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DISSERTATION PAPERDISSERTATION PAPER
Modeling and Modeling and FForecasting the orecasting the VVolatility of the olatility of the EUREUR//ROLROL EExchange xchange RRate ate UUsing sing GARCHGARCH
MModels.odels.
Student :Student :Becar IulianaBecar IulianaSupervisor: Professor Supervisor: Professor MoisaMoisa Altar Altar
Table of ContentsTable of Contents
• The importance of forecasting exchange rate The importance of forecasting exchange rate
volatility.volatility.
• Data description.Data description.
• Model estimates and forecasting Model estimates and forecasting
performances.performances.
• Concluding remarks.Concluding remarks.
Why model and forecast volatility?Why model and forecast volatility?
Volatility is one of the most important concepts in the whole of Volatility is one of the most important concepts in the whole of finance.finance.
ARCH models offered new tools for measuring risk, and its ARCH models offered new tools for measuring risk, and its impact on return. impact on return.
Volatility of exchange rates is of importance because of the Volatility of exchange rates is of importance because of the uncertainty it creates for prices of exports and imports, for the uncertainty it creates for prices of exports and imports, for the value of international reserves and for open positions in foreign value of international reserves and for open positions in foreign currency.currency.
Volatility Models.Volatility Models.
ARCH/GARCH models.ARCH/GARCH models.
Engle(1982)Engle(1982)
Bollerslev(1986)Bollerslev(1986)
Baillie, Baillie, BollerslevBollerslev and and MikkelsenMikkelsen (1996) (1996)
ARFIMA models.ARFIMA models.
Granger (1980)Granger (1980)
Data descriptionData description Data series: nominal daily EUR/ROL exchange ratesData series: nominal daily EUR/ROL exchange rates Time length: 04:01:1999-11:06:2004Time length: 04:01:1999-11:06:2004 1384 nominal percentage returns1384 nominal percentage returns
)]ln()[ln(100 1 ttt ssy
0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300
15000
20000
25000
30000
35000
40000 Time Series of The Exchange RateExchange Rate
Descriptive Statistics for the return series.Descriptive Statistics for the return series.
-2 -1 0 1 2 3 4 5 6 7
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8 DensityHistogram of Returns together with the Normal and Return Density
Statistic t-Test P-Value
Skewness 1.0472 15.922 4.4605e-057
Excess Kurtosis
8.5138 64.769 0.00000
Jarque-Bera
4432.9
HeteroscedasticityHeteroscedasticity
0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300
-2
-1
0
1
2
3
4
5
6
7
Autocorrelation and Partial autocorrelation of the Return Series
0 5 10 15 20
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
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].
6.37220
Autocorrelation and Partial Autocorrelation Autocorrelation and Partial Autocorrelation of Squared Returnsof Squared Returns
0 5 10 15 20
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
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]
StationarityStationarityUnit Root Tests for EUR/ROL return series.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.
Model estimates and forecasting performances.Model estimates and forecasting performances. Methodology.Methodology. Ox Professional 3.30 Ox Professional 3.30 [email protected]@RCH4.0
4.01.1999-30.12.2002 (1018 observations) for model 4.01.1999-30.12.2002 (1018 observations) for model estimation estimation
06.01.2003-11.06.2004 (366 observations) for out of 06.01.2003-11.06.2004 (366 observations) for out of sample forecast evaluation.sample forecast evaluation.
The Models.The Models.Two distributions: Student, Skewed Student, QMLE.Two distributions: Student, Skewed Student, QMLE.
The Mean Equations:The Mean Equations: 1. A constant mean1. A constant mean
2. An ARFIMA(1,d2. An ARFIMA(1,daa,0) mean,0) mean
3. An ARFIMA(0, d3. An ARFIMA(0, daa,1) mean,1) mean
The variance equations.The variance equations. GARCH(1,1) and FIGARCH(1,d,1) without the constant term and with a non-GARCH(1,1) and FIGARCH(1,d,1) without the constant term and with a non-
trading day dummy variable.trading day dummy variable.
The estimated twelve models.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 which implies stationarity.stationarity.• all variance coefficients are positiveall variance coefficients are positive andand
5.0;5.0
1
In-sample model evaluation. Residual tests. GARCH models.In-sample model evaluation. Residual tests. GARCH models.
Model SBC Skewness EK1 Q* Q2** ARCH*** Nyblom
ARMA (0,0)GARCH(1,1)Skewed-Student
2.210463 0.75224 3.9543 37.5958 [0.9019571]
30.3204[0.9783154]
1.1358[0.3395]
1.96933
ARMA (0,0)GARCH(1,1)Student
2.212901 0.74033 3.8319 37.5877[0.9021277]
30.3145[0.9783579]
1.1238 [0.3458]
1.58334
ARFIMA (1,d,0)GARCH(1,1)Skewed-Student
2.214579 0.76024 4.1028 36.4188 [0.9083405]
31.7529[0.9659063]
1.2484 [0.2843]
2.24209
ARFIMA (1,d,0)GARCH(1,1)Student
2.216388 0.73353 3.857 36.0009[0.9165657]
31.8411[0.9649974]
1.1801[0.3169]
1.89543
ARFIMA (0,d,1)GARCH(1,1)Skewed-Student
2.215735 0.75909 4.1153 36.1425[0.9138359]
31.3112 [0.9701942]
1.2084 [0.3030]
2.2612
ARFIMA (0,d,1)GARCH(1,1)Student
2.217401 0.73390 3.8852 35.8043[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.
In-sample model evaluation. Residual tests. FIGARCH models.In-sample model evaluation. Residual tests. FIGARCH models.
Model SBC Skewness EK1 Q* Q2** ARCH*** Nyblom
ARMA (0,0)FIGARCH(1,d,1)Skewed-Student
2.222089 0.76305 3.8723 37.4681[0.9046084]
28.4572[0.9888560]
1.2601 [0.2790]
1.56799
ARMA (0,0)FIGARCH(1,d,1)Student*
2.22472 0.74698 3.7313 37.7303[0.8991133]
28.9803[0.9864387]
1.3297[0.2491]
1.37719
ARFIMA (1,d,0)FIGARCH(1,d,1)Skewed-Student
2.226549 0.757 3.9242 36.3540[0.9096502]
29.8994[0.9811947]
1.3204[0.2529]
2.05757
ARFIMA (1,d,0)FIGARCH(1,d,1)Student
2.228334 0.73378 3.7256 36.1801[0.9131002]
30.4315[0.9775013]
1.3272 [0.2501]
1.82764
ARFIMA (0,d,1)FIGARCH(1,d,1)Skewed-Student
2.227516 0.75901 3.96 36.2611[0.9115043]
29.2088[0.9852596]
1.2729 [0.2733]
2.0233
ARFIMA (0,d,1)FIGARCH(1,d,1)Student
2.229199 0.73799 3.7813 36.1313[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.
Out-of-sample Forecast EvaluationOut-of-sample Forecast Evaluation
Forecast methodologyForecast 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 meanfor the conditional variance and the conditional mean
-dynamic forecast is programmed in OxEdit -dynamic forecast is programmed in OxEdit
[email protected]@RCH3.0 package package
Benchmark: ex-post volatility = squared returns. Benchmark: ex-post volatility = squared returns.
Measuring Forecast Accuracy.Measuring Forecast Accuracy.
The The Mincer-ZarnowitzMincer-Zarnowitz regressionregression::
The Mean Absolute The Mean Absolute Error:Error:
Root Mean Square Error Root Mean Square Error (standard error):(standard error):
Theil's inequality Theil's inequality coefficient -Theil's Ucoefficient -Theil's U::
n
tttn
MAE1
22 ˆ1
nRMSE
n
ttt
1
222 )ˆ(
n
ttt
n
ttt
U
1
2221
1
222
)(
)ˆ(
ttt ubetaalfa 22 ̂
One Step Ahead Forecast Evaluation Measures.One Step Ahead Forecast Evaluation Measures.
Model alfa beta R2 Model alfa beta R2
ARMA (0,0)GARCH(1,1)Skewed-Student
-0.104961 [0.0699]
0.624769[0.0006]
0.0533211 ARMA (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]
0.0530545 ARMA (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]
0.0518779 ARFIMA (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]
0.0522936 ARFIMA (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]
0.052295 ARFIMA (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]
0.0527494 ARFIMA (0,d,1)FIGARCH(1,d,1)Student
-0.043431[0.2715]
0.70931[0.0006]
0.0742364
1. The Mincer-Zarnowitz regression
2. Forecasting the conditional mean. Loss functions.2. Forecasting the conditional mean. Loss functions.
Model MAE RMSE TIC Model MAE RMSE TIC
ARMA (0,0)GARCH(1,1)Skewed-Student
0.2601 0.3412 0.7895 ARMA (0,0)FIGARCH(1,d,1)Skewed-Student
0.2606 0.3416 0.7861
ARMA (0,0)GARCH(1,1)Student
0.2576 0.3395 0.812 ARMA (0,0)FIGARCH(1,d,1)Student
0.258 0.3397 0.8086
ARFIMA (1,d,0)GARCH(1,1)Skewed-Student
0.2724 0.3521 0.7527 ARFIMA (1,d,0)FIGARCH(1,d,1)Skewed-Student
0.2726 0.3522 0.7518
ARFIMA (1,d,0)GARCH(1,1)Student
0.2694 0.3493 0.77 ARFIMA (1,d,0)FIGARCH(1,d,1)Student
0.2697 0.3496 0.7684
ARFIMA (0,d,1)GARCH(1,1)Skewed-Student
0.2722 0.352 0.7548 ARFIMA (0,d,1)FIGARCH(1,d,1)Skewed-Student
0.2724 0.3522 0.7536
ARFIMA (0,d,1)GARCH(1,1)Student
0.2691 0.3493 0.7729 ARFIMA (0,d,1)FIGARCH(1,d,1)Student
0.2694 0.3495 0.7711
3. Forecasting the conditional variance. Loss functions.3. Forecasting the conditional variance. Loss functions.
Model MAE RMSE TIC Model MAE RMSE TIC
ARMA (0,0)GARCH(1,1)Skewed-Student
0.2844 0.3148 0.5253 ARMA (0,0)FIGARCH(1,d,1)Skewed-Student
0.17 0.2234 0.484
ARMA (0,0)GARCH(1,1)Student
0.2824 0.3131 0.5244 ARMA (0,0)FIGARCH(1,d,1)Student
0.1726 0.2253 0.4845
ARFIMA (1,d,0)GARCH(1,1)Skewed-Student
0.2907 0.3204 0.5286 ARFIMA (1,d,0)FIGARCH(1,d,1)Skewed-Student
0.1802 0.2299 0.4856
ARFIMA (1,d,0)GARCH(1,1)Student
0.2866 0.3168 0.5265 ARFIMA (1,d,0)FIGARCH(1,d,1)Student
0.1832 0.2322 0.4861
ARFIMA (0,d,1)GARCH(1,1)Skewed-Student
0.2903 0.32 0.5283 ARFIMA (0,d,1)FIGARCH(1,d,1)Skewed-Student
0.1794 0.2294 0.4854
ARFIMA (0,d,1)GARCH(1,1)Student
0.2862 0.3164 0.5263 ARFIMA (0,d,1)FIGARCH(1,d,1)Student
0.1822 0.2315 0.4859
ConcludingConcluding remarks.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.
ConcludingConcluding remarks.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
BibliographyBibliography Alexander, Carol (2001) – Alexander, Carol (2001) – Market Models - A Guide to Financial Data Analysis, Market Models - A Guide to Financial Data Analysis, John Wiley John Wiley
&Sons, Ltd.;&Sons, Ltd.; Andersen, T. G. and T. Bollerslev (1997) - Andersen, T. G. and T. Bollerslev (1997) - Answering the Skeptics: Yes, Standard Volatility Answering the Skeptics: Yes, Standard Volatility
Models Do Provide Accurate Forecasts,Models Do Provide Accurate Forecasts, International Economic Review; International Economic Review; Andersen, T. G., T. Bollerslev, Andersen, T. G., T. Bollerslev, Francis X. Diebold and Paul Labys (2000)- MFrancis X. Diebold and Paul Labys (2000)- Modeling and odeling and
Forecasting Realized Volatility, Forecasting Realized Volatility, the June 2000 Meeting of the Western Finance Association. the June 2000 Meeting of the Western Finance Association. Andersen, T. G., T. Bollerslev and Andersen, T. G., T. Bollerslev and Francis X. Diebold (2002)- Francis X. Diebold (2002)- Parametric and Parametric and
Nonparametric Volatility MeasurementNonparametric Volatility Measurement, Prepared for Yacine Aït-Sahalia and Lars Peter , Prepared for Yacine Aït-Sahalia and Lars Peter Hansen (eds.), Handbook of Financial Econometrics,Hansen (eds.), Handbook of Financial Econometrics, North Holland. North Holland.
Andersen, T. G., T. Bollerslev and Andersen, T. G., T. Bollerslev and Peter Christoffersen (2004)-Peter Christoffersen (2004)-Volatility ForecastingVolatility Forecasting, Rady , Rady School of Management at UCSDSchool of Management at UCSD
Baillie, R.T., Bollerslev T., Mikkelsen H.O. (1996)- Baillie, R.T., Bollerslev T., Mikkelsen H.O. (1996)- Fractionally Integrated Generalized Fractionally Integrated Generalized Autoregressive Conditional HeteroskedasticityAutoregressive Conditional Heteroskedasticity , Journal of Econometrics, Vol. 74, No.1, pp. , Journal of Econometrics, Vol. 74, No.1, pp. 3-30.3-30.
Bollerslev, Tim, Robert F. Engle and Daniel B. Nelson (1994)– Bollerslev, Tim, Robert F. Engle and Daniel B. Nelson (1994)– ARCH Models, ARCH Models, Handbook of Handbook of Econometrics, Volume 4, Chapter 49, North Holland;Econometrics, Volume 4, Chapter 49, North Holland;
Diebold, Francis and Marc Nerlove (1989)-Diebold, Francis and Marc Nerlove (1989)-The Dynamics of Exchange Rate Volatility: A The Dynamics of Exchange Rate Volatility: A Multivariate Latent factor Arch ModelMultivariate Latent factor Arch Model, Journal of Applied Econometrics, Vol. 4, No.1., Journal of Applied Econometrics, Vol. 4, No.1.
Diebold, Francis and Jose A. Lopez (1995)-Diebold, Francis and Jose A. Lopez (1995)- Forecast Evaluation and CombinationForecast Evaluation and Combination, Prepared , Prepared for G.S. Maddala and C.R. Rao (eds.), Handbook of Statistics, for G.S. Maddala and C.R. Rao (eds.), Handbook of Statistics, North HollandNorth Holland..
Enders WEnders W. . (1995)- (1995)- Applied Econometric Time SeriesApplied Econometric Time Series, 1st Edition, New York: Wiley., 1st Edition, New York: Wiley.
BibliographyBibliography Engle, R.F. (1982) – Engle, R.F. (1982) – Autoregressive conditional heteroskedasticity with estimates of the Autoregressive conditional heteroskedasticity with estimates of the
variance of UK inflation, variance of UK inflation, Econometrica, 50, pp. 987-1007;Econometrica, 50, pp. 987-1007; Engle, R.F. and Victor K. Ng (1993) – Engle, R.F. and Victor K. Ng (1993) – Measuring and Testing the Impact of News on Measuring and Testing the Impact of News on
Volatility, Volatility, The Journal of Finance, Vol. XLVIII, No. 5;The Journal of Finance, Vol. XLVIII, No. 5; Engle, R. (2001) – Engle, R. (2001) – Garch 101:Garch 101: The Use of ARCH/GARCH Models in Applied The Use of ARCH/GARCH Models in Applied
EconometricsEconometrics, Journal of Economic Perspectives – Volume 15, Number 4 – Fall 2001 – , Journal of Economic Perspectives – Volume 15, Number 4 – Fall 2001 – Pages 157-168;Pages 157-168;
Engle, R. and A. J. Patton (2001) – Engle, R. and A. J. Patton (2001) – What good is a volatility model?What good is a volatility model?, Research Paper, , Research Paper, Quantitative Finance, Volume 1, 237-245;Quantitative Finance, Volume 1, 237-245;
Engle, R. (2001) – Engle, R. (2001) – New Frontiers for ARCH ModelsNew Frontiers for ARCH Models, prepared for Conference on , prepared for Conference on Volatility Modelling and Forecasting, Perth, Australia, September 2001;Volatility Modelling and Forecasting, Perth, Australia, September 2001;
Hamilton, J.D. (1994) – Hamilton, J.D. (1994) – Time Series AnalysisTime Series Analysis, Princeton University Press;, Princeton University Press; Lopez, J.A.(1999) – Lopez, J.A.(1999) – Evaluating the Predictive Accuracy of Volatility Models, Evaluating the Predictive Accuracy of Volatility Models,
Economic Research Deparment, Federal Reserve Bank of San Francisco;Economic Research Deparment, Federal Reserve Bank of San Francisco; Peters, J. and S. Laurent (2001) – Peters, J. and S. Laurent (2001) – A Tutorial for G@RCH 2.3, a Complete Ox Package A Tutorial for G@RCH 2.3, a Complete Ox Package
for Estimating and Forecasting ARCH Models;for Estimating and Forecasting ARCH Models; Peters, J. and S. Laurent (2002) – Peters, J. and S. Laurent (2002) – A Tutorial for G@RCH 2.3, a Complete Ox Package A Tutorial for G@RCH 2.3, a Complete Ox Package
for Estimating and Forecasting ARCH Models;for Estimating and Forecasting ARCH Models; West, Kenneth and Dongchul Cho (1994)-West, Kenneth and Dongchul Cho (1994)-The Predictive Ability of Several Models of The Predictive Ability of Several Models of
Exchange Rate Volatility,Exchange Rate Volatility, NBER Technical Working Paper #152. NBER Technical Working Paper #152.
Appendix 1.Appendix 1.
The ARMA (0, 0), GARCH (1, 1) Skewed Student model. Robust Standard Errors (Sandwich formula)
Coefficient Std.Error t-value Probability
Constant(Mean) 0.091930 0.021613 4.253 0.0000
dummyFriday (V) 0.048977 0.019781 2.476 0.0134
ARCH(Alpha1) 0.036076 0.011561 3.121 0.0019
GARCH(Beta1) 0.924490 0.018052 51.21 0.0000
Asymmetry 0.145722 0.047250 3.084 0.0021
Tail 9.872213 3.3488 2.948 0.0033
For more details see Appendix 1, page 45.
Appendix 2Appendix 2
The ARMA (0, 0), GARCH (1, 1) Student model.Robust Standard Errors (Sandwich formula)
Coefficient Std.Error t-alue Probability
Constant(Mean) 0.077795 0.021673 3.589 0.0003
dummyFriday (V) 0.049240 0.020163 2.442 0.0148
ARCH(Alpha1) 0.037186 0.011975 3.105 0.0020
GARCH(Beta1) 0.923353 0.018479 49.97 0.0000
Student(DF) 8.921340 2.8119 3.173 0.0016
For more details, see Appendix 2, page 47.
Appendix 3Appendix 3 The ARFIMA (1, da, 0),GARCH (1, 1) Skewed Student model.
Robust Standard Errors (Sandwich formula)
Coefficient Std.Error t-value Probability
Constant(Mean) 0.089939 0.010527 8.544 0.0000
d-Arfima -0.128224 0.045067 -2.845 0.0045
AR(1) 0.123269 0.054553 2.260 0.0241
dummyFriday (V) 0.048860 0.019703 2.480 0.0133
ARCH(Alpha1) 0.033897 0.011677 2.903 0.0038
GARCH(Beta1) 0.926283 0.018096 51.19 0.0000
Asymmetry 0.139771 0.047194 2.962 0.0031
Tail 9.189523 2.9091 3.159 0.0016
For more details, see Appendix 3, page 49.
Appendix 4Appendix 4The ARFIMA (1, da, 0),GARCH (1, 1) Student
model.Robust Standard Errors (Sandwich formula)
Coefficient Std.Error t-value Probabilty
Constant(Mean) 0.082711 0.010237 8.080 0.0000
d-Arfima -0.136317 0.045875 -2.971 0.0030
AR(1) 0.140455 0.055832 2.516 0.0120
dummyFriday (V) 0.049635 0.020117 2.467 0.0138
ARCH(Alpha1) 0.036517 0.012510 2.919 0.0036
GARCH(Beta1) 0.923503 0.018602 49.64 0.0000
Student(DF) 8.436809 2.5257 3.340 0.0009
For more details, see Appendix 4, page 52.
Appendix 5Appendix 5
The ARFIMA (0, da,1),GARCH (1, 1) Skewed Student model.
Robust Standard Errors (Sandwich formula) Coefficient Std.Error t-value Probability
Constant(Mean) 0.090415 0.011041 8.189 0.0000
d-Arfima -0.117757 0.037429 -3.146 0.0017
MA(1) 0.114844 0.046060 2.493 0.0128
dummyFriday (V) 0.048681 0.019787 2.460 0.0140
ARCH(Alpha1) 0.033847 0.011641 2.908 0.0037
GARCH(Beta1) 0.926414 0.018172 50.98 0.0000
Asymmetry 0.138631 0.047049 2.947 0.0033
Tail 9.279306 2.9613 3.134 0.0018
For more details, see Appendix 5, page 54.
Appendix 6Appendix 6
The ARFIMA (0, da,1),GARCH (1, 1) Student model.
Robust Standard Errors (Sandwich formula)
Coefficient Std.Error t-value Probability
Constant(Mean) 0.082822 0.010833 7.645 0.0000
d-Arfima -0.122519 0.036843 -3.325 0.0009
MA(1) 0.128311 0.045146 2.842 0.0046
dummyFriday (V) 0.049380 0.020207 2.444 0.0147
ARCH(Alpha1) 0.036344 0.012449 2.919 0.0036
GARCH(Beta1) 0.923788 0.018703 49.39 0.0000
Student(DF) 8.516429 2.5689 3.315 0.0009
For more details, see Appendix 6, page 56.
Appendix 7Appendix 7The ARMA (0, 0), FIGARCH-BBM (1,d,1) Skewed Student model.Robust Standard Errors (Sandwich formula)
Coefficient Std.Error t-value Probability
Constant(Mean) 0.094259 0.021931 4.298 0.0000
dummyFriday (V) 0.047278 0.025975 1.820 0.0690
d-Figarch 0.358622 0.098899 3.626 0.0003
ARCH(Alpha1) 0.288896 0.094598 3.054 0.0023
GARCH(Beta1) 0.635309 0.058513 10.86 0.0000
Asymmetry 0.147588 0.046529 3.172 0.0016
Tail 9.545031 3.0964 3.083 0.0021
For more details, see Appendix 7, page 59.
Appendix 8Appendix 8
The ARMA (0, 0), FIGARCH-BBM (1,d,1) Student model.Robust Standard Errors (Sandwich formula)
Coefficient Std.Error t-value Probability
Constant(Mean) 0.079807 0.021915 3.642 0.0003
dummyFriday (V) 0.049310 0.027926 1.766 0.0777
d-Figarch 0.351448 0.10506 3.345 0.0009
ARCH(Alpha1) 0.312018 0.11026 2.830 0.0047
GARCH(Beta1) 0.644842 0.057580 11.20 0.0000
Student(DF) 8.596805 2.6044 3.301 0.0010
For more details, see Appendix 8, page 61.
Appendix 9Appendix 9The ARFIMA (1,da,0), FIGARCH-BBM (1,d,1) Skewed Student model.
Robust Standard Errors (Sandwich formula)
Coefficient Std.Error t-value Probability
Constant(Mean) 0.090400 0.010719 8.434 0.0000
d-Arfima -0.126724 0.046241 -2.741 0.0062
AR(1) 0.119364 0.054745 2.180 0.0295
dummyFriday (V)
0.052164 0.030787 1.694 0.0905
d-Figarch 0.332074 0.10662 3.115 0.0019
ARCH(Alpha1) 0.339292 0.13642 2.487 0.0130
GARCH(Beta1) 0.649620 0.053779 12.08 0.0000
Asymmetry 0.139501 0.046638 2.991 0.0028
Tail 8.871259 2.6840 3.305 0.0010
For more details, see Appendix 9, page 63.
Appendix 10Appendix 10
The ARFIMA (1,da,0), FIGARCH-BBM (1,d,1) Student model.
Robust Standard Errors (Sandwich formula)Coefficient Std.Error t-value Probability
Constant(Mean) 0.083221 0.010263 8.109 0.0000
d-Arfima -0.136270 0.047181 -2.888 0.0040
AR(1) 0.138494 0.056208 2.464 0.0139
dummyFriday (V) 0.054562 0.034015 1.604 0.1090
d-Figarch 0.328545 0.12291 2.673 0.0076
ARCH(Alpha1) 0.360347 0.17155 2.101 0.0359
GARCH(Beta1) 0.659966 0.057997 11.38 0.0000
Student(DF) 8.093551 2.3226 3.485 0.0005
For more details, see Appendix 10, page 66.
Appendix 11Appendix 11
The ARFIMA (0,da,1), FIGARCH-BBM (1,d,1) Skewed Student model.
Robust Standard Errors (Sandwich formula)Coefficient Std.Error t-value Probability
Constant(Mean) 0.090938 0.011202 8.118 0.0000
d-Arfima -0.117093 0.039118 -2.993 0.0028
MA(1) 0.112312 0.047184 2.380 0.0175
dummyFriday (V) 0.051724 0.030327 1.706 0.0884
d-Figarch 0.332759 0.10397 3.200 0.0014
ARCH(Alpha1) 0.334340 0.12765 2.619 0.0089
GARCH(Beta1) 0.647135 0.052822 12.25 0.0000
Asymmetry 0.138659 0.046925 2.955 0.0032
Tail 8.973744 2.7438 3.270 0.0011
For more details, see Appendix 11, page 68.
Appendix 12Appendix 12
The ARFIMA (0,da,1), FIGARCH-BBM (1,d,1) Student model.
Robust Standard Errors (Sandwich formula)Coefficient Std.Error t-value Probability
Cst(M) 0.083434 0.010870 7.675 0.0000
d-Arfima -0.122620 0.038276 -3.204 0.0014
MA(1) 0.126887 0.045925 2.763 0.0058
dummyFriday (V) 0.054060 0.033155 1.631 0.1033
d-Figarch 0.329579 0.11765 2.801 0.0052
ARCH(Alpha1) 0.353442 0.15661 2.257 0.0242
GARCH(Beta1) 0.656630 0.055867 11.75 0.0000
Student(DF) 8.182206 2.3695 3.453 0.0006
For more details, see Appendix 12, page 70.
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
Variable Coefficient Std. Error t-Statistic Prob.
RETURNS(-1) -0.957262 0.026883 -35.60834 0.0000
C 0.078392 0.018264 4.292148 0.0000
R-squared 0.478843 Mean dependent var -0.000589
Adjusted R-squared 0.478465 S.D. dependent var 0.933223
S.E. of regression 0.673949 Akaike info criterion 2.050121
Sum squared resid 626.8057 Schwarz criterion 2.057692
Log likelihood -1414.634 F-statistic 1267.954
Durbin-Watson stat 1.994863 Prob(F-statistic) 0.000000
StationarityStationarity tests. Appendix 13. tests. Appendix 13. 1. Dickey-Fuller Test.1. Dickey-Fuller Test.
ADFTest
-17.25675 1% CriticalValue*
-3.4380
5%CriticalValue
-2.8641
10% CriticalValue
-2.5681
*MacKinnon critical values forrejection of hypothesis of a unit root.
Dependent Variable: D(RETURNS)
Method: Least Squares
Sample(adjusted): 7 1384
Included observations: 1378 after adjusting endpoints
Variable Coefficient Std. Error t-Statistic Prob.
RETURNS(-1) -1.047183 0.060682 -17.25675 0.0000
D(RETURNS(-1)) 0.091319 0.053927 1.693396 0.0906
D(RETURNS(-2)) 0.039379 0.046166 0.852989 0.3938
D(RETURNS(-3)) 0.009635 0.037319 0.258186 0.7963
D(RETURNS(-4)) 0.015333 0.026967 0.568585 0.5697
C 0.086684 0.018835 4.602399 0.0000
R-squared 0.480683 Mean dependent var 0.000495
Adjusted R-squared 0.478791 S.D. dependent var 0.933787
S.E. of regression 0.674146 Akaike info criterion 2.053604
Sum squared resid 623.5364 Schwarz criterion 2.076369
Log likelihood -1408.933 F-statistic 253.9867
Durbin-Watson stat 1.998880 Prob(F-statistic) 0.000000Appendix 14.
ADF Test.
Appendix 15.Phillips-Appendix 15.Phillips-Perron Perron Test.Test.
Lag truncation for Bartlett kernel: 7 ( Newey-West suggests: 7 )
Residual variance with no correction 0.453550
Residual variance with correction 0.407637
Phillips-Perron Test Equation
Dependent Variable: D(RETURNS)
Method: Least Squares
Sample(adjusted): 3 1384
Included observations: 1382 after adjusting endpoints
Variable Coefficient Std. Error t-Statistic Prob.
RETURNS(-1) -0.957262 0.026883 -35.60834 0.0000
C 0.078392 0.018264 4.292148 0.0000
R-squared 0.478843 Mean dependent var -0.000589
Adjusted R-squared 0.478465 S.D. dependent var 0.933223
S.E. of regression 0.673949 Akaike info criterion 2.050121
Sum squared resid 626.8057 Schwarz criterion 2.057692
Log likelihood -1414.634 F-statistic 1267.954
Durbin-Watson stat 1.994863 Prob(F-statistic) 0.000000