Particle Gibbs with ancestor sampling for stochastic volatility models with: heavy tails, in mean effects, leverage, serial dependence and structural breaks
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Nima Nonejad
Abstract
Particle Gibbs with ancestor sampling (PG-AS) is a new tool in the family of sequential Monte Carlo methods. We apply PG-AS to the challenging class of stochastic volatility models with increasing complexity, including leverage and in mean effects. We provide applications that demonstrate the flexibility of PG-AS under these different circumstances and justify applying it in practice. We also combine discrete structural breaks within the stochastic volatility model framework. For instance, we model changing time series characteristics of monthly postwar US core inflation rate using a structural break autoregressive fractionally integrated moving average (ARFIMA) model with stochastic volatility. We allow for structural breaks in the level, long and short-memory parameters with simultaneous breaks in the level, persistence and the conditional volatility of the volatility of inflation.
A Appendix
A.1 Prior sensitivity analysis
In this section, sensitivity of the results to prior specification is evaluated by investigating alternative prior hyperparameter values on the transition probabilities, pkk∼Beta(a0, b0), keeping prior hyperparameter values of the other parameters the same as in the main text. pkk, k=1, …, m–1 is one of the key parameters of the model because it controls the duration of each regime in S.
We experiment with different hyperparameter values on pkk in Table 6. We report the break dates for each of them by estimating CP(2)-ARFIMA-SV using the corresponding values of a0 and b0. For instance, the first alternative prior is pkk∼Beta(0.1,0.1), which is relatively flat. With this prior, we still find that the change-point dates correspond to 1973:7 and 1984:2. In fact, regardless of the values of a0 and b0, we still find that the change-point dates for each of these specifications correspond to 1973:7 and 1984:2. We also report logBF of CP(2)-ARFIMA-SV versus ARFIMA-SV using the corresponding values of α, along with the difference in DIC between CP(2)-ARFIMA-SV and ARFIMA-SV, see Table 6. These results overwhelmingly suggest existence of structural breaks. More importantly, we find that the choice of prior hyperparameter values on p is of relatively limited importance.
Prior sensitivity analysis, CP(2)-ARFIMA-SV.
| Prior | Break dates | logBFα=0.50 | logBFα=0.75 | logBFα=0.95 | logBFα=0.99 | Diff(DIC) |
|---|---|---|---|---|---|---|
| Beta(0.1, 0.1) | 1973:7, 1984:2 | 59.345 | 59.347 | 59.346 | 59.346 | –33.251 |
| Beta(8, 0.1) | 1973:7, 1984:2 | 67.807 | 67.808 | 67.808 | 67.808 | –33.385 |
| Beta(20, 0.1) | 1973:7, 1984:2 | 74.911 | 74.913 | 74.912 | 74.912 | –27.454 |
| Beta(100, 0.1) | 1973:7, 1984:2 | 60.123 | 60.125 | 60.124 | 60.124 | –29.880 |
This table compares the performance of CP(2)-ARFIMA-SV for different values of a0 and b0, where pkk∼Beta(a0, b0). The priors of the other parameters are set according to the main text. logBF, logarithm of the Bayes factor of CP(2)-ARFIMA-SV versus ARFIMA-SV using the corresponding value of α. diff(DIC), difference in DIC between CP(2)-ARFIMA-SV and ARFIMA-SV.
A.2 Sensitivity of PG-AS with respect to M
We often find that the choice of M is important because it ensures that the estimate of h1:T is not too jittery or imprecise. Furthermore, increasing M also increases the computation time. Therefore, it is important to find a reasonable value for M that avoids the above mentioned problems. In the following, we experiment with different values of M to find out its effects on estimation results. We do this by re-estimating the SV model using the DJIA data for M=2, 10, 100 and 1000. We report parameter estimates of the SV model using the above mentioned number of particles in Table 7. Besides these estimates, we also report the inefficiency factors (RB) of the parameters and h1:T for each case, see Figure 5. Furthermore, we compute Geweke’s convergence statistics and present estimation time in seconds for each M. In each case, we sample N=20,000 draws from p(θ,h1:T|YT) after a burn-in of 1000.
Sensitivity of the PG-AS sampler with respect to M.
Graphs (A)–(D): Box plots of the inefficiency factors of h1:T using the corresponding number of particles.
Sensitivity of PG-AS with respect to M, DJIA daily returns.
| Parameter | Mean | Std. dev | 5%-tile | 95%-tile | RB | Geweke |
|---|---|---|---|---|---|---|
| PG-AS, M=2 | ||||||
| μ | 0.088 | (0.019) | 0.057 | 0.119 | 6.091 | 0.800 |
| μh | –0.126 | (0.289) | –0.587 | 0.342 | 1.887 | –0.532 |
| ϕh | 0.980 | (0.006) | 0.969 | 0.989 | 22.883 | 1.175 |
| | 0.051 | (0.011) | 0.035 | 0.070 | 61.345 | –1.446 |
| PG-AS, M=10 | ||||||
| μ | 0.089 | (0.019) | 0.057 | 0.120 | 2.458 | 0.186 |
| μh | –0.120 | (0.299) | –0.589 | 0.362 | 1.280 | 0.309 |
| ϕh | 0.980 | (0.006) | 0.969 | 0.989 | 17.867 | –0.134 |
| | 0.050 | (0.011) | 0.034 | 0.069 | 48.682 | 0.629 |
| PG-AS, M=100 | ||||||
| μ | 0.089 | (0.019) | 0.058 | 0.120 | 1.804 | 1.103 |
| μh | –0.123 | (0.293) | –0.590 | 0.338 | 1.050 | 0.757 |
| ϕh | 0.979 | (0.006) | 0.968 | 0.989 | 17.703 | –1.691 |
| | 0.052 | (0.012) | 0.035 | 0.073 | 46.176 | 1.719 |
| PG-AS, M=1000 | ||||||
| μ | 0.088 | (0.019) | 0.057 | 0.119 | 1.924 | –0.899 |
| μh | –0.120 | (0.296) | –0.587 | 0.352 | 1.002 | –1.522 |
| ϕh | 0.979 | (0.006) | 0.969 | 0.989 | 17.550 | 0.021 |
| | 0.051 | (0.011) | 0.034 | 0.071 | 45.178 | 0.331 |
| Gibbs sampling, Kim et al. (1998) | ||||||
| μ | 0.086 | (0.019) | 0.055 | 0.117 | 2.210 | –0.498 |
| μh | –0.124 | (0.338) | –0.764 | 0.285 | 1.340 | –1.422 |
| ϕh | 0.983 | (0.006) | 0.973 | 0.992 | 17.089 | 0.871 |
| | 0.048 | (0.009) | 0.029 | 0.069 | 47.383 | –0.905 |
RB, Inefficiency factor (using a bandwidth, B, of 100); Geweke, Geweke’s convergence statistic; PG-AS, M=2, Particle Gibbs with ancestor sampling using M=2 particles, etc; Gibbs sampling, Estimation of the SV model using the method of Kim et al. (1998). Total number of observations, T=1740.
Overall, we see that PG-AS performs very well as parameter estimates are very similar regardless the values of M. In fact, we get almost identical results for M=10 and M=100. However, the RBs for both θ and h1:T decrease for M=100. In Figure 5, for M=10, 75% of h1:Ts have inefficiency factors <8, while for M=100 this number is close to 4. Compared to M=100, we do not obtain any significant gains in RB for M=1000. However, as M increases the computation time also increases. From this point of view, M=1000 seems computationally very demanding. For instance, for M=100 the computation time is around 3.5 h, whereas for M=1000 the computation time is around 11 h.
Finally, since the SV model is a relatively simple model, we also report estimation results for this model using the algorithm of Kim et al. (1998) in the bottom part of Table 7. We sample N=20,000 draws (after a burn-in of 1000) from p(θ, h1:T, z1:T|YT), where z1:T are the mixture component indicators, see Kim et al. (1998). As expected, the Gibbs sampler estimates of Kim et al. (1998) are very similar to the PG-AS estimates. Furthermore, we see that PG-AS provides very similar mixing compared to Kim et al. (1998) for the model parameters.
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Supplemental Material
The online version of this article (DOI: 10.1515/snde-2014-0043) offers supplementary material, available to authorized users.
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Articles in the same Issue
- Frontmatter
- Fourier inversion formulas for multiple-asset option pricing
- Particle Gibbs with ancestor sampling for stochastic volatility models with: heavy tails, in mean effects, leverage, serial dependence and structural breaks
- Testing the relationships between shadow economy and unemployment: empirical evidence from linear and nonlinear tests
- Business cycle (de)synchronization in the aftermath of the global financial crisis: implications for the Euro area
- Amplitude and phase synchronization of European business cycles: a wavelet approach
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Articles in the same Issue
- Frontmatter
- Fourier inversion formulas for multiple-asset option pricing
- Particle Gibbs with ancestor sampling for stochastic volatility models with: heavy tails, in mean effects, leverage, serial dependence and structural breaks
- Testing the relationships between shadow economy and unemployment: empirical evidence from linear and nonlinear tests
- Business cycle (de)synchronization in the aftermath of the global financial crisis: implications for the Euro area
- Amplitude and phase synchronization of European business cycles: a wavelet approach
- On the relationship between oil and gold before and after financial crisis: linear, nonlinear and time-varying causality testing
- Stock market’s reaction to money supply: a nonparametric analysis
