Repositório ISCTE-IUL

In this paper, using daily data for six major international stock market indexes and a modi(cid:28)ed EGARCH speci(cid:28)cation, the links between stock market returns, volatility and trading volume are investigated in a new nonlinear conditional variance framework with multiple regimes and volume e(cid:27)ects. Volatility forecast comparisons, using the Harvey-Newbold test for multiple forecasts encompassing, seem to demonstrate that the MSV-EGARCH complex threshold structure is able to correctly (cid:28)t GARCH-type dynamics of the series under study and dominates competing standard asymmetric models in several of the considered stock indexes.


Introduction
A frequently documented feature of stock market data is that returns appear to be drawn from a time-dependent heteroskedastic distribution. As early noted in the pioneering studies of Mandelbort (1963) and Fama (1965), nancial time series vary systematically with time and tend to display periods of unusually large volatility, followed by periods of low volatility.
Despite these early studies, eorts to model volatility dynamics have only been developed in the last decades. In fact, until recently, the variance of the disturbance term was assumed to be constant in conventional econometric models, i.e., nancial time series modelling centered on the conditional rst moment, with any temporal dependencies in the higher order moments treated as a nuisance.
However, the increased importance played by risk and uncertainty considerations has recently spurred a vast literature on modelling and forecasting return's volatility. The trade-o between risk and return, where risk is associated with the variability of the random (unforeseen) component of a time series (volatility), constitutes one of the cornerstones of modern nance. In eect, nance is nowadays a eld where the explicit modelling of uncertainty takes on a particularly signicant role, since valuation models for the majority of assets are essentially based on the rst two moments of the return series: mean, variance and covariances. Moreover, due to the compelling theoretical and empirical results supporting the ecient market theory, academicians and practitioners have to some extent ignored the question of return's forecastibility in recent decades; concentrating, instead, on exploring the question of risk. Understanding the statistical properties of volatility is currently considered an important area of interest, given the impact of volatility changes, namely, in risk analysis, portfolio selection, market timing, and derivative pricing.
Recent studies on stock return's volatility have been dominated by ARCH models (Engle, 1982;Bollerslev, 1986), which stand for autoregressive conditional heteroskedasticity. The conditional heteroskedasticity framework is a stationary, parametric, conditional approach which postulates that the main time-varying feature of returns is the conditional volatility, while it assumes that the unconditional volatility remains unchanged through time.
The popular autoregressive conditional heteroskedasticity models have been extremely successful in accounting for the main characteristics of nancial data time series. Nevertheless, in some applications it has been found that ARCH models with conditional normal distributions 1 fail to fully capture the leptokurtosis present in high frequency data. The empirical evidence against the normality assumption, pointed rstly by Mandelbort (1963) and Fama (1965), has led to the use of non-normal distributions capable of modelling the excess of kurtosis, such as the Student's t distribution in Bollerslev (1987), the Generalised Error Distribution in Nelson (1991), the Laplace Distribution in Granger and Ding (1995) and the Stable Paretian Distribution in Liu and Brorsen (1995), Panorska et al. (1995), Mittnik et al. (1998), Curto et al. (1998 and Tavares et al. (2007).
The Student's t distribution, in particular, has a long tradition in the econometrics literature as a popular choice of a fat-tailed distribution, since it has nite second moment (in contrast to stable non-Gaussian distributions), its mathematical properties are well known, it is undemanding to estimate, and is often found capable of capturing the excess of kurtosis observed in nancial time-series.
In some aspects ARCH relative success has made it less interesting to continue research on volatility models. Nonetheless, there are key aspects that warrant further investigation. Using daily data for six major international stock market indexes from January 1995, through April 2008, this paper analyses the links between stock market returns, volatility and unexpected trading volume in a new nonlinear conditional variance framework. First, a new multiple regime model is proposed, in opposition to the standard single zero threshold adopted by nonlinear GARCH models. This provides increased exibility to the proposed MSV-EGARCH specication, allowing it to capture individual irregular bursts in the volatility time series that, otherwise, could be treated as mere outliers. Second, as part of the literature on volatility clustering suggests that ARCH eects in stock returns can be explained by temporal dependence in trading volume, unexpected volume, dened as above-average trading activity, is used as a variance regressor variable, helping to bridge the gap between theory and practice in volatility modelling. This paper is organized as follows. Next section presents the MSV-EGARCH model specication and section 3 describes the data sets. Section 4 discusses estimation results, compares goodness-of-t and presents out-of-sample evaluation results for the MSV-EGARCH, GJR and EGARCH models. Finally, section 5 presents some concluding remarks. 2 2 The Multiple Sign-Volume sensitive regime EGARCH model (MSV-EGARCH) In GARCH models the autoregressive structure in the variance specication allows for the persistence of volatility shocks, enabling to capture the frequently observed clustering of similar-sized price changes, the so-called ARCH eects. In this paper, using an EGARCH specication, the relationship between volatility, bad news and trading volume is re-examined through the modelling of multiple sign-volume-sensitive regimes in the conditional variance behaviour. This yields a distinctive EGARCH model specication that extends previous research by combining multiple news and volume asymmetric dynamics in a new conditional variance formulation.
In the original EGARCH(p, q) model, introduced by Nelson (1991), the conditional variance σ 2 t is an asymmetric function of past unpredictable returns ε t 's: Unlike the symmetric GARCH(p, q) model in the EGARCH(p, q) specication no parameters restrictions are needed to ensure the non-negativity of the conditional variance and bad news (unexpected decreases in returns) are allowed to have a greater impact on future volatility when compared to good news (unexpected increases in returns) if the asymmetry parameters γ i are negative. As most of the empirical papers in the nancial econometrics literature deal only with (1,1)type and due to its noteworthy success in nancial volatility modelling, in this paper the simplest asymmetric EGARCH(1, 1) specication is adopted.
The proposed multiple sign-volume sensitive EGARCH model, MSV-EGARCH(1,1), is described by the following equations: where r t is the continuously compounded daily rate of return at period t, ε t is the conditional 3 error term and V t−1 is a high/low volume indicator variable: where σ is the unconditional standard deviation of ε t . Thus, the indicator variable V t−1 is one if the lagged volume is above its fty days 1 lagged moving average, and is zero otherwise.
Equation (3) mimics many of the well known time series properties of traditional GARCH models and takes into account additional dynamic asymmetries. The conditional variance is assumed to be predicted by the previous conditional variance, the lagged shock terms and the above-average trading volume. The previous negative shocks are dierentiated using indicator variables, which depend on the sign and intensity of the shocks.

Multiple Regimes
A fundamental idea in the proposed model specication is the existence of multiple thresholds in the conditional variance equation. Threshold parameters determine abrupt changes in the dynamics of the process as it moves through regions of the state space.
Financial time series present a non-negligible probability of occurrence of violent market movements. These signicant market movements, far from being discardable as mere outliers, focus the attention of market participants since their magnitude may be such that they may account for an important fraction of the return accumulated over a large period of time. However, ARCH type models often fail to fully capture the nonlinearity in stock returns. A natural approach to address such nonlinearity is to dene dierent regimes and to allow the dynamic behaviour of volatility to depend on the regime that occurs at any given point in time.
A pioneering eort to allow the data to estimate the shape of the conditional volatility equation was proposed in Engle and Ng (1993) Partially Non-Parametric (PNP) model: 1 We follow Wagner and Marsh (2005) to determine the length of the moving average.
where the range of {ε t } was divided into m intervals with break points at iσ (m + positive intervals and m − negative intervals) and This model, which regarded the long memory component as being parametric while the relationship between news and volatility was considered nonparametric, used equally spaced bins with knots at ε t−1 equal to 0, ±σ, ±2σ, ±3σ and ±4σ (where σ is the unconditional standard deviation) to estimate the news impact curve.
In the empirical analysis of the PNP model, conducted for the Topix Index for the period ranging from 1980 to 1988, Engle and Ng found that the non-parametric approach was able to capture both the leverage and size eects and to outperform all the other estimated models: GARCH(1,1), EGARCH(1,1), Asymmetric-GARCH(1,1), VGARCH(1,1), Nonlinear-Asymmetric-GARCH(1,1) and GJR-GARCH(1,1). Nevertheless, several of the estimated parameters presented unexpected signs and magnitudes, and were found statistically insignicant, regarding the robust standard errors.
Hence, in contrast to Engle and Ng, in this paper the indicator intervals are chosen as less extreme multiples of the unconditional standard deviation of the unpredictable index returns series ε t−1 .
The conditional variance specication now proposed, accommodates both the sign and magnitude of return innovations. Levels of lagged ε t−1 are employed to capture the perception that volatility is related in an asymmetric way to lagged return innovations, with sharp drops in stock prices causing more future volatility than upturns cause. Furthermore, by allowing the existence of more than two regimes, the model specication extends the asymmetry of the EGARCH specication where the threshold is predetermined and equal to zero.
A fundamental idea in the proposed model specication is also the principle of parsimony.
The aim of the model is to approximate the true data-generating process without incorporating an excessive number of coecients. It would be possible to present multiple regimes with innite threshold values. However, it would prove to be statistically unfeasible. The sample space of ε t is therefore partitioned into m − news intervals below zero. The model is estimated for m − = 3 with kinks equal to 0, −1.25σ and −2.5σ.
Despite the groundwork of Engle and Ng, time series models that incorporate multiple thresholds in the conditional variance equation are rare. In fact, GARCH models tend to assume a 5 rather stable environment, failing to capture irregular phenomena. One of the few exceptions is Medeiros and Veiga (2008) that found strong evidence of the existence of more than two regimes for most of the worldwide stock indexes analysed.

Volume
Trading volume can be considered as an important source of information in the context of the future volatility process, providing information which is not available from historical prices. In fact, whereas returns reect average changes in market expectations as a whole, trading occurs when market participants value an asset dierently. Thus, trading volume reects the sum of the distinct investors' reactions, preserving the dierences among individual investors that are averaged out in return data.
The inclusion of unexpected trading volume in equation (3) allows low and high volatilities to be triggered by positive and negative shocks and by the associated trading activity that ows into the market a behaviour which standard GARCH models fail to accommodate. In fact, although the literature on the GARCH models is quite extensive, asymmetric GARCH models rely primarily on news shocks but tend to be silent on the role trading volume plays in market volatility.
We propose the use of surprise volume (Wagner and Marsh, 2005) as a volume variable which is dened as unexpected above-average trading activity. In contrast to those authors, the conditional volatility structure we propose does not consider a contemporaneous relation between volume and volatility, focusing, instead, in a lagged relation that is more suitable for forecasting. The reasoning is that portfolio reallocations for the market as a whole tend to be somewhat sticky, i.e., due to market uncertainty, non-trivial trading costs, short-sale restrictions, liquidity and time constraints; investors tend to take time to update their beliefs about the private information ows, to reassess their daily investment performance and then to restructure their portfolios, often adopting trend-following trading strategies.
Since high trading volume is usually associated with an inux of informed traders, prices tend to become more informative in these periods. In the MSV-EGARCH model specication, if period t − 1 unexpected trading volume is positive, period t variance equation will include lagged unexpected trading volume as a regressor (V t−1 =1). When this occurs, the market values the sign (ε t−1 > 0 or ε t−1 < 0) and the size of the information reected in the market in the preceding period, leading to an upward revision of period t conditional volatility. The model 6 specication adopted assumes that more information arrives to investors when return moves are large and therefore allows bad news followed by signicant volume to have much more impact on volatility than the traditional EGARCH model, whose asymmetry is based only on the sign of the lagged return innovations.
The adopted volume-volatility relation provides a market information aggregate-based explanation for both the volatility clustering and volatility persistence phenomena. Given that dierences in the price reactions of investors to both good and bad news are partially lost by the averaging of prices, but are preserved in trading volume, the proposed conditional volatility model takes into account the sign of the shock, the size of the shock and the associated trading volume.
In addition, since information arrives at an uneven rate, periods of high and low volatility will tend to cluster. Lastly, given that nancial asset trades must, at some future date, be reversed, volatility persistence will arise.

Statistical properties of returns
The empirical analysis is based on daily closing prices and trading volume for six major inter- Following the conventional approach, daily stock returns (r t ) are obtained by taking the logarithmic dierence of daily stock index price data: Table 1 provides a general overview of the data used and presents preliminary descriptive statistics and diagnostics for the return series of each of the six stock indexes.

7
The sample moments for all return series indicate empirical distributions with heavy tails relative to the normal. Not surprisingly, the Jarque-Bera statistic (Jarque and Bera, 1987) rejects the normality assumption for each of the series. The rst striking feature is that the mean of daily returns is higher in NASDAQ and DAX and higher returns go hand in hand with higher standard deviations. The unfavourable outcome of Japanese stock returns is attributable to the fact that the Japanese market has been a bear market since 1989.
The excess of kurtosis ranges from 1.876 for the NIKKEI to 4.1768 for the NASDAQ, suggesting that big shocks, of either sign, are likely to be present.
According to the Ljung-Box statistics on returns, computed at a tenth-order lag, there is relevant autocorrelation in all of the stock indexes. The Ljung-Box statistic on the squared returns and the LM test for a tenth-order linear ARCH eect strongly suggest the presence of time-varying volatility. Since variations in ε 2 t are not purely random, the variance is predictable, by conditioning the volatility of the process on past information. The structure of the MSV-EGARCH model is such that it can adapt exibly to capture dierent features of the conditional distribution of returns. However, altering the EGARCH specication through the introduction of additional parameters involves the risk of in-sample over-tting. Therefore, it is essential to determine whether or not the improved in-sample t is useful for forecasting out-of-sample.
The sample is partitioned in two distinct periods: the rst 3/4 of the sample is retained for the estimation of parameters while the remaining 1/4 is considered as the forecast period. To compare the conditional in-sample tted MSV-EGARCH with the standard GJR (Glosten et al., 1993) and EGARCH asymmetric models, three likelihood based goodness-of-t criteria are used. The rst is the maximum log-likelihood value obtained from ML estimation. The second is the AIC: Akaike information criteria (Akaike, 1978) and the third is the SBC: Schwarz Bayesian criteria (Schwarz, 1978).
Out-of-sample volatility forecast evaluation is conducted by applying the modied Diebold-Mariano (Diebold and Mariano, 1995) test proposed by Harvey and Newbold (2000) to gauge 2 Estimates are obtained using EVIEWS 5.0-based custom software. 9 whether the MSV-EGARCH encompasses the standard GJR and EGARCH models. According to Hansen (2005), when the comparison involves nested models (the MSV-EGARCH model nests EGARCH) it is more appropriate to apply a test for equal predictive accuracy (EPA), such as that of Harvey and Newbold (2000). In the null we state that each particular model (MSV-EGARCH, EGARCH and GJR) encompasses its competitors, in the sense that they do not contain useful information not present in the forecasts resulting from the model considered in the null.
Since volatility itself is not directly observable, establishing the eectiveness of the volatility forecast involves the use of a volatility proxy that may constitute an imperfect estimate of the true volatility, as mentioned by Andersen and Bollerslev (1998), Hansen and Lunde (2003) and Hansen and Lunde (2005). Following the conventional approach, squared returns are used as a proxy for the latent volatility process. However, as those authors argue, this volatility proxy can constitute a noisy estimator of the actual variance dynamics that can compromise the inference regarding the forecast accuracy. Yet, more recently, Patton (2006) showed that the squared daily returns constitutes in fact a valid volatility proxy.

Results
Tables 2 to 7 report in-sample results for the six stock indexes. The in-sample estimation results conrm that markets become more volatile in response to bad news (negative return surprises) as the sign of the parameters estimates proxying for asymmetry in the three regimes is always negative (the exception is NIKKEI's second regime). According to the Wald test, the dierences among the three regimes estimates are statistically signicant in the case of CAC, FTSE and NIKKEI, pointing to multiple regimes in nancial volatility.
The results also conrm that above-average trading volume is an important factor to consider in explaining volatility, with volume playing the role of a switching variable between states. The estimates for the unexpected volume indicator variable are always positive and statistically signicant in the case of DAX, NASDAQ, NIKKEI and S&P500. Thus, nancial volatility increases with lagged above-average trading volume.
The largest log-likelihood values indicate that the MSV-EGARCH is the model more prone to have generated the data.
Regarding the information criteria, the proposed MSV model presents lower AIC values in the case of CAC, FTSE and NIKKEI. When the SBC is used instead, the standard EGARCH domi-nates the other two models: GJR and MSV-EGARCH. This is due to the fact that although both the Akaike and Schwarz criterion are based on parsimony, the Schwarz criterion imposes a larger penalty for additional coecients, which penalizes in particular the additional complexity of the MSV specication. Thus, AIC and SBC provide inconclusive results in the French, English and Japanese stock indexes whereas both information criteria favour the standard EGARCH model in the case of DAX, NASDAQ and S&P500. The MSV-EGARCH outperforms the standard GJR in most of the six stock indexes whatever the information criteria considered.

INCLUDE TABLES 2 TO 7 HERE
In the out-of-sample analysis, based on Harvey-Newbold (HN) test (table 8), we fail to reject the null that the MSV-EGARCH forecasts encompass, or cannot be improved by combination with, the corresponding EGARCH and GJR volatility predictions at the 10% signicance level in the case of CAC, DAX, FTSE and NASDAQ. The null is rejected at this signicance level in the case of the NIKKEI and S&P, implying that combination of the EGARCH and/or GJR predictions with those of MSV-EGARCH would lead to an improvement in the NIKKEI and S&P forecast performance. Excluding the case of the FTSE index, the HN test results point to the same conclusions when one tests if EGARCH forecasts encompass those of the competing MSV-EGARCH and GJR. In contrast, the hypothesis that the GJR forecasts encompass its rivals is rejected in four of the six stock indexes.
Thus, even if the failure to reject the null hypothesis of forecast encompassing among multiple forecasts does not necessarily imply that the forecast under the null is superior and dominant with respect to its competitors, this constitutes one legitimate possibility (Harvey and Newbold, 2000). Based on this, along with the fact that the number of non null hypothesis rejections in the HN test is higher when compared to the standard EGARCH and GJR models, we can admit the superior predictability of the MSV-EGARCH model.

INCLUDE TABLE 8 HERE 5 Conclusions
Using daily data for six major international stock market indexes from January 1995, through April 2008, in this paper, the links between stock market returns, volatility and trading volume are analysed in a new nonlinear conditional variance framework. An innovative multiple regime EGARCH model is proposed, in opposition to the single zero threshold adopted by the conventional model. In this modied model, the asymmetry of the EGARCH is decomposed into multiple regimes, with the transition across regimes being controlled by threshold variables, related to the level of the unconditional standard deviation of the return series. In addition, the model smoothes the gap between theory and practice in volatility modelling by incorporating an on-o volume eect, with above-average trading volume playing the role of a switching variable between states.
An empirical example shows that multiple regimes are statistically signicant for three of the return series analysed and also that above-average trading volume is an important factor to consider when explaining volatility.
A comparison between the increased exibility of the proposed model and the parsimony of the conventional GJR and EGARCH specications leads to goodness-of-t statistics that are not unanimous regarding the in-sample superiority of the MSV-EGARCH model. Yet, when the predictive performance is compared, based on Harvey-Newbold encompassing test, there is evidence that MSV-EGARCH dominates the competing standard asymmetric models in several of the considered stock indexes.  1.5186 *, **, *** denote signicant at the 1%, 5% and 10% level, respectively a TDF denotes the degrees of freedom for the Student's t distribution b Wald testS the restriction that γ 1 = γ 2 = γ 3 3.8844** *, **, *** denote signicant at the 1%, 5% and 10% level, respectively a TDF denotes the degrees of freedom for the Student's t distribution b Wald tests the restriction that γ 1 = γ 2 = γ 3 1.5169 *, **, *** denote signicant at the 1%, 5% and 10% level, respectively a TDF denotes the degrees of freedom for the Student's t distribution b Wald tests the restriction that γ 1 = γ 2 = γ 3 16.6815* *, **, *** denote signicant at the 1%, 5% and 10% level, respectively a TDF denotes the degrees of freedom for the Student's t distribution b Wald tests the restriction that γ 1 = γ 2 = γ 3 1.5216 *, **, *** denote signicant at the 1%, 5% and 10% level, respectively a TDF denotes the degrees of freedom for the Student's t distribution b Wald tests the restriction that γ 1 = γ 2 = γ 3