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

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.

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.

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)

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

Descriptive Statistics for the return series. Statistict-TestP-Value Skewness e-057 Excess Kurtosis Jarque- Bera

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 [0.125].

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

Stationarity Unit Root Tests for EUR/ROL return series. ADF Test Statistic % Critical Value* % Critical Value % Critical Value *MacKinnon critical values for rejection of hypothesis of a unit root. PP Test Statistic % Critical Value* % Critical Value % Critical Value *MacKinnon critical values for rejection of hypothesis of a unit root.

Model estimates and forecasting performances. Methodology. Methodology. Ox Professional 3.30 Ox Professional (1018 observations) for model estimation (1018 observations) for model estimation (366 observations) for out of sample forecast evaluation (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

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

In-sample model evaluation. Residual tests. GARCH models. ModelSBCSkewnessEK 1 Q*Q2**ARCH***Nyblom ARMA (0,0) GARCH(1,1) Skewed-Student [ ] [ ] [0.3395] ARMA (0,0) GARCH(1,1) Student [ ] [ ] [0.3458] ARFIMA (1,d,0) GARCH(1,1) Skewed-Student [ ] [ ] [0.2843] ARFIMA (1,d,0) GARCH(1,1) Student [ ] [ ] [0.3169] ARFIMA (0,d,1) GARCH(1,1) Skewed-Student [ ] [ ] [0.3030] ARFIMA (0,d,1) GARCH(1,1) Student [ ] [ ] [0.3394] 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. ModelSBCSkewnessEK 1 Q*Q2**ARCH***Nyblom ARMA (0,0) FIGARCH(1,d,1) Skewed-Student [ ] [ ] [0.2790] ARMA (0,0) FIGARCH(1,d,1) Student* [ ] [ ] [0.2491] ARFIMA (1,d,0) FIGARCH(1,d,1) Skewed-Student [ ] [ ] [0.2529] ARFIMA (1,d,0) FIGARCH(1,d,1) Student [ ] [ ] [0.2501] ARFIMA (0,d,1) FIGARCH(1,d,1) Skewed-Student [ ] [ ] [0.2733] ARFIMA (0,d,1) FIGARCH(1,d,1) Student [ ] [ ] [0.2777] 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 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 package Benchmark: ex-post volatility = squared returns. Benchmark: ex-post volatility = squared returns.

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:

One Step Ahead Forecast Evaluation Measures. ModelalfabetaR2R2 ModelalfabetaR2R2 ARMA (0,0) GARCH(1,1) Skewed-Student [0.0699] [0.0006] ARMA (0,0) FIGARCH(1,d,1) Skewed-Student [0.3070] [0.0005] ARMA (0,0) GARCH(1,1) Student [0.0766] [0.0007] ARMA (0,0) FIGARCH(1,d,1) Student [0.3143] [0.0005] ARFIMA (1,d,0) GARCH(1,1) Skewed-Student [0.0607] [0.0006] ARFIMA (1,d,0) FIGARCH(1,d,1) Skewed-Student [0.2517] [0.0006] ARFIMA (1,d,0) GARCH(1,1) Student [0.0698] [0.0006] ARFIMA (1,d,0) FIGARCH(1,d,1) Student [0.2681] [0.0006] ARFIMA (0,d,1) GARCH(1,1) Skewed-Student [0.0596] [0.0006] ARFIMA (0,d,1) FIGARCH(1,d,1) Skewed-Student [0.254] [0.0006] ARFIMA (0,d,1) GARCH(1,1) Student [0.0680] [0.0006] ARFIMA (0,d,1) FIGARCH(1,d,1) Student [0.2715] [0.0006] The Mincer-Zarnowitz regression

2. Forecasting the conditional mean. Loss functions. ModelMAERMSETICModelMAERMSETIC ARMA (0,0) GARCH(1,1) Skewed-Student ARMA (0,0) FIGARCH(1,d,1) Skewed-Student ARMA (0,0) GARCH(1,1) Student ARMA (0,0) FIGARCH(1,d,1) Student ARFIMA (1,d,0) GARCH(1,1) Skewed-Student ARFIMA (1,d,0) FIGARCH(1,d,1) Skewed-Student ARFIMA (1,d,0) GARCH(1,1) Student ARFIMA (1,d,0) FIGARCH(1,d,1) Student ARFIMA (0,d,1) GARCH(1,1) Skewed-Student ARFIMA (0,d,1) FIGARCH(1,d,1) Skewed-Student ARFIMA (0,d,1) GARCH(1,1) Student ARFIMA (0,d,1) FIGARCH(1,d,1) Student

3. Forecasting the conditional variance. Loss functions. ModelMAERMSETICModelMAERMSETIC ARMA (0,0) GARCH(1,1) Skewed-Student ARMA (0,0) FIGARCH(1,d,1) Skewed-Student ARMA (0,0) GARCH(1,1) Student ARMA (0,0) FIGARCH(1,d,1) Student ARFIMA (1,d,0) GARCH(1,1) Skewed-Student ARFIMA (1,d,0) FIGARCH(1,d,1) Skewed-Student ARFIMA (1,d,0) GARCH(1,1) Student ARFIMA (1,d,0) FIGARCH(1,d,1) Student ARFIMA (0,d,1) GARCH(1,1) Skewed-Student ARFIMA (0,d,1) FIGARCH(1,d,1) Skewed-Student ARFIMA (0,d,1) GARCH(1,1) Student ARFIMA (0,d,1) FIGARCH(1,d,1) Student

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.

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

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Appendix 1. The ARMA (0, 0), GARCH (1, 1) Skewed Student model. Robust Standard Errors (Sandwich formula) CoefficientStd.Errort-valueProbability Constant(Mean) dummyFriday (V) ARCH(Alpha1) GARCH(Beta1) Asymmetry Tail For more details see Appendix 1, page 45.

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

Appendix 3 The ARFIMA (1, d a, 0),GARCH (1, 1) Skewed Student model. Robust Standard Errors (Sandwich formula) CoefficientStd.Errort-valueProbability Constant(Mean) d-Arfima AR(1) dummyFriday (V) ARCH(Alpha1) GARCH(Beta1) Asymmetry Tail For more details, see Appendix 3, page 49.

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

Appendix 5 The ARFIMA (0, d a,1),GARCH (1, 1) Skewed Student model. Robust Standard Errors (Sandwich formula) CoefficientStd.Errort-valueProbability Constant(Mean) d-Arfima MA(1) dummyFriday (V) ARCH(Alpha1) GARCH(Beta1) Asymmetry Tail For more details, see Appendix 5, page 54.

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

Appendix 7 The ARMA (0, 0), FIGARCH-BBM (1,d,1) Skewed Student model. Robust Standard Errors (Sandwich formula) CoefficientStd.Errort-valueProbability Constant(Mean) dummyFriday (V) d-Figarch ARCH(Alpha1) GARCH(Beta1) Asymmetry Tail For more details, see Appendix 7, page 59.

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

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) d-Arfima AR(1) dummyFriday (V) d-Figarch ARCH(Alpha1) GARCH(Beta1) Asymmetry Tail For more details, see Appendix 9, page 63.

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) d-Arfima AR(1) dummyFriday (V) d-Figarch ARCH(Alpha1) GARCH(Beta1) Student(DF) For more details, see Appendix 10, page 66.

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) d-Arfima MA(1) dummyFriday (V) d-Figarch ARCH(Alpha1) GARCH(Beta1) Asymmetry Tail For more details, see Appendix 11, page 68.

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) d-Arfima MA(1) dummyFriday (V) d-Figarch ARCH(Alpha1) GARCH(Beta1) Student(DF) 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): Included observations: 1382 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. RETURNS(-1) C R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Stationarity tests. Appendix Dickey-Fuller Test.

ADF Test % Critical Value* % Critical Value % Critical Value *MacKinnon critical values for rejection of hypothesis of a unit root. Dependent Variable: D(RETURNS) Method: Least Squares Sample(adjusted): Included observations: 1378 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. RETURNS(-1) D(RETURNS(-1)) D(RETURNS(-2)) D(RETURNS(-3)) D(RETURNS(-4)) C R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Appendix 14. ADF Test.

Appendix 15.Phillips-Perron Test. Lag truncation for Bartlett kernel: 7 ( Newey-West suggests: 7 ) Residual variance with no correction Residual variance with correction Phillips-Perron Test Equation Dependent Variable: D(RETURNS) Method: Least Squares Sample(adjusted): Included observations: 1382 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. RETURNS(-1) C R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)