"Frontmatter". In: Analysis of Financial Time Series

REFERENCES 297a Gaussian GARCH(1, 1) model. Perform a diagnostic check on the fittedmodel.(g) Repeat the prior two-dimensional nonhomogeneous Poisson process withthreshold 2.5% or 3%. Comment on the selection of threshold.3. Use Hill’s estimator and the data “d-csco9199.dat” to estimate the tail index fordaily stock returns of Cisco Systems.4. The file “d-hwp3dx8099.dat” contains the daily log returns of Hewlett-Packard,CRSP value-weighted index, equal-weighted index, and S&P 500 index from1980 to 1999. All returns are in percentages and include dividend distributions.Assume that the tail probability of interest is 0.01. Calculate Value at Risk for thefollowing financial positions for the first trading day of year 2000.(a) Long on Hewlett-Packard stock of $1 million and S&P 500 index of $1 million,using RiskMetrics. The α coefficient of the IGARCH(1, 1) model foreach series should be estimated.(b) The same position as part (a), but using a univariate ARMA-GARCH modelfor each return series.(c) A long position on Hewlett-Packard stock of $1 million using a twodimensionalnonhomogeneous Poisson model with the following explanatoryvariables: (1) an annual time trend, (2) a fitted volatility based on a GaussianGARCH model for Hewlett-Packard stock, (3) a fitted volatility based on aGaussian GARCH model for S&P 500 index returns, and (4) a fitted volatilitybased on a Gaussian GARCH model for the value-weighted index return.Perform a diagnostic check for the fitted models. Are the market volatilityas measured by S&P 500 index and value-weighted index returns helpful indetermining the tail behavior of stock returns of Hewlett-Packard? You maychoose several thresholds.REFERENCESBerman, S. M. (1964), “Limiting theorems for the maximum term in stationary sequences,”Annals of Mathematical Statistics, 35, 502–516.Cox, D. R., and Hinkley, D. V. (1974), Theoretical Statistics, London: Chapman and Hall.Danielsson, J., and De Vries, C. G. (1997a), “Value at risk and extreme returns,” workingpaper, London School of Economics, London, U.K.Danielsson, J., and De Vries, C. G. (1997b), “Tail index and quantile estimation with veryhigh frequency data,” Journal of Empirical Finance,4,241–257.Davison, A. C., and Smith, R. L. (1990), “Models for exceedances over high thresholds,” (withdiscussion), Journal of the Royal Statistical Society, Series B, 52, 393–442.De Haan L., Resnick, I. S., Rootzén, and De Vries, C. G. (1989), “Extremal behavior of solutionsto a stochastic difference equation with applications to ARCH process,” StochasticProcesses and Their Applications, 32, 213–224.Dekkers, A. L. M., and De Haan, L. (1989), “On the estimation of extreme value index andlarge quantile estimation,” Annals of Statistics, 17, 1795–1832.