1. DISSERTATION PAPER Modeling and Forecasting the Volatility of the EUR/ROL Exchange Rate Using GARCH Models. Student :Becar Iuliana Student :Becar Iuliana Supervisor: Professor Moisa Altar

2. Table of Contents The importance of forecasting exchange rate volatility. The importance of forecasting exchange rate volatility. Data description. Data description. Model estimates and forecasting performances. Model estimates and forecasting performances. Concluding remarks. Concluding remarks.

3. Why model and forecast volatility? Volatility is one of the most important concepts in the whole of finance. Volatility is one of the most important concepts in the whole of finance. ARCH models offered new tools for measuring risk, and its impact on return. ARCH models offered new tools for measuring risk, and its impact on return. Volatility of exchange rates is of importance because of the uncertainty it creates for prices of exports and imports, for the value of international reserves and for open positions in foreign currency. Volatility of exchange rates is of importance because of the uncertainty it creates for prices of exports and imports, for the value of international reserves and for open positions in foreign currency.

4. Volatility Models. ARCH/GARCH models. ARCH/GARCH models. Engle(1982) Engle(1982) Bollerslev(1986) Bollerslev(1986) Baillie, Bollerslev and Mikkelsen (1996) Baillie, Bollerslev and Mikkelsen (1996) ARFIMA models. ARFIMA models. Granger (1980) Granger (1980)

5. Data description  Data series: nominal daily EUR/ROL exchange rates  Time length: 04:01:1999-11:06:2004  1384 nominal percentage returns

6. Descriptive Statistics for the return series. Statistict-TestP-Value Skewness1.047215.9224.4605e-057 Excess Kurtosis 8.513864.7690.00000 Jarque- Bera 4432.9

7. Heteroscedasticity Autocorrelation and Partial autocorrelation of the Return Series The Daily Return Series  The returns are not homoskedastic.  Low serial dependence in returns.  The Ljung-Box statistic for 20 lags equals 27.392 [0.125].

8. Autocorrelation and Partial Autocorrelation of Squared Returns ARCH 1 test: 17.955 [0.0000]** ARCH 2 test: 18.847 [0.0000]** The Ljung-Box statistic for 20 lags equals 151.01[0.000]

9. Stationarity Unit Root Tests for EUR/ROL return series. ADF Test Statistic -35.60834 1% Critical Value* -3.4380 5% Critical Value -2.8641 10% Critical Value -2.5681 *MacKinnon critical values for rejection of hypothesis of a unit root. PP Test Statistic -35.57805 1% Critical Value* -3.4380 5% Critical Value -2.8641 10% Critical Value -2.5681 *MacKinnon critical values for rejection of hypothesis of a unit root.

10. Model estimates and forecasting performances. Methodology. Methodology. Ox Professional 3.30 G@RCH4.0 Ox Professional 3.30 G@RCH4.0G@RCH4.0 4.01.1999-30.12.2002 (1018 observations) for model estimation 4.01.1999-30.12.2002 (1018 observations) for model estimation 06.01.2003-11.06.2004 (366 observations) for out of sample forecast evaluation. 06.01.2003-11.06.2004 (366 observations) for out of sample forecast evaluation. The Models. The Models. Two distributions: Student, Skewed Student, QMLE. The Mean Equations: The Mean Equations: 1. A constant mean 1. A constant mean 2. An ARFIMA(1,d a,0) mean 2. An ARFIMA(1,d a,0) mean 3. An ARFIMA(0, d a,1) mean 3. An ARFIMA(0, d a,1) mean

11. The variance equations. GARCH(1,1) and FIGARCH(1,d,1) without the constant term and with a non-trading day dummy variable. GARCH(1,1) and FIGARCH(1,d,1) without the constant term and with a non-trading day dummy variable. The estimated twelve models. Examining the models page 30 to 34 the conclusions are: Examining the models page 30 to 34 the conclusions are: The estimated coefficients are significantly different from zero at the 10% level. The estimated coefficients are significantly different from zero at the 10% level. the ARFIMA coefficient lies between the ARFIMA coefficient lies between which implies stationarity. which implies stationarity. all variance coefficients are positive and all variance coefficients are positive and

12. In-sample model evaluation. Residual tests. GARCH models. ModelSBCSkewnessEK 1 Q*Q2**ARCH***Nyblom ARMA (0,0) GARCH(1,1) Skewed-Student 2.2104630.752243.954337.5958 [0.9019571] 30.3204 [0.9783154] 1.1358 [0.3395] 1.96933 ARMA (0,0) GARCH(1,1) Student 2.2129010.740333.831937.5877 [0.9021277] 30.3145 [0.9783579] 1.1238 [0.3458] 1.58334 ARFIMA (1,d,0) GARCH(1,1) Skewed-Student 2.2145790.760244.102836.4188 [0.9083405] 31.7529 [0.9659063] 1.2484 [0.2843] 2.24209 ARFIMA (1,d,0) GARCH(1,1) Student 2.2163880.733533.85736.0009 [0.9165657] 31.8411 [0.9649974] 1.1801 [0.3169] 1.89543 ARFIMA (0,d,1) GARCH(1,1) Skewed-Student 2.2157350.759094.115336.1425 [0.9138359] 31.3112 [0.9701942] 1.2084 [0.3030] 2.2612 ARFIMA (0,d,1) GARCH(1,1) Student 2.2174010.733903.885235.8043 [0.9202571] 31.3087 [0.9702172] 1.1360 [0.3394] 1.9047 1 EK-Excess Kurtosis;* Q-Statistics on Standardized Residuals with 50 lags; ** Q-Statistics on Squared Standardized Residuals 50 lags; *** ARCH test with 5 lags; P-values in brackets.

13. In-sample model evaluation. Residual tests. FIGARCH models. ModelSBCSkewnessEK 1 Q*Q2**ARCH***Nyblom ARMA (0,0) FIGARCH(1,d,1) Skewed-Student 2.2220890.763053.872337.4681 [0.9046084] 28.4572 [0.9888560] 1.2601 [0.2790] 1.56799 ARMA (0,0) FIGARCH(1,d,1) Student* 2.224720.746983.731337.7303 [0.8991133] 28.9803 [0.9864387] 1.3297 [0.2491] 1.37719 ARFIMA (1,d,0) FIGARCH(1,d,1) Skewed-Student 2.2265490.7573.924236.3540 [0.9096502] 29.8994 [0.9811947] 1.3204 [0.2529] 2.05757 ARFIMA (1,d,0) FIGARCH(1,d,1) Student 2.2283340.733783.725636.1801 [0.9131002] 30.4315 [0.9775013] 1.3272 [0.2501] 1.82764 ARFIMA (0,d,1) FIGARCH(1,d,1) Skewed-Student 2.2275160.759013.9636.2611 [0.9115043] 29.2088 [0.9852596] 1.2729 [0.2733] 2.0233 ARFIMA (0,d,1) FIGARCH(1,d,1) Student 2.2291990.737993.781336.1313 [0.9140531] 29.5586 [0.9832983] 1.2630 [0.2777] 1.79097 1 EK-Excess Kurtosis;* Q-Statistics on Standardized Residuals with 50 lags; ** Q-Statistics on Squared Standardized Residuals 50 lags; *** ARCH test with 5 lags; P-values in brackets.

14. Out-of-sample Forecast Evaluation Forecast methodology Forecast methodology - sample window: 1018 observations - sample window: 1018 observations - at each step, the 1 step ahead dynamic forecast is stored - at each step, the 1 step ahead dynamic forecast is stored for the conditional variance and the conditional mean for the conditional variance and the conditional mean -dynamic forecast is programmed in OxEdit -dynamic forecast is programmed in OxEdit G@RCH3.0 package G@RCH3.0 packageG@RCH3.0 Benchmark: ex-post volatility = squared returns. Benchmark: ex-post volatility = squared returns.

15. Measuring Forecast Accuracy. The Mincer-Zarnowitz regression: The Mincer-Zarnowitz regression: The Mean Absolute Error: The Mean Absolute Error: Root Mean Square Error (standard error): Root Mean Square Error (standard error): Theil's inequality coefficient -Theil's U: Theil's inequality coefficient -Theil's U:

16. One Step Ahead Forecast Evaluation Measures. ModelalfabetaR2R2 ModelalfabetaR2R2 ARMA (0,0) GARCH(1,1) Skewed-Student -0.104961 [0.0699] 0.624769 [0.0006] 0.0533211ARMA (0,0) FIGARCH(1,d,1) Skewed-Student -0.038611 [0.3070] 0.741465 [0.0005] 0.0822328 ARMA (0,0) GARCH(1,1) Student -0.100843 [0.0766] 0.617284 [0.0007] 0.0530545ARMA (0,0) FIGARCH(1,d,1) Student -0.037921 [0.3143] 0.725906 [0.0005] 0.0793558 ARFIMA (1,d,0) GARCH(1,1) Skewed-Student -0.112153 [0.0607] 0.631864 [0.0006] 0.0518779ARFIMA (1,d,0) FIGARCH(1,d,1) Skewed-Student -0.046087 [0.2517] 0.730264 [0.0006] 0.0759213 ARFIMA (1,d,0) GARCH(1,1) Student -0.104983 [0.0698] 0.620363 [0.0006] 0.0522936ARFIMA (1,d,0) FIGARCH(1,d,1) Student -0.043940 [0.2681] 0.707455 [0.0006] 0.0735089 ARFIMA (0,d,1) GARCH(1,1) Skewed-Student -0.112613 [0.0596] 0.634110 [0.0006] 0.052295ARFIMA (0,d,1) FIGARCH(1,d,1) Skewed-Student -0.045701 [0.254] 0.731791 [0.0006] 0.0765561 ARFIMA (0,d,1) GARCH(1,1) Student -0.105667 [0.0680] 0.623092 [0.0006] 0.0527494ARFIMA (0,d,1) FIGARCH(1,d,1) Student -0.043431 [0.2715] 0.70931 [0.0006] 0.0742364 1. The Mincer-Zarnowitz regression

17. 2. Forecasting the conditional mean. Loss functions. ModelMAERMSETICModelMAERMSETIC ARMA (0,0) GARCH(1,1) Skewed-Student 0.26010.34120.7895ARMA (0,0) FIGARCH(1,d,1) Skewed-Student 0.26060.34160.7861 ARMA (0,0) GARCH(1,1) Student 0.25760.33950.812ARMA (0,0) FIGARCH(1,d,1) Student 0.2580.33970.8086 ARFIMA (1,d,0) GARCH(1,1) Skewed-Student 0.27240.35210.7527ARFIMA (1,d,0) FIGARCH(1,d,1) Skewed-Student 0.27260.35220.7518 ARFIMA (1,d,0) GARCH(1,1) Student 0.26940.34930.77ARFIMA (1,d,0) FIGARCH(1,d,1) Student 0.26970.34960.7684 ARFIMA (0,d,1) GARCH(1,1) Skewed-Student 0.27220.3520.7548ARFIMA (0,d,1) FIGARCH(1,d,1) Skewed-Student 0.27240.35220.7536 ARFIMA (0,d,1) GARCH(1,1) Student 0.26910.34930.7729ARFIMA (0,d,1) FIGARCH(1,d,1) Student 0.26940.34950.7711

18. 3. Forecasting the conditional variance. Loss functions. ModelMAERMSETICModelMAERMSETIC ARMA (0,0) GARCH(1,1) Skewed-Student 0.28440.31480.5253ARMA (0,0) FIGARCH(1,d,1) Skewed-Student 0.170.22340.484 ARMA (0,0) GARCH(1,1) Student 0.28240.31310.5244ARMA (0,0) FIGARCH(1,d,1) Student 0.17260.22530.4845 ARFIMA (1,d,0) GARCH(1,1) Skewed-Student 0.29070.32040.5286ARFIMA (1,d,0) FIGARCH(1,d,1) Skewed-Student 0.18020.22990.4856 ARFIMA (1,d,0) GARCH(1,1) Student 0.28660.31680.5265ARFIMA (1,d,0) FIGARCH(1,d,1) Student 0.18320.23220.4861 ARFIMA (0,d,1) GARCH(1,1) Skewed-Student 0.29030.320.5283ARFIMA (0,d,1) FIGARCH(1,d,1) Skewed-Student 0.17940.22940.4854 ARFIMA (0,d,1) GARCH(1,1) Student 0.28620.31640.5263ARFIMA (0,d,1) FIGARCH(1,d,1) Student 0.18220.23150.4859

19. Concluding remarks. In-sample analysis: In-sample analysis: Residual tests: Residual tests: -all models may be appropriate. -all models may be appropriate. -the Student distribution is better than the Skewed Student. -the Student distribution is better than the Skewed Student. Out-of-sample analysis: Out-of-sample analysis: -the FIGARCH models are superior. -the FIGARCH models are superior. -for the conditional mean the Student distribution is -for the conditional mean the Student distribution is superior. superior. -the two ARFIMA mean equations don't provide a better -the two ARFIMA mean equations don't provide a better forecast of the conditional mean. forecast of the conditional mean. - for the conditional variance the Skewed Student - for the conditional variance the Skewed Student distribution is superior. distribution is superior.

20. Concluding remarks. Model construction problems; Model construction problems; Further research: Further research: -option prices, which reflect the market’s expectation -option prices, which reflect the market’s expectation of volatility over the remaining life span of the option. of volatility over the remaining life span of the option. -daily realized volatility can be computed as the sum of -daily realized volatility can be computed as the sum of squared intraday returns squared intraday returns

21. Bibliography Alexander, Carol (2001) – Market Models - A Guide to Financial Data Analysis, John Wiley &Sons, Ltd.; Alexander, Carol (2001) – Market Models - A Guide to Financial Data Analysis, John Wiley &Sons, Ltd.; Andersen, T. G. and T. Bollerslev (1997) - Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts, International Economic Review; Andersen, T. G. and T. Bollerslev (1997) - Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts, International Economic Review; Andersen, T. G., T. Bollerslev, Francis X. Diebold and Paul Labys (2000)- Modeling and Forecasting Realized Volatility, the June 2000 Meeting of the Western Finance Association. Andersen, T. G., T. Bollerslev, Francis X. Diebold and Paul Labys (2000)- Modeling and Forecasting Realized Volatility, the June 2000 Meeting of the Western Finance Association. Andersen, T. G., T. Bollerslev and Francis X. Diebold (2002)- Parametric and Nonparametric Volatility Measurement, Prepared for Yacine Aït-Sahalia and Lars Peter Hansen (eds.), Handbook of Financial Econometrics, North Holland. Andersen, T. G., T. Bollerslev and Francis X. Diebold (2002)- Parametric and Nonparametric Volatility Measurement, Prepared for Yacine Aït-Sahalia and Lars Peter Hansen (eds.), Handbook of Financial Econometrics, North Holland. Andersen, T. G., T. Bollerslev and Peter Christoffersen (2004)-Volatility Forecasting, Rady School of Management at UCSD Andersen, T. G., T. Bollerslev and Peter Christoffersen (2004)-Volatility Forecasting, Rady School of Management at UCSD Baillie, R.T., Bollerslev T., Mikkelsen H.O. (1996)- Fractionally Integrated Generalized Autoregressive Conditional Heteroskedasticity, Journal of Econometrics, Vol. 74, No.1, pp. 3-30. Baillie, R.T., Bollerslev T., Mikkelsen H.O. (1996)- Fractionally Integrated Generalized Autoregressive Conditional Heteroskedasticity, Journal of Econometrics, Vol. 74, No.1, pp. 3-30. Bollerslev, Tim, Robert F. Engle and Daniel B. Nelson (1994)– ARCH Models, Handbook of Econometrics, Volume 4, Chapter 49, North Holland; Bollerslev, Tim, Robert F. Engle and Daniel B. Nelson (1994)– ARCH Models, Handbook of Econometrics, Volume 4, Chapter 49, North Holland; Diebold, Francis and Marc Nerlove (1989)-The Dynamics of Exchange Rate Volatility: A Multivariate Latent factor Arch Model, Journal of Applied Econometrics, Vol. 4, No.1. Diebold, Francis and Marc Nerlove (1989)-The Dynamics of Exchange Rate Volatility: A Multivariate Latent factor Arch Model, Journal of Applied Econometrics, Vol. 4, No.1. Diebold, Francis and Jose A. Lopez (1995)- Forecast Evaluation and Combination, Prepared for G.S. Maddala and C.R. Rao (eds.), Handbook of Statistics, North Holland. Diebold, Francis and Jose A. Lopez (1995)- Forecast Evaluation and Combination, Prepared for G.S. Maddala and C.R. Rao (eds.), Handbook of Statistics, North Holland. Enders W. (1995)- Applied Econometric Time Series, 1st Edition, New York: Wiley. Enders W. (1995)- Applied Econometric Time Series, 1st Edition, New York: Wiley.

22. Bibliography Engle, R.F. (1982) – Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation, Econometrica, 50, pp. 987-1007; Engle, R.F. (1982) – Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation, Econometrica, 50, pp. 987-1007; Engle, R.F. and Victor K. Ng (1993) – Measuring and Testing the Impact of News on Volatility, The Journal of Finance, Vol. XLVIII, No. 5; Engle, R.F. and Victor K. Ng (1993) – Measuring and Testing the Impact of News on Volatility, The Journal of Finance, Vol. XLVIII, No. 5; Engle, R. (2001) – Garch 101: The Use of ARCH/GARCH Models in Applied Econometrics, Journal of Economic Perspectives – Volume 15, Number 4 – Fall 2001 – Pages 157-168; Engle, R. (2001) – Garch 101: The Use of ARCH/GARCH Models in Applied Econometrics, Journal of Economic Perspectives – Volume 15, Number 4 – Fall 2001 – Pages 157-168; Engle, R. and A. J. Patton (2001) – What good is a volatility model?, Research Paper, Quantitative Finance, Volume 1, 237-245; Engle, R. and A. J. Patton (2001) – What good is a volatility model?, Research Paper, Quantitative Finance, Volume 1, 237-245; Engle, R. (2001) – New Frontiers for ARCH Models, prepared for Conference on Volatility Modelling and Forecasting, Perth, Australia, September 2001; Engle, R. (2001) – New Frontiers for ARCH Models, prepared for Conference on Volatility Modelling and Forecasting, Perth, Australia, September 2001; Hamilton, J.D. (1994) – Time Series Analysis, Princeton University Press; Hamilton, J.D. (1994) – Time Series Analysis, Princeton University Press; Lopez, J.A.(1999) – Evaluating the Predictive Accuracy of Volatility Models, Economic Research Deparment, Federal Reserve Bank of San Francisco; Lopez, J.A.(1999) – Evaluating the Predictive Accuracy of Volatility Models, Economic Research Deparment, Federal Reserve Bank of San Francisco; Peters, J. and S. Laurent (2001) – A Tutorial for G@RCH 2.3, a Complete Ox Package for Estimating and Forecasting ARCH Models; Peters, J. and S. Laurent (2001) – A Tutorial for G@RCH 2.3, a Complete Ox Package for Estimating and Forecasting ARCH Models; Peters, J. and S. Laurent (2002) – A Tutorial for G@RCH 2.3, a Complete Ox Package for Estimating and Forecasting ARCH Models; Peters, J. and S. Laurent (2002) – A Tutorial for G@RCH 2.3, a Complete Ox Package for Estimating and Forecasting ARCH Models; West, Kenneth and Dongchul Cho (1994)-The Predictive Ability of Several Models of Exchange Rate Volatility, NBER Technical Working Paper #152. West, Kenneth and Dongchul Cho (1994)-The Predictive Ability of Several Models of Exchange Rate Volatility, NBER Technical Working Paper #152.

23. Appendix 1. The ARMA (0, 0), GARCH (1, 1) Skewed Student model. Robust Standard Errors (Sandwich formula) CoefficientStd.Errort-valueProbability Constant(Mean)0.0919300.0216134.2530.0000 dummyFriday (V)0.0489770.0197812.4760.0134 ARCH(Alpha1)0.0360760.0115613.1210.0019 GARCH(Beta1)0.9244900.01805251.210.0000 Asymmetry0.1457220.0472503.0840.0021 Tail9.8722133.34882.9480.0033 For more details see Appendix 1, page 45.

24. Appendix 2 The ARMA (0, 0), GARCH (1, 1) Student model. Robust Standard Errors (Sandwich formula) CoefficientStd.Errort-alueProbability Constant(Mean)0.0777950.0216733.5890.0003 dummyFriday (V)0.0492400.0201632.4420.0148 ARCH(Alpha1)0.0371860.0119753.1050.0020 GARCH(Beta1)0.9233530.01847949.970.0000 Student(DF)8.9213402.81193.1730.0016 For more details, see Appendix 2, page 47.

25. Appendix 3 The ARFIMA (1, d a, 0),GARCH (1, 1) Skewed Student model. Robust Standard Errors (Sandwich formula) CoefficientStd.Errort-valueProbability Constant(Mean)0.0899390.0105278.5440.0000 d-Arfima-0.1282240.045067-2.8450.0045 AR(1)0.1232690.0545532.2600.0241 dummyFriday (V)0.0488600.0197032.4800.0133 ARCH(Alpha1)0.0338970.0116772.9030.0038 GARCH(Beta1)0.9262830.01809651.190.0000 Asymmetry0.1397710.0471942.9620.0031 Tail9.1895232.90913.1590.0016 For more details, see Appendix 3, page 49.

26. Appendix 4 The ARFIMA (1, d a, 0),GARCH (1, 1) Student model. Robust Standard Errors (Sandwich formula) CoefficientStd.Errort-valueProbabilty Constant(Mean)0.0827110.0102378.0800.0000 d-Arfima-0.1363170.045875-2.9710.0030 AR(1)0.1404550.0558322.5160.0120 dummyFriday (V)0.0496350.0201172.4670.0138 ARCH(Alpha1)0.0365170.0125102.9190.0036 GARCH(Beta1)0.9235030.01860249.640.0000 Student(DF)8.4368092.52573.3400.0009 For more details, see Appendix 4, page 52.

27. Appendix 5 The ARFIMA (0, d a,1),GARCH (1, 1) Skewed Student model. Robust Standard Errors (Sandwich formula) CoefficientStd.Errort-valueProbability Constant(Mean)0.0904150.0110418.1890.0000 d-Arfima-0.1177570.037429-3.1460.0017 MA(1)0.1148440.0460602.4930.0128 dummyFriday (V)0.0486810.0197872.4600.0140 ARCH(Alpha1)0.0338470.0116412.9080.0037 GARCH(Beta1)0.9264140.01817250.980.0000 Asymmetry0.1386310.0470492.9470.0033 Tail9.2793062.96133.1340.0018 For more details, see Appendix 5, page 54.

28. Appendix 6 The ARFIMA (0, d a,1),GARCH (1, 1) Student model. Robust Standard Errors (Sandwich formula) CoefficientStd.Errort-valueProbability Constant(Mean)0.0828220.0108337.6450.0000 d-Arfima-0.1225190.036843-3.3250.0009 MA(1)0.1283110.0451462.8420.0046 dummyFriday (V)0.0493800.0202072.4440.0147 ARCH(Alpha1)0.0363440.0124492.9190.0036 GARCH(Beta1)0.9237880.01870349.390.0000 Student(DF)8.5164292.56893.3150.0009 For more details, see Appendix 6, page 56.

29. Appendix 7 The ARMA (0, 0), FIGARCH-BBM (1,d,1) Skewed Student model. Robust Standard Errors (Sandwich formula) CoefficientStd.Errort-valueProbability Constant(Mean)0.0942590.0219314.2980.0000 dummyFriday (V)0.0472780.0259751.8200.0690 d-Figarch0.3586220.0988993.6260.0003 ARCH(Alpha1)0.2888960.0945983.0540.0023 GARCH(Beta1)0.6353090.05851310.860.0000 Asymmetry0.1475880.0465293.1720.0016 Tail9.5450313.09643.0830.0021 For more details, see Appendix 7, page 59.

30. Appendix 8 The ARMA (0, 0), FIGARCH-BBM (1,d,1) Student model. Robust Standard Errors (Sandwich formula) CoefficientStd.Errort-valueProbability Constant(Mean)0.0798070.0219153.6420.0003 dummyFriday (V)0.0493100.0279261.7660.0777 d-Figarch0.3514480.105063.3450.0009 ARCH(Alpha1)0.3120180.110262.8300.0047 GARCH(Beta1)0.6448420.05758011.200.0000 Student(DF)8.5968052.60443.3010.0010 For more details, see Appendix 8, page 61.

31. Appendix 9 The ARFIMA (1,d a,0), FIGARCH-BBM (1,d,1) Skewed Student model. Robust Standard Errors (Sandwich formula) CoefficientStd.Errort-valueProbability Constant(Mean)0.0904000.0107198.4340.0000 d-Arfima-0.1267240.046241-2.7410.0062 AR(1)0.1193640.0547452.1800.0295 dummyFriday (V) 0.0521640.0307871.6940.0905 d-Figarch0.3320740.106623.1150.0019 ARCH(Alpha1)0.3392920.136422.4870.0130 GARCH(Beta1)0.6496200.05377912.080.0000 Asymmetry0.1395010.0466382.9910.0028 Tail8.8712592.68403.3050.0010 For more details, see Appendix 9, page 63.

32. Appendix 10 The ARFIMA (1,d a,0), FIGARCH-BBM (1,d,1) Student model. Robust Standard Errors (Sandwich formula) CoefficientStd.Errort-valueProbability Constant(Mean)0.0832210.0102638.1090.0000 d-Arfima-0.1362700.047181-2.8880.0040 AR(1)0.1384940.0562082.4640.0139 dummyFriday (V)0.0545620.0340151.6040.1090 d-Figarch0.3285450.122912.6730.0076 ARCH(Alpha1)0.3603470.171552.1010.0359 GARCH(Beta1)0.6599660.05799711.380.0000 Student(DF)8.0935512.32263.4850.0005 For more details, see Appendix 10, page 66.

33. Appendix 11 The ARFIMA (0,d a,1), FIGARCH-BBM (1,d,1) Skewed Student model. Robust Standard Errors (Sandwich formula) CoefficientStd.Errort-valueProbability Constant(Mean)0.0909380.0112028.1180.0000 d-Arfima-0.1170930.039118-2.9930.0028 MA(1)0.1123120.0471842.3800.0175 dummyFriday (V)0.0517240.0303271.7060.0884 d-Figarch0.3327590.103973.2000.0014 ARCH(Alpha1)0.3343400.127652.6190.0089 GARCH(Beta1)0.6471350.05282212.250.0000 Asymmetry0.1386590.0469252.9550.0032 Tail8.9737442.74383.2700.0011 For more details, see Appendix 11, page 68.

34. Appendix 12 The ARFIMA (0,d a,1), FIGARCH-BBM (1,d,1) Student model. Robust Standard Errors (Sandwich formula) CoefficientStd.Errort-valueProbability Cst(M)0.0834340.0108707.6750.0000 d-Arfima-0.1226200.038276-3.2040.0014 MA(1)0.1268870.0459252.7630.0058 dummyFriday (V)0.0540600.0331551.6310.1033 d-Figarch0.3295790.117652.8010.0052 ARCH(Alpha1)0.3534420.156612.2570.0242 GARCH(Beta1)0.6566300.05586711.750.0000 Student(DF)8.1822062.36953.4530.0006 For more details, see Appendix 12, page 70.

35. Augmented Dickey-Fuller Test Equation Dependent Variable: D(RETURNS) Method: Least Squares Date: 06/26/04 Time: 07:50 Sample(adjusted): 3 1384 Included observations: 1382 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. RETURNS(-1)-0.9572620.026883-35.608340.0000 C0.0783920.0182644.2921480.0000 R-squared0.478843 Mean dependent var-0.000589 Adjusted R-squared0.478465 S.D. dependent var0.933223 S.E. of regression0.673949 Akaike info criterion2.050121 Sum squared resid626.8057 Schwarz criterion2.057692 Log likelihood-1414.634 F-statistic1267.954 Durbin-Watson stat1.994863 Prob(F-statistic)0.000000 Stationarity tests. Appendix 13. 1. Dickey-Fuller Test.

36. ADF Test -17.256751% Critical Value* -3.4380 5% Critical Value -2.8641 10% Critical Value -2.5681 *MacKinnon critical values for rejection of hypothesis of a unit root. Dependent Variable: D(RETURNS) Method: Least Squares Sample(adjusted): 7 1384 Included observations: 1378 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. RETURNS(-1)-1.0471830.060682-17.256750.0000 D(RETURNS(-1))0.0913190.0539271.6933960.0906 D(RETURNS(-2))0.0393790.0461660.8529890.3938 D(RETURNS(-3))0.0096350.0373190.2581860.7963 D(RETURNS(-4))0.0153330.0269670.5685850.5697 C0.0866840.0188354.6023990.0000 R-squared0.480683 Mean dependent var0.000495 Adjusted R-squared0.478791 S.D. dependent var0.933787 S.E. of regression0.674146 Akaike info criterion2.053604 Sum squared resid623.5364 Schwarz criterion2.076369 Log likelihood-1408.933 F-statistic253.9867 Durbin-Watson stat1.998880 Prob(F-statistic)0.000000 Appendix 14. ADF Test.

37. Appendix 15.Phillips-Perron Test. Lag truncation for Bartlett kernel: 7 ( Newey-West suggests: 7 ) Residual variance with no correction0.453550 Residual variance with correction0.407637 Phillips-Perron Test Equation Dependent Variable: D(RETURNS) Method: Least Squares Sample(adjusted): 3 1384 Included observations: 1382 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. RETURNS(-1)-0.9572620.026883-35.608340.0000 C0.0783920.0182644.2921480.0000 R-squared0.478843 Mean dependent var-0.000589 Adjusted R-squared0.478465 S.D. dependent var0.933223 S.E. of regression0.673949 Akaike info criterion2.050121 Sum squared resid626.8057 Schwarz criterion2.057692 Log likelihood-1414.634 F-statistic1267.954 Durbin-Watson stat1.994863 Prob(F-statistic)0.000000