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 = 100,
  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 variances on the constant term. A single number will be recycled accordingly. Default is prior_intercept=100.

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: Numerical indicating the average time it took to generate one single draw of the joint posterior distribution of all parameters.

• 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. (2023). Forecasting macroeconomic data with Bayesian VARs: Sparse or dense? It depends! arXiv:2206.04902.

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 .

See also

• Helpers for prior configuration: specify_prior_phi(), specify_prior_sigma().

• Plotting: plot.bayesianVARs_bvar().

• Extractors: coef.bayesianVARs_bvar(), vcov.bayesianVARs_bvar().

• 'stable' bvar: stable_bvar().

• summary method: summary.bayesianVARs_bvar().

• predict method: predict.bayesianVARs_bvar().

• fitted method: fitted.bayesianVARs_bvar().

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     0.236588 0.006926  2.109e-02
#> CPIAUCSL.l1 -0.053390 0.616162 -3.646e-03
#> FEDFUNDS.l1  0.007874 0.037088  1.001e+00
#> intercept    0.005874 0.001291 -8.055e-05
#> 
#> Posterior interquartile range of of reduced-form coefficients:
#>                GDPC1  CPIAUCSL  FEDFUNDS
#> GDPC1.l1    0.089598 0.0290897 0.0252782
#> CPIAUCSL.l1 0.111515 0.0904426 0.0150976
#> FEDFUNDS.l1 0.018416 0.0141909 0.0076726
#> intercept   0.001135 0.0007528 0.0001651