Markov Chain Monte Carlo Sampling for Bayesian Vectorautoregressions

bvar simulates from the joint posterior distribution of the parameters and latent variables and returns the posterior draws.

Usage

bvar(
  data,
  lags = 1L,
  draws = 1000L,
  burnin = 1000L,
  thin = 1L,
  prior_intercept = 10,
  prior_phi = specify_prior_phi(data = data, lags = lags, prior = "HS"),
  prior_sigma = specify_prior_sigma(data = data, type = "factor", quiet = TRUE),
  sv_keep = "last",
  quiet = FALSE,
  startvals = list(),
  expert = list()
)

Arguments

data: Data matrix (can be a time series object). Each of \(M\) columns is assumed to contain a single time-series of length \(T\).
lags: Integer indicating the order of the VAR, i.e. the number of lags of the dependent variables included as predictors.
draws: single integer indicating the number of draws after the burnin
burnin: single integer indicating the number of draws discarded as burnin
thin: single integer. Every \(thin\)th draw will be stored. Default is thin=1L.
prior_intercept: Either prior_intercept=FALSE and no constant term (intercept) will be included. Or a numeric vector of length \(M\) indicating the (fixed) prior standard deviations on the constant term. A single number will be recycled accordingly. Default is prior_intercept=10.
prior_phi: bayesianVARs_prior_phi object specifying prior for the reduced form VAR coefficients. Best use constructor specify_prior_phi.
prior_sigma: bayesianVARs_prior_sigma object specifying prior for the variance-covariance matrix of the VAR. Best use constructor specify_prior_sigma.
sv_keep: String equal to "all" or "last". In case of sv_keep = "last", the default, only draws for the very last log-variance \(h_T\) are stored.
quiet: logical value indicating whether information about the progress during sampling should be displayed during sampling (default is TRUE).
startvals: optional list with starting values.
expert: optional list with expert settings.

Value

An object of type bayesianVARs_bvar, a list containing the following objects:

PHI: A bayesianVARs_coef object, an array, containing the posterior draws of the VAR coefficients (including the intercept).
U: A bayesianVARs_draws object, a matrix, containing the posterior draws of the contemporaneous coefficients (if cholesky decomposition for sigma is specified).
logvar: A bayesianVARs_draws object containing the log-variance draws.
sv_para: A baysesianVARs_draws object containing the posterior draws of the stochastic volatility related parameters.
phi_hyperparameter: A matrix containing the posterior draws of the hyperparameters of the conditional normal prior on the VAR coefficients.
u_hyperparameter: A matrix containing the posterior draws of the hyperparameters of the conditional normal prior on U (if cholesky decomposition for sigma is specified).
bench: proc_time object containing the run time of the sampler.
V_prior: An array containing the posterior draws of the variances of the conditional normal prior on the VAR coefficients.
facload: A bayesianVARs_draws object, an array, containing draws from the posterior distribution of the factor loadings matrix (if factor decomposition for sigma is specified).
fac: A bayesianVARs_draws object, an array, containing factor draws from the posterior distribution (if factor decomposition for sigma is specified).
Y: Matrix containing the dependent variables used for estimation.
X matrix containing the lagged values of the dependent variables, i.e. the covariates.
lags: Integer indicating the lag order of the VAR.
intercept: Logical indicating whether a constant term is included.
heteroscedastic logical indicating whether heteroscedasticity is assumed.
Yraw: Matrix containing the dependent variables, including the initial 'lags' observations.
Traw: Integer indicating the total number of observations.
sigma_type: Character specifying the decomposition of the variance-covariance matrix.
datamat: Matrix containing both 'Y' and 'X'.
config: List containing information on configuration parameters.

Details

The VAR(p) model is of the following form: \( \boldsymbol{y}^\prime_t = \boldsymbol{\iota}^\prime + \boldsymbol{x}^\prime_t\boldsymbol{\Phi} + \boldsymbol{\epsilon}^\prime_t\), where \(\boldsymbol{y}_t\) is a \(M\)-dimensional vector of dependent variables and \(\boldsymbol{\epsilon}_t\) is the error term of the same dimension. \(\boldsymbol{x}_t\) is a \(K=pM\)-dimensional vector containing lagged/past values of the dependent variables \(\boldsymbol{y}_{t-l}\) for \(l=1,\dots,p\) and \(\boldsymbol{\iota}\) is a constant term (intercept) of dimension \(M\times 1\). The reduced-form coefficient matrix \(\boldsymbol{\Phi}\) is of dimension \(K \times M\).

bvar offers two different specifications for the errors: The user can choose between a factor stochastic volatility structure or a cholesky stochastic volatility structure. In both cases the disturbances \(\boldsymbol{\epsilon}_t\) are assumed to follow a \(M\)-dimensional multivariate normal distribution with zero mean and variance-covariance matrix \(\boldsymbol{\Sigma}_t\). In case of the cholesky specification \(\boldsymbol{\Sigma}_t = \boldsymbol{U}^{\prime -1} \boldsymbol{D}_t \boldsymbol{U}^{-1}\), where \(\boldsymbol{U}^{-1}\) is upper unitriangular (with ones on the diagonal). The diagonal matrix \(\boldsymbol{D}_t\) depends upon latent log-variances, i.e. \(\boldsymbol{D}_t=diag(exp(h_{1t}),\dots, exp(h_{Mt})\). The log-variances follow a priori independent autoregressive processes \(h_{it}\sim N(\mu_i + \phi_i(h_{i,t-1}-\mu_i),\sigma_i^2)\) for \(i=1,\dots,M\). In case of the factor structure, \(\boldsymbol{\Sigma}_t = \boldsymbol{\Lambda} \boldsymbol{V}_t \boldsymbol{\Lambda}^\prime + \boldsymbol{G}_t\). The diagonal matrices \(\boldsymbol{V}_t\) and \(\boldsymbol{G}_t\) depend upon latent log-variances, i.e. \(\boldsymbol{G}_t=diag(exp(h_{1t}),\dots, exp(h_{Mt})\) and \(\boldsymbol{V}_t=diag(exp(h_{M+1,t}),\dots, exp(h_{M+r,t})\). The log-variances follow a priori independent autoregressive processes \(h_{it}\sim N(\mu_i + \phi_i(h_{i,t-1}-\mu_i),\sigma_i^2)\) for \(i=1,\dots,M\) and \(h_{M+j,t}\sim N(\phi_ih_{M+j,t-1},\sigma_{M+j}^2)\) for \(j=1,\dots,r\).

MCMC algorithm

To sample efficiently the reduced-form VAR coefficients assuming a factor structure for the errors, the equation per equation algorithm in Kastner & Huber (2020) is implemented. All parameters and latent variables associated with the factor-structure are sampled using package factorstochvol-package's function update_fsv callable on the C-level only.

To sample efficiently the reduced-form VAR coefficients, assuming a cholesky-structure for the errors, the corrected triangular algorithm in Carriero et al. (2021) is implemented. The SV parameters and latent variables are sampled using package stochvol's update_fast_sv function. The precision parameters, i.e. the free off-diagonal elements in \(\boldsymbol{U}\), are sampled as in Cogley and Sargent (2005).

References

Gruber, L. and Kastner, G. (2025). Forecasting macroeconomic data with Bayesian VARs: Sparse or dense? It depends! International Journal of Forecasting. doi:10.1016/j.ijforecast.2025.02.001 .

Kastner, G. and Huber, F. Sparse (2020). Bayesian vector autoregressions in huge dimensions. Journal of Forecasting. 39, 1142–1165, doi:10.1002/for.2680 .

Kastner, G. (2019). Sparse Bayesian Time-Varying Covariance Estimation in Many Dimensions Journal of Econometrics, 210(1), 98–115, doi:10.1016/j.jeconom.2018.11.007 .

Carriero, A. and Chan, J. and Clark, T. E. and Marcellino, M. (2021). Corrigendum to “Large Bayesian vector autoregressions with stochastic volatility and non-conjugate priors” [J. Econometrics 212 (1) (2019) 137–154]. Journal of Econometrics, doi:10.1016/j.jeconom.2021.11.010 .

Cogley, S. and Sargent, T. (2005). Drifts and volatilities: monetary policies and outcomes in the post WWII US. Review of Economic Dynamics, 8, 262–302, doi:10.1016/j.red.2004.10.009 .

Hosszejni, D. and Kastner, G. (2021). Modeling Univariate and Multivariate Stochastic Volatility in R with stochvol and factorstochvol. Journal of Statistical Software, 100, 1–-34. doi:10.18637/jss.v100.i12 .

Examples

# Access a subset of the usmacro_growth dataset
data <- usmacro_growth[,c("GDPC1", "CPIAUCSL", "FEDFUNDS")]

# Estimate a model
mod <- bvar(data, sv_keep = "all", quiet = TRUE)

# Plot
plot(mod)


# Summary
summary(mod)
#> 
#> Posterior median of reduced-form coefficients:
#>              GDPC1 CPIAUCSL FEDFUNDS
#> GDPC1.l1     0.153   -0.003    0.020
#> CPIAUCSL.l1 -0.098    0.607   -0.001
#> FEDFUNDS.l1  0.014    0.043    1.001
#> intercept    0.006    0.001    0.000
#> 
#> Posterior interquartile range of of reduced-form coefficients:
#>             GDPC1 CPIAUCSL FEDFUNDS
#> GDPC1.l1    0.111    0.028    0.018
#> CPIAUCSL.l1 0.140    0.086    0.017
#> FEDFUNDS.l1 0.020    0.014    0.008
#> intercept   0.001    0.001    0.000
#> 
#> Posterior median of factor loadings:
#>          factor1
#> GDPC1      0.000
#> CPIAUCSL   0.000
#> FEDFUNDS   0.001
#> 
#> Posterior interquartile range of factor loadings:
#>          factor1
#> GDPC1      0.000
#> CPIAUCSL   0.000
#> FEDFUNDS   0.001