"Frontmatter". In: Analysis of Financial Time Series

EXERCISES 441eter uncertainty in producing forecasts. In contrast, the GARCH model assumes thatthe parameters are fixed and given in Eq. (10.26). This is an important differenceand is one of the reasons that GARCH models tend to underestimate the volatility incomparison with the implied volatility obtained from derivative pricing.Remark: Besides the advantage of taking into consideration parameter uncertaintyin forecast, the MCMC method produces in effect a predictive distribution ofthe volatility of interest. The predictive distribution is more informative than a simplepoint forecast. It can be used, for instance, to obtain the quantiles needed in Value atRisk calculation.10.10 OTHER APPLICATIONSThe MCMC method is applicable to many other financial problems. For example,Zhang, Russell, and Tsay (2000) use it to analyze information determinants of bidand ask quotes, McCulloch and Tsay (2000) use the method to estimate a hierarchicalmodel for IBM transaction data, and Eraker (2001) and Elerian, Chib and Shephard(2001) use it to estimate diffusion equations. The method is also useful in Value atRisk calculation because it provides a natural way to evaluate predictive distributions.The main question is not whether the methods can be used in most financialapplications, but how efficient the methods can become. Only time and experiencecan provide an adequate answer to the question.EXERCISES1. Suppose that x is normally distributed with mean µ and variance 4. Assume thatthe prior distribution of µ is also normal with mean 0 and variance 25. What isthe posterior distribution of µ given the data point x?2. Consider the linear regression model with time-series errors in Section 10.5.Assume that z t is an AR(p) process (i.e., z t = φ 1 z t−1 + ··· +φ p z t−p + a t ).Let φ = (φ 1 ,...,φ p ) ′ be the vector of AR parameters. Derive the conditionalposterior distributions of f (β | Y, X, φ,σ 2 ), f (φ | Y, X, β,σ 2 ),and f (σ 2 |Y, X, β, φ) assuming that conjugate prior distributions are used—that is,β ∼ N(β o , Σ o ), φ ∼ N(φ o , A o ), (vλ)/σ 2 ∼ χ 2 v .3. Consider the linear AR(p) model in Subsection 10.6.1. Suppose that x h and x h+1are two missing values with a joint prior distribution being multivariate normalwith mean µ o and covariance matrix Σ o . Other prior distributions are the same asthat in the text. What is the conditional posterior distribution of the two missingvalues?