004),Taylor (1994).

For example, in the case of the FTSE20 index daily returns, the conditional variance of the DAX30 and SP500 returns were regarded as exogenous variables. In order to estimate the conditional variance of the DAX30 and SP500 indices, their daily returns were used for the period of January 1992 to July 2002, or 1000 trading days prior January 3 rd , 1996.

1 2 -

All the simulation studies are available to the readers upon request.

De Vries (1991),Ghose and Kroner (1995) andGroenendijk et al. (1995) demonstrate that ARCH models share many of the properties of Levy-stable distribution but the true data generating process for an examined set of financial data is more likely ARCH than Levy-stable. A number of studies, such asLiu and Brorsen (1995),Mittnik et al. (1999),Panorska et al. (1995),[START_REF] Tsionas | Likelihood-Based Comparison of Stable Paretian and Competing Models: Evidence from Daily Exchange Rates[END_REF], examined the properties of ARCH models with Levy-stable distributed innovations.

Figures depicting the rolling-sampled estimates of the parameters for the DAX30, FTSE100 and SP500 indices are available upon request.

Figure 2.b plots the one-day-ahead 95% prediction interval, which is constructed as the one-day-ahead conditional mean ± 2.06 times the conditional standard deviation, both measurable to information set, or

is the quantile of the GED distribution. Hence, each trading day, ( ), the next trading day's, ( ), prediction intervals are constructed, using only information available at current trading day, t . There are 29, 22, 21 and 32 observations or 4.67%, 3.46%, 3.29% and 5.04% for the FTSE20, DAX30, FTSE100 and SP500 indices, respectively, outside the 95% prediction intervals

INSERT FIGURE 2 ABOUT HERE

For a more formal method of evaluating forecasting adequacy, we apply two hypotheses tests that measure the forecasting accuracy in a VaR framework. One-day-ahead VaR at a given probability level, , is the next trading day's predicted amount of financial loss of a portfolio, or a ( ) Kupiec (1995) introduced a likelihood ratio statistic for testing the null hypothesis that the proportion of confidence interval violations is not larger than the VaR forecast. The test statistic, which is asymptotically distributed, is computed as

is the number of trading days over 9 Figures, similar to Figure 2, that depict the in-sample 95% confidence interval and the one-day-ahead 95% prediction intervals for the DAX30, FTSE100 and SP500 indices are also available upon request.