Download - DISSERTATION PAPER Modeling and Forecasting the Volatility of the EUR/ROL Exchange Rate Using GARCH Models. Student :Becar Iuliana Student :Becar Iuliana

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

mailto:[email protected]

Becar Iuliana

QUASI MAXIMUM LIKELIHOOD ESTIMATES robust standard errors

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


2.215735 0.75909 4.1153 36.1425[0.9138359]

31.3112 [0.9701942]

1.2084 [0.3030]

2.2612


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.

Becar Iuliana

76,154

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.

mailto:[email protected]

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


-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


-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


-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


-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


-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


-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


0.2601 0.3412 0.7895 ARMA (0,0)FIGARCH(1,d,1)Skewed-Student

0.2606 0.3416 0.7861


0.2576 0.3395 0.812 ARMA (0,0)FIGARCH(1,d,1)Student

0.258 0.3397 0.8086


0.2724 0.3521 0.7527 ARFIMA (1,d,0)FIGARCH(1,d,1)Skewed-Student

0.2726 0.3522 0.7518


0.2694 0.3493 0.77 ARFIMA (1,d,0)FIGARCH(1,d,1)Student

0.2697 0.3496 0.7684



0.2724 0.3522 0.7536



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


0.2844 0.3148 0.5253 ARMA (0,0)FIGARCH(1,d,1)Skewed-Student

0.17 0.2234 0.484


0.2824 0.3131 0.5244 ARMA (0,0)FIGARCH(1,d,1)Student

0.1726 0.2253 0.4845



0.1802 0.2299 0.4856



0.1832 0.2322 0.4861



0.1794 0.2294 0.4854



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)


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


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



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



The ARFIMA (0, da,1),GARCH (1, 1) Student model.



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


Appendix 7Appendix 7The ARMA (0, 0), FIGARCH-BBM (1,d,1) Skewed Student model.Robust Standard Errors (Sandwich formula)


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



The ARMA (0, 0), FIGARCH-BBM (1,d,1) Student model.Robust Standard Errors (Sandwich formula)


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


Appendix 9Appendix 9The ARFIMA (1,da,0), FIGARCH-BBM (1,d,1) Skewed Student model.



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



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



The ARFIMA (0,da,1), FIGARCH-BBM (1,d,1) Skewed Student model.


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



The ARFIMA (0,da,1), FIGARCH-BBM (1,d,1) Student model.


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


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.






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





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






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





Durbin-Watson stat 1.994863 Prob(F-statistic) 0.000000

Download - DISSERTATION PAPER Modeling and Forecasting the Volatility of the EUR/ROL Exchange Rate Using GARCH Models. Student :Becar Iuliana Student :Becar Iuliana

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