Long Range Dependence in the Factor of Implied Volatility Strings

Transcription

1 Long Range Dependence in the Factor of Implied Volatility Strings Julius Mungo Wolfgang Härdle Center for Applied Statistics and Economics (C.A.S.E.), School of Business and Economics, Humboldt-Universität zu Berlin

2 Motivation 1-1 Many economic and financial time series show evidence of neither stationarity nor non-stationarity, presenting difficulty of distinguishing between them. An intermediate is the fractional integrated process (1 L) d X t = ε t (1) (1 L) d = 1 dl 1 2 d(1 d)l2 1 6 d(1 d)(2 d)l3... For 0 < d < 1, current values of X t are influenced by the immediate past values and values from previous time periods. [Granger and Joyeux (1980)], [Hosking (1981)]

3 Motivation 1-2 Long-range dependence a hyperbolic rate of decay of correlations such that lim T T j= T ρ(j) = (are not summable), T is number observations. [McLeod and Hipel (1978)] the correlation function ρ k behaves for s, k as: lim k ρ(k) = 1 (2) c ρ k2d 1 c ρ > 0, k = 1, 2,... and d (0, 0.5) is the memory parameter. [Beran (1994)]

4 Motivation 1-3 Long-range dependence the spectral density f (λ) = 1 2π k= e(tkλ) γ(k) behaves for λ 0 as lim λ 0 where γ k = cov(x t, X t+k ). f (λ) = 1 (3) 2d c f λ as k, γ k ck 2d 1 for some c > 0 and j= γ k =. For d 0, long memory, For d ( 0.5, 0.5), then X t is stationary and invertible, For d = 1, indicates a unit root process. [Mandelbrot (1983)]

5 Motivation 1-4 Does Long memory in volatility matter? Volatility is directly related to information arrival at the financial markets at a given time. A rise (fall) in information arrival increases (decreases) the variance of returns and mean trading volume. with long memory the arrival of new market information cannot be fully arbitraged away. If the market is weakly efficient, stock prices should behave as a martingale process. [Mandelbrot (1971)]

6 Motivation 1-5 Predictability in volatility leads to improve forecasts of the movements of asset prices, contradicting efficient market hypotheses, (all relevant information is fully reflected into prices). 0ption price are significantly different when standard models are applied as compared to models allowing for long memory. [Bollerslev and Mikkelsen (1996)], [Sibbertsen (2004)]

7 Motivation 1-6 portfolio diversification decisions in strategic asset allocation may become extremely sensitive to the investment horizon. there may be diversification benefits in the short and medium term, but not if the assets are held together over the long term if long memory is present. e.g., in a market that exhibits antipersistence, asset prices tend to reverse its trend in the short term thus creating short-term trading opportunities. [Cheung and Lai (1995)], [Wilson and Okunev (1999)]

8 Motivation 1-7 The semiparametric factor model Log-implied volatility Y t,j Y t,j = L z t,l m l (X t,j ) + ε t,j (4) l=0 z t,0 = 1, j = 1,..., J t (t = 1,..., T ) is number of IV observations on day t, L is number of basis functions X t,j is two-dimensional containing moneyness and maturity z t,l are time dependent factor loadings and m l are smooth basis functions. [Borak, Härdle and Fengler (2005)], [Borak et al. (2007)]

9 Motivation 1-8 First loading series: Z1 Second loading series: Z Third loading series: Z Figure 1: Time series plots in levels of three loading series from a DSFM fit for the DAX-Option analyzed from Figure 1: Factor Loading Time Series from Dynamic Semiparametric Model for Implied Figure 2: Factor Loading Time Series from Dynamic Semiparametric Model for Implied Volatility String Dynamics (left Volatility String Dynamics. column).

10 Motivation 1-9 Aim study persistence in the factor loading series as proxi of volatility, search for economic explanations of the movements in asset returns, the possibility of improve price forecasting or application in risk management.

11 Motivation 1-10 Overview 1. Motivation 2. Testing and Measuring Long Memory 3. Long Memory Models 4. Empirical Analysis 5. Conclusion

12 Testing and Measuring Long Memory 2-1 Rescale Variance method: V/S By centering of the KPSS statistic based on the partial sum of the deviations from the mean.! kx 2! TX kx 2 3 V /S(q) = (X j X T ) 1 (X j X T ) 5 T X T T 2ˆσ T 2 (q) k=1 j=1 k=1 j=1 (5) where k j=1 (X j X T ) are the partial sums of the observations and ˆσ T 2 (q) = ˆγ ( ) q j=1 1 j 1+q ˆγ j, q < T is the [Newey and West (1994)] Heteroscedastic and Autocorrelation Consistent estimator of the variance at truncation q. ˆγ 0 is the variance of the process. [Giraitis, Kokoszka, Leipus (2001)]

13 Testing and Measuring Long Memory 2-2 Lobato and Robinson: LobRob Based on lim λi 0 + f (λ i) = Cλ 2d i t LR = (m)ĉ1 Ĉ 0 (6) with Ĉk = 1 m m j=1 ζk j I (λ j) and ζ j = log(j) 1 m m i=1 log(i), where I (λ) = 1 2πT T t=1 X te itλ 2 is the periodogram estimated for degenerate band of Fourier frequencies λ j = 2πj T, j = 1,..., m << [T /2] with bandwidth parameter m. Under null hypothesis of I(0), t LR is asymptotically normally distributed. [Lobato and Robinson (1998)]

14 Testing and Measuring Long Memory 2-3 Log-periodogram Regression (GPH) Uses an approximation of the spectrum near the zero frequency: f (λ) = C { 4sin 2 (λ j /2) } d. Estimate d from the log-periodogram regression: log{i (λ j )} = log C dlog { 4sin 2 (λ j /2 } + ε j at harmonic frequencies, λ j = 2πj T with j (l; m], where l is a trimming parameter discarding the lowest frequencies and m is a bandwidth parameter. m j=1 d = (X j X )log{i (λ j )} 2 m j=1 (X (7) j X ) where X j = log { 4sin 2 (λ j /2 } [Geweke and Porter-Hudax (1983)]

15 Testing and Measuring Long Memory 2-4 Gaussian Semiparametric Estimator (GSP) Based on the approximation lim f (λ i) = Cλ 2d λ i 0 + i d is obtained by solving the minimization { {Ĉ, d} = arg min L(C, d) = 1 m log(cλ 2d j ) + I (λ j) C,d m Cλ 2d j=1 j d = arg min d [Robinson (1995a)] log 1 m m j=1 I (λ j ) Cλ 2d j 2d m } m log(λ j ) (8) j=1

16 Long Memory Models 3-1 ARFIMA(p,d,q) Extends the integration order of the conventional ARMA model to non-integer value between 0 and 1, j=0 Φ(L)(1 L) d (z t µ) = Θ(L)ε t (9) ε t i.i.d(0, σε). 2 (1 L) d Γ(d + 1) = Γ(j + 1)Γ(d j + 1) Lj = 1 dl 1 2 d(1 d)l2 1 6 d(1 d)(2 δ)l3... Persistence for 0 < d < 0.5, anti-persistence for 0.5 < d < 0, non-stationary for d > 0.5 [Granger and Joyeux (1980)], [Hosking (1981)]

17 Long Memory Models 3-2 FIGARCH(p,δ,q) Combines the I (δ) process for the mean with regular GARCH process for conditional variance, [Baillie et al. (1996)] Φ(L)(1 L) δ ε 2 t = ω + Θ(L)(ε 2 t σ 2 t ) (10) σ 2 t = ω [1 1 θ(l) + ] φ(l)(1 L)δ ε 2 t 1 θ(l) with 0 δ 1. [Chung (1999)], parametrization: Φ(L)(1 L) δ (ε 2 t σ 2 ) = Θ(L)(ε 2 t σ 2 ) (11) where σ 2 is the unconditional variance of ε t. σt 2 = σ 2 + [1 ] φ(l)(1 L)δ (ε 2 t σ 2 ) 1 θ(l)

18 Long Memory Models 3-3 HYGARCH(p,α,d,q) Extends the conditional variance of FIGARCH(p, δ, q) by introducing weights to the difference operator, σt 2 ω = [1 1 θ(l) + φ(l) { 1 + α(1 L) d} ] ε 2 t (12) 1 θ(l) where α are weights to (1 L) d. for α = 0, GARCH for α = d = 1, IGARCH for α = 1 (log α = 0), FIGARCH for α < 1 (log α < 0), a stationary process [Davidson (2004)]

19 Data and Empirical Analysis 4-1 ACF-z1 ACF-z2 ACF-z Spectrum z Spectrum z Spectrum z3 density density*e density*e frequency frequency frequency Figure 2: Plots of the sample autocorrelation functions with length 300 and spectrum of the loadings series in levels.

20 Data and Empirical Analysis 4-2 level q z z z r t q z z z r t q z z z Table 1: The rescaled variance V /S test for I (0) against I (d) for series in levels, return and absolute return. q is the truncation. If the evaluated statistics are over the critical value, for I(0), we fail to reject the alternative hypothesis that the series display long memory.

21 Data and Empirical Analysis 4-3 level m z z z r t m z z z r t m z z z Table 2: LobRob: Semiparametric test for I(0) of a time series against long-memory and antipersistence. m is the bandwidth parameter. Short memory is rejected against long-memory if the test statistic is in the lower tail of the standard normal distribution. If statistic is in upper tail of the standard normal distribution, short memory is rejected against antipersistent.

22 Data and Empirical Analysis 4-4 b dgph : z1 b dgsp : z1 m level r t r t m level r t r t b dgph : z2 b dgsp : z2 m level r t r t m level r t r t b dgph : z3 b dgsp : z3 m level r t r t m level r t r t Table 3: The log periodogram (ˆd GPH ) and the Gaussian semiparametric (ˆd GSP ) estimates of d for levels, returns and absolute returns. Bandwidth m for GPH estimator is m = T α with α = 0.5, 0.525, 0.575, 0.60, 0.8 and T = 1052 is the sample size. For the GSP estimator the bandwidth is chosen such that m = [ T 4 ], [ T 8 ], [ T 16 ], [ T 32 ], [ T 64 ].

23 Data and Empirical Analysis 4-5 Parameter Estimation Level z1 z2 z3 ARFIMA (5, d, 4) (3, d, 3) (1, d, 5) d 0.29 ( 1.78) 0.53 (4.87) 0.29 ( 0.74) φ ( 3.13) (-0.60) 0.96 (26.10) φ ( 0.31) 0.74 (4.01) φ ( 1.84) 0.34 (0.99) φ ( 2.71) φ (-3.92) θ ( 0.94) 0.17 (0.35) (-1.41 ) θ ( 0.29) (-3.49) (-0.24) θ (-1.37) (-0.64) (-2.62) θ (-4.78) 0.03 ( 0.79) θ (-2.44) constant Ln(l) AIC Table 4: ARFIMA estimation of factor loading series in levels from to φ and θ correspond to the AR and MA coefficients respectively. t-value of the estimated parameters in brackets, Ln(l) is the log-likelihood and (AIC) Akaike Information Criterion.

24 Data and Empirical Analysis 4-6 Absolute returns z1 z2 z3 ARFIMA (2, d, 2) (1, d, 5) (1, d, 2) d 0.30 ( 4.73) (-0.71) 0.24 ( 2.34) φ (-4.65) 0.89 ( 5.89) 0.57 ( 4.28) φ ( 0.02) θ ( 2.90) 0.06 (-0.31) (-2.11) θ (-1.09) (-5.59) (-8.40) θ (-0.61) θ (-0.40) θ (-0.07) const. 0.3 Ln(l) AIC Table 5: ARFIMA estimation of factor loading series in absolute returns from to φ and θ correspond to the AR and MA coefficients respectively. t-value of the estimated parameters in brackets, Ln(l) is the log-likelihood and (AIC) Akaike Information Criterion.

25 Data and Empirical Analysis 4-7 z1 z2 z3 z1 z2 z3 µ (11.810) (-0.257) (-0.241) (25.460) (22.660) (21.240) ω (2.379) (2.566) (7.990) (2.336) (3.363) (0.932) δ (4.281) (-1.364) (7.557) (2.693) (-1.344) (9.259) φ (1.540) (10.100) ( ) (0.636) (14.870) (5.356) β (3.278) (-1.111) ( ) (1.586) (-1.830) (14.400) ν (3.001) (11.320) (9.100) (6.016) (22.710) (18.180) Ln(l) Q 2 (24) [0.743] [1.000] [0.995] [0.961] [1.000] [0.999] Table 6: FIGARCH estimation in levels and absolute returns, (t statistics in parentheses). Significance is at 5% level. Estimation is with the Student distribution with ν degrees of freedom. Ln(l) is the value of the maximized likelihood. Q 2 (24) is the Box-Pierce statistic for remaining serial correlation in the squared standardized residuals using 24 s, ( p-values in square brackets). The critical value at significant level of 5% is 36.4.

26 Data and Empirical Analysis 4-8 z1 z2 z3 z1 z2 z3 µ (11.520) (0.703) (-2.061) (26.290) (20.230) (19.900) ω (-0.203) (5.055) (3.344) (1.885) (2.267) (2.120) d (1.164) (32.630) (13.220) (7.073) (46.840) (11.610) φ ( ) (9.161) ( ) (-0.256) (2.820) (1.663) β ( ) (0.037) (-1.791) (5.715) (4.431) (-0.776) ν (3.059) (7.918) (1.413) (5.538) (9.835) (8.027) log (α) (1.003) (1.586) (-7.471) (-1.699) (-2.553) (10.200) Ln(l) Q 2 (24) [0.610] [0.930] [0.999] [0.940] [1.000] [1.000] Table 7: HYGARCH estimation in levels and absolute returns, (t statistics in parentheses). Significance is at 5% level. ν is the degrees of freedom for Student-t distribution. log (α) is the log of weight α, to the difference operator (1 L) d. Ln(l) is the value of the maximized likelihood. Q 2 (24) is the Box-Pierce statistic for remaining serial correlation in the squared standardized residuals using 24 s, ( p-values in square brackets) with critical value of 36.4 at 5% significant level.

27 PcGive Graphics 14:01:43 31-Jan-2007 PcGive Graphics 14:17:27 31-Jan-2007 PcGive Graphics 19:01:26 25-Jan-2007 PcGive Graphics 19:08:58 25-Jan-2007 Data and Empirical Analysis 4-9 ARFIMA(5,0.29,4) fit z1 series ARFIMA(2,0.3,2) forecasts z1 series ARFIMA(3,0.53,3) fit z2 series ARFIMA(1,-0.32,5) forecasts z2 series ARFIMA(1,0.29,5) fit z3 series ARFIMA(1,0.24,2) forecasts z3 series Figure 3: Actual series (red) and in-sample fit (blue) for the estimated ARFIMA(p, d, q) model in levels and Figure 5: Actual series (red) and in-sample fit (blue) for the estimated ARF IMA(p, d, q) model in levels and absolute returns. Time interval from absolute returns. Time , interval from with , observations. with 1039 observations estimated models is that the F IGARCH model show better forecast per-

28 Data and Empirical Analysis FIGARCH (1,0.46,1) Cond. Var. Forecasts for z1 FIGARCH (1,0.29,1) Cond. Var. Forecasts for z Forecasting /31/07 16:59: Forecasting 01/28/07 11:05: FIGARCH (1,-0.03,1) Cond. Var. Forecasts for z2 FIGARCH (1,-0.04,1) Cond. Var. Forecasts for z G@RCH Forecasting 01/31/07 18:54: G@RCH Forecasting 02/06/07 21:19: FIGARCH (1,0.29,1) Cond. Var. Forecasts for z FIGARCH (1,0.98,1) Cond. Var. Forecasts for z Page: 1 of 1 Figure 4: Left-Right panels: FIGARCH conditional variance forecast for factor loading in levels and absolute Page: 1 of 1 returns. Time interval from , with 1039 observations.

29 Data and Empirical Analysis HYGARCH (1,0.89,1) Cond. Var. Forecasts for z G@RCH Forecasting 01/31/07 18:07: HYGARCH (1,0.97,1) Cond. Var. Forecasts for z G@RCH Forecasting 02/06/07 21:41: HYGARCH (1,0.98,1) Cond. Var. Forecasts for z2 HYGARCH (1,0.98,1) Cond. Var. Forecasts for z G@RCH Forecasting 01/28/07 10:03: G@RCH 0.04 Forecasting 01/28/07 11:42: HYGARCH (1,0.91,1) Cond. Var. Forecasts for z HYGARCH (1,0.001,1) Cond. Var. Forecasts for z Figure Page: 1 of 1 5: Left-Right panels: HYGARCH conditional variance Page: 1 of 1 forecast for factor loading in levels and absolute returns. Time interval from , with 1039 observations.

30 Data and Empirical Analysis 4-12 ARFIMA FIGARCH HYGARCH RMSE z z z z z z MAPE z z z z z z Table 8: In-sample performance of the five-step ahead forecast of the estimated ARFIMA, FIGARCH and HYGARCH models for the factor loading series in levels and absolute returns. The measures of forecast accuracy are the Root Mean Square Error (RMSE) and the Mean Absolute Prediction Error (MAPE).

31 Conclusion 5-1 Conclusion Factors of Implied volatility dynamics exhibit long range dependence in levels and absolute returns. The class of fractional integrated model can better describe the long-run behavior of the loading series in a flexible way, therefore should be taken into account when pricing DAX options.. Estimation of long memory models, FIGARCH(1, d, 1 and HYGARCH(1, d, 1) for the Factor loading series can be applied to risk measures such as Value at Risk, expected shortfall.

32 References 6-1 Baillie, R. T., Bollerslev, T., Mikkelson, H. Fractionally Integrated Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics, 14: 3 30, Beran, J. Statistics of long-memory processes Chapman & Hall, New York, Bollerslev, T. and Mikkelsen, H.O. Modeling and pricing long memory in stock market volatility, Journal of Econometics, 73: , 1996 Borak, S. and Härdle, W. & and Fengler, M. DSFM fitting of Implied Volatility Surfaces, Proceedings 5th International Conference on Intelligent System Design and Applications, IEEE Computer Soceity Number P2286, Library of Congress Number

33 References 6-2 Borak, S., Härdle, W., Mammen, E. and Park, B. U. Time series modeling with semiparametric factor dynamics, SFB Discussion Papers, Humboldt Universitä zu Berlin. Cheung, Y. W. and Lai, K. S. A search for long memory in international stock market returns. Journal of International Money and Finance, 14, , Chung, C. F. Estimating the Fractionally Integrated GARCH Model. working paper, National Taïwan Univerity, Davidson, J. Moments and memory properties of linear conditional heteroskedasticity models, and a new model. Journal of Business and Economic Statistics, 22, 16-29, 2004.

34 References 6-3 Diebold, F. X. and Inoue, A. Long memory and structural change Stern School of Business discussion paper, 1999 Doornik, J.A. and Ooms, M. A package for estimating, forecasting and simulating ARFIMA models: Arfima package 1.0 for Ox, Discussion paper, Nuffield College, Oxford. Elliott, G., Rothenberg, T. J & Stock, J. H. Efficient tests for an autoregressive unit root. Econometrica, 64: , 1996 Geweke, J.F. and Porter-Hudax, S. The estimation and application of long memory time series models. Journal of Time Series Analysis, 4(4): , 1983.

35 References 6-4 Giraitis, L., Kokoszka, P. and Leipus, R. Testing for long memory in the presence of a general trend. Journal of Applied Probability, 38: , Granger, C.W.J and Joyuex, R. An introduction to long memory time series models and fractional differencing. Journal of Time Series Analysis, 1: 15 39, Hall, P., Härdle, W., Kleinow, T. and Schmidt, P. On Semiparametric Bootstrap Approach to Hypothesis Tests and Confidence Intervals for Hurst Coefficients, Statistics of Stochastic Processes, 3: , Hosking, J.R.M. Fractional differencing, Biometrika, 68: , 1981.

36 References 6-5 Hurvich, C., Doe, R. and Brodsky, J. The Mean Square Error of Geweke and Porter-Hudak s estimator of the Memory Parameter of a Long-Memory time series, Journal of Time Series Analysis, 19: 19-46, Kwiatkowski, D., Peter C. B. Phillips, Peter Schmidt and Yongcheol Shin Testing the Null Hypothesis of Stationarity Against the Alternative of a Unit Root Journal of Econometrics, 54: , Liu, M. Modelling long memory in stock market volatility, Journal of Econometrics, 99: , Lobato, I and Robinson,P.M A Nonparametric Test for I (0) Review of Economic Studies,forthcoming. Lobato, I. and Savin, N.E

37 References 6-6 MacKinnon, J. G. Critical values for cointegration tests, in Granger, C. W. J. and Engle, R. F. (eds), Long-Run Economic Relationships, Oxford University Press, Oxford, Mandelbrot, B.B. When a price be arbitraged efficiently? A limit to the validity of the random walk and Martingale models. Review of Economics and Statistics, 53, , Mandelbrot, B.B. The fractal geometry of nature, Freeman, New York, McLeod, A.I. and Hipel, K.W. Preservation of the rescaled adjusted range, 1. A reassessment of the Hurst phenomenon. Water Resources Research, 14: , 1978.

38 References 6-7 Newey, W. and West, K. Automatic Lag Selection in Covariance Matrix Estimation Review of Economic Studies, 61: , Robinson, P.M. Gaussian Semiparametric Estimation of Long-Range Dependence. The Annals of Statistics, 23: , Sibbertsen, P. Long-memory in volatilities of German stock returns Empirical Economics, 29, Wilson, P. J., and Okunev, J. Long-Term Dependencies and Long Run Non-Periodic co-cycles: Real Estate and Stock Markets. Journal of Real Estate Research, 18, , 1999.

39 Appendix 7-1 Long Range Dependence (LRD) Series Memory Mean Variance ACF reversion 0.5 < d < 0 fractional antipersistent finite hyperbolic integrated d = 0 stationary short finite exponential 0 < d < 0.5 fractional long finite hyperbolic integrated 0.5 d < 1 fractional long infinite hyperbolic integrated d = 1 integrated infinite x infinite linear Table 9: Time series long memory characteristics.

40 Appendix 7-2 Unit root tests The Augmented Dickey-Fuller (ADF) test refers to the regression equation p z t,k = φz t 1,k + α i z t i,k + ε t,k, (13) where p is the number of s of z t,k by which the regression equation (13) is augmented in order to get residuals free of autocorrelation. The size of the test is better when p is large but causes the test to lose power. Under H 0, the unit root the parameter φ should be zero. Hence, the t-statistic of the OLS estimator of φ is used as the ADF test statistic. i=1

41 Appendix 7-3 Unit Root Tests 2 The KPSS test is based on the statistic ν b = T 2 T t=1 S t 2 ω b 2, (14) where S t = t i=1 êt and ê t are residuals from a regression of the time series on a constant. ω 2 b is the spectral density estimator for ê t at frequency zero. Under the null hypothesis the time series is assumed to be stationary

42 Appendix 7-4 The limiting distribution of the test statistic is nonstandard. Critical or p-values have to be derived by the help of simulation methods. The critical values (Mackinnon, J.G 1991) are 2.57 (10%), 2.86 (5%), and 3.44 (1%). Lag order p is determined by the AIC, HQ, and SC information criteria. ADF test suffers from low power, therefore may fail to detect a stationary time series

43 Appendix 7-5 Unit root test statistics Series ADF-AIC ˆp ADF-HQ ˆp ERS-AIC ˆb ERS-HQ ˆb z t [0.29] [0.19] z t [0.01] [0.01] z t [0.05] [0.05] Table 10: ADF-AIC and ADF-HQ refer to ADF tests using AIC and HQ criteria respectively to estimate length p. ERS-AIC and ERS-SC criteria used, refer to the length b chosen for the estimation regression of the autoregressive spectral density estimator. Critical values for ADF test are 2.57 (10%), 2.86 (5%), and 3.44 (1%) (see, [MacKinnon (1991)]). The p-values for the ADF tests are given in brackets. Critical values for ERS test (see, [Elliott, Rothenberg and Stock (1996)]) are 4.48 (10%), 3.26 (5%) and 1.99 (1%)., and denote significance at 1%, 5%, and 10% level respectively.

44 Appendix 7-6 Point-optimal unit root test: (ERS) Elliott, Rothenberg and Stock (1996). Superior to ADF in case of processes affected by conditional heteroscedasticity. Test is based on quasi-differences of z t,k which are defined by { 1 if t = 1 d(z t,k a) = z t,k az t 1,k if t > 1, a is the point alternative against which the null of a unit root is tested. Following the suggestion of Elliott et al. (1996), we use a = ā = 1 7/T since only a constant term is considered.

45 Appendix 7-7 Let ê t be the residuals from a regression of the time series on a quasi-differenced constant and let S(ā) and S(1) be the sums of squared residuals for the cases a = ā and a = 1 respectively. Then the test is defined by ERS = (S(a) as(1))/ˆω b, (15) where ˆω b is the spectral density estimator of ê t at frequency zero. We apply the autoregressive spectral density estimator as proposed by Elliott et al. (1996). Critical values are 4.48 (10%), 3.26 (5%) and 1.99 (1%).

46 Appendix 7-8 Parameter Estimation Level z1 z2 z3 ARFIMA (5, d, 4) (2, d, 1) (5, d, 3) d 0.14 ( 1.05) 0.45 ( 6.94) 0.01 ( 0.05) φ ( 0.99) 0.87 (10.60) 0.47 ( 1.78) φ (-0.07) 0.05 ( 1.02) 0.26 ( 4.31) φ ( 1.84) 0.92 ( 4.39) φ ( 0.06) (-2.12) φ (-1.08) (-2.48) θ (0.62) (-10.9 ) 0.23 (2.25) θ (1.59) 0.13 (1.02) θ (-2.05) (-6.54) θ (-1.15) const InL AIC Table 11: Maximum likelihood estimation of ARFIMA model for series in level from to φ is the AR coefficient and θ is the coefficients to the moving average part. t-value of the estimated parameters in brackets, InL is the log-likelihood and (AIC) Akaike Information Criterion. AIC = 2l + 2s (l is the log-likelihood and s = p + q + 1 is the number of estimated parameters)

47 Appendix 7-9 Absolute returns z1 z2 z3 ARFIMA (2, d, 2) (1, d, 5) (1, d, 2) d 0.29 ( 4.96) (-0.77) 0.24 ( 2.51) φ (-4.57) 0.92 ( 10.80) 0.56 ( 2.51) φ (-0.09) θ ( 3.01) 0.02 ( 0.07) (-1.83) θ (-0.93) (-2.99) (-7.15) θ (-0.96) θ (-0.67) θ (-0.15) const. InL AIC Table 12: Maximum likelihood estimation of ARFIMA model for absolute returns of the factor loadings z 1, z 2 and z 3 from to The φ coefficients correspond to the autoregressive part and the θ coefficients to the moving average part. t-value of the estimated parameters in brackets, InL is the log-likelihood and (AIC) Akaike Information Criterion. AIC = 2l + 2s (l is the log-likelihood and s = p + q + 1 is the number of estimated parameters)