"Frontmatter". In: Analysis of Financial Time Series

122 CONDITIONAL HETEROSCEDASTIC MODELScompute c0 = 0.01, p1 = 0.01, th = 0.1, ga = 0.1compute a0 = 0.01, a1 = 0.5maximize(method=bhhh,recursive,iterations=150) garchlnset fv = gvar(t)set resid = epsi(t)set residsq = resid(t)*resid(t)cor(qstats,number=20,span=10) residcor(qstats,number=20,span=10) residsqEXERCISES1. Derive multistep ahead forecasts for a GARCH(1, 2) model at the forecast originh.2. Derive multistep ahead forecasts for a GARCH(2, 1) model at the forecast originh.3. Suppose that r 1 ,...,r n are observations of a return series that follows theAR(1)-GARCH(1, 1) modelr t = µ + φ 1 r t−1 + a t , a t = σ t ɛ t , σ 2t = α 0 + α 1 a 2 t−1 + β 1σ 2t−1 ,where ɛ t is a standard Gaussian white noise series. Derive the conditional loglikelihood function of the data.4. In the previous equation, assume that ɛ t follows a standardized Student-t distributionwith v degrees of freedom. Derive the conditional log likelihood functionof the data.5. Consider the monthly log return of Intel stock from 1973 to 1997. Build aGARCH model for the series and compute 1 to 5-step ahead volatility forecastsat the forecast origin December 1997.6. The file “m-mrk.dat” contains monthly simple returns of Merck stock. There arethree columns—namely, monthly simple returns, years, and months. Transformthe simple returns to log returns.• Is there evidence of ARCH effects in the log returns? Use Ljung–Box statisticsfor the squared returns with 5 and 10 lags of autocorrelation and 5% significancelevel to answer the question.• Use the PACF of the squared log returns to identify an ARCH model for thedata and fit the identified model. Write down the fitted model.7. The file “m-mmm.dat” contains two columns. They are monthly simple returnand date for 3M stock. Transform the returns to log returns.