Markov Chain Monte Carlo (MCMC) Sampling for the Stochastic Volatility (SV) Model

Source: R/wrappers.R

svsample simulates from the joint posterior distribution of the SV parameters mu, phi, sigma (and potentially nu and rho), along with the latent log-volatilities h_0,...,h_n and returns the MCMC draws. If a design matrix is provided, simple Bayesian regression can also be conducted. For similar functionality with a formula interface, see svlm.

svsample(
  y,
  draws = 10000,
  burnin = 1000,
  designmatrix = NA,
  priormu = c(0, 100),
  priorphi = c(5, 1.5),
  priorsigma = 1,
  priornu = 0,
  priorrho = NA,
  priorbeta = c(0, 10000),
  priorlatent0 = "stationary",
  priorspec = NULL,
  thin = 1,
  thinpara = thin,
  thinlatent = thin,
  keeptime = "all",
  quiet = FALSE,
  startpara = NULL,
  startlatent = NULL,
  parallel = c("no", "multicore", "snow"),
  n_cpus = 1L,
  cl = NULL,
  n_chains = 1L,
  print_progress = "automatic",
  expert = NULL,
  ...
)

svtsample(
  y,
  draws = 10000,
  burnin = 1000,
  designmatrix = NA,
  priormu = c(0, 100),
  priorphi = c(5, 1.5),
  priorsigma = 1,
  priornu = 0.1,
  priorrho = NA,
  priorbeta = c(0, 10000),
  priorlatent0 = "stationary",
  priorspec = NULL,
  thin = 1,
  thinpara = thin,
  thinlatent = thin,
  keeptime = "all",
  quiet = FALSE,
  startpara = NULL,
  startlatent = NULL,
  parallel = c("no", "multicore", "snow"),
  n_cpus = 1L,
  cl = NULL,
  n_chains = 1L,
  print_progress = "automatic",
  expert = NULL,
  ...
)

svlsample(
  y,
  draws = 20000,
  burnin = 2000,
  designmatrix = NA,
  priormu = c(0, 100),
  priorphi = c(5, 1.5),
  priorsigma = 1,
  priornu = 0,
  priorrho = c(4, 4),
  priorbeta = c(0, 10000),
  priorlatent0 = "stationary",
  priorspec = NULL,
  thin = 1,
  thinpara = thin,
  thinlatent = thin,
  keeptime = "all",
  quiet = FALSE,
  startpara = NULL,
  startlatent = NULL,
  parallel = c("no", "multicore", "snow"),
  n_cpus = 1L,
  cl = NULL,
  n_chains = 1L,
  print_progress = "automatic",
  expert = NULL,
  ...
)

svtlsample(
  y,
  draws = 20000,
  burnin = 2000,
  designmatrix = NA,
  priormu = c(0, 100),
  priorphi = c(5, 1.5),
  priorsigma = 1,
  priornu = 0.1,
  priorrho = c(4, 4),
  priorbeta = c(0, 10000),
  priorlatent0 = "stationary",
  priorspec = NULL,
  thin = 1,
  thinpara = thin,
  thinlatent = thin,
  keeptime = "all",
  quiet = FALSE,
  startpara = NULL,
  startlatent = NULL,
  parallel = c("no", "multicore", "snow"),
  n_cpus = 1L,
  cl = NULL,
  n_chains = 1L,
  print_progress = "automatic",
  expert = NULL,
  ...
)

svsample2(
  y,
  draws = 10000,
  burnin = 1000,
  designmatrix = NA,
  priormu = c(0, 100),
  priorphi = c(5, 1.5),
  priorsigma = 1,
  priornu = 0,
  priorrho = NA,
  priorbeta = c(0, 10000),
  priorlatent0 = "stationary",
  thinpara = 1,
  thinlatent = 1,
  keeptime = "all",
  quiet = FALSE,
  startpara = NULL,
  startlatent = NULL
)

Arguments

y

numeric vector containing the data (usually log-returns), which must not contain zeros. Alternatively, y can be an svsim object. In this case, the returns will be extracted and a message is signalled.

draws

single number greater or equal to 1, indicating the number of draws after burn-in (see below). Will be automatically coerced to integer. The default value is 10000.

burnin

single number greater or equal to 0, indicating the number of draws discarded as burn-in. Will be automatically coerced to integer. The default value is 1000.

designmatrix

regression design matrix for modeling the mean. Must have length(y) rows. Alternatively, designmatrix may be a string of the form "arX", where X is a nonnegative integer. To fit a constant mean model, use designmatrix = "ar0" (which is equivalent to designmatrix = matrix(1, nrow = length(y))). To fit an AR(1) model, use designmatrix = "ar1", and so on. If some elements of designmatrix are NA, the mean is fixed to zero (pre-1.2.0 behavior of stochvol).

priormu

numeric vector of length 2, indicating mean and standard deviation for the Gaussian prior distribution of the parameter mu, the level of the log-volatility. The default value is c(0, 100), which constitutes a practically uninformative prior for common exchange rate datasets, stock returns and the like.

priorphi

numeric vector of length 2, indicating the shape parameters for the Beta prior distribution of the transformed parameter (phi + 1) / 2, where phi denotes the persistence of the log-volatility. The default value is c(5, 1.5), which constitutes a prior that puts some belief in a persistent log-volatility but also encompasses the region where phi is around 0.

priorsigma

single positive real number, which stands for the scaling of the transformed parameter sigma^2, where sigma denotes the volatility of log-volatility. More precisely, sigma^2 ~ priorsigma * chisq(df = 1). The default value is 1, which constitutes a reasonably vague prior for many common exchange rate datasets, stock returns and the like.

priornu

single non-negative number, indicating the rate parameter for the exponential prior distribution of the parameter; can be Inf nu, the degrees-of-freedom parameter of the conditional innovations t-distribution. The default value is 0, fixing the degrees-of-freedom to infinity. This corresponds to conditional standard normal innovations, the pre-1.1.0 behavior of stochvol.

priorrho

either NA for the no-leverage case or a numeric vector of length 2 that specify the beta prior distribution for (rho+1)/2

priorbeta

numeric vector of length 2, indicating the mean and standard deviation of the Gaussian prior for the regression parameters. The default value is c(0, 10000), which constitutes a very vague prior for many common datasets. Not used if designmatrix is NA.

priorlatent0

either a single non-negative number or the string 'stationary' (the default, also the behavior before version 1.3.0). When priorlatent0 is equal to 'stationary', the stationary distribution of the latent AR(1)-process is used as the prior for the initial log-volatility h_0. When priorlatent0 is equal to a number \(B\), we have \(h_0 \sim N(\mu, B\sigma^2)\) a priori.

priorspec

in case one needs different prior distributions than the ones specified by priormu, ..., priorrho, a priorspec object can be supplied here. A smart constructor for this usecase is specify_priors.

thin

single number greater or equal to 1, coercible to integer. Every thinparath parameter and latent draw is kept and returned. The default value is 1, corresponding to no thinning of the parameter draws i.e. every draw is stored.

thinpara

single number greater or equal to 1, coercible to integer. Every thinparath parameter draw is kept and returned. The default value is thin.

thinlatent

single number greater or equal to 1, coercible to integer. Every thinlatentth latent variable draw is kept and returned. The default value is thin

keeptime

Either 'all' (the default) or 'last'. Indicates which latent volatility draws should be stored.

quiet

logical value indicating whether the progress bar and other informative output during sampling should be omitted. The default value is FALSE, implying verbose output.

startpara

optional named list, containing the starting values for the parameter draws. If supplied, startpara may contain elements named mu, phi, sigma, nu, rho, beta, and latent0. The default value is equal to the prior mean. In case of parallel execution with cl provided, startpara can be a list of named lists that initialize the parallel chains.

startlatent

optional vector of length length(y), containing the starting values for the latent log-volatility draws. The default value is rep(-10, length(y)). In case of parallel execution with cl provided, startlatent can be a list of numeric vectors that initialize the parallel chains.

parallel

optional one of "no" (default), "multicore", or "snow", indicating what type of parallellism is to be applied. Option "multicore" is not available on Windows.

n_cpus

optional positive integer, the number of CPUs to be used in case of parallel computations. Defaults to 1L. Ignored if parameter cl is supplied and parallel != "snow".

cl

optional so-called SNOW cluster object as implemented in package parallel. Ignored unless parallel == "snow".

n_chains

optional positive integer specifying the number of independent MCMC chains

print_progress

optional one of "automatic", "progressbar", or "iteration", controls the output. Ignored if quiet is TRUE.

expert

optional named list of expert parameters. For most applications, the default values probably work best. Interested users are referred to the literature provided in the References section. If expert is provided, it may contain the following named elements:

interweave

Logical value. If TRUE (the default), then ancillarity-sufficiency interweaving strategy (ASIS) is applied to improve on the sampling efficiency for the parameters. Otherwise one parameterization is used.

correct_model_misspecification

Logical value. If FALSE (the default), then auxiliary mixture sampling is used to sample the latent states. If TRUE, extra computations are made to correct for model misspecification either ex-post by reweighting or on-line using a Metropolis-Hastings step.

...

Any extra arguments will be forwarded to updatesummary, controlling the type of statistics calculated for the posterior draws.

Value

The value returned is a list object of class svdraws holding

para

mcmc.list object containing the parameter draws from the posterior distribution.

latent

mcmc.list object containing the latent instantaneous log-volatility draws from the posterior distribution.

latent0

mcmc.list object containing the latent initial log-volatility draws from the posterior distribution.

tau

mcmc.list object containing the latent variance inflation factors for the sampler with conditional t-innovations (optional).

beta

mcmc.list object containing the regression coefficient draws from the posterior distribution (optional).

y

the left hand side of the observation equation, usually the argument y. In case of an AR(k) specification, the first k elements are removed.

runtime

proc_time object containing the run time of the sampler.

priors

a priorspec object containing the parameter values of the prior distributions for mu, phi, sigma, nu, rho, and betas, and the variance of specification for latent0.

thinning

list containing the thinning parameters, i.e. the arguments thinpara, thinlatent and keeptime.

summary

list containing a collection of summary statistics of the posterior draws for para, latent, and latent0.

meanmodel

character containing information about how designmatrix was employed.

To display the output, use print, summary and plot. The print method simply prints the posterior draws (which is very likely a lot of output); the summary method displays the summary statistics currently stored in the object; the plot method plot.svdraws gives a graphical overview of the posterior distribution by calling volplot, traceplot and densplot and displaying the results on a single page.

Details

Functions svtsample, svlsample, and svtlsample are wrappers around svsample with convenient default values for the SV model with t-errors, leverage, and both t-errors and leverage, respectively.

For details concerning the algorithm please see the paper by Kastner and Frühwirth-Schnatter (2014) and Hosszejni and Kastner (2019).

Note

If y contains zeros, you might want to consider de-meaning your returns or use designmatrix = "ar0".

svsample2 is deprecated.

References

Kastner, G. and Frühwirth-Schnatter, S. (2014). Ancillarity-sufficiency interweaving strategy (ASIS) for boosting MCMC estimation of stochastic volatility models. Computational Statistics & Data Analysis, 76, 408--423, doi:10.1016/j.csda.2013.01.002 .

Hosszejni, D. and Kastner, G. (2019). Approaches Toward the Bayesian Estimation of the Stochastic Volatility Model with Leverage. Springer Proceedings in Mathematics & Statistics, 296, 75--83, doi:10.1007/978-3-030-30611-3_8 .

Examples

###############
# Full examples
###############

# Example 1
## Simulate a short and highly persistent SV process 
sim <- svsim(100, mu = -10, phi = 0.99, sigma = 0.2)

## Obtain 5000 draws from the sampler (that's not a lot)
draws <-
  svsample(sim, draws = 5000, burnin = 100,
           priormu = c(-10, 1), priorphi = c(20, 1.5), priorsigma = 0.2)
#> Extracted data vector from 'svsim'-object.
#> Done!
#> Summarizing posterior draws...

## Check out the results
summary(draws)
#> 
#> Summary of 'svdraws' object
#> 
#> Prior distributions:
#> mu        ~ Normal(mean = -10, sd = 1)
#> (phi+1)/2 ~ Beta(a = 20, b = 1.5)
#> sigma^2   ~ Gamma(shape = 0.5, rate = 2.5)
#> nu        ~ Infinity
#> rho       ~ Constant(value = 0)
#> 
#> Stored 5000 MCMC draws after a burn-in of 100.
#> No thinning.
#> 
#> Posterior draws of SV parameters (thinning = 1):
#>               mean      sd       5%      50%      95% ESS
#> mu        -11.3240 0.32117 -1.2e+01 -11.3599 -10.8211 951
#> phi         0.8443 0.10176  6.4e-01   0.8620   0.9763 416
#> sigma       0.2152 0.13545  2.3e-02   0.2011   0.4625 255
#> exp(mu/2)   0.0035 0.00073  2.8e-03   0.0034   0.0045 951
#> sigma^2     0.0646 0.07473  5.3e-04   0.0404   0.2139 255
#> 
plot(draws)
#> Simulation object extracted from input



# \donttest{
# Example 2
## Simulate an asymmetric and conditionally heavy-tailed SV process
sim <- svsim(150, mu = -10, phi = 0.96, sigma = 0.3, nu = 10, rho = -0.3)

## Obtain 10000 draws from the sampler
## Use more advanced prior settings
## Run two parallel MCMC chains
advanced_draws <-
  svsample(sim, draws = 10000, burnin = 5000,
           priorspec = specify_priors(mu = sv_normal(-10, 1),
                                      sigma2 = sv_gamma(0.5, 2),
                                      rho = sv_beta(4, 4),
                                      nu = sv_constant(5)),
           parallel = "snow", n_chains = 2, n_cpus = 2)
#> Extracted data vector from 'svsim'-object.
#> Done!
#> Summarizing posterior draws...

## Check out the results
summary(advanced_draws)
#> 
#> Summary of 'svdraws' object
#> 
#> Prior distributions:
#> mu        ~ Normal(mean = -10, sd = 1)
#> (phi+1)/2 ~ Beta(a = 5, b = 1.5)
#> sigma^2   ~ Gamma(shape = 0.5, rate = 2)
#> nu        ~ Constant(value = 5)
#> rho       ~ Beta(a = 4, b = 4)
#> 
#> Stored 10000 MCMC draws after a burn-in of 5000.
#> No thinning.
#> 
#> Posterior draws of SV parameters (thinning = 1):
#>               mean      sd       5%      50%     95%  ESS
#> mu        -10.3541 0.25921 -10.7697 -10.3508 -9.9495 2987
#> phi         0.7439 0.17158   0.4016   0.7863  0.9317  129
#> sigma       0.4873 0.17549   0.2261   0.4741  0.7879  171
#> rho        -0.0196 0.22258  -0.3994  -0.0116  0.3259  183
#> exp(mu/2)   0.0057 0.00074   0.0046   0.0057  0.0069 2987
#> sigma^2     0.2683 0.18948   0.0511   0.2248  0.6208  171
#> 
plot(advanced_draws)
#> Simulation object extracted from input



# Example 3
## AR(1) structure for the mean
data(exrates)
len <- 3000
ahead <- 100
y <- head(exrates$USD, len)

## Fit AR(1)-SVL model to EUR-USD exchange rates
res <- svsample(y, designmatrix = "ar1")
#> Done!
#> Summarizing posterior draws...

## Use predict.svdraws to obtain predictive distributions
preddraws <- predict(res, steps = ahead)

## Calculate predictive quantiles
predquants <- apply(predy(preddraws), 2, quantile, c(.1, .5, .9))

## Visualize
expost <- tail(head(exrates$USD, len+ahead), ahead)
ts.plot(y, xlim = c(length(y)-4*ahead, length(y)+ahead),
         ylim = range(c(predquants, expost, tail(y, 4*ahead))))
for (i in 1:3) {
  lines((length(y)+1):(length(y)+ahead), predquants[i,],
        col = 3, lty = c(2, 1, 2)[i])
}
lines((length(y)+1):(length(y)+ahead), expost,
      col = 2)



# Example 4
## Predicting USD based on JPY and GBP in the mean
data(exrates)
len <- 3000
ahead <- 30
## Calculate log-returns
logreturns <- apply(exrates[, c("USD", "JPY", "GBP")], 2,
                    function (x) diff(log(x)))
logretUSD <- logreturns[2:(len+1), "USD"]
regressors <- cbind(1, as.matrix(logreturns[1:len, ]))  # lagged by 1 day

## Fit SV model to EUR-USD exchange rates
res <- svsample(logretUSD, designmatrix = regressors)
#> Done!
#> Summarizing posterior draws...

## Use predict.svdraws to obtain predictive distributions
predregressors <- cbind(1, as.matrix(logreturns[(len+1):(len+ahead), ]))
preddraws <- predict(res, steps = ahead,
                     newdata = predregressors)
predprice <- exrates[len+2, "USD"] * exp(t(apply(predy(preddraws), 1, cumsum)))

## Calculate predictive quantiles
predquants <- apply(predprice, 2, quantile, c(.1, .5, .9))

## Visualize
priceUSD <- exrates[3:(len+2), "USD"]
expost <- exrates[(len+3):(len+ahead+2), "USD"]
ts.plot(priceUSD, xlim = c(len-4*ahead, len+ahead+1),
         ylim = range(c(expost, predquants, tail(priceUSD, 4*ahead))))
for (i in 1:3) {
  lines(len:(len+ahead), c(tail(priceUSD, 1), predquants[i,]),
        col = 3, lty = c(2, 1, 2)[i])
}
lines(len:(len+ahead), c(tail(priceUSD, 1), expost),
      col = 2)

# }

########################
# Further short examples
########################

y <- svsim(50, nu = 10, rho = -0.1)$y

# Supply initial values
res <- svsample(y,
                startpara = list(mu = -10, sigma = 1))
#> Done!
#> Summarizing posterior draws...

# \donttest{
# Supply initial values for parallel chains
res <- svsample(y,
                startpara = list(list(mu = -10, sigma = 1),
                                 list(mu = -11, sigma = .1, phi = 0.9),
                                 list(mu = -9, sigma = .3, phi = 0.7)),
                parallel = "snow", n_chains = 3, n_cpus = 2)
#> Done!
#> Summarizing posterior draws...

# Parallel chains with with a pre-defined cluster object
cl <- parallel::makeCluster(2, "PSOCK", outfile = NULL)  # print to console
res <- svsample(y,
                parallel = "snow", n_chains = 3, cl = cl)
#> Done!
#> Summarizing posterior draws...
parallel::stopCluster(cl)
# }

# Turn on correction for model misspecification
## Since the approximate model is fast and it is working very
##   well in practice, this is turned off by default
res <- svsample(y,
                expert = list(correct_model_misspecification = TRUE))
#> Done!
#> Summarizing posterior draws...
#> No computation of effective sample size after re-sampling
print(res)
#> 
#> Summary of 'svdraws' object
#> 
#> Prior distributions:
#> mu        ~ Normal(mean = 0, sd = 100)
#> (phi+1)/2 ~ Beta(a = 5, b = 1.5)
#> sigma^2   ~ Gamma(shape = 0.5, rate = 0.5)
#> nu        ~ Infinity
#> rho       ~ Constant(value = 0)
#> 
#> Stored 10000 MCMC draws after a burn-in of 1000.
#> No thinning.
#> 
#> Posterior draws of SV parameters (thinning = 1):
#>               mean     sd       5%      50%     95%
#> mu        -10.1530 0.3265 -10.6877 -10.1392 -9.6695
#> phi         0.3689 0.3128  -0.1722   0.3838  0.8502
#> sigma       0.5682 0.3590   0.0568   0.5309  1.2278
#> exp(mu/2)   0.0063 0.0011   0.0048   0.0063  0.0079
#> sigma^2     0.4517 0.5209   0.0032   0.2818  1.5075
#> 
#> Posterior draws from the auxiliary SV model were re-sampled after the MCMC
#> procedure ended to correct for model mis-specification.
#>   Re-sampling of parameters and latents was of little practical importance:
#>     - max reachable entropy for this sample size: 9.21034,
#>     - entropy of the re-sampling weight distribution: 9.209905
#> 

if (FALSE) {
# Parallel multicore chains (not available on Windows)
res <- svsample(y, draws = 30000, burnin = 10000,
                parallel = "multicore", n_chains = 3, n_cpus = 2)

# Plot using a color palette
palette(rainbow(coda::nchain(para(res, "all"))))
plot(res)

# Use functionality from package 'coda'
## E.g. Geweke's convergence diagnostics
coda::geweke.diag(para(res, "all")[, c("mu", "phi", "sigma")])

# Use functionality from package 'bayesplot'
bayesplot::mcmc_pairs(res, pars = c("sigma", "mu", "phi", "h_0", "h_15"))
}