# 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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• DISSERTATION PAPER

Modeling and Forecasting the Volatility of the EUR/ROL Exchange Rate Using GARCH Models.

Student :Becar IulianaSupervisor: Professor Moisa Altar

• Why model and forecast volatility?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.

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. Engle(1982) Bollerslev(1986) Baillie, Bollerslev and Mikkelsen (1996)

ARFIMA models. Granger (1980)

• Data descriptionData series: nominal daily EUR/ROL exchange ratesTime length: 04:01:1999-11:06:2004 1384 nominal percentage returns

• Descriptive Statistics for the return series.

• Heteroscedasticity

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

• 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]

• StationarityUnit Root Tests for EUR/ROL return series.

• Model estimates and forecasting performances.Methodology. Ox Professional 3.30 G@RCH4.0 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.The Models.Two distributions: Student, Skewed Student, QMLE. The Mean Equations: 1. A constant mean 2. An ARFIMA(1,da,0) mean 3. An ARFIMA(0, da,1) mean

Becar Iuliana - QUASI MAXIMUM LIKELIHOOD ESTIMATES robust standard errors

• The variance equations.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:The estimated coefficients are significantly different from zero at the 10% level.the ARFIMA coefficient lies between which implies stationarity.all variance coefficients are positive and

• In-sample model evaluation. Residual tests. GARCH models.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.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 EvaluationForecast methodology - sample window: 1018 observations - at each step, the 1 step ahead dynamic forecast is stored for the conditional variance and the conditional mean -dynamic forecast is programmed in OxEdit G@RCH3.0 package

Benchmark: ex-post volatility = squared returns.

• Measuring Forecast Accuracy.The Mincer-Zarnowitz regression:

The Mean Absolute Error:

Root Mean Square Error (standard error):

Theil's inequality coefficient -Theil's U:

• One Step Ahead Forecast Evaluation Measures. 1. The Mincer-Zarnowitz regression

• 2. Forecasting the conditional mean. Loss functions.

• 3. Forecasting the conditional variance. Loss functions.

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

• Concluding remarks.Model construction problems;

Further research: -option prices, which reflect the markets expectation of volatility over the remaining life span of the option. -daily realized volatility can be computed as the sum of squared intraday returns

• BibliographyAlexander, 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., 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 At-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 UCSDBaillie, 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;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.Enders W. (1995)- Applied Econometric Time Series, 1st Edition, New York: Wiley.

• BibliographyEngle, 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. (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. (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;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 (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.

• Appendix 1. The ARMA (0, 0), GARCH (1, 1) Skewed Student model. Robust Standard Errors (Sandwich formula)

For more details see Appendix 1, page 45.

CoefficientStd.Errort-valueProbabilityConstant(Mean)0.0919300.0216134.2530.0000dummyFriday (V)0.0489770.0197812.4760.0134ARCH(Alpha1)0.0360760.0115613.1210.0019GARCH(Beta1)0.9244900.01805251.210.0000Asymmetry0.1457220.0472503.0840.0021Tail9.8722133.34882.9480.0033

• Appendix 2The ARMA (0, 0), GARCH (1, 1) Student model.Robust Standard Errors (Sandwich formula)

For more details, see Appendix 2, page 47.

CoefficientStd.Errort-alueProbabilityConstant(Mean)0.0777950.0216733.5890.0003dummyFriday (V)0.0492400.0201632.4420.0148ARCH(Alpha1)0.0371860.0119753.1050.0020GARCH(Beta1)0.9233530.01847949.970.0000Student(DF)8.9213402.81193.1730.0016

• Appendix 3 The ARFIMA (1, da, 0),GARCH (1, 1) Skewed Student model. Robust Standard Errors (Sandwich formula)

For more details, see Appendix 3, page 49.

CoefficientStd.Errort-valueProbabilityConstant(Mean)0.0899390.0105278.5440.0000d-Arfima-0.1282240.045067-2.8450.0045AR(1)0.1232690.0545532.2600.0241dummyFriday (V)0.0488600.0197032.4800.0133ARCH(Alpha1)0.0338970.0116772.9030.0038GARCH(Beta1)0.9262830.01809651.190.0000Asymmetry0.1397710.0471942.9620.0031Tail9.1895232.90913.1590.0016

• Appendix 4The ARFIMA (1, da, 0),GARCH (1, 1) Student model.Robust Standard Errors (Sandwich formula)

For more details, see Appendix 4, page 52.

CoefficientStd.Errort-valueProbabiltyConstant(Mean)0.0827110.0102378.0800.0000d-Arfima-0.1363170.045875-2.9710.0030AR(1)0.1404550.0558322.5160.0120dummyFriday (V)0.0496350.0201172.4670.0138ARCH(Alpha1)0.0365170.0125102.9190.0036GARCH(Beta1)0.9235030.01860249.640.0000Student(DF)8.4368092.52573.3400.0009

• Appendix 5The ARFIMA (0, da,1),GARCH (1, 1) Skewed Student model.Robust Standard Errors (Sandwich formula)

For more details, see Appendix 5, page 54.

CoefficientStd.Errort-valueProbabilityConstant(Mean)0.0904150.0110418.1890.0000d-Arfima-0.1177570.037429-3.1460.0017MA(1)0.1148440.0460602.4930.0128dummyFriday (V)0.0486810.0197872.4600.0140ARCH(Alpha1)0.0338470.0116412.9080.0037GARCH(Beta1)0.9264140.01817250.980.0000Asymmetry0.1386310.0470492.9470.0033Tail9.2793062.96133.1340.0018

• Appendix 6 The ARFIMA (0, da,1),GARCH (1, 1) Student model. Robust Standard Errors (Sandwich formula)

For more details, see Appendix 6, page 56.

CoefficientStd.Errort-valueProbabilityConstant(Mean)0.0828220.0108337.6450.0000d-Arfima-0.1225190.036843-3.3250.0009MA(1)0.1283110.0451462.8420.0046dummyFriday (V)0.0493800.0202072.4440.0147ARCH(Alpha1)0.0363440.0124492.9190.0036GARCH(Beta1)0.9237880.01870349.390.0000Student(DF)8.5164292.56893.3150.0009

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

For more details, see Appendix 7, page 59.

CoefficientStd.Errort-valueProbabilityConstant(Mean)0.0942590.0219314.2980.0000dummyFriday (V)0.0472780.0259751.8200.0690d-Figarch0.3586220.0988993.6260.0003ARCH(Alpha1)0.2888960.0945983.0540.0023GARCH(Beta1)0.6353090.05851310.860.0000Asymmetry0.1475880.0465293.1720.0016Tail9.5450313.09643.0830.0021

• Appendix 8The ARMA (0, 0), FIGARCH-BBM (1,d,1) Student model.Robust Standard Errors (Sandwich formula)

For more details, see Appendix 8, page 61.

CoefficientStd.Errort-valueProbabilityConstant(Mean)0.0798070.0219153.6420.0003dummyFriday (V)0.0493100.0279261.7660.0777d-Figarch0.3514480.105063.3450.0009ARCH(Alpha1)0.3120180.110262.8300.0047GARCH(Beta1)0.6448420.05758011.200.0000Student(DF)8.5968052.60443.3010.0010

• Appendix 9The ARFIMA (1,da,0), FIGARCH-BBM (1,d,1) Skewed Student model.Robust Standard Errors (Sandwich formula)

For more details, see Appendix 9, page 63.

CoefficientStd.Errort-valueProbabilityConstant(Mean)0.0904000.0107198.4340.0000d-Arfima-0.1267240.046241-2.7410.0062AR(1)0.1193640.0547452.1800.0295dummyFriday (V)0.0521640.0307871.6940.0905d-Figarch0.3320740.106623.1150.0019ARCH(Alpha1)0.3392920.136422.4870.0130GARCH(Beta1)0.6496200.05377912.080.0000Asymmetry0.1395010.0466382.9910.0028Tail8.8712592.68403.3050.0010

• Appendix 10The ARFIMA (1,da,0), FIGARCH-BBM (1,d,1) Student model.Robust Standard Errors (Sandwich formula)

For more details, see Appendix 10, page 66.

CoefficientStd.Errort-valueProbabilityConstant(Mean)0.0832210.0102638.1090.0000d-Arfima-0.1362700.047181-2.8880.0040AR(1)0.1384940.0562082.4640.0139dummyFriday (V)0.0545620.0340151.6040.1090d-Figarch0.3285450.122912.6730.0076ARCH(Alpha1)0.3603470.171552.1010.0359GARCH(Beta1)0.6599660.05799711.380.0000Student(DF)8.0935512.32263.4850.0005

• Appendix 11The ARFIMA (0,da,1), FIGARCH-BBM (1,d,1) Skewed Student model.Robust Standard Errors (Sandwich formula)

For more details, see Appendix 11, page 68.

CoefficientStd.Errort-valueProbabilityConstant(Mean)0.0909380.0112028.1180.0000d-Arfima-0.1170930.039118-2.9930.0028MA(1)0.1123120.0471842.3800.0175dummyFriday (V)0.0517240.0303271.7060.0884d-Figarch0.3327590.103973.2000.0014ARCH(Alpha1)0.3343400.127652.6190.0089GARCH(Beta1)0.6471350.05282212.250.0000Asymmetry0.1386590.0469252.9550.0032Tail8.9737442.74383.2700.0011

• Appendix 12The ARFIMA (0,da,1), FIGARCH-BBM (1,d,1) Student model.Robust Standard Errors (Sandwich formula)

For more details, see Appendix 12, page 70.

CoefficientStd.Errort-valueProbabilityCst(M)0.0834340.0108707.6750.0000d-Arfima-0.1226200.038276-3.2040.0014MA(1)0.1268870.0459252.7630.0058dummyFriday (V)0.0540600.0331551.6310.1033d-Figarch0.3295790.117652.8010.0052ARCH(Alpha1)0.3534420.156612.2570.0242GARCH(Beta1)0.6566300.05586711.750.0000Student(DF)8.1822062.36953.4530.0006

• Stationarity tests. Appendix 13. 1. Dickey-Fuller Test.