shrinkTVP {shrinkTVP}

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

Description

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

Usage

shrinkTVP(
  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,
  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.

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.

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.

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.

Details

For details concerning the algorithms please refer to the papers by Bitto and Frühwirth-Schnatter (2019) and Cadonna et al. (2020). 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.

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".

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>

See Also

plot.shrinkTVP, plot.mcmc.tvp

Examples

## Example 1, learn everything
set.seed(123)
sim <- simTVP(theta = c(0.2, 0, 0), beta_mean = c(1.5, -0.3, 0))
data <- sim$data

res <- shrinkTVP(y ~ x1 + x2, data = data)
# summarize output
summary(res)


## Example 2, hierarchical Bayesian Lasso
res <- shrinkTVP(y ~ x1 + x2, data = data,
                learn_a_xi = FALSE, learn_a_tau = FALSE,
                a_xi = 1, a_tau = 1)


## Example 3, non-hierarchical Bayesian Lasso
res <- shrinkTVP(y ~ x1 + x2, data = data,
                learn_a_xi = FALSE, learn_a_tau = FALSE,
                a_xi = 1, a_tau = 1,
                learn_kappa2 = FALSE, learn_lambda2 = FALSE)


## Example 4, adding stochastic volatility
res <- shrinkTVP(y ~ x1 + x2, data = data,
                sv = TRUE)


## Example 4, changing some of the default hyperparameters
res <- shrinkTVP(y ~ x1 + x2, data = data,
                hyperprior_param = list(beta_a_xi = 5,
                                        alpha_a_xi = 10))

## Example 5, using the triple gamma prior
res <- shrinkTVP(y ~ x1 + x2, data = data,
                mod_type = "triple")