Final Exam Financial Data Analysis at the University of Freiburg (Winter Semester 2008/2009) Friday, November 14, 2008,

Transcription

1 Professor Dr. Roman Liesenfeld Final Exam Financial Data Analysis at the University of Freiburg (Winter Semester 2008/2009) Friday, November 14, 2008, am 1

2 Part 1 (38 Points) Consider the following model for log returns r t : r t = c + ψ 1 r t 1 + ɛ t, ψ < 1, (1) where ɛ t is a white noise term with E(ɛ t ) = 0 and E(ɛ 2 t ) = σ 2. a) (4 pts) Derive rt+1=e[r t+1 I t ] with I t = {r t, r t 1,...}. Interpret the obtained quantity. b) (8 pts) Show that model (1) can be transformed into the following form: r t = φ 0 + ψ1ɛ i t i. (2) i=0 Give an interpretation of this representation of r t. c) (6 pts) Derive the unconditional mean E[r t ] and the unconditional variance Var[r t ] under model (1). Is r t under model (1) weakly stationary? Why? d) (14 pts) Table 1 in the Appendix provides results (p-value) of the Ljung-Box test for the daily log returns of SIEMENS stock for the period of 1998 to 2008 as well as for the residuals of an AR(1) model fitted to the SIEMENS log return data. Do these results provide evidence against the model (1)? Explain your answer. e) (6 pts) What is the definition of Market Efficiency in its weak form? Table 1 in the Appendix provides results (p-value) of the Cowles-Jones-test for the daily log returns of SIEMENS stock for the period of 1998 to 2008 as well as for the residuals of an AR(1) model fitted to the SIEMENS log return data. Do those results provide evidence for/against the hypothesis of weak-form Market Efficiency? Explain your answer briefly. 2

3 Part 2 (8 Points) Consider the following GARCH(2,1) model: ɛ t = σ t u t, σ 2 t = α 0 + α 1 ɛ 2 t 1 + α 2 ɛ 2 t 2 + β 1 σ 2 t 1 where u t iid, E[u t ] = 0, E[u 2 t ] = 1. On the parameters the following restrictions are imposed: α 0 > 0, α 1 0, α 2 0, β 1 0, (α 1 + α 2 + β 1 ) < 1. a) (4 pts) Derive the unconditional mean and unconditional variance of ɛ t. b) (4 pts) Why is it necessary to impose the above restrictions on the parameters? Part 3 (16 Points) An ARMA(p,q)-GARCH(r,s) model is fitted to the daily log returns of the GREED stock. a) (8 pts) Estimation results from EVIEWS are reported in Table 2 and Table 3 in the Appendix. Write down the fitted model (round the figures to 0.01). Provide a measure for the strength of persistence in the volatility process. Is the fitted model adequate? b) (8 pts) Assume that ɛ 2 T 1 = 0, σ2 T 1 = 1 and ɛ2 T = 1. Use your fitted model in a) to calculate the 1-step and 2-step ahead forecast of period T for the conditional variance of the returns of the GREED stock. (round the figures to 0.01). What is the h-step ahead forecast for h? 3

4 Part 4 (28 Points) Consider the following ordered probit model for the transaction price change data of stocks: y i = s 1, if α 0 < y i α 1 s 2, if α 1 < yi α 2, s 3, if α 2 < yi < α 3 y i = TDIFF i β 1 + y i 1 β 2 + y i 2 β 3 + y i 3 β 4 + ɛ i, ɛ i N(0, σ 2 ), (3) where yi : unobservable continuous price change at time t i, i = 1,..., n; y i : observable discrete price change at time t i with three possible categories; y i l : observable discrete price change at time t i l, l = 1, 2, 3; TDIFF i : t i t i 1, time elapsed between trade i 1 and i, measured in seconds. a) (2 pts) What is the motivation to assume for transaction price changes such an ordered probit model? b) (8 pts) What signs would the Roll-model (for the bid-ask bounce) predict for the coefficients β 2, β 3, β 4 in model (3)? Explain your answer briefly. c) (4 pts) Table 4 in the Appendix contains the estimation results from EVIEWS for the IBM stock using the ordered probit model described above. What are the estimated values of (α 0, α 1, α 2, α 3 )? Give a brief interpretation of those estimates. d) (4 pts) You do not find the estimated value of σ in the EVIEWS outputs, why? e) (10 pts) Consider the following statement: If the elapsed time between two trades increases, the probability that the observed price change falls into the categories of s 1 would be (ceteris paribus) higher. Comment on this statement based on the results from Table 4. 4

5 Appendix Table 1: Results (p value) of the Ljung Box (LB) test and the Cowles Jones (CJ) test for the daily log returns of SIEMENS and for the residuals of a fitted AR(1) model. Series LB(8) - Test LB(16) -Test CJ -Test p-value p-value p-value Log returns Residuals of AR(1) Table 2: EVIEWS output of a ARMA(p,q) GARCH(r,s) model for the daily log returns of stock GREED. Dependent Variable: lr_greed Method: ML - ARCH (Marquardt) - Normal distribution Date: 10/16/08 Time: 20:23 Sample (adjusted): Included observations: 1999 after adjustments Convergence achieved after 23 iterations Presample variance: backcast (parameter = 0.7) GARCH = C(2) + C(3)*RESID(-1)^2 + C(4)*GARCH(-1) Coefficient Std. Error z-statistic Prob. C Variance Equation C 5.07E E RESID(-1)^ GARCH(-1) 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 Hannan-Quinn criter Durbin-Watson stat

6 Table 3: Results of the Ljung Box (LB) test including h lags for the estimated ARMA(p,q) GARCH(r,s) model presented in Table 2. a) Standardized residuals b) Squared of standardized residuals h LB(h) Test Statistic p-value h LB(h) Test Statistic p-value

7 Table 4: Estimation results from EVIEWS for the IBM stock using the ordered probit model described in Question 4. Dependent Variable: Y Method: ML - Ordered Probit (Quadratic hill climbing) Date: 10/16/08 Time: 16:53 Sample: IF PC<100 Included observations: Number of ordered indicator values: 3 Convergence achieved after 4 iterations Covariance matrix computed using second derivatives Coefficient Std. Error z-statistic Prob. TDIFF Y(-1) Y(-2) Y(-3) Limit Points LIMIT_0:C(5) LIMIT_1:C(6) Pseudo R-squared Akaike info criterion Schwarz criterion Log likelihood Hannan-Quinn criter Restr. log likelihood LR statistic Avg. log likelihood Prob(LR statistic)