A dynamic regression model that differs from the dynamic
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A dynamic regression model that differs from the dynamic linear model in Section 5.3.2 assumes piecewise constant regres
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A dynamic regression model that differs from the dynamic linear model in Section 5.3.2 assumes piecewise constant regression parameter vectors in yt = ?T t xt + ?t, t = 1, 2, . . . , n, (9.84) in which the t are assumed to be i.i.d. standard normal and xt is an observed regressor that is determined by the events up to time t?1. This is the same as the Bayesian change-point model (9.45) except that ?t in (9.45) is replaced by the constant ?, which is tautamount to letting the variance of the gamma prior distribution in (9.46) approach 0. The posterior distribution of ?t given x1, y1, . . . ,xt, yt in this case is the normal mixture t j=1 pj,tN(zj,t, ?2Vj,t), in which pj,t = p? j,t t i=1 p? i,t and p? j,t is given by (9.50) with f00 = (det(V))1/2 exp zTV?1z (2?2) , fij = (det(Vi,j))1/2 exp zT i,jVi,j ?1zi,j (2?2) . Apply the change-point regression model (9.84), with univariate ?t and xt, to re-analyze the monthly log returns of the Apple Computer stock (the first column in the file m logret 10stocks.txt) in Exercise 5.10, which also uses related information on CAPM contained in the the file m sp500ret 3mtcm.txt. (a) As in Exercise 5.10, we use January 1994 to June 1998 as the training period in which ?t is assumed to remain constant. Estimate z, ?2, and V by ? , ? 2, and V which is the estimated variance of ? , respectively; see Section 1.1.4. (b) For the period July 1998 to December 2006, we use the posterior mean in the Bayesian change-point model (9.84) to estimate ?t based on observations up to time t. The posterior mean, however, requires specification of the hyperparameter ? := (p, z, ?2, V ). With the estimates z , ? 2, and V obtained from (a), we can calculate the log-likelihood function of p based on the training period from January 1994 to June 1998. Maximize this log-likelihood function over p = 10?4i, 1 ? i ? 500, to obtain the estimated hyperparameter ? . (c) Substituting ? by ? in the posterior mean of ?t given the observations up to time t, obtain the estimated ? t as t varies from July 1998 to December 2006. Plot ? t versus t and compare these sequential estimates with the beta obtained by fitting CAPM to the period July 1998 to December 2006.
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