To conclude this Appendix, we comment briefly on the performance of the MCMC algorithm described in Section A.7 and used for simulating λ1. Im-
plementing this algorithm requires the choice of three tuning parameters: the number N of filter particles; the number M of particle smoother paths; and the number K + 1 of blocks in λ1. For all the simulations described in Sections 5.1 and 5.2, involving three different leverage formulations and five different sample sizes, the same tuning parameter values of N = 2000 and M = 100 were used. The number of blocks was chosen as the integer nearest to T /B, with B = 10 for the first 100 MCMC sweeps and B = 50 otherwise. The average MH acceptance probability crucially depends on B, which is an approximate expected block size. Table 8 shows the average acceptance probabilities (after burn-in) for the three stochastic volatility models and for four different sample sizes. They are remark- ably similar across models and sample sizes, in spite of the fact that the same values of N, M, and B were used throughout the simulations. This suggests that the method of Section A.7 is generally applicable beyond the models estimated in this paper.
Appendix B. A description of the particle filter used for marginal likelihood estimation
Table 8. Average acceptance probabilities
Data Model T K λ1(MH) λ1(AR) φ1 φ2 (ν, β) (ρ, σ1)
SV-SL 1041 20 0.83 0.09 0.97 0.80 SMI 93–96 SV-HL 1041 20 0.84 0.09 0.97 0.97 0.80 SV-CL 1041 20 0.81 0.08 0.96 0.79 0.80 SV-SL 2667 52 0.79 0.07 0.97 0.79 S&P500 04–14 SV-HL 2667 52 0.78 0.07 0.98 0.98 0.80 SV-CL 2667 52 0.79 0.08 0.97 0.80 0.80 SV-SL 5439 108 0.79 0.07 0.95 0.80 S&P500 93–14 SV-HL 5439 108 0.78 0.06 0.96 0.99 0.81 SV-CL 5439 108 0.79 0.08 0.97 0.80 0.81 SV-SL 5682 113 0.80 0.07 0.98 0.80 SMI 93–14 SV-HL 5682 113 0.79 0.07 0.99 0.99 0.81 SV-CL 5682 113 0.80 0.08 0.98 0.80 0.81
T + 1: sample size; K + 1: number of blocks.
now form the vector αit = (λi1t, λi2t, Zti). However, λ2,t+1 can be integrated out
of (A.7), yielding the prediction density:
π(λ1,t+1 | λ1t, λ2t, yt) = fN λ1,t+1; μ1+ φ1λ1t+ (μ2+ φ2λ2t)yt, σ12+ yt2σ22
.
(B.1) The observation density is now written as:
p(yt | αt) = p(yt | λ1t, Zt) = fN yt; β Zt− δ 2 ν− 2 exp λ1t 2 , Ztexp(λ1t) . (B.2) The filter is initialized by drawing N independent particles (λi10, λi20, Z0i) from (2.5), (2.7), and (2.10). The first term in (4.3) is computed as:
p(y0 | θ) = 1 N N $ i=1 p(y0 | λi10, Z0i)
and the filter weights at time t = 0 are computed as: wi0 = p(y0 | λi10, Z0i), π0i = w i 0 N j=1wj0 .
For t = 1, . . . , T , p(yt | y0:t−1,θ) is computed by the following algorithm,
which uses as input N ancestor particle vectors (λi1,t−1, λi2,t−1) with associated normalized importance weights πit−1.
(1) Draw N independent particles Zti from (2.5). (2) Compute, for i = 1, . . . , N:
˜
λi1t = μ1 + φ1λi1,t−1 + (μ2+ φ2λi2,t−1)yt−1 (B.3)
and compute the first stage weights:
wit−1|t= p(yt | ˜λi1t, Zti)× πit−1, πt−1|ti = wit−1|t N j=1wt−1|tj . (B.4) (3) For i = 1, . . . , N:
3.1 Sample an index k ∈ {1, . . . , N} with probability πt−1|tk .
3.2 Draw λi2tfrom a Normal distribution with expectation μ2+φ2λk2,t−1
and variance σ2 2.
3.3 Draw λi1t from an importance density derived along the lines of (A.15):
g(λ1t| λk1,t−1, λi2t, Zti, yt, yt−1) =
fN λ1t; m(λk1,t−1, λi2t, Zti, yt, yt−1), v(Zti, yt)
(B.5) where the moments m(•) and v(•) are obtained from the right- hand sides of (A.12), (A.16) and (A.17), upon replacing mt(λ1,t−1)
by μ1+ φ1λk1,t−1+ λi2tyt−1, Zt by Zti, and θt by β(Zti− δ2/(ν− 2)).
3.4 Compute the second stage weights:
wit = p(yt | λ
i
1t, Zti)p(λi1t | λk1,t−1, λi2t, yt−1)
g(λi1t| λk1,t−1, λi2t, Zti, yt, yt−1)p(yt | ˜λk1t, Zti)
(B.6) where k is the index drawn in step 3.1 and ˜λk1t is given by (B.3) with
i = k; the densities appearing in (B.6) are given by (B.2), (B.5),
and:
(4) Normalize the second stage weights to obtain: πti = w i t N j=1wtj .
(5) From Pitt et al. (2012), the estimated predictive density is: p(yt | y0:t−1,θ) = N $ i=1 wit−1|t N i=1wit N .
(6) Resample (λi1t, λi2t) if necessary, using the weights πit. References
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