shrinkDTVP: Markov Chain Monte Carlo (MCMC) for time-varying parameter...

In shrinkTVP: Efficient Bayesian Inference for Time-Varying Parameter Models with Shrinkage

Markov Chain Monte Carlo (MCMC) for time-varying parameter models with dynamic shrinkage

Description

shrinkTVP samples from the joint posterior distribution of the parameters of a time-varying parameter model with dynamic triple gamma shrinkage, potentially including stochastic volatility (SV), and returns the MCMC draws.

Usage

shrinkDTVP(
  formula,
  data,
  mod_type = "double",
  niter = 10000,
  nburn = round(niter/2),
  nthin = 1,
  learn_a_xi = TRUE,
  learn_a_tau = TRUE,
  a_xi = 0.1,
  a_tau = 0.1,
  learn_c_xi = TRUE,
  learn_c_tau = TRUE,
  c_xi = 0.1,
  c_tau = 0.1,
  a_eq_c_xi = FALSE,
  a_eq_c_tau = FALSE,
  learn_kappa2_B = TRUE,
  learn_lambda2_B = TRUE,
  kappa2_B = 20,
  lambda2_B = 20,
  a_psi = 0.5,
  c_psi = 0.5,
  iid = FALSE,
  shrink_inter = TRUE,
  hyperprior_param,
  display_progress = TRUE,
  sv = FALSE,
  sv_param,
  MH_tuning,
  starting_vals
)

Arguments

formula	object of class "formula": a symbolic representation of the model, as in the function lm. For details, see formula.
data	optional data frame containing the response variable and the covariates. If not found in data, the variables are taken from environment(formula), typically the environment from which shrinkTVP is called. No NAs are allowed in the response variable and the covariates.
mod_type	character string that reads either "triple", "double" or "ridge". Determines whether the triple gamma, double gamma or ridge prior are used for theta_sr and beta_mean. The default is "double".
niter	positive integer, indicating the number of MCMC iterations to perform, including the burn-in. Has to be larger than or equal to nburn + 2. The default value is 10000.
nburn	non-negative integer, indicating the number of iterations discarded as burn-in. Has to be smaller than or equal to niter - 2. The default value is round(niter / 2).
nthin	positive integer, indicating the degree of thinning to be performed. Every nthin draw is kept and returned. The default value is 1, implying that every draw is kept.
learn_a_xi	logical value indicating whether to learn a_xi, the spike parameter of the state variances. Ignored if mod_type is set to "ridge". The default value is TRUE.
learn_a_tau	logical value indicating whether to learn a_tau, the spike parameter of the mean of the initial values of the states. Ignored if mod_type is set to "ridge". The default value is TRUE.
a_xi	positive, real number, indicating the (fixed) value for a_xi. Ignored if learn_a_xi is TRUE or mod_type is set to "ridge". The default value is 0.1.
a_tau	positive, real number, indicating the (fixed) value for a_tau. Ignored if learn_a_tau is TRUE or mod_type is set to "ridge". The default value is 0.1.
learn_c_xi	logical value indicating whether to learn c_xi, the tail parameter of the state variances. Ignored if mod_type is not set to "triple" or a_eq_c_xi is set to TRUE. The default value is TRUE.
learn_c_tau	logical value indicating whether to learn c_tau, the tail parameter of the mean of the initial values of the states. Ignored if mod_type is not set to "triple" or a_eq_c_tau is set to TRUE. The default value is TRUE.
c_xi	positive, real number, indicating the (fixed) value for c_xi. Ignored if learn_c_xi is TRUE, mod_type is not set to "triple" or a_eq_c_xi is set to TRUE. The default value is 0.1.
c_tau	positive, real number, indicating the (fixed) value for c_tau. Ignored if learn_c_xi is TRUE, mod_type is not set to "triple" or a_eq_c_tau is set to TRUE. The default value is 0.1.
a_eq_c_xi	logical value indicating whether to force a_xi and c_xi to be equal. If set to TRUE, beta_a_xi and alpha_a_xi are used as the hyperparameters and beta_c_xi and alpha_c_xi are ignored. Ignored if mod_type is not set to "triple". The default value is FALSE.
a_eq_c_tau	logical value indicating whether to force a_tau and c_tau to be equal. If set to TRUE, beta_a_tau and alpha_a_tau are used as the hyperparameters and beta_c_tau and alpha_c_tau are ignored. Ignored if mod_type is not set to "triple". The default value is FALSE.
learn_kappa2_B	logical value indicating whether to learn kappa2_B, the global level of shrinkage for the state variances. The default value is TRUE.
learn_lambda2_B	logical value indicating whether to learn the lambda2_B parameter, the global level of shrinkage for the mean of the initial values of the states. The default value is TRUE.
kappa2_B	positive, real number, indicating the (fixed) value for kappa2_B. Ignored if learn_kappa2_B is TRUE. The default value is 20.
lambda2_B	positive, real number, indicating the (fixed) value for lambda2_B. Ignored if learn_lambda2_B is TRUE. The default value is 20.
a_psi	positive, real number, or a vector of length equal to the number of covariates containing positive, real numbers. Indicates the value for a_psi, which is the pole parameter of the dynamic triple gamma. The default value is 0.5.
c_psi	positive, real number, or a vector of length equal to the number of covariates containing positive, real numbers. Indicates the value for c_psi, which is the tail parameter of the dynamic triple gamma. The default value is 0.5.
iid	logical value indicating whether the innovations are assumed to be independent and identically distributed. If set to TRUE, the innovations are assumed to be a priori iid triple gamma. If set to FALSE, the prior on the innovations is the dynamic triple gamma specification of Knaus and Frühwirth-Schnatter (2023). The default value is FALSE.
shrink_inter	logical value indicating whether to dynamically shrink the intercept. Note that shrinkage is still applied to the theta_sr and beta_mean associated with the intercept. The intercept column is automatically determined by the function and does not have to be included in the formula. This is done by finding the column that contains only 1s. The default value is TRUE.
hyperprior_param	optional named list containing hyperparameter values. Not all have to be supplied, with those missing being replaced by the default values. Any list elements that are misnamed will be ignored and a warning will be thrown. All hyperparameter values have to be positive, real numbers. The following hyperparameters can be supplied: c0: The default value is 2.5. g0: The default value is 5. G0: The default value is 5 / (2.5 - 1). e1: The default value is 0.001. e2: The default value is 0.001. d1: The default value is 0.001. d2: The default value is 0.001. alpha_a_xi: The default value is 5. alpha_a_tau: The default value is 5. beta_a_xi: The default value is 10. beta_a_tau: The default value is 10. alpha_c_xi: The default value is 5. alpha_c_tau: The default value is 5. beta_c_xi: The default value is 2. beta_c_tau: The default value is 2. a_rho: The default value is 2. b_rho: The default value is 0.95. alpha_rho: The default value is 0.5. beta_rho: The default value is 3.
display_progress	logical value indicating whether the progress bar and other informative output should be displayed. The default value is TRUE.
sv	logical value indicating whether to use stochastic volatility for the error of the observation equation. For details please see stochvol, in particular svsample. The default value is FALSE.
sv_param	optional named list containing hyperparameter values for the stochastic volatility parameters. Not all have to be supplied, with those missing being replaced by the default values. Any list elements that are misnamed will be ignored and a warning will be thrown. Ignored if sv is FALSE. The following elements can be supplied: Bsigma_sv: positive, real number. The default value is 1. a0_sv: positive, real number. The default value is 5. b0_sv: positive, real number. The default value is 1.5. bmu: real number. The default value is 0. Bmu: real number. larger than 0. The default value is 1.
MH_tuning	optional named list containing values used to tune the MH steps for a_xi, a_tau, c_xi, and c_tau. Not all have to be supplied, with those missing being replaced by the default values. Any list elements that are misnamed will be ignored and a warning will be thrown. The arguments for a_xi(a_tau) are only used if learn_a_xi(learn_a_tau) is set to TRUE and mod_type is not equal to "ridge". The arguments for c_xi(c_tau) are only used if learn_c_xi(learn_c_tau) is set to TRUE and mod_type is equal to "triple". Arguments ending in "adaptive" are logical values indicating whether or not to make the MH step for the respective parameter adaptive. Arguments ending in "tuning_par" serve two different purposes. If the respective MH step is not set to be adaptive, it acts as the standard deviation of the proposal distribution. If the respective MH step is set to be adaptive, it acts as the initial standard deviation. Arguments ending in "target_rate" define the acceptance rate the algorithm aims to achieve. Arguments ending in "max_adapt" set the maximum value by which the logarithm of the standard deviation of the proposal distribution is adjusted. Finally, arguments ending in "batch_size" set the batch size after which the standard deviation of the proposal distribution is adjusted. The following elements can be supplied: a_xi_adaptive: logical value. The default is TRUE. a_xi_tuning_par: positive, real number. The default value is 1. a_xi_target_rate: positive, real number, between 0 and 1. The default value is 0.44. a_xi_max_adapt: positive, real number. The default value is 0.01. a_xi_batch_size: positive integer. The default value is 50. a_tau_adaptive: logical value. The default is TRUE. a_tau_tuning_par: positive, real number. The default value is 1. a_tau_target_rate: positive, real number, between 0 and 1. The default value is 0.44. a_tau_max_adapt: positive, real number. The default value is 0.01. a_tau_batch_size: positive integer. The default value is 50. c_xi_adaptive: logical value. The default is TRUE. c_xi_tuning_par: positive, real number. The default value is 1. c_xi_target_rate: positive, real number, between 0 and 1. The default value is 0.44. c_xi_max_adapt: positive, real number. The default value is 0.01. c_xi_batch_size: positive integer. The default value is 50. c_tau_adaptive: logical value. The default is TRUE. c_tau_tuning_par: positive, real number. The default value is 1. c_tau_target_rate: positive, real number, between 0 and 1. The default value is 0.44. c_tau_max_adapt: positive, real number. The default value is 0.01. c_tau_batch_size: positive integer. The default value is 50. rho_adaptive: logical value. The default is TRUE. rho_tuning_par: positive, real number. The default value is 1. rho_target_rate: positive, real number, between 0 and 1. The default value is 0.44. rho_max_adapt: positive, real number. The default value is 0.01. rho_batch_size: positive integer. The default value is 50.
starting_vals	optional named list containing the values at which the MCMC algorithm will be initialized. In the following d refers to the number of covariates, including the intercept and expanded factors. Not all have to be supplied, with those missing being replaced by the default values. Any list elements that are misnamed will be ignored and a warning will be thrown. The following elements can be supplied: beta_mean_st: vector of length d containing single numbers. The default is rep(0, d). theta_sr_st: vector of length d containing single, positive numbers. The default is rep(1, d). tau2_st: vector of length d containing single, positive numbers. The default is rep(1, d). xi2_st: vector of length d containing single, positive numbers. The default is rep(1, d). kappa2_st: vector of length d containing single, positive numbers. The default is rep(1, d). lambda2_st: vector of length d containing single, positive numbers. The default is rep(1, d). kappa2_B_st: positive, real number. The default value is 20. lambda2_B_st: positive, real number. The default value is 20. a_xi_st: positive, real number. The default value is 0.1. a_tau_st: positive, real number. The default value is 0.1. c_xi_st: positive, real number. The default value is 0.1. Note that the prior for c_xi is restricted to (0, 0.5). c_tau_st: positive, real number. The default value is 0.1. Note that the prior for c_tau is restricted to (0, 0.5). sv_mu_st: real number. The default value is -10. sv_phi_st: positive, real number between -1 and 1. The default value is 0.5. sv_sigma2_st : positive, real number. The default value is 1. C0_st: positive, real number. The default value is 1. sigma2_st: positive, real number if sv is FALSE, otherwise a vector of positive, real numbers of length N. The default value is 1 or a vector thereof. h0_st: real number. The default value is 0. lambda_0_st vector of length d containing positive, real numbers. The default value is rep(1, d). rho_st: vector of length d containing real numbers between 0 and b_rho. The default value is rep(max(0.1, hyperprior_param$b_rho - 0.1), d).

Details

For details concerning the algorithms please refer to the papers by Bitto and Frühwirth-Schnatter (2019), Cadonna et al. (2020) and Knaus and Frühwirth-Schnatter (2023). For more details on the package and the usage of the functions, see Knaus et al. (2021).

Value

The value returned is a list object of class shrinkTVP containing

beta	list object containing an mcmc.tvp object for the parameter draws from the posterior distribution of the centered states, one for each covariate. In the case that there is only one covariate, this becomes just a single mcmc.tvp object.
beta_mean	mcmc object containing the parameter draws from the posterior distribution of beta_mean.
theta_sr	mcmc object containing the parameter draws from the posterior distribution of the square root of theta.
tau2	mcmc object containing the parameter draws from the posterior distribution of tau2.
xi2	mcmc object containing the parameter draws from the posterior distribution of xi2.
psi	list object containing an mcmc.tvp object for the parameter draws from the posterior distribution of psi, one for each covariate. In the case that there is only one covariate, this becomes just a single mcmc.tvp object.
lambda_p	list object containing an mcmc.tvp object for the parameter draws from the posterior distribution of lambda_p, one for each covariate. In the case that there is only one covariate, this becomes just a single mcmc.tvp object.
kappa_p	(optional) list object containing an mcmc.tvp object for the parameter draws from the posterior distribution of kappa_p, one for each covariate. In the case that there is only one covariate, this becomes just a single mcmc.tvp object. Not returned if iid is not TRUE.
lambda2	(optional) mcmc object containing the parameter draws from the posterior distribution of lambda2. Not returned if mod_type is not "triple".
kappa2	(optional) mcmc object containing the parameter draws from the posterior distribution of kappa2. Not returned if mod_type is not "triple".
a_xi	(optional) mcmc object containing the parameter draws from the posterior distribution of a_xi. Not returned if learn_a_xi is FALSE or mod_type is "ridge".
a_tau	(optional) mcmc object containing the parameter draws from the posterior distribution of a_tau. Not returned if learn_a_tau is FALSE or mod_type is "ridge".
c_xi	(optional) mcmc object containing the parameter draws from the posterior distribution of c_xi. Not returned if learn_c_xi is FALSE or mod_type is not "triple".
c_tau	(optional) mcmc object containing the parameter draws from the posterior distribution of c_tau. Not returned if learn_c_tau is FALSE or mod_type is not "triple".
lambda2_B	(optional) mcmc object containing the parameter draws from the posterior distribution of lambda2_B. Not returned if learn_lambda2_B is FALSE or mod_type is "ridge".
kappa2_B	(optional) mcmc object containing the parameter draws from the posterior distribution of kappa2_B. Not returned if learn_kappa2_B is FALSE or mod_type is "ridge".
rho	(optional) mcmc object containing the parameter draws from the posterior distribution of rho. Not returned if iid is not TRUE.
sigma2	mcmc object containing the parameter draws from the posterior distribution of sigma2. If sv is TRUE, sigma2 is additionally an mcmc.tvp object.
C0	(optional) mcmc object containing the parameter draws from the posterior distribution of C0. Not returned if sv is TRUE.
sv_mu	(optional) mcmc object containing the parameter draws from the posterior distribution of the mu parameter for the stochastic volatility model on the errors. Not returned if sv is FALSE.
sv_phi	(optional) mcmc object containing the parameter draws from the posterior distribution of the phi parameter for the stochastic volatility model on the errors. Not returned if sv is FALSE.
sv_sigma2	(optional) mcmc object containing the parameter draws from the posterior distribution of the sigma2 parameter for the stochastic volatility model on the errors. Not returned if sv is FALSE.
MH_diag	(optional) named list containing statistics for assessing MH performance. Not returned if no MH steps are required or none of them are specified to be adaptive.
internals	list object containing two arrays that are required for calculating the LPDS.
priorvals	list object containing hyperparameter values of the prior distributions, as specified by the user.
model	list object containing the model matrix, model response and formula used.
summaries	list object containing a collection of summary statistics of the posterior draws.

To display the output, use plot and summary. The summary method displays the specified prior values stored in priorvals and the posterior summaries stored in summaries, while the plot method calls coda's plot.mcmc or the plot.mcmc.tvp method. Furthermore, all functions that can be applied to coda::mcmc objects (e.g. coda::acfplot) can be applied to all output elements that are coda compatible.

Author(s)

Peter Knaus [e-mail]

References

Bitto, A., & Frühwirth-Schnatter, S. (2019). "Achieving shrinkage in a time-varying parameter model framework." Journal of Econometrics, 210(1), 75-97. <doi:10.1016/j.jeconom.2018.11.006>

Cadonna, A., Frühwirth-Schnatter, S., & Knaus, P. (2020). "Triple the Gamma—A Unifying Shrinkage Prior for Variance and Variable Selection in Sparse State Space and TVP Models." Econometrics, 8(2), 20. <doi:10.3390/econometrics8020020>

Knaus, P., Bitto-Nemling, A., Cadonna, A., & Frühwirth-Schnatter, S. (2021) "Shrinkage in the Time-Varying Parameter Model Framework Using the R Package shrinkTVP." Journal of Statistical Software 100(13), 1–32. <doi:10.18637/jss.v100.i13>

Knaus, P., & Frühwirth-Schnatter, S. (2023). "The Dynamic Triple Gamma Prior as a Shrinkage Process Prior for Time-Varying Parameter Models." arXiv preprint arXiv:2312.10487. <doi:10.48550/arXiv.2312.10487>

See Also

plot.shrinkTVP, plot.mcmc.tvp

Examples

set.seed(123)
sim <- simTVP(DTG = TRUE, theta = c(0, 1, 0), beta_mean = c(1, 1, 0), rho = 0.95, c_psi = 2)
data <- sim$data

## Example 1, match the true underlying process
res <- shrinkDTVP(y ~ x1 + x2, data = data, c_psi = 2)
# summarize output
summary(res)

## Example 2, dynamic horseshoe
res <- shrinkDTVP(y ~ x1 + x2, data = data)


## Example 3, modify hyperparameters
res <- shrinkDTVP(y ~ x1 + x2, data = data,
                 hyperprior_param = list(a_rho = 1,
                                         alpha_rho = 0.5,
                                         beta_rho = 0.5))

shrinkTVP documentation built on May 29, 2024, 7:24 a.m.