fsvsample: Markov Chain Monte Carlo (MCMC) Sampling for the Factor...

In factorstochvol: Bayesian Estimation of (Sparse) Latent Factor Stochastic Volatility Models

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

Description

fsvsample simulates from the joint posterior distribution and returns the MCMC draws. It is the main workhorse to conduct inference for factor stochastic volatility models in this package.

Usage

fsvsample(
  y,
  factors = 1,
  draws = 1000,
  thin = 1,
  burnin = 1000,
  restrict = "none",
  zeromean = TRUE,
  priorfacloadtype = "rowwiseng",
  priorfacload = 0.1,
  facloadtol = 1e-18,
  priorng = c(1, 1),
  priormu = c(0, 10),
  priorphiidi = c(10, 3),
  priorphifac = c(10, 3),
  priorsigmaidi = 1,
  priorsigmafac = 1,
  priorh0idi = "stationary",
  priorh0fac = "stationary",
  priorbeta = c(0, 10000),
  keeptime = "last",
  heteroskedastic = TRUE,
  priorhomoskedastic = NA,
  runningstore = 6,
  runningstorethin = 10,
  runningstoremoments = 2,
  signident = TRUE,
  signswitch = FALSE,
  interweaving = 4,
  quiet = FALSE,
  samplefac = TRUE,
  startfac,
  startpara,
  startlogvar,
  startlatent,
  startlogvar0,
  startlatent0,
  startfacload,
  startfacloadvar,
  expert
)

Arguments

y	Data matrix. Each of m columns is assumed to contain a single (univariate) series of length n.
factors	Number of latent factors to be estimated.
draws	Number of MCMC draws kept after burn-in.
thin	Single number greater or equal to 1, coercible to integer. Every thinth MCMC draw is kept and returned. The default value is 1, corresponding to no thinning of the draws, i.e. every draw is stored.
burnin	Number of initial MCMC draws to be discarded.
restrict	Either "upper", "none", or "auto", indicating whether the factor loadings matrix should be restricted to have zeros above the diagonal ("upper"), whether all elements should be estimated from the data ("none"), or whether the function findrestrict should be invoked for a priori finding suitable zeros. Setting restrict to "upper" or "auto" often stabilizes MCMC estimation and can be important for identifying the factor loadings matrix, however, it generally is a strong prior assumption. Setting restrict to "none" is usually the preferred option if identification of the factor loadings matrix is of less concern but covariance estimation or prediction is the goal. Alternatively, restrict can be a logical matrix of dimension c(m, r) indicating which elements should be unrestricted (where restrict is FALSE) or zero (where restrict is TRUE).
zeromean	Logical. If FALSE, a constant mean is included in the model for each of the m univariate series. If TRUE, the mean is not modeled. Defaults to TRUE.
priorfacloadtype	Can be "normal", "rowwiseng", "colwiseng". "normal": Normal prior. The value of priorfacload is interpreted as the standard deviations of the Gaussian prior distributions for the factor loadings. "rowwiseng": Row-wise Normal-Gamma prior. The value of priorfacload is interpreted as the shrinkage parameter a. "colwiseng": Column-wise Normal-Gamma prior. The value of priorfacload is interpreted as the shrinkage parameter a. For details please see Kastner (2019).
priorfacload	Either a matrix of dimensions m times factors with positive elements or a single number (which will be recycled accordingly). The meaning of priorfacload depends on the setting of priorfacloadtype and is explained there.
facloadtol	Minimum number that the absolute value of a factor loadings draw can take. Prevents numerical issues that can appear when strong shrinkage is enforced if chosen to be greater than zero.
priorng	Two-element vector with positive entries indicating the Normal-Gamma prior's hyperhyperparameters c and d.
priormu	Vector of length 2 denoting prior mean and standard deviation for unconditional levels of the idiosyncratic log variance processes.
priorphiidi	Vector of length 2, indicating the shape parameters for the Beta prior distributions of the transformed parameters (phi+1)/2, where phi denotes the persistence of the idiosyncratic log variances.
priorphifac	Vector of length 2, indicating the shape parameters for the Beta prior distributions of the transformed parameters (phi+1)/2, where phi denotes the persistence of the factor log variances.
priorsigmaidi	Vector of length m containing the prior volatilities of log variances. If priorsigmaidi has exactly one element, it will be recycled for all idiosyncratic log variances.
priorsigmafac	Vector of length factors containing the prior volatilities of log variances. If priorsigmafac has exactly one element, it will be recycled for all factor log variances.
priorh0idi	Vector of length 1 or m, containing information about the Gaussian prior for the initial idiosyncratic log variances. If an element of priorh0idi is a nonnegative number, the conditional prior of the corresponding initial log variance h0 is assumed to be Gaussian with mean 0 and standard deviation priorh0idi times $sigma$. If an element of priorh0idi is the string 'stationary', the prior of the corresponding initial log volatility is taken to be from the stationary distribution, i.e. h0 is assumed to be Gaussian with mean 0 and variance $sigma^2/(1-phi^2)$.
priorh0fac	Vector of length 1 or factors, containing information about the Gaussian prior for the initial factor log variances. If an element of priorh0fac is a nonnegative number, the conditional prior of the corresponding initial log variance h0 is assumed to be Gaussian with mean 0 and standard deviation priorh0fac times $sigma$. If an element of priorh0fac is the string 'stationary', the prior of the corresponding initial log volatility is taken to be from the stationary distribution, i.e. h0 is assumed to be Gaussian with mean 0 and variance $sigma^2/(1-phi^2)$.
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 zeromean is TRUE.
keeptime	Either a number coercible to a positive integer, or a string equal to "all" or "last". If a number different from 1 is provided, only every keeptimeth latent log-volatility is being monitored. If, e.g., keeptime = 3, draws for the latent log variances h_1,h_4,h_7,... will be kept. If keeptime is set to "all", this is equivalent to setting it to 1. If keeptime is set to "last" (the default), only draws for the very last latent log variances h_n are kept.
heteroskedastic	Vector of length 1, 2, or m + factors, containing logical values indicating whether time-varying (heteroskedastic = TRUE) or constant (heteroskedastic = FALSE) variance should be estimated. If heteroskedastic is of length 2 it will be recycled accordingly, whereby the first element is used for all idiosyncratic variances and the second element is used for all factor variances.
priorhomoskedastic	Only used if at least one element of heteroskedastic is set to FALSE. In that case, priorhomoskedastic must be a matrix with positive entries and dimension c(m, 2). Values in column 1 will be interpreted as the shape and values in column 2 will be interpreted as the rate parameter of the corresponding inverse gamma prior distribution of the idisyncratic variances.
runningstore	Because most machines these days do not have enough memory to store all draws for all points in time, setting runningstore to an integer greater than 0 will cause fsvsample to store the first runningstoremoments ergodic moments of certain variables of interest. More specifically, mean, variance, skewness, etc. will be stored for certain variables if runningstore is set to a value... >= 1: Latent log variances h_1,h_2,...,h_(n+r). >= 2: Latent factors f_1,...,f_r. >= 3: Latent volatilities sqrt(exp(h_1,h_2,...,h_(n+r))). >= 4: Conditional covariance matrix and the square roots of its diagonal elements. >= 5: Conditional correlation matrix. >= 6: Communalities, i.e. proportions of variances explained through the common factors.
runningstorethin	How often should the calculation of running moments be conducted? Set to a value > 1 if you want to avoid time consuming calculations at every MCMC iteration.
runningstoremoments	Selects how many running moments (up to 4) should be calculated.
signident	If set to FALSE, no ex-post sign-identification is performed. Defaults to TRUE.
signswitch	Set to TRUE to turn on a random sign switch of factors and loadings. Note that the signs of each factor loadings matrix column and the corresponding factor cannot be identified from the likelihood.
interweaving	The following values for interweaving the factor loadings are accepted: 0: No interweaving. 1: Shallow interweaving through the diagonal entries. 2: Deep interweaving through the diagonal entries. 3: Shallow interweaving through the largest absolute entries in each column. 4: Deep interweaving through the largest absolute entries in each column. For details please see Kastner et al. (2017). A value of 4 is the highly recommended default.
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.
samplefac	If set to FALSE, the factors are not sampled (but remain at their starting values forever). This might be useful if one wants to include observed factors instead of latent ones.
startfac	optional numeric matrix of dimension c(factors, n), containing the starting values of the latent factors. In case of a single factor model, a numeric vector of length n is also accepted.
startpara	optional numeric matrix of dimension c(3, m + factors), containing the starting values for the parameter draws. The first m columns must contain parameters values corresponding to the idiosyncratic volatilities, the subsequent factor columns must contain parameter values corresponding to the factor volatilities. The first row of startpara corresponds to mu, the level of the log variances (can be arbitrary numerical values), the second row corresponds to phi, the persistence parameters of the log variances (numeric values between -1 and 1), and the third row corresponds to sigma (positive numeric values).
startlogvar	optional numeric matrix of dimension c(n, m + factors), containing the starting values of the latent log variances. The first m rows correspond to the idiosyncratic log variances, the subsequent factor rows correspond to the factor log variances. Was previously called startlatent.
startlatent	Deprecated. Please use startlogvar instead.
startlogvar0	optional numeric vector of length m + factors, containing the starting values of the initial latent log variances. The first m elements correspond to the idiosyncratic log variances, the subsequent factor elements correspond to the factor log variances. Was previously called startlatent0.
startlatent0	Deprecated. Please use startlogvar0 instead.
startfacload	optional numeric matrix of dimension c(m, factors), containing the starting values of the factor loadings. In case of a single factor model, a numeric vector of length n is also accepted.
startfacloadvar	optional numeric matrix of dimension c(m, factors), containing the starting values of the factor loadings variances \tau_{ij}^2. Used only when the normal-gamma prior is employed (priorfacloadtype != "normal") while ignored when static loadings variances are used (priorfacloadtype == "normal").
expert	optional named list of expert parameters for the univariate SV models (will be transformed and passed to the stochvol package). For most applications, the default values probably work best. Interested users are referred to Kastner and Frühwirth-Schnatter (2014), the package vignette, and Kastner (2016). If expert is provided, it may contain the following named elements: parameterization: Character string equal to "centered", "noncentered", "GIS_C", or "GIS_NC". Defaults to "GIS_C". mhcontrol: Single numeric value controlling the proposal density of a Metropolis-Hastings (MH) update step when sampling sigma. If mhcontrol is smaller than 0, an independence proposal will be used, while values greater than zero control the stepsize of a log-random-walk proposal. Defaults to -1. gammaprior: Single logical value indicating whether a Gamma prior for sigma^2 should be used. If set to FALSE, an Inverse Gamma prior is employed. Defaults to TRUE. truncnormal: Single logical value indicating whether a truncated Gaussian distribution should be used as proposal for draws of phi. If set to FALSE, a regular Gaussian prior is employed and the draw is immediately discarded when values outside the unit ball happen to be drawn. Defaults to FALSE. mhsteps: Either 1, 2, or 3. Indicates the number of blocks used for drawing from the posterior of the parameters. Defaults to 2. proposalvar4sigmaphi: Single positive number indicating the conditional prior variance of sigma*phi in the ridge proposal density for sampling (mu, phi). Defaults to 10^8. proposalvar4sigmatheta: Single positive number indicating the conditional prior variance of sigma*theta in the ridge proposal density for sampling (mu, phi). Defaults to 10^12.

Details

For details concerning the factor SV algorithm please see Kastner et al. (2017), details about the univariate SV estimation can be found in Kastner and Frühwirth-Schnatter (2014).

Value

The value returned is a list object of class fsvdraws holding

facload:

Array containing draws from the posterior distribution of the factor loadings matrix.

fac:

Array containing factor draws from the posterior distribution.

logvar:

Array containing idiosyncratic and factor initial log variance draws.

logvar0:

Array containing idiosyncratic and factor log variance draws.

para:

Array containing parameter draws form the posterior distribution.

y:

Matrix containing the data supplied.

latestauxiliary:

List containing the latest draws of auxiliary quantities used for sampling the factor loadings matrix.

runningstore:

List whose elements contain ergodic moments of certain variables of interest. See argument runningstore for details about what is being stored here.

config:

List containing information on configuration parameters.

priors:

List containing prior hyperparameter values.

identifier:

Matrix containing the indices of the series used for ex-post sign-identification along with the corresponding minimum distances to zero. See signident for details.

To display the output, use print, plot, and in particular specialized extractors and printing functions. The print method prints a high-level overview; specialized extractors such as covmat or runningcovmat are also available. The plot method invokes a simple covariance matrix plot; specialized plotting functions are linked in the documentation of plot.fsvdraws.

References

Kastner, G., Frühwirth-Schnatter, S., and Lopes, H.F. (2017). Efficient Bayesian Inference for Multivariate Factor Stochastic Volatility Models. Journal of Computational and Graphical Statistics, 26(4), 905–917, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/10618600.2017.1322091")}.

Kastner, G. (2019). Sparse Bayesian Time-Varying Covariance Estimation in Many Dimensions Journal of Econometrics, 210(1), 98–115, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.jeconom.2018.11.007")}

Kastner, G. (2016). Dealing with stochastic volatility in time series using the R package stochvol. Journal of Statistical Software, 69(5), 1–30, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v069.i05")}.

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, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.csda.2013.01.002")}.

Examples

# Load exchange rate data (ships with stochvol):
data(exrates, package = "stochvol")
exrates$date <- NULL

# Compute the percentage log returns:
dat <- 100 * logret(exrates)

# We are going to fit a one-factor model so the ordering is irrelevant
# NOTE that these are very few draws, you probably want more...
res <- fsvsample(dat, factors = 2, draws = 2000, burnin = 1000,
  runningstore = 6, zeromean = FALSE)

voltimeplot(res)

corimageplot(res, nrow(dat), plotCI = 'circle')

oldpar <- par(ask = TRUE)
plot(res)
par(oldpar)
pairs(t(res$beta[1:4, ]))

factorstochvol documentation built on Nov. 24, 2023, 5:08 p.m.