Dr. Roland Füss Winter Term 2005/2006 Final Exam Financial Data Analysis (6 Credit points/imp Students) March 2, 2006 Note the following important information: 1. The total disposal time is 60 minutes. 2. You have to answer all 3 questions. 3. Allocate your limited time; keep in mind how the points are distributed among the questions. 4. Read the questions and instructions carefully. Good luck! Question 1 (10 points): ARIMA modelling Consider the random variable Y t, which is defined by equation Y t = 0.8Y t-1 + u t with t = 1,,T and a given initial condition Y 0 = 2. For simplicity it is assumed that σ 2 1. a) Calculate the expected values E[Y 1 ] and E[Y 2 ]. b) Determine the variance of Y 1 and the autocovariance of Y 1 and Y 2. c) Determine in short and verbally the approximated mean and variance of Y 100. Thereby, keep in mind that it is about a covariance stationary process. u = Question 2 (20 points): Dickey-Fuller test a) Specify the test design of the simple Dickey-Fuller t-test. Thereby, refer to the terms pure random walk, random walk with drift, and random walk with drift and trend. b) Describe the sequential procedure of the augmented Dickey-Fuller t-test considering the example in the following tables. Page 1 of 5
Augmented Dickey-Fuller test - FTSE 100 Null Hypothesis: LFTSE100 has a unit root Exogenous: Constant Lag Length: 0 Augmented Dickey-Fuller test - FTSE 100 Null Hypothesis: D(LFTSE100) has a unit root Exogenous: Constant Lag Length: 0 t-statistic Prob.* t-statistic Prob.* Augmented Dickey-Fuller test statistic -1.605870 0.4744 Test critical values: 1% level -3.524233 5% level -2.902358 10% level -2.588587 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller test statistic -8.839974 0.0000 Test critical values: 1% level -3.525618 5% level -2.902953 10% level -2.588902 *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(LFTSE100) Method: Least Squares Date: 03/01/05 Time: 14:12 Sample (adjusted): 1987Q1 2004Q4 Included observations: 72 after adjustments Variable Coefficient Std. Error t-statistic Prob. Augmented Dickey-Fuller Test Equation Dependent Variable: D(LFTSE100,2) Method: Least Squares Date: 03/01/05 Time: 14:19 Sample (adjusted): 1987Q2 2004Q4 Included observations: 71 after adjustments Variable Coefficient Std. Error t-statistic Prob. LFTSE100(-1) -0.028051 0.017468-1.605870 0.1128 C 0.226066 0.126131 1.792311 0.0774 R-squared 0.035531 Mean dependent var 0.024195 Adjusted R-squared 0.021753 S.D. dependent var 0.088503 S.E. of regression 0.087535 Akaike info criterion -2.006171 Sum squared resid 0.536367 Schwarz criterion -1.942930 Log likelihood 74.22214 F-statistic 2.578819 Durbin-Watson stat 2.044422 Prob(F-statistic) 0.112805 D(LFTSE100(-1)) -1.039088 0.117544-8.839974 0.0000 C 0.022860 0.010760 2.124484 0.0372 R-squared 0.531075 Mean dependent var -0.001797 Adjusted R-squared 0.524279 S.D. dependent var 0.126964 S.E. of regression 0.087570 Akaike info criterion -2.004988 Sum squared resid 0.529129 Schwarz criterion -1.941251 Log likelihood 73.17708 F-statistic 78.14513 Durbin-Watson stat 2.051644 Prob(F-statistic) 0.000000 Page 2 of 5
Question 3 (30 points): ARCH/GARCH modelling The following tables 1 to 9 present the results of volatility estimations for the stock markets of the developing countries Korea, Malaysia and Thailand. The log returns are calculated from daily prices of the corresponding stock market indices taken from Datastream. Describe the several steps of estimation and forecasting GARCH(p,q) models! Furthermore, there exist asymmetric volatility models such as the E- and TGARCH model. What are the differences in comparison to the GARCH(p,q) model. In this context, please explain the leverage effect. Table 1: Normality test of daily returns Indices Period Jarque-Bera test (Korea) 03.90-12.90 38.4178 *** (Malaysia) 02.87-11.87 2,155.8200 *** (Thailand) 02.87-11.87 331.0944 *** Table 2: Ljung-Box test of squared returns Indices L=10 L=20 L=40 (Korea) 56.69 *** 75.03 *** 94.49 *** (Malaysia) 160.24 *** 161.10 *** 162.92 *** (Thailand) 113.91 *** 127.88 *** 130.32 *** Note: L = lag length Table 3: Autocorrelation test of squared residuals LB test ARCH-LM test Indices Period L=10 L =20 L=40 L=1 (Korea) 03.90-12.90 58.23 *** 76.77 *** 94.87 *** 32.2420 *** (Malaysia) 02.87-11.87 188.87 *** 189.94 *** 191.37 *** 55.2778 *** (Thailand) 02.87-11.87 99.46 *** 113.08 *** 116.78 *** 25.4980 *** Note: L = lag length Page 3 of 5
Parameter ĉ ωˆ ˆα 1 1 ˆβ Table 4: GARCH(p,q) estimations (Korea) (Malaysia) (Thailand) GARCH(1,1) GARCH(2,1) GARCH(1,1) 03.90-12.90 02.87-11.87 02.87-11.87-0.1449 (0.1104) 1.4130 (0.2831) 0.3752 (0.1182) 0.2339 (0.1325) ˆβ 2 - Mean equation 0.4249 (0.1168) Variance equation 0.7753 (0.2938) 0.6989 (0.1446) 0.0865 (0.1219) 0.2494 (0.1094) Note: Semi-robust standard errors in parenthesis. 0.3520 (0.1191) 0.3011 (0.1367) 0.1788 (0.0306) 0.7587 (0.0460) - Table 5: Statistical description of the standardized residuals Indices Period Model Mean Std.-dev. Skewness Kurtosis (Korea) 03.90-12.90 GARCH(1,1) -0.0072 1.0023 0.0113 4.8962 (Malaysia) 02.87-11.87 GARCH(2,1) -0.1934 0.9865-1.8699 14.6946 (Thailand) 02.87-11.87 GARCH(1,1) -0.0483 1.0006-0.7356 6.1683 Table 6: Normality test of the standardized residuals Indices Model Jarque-Bera test (Korea) GARCH(1,1) 32.66 *** (Malaysia) GARCH(2,1) 1,356.73 *** (Thailand) GARCH(1,1) 109.82 *** Page 4 of 5
Table 7: Autocorrelation test of squared standardized residuals LB-Test ARCH- LM test Indices Period Model L=10 L =20 L=40 L=1 (Korea) 03.90-12.90 GARCH(1,1) 7.22 34.96 ** 45.16 0.0030 (Malaysia) 02.87-11.87 GARCH(2,1) 1.40 6.18 12.73 0.0256 (Thailand) 02.87-11.87 GARCH(1,1) 9.30 21.83 29.60 4.6994 ** Table 8: Estimation results Indices Period Model LogL AIC SIC (Korea) 03.90-12.90 GARCH(1,1) -430.2180 3.9837 4.0458 (Malaysia) 02.87-11.87 GARCH(2,1) -438.7908 4.1092 4.1873 (Thailand) 02.87-11.87 GARCH(1,1) -434.9631 4.0645 4.1270 Table 9: Forecast results Indices Period Model RMSE MAE MAPE TIC (Korea) 03.90-12.90 GARCH(1,1) 2.1547 1.7247 623.08 0.9208 (Malaysia) 02.87-11.87 GARCH(2,1) 2.5862 2.1191 146.21 0.9664 (Thailand) 02.87-11.87 GARCH(1,1) 3.3046 2.5810 159.08 0.9641 TIC = Theil s inequality coefficient Page 5 of 5