Stochastic Volatility Models • bvhar

library(bvhar)

etf <- etf_vix[1:55, 1:3]
# Split-------------------------------
h <- 5
etf_eval <- divide_ts(etf, h)
etf_train <- etf_eval$train
etf_test <- etf_eval$test

Models with Stochastic Volatilities

By specifying cov_spec = set_sv(), var_bayes() and vhar_bayes() fits VAR-SV and VHAR-SV with shrinkage priors, respectively.

Three different prior for innovation covariance, and specify through bayes_spec
- Minneosta prior
  - BVAR: set_bvar()
  - BVHAR: set_bvhar() and set_weight_bvhar()
- SSVS prior: set_ssvs()
- Horseshoe prior: set_horseshoe()
- NG prior: set_ng()
- DL prior: set_dl()
sv_spec: prior settings for SV, set_sv()
intercept: prior for constant term, set_intercept()

set_sv()
#> Model Specification for SV with Cholesky Prior
#> 
#> Parameters: Contemporaneous coefficients, State variance, Initial state
#> Prior: Cholesky
#> ========================================================
#> Setting for 'shape':
#> [1]  rep(3, dim)
#> 
#> Setting for 'scale':
#> [1]  rep(0.01, dim)
#> 
#> Setting for 'initial_mean':
#> [1]  rep(1, dim)
#> 
#> Setting for 'initial_prec':
#> [1]  0.1 * diag(dim)

SSVS

(fit_ssvs <- vhar_bayes(etf_train, num_chains = 2, num_iter = 20, bayes_spec = set_ssvs(), cov_spec = set_sv(), include_mean = FALSE, minnesota = "longrun"))
#> Call:
#> vhar_bayes(y = etf_train, num_chains = 2, num_iter = 20, bayes_spec = set_ssvs(), 
#>     cov_spec = set_sv(), include_mean = FALSE, minnesota = "longrun")
#> 
#> BVHAR with Stochastic Volatility
#> Fitted by Gibbs sampling
#> Number of chains: 2
#> Total number of iteration: 20
#> Number of burn-in: 10
#> ====================================================
#> 
#> Parameter Record:
#> # A draws_df: 10 iterations, 2 chains, and 177 variables
#>      phi[1]  phi[2]   phi[3]   phi[4]   phi[5]   phi[6]  phi[7]   phi[8]
#> 1   -0.2154  -0.918  -0.7346   2.1794   1.7256   0.2252  -1.092   0.5860
#> 2   -0.2785   0.361  -0.1798  -0.0515   0.2544  -0.1548  -0.450  -0.3199
#> 3    0.0353   0.931   0.2222   0.0171  -0.2469  -0.0554   0.117   1.4097
#> 4    0.5949   0.525   0.1028   0.0410   0.0271  -0.0455  -0.951   0.7377
#> 5    0.3498   0.290   0.0491   0.2809  -0.0988  -0.1612  -2.714   0.2611
#> 6    0.2042  -0.608  -0.7952   1.0893  -0.2569   0.1259  -0.668   0.0569
#> 7    2.8408   0.429   0.8159  -0.1145  -0.4483  -0.0771  -1.257   0.3496
#> 8    1.2273   0.108  -0.2173   0.6008   0.2836  -0.5716   0.109   0.5889
#> 9    0.8396  -0.286   0.4581   1.8056  -1.0432   0.7261   0.666   0.0355
#> 10  -1.5707  -2.630   2.8384   2.5789  -0.1900  -0.6488   0.906  -0.0653
#> # ... with 10 more draws, and 169 more variables
#> # ... hidden reserved variables {'.chain', '.iteration', '.draw'}

Horseshoe

(fit_hs <- vhar_bayes(etf_train, num_chains = 2, num_iter = 20, bayes_spec = set_horseshoe(), cov_spec = set_sv(), include_mean = FALSE, minnesota = "longrun"))
#> Call:
#> vhar_bayes(y = etf_train, num_chains = 2, num_iter = 20, bayes_spec = set_horseshoe(), 
#>     cov_spec = set_sv(), include_mean = FALSE, minnesota = "longrun")
#> 
#> BVHAR with Stochastic Volatility
#> Fitted by Gibbs sampling
#> Number of chains: 2
#> Total number of iteration: 20
#> Number of burn-in: 10
#> ====================================================
#> 
#> Parameter Record:
#> # A draws_df: 10 iterations, 2 chains, and 211 variables
#>       phi[1]   phi[2]   phi[3]    phi[4]   phi[5]    phi[6]   phi[7]     phi[8]
#> 1    0.09387  -0.2328   0.4597  -0.18840   0.8345   0.01655  -0.9590   0.038319
#> 2   -0.09961  -0.6774   0.2270   0.16758   0.5573   0.00551   0.2524   0.064362
#> 3   -0.19695  -0.7337   0.0183   0.40152  -0.2635  -0.00795   0.0042   0.057482
#> 4   -0.09827  -0.1309  -0.1432  -0.00451   0.1331   0.00148   0.0495   0.020706
#> 5    0.11908   0.0935  -0.2156   0.22296  -0.1274  -0.00820  -0.0373  -0.005457
#> 6   -0.02723   0.0708   0.4767  -0.20709   0.1779  -0.00173   0.0112  -0.044321
#> 7    0.03845  -0.0797   0.4980  -0.24877   0.1477   0.02493   0.6637  -0.011320
#> 8   -0.00283   0.1016   0.4239   0.12432   0.0137  -0.01699   0.6483  -0.000937
#> 9   -0.06435   0.1563   0.4189   0.61074   0.1722  -0.01553   0.2496  -0.000081
#> 10   0.07823   0.1444   0.1609   0.70138   0.0250  -0.01597   0.1058  -0.001969
#> # ... with 10 more draws, and 203 more variables
#> # ... hidden reserved variables {'.chain', '.iteration', '.draw'}

Normal-Gamma prior

(fit_ng <- vhar_bayes(etf_train, num_chains = 2, num_iter = 20, bayes_spec = set_ng(), cov_spec = set_sv(), include_mean = FALSE, minnesota = "longrun"))
#> Call:
#> vhar_bayes(y = etf_train, num_chains = 2, num_iter = 20, bayes_spec = set_ng(), 
#>     cov_spec = set_sv(), include_mean = FALSE, minnesota = "longrun")
#> 
#> BVHAR with Stochastic Volatility
#> Fitted by Metropolis-within-Gibbs
#> Number of chains: 2
#> Total number of iteration: 20
#> Number of burn-in: 10
#> ====================================================
#> 
#> Parameter Record:
#> # A draws_df: 10 iterations, 2 chains, and 184 variables
#>       phi[1]   phi[2]    phi[3]  phi[4]    phi[5]     phi[6]   phi[7]   phi[8]
#> 1   -0.03503   0.4053   0.80020   2.371  -0.00209  -0.005311   0.3870  -0.1045
#> 2    0.39979   0.2931   0.91785   1.872   0.00119   0.027651   3.3153  -0.1662
#> 3   -0.35569  -0.0641   0.66248   0.597   0.00706   0.002321   1.6667  -0.3047
#> 4    0.06540  -0.0491   0.85497   0.114   0.04083  -0.000544   2.7827  -0.3749
#> 5   -0.02658   0.0935   0.23674   1.310  -0.00705  -0.319279   1.6540  -0.8789
#> 6    0.10168  -0.0181   0.16811   1.184   0.01823  -0.329084   0.7259   0.2146
#> 7    0.10309   0.0822  -0.01365   0.860  -0.01033   0.122883  -0.2532   0.3278
#> 8   -0.00633   0.3103  -0.03168  -0.237   0.04021   0.247472  -0.6210  -0.0877
#> 9    0.01719   0.1843  -0.00242   0.297   0.13990   0.005749   0.8780   0.0404
#> 10   0.00985  -0.0215  -0.09928   0.311  -0.13098  -0.117641   0.0465  -0.0343
#> # ... with 10 more draws, and 176 more variables
#> # ... hidden reserved variables {'.chain', '.iteration', '.draw'}

Dirichlet-Laplace prior

(fit_dl <- vhar_bayes(etf_train, num_chains = 2, num_iter = 20, bayes_spec = set_dl(), cov_spec = set_sv(), include_mean = FALSE, minnesota = "longrun"))
#> Call:
#> vhar_bayes(y = etf_train, num_chains = 2, num_iter = 20, bayes_spec = set_dl(), 
#>     cov_spec = set_sv(), include_mean = FALSE, minnesota = "longrun")
#> 
#> BVHAR with Stochastic Volatility
#> Fitted by Gibbs sampling
#> Number of chains: 2
#> Total number of iteration: 20
#> Number of burn-in: 10
#> ====================================================
#> 
#> Parameter Record:
#> # A draws_df: 10 iterations, 2 chains, and 178 variables
#>       phi[1]    phi[2]     phi[3]  phi[4]  phi[5]    phi[6]    phi[7]    phi[8]
#> 1   -0.00311   0.37352  -0.082172  -0.807  1.1321   0.08043   0.12454   0.91372
#> 2   -0.01038  -0.23417  -0.007496   2.195  0.1480  -0.01890  -0.00529   0.27286
#> 3    0.00177   0.03663   0.001182   1.423  0.0979   0.00538   0.08415   0.41596
#> 4    0.01179   0.00553   0.000538   0.280  0.6115   0.02157   0.04629   0.01366
#> 5   -0.02635  -0.28482   0.080125   0.844  0.3990   0.10108  -0.03438  -0.00469
#> 6    0.03827  -0.25505   0.043283   1.368  1.6705  -0.00775   0.05829  -0.07362
#> 7   -0.02251  -0.23306  -0.001043   1.185  1.1578   0.03061   0.02282   0.10891
#> 8   -0.06679  -0.38600  -0.027257   0.849  1.7294  -0.01657   0.02059  -0.14193
#> 9    0.22481  -0.16968  -0.007867   0.945  0.1527   0.13183   0.02438   0.09854
#> 10   0.04347   0.18982  -0.044606   0.471  0.2496   0.06070  -0.02120  -0.14466
#> # ... with 10 more draws, and 170 more variables
#> # ... hidden reserved variables {'.chain', '.iteration', '.draw'}

Bayesian visualization

autoplot() also provides Bayesian visualization. type = "trace" gives MCMC trace plot.

autoplot(fit_hs, type = "trace", regex_pars = "tau")

type = "dens" draws MCMC density plot.

autoplot(fit_hs, type = "dens", regex_pars = "tau")