Asymmetric long memory volatility in the PIIGS economies

The current issue and full text archive of this journal is available at www.emeraldinsight.com/1475-7702.htm Asymmetric long memory volatility in the PIIGS economies Asymmetric long memory volatility Dilip Kumar Institute for Financial Management and Research, Chennai, India, and 23 S. Maheswaran Centre for Advanced Financial Studies, Institute for Financial Management and Research, Chennai, India Received 29 November 2011 Revised 29 March 2012 25 June 2012 Accepted 28 July 2012 Abstract Purpose – The main purpose of this paper is to examine the asymmetry and long memory properties in the volatility of the stock indices of the PIIGS economies (Portugal, Ireland, Italy, Greece and Spain). Design/methodology/approach – The paper utilizes the wavelets approach (based on Haar, Daubechies-4, Daubechies-12 and Daubechies-20 wavelets) and the GARCH class of models (namely, ARFIMA (p,d0 ,q)-GARCH (1,1), IGARCH (1,1), FIGARCH (1,d,0), FIGARCH (1,d,1), EGARCH (1,1) and FIEGARCH (1,d,1)) to accomplish the desired goals. Findings – The findings provide evidence in support of the presence of long range dependence in the various proxies of volatility of the PIIGS economies. The results from the wavelet approach also support the Taylor effect in the volatility proxies. The results show that ARFIMA (p,d0 ,q)-FIGARCH (1,d,0) model specification is better able to capture the long memory property of conditional volatility than the conventional GARCH and IGARCH models. In addition, the ARFIMA (p,d0 ,q)-FIEGARCH (1,d,1) model is better able to capture the asymmetric long memory feature in the conditional volatility. Originality/value – This paper has both methodological and empirical originality. On the methodological side, the study applies the wavelet technique on the major proxies of volatility (squared returns, absolute returns, logarithm squared returns and the range) because the wavelet-based estimator exhibits superior properties in modeling the behavior of the volatility of stock returns. On the empirical side, the paper finds asymmetry and long range dependence in the conditional volatility of the stock returns in PIIGS economies using the GARCH family of models. Keywords Long-range dependence, Hurst exponent, Wavelet analysis, FIGARCH, FIEGARCH, Stock markets, Portugal, Ireland, Italy, Greece, Spain Paper type Research paper 1. Introduction In economics and finance, the study of asymmetric long range dependence in the volatility of asset returns is an important area to explore in research because of its importance for capital market theories (Baillie et al., 1996; Vilasuso, 2002). The analysis related to the long memory property can be realized through the estimation of the fractional integration parameter or the Hurst exponent. The subject of detecting long memory in a given time series was first studied by Hurst (1951), an English hydrologist, who proposed the concept of the Hurst exponent based on Einstein’s contributions to the Brownian motion in physics to deal with the obstacles related to the reservoir control near the Nile river dam. The Hurst exponent lies in the range 0 # H # 1. If the Hurst exponent is 0.5 then the process does not exhibit any significant long range or short range dependence in the series. On the other hand, the Hurst exponent being greater than 0.5 indicates the presence of positive long range dependence in the series. If the Hurst exponent is smaller than 0.5, it indicates negative long range dependence in the series. Review of Accounting and Finance Vol. 12 No. 1, 2013 pp. 23-43 q Emerald Group Publishing Limited 1475-7702 DOI 10.1108/14757701311295818 RAF 12,1 24 The Hurst exponent has characteristics that reflect facts having a bearing on market efficiency. Market inefficiency refers to the phenomenon that the market does not always react immediately when new information flows in but responds to it gradually over a period of time. It has been observed that the squared return, the absolute return, the logarithm of the squared return and the range of financial assets exhibit serial correlations that show hyperbolic decay similar to those of an I(d) process (Taylor, 1986). The fractional integration parameter can be estimated by fitting an ARFIMA(p,d,q) model to the volatility series or by applying the fractionally integrated GARCH class models to the log-differenced series. The fractional integration parameter d equals H 2 0.5, where H is the Hurst exponent. It becomes interesting to investigate empirically what the estimated value of the Hurst exponent or the fractional integration parameter is in the data, because it will help us understand the nature of long range dependence that might be present in it. After the autoregressive conditional heteroskedasticity (ARCH) model and the Generalized ARCH (GARCH) model were introduced by Engle (1982) and Bollerslev (1986), respectively, numerous extensions of ARCH models have been proposed in the literature, by specifying the conditional mean and conditional variance equations, which are potentially helpful in forecasting the future volatility of stock prices. Engle and Bollerslev (1986) propose the Integrated GARCH (IGARCH) model to capture the impact of a shock on the future volatility over an infinite horizon. However, these GARCH and IGARCH models are not able to capture the long memory property of volatility satisfactorily. To deal with this shortcoming, Baillie et al. (1996) propose the fractionally integrated GARCH (FIGARCH) model to allow for fractional orders I(d) of integration, where 0 , d , 1. This model estimates an intermediate process between GARCH and IGARCH. It has also been observed that there is an asymmetric response of volatility to news indicating that a negative price change results in greater volatility than a positive price change of the same magnitude. The EGARCH model (Nelson, 1991), GJR-GARCH model (Glosten et al., 1993) and APARCH model (Ding et al., 1993) are popular asymmetric GARCH class models. This asymmetry has a major impact on the future volatility of the market under the influence of shocks. To consider the long memory effect along with the asymmetry feature in volatility, the literature provides various approaches such as Bollerslev and Mikkelsen’s (1996) fractionally integrated exponential GARCH (FIEGARCH) model and Tse’s (1998) fractionally integrated asymmetric power ARCH (FIAPARCH) model. The motivation for this study is that asymmetry and long memory in the volatility of stock returns are very interesting aspects of the behavior of financial markets. Economists, regulators and practitioners have a keen interest in analyzing the pattern of stock market volatility because of its important implications. Volatility is widely regarded as a proxy of investment risk. The persistence in volatility has a major impact on the future volatility of the market under the influence of shocks. In addition, long memory in volatility can help in predicting future volatility which in turn is helpful in forecasting economic variables. It is also important for option traders who need to value options with expirations stretching into the future. The central aim of this paper is to examine the asymmetric long memory characteristics in the volatility of the indices in the PIIGS economies. We apply the wavelets approach (based on Haar, Daubechies-4, Daubechies-12 and Daubechies-20 wavelets) and the GARCH class of models (namely, ARFIMA (p,d0 ,q)-GARCH (1,1), IGARCH (1,1), FIGARCH (1,d,0), FIGARCH (1,d,1), EGARCH (1,1) and FIEGARCH (1,d,1)) to accomplish our goals. Modeling asymmetric long memory in the volatility of the stock markets in the PIIGS economies has been a neglected area of research in that very few studies exist in the literature that are mainly focused on the long memory property of volatility in the PIIGS economies. We provide two contributions through this study. First, our study makes a contribution to this topic by way of an analysis of the main proxies of volatility and the asymmetric behavior of the conditional volatility in the PIIGS economies. Second, using econometric and econophysics techniques, our study contributes to an improvement of the robustness of the results for asymmetric long memory against short memory in the volatility of the PIIGS economies. The remainder of this paper is organized as follows: Section 2 discusses the literature review on the issue. Section 3 introduces the tests we will use in this study. Section 4 describes the data and discusses the computational details. Section 5 reports the empirical results and Section 6 concludes with a summary of our main findings. 2. Literature review Since the pioneering work of Engle (1982) and Bollerslev (1986), a considerable number of extensions of ARCH models have been developed to capture the dynamics in volatility. Baillie et al. (1996) have proposed the FIGARCH model, Bollerslev and Mikkelsen (1996) the FIEGARCH model, Tse (1998) the FIAPARCH model and Christensen et al. (2010) the FIEGARCH-in-mean (FIEGARCH-M) model to capture symmetric and asymmetric long memory volatility dynamics in time series. Such models have been used extensively in modeling the long memory properties of volatility. Vilasuso (2002) obtains exchange rate volatility forecasts by using the FIGARCH model and finds that the FIGARCH model produces significantly better volatility forecasts (for one-day and ten-days ahead) compared to the GARCH and IGARCH models. Zumbach (2004) has obtained a similar inference for the FIGARCH model, as observed by Vilasuso (2002) using high frequency data. Kang and Yoon (2006, 2007) study the asymmetric long memory feature in the volatility of Asian stock markets and, in particular, the Korean stock market and find that the FIGARCH model better explains the dynamics of volatility. Cheong et al. (2007) investigate the asymmetry and long memory in the volatility of the Malaysian Stock Exchange daily data by considering the financial crisis between 1991 and 2006 on various sub-periods (pre-crisis, crisis and post-crisis) and find mixed results for long memory behavior in different sub-periods. Dionisio et al. (2007) prefer the FIGARCH model over the GARCH and IGARCH models to capture the dynamics of volatility for various international stock market indices. Bentes et al. (2008) examine the long memory properties in the volatility of S&P 500, NASDAQ 100 and Stoxx 50 indices to compare the US and European markets using the FIGARCH model and entropy measures. Oh et al. (2008) study the long-term memory in the KOSPI 1 – minute market index and the exchange rates of six countries using detrended fluctuation analysis (DFA) and the FIGARCH model and find evidence of long memory in the volatility series. Kasman et al. (2009) examine the presence of long memory in the eight Central and Eastern European (CEE) countries’ stock markets and find strong evidence thereof in the conditional volatility of these markets. Fleming and Kirby (2011) examine the joint dynamics of volatility and traded volume using high frequency data and find that both Asymmetric long memory volatility 25 RAF 12,1 26 volatility and volume exhibit long memory. They suggest the use of trading volume to obtain more precise estimates of daily volatility if high frequency data is unavailable. The hyperbolic decay pattern of the autocorrelation function (ACF) in absolute returns, squared returns and logarithm of squared returns can be used to capture the time – series dependence in volatility. Kilic (2004) and Assaf and Cavalcante (2005) investigate the presence of long memory in unconditional volatility (the absolute, squared, and log squared returns) and conditional volatility using both parametric and nonparametric methods in the Istanbul stock market and the Brazilian stock market, respectively, and find strong evidence of long memory in conditional volatility. Gu and Zhou (2007) apply DFA, R/S analysis and modified R/S analysis to study the long memory property of the unconditional volatility of 500 stocks traded on the Shanghai Stock Exchange (SHSE) and Shenzhen Stock Exchange (SZSE) and find strong evidence in support of long memory in the volatility of the 500 stocks. DiSario et al. (2008) apply methods based on wavelets and aggregate series to test the long memory aspect in various proxies of unconditional volatility (absolute returns, squared returns and log squared returns) of ISE National-100 and find strong evidence in favor of the presence of long memory in all the proxies of volatility. Kang et al. (2010) utilize two semi-parametric tests, the Geweke and Porter-Hudak (GPH) test and the Local Whittle (LW) test, and the FIGARCH model to examine the long memory property in the volatility of the Chinese stock market and find evidence of long memory in the volatility time series. They suggest that the assumption of non-normality provides better specifications regarding long memory volatility processes. 3. Methodology 3.1 Long memory in a financial time series Both time domain and frequency domain measures are available to detect the presence of long memory in a given time series. In the time domain, a hyperbolically decaying autocovariance function characterizes the presence of long memory. Suppose xt is a stationary process and lt is its autocovariance function at lag t, then, the asymptotic property of the autocovariance function is given as: lt < jtj2H 22 as jtj ! 1 ð1Þ where H [ (0,1) is a long memory parameter and called the Hurst exponent. In the frequency domain, long memory is present when the spectral density function approaches infinity at low frequencies. Suppose f(l) is the spectral density function. The series xt is said to exhibit long memory if: 122H f ðlÞ , C f jlj as l!0 where Cf . 0 and H [ (0,1). The hypotheses to be tested (stated as the null) are given below: H1. The unconditional volatility series exhibits short term dependence. Null hypothesis, H10: H ¼ 0.5, and alternative hypothesis, H11: H . 0.5: H2. The conditional volatility series exhibits short term dependence. ð2Þ Null hypothesis, H20: d ¼ 0, and alternative hypothesis, H21: d . 0: H3. The conditional volatility series exhibits symmetric long range dependence. Null hypothesis, H30: u1 ¼ 0 and d ¼ 0, and alternative hypothesis, H31: u1 – 0 and d . 0. It is important to analyze for how long the shocks to volatility persist and whether conditional volatility exhibits asymmetric behavior, because of its implications for option pricing. We have considered four proxies of unconditional volatility, i.e. squared daily returns, log squared daily returns, absolute value of daily returns and the range to test H1. We apply FIGARCH and FIEGARCH models to test H2 and H3. 3.2 Discrete wavelet transform Wavelets are based on multi-resolution analysis where by the spectral representation theorem, any covariance-stationary process xt can be represented as a linear combination of sine and cosine functions. The Fourier series of any real-valued function f(x) on the [0,1] interval is expressed as: 1 X f ðxÞ ¼ b0 þ ð3Þ ½bk cos 2pkx þ ak sin 2pkx
k¼1 where the parameters b0, bk and ak, ;k, can be estimated by using least squares. In the wavelet domain, the function f(x) can be expressed as: f ðxÞ ¼ c0 þ 2j21 1 X X cjk cð2j x 2 kÞ ð4Þ j¼0 k¼0 where C(x) is known as the mother wavelet, as it is the mother to all dilations and translations of C in equation (4). The functions of Cjk(x) ¼ C(2jx – k) for j $ 0 and 0 # k , 2 j are orthogonal and they form a basis for square integrable functions L 2 along the [0, 1] interval 2001). Hence, a wavelet is a function C [ L 2 such that R R (Tkacz, 2 C(t)dt ¼ 0 and jC(t)j dt , 1. Haar and Daubechies wavelets are the most commonly used wavelets for economics and finance applications (Tkacz, 2001). Daubechies (1988) proposes a system of wavelets where different wavelets represent different degrees of smoothing of the step function. In this paper, we use the Haar wavelet and three representative Daubechies wavelets to investigate the robustness of the results to varying degrees of smoothing. The Haar wavelet is the least smooth wavelet, followed by the Daubechies-4 and the Daubechies-12, and the Daubechies-20 is the smoothest wavelet used in this study. We use multi-resolution analysis (Mallat, 1989) to obtain the coefficients corresponding to the wavelet transform of the observed time series. To identify the long memory properties using wavelet analysis, we apply the techniques proposed by Jensen (1999) and Tkacz (2001), which is explained as follows. Suppose xt is a random process: ð1 2 LÞd xt ¼ 1t ð5Þ where L is the lag operator and 1t is independent and identically distributed (i.i.d.) normal with mean zero and variance s 2 and d is the differencing parameter. Jensen (1999) empirically shows (following Tevfik and Kim, 1992 and McCoy and Asymmetric long memory volatility 27 RAF 12,1 Walden, 1996) that for a fractionally integrated I(d) process xt with d , 0.5, the autocovariance function shows that the detail coefficients cjk are distributed as N(0, s 222 2( J 2 j)d), where j is the scaling parameter of the wavelets ( j ¼ 1, . . . , J). The fractional integration parameter d can be estimated by using an ordinary least square regression as follows: ln Varðcjk Þ ¼ ln s 2 þ d ln 222ðJ 2jÞ 28 ð6Þ where ln Var(cjk) is the logarithmic transformation of the variance of the detail coefficients cjk. The variance of the detail coefficients decomposes the variance of the original series across different scales and helps us to investigate the behavior of the time series at each scale. The Hurst exponent can be computed as, H ¼ (1 þ d )/2. Note that H ¼ 0.5 for white noise. When the process is persistent (i.e. has long memory), then H . 0.5 and for an anti-persistent process (i.e. with mean reversion), H , 0.5. 3.3 FIGARCH model The conditional mean equation of the stock returns of PIIGS economies is taken to follow an ARFIMA(p,d0 ,q) model of {yt} for all the sub-periods under study. The log returns are calculated from the stock price indices; i.e.:   Pt yt ¼ ln *100 P t21 where Pt is a value of the index at time t and ln is the natural logarithm. The ARFIMA(p,d0 ,q) model is given as below: 0 fðLÞð1 2 LÞd yt ¼ uðLÞ1t ; where 1t is serially uncorrelated, but dependent to its lagged values, d 0 is the fractional difference parameter which measures the degree of long memory, f(L) and u(L) are polynomials in the lag operator of orders p and q, respectively. The standard generalized autoregressive conditional heteroskedasticity (GARCH) model is given as: 1t ¼ z t s t ; zt , N ð0; 1Þ ð7Þ st2 ¼ v þ aðLÞ12t þ bðLÞst2 ; ð8Þ i where v . 0, and a(L) and b(L) are polynomials in the backshift operator L (L xt ¼ xt2 i) of order q and p, respectively. Equation (8) can be rewritten as infinite-order ARCH process (assuming that ai $ 0 and bi $ 0 for all i ): fðLÞ12t ¼ v þ ½1 2 bðLÞ
vt ; 12t s2t ð9Þ where vt ¼ 2 is interpreted as an innovation for the conditional variance, has a zero mean and is serially uncorrelated, and f(L) ¼ [1 2 a(L) 2 b(L)]. The GARCH model has short memory and volatility shocks decay fast at a geometric rate and so, to capture the observed persistence in the volatility of the return series, we test for a unit root in the conditional volatility series. The resulting integrated generalized autoregressive conditional heteroskedasticity (IGARCH) model of Engle and Bollerslev (1986) is given as: fðLÞð1 2 LÞ12t ¼ v þ ½1 2 bðLÞ
vt ð10Þ In the IGARCH model, volatility shocks never die out. Hence, the IGARCH model cannot be used for modeling long memory in the volatility. To overcome this drawback, Baillie and Bollerslev (1996) propose the fractionally integrated generalized autoregressive conditional heteroskedasticity (FIGARCH) model. The FIGARCH ( p, d, q) model is given as: fðLÞð1 2 LÞd 12t ¼ v þ ½1 2 bðLÞ
vt ð11Þ where 0 # d # 1 is the fractional difference parameter which measures the degree of long memory. The FIGARCH approach produces a more flexible class of processes for the conditional variance that accommodates covariance-stationary GARCH process (when d ¼ 0) and IGARCH process (when d ¼ 1). 3.4 Asymmetric volatility models To study the asymmetry in volatility, we apply Nelson’s (1991) EGARCH model which captures the asymmetric response of volatility to news. Bollerslev and Mikkelsen (1996) re-expressed the EGARCH model and its specification can be expressed as follows: log st2 ¼ v þ ½1 2 bðLÞ
21 ½1 þ aðLÞ
gðzt21 Þ; ð12Þ where a(L) and b(L) are real, non-stochastic, scalar sequences in the backshift operator L (L i xt ¼ xt2 i) of order q and p, respectively. Nelson (1991) finds that to accommodate the asymmetric relation between stock returns and volatility changes, the value of g(zt) must be a function of both the magnitude and the sign of zt. Nelson (1991) chooses g(zt) to be a linear combination of zt and j zt j, which is given as follow: gðzt Þ ¼ u1 zt þ u2 ½jzt j 2 Ejzt j
; ð13Þ The two components of g(zt) are the sign effect (u1zt) and the magnitude effect ðu2 ½jzt j 2 Ejzt j
Þ, each with mean zero. The log specification form of the EGARCH model ensures that the conditional variance remains positive even if the parameters are negative. Hence, it is not necessary to impose non-negativity constraints on the model parameters. Also, the EGARCH model allows positive news (positive return shocks) and negative news (negative return shocks) to have different impacts on the conditional volatility. The EGARCH model specified in equation (12) can be extended to account for long memory by factorizing the autoregressive polynomial [1 – b(L) ] ¼ w(L)(1 – L)d where all the roots of w(z) ¼ 0 lie outside the unit circle. The FIEGARCH (p,d,q) model is specified as follows:
log st2 ¼ v þ BðLÞ21 ð1 2 LÞ2d ½1 þ aðLÞ
gðzt21 Þ; ð14Þ The most common approach for estimating the parameters of the GARCH family of models is to maximize the likelihood function based on the quasi maximum likelihood estimation (QMLE) technique of Bollerslev and Wooldridge (1992). Engle and Ng (1993) propose the sign bias, negative size bias, positive size bias and joint tests in standardized residuals to determine the response of the asymmetry Asymmetric long memory volatility 29 RAF 12,1 volatility models to news. From equation (7), zt ¼ 1t/st. Suppose S 2 t is a dummy variable which takes value 1 if 1t2 1 is negative and Sþ t is a dummy variable that takes value 1 if 1t2 1 is positive and zero otherwise. Hence, the regression equations for the sign bias, negative size bias, positive size bias and joint tests are as follow: Sign bias test : z2t ¼ a þ bS 2 t þ et 30 Negative size bias test : z2t ¼ a þ bS 2 t 1t21 þ et 2 þ Joint test : z2t ¼ a þ bS 2 t þ cS t 1t21 þ dS t 1t21 þ et where a, b, c and d are constants. et is the residual series of the regression equations. 4. Data and computational details 4.1 Data In order to test for the long memory property in the volatility of the stock returns of PIIGS economies, we have used daily price data of fives indices associated with the respective economies. The indices used are PSI-20 (the Portuguese stock index, composed of 20 firms with the largest market capitalization and share turnover), ISEQ overall index (also known as Irish Stock Exchange Quotient, is a capitalization weighted index of all equities listed on Irish Stock Exchange), ATG index (also known as the Athens composite share price index, composed of 40 firms with the largest market capitalization), IBEX-35 index (composed of the 35 most liquid securities listed on the stock exchange interconnection system of the four Spanish stock exchanges), FTSE-MIB index (includes 40 Italian stocks that capture 80 percent of the total market capitalization). All the data have been obtained from the Reuters database. The period of study for all the indices is from 22 August 2003 to 30 June 2011 (2,048 observations for each index). We have used the country name to represent the index, i.e. Portugal for the PSI-20 index, Ireland for the ISEQ Overall index, Italy for the FTSE-MIB index, Greece for the ATG index and Spain for the IBEX-35 index. In this paper, we have taken five measures of volatility to accomplish our goals. Our focus is on squared daily returns, X2t , log squared daily returns, ln(X2t ), absolute value of daily returns, jXtj, the range and the conditional volatility estimated using the GARCH family of models as proxies for the volatility of all the indices under study. When returns are zero, then the ln(X2t ) values will be not defined. So, to overcome this problem, we use the transformation proposed by Fuller (1996), which is given as follow:   ls 2 y0t ¼ ln X 2t þ ls 2 2 2 ð15Þ X t þ ls 2 where l is a constant and its value is taken as 0.02 (Ray and Tsay, 2000), s 2 denotes the sample variance of the daily returns and y’t represent the transformed value for ln(X2t ). Range is computed as: Range ¼ max{log H t ; log P t21 } 2 min{log Lt ; log P t21 } ð16Þ where Ht, Lt and Pt are high, low and close price of the stock at time t. 4.2 Descriptive statistics Table I provides the descriptive statistics of the daily returns of all the indices under study. The median daily return is higher for Spain and Italy but the indices in Ireland, Mean Median Stdev Min Max Quartile 1 Quartile 3 Skewness Kurtosis JB-stat. Shapiro Wilk ARCH LM n Q(20) ADF KPSS Portugal Ireland Italy Greece Spain 0.010 0.038 1.136 210.379 10.196 20.414 0.507 20.035 13.549 15,703.582 * 0.870 * 292.908 * 2,048 49.241 * 211.872 * 0.390 * * * 20.021 0.020 1.594 213.964 9.733 20.620 0.689 20.615 8.185 5,861.349 * 0.896 * 512.301 * 2,048 56.291 * 212.456 * 0.463 * * 2 0.011 0.050 1.393 2 8.599 10.874 2 0.548 0.612 0.057 9.302 7,404.348 * 0.888 * 388.351 * 2,048 80.084 * 2 11.848 * 0.287 2 0.028 0.007 1.622 2 10.214 9.114 2 0.756 0.787 2 0.225 4.271 1,579.285 * 0.940 * 369.975 * 2,048 47.539 * 2 11.968 * 0.590 * * 0.018 0.060 1.417 2 9.586 13.484 2 0.571 0.653 0.208 10.724 9,852.828 * 0.892 * 320.284 * 2,048 47.906 * 2 12.189 * 0.247 Notes: Significant at: *1, * *5 and * * *10 percent levels, respectively; Stdev represents the standard deviation of the series; ARCH LM indicates the Lagrange multiplier test for conditional heteroskedasticity with 10 lags; JB-stat. indicates the Jarque-Bera statistics; Q(20) indicates the BoxPierce statistics for 20 lags Italy and Greece exhibit decline in their indices value over the study period and hence we observe negative average daily returns in these indices. ATG index (Greece) seems to be more volatile than the other indices and PSI-20 (Portugal) shows the least volatility in its index values. Jarque-Bera and Shapiro-Wilk statistics confirm the significant non-normality in the daily returns of all the indices. The ARCH-LM test provides evidence in support of the presence of conditional heteroskedasticity in the return series. The indices associated with Portugal, Ireland and Greece show significant negative skewness. In addition, there is evidence of excess kurtosis which confirms the leptokurtosis in the distribution of returns of all the indices. The Box-Pierce Q-test strongly rejects the presence of no significant autocorrelations in the first 20 lags for all the return series. Insignificant KPSS statistics for all the indices support the non-rejection of the null hypothesis of stationarity of the series except for Portugal, Ireland and Greece which show moderate signs of violation of the null hypothesis of the stationarity. Also, the ADF test rejects the null hypothesis of a unit root in all the return series. Figure 1 shows the time plots of the daily prices and log returns for all the indices. It can clearly be observed that all the indices display a great deal of momentum in their returns and prices series which includes a steep rise in the index value from 2005 to the beginning of 2007 and a sudden drop from the beginning of 2007 to the middle of 2008. We also observe volatility clustering during the period 2007-2009 for all the indices. Figure 2 shows the ACFs up to 200 lags of the daily returns xt, squared daily returns, x2t , log squared daily returns, ln(x2t ), absolute value of daily returns, jxtj, and the range (Range) for all the indices with two-tailed 5 percent critical values. We find that the ACF for daily returns decays very fast to zero and fluctuates around zero at different lags, but the ACF for the absolute returns, the squared returns, Asymmetric long memory volatility 31 Table I. Descriptive statistics of daily stock returns for all the indices 32 5 15 14,060 10 12,060 5 10,060 0 8,060 –5 Ireland 10,060 9,060 8,060 0 7,060 –5 6,060 5,060 6,060 –10 –10 4,060 –15 3,060 –15 2003 2004 2005 2006 2007 2008 2009 2010 2,060 –20 2003 2004 2005 2006 2007 2008 2009 2010 2,060 15 Italy 4,060 47,060 Greece 15 42,060 10 10 32,060 5 27,060 22,060 0 17,060 Return (%) 37,060 Price Return (%) 11,060 12,060 –5 15 5 0 –5 –10 7,060 –10 2003 2004 2005 2006 2007 2008 2009 2010 Price Return (%) 10 16,060 –15 2003 2004 2005 2006 2007 2008 2009 2010 2,060 6,000 5,500 5,000 4,500 4,000 3,500 3,000 2,500 2,000 1,500 1,000 Price Portugal Return (%) 15 Price RAF 12,1 18,060 Spain 16,060 10 12,060 10,060 0 Price Return (%) 14,060 5 8,060 –5 6,060 –10 Figure 1. Price and return plots for all the indices 4,060 –15 2003 2004 2005 2006 2007 2008 2009 2010 Return 2,060 Price the logarithm of squared returns and the range decay at a very slow rate which indicates a high degree of persistence in the volatility of stock returns of the PIIGS economies. 5. Empirical results 5.1 Results from wavelet analysis Table II reports the estimates of the Hurst exponent for the PIIGS economies over the whole sample period based on wavelet analysis for the four proxies of volatility, that is, the squared return, the absolute return, the logarithm of squared return and the range. The Hurst exponents for all the cases are significantly more than 0.5. The results obtained from the Haar wavelet analysis provide evidence of long memory for all the volatility series for all the indices under study. We have also used the Daubechies-4, Daubechies-12 and Daubechies-20 wavelets in our analysis to test the robustness of the results obtained from the Haar wavelet analysis. All the Daubechies wavelets (Daubechies-4, Daubechies-12 and Daubechies-20) support the same inference of significance of the Hurst exponent as suggested by the xt 0.7 xt^2 0.6 0.6 |xt| 0.5 0.5 log (xt^2) 0.4 Range Portugal ACF 0.7 0.3 0.2 0.1 0.1 0 0 21 41 61 81 101 121 141 161 181 –0.1 log (xt^2) xt 0.7 xt^2 0.6 0.6 |xt| 0.5 log (xt^2) 0.4 Range Italy Range 33 1 21 41 61 81 101 121 141 161 181 Greece 0.3 |xt| log (xt^2) 0.4 0.2 xt xt^2 0.5 ACF ACF Asymmetric long memory volatility Lag 0.7 Range 0.3 0.2 0.1 0.1 0 0 –0.1 1 –0.2 xt^2 |xt| Lag 0.8 xt 0.3 0.2 –0.1 1 Ireland 0.4 ACF 0.8 21 41 61 –0.1 1 –0.2 81 101 121 141 161 181 Lag 0.8 Spain 0.7 41 61 81 101 121 141 161 181 Lag xt xt^2 0.6 |xt| 0.5 ACF 21 log (xt^2) 0.4 Range 0.3 0.2 0.1 0 –0.1 1 –0.2 21 41 61 81 101 121 141 161 181 Lag Haar wavelet analysis. The value of the long memory parameter (Hurst exponent) that we obtain from Daubechies-4, Daubechies-12 and Daubechies-20 wavelets are very similar to those obtained from the Haar wavelet. The results related to the persistence of volatility proxies confirm the same pattern as what we obtained from the ACF of the volatility proxies (Figure 2). The results are in confirmation with findings of Gu and Zhou (2007), DiSario et al. (2008) and Kang et al. (2010) for various proxies of unconditional volatility. Overall, Table II provides evidence in support of long memory in the volatility of the PIIGS economies. 5.2 Results of GARCH family models First, we determine the order of ARFIMA(p,d0 ,q)-GARCH(1,1) model for all the indices based on the minimum value of the Schwarz Bayesian Information Criteria (SIC). We find that the ARFIMA (2,d0 ,2), ARFIMA (1,d0 ,1), ARFIMA (3,d0 ,1), ARFIMA (1,d0 ,2) and ARFIMA (1,d0 ,1) specification to be suitable for the mean equation of Portugal, Ireland, Italy, Greece and Spain, respectively. Tables III-VII report the estimation results of ARFIMA (p,d0 ,q)-GARCH(1,1), ARFIMA (p,d0 ,q)-IGARCH(1,1), Figure 2. ACF plots for returns and various proxies of volatility for all indices RAF 12,1 34 Table II. Estimated Hurst exponents with t-statistics based on wavelet analysis Portugal Haar Dau-4 Dau-12 Dau-20 Ireland Haar Dau-4 Dau-12 Dau-20 Italy Haar Dau-4 Dau-12 Dau-20 Greece Haar Dau-4 Dau-12 Dau-20 Spain Haar Dau-4 Dau-12 Dau-20 x2t jxtj Log(x2t ) Range 0.643 * (11.813) 0.640 * (11.917) 0.645 * (11.729) 0.647 * (11.649) 0.675 * (9.463) 0.686 * (8.230) 0.687 * (8.279) 0.685 * (8.422) 0.660 * (6.069) 0.680 * (4.760) 0.677 * (4.934) 0.673 * (5.151) 0.718 * (5.331) 0.735 * (4.420) 0.736 * (4.439) 0.733 * (4.531) 0.691 * (7.810) 0.653 * (9.059) 0.673 * (8.497) 0.681 * (8.133) 0.709 * (5.092) 0.688 * (6.174) 0.697 * (5.780) 0.702 * (5.484) 0.687 * (4.685) 0.678 * (4.926) 0.681 * (4.863) 0.684 * (4.752) 0.744 * (2.970) 0.726 * (3.489) 0.735 * (3.268) 0.739 * (3.102) 0.663 * (9.592) 0.651 * (10.180) 0.671 * (9.440) 0.675 * (9.141) 0.683 * (7.406) 0.686 * (7.246) 0.701 * (6.281) 0.704 * (6.057) 0.667 * (5.653) 0.678 * (5.104) 0.689 * (4.561) 0.689 * (4.539) 0.738 * (4.011) 0.746 * (3.780) 0.759 * (3.305) 0.760 * (3.224) 0.616 * (11.785) 0.662 * (9.528) 0.666 * (9.476) 0.662 * (9.655) 0.633 * (10.951) 0.678 * (6.833) 0.680 * (6.785) 0.671 * (7.411) 0.639 * (7.349) 0.664 * (5.673) 0.664 * (5.665) 0.654 * (6.313) 0.694 * (5.339) 0.737 * (3.417) 0.738 * (3.381) 0.729 * (3.734) 0.644 * (3.734) 0.638 * (10.078) 0.644 * (10.000) 0.645 * (9.801) 0.676 * (7.572) 0.677 * (7.491) 0.681 * (7.393) 0.683 * (7.190) 0.658 * (5.995) 0.670 * (5.269) 0.667 * (5.586) 0.666 * (5.654) 0.730 * (4.110) 0.743 * (3.670) 0.743 * (3.739) 0.744 * (3.646) Note: Significant at *1 percent level ARFIMA (p,d0 ,q)-FIGARCH(1,d,0), ARFIMA (p,d0 ,q)-FIGARCH(1,d,1), ARFIMA (p,d0 ,q)-EGARCH(1,1) and ARFIMA (p,d0 ,q)-FIEGARCH(1,d,1) for Portugal, Ireland, Italy, Greece and Spain, respectively, to compare and demonstrate the empirical properties of symmetric and asymmetric models. For the ARFIMA (p,d0 ,q)-GARCH(1,1) model, the estimates of a1 and b1 are significantly different from zero and (a1 þ b1) , 1 for all the indices, which indicates that the GARCH (1,1) model is valid to investigate the volatility clustering in Portugal, Ireland, Italy, Greece and Spain. Also, the asymmetric coefficient (u1) for the EGARCH model is highly significant at 5 percent level of significance for all the indices except for the case of Greece. The significance of the asymmetry coefficient implies that an unexpected negative shock is followed by greater volatility than an unexpected positive shock of the same magnitude. Hence, the standard GARCH model overestimates volatility under the impact of positive shocks and underestimates volatility due to negative shocks. 5.3 Long memory characteristics in volatility This section discusses the estimation and performance evaluation of parametric GARCH model specifications (ARFIMA (p,d0 ,q)-GARCH (1, 1), IGARCH (1, 1), FIGARCH (1, d, 0) and FIGARCH (1, d, 1)) to study the long memory characteristics in the conditional volatility of the various indices of the PIIGS economies. GARCH (1,1) m d1 d2 C1 C2 d0 v a1 b1 u1 u2 d Log-Likelihood SIC JB-stat. Q(10) Qs(10) ARCH-LM(10) Sign bias test Negative size bias Joint test 0.085 * (0.018) 2 1.065 * (0.324) 2 0.085 (0.317) 1.096 * (0.312) 0.121 (0.304) 0.024 (0.030) 0.016 * * (0.007) 0.154 * (0.030) 0.841 * (0.029) – – – – – – 2 2,584.260 2.557 298.050 * 10.782 * * * 3.922 0.404 1.635 1.205 17.753 * Symmetric models IGARCH (1,1) FIGARCH (1,d,0) FIGARCH (1,d,1) Asymmetric models EGARCH (1,1) FIEGARCH (1,1) 0.084 * (0.017) 2 0.897 * (0.034) 2 0.877 * (0.055) 0.933 * (0.026) 0.919 * (0.046) 0.022 (0.022) 0.015 * (0.006) 0.159 * (0.030) 0.841 – – – – – – – 2 2,581.104 2.550 273.260 * 8.116 4.352 0.447 1.477 1.237 17.368 * 0.085 * (0.017) 2 0.897 * (0.037) 2 0.881 * (0.056) 0.932 * (0.027) 0.919 * (0.048) 0.022 (0.023) 0.029 * (0.011) 0.060 (0.076) 0.510 * (0.100) – – – – 0.582 * (0.090) 2 2,571.116 2.548 311.410 * 7.527 4.524 0.458 1.272 1.885 * * * 19.296 * 0.061 (0.041) 2 0.408 * (0.098) 0.573 * (0.097) 0.267 * (0.088) 2 0.708 * (0.084) 0.199 (0.125) 0.024 (0.208) 2 0.226 (0.163) 0.976 * (0.007) 2 0.124 * (0.034) 0.269 * (0.049) – – 2 2,567.146 2.548 279.760 * 10.748 * * * 7.750 0.758 0.285 0.861 3.914 0.085 * (0.017) 2 0.896 * (0.039) 2 0.880 * (0.059) 0.931 * (0.029) 0.918 * (0.051) 0.022 (0.023) 0.033 * (0.010) – – 0.456 * (0.108) – – – – 0.576 * (0.097) 2 2,571.585 2.545 312.910 * 7.139 5.054 0.508 1.256 2.115 * * 20.488 * 0.089 (0.169) 2 0.374 (0.383) 0.597 * * * (0.361) 0.228 (0.411) 2 0.733 * * (0.373) 0.207 (0.159) 0.036 (0.460) 2 0.477 * (0.159) 0.878 * (0.100) 2 0.130 * (0.041) 0.257 * (0.047) 0.357 * * * (0.193) 2 2,562.468 2.547 328.820 * 11.665 * * * 8.189 0.813 0.371 0.768 3.169 Notes: Significant at: *1, * *5 and * * *10 percent levels, respectively; the values in the parentheses represent the standard errors Asymmetric long memory volatility 35 Table III. Estimation results for the GARCH class models for Portugal m d1 C1 d0 v a1 b1 u1 u2 d Log-Likelihood SIC JB-stat. Q(10) Qs(10) ARCH-LM(10) Sign bias test Negative size bias Joint test 0.084 * (0.022) 20.606 (0.545) 0.615 (0.540) 0.001 (0.022) 0.021 * * * (0.011) 0.108 * (0.026) 0.886 * (0.026) – – – – – – 23,276.729 3.226 1,475.900 * 5.627 2.581 0.261 0.821 0.650 2.307 Symmetric models IGARCH (1,1) FIGARCH (1,d,0) 0.092 * (0.030) 0.762 * (0.219) 2 0.799 * (0.192) 0.047 (0.057) 0.019 * * * (0.010) 0.116 * (0.028) 0.884 – – – – – – – 2 3,276.938 3.222 1,607.700 * 7.249 2.991 0.304 0.933 0.870 2.919 0.109 * (0.035) 0.710 * (0.183) 2 0.774 * (0.148) 0.079 (0.063) 0.056 * * (0.023) – – 0.487 * (0.118) – – – – 0.545 * (0.104) 2 3,268.349 3.218 1,231.400 * 7.872 2.661 0.270 0.878 0.016 2.126 FIGARCH (1,d,1) 0.093 * (0.026) 20.970 * (0.026) 0.976 * (0.023) 0.008 (0.018) 0.051 (0.039) 0.034 (0.138) 0.533 * * (0.236) – – – – 0.561 * (0.135) 23,268.396 3.222 1,313.400 * 6.290 2.460 0.250 1.009 0.159 2.215 RAF 12,1 36 Table IV. Estimation results for the GARCH class models for Ireland GARCH (1,1) Asymmetric models EGARCH (1,1) FIEGARCH (1,1) 0.062 * * * (0.032) 0.317 (0.693) 2 0.394 (0.755) 0.082 (0.099) 0.747 * (0.272) 0.692 (0.799) 0.983 * (0.009) 2 0.046 * * (0.021) 0.110 * * (0.045) – – 23,257.237 3.214 840.270 * 9.420 2.863 0.277 0.809 0.287 0.922 Notes: Significant at: *1, * *5 and * * *10 percent levels, respectively; the values in the parentheses represent the standard errors 0.134 * (0.042) 0.575 * (0.088) 2 0.730 * (0.054) 0.173 * (0.061) 0.074 (0.363) 1.647 (1.466) 0.319 * * (0.125) 2 0.057 * * (0.027) 0.087 * * (0.042) 0.604 * (0.051) 2 3,240.623 3.202 695.090 * 14.344 3.252 0.318 0.900 0.334 1.201 GARCH (1,1) m d1 d2 d3 C1 d0 v a1 b1 u1 u2 d Log-Likelihood SIC JB-stat. Q(10) Qs(10) ARCH-LM(10) Sign bias test Negative size bias Joint test 0.046 * * (0.018) 2 0.649 (0.460) 2 0.022 (0.053) 2 0.031 (0.029) 0.621 (0.466) 2 0.009 (0.042) 0.016 * * (0.007) 0.102 * (0.022) 0.891 * (0.022) – – – – – – 2 3,037.202 3.000 226.550 * 4.299 13.644 * * * 1.382 1.393 0.302 17.869 * Symmetric models IGARCH (1,1) FIGARCH (1,d,0) 0.045 * * (0.019) 2 0.652 (0.438) 2 0.023 (0.052) 2 0.032 (0.028) 0.623 (0.444) 2 0.008 (0.041) 0.013 * * (0.006) 0.107 * (0.022) 0.893 – – – – – – – 2 3,038.004 2.997 237.710 * 4.199 12.656 1.275 1.437 0.046 18.072 * 0.049 * (0.019) 2 0.569 (0.396) 2 0.025 (0.050) 2 0.034 (0.034) 0.552 (0.396) 2 0.008 (0.042) 0.044 * (0.014) – – 0.530 * (0.127) – – – – 0.567 * (0.123) 2 3,026.136 2.989 208.230 * 4.401 8.441 0.875 1.377 1.256 19.700 * FIGARCH (1,d,1) 0.048 * * (0.019) 2 0.577 (0.398) 2 0.023 (0.050) 2 0.033 (0.033) 0.558 (0.398) 2 0.008 (0.042) 0.038 * * (0.015) 0.048 (0.066) 0.556 * (0.088) – – – – 0.554 * (0.098) 2 3,025.673 2.992 210.410 * 4.181 8.058 0.825 1.372 1.121 19.272 * Asymmetric models EGARCH (1,1) FIEGARCH (1,1) 2 0.036 (0.054) 2 0.489 (0.851) 2 0.093 (0.110) 2 0.042 (0.053) 0.361 (0.890) 0.112 (0.069) 0.434 * * * (0.243) 2 0.144 (0.210) 0.987 * (0.004) 2 0.144 * (0.043) 0.120 * (0.025) – – 2 2,994.147 2.965 219.340 * 7.237 11.762 1.227 1.056 0.854 7.276 * * * 0.036 (0.027) 2 0.384 (0.353) 2 0.045 (0.119) 2 0.021 (0.040) 0.327 (0.438) 0.037 (0.129) 2 0.080 (0.342) 2 0.030 (0.747) 0.541 (0.383) 2 0.155 * (0.051) 0.107 * (0.023) 0.575 * (0.082) 2 2,983.729 2.958 207.710 * 5.257 8.790 0.929 0.769 0.936 6.547 * * * Notes: Significant at: *1, * *5 and * * *10 percent levels, respectively; the values in the parentheses represent the standard errors Asymmetric long memory volatility 37 Table V. Estimation results for the GARCH class models for Italy m d1 C1 C2 d0 v a1 b1 0.086 * (0.027) 0.019 (0.167) 0.025 (0.188) 20.029 (0.035) 0.014 (0.043) 0.019 * (0.007) 0.103 * (0.018) 0.893 * (0.017) Symmetric models IGARCH (1,1) FIGARCH (1,d,0) 0.086 * (0.027) 0.018 (0.165) 0.025 (0.185) 20.029 (0.035) 0.014 (0.042) 0.017 * (0.006) 0.106 * (0.017) 0.894 0.079 * (0.027) 0.044 (0.168) 0.005 (0.190) 20.035 (0.037) 0.014 (0.043) 0.059 * (0.020) 0.443 * (0.098) FIGARCH (1,d,1) 0.079 * (0.027) 0.039 (0.170) 0.010 (0.192) 20.034 (0.036) 0.013 (0.043) 0.049 * * (0.021) 0.054 (0.057) 0.508 * (0.101) u1 u2 d Log-Likelihood SIC JB-stat. Q(10) Qs(10) ARCH-LM(10) Sign bias test Negative size bias Joint test 23,500.188 3.448 71.034 * 9.185 13.190 1.411 2.941 * 1.299 24.765 * 23,500.355 3.444 70.552 * 9.248 13.041 1.396 2.944 * 1.417 25.220 * 0.469 * (0.090) 23,488.280 3.436 52.477 * 8.019 2.071 0.207 2.746 * 0.612 24.772 * 0.487 * (0.085) 23,487.745 3.440 53.162 * 8.232 2.670 0.267 2.783 * 0.357 24.462 * RAF 12,1 38 Table VI. Estimation results for the GARCH class models for Greece GARCH (1,1) Asymmetric models EGARCH (1,1) FIEGARCH (1,1) 0.053 * (0.039) 20.012 (0.164) 0.035 (0.194) 20.024 (0.037) 0.042 (0.052) 0.933 (0.250) 0.635 (0.826) 0.984 * (0.006) 20.039 (0.025) 0.113 * * (0.048) 23,494.050 3.449 45.083 * 8.369 11.946 1.177 3.196 * 1.114 19.919 * Notes: Significant at: *1, * *5 and * * *10 percent levels, respectively; the values in the parentheses represent the standard errors 0.062 * * (0.031) 20.016 (0.162) 0.032 (0.188) 20.034 (0.036) 0.052 (0.048) 0.689 * (0.267) 0.125 (0.562) 0.670 * (0.175) 20.071 * * (0.034) 0.120 * (0.036) 0.529 * (0.062) 23,480.648 3.440 41.383 * 6.262 9.531 0.931 3.038 * 1.383 17.227 * GARCH (1,1) m d1 C1 d0 v a1 b1 0.077 * (0.015) 0.122 (0.616) 2 0.074 (0.600) 2 0.063 (0.043) 0.027 * (0.010) 0.128 * (0.029) 0.862 * (0.028) Symmetric models IGARCH (1,1) FIGARCH (1,d,0) 0.077 * (0.015) 0.130 (0.675) 20.083 (0.659) 20.062 (0.043) 0.023 * * (0.009) 0.137 * (0.028) 0.863 0.078 * (0.015) 2 0.042 (0.358) 0.095 (0.344) 2 0.060 * * * (0.034) 0.060 * (0.018) 0.492 * (0.112) FIGARCH (1,d,1) 0.078 * (0.015) 2 0.011 (0.393) 0.063 (0.379) 2 0.060 * * * (0.035) 0.053 * (0.020) 0.042 (0.071) 0.518 * (0.083) u1 u2 d Log-Likelihood SIC JB-stat. Q(10) Qs(10) ARCH-LM(10) Sign bias Negative size bias Joint test 23,129.634 3.082 329.660 * 5.619 26.152 * 2.742 * 2.342 * * 0.014 18.010 * 23,130.725 3.080 344.320 * 5.696 24.497 * 2.555 * 2.502 * * 0.344 18.675 * 0.557 * (0.112) 2 3,117.251 3.070 332.640 * 5.192 19.484 * * 1.991 * * 2.087 * * 0.944 18.017 * 0.549 * (0.102) 2 3,116.968 3.074 343.800 * 5.163 19.832 * * 2.034 * * 2.146 * * 0.790 17.971 * Asymmetric models EGARCH (1,1) FIEGARCH (1,1) 0.087 * (0.022) 0.605 * (0.173) 2 0.686 * (0.157) 0.084 (0.057) 2 0.276 (0.363) 2 0.022 (0.313) 0.561 * (0.161) 2 0.174 * (0.046) 0.119 * (0.027) 0.547 * (0.060) 2 3,060.439 3.026 201.300 * 5.264 12.239 1.254 0.987 0.708 3.426 20.004 (0.062) 0.415 * (0.105) 20.561 * (0.128) 0.146 (0.121) 0.427 * * (0.206) 20.101 (0.237) 0.982 * (0.008) 20.157 * (0.042) 0.129 * (0.037) 23,075.912 3.037 200.240 * 8.028 15.013 * * * 1.593 1.327 0.538 3.790 Notes: Significant at *1, * *5 and * * *10 percent levels, respectively; the values in the parentheses represent the standard errors Asymmetric long memory volatility 39 Table VII. Estimation results for the GARCH class models for Spain RAF 12,1 40 Tables III-VII present the results for the GARCH class of models (as mentioned above) that have been estimated by maximizing the Gaussian likelihood function for the indices of the PIIGS economies. The results indicate that the fractional differencing parameter d obtained under FIGARCH models is statistically significant for all the indices and lie between 0.469 and 0.582. We also find that the log-likelihood values for the IGARCH model (which assumes d ¼ 1) is the least of other GARCH class models followed by the GARCH model (which assumes d ¼ 0) except for the case of Portugal in which case the GARCH model has the least log-likelihood value. The log-likelihood values for FIGARCH (1,d,1) are the highest when compared with other symmetric GARCH family models except for the case of Ireland where FIGARCH (1,d,0) provides the highest log-likelihood value. Overall, we can see that the log-likelihood values of FIGARCH models are very close to each other compared to GARCH and IGARCH models. Also, the SIC values are lower for the FIGARCH (1,d,0) models than the GARCH, the IGARCH model and the FIGARCH (1,d,1) model and the parameters estimated by GARCH (1, 1) model and IGARCH (1, 1) model are not very different from each other. This significantly affects the validity of the null hypothesis for the GARCH and IGARCH models at conventional levels of significance. It also shows that the FIGARCH models better capture the dynamics of conditional volatility than do the GARCH and IGARCH models (Kilic, 2004; Assaf and Cavalcante, 2005; Kang and Yoon, 2007; Oh et al., 2008; Bentes et al., 2008; Kang et al., 2010). Hence, our results indicate that conditional volatility also exhibits the long memory property which implies a non-zero correlation between distant observations and this feature can be used to forecast volatility values. If we compare the results of FIGARCH (1,d,0) and FIGARCH (1,d,1) models, we find that the FIGARCH (1,d,0) specification is better able to describe the long memory in conditional volatility due to comparable values of the log-likelihood function, insignificant values of Ljung-Box statistic for the standardized residuals, Q(10) and the squared standardized residuals, Qs(10), up to 10 lags and ARCH-LM (up to 10 lags) test of heteroskedasticity at 1 percent level of significance, and lower values of SIC. We also do not find any significant bias from the perspective of the sign bias, negative size bias and joint tests (as proposed by Engle and Ng, 1993) in the standardized residuals of Ireland for all the estimated GARCH class models. Portugal and Italy show some bias as per the joint test and Greece and Spain show both the sign bias and the joint test bias. 5.4 Asymmetry and long memory characteristics in volatility In this sub-section, we investigate the asymmetric long memory characteristics in the volatility of the PIIGS economies. To account for the asymmetric long memory characteristics in the conditional volatility of the indices of the PIIGS economies, we apply the FIEGARCH model. The condition mean equation is kept the same as discussed in the previous sub-sections. The ARFIMA(p,d0 ,q)-FIEGARCH(1,d,1) model does well at capturing the asymmetric long memory properties of the index returns. The asymmetry coefficient (u1) shows results that are similar to those obtained in ARFIMA(p,d0 ,q)-EGARCH(1,1) model, which confirms the asymmetry in returns of all the indices under study. The significant value of the fractional differencing parameter d at conventional levels of significance also indicates that conditional volatility is a long memory process as well. Comparing the EGARCH and FIEGARCH models, we find that FIEGARCH model can better explain the asymmetric properties of the returns with the additional long memory feature of conditional volatility (Bekaert and Wu, 2000; Kang and Yoon, 2006; Cheong et al., 2007) because of lower values of SIC and comparable values of log-likelihood function. In addition, we do not find any serial correlation in the standardized residuals (insignificant values of Q(10)) and the squared standardized residuals (insignificant values of Qs(10)) at 5 percent level of significance. Insignificant values of the ARCH-LM statistic also confirm the absence of heteroskedasticity in the residual series up to 10 lags for both the EGARCH and FIEGARCH models. Also, the insignificant values of the sign bias test, the negative size bias test and the joint test for Portugal, Ireland, Italy and Spain indicate that the asymmetric properties of returns are captured well by the asymmetric GARCH family models considered here. 6. Conclusion In this paper, we have examined the asymmetry and long memory properties in the volatility of the stock indices in the PIIGS economies. We have applied the wavelets approach (based on Haar, Daubechies-4, Daubechies-12 and Daubechies-20 wavelets) and the GARCH class of models (namely, ARFIMA (p,d0 ,q)-GARCH (1,1), IGARCH (1,1), FIGARCH (1,d,0), FIGARCH (1,d,1), EGARCH (1,1) and FIEGARCH (1,d,1)) to accomplish our goals. Our results support the presence of long memory in the volatility of the indices. The estimated value of the Hurst exponent from the wavelet approach also supports the Taylor effect in the volatility proxies used. On the other hand, the results from the GARCH family models also provide evidence in favor of long memory in the conditional volatility of the PIIGS economies. The results indicate that the ARFIMA (p,d0 ,q)-FIGARCH (1,d,0) model specification is better able to capture the long memory property of conditional volatility than the conventional GARCH and IGARCH models. To examine the asymmetric long memory properties in the conditional volatility of PIIGS economies, we apply the ARFIMA (p,d0 ,q)-FIEGARCH (1,1) model specification. The significant value of the fractional differencing parameter d and the asymmetry coefficient provide evidence of asymmetric long memory characteristics of conditional volatility for the indices under examination. We observe no significant bias in any of the sign bias, negative size bias and joint tests in the standardized residuals from the EGARCH and FIEGARCH models for any of the indices except for the case of Greece. Our study contributes to the literature in documenting the asymmetric long memory properties of the volatility of the stock markets of the PIIGS economies using various proxies including conditional and unconditional volatility measures. Our study also contributes to the literature in examining the long memory effects of range-based volatility estimation. A natural implication of our empirical findings is that models used for deriving option prices, valuing futures contracts and other derivative securities should incorporate asymmetry and long memory components in volatility. Our findings suggest that economists, regulators, policy makers and financial analysts should consider long memory characteristics while modeling and forecasting volatility. In particular, our findings will be of value to economists, regulators and policy makers who are concerned about excess volatility in the market which can help in identifying possible bubbles in an asset market. 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Corresponding author Dilip Kumar can be contacted at: [email protected] To purchase reprints of this article please e-mail: [email protected] Or visit our web site for further details: www.emeraldinsight.com/reprints Asymmetric long memory volatility 43