The MCMC method is applicable to many other financial problems. For example, Zhang, Russell, and Tsay (2008) use it to analyze information determinants of bid and ask quotes, McCulloch and Tsay (2001) use the method to estimate a hierarchical model 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 at risk calculation because it provides a natural way to evaluate predictive distributions. The main question is not whether the methods can be used in most financial applications, but how efficient the methods can become. Only time and experience can provide an adequate answer to the question.
Exercises
12.1 Suppose that x is normally distributed with mean μ and variance 4. Assume that the prior distribution of μ is also normal with mean 0 and variance 25. What is the posterior distribution of μ given the data point x?
12.2 Consider the linear regression model with time series errors in Section 12.5. Assume that zt is an AR(p) process (i.e., zt = ϕ1zt−1 + ⋯ + ϕpzt−p + at). Let 
be the vector of AR parameters. Derive the conditional posterior distributions of 
, 
, and 
using the conjugate prior distributions, that is, the priors are
12.3 Consider the linear AR(p) model in Section 12.6.1. Suppose that xh and xh+1 are two missing values with a joint prior distribution being multivariate normal with mean 
and covariance matrix 
. Other prior distributions are the same as that in the text. What is the conditional posterior distribution of the two missing values?
12.4 Consider the monthly log returns of Ford Motors stock from January 1965 to December 2008: (a) Build a GARCH model for the series, (b) build a stochastic volatility model for the series, and (c) compare and discuss the two volatility models. The simple returns of the stock are in the file m-fsp6508.txt.
12.5 Build a stochastic volatility model for the daily log return of Cisco Systems stock from January 2001 to December 2008. You may download the simple return of the stock from the CRSP database or the file d-csco0108.txt. Transform the data into log returns in percentage. Use the model to obtain a predictive distribution for 1-step-ahead volatility forecast at the forecast origin December 31, 2008. Finally, use the predictive distribution to compute the value at risk of a long position worth $1 million with probability 0.01 for the next trading day.
12.6 Build a bivariate stochastic volatility model for the monthly log returns of Ford Motors stock and the S&P composite index for the sample period from January 1965 to December 2008. Discuss the relationship between the two volatility processes and compute the time-varying beta for the Ford stock.
12.7 Consider the monthly log returns of Procter & Gamble stock and the value-weighted index from January 1965 to December 2008. The simple returns are given in the file m-pgvw6508.txt. Transform the data into log returns in percentages. (a) Build a bivariate stochastic volatility model for the two return series. (b) Build a BEKK(1,1) model for the two series. (c) Compare and discuss the two models.
12.8 Consider the monthly data of 30-year mortgage rate and the 3-month Treasury Bill rate of the secondary market from April 1971 to September 2009. The data are in m-mort3mtb7109.txt. (a) Build a regression model with time series error to study the effect of 3-month Treasury Bill rate on the mortgage rate. (b) Reestimate the model using MCMC method. (c) Compare and discuss the two fitted models.
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