"Frontmatter". In: Analysis of Financial Time Series

REGRESSION WITH SERIAL CORRELATIONS 409where the original interest rates r it are measured in percentages. In Chapter 2, weemployed a linear regression model with an MA(1) error for the data. Here we consideran AR(2) model for the error process. Using the traditional approach, we obtainthe modelc 3t = 0.0002 + 0.782c 1t + z t , z t = 0.205z t−1 − 0.068z t−2 + a t , (10.12)where ̂σ a = 0.067. Standard errors of the coefficient estimates of Eq. (10.12) are0.0017, 0.008, 0.023, and 0.023, respectively. Except for a marginally significantresidual ACF at lag 6, the prior model seems adequate.Writing the model asc 3t = β 0 + β 1 c 1t + z t , z t = φ 1 z t−1 + φ 2 z t−2 + a t , (10.13)where {a t } is an independent sequence of N(0,σ 2 ) random variables, we estimatethe parameters by Gibbs sampling. The prior distributions used areβ ∼ N(0, 4I 2 ), φ ∼ N[0, diag(0.25, 0.16)], (vλ)/σ 2 = (10 × 0.1)/σ 2 ∼ χ 2 10 ,where I 2 is the 2 × 2 identity matrix. The initial parameter estimates are obtained bythe ordinary least squares method (i.e., by using a two-step procedure of fitting thelinear regression model first, then fitting an AR(2) model to the regression residuals).Since the sample size 1966 is large, the initial estimates are close to those given inEq. (10.12). We iterated the Gibbs sampling for 2100 iterations, but discard results ofthe first 100 iterations. Table 10.1 gives the posterior means and standard errors of theparameters. Figure 10.1 shows the histogram of the marginal posterior distributionof each parameter.We repeated the Gibbs sampling with different initial values, but obtained similarresults. The Gibbs sampling appears to have converged. From Table 10.1, the posteriormeans are close to the estimates of Eq. (10.12) except for the coefficient of z t−2 .However, the posterior standard errors of φ 1 and φ 2 are relatively large, indicatinguncertainty in these two estimates. The histograms of Figure 10.1 are informative.In particular, they show that the distributions of ̂φ 1 and ̂φ 2 have not converged to theasymptotic normality; the distributions are skewed to the right. However, the asymptoticnormality of ̂β 0 and ̂β 1 seems reasonable.Table 10.1. Posterior Means and Standard Errors of Model(10.13) Estimated by a Gibbs Sampling with 2100 Iterations.The Results Are Based on the Last 2000 Iterations and thePrior Distributions Are Given in the Text.Parameter β 0 β 1 φ 1 φ 2 σMean 0.025 0.784 0.305 0.032 0.074St. Error 0.024 0.009 0.089 0.087 0.003