Abstract
A Bayesian analysis of the stochastic volatility model with regime-switching skewness in heavy-tailed errors is proposed using a generalized hyperbolic (GH) skew Student’s t-distribution. The skewness parameter is allowed to shift according to a first-order Markov switching process. We summarize Bayesian methods for model fitting and discuss analyses of exchange rate return time series. Empirical results show that interpretable regime-switching skewness can improve model fit and Value-at-Risk performance in a comparison against several other SV models with constant skewness or jump diffusions.
Appendix A MCMC algorithm for the SVRSt model
Generation of the SV parameters (ϕ, σ, ρ, μ, h), (Steps 1–4)
Step 1. The conditional posterior probability density of φ is given by
where
To generate sample from this conditional posterior distribution, we use the Metropolis-Hastings (MH) algorithm [see, e.g., Koop (2003)]. The candidate, denoted by φ*, is generated from the proposal distribution TN(-1,1)
where TN(c,d)(m, ν2) denotes the normal distribution with mean m and variance ν2 truncated on the interval (c, d). The candidate is accepted with the MH acceptance probability,where φ0 denotes the current point.
Step 2. The joint conditional posterior probability density of γ=(σ, ρ)′ is given by
Because this conditional posterior density is not easy to sample from, we use the MH algorithm with the proposal distribution computed by the normal-approximated density using the mode of the posterior. First, for the constraint of parameter space, i.e., R={γ: σ>0, |ρ|<1}, we consider the transformation of γ to λ=(λ1, λ2)′, where λ1=log σ, and λ2=log (1+ρ)–log(1–ρ), to facilitate the sample generation. Second, we search for the point, denoted by
that maximizes (or approximately maximizes) g(γ|‧), and obtain its transformed value Third, the candidate, denoted by λ*, is generated from the proposal distribution N(λ*, Σ*), whereand
denotes the transformed conditional posterior density. Finally, the candidate is accepted with the MH acceptance probabilitywhere γ0 and λ0 denote the current points, and J(γ) denotes the Jacobian for the above transformation evaluated at γ.
Step 3. The conditional posterior probability density of μ is equal to
whereStep 4. To generate sample of the log-volatility h, we utilize the so-called multi-move sampler, developed by Shephard and Pitt (1997) and Watanabe and Omori (2004). The idea is to divide the process (h1, …, hn) into several blocks and generate sample of each block given the other blocks. It can produce efficient draws in comparison with a single-move sampler which generates sample of one state, say ht, at a time given the others, {hk;1≤k≤n, k≠t}. Omori and Watanabe (2008) utilize the multi-move sampler for the SV model with leverage [see also Takahashi, Omori, and Watanabe (2009)]. Nakajima and Omori (2012) extend it to the case of the SV model with the GH skew Student’s t-distributed errors. Their algorithm can be straightforwardly utilized for the SVRSt model. The detail of the multi-move sampler is described in Appendix B.
Generation of skew-t parameters (β, ν, z) (Steps 5–7)
Step 5. The conditional posterior probability density of βi is equal to
wherewhere Ti denotes the set {t;1≤t≤n, st=i}, for i=1, …, K. This conditional posterior distribution is truncated on the interval that satisfies the identification constraint of β. For instance, when K=3, the parameter spaces for the sample generation are given by β1∈(–∞, β2), β2∈(β1, β3) and β3∈(β2, ∞).
Step 6. The conditional posterior probability density of ν is given by
a probability density from which is not easy to sample. We use the MH algorithm with the proposal distribution computed by a normal-approximated density using the mode of the posterior as in Step 2. Note that the candidate is generated from the truncated normal distribution for ν∈(4, ∞).
Step 7. The conditional posterior probability density of zt is given by
where
for t=1, …, n. We use the MH algorithm with the candidate, denoted by
generated from IG((ν+1)/2, ν/2). The candidate is accepted with the MH acceptance probability, where denotes the current point.Generation of Markov-switching parameters (s, p) (Steps 8–9)
Step 8. To generate the sample of (s1, …, sn) from the joint conditional posterior probability density, we use the multi-move sampler for the Markov-switching model [e.g., Carter and Kohn (1994), Chib (1996)]. The multi-move sampler primarily provides more efficient draws than a single-move sampler which generates the sample of st at a time given the others, {sk;1≤t≤n, k≠t}.
Define ϑ≡(θ, p, h, z) and
First, we recursively compute the following two steps:update step: π(st|Yt, ϑ)∝π(st|Yt–1, ϑ) f(yt|st, ϑ),
where
for t=1, …, n. We next generate sn from π(sn|Yn, ϑ), and then, recursively generate st for t=n–1, …, 1, following the probability, π(st|Yt, ϑ)×π(st+1|st, ϑ).
Step 9. The conditional posterior probability density of pi is proportional to
where nij denotes the size of the set {t;st=j, st–1=i, 2≤t≤n} for j=1, …, K. Because this conditional posterior distribution is not directly generated from, we use the MH algorithm with the candidate, denoted by generated from Dirichlet (ωi+ni), where ni=(ni1, …, niK). The candidate is accepted with the MH acceptance probability, where denotes the current point.Appendix B Multi-move sampler for the volatility in the SVRSt model
Extending the algorithm of Omori and Watanabe (2008), we describe the multi-move sampler for sampling the volatility h in the SVRSt model. Defining αt=ht–μ, for t=0, …, n and γ=exp(μ/2), we consider the state space model with respect to
asLet
To sample a block (αr+1, …, αr+d) from its joint conditional posterior density using MH algorithm, (r≥0, d≥1, r+d≤n), we sample disturbanceswhere
and ρt=ρ·I[r+d<n]. To determine the block (r and d), we use the stochastic knots of Shephard and Pitt (1997).
Let
and To construct a proposal density based on the normal approximation of the posterior density of we first definefor t=r+2, …, r+d, and Br+1=0. For the second derivatives, we take the expectations with respect to yt’s and obtain
Applying the second-order Taylor expansion to the log of the posterior density around the mode,
we obtain an approximate normal density as follows:where
and is the value of L, δ and Q at (or, equivalently at ). It can be shown that the proposal density is the posterior density of for a linear Gaussian state space model derived below. The mode can be obtained by repeating the following algorithm until it converges.Initialize
and compute at using the state equation (7) recursively.Evaluate
’s, ’s and ’s atLet
and Compute the following variables recursively for t=r+2, …, r+d:Define an auxiliary variable
where for t=r+1, …, r+d, andConsider the linear Gaussian state space model,
where ζt~N(0, I2),
Ht=(0,σ), for t=r+1, …, r+d, and Apply the Kalman filter and the disturbance smoother to this state space, and obtain the posterior mode andGo to 2.
In the MCMC sampling procedure, the current sample of
may be taken as an initial value of the in Step 1. To sample from the conditional posterior density, we implement the AR (Accept-Reject)-MH algorithm via the simulation smoother [e.g., de Jong and Shephard (1995), Durbin and Koopman (2002)] using the mode to obtain the approximated linear Gaussian state space model. See Omori and Watanabe (2008), Takahashi, Omori, and Watanabe (2009) for the detail of this AR-MH algorithm.Appendix C Priors
Prior distributions assumed in this paper are listed in Table 10.
Model | Prior-1 SVRSt-2 | Prior-2 SVRSt-3 | SVRSt-3s | |
---|---|---|---|---|
(φ+1)/2 | Beta(20, 1.5) | – | – | |
σ–2 | Gamma(2.5, 0.025) | – | – | |
ρ | U(–1, 1) | – | – | |
μ | N(–10, 1) | – | – | |
ν | Gamma(16, 0.8) | – | – | |
β1 | TN(–∞, 0) (–1, 2) | |||
β2 | ||||
β3 | TN(0,∞) (1, 2) | |||
p1 | Beta(499, 1) | – | – | |
p2 | Beta(499, 1) | – | – | |
Model | Prior-3SVt | SVSt | SVJt | |
(φ+1)/2 | Beta(20, 1.5) | – | – | |
σ–2 | Gamma(2.5, 0.025) | – | – | |
ρ | U(–1, 1) | – | – | |
μ | N(–10, 1) | – | – | |
ν | Gamma(16, 0.8) | – | – | |
β | N(0, 1) | |||
κ | Beta(2, 100) | |||
log (δ) | N(–2.5, 0.15) | |||
Model | Prior-4SVRSt | Prior-5SVRSt | Prior-6SVRSt | SVSt |
(φ+1)/2 | Beta(20, 1.5) | – | – | – |
σ–2 | Gamma(2.5, 0.025) | – | – | – |
ρ | U(–1, 1) | – | – | – |
μ | N(–10, 1) | – | – | – |
ν | Gamma(16, 0.8) | – | Gamma(24, 0.6) | – |
β1 | TN(–∞, 0) (–5, 10) | – | TN(–∞, 0) (–1, 2) | |
β2 | TN(0,∞) (5, 10) | – | TN(0,∞) (1, 2) | |
p1 | Beta(499, 1) | Beta(199, 1) | Beta(499, 1) | |
p2 | Beta(499, 1) | Beta(199, 1) | Beta(499, 1) |
“–” refers to the same distribution as in its left column.
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