Hyperparameters for Bayesian Models

Set hyperparameters of Bayesian VAR and VHAR models.

Usage

set_bvar(sigma, lambda = 0.1, delta, eps = 1e-04)

set_bvar_flat(U)

set_bvhar(sigma, lambda = 0.1, delta, eps = 1e-04)

set_weight_bvhar(sigma, lambda = 0.1, eps = 1e-04, daily, weekly, monthly)

# S3 method for bvharspec
print(x, digits = max(3L, getOption("digits") - 3L), ...)

# S3 method for bvharspec
knit_print(x, ...)

Arguments

sigma: Standard error vector for each variable (Default: sd)
lambda: Tightness of the prior around a random walk or white noise (Default: .1)
delta: Persistence (Default: Litterman sets 1 = random walk prior, White noise prior = 0)
eps: Very small number (Default: 1e-04)
U: Positive definite matrix. By default, identity matrix of dimension ncol(X0)
daily: Same as delta in VHAR type (Default: 1 as Litterman)
weekly: Fill the second part in the first block (Default: 1)
monthly: Fill the third part in the first block (Default: 1)
x: bvharspec object
digits: digit option to print
...: not used

Value

Every function returns bvharspec

class. It is the list of which the components are the same as the arguments provided. If the argument is not specified, NULL is assigned here. The default values mentioned above will be considered in each fitting function.

process: Model name: BVAR, BVHAR
prior: Prior name: Minnesota (Minnesota prior for BVAR), Hierarchical (Hierarchical prior for BVAR), MN_VAR (BVHAR-S), MN_VHAR (BVHAR-L), Flat (Flat prior for BVAR)
sigma: Vector value (or bvharpriorspec class) assigned for sigma
lambda: Value (or bvharpriorspec class) assigned for lambda
delta: Vector value assigned for delta
eps: Value assigned for epsilon

set_weight_bvhar() has different component with delta due to its different construction.

daily: Vector value assigned for daily weight
weekly: Vector value assigned for weekly weight
monthly: Vector value assigned for monthly weight

Details

Missing arguments will be set to be default values in each model function mentioned above.
set_bvar() sets hyperparameters for bvar_minnesota().
Each delta (vector), lambda (length of 1), sigma (vector), eps (vector) corresponds to \(\delta_j\), \(\lambda\), \(\delta_j\), \(\epsilon\).

\(\delta_i\) are related to the belief to random walk.

If \(\delta_i = 1\) for all i, random walk prior
If \(\delta_i = 0\) for all i, white noise prior

\(\lambda\) controls the overall tightness of the prior around these two prior beliefs.

If \(\lambda = 0\), the posterior is equivalent to prior and the data do not influence the estimates.
If \(\lambda = \infty\), the posterior mean becomes OLS estimates (VAR).

\(\sigma_i^2 / \sigma_j^2\) in Minnesota moments explain the data scales.

set_bvar_flat sets hyperparameters for bvar_flat().

set_bvhar() sets hyperparameters for bvhar_minnesota() with VAR-type Minnesota prior, i.e. BVHAR-S model.

set_weight_bvhar() sets hyperparameters for bvhar_minnesota() with VHAR-type Minnesota prior, i.e. BVHAR-L model.

Note

By using set_psi() and set_lambda() each, hierarchical modeling is available.

References

Bańbura, M., Giannone, D., & Reichlin, L. (2010). Large Bayesian vector auto regressions. Journal of Applied Econometrics, 25(1).

Litterman, R. B. (1986). Forecasting with Bayesian Vector Autoregressions: Five Years of Experience. Journal of Business & Economic Statistics, 4(1), 25.

Ghosh, S., Khare, K., & Michailidis, G. (2018). High-Dimensional Posterior Consistency in Bayesian Vector Autoregressive Models. Journal of the American Statistical Association, 114(526).

Kim, Y. G., and Baek, C. (2023+). Bayesian vector heterogeneous autoregressive modeling. Journal of Statistical Computation and Simulation.

Examples

# Minnesota BVAR specification------------------------
bvar_spec <- set_bvar(
  sigma = c(.03, .02, .01), # Sigma = diag(.03^2, .02^2, .01^2)
  lambda = .2, # lambda = .2
  delta = rep(.1, 3), # delta1 = .1, delta2 = .1, delta3 = .1
  eps = 1e-04 # eps = 1e-04
)
class(bvar_spec)
#> [1] "bvharspec"
str(bvar_spec)
#> List of 6
#>  $ process: chr "BVAR"
#>  $ prior  : chr "Minnesota"
#>  $ sigma  : num [1:3] 0.03 0.02 0.01
#>  $ lambda : num 0.2
#>  $ delta  : num [1:3] 0.1 0.1 0.1
#>  $ eps    : num 1e-04
#>  - attr(*, "class")= chr "bvharspec"
# Flat BVAR specification-------------------------
# 3-dim
# p = 5 with constant term
# U = 500 * I(mp + 1)
bvar_flat_spec <- set_bvar_flat(U = 500 * diag(16))
class(bvar_flat_spec)
#> [1] "bvharspec"
str(bvar_flat_spec)
#> List of 3
#>  $ process: chr "BVAR"
#>  $ prior  : chr "Flat"
#>  $ U      : num [1:16, 1:16] 500 0 0 0 0 0 0 0 0 0 ...
#>  - attr(*, "class")= chr "bvharspec"
# BVHAR-S specification-----------------------
bvhar_var_spec <- set_bvhar(
  sigma = c(.03, .02, .01), # Sigma = diag(.03^2, .02^2, .01^2)
  lambda = .2, # lambda = .2
  delta = rep(.1, 3), # delta1 = .1, delta2 = .1, delta3 = .1
  eps = 1e-04 # eps = 1e-04
)
class(bvhar_var_spec)
#> [1] "bvharspec"
str(bvhar_var_spec)
#> List of 6
#>  $ process: chr "BVHAR"
#>  $ prior  : chr "MN_VAR"
#>  $ sigma  : num [1:3] 0.03 0.02 0.01
#>  $ lambda : num 0.2
#>  $ delta  : num [1:3] 0.1 0.1 0.1
#>  $ eps    : num 1e-04
#>  - attr(*, "class")= chr "bvharspec"
# BVHAR-L specification---------------------------
bvhar_vhar_spec <- set_weight_bvhar(
  sigma = c(.03, .02, .01), # Sigma = diag(.03^2, .02^2, .01^2)
  lambda = .2, # lambda = .2
  eps = 1e-04, # eps = 1e-04
  daily = rep(.2, 3), # daily1 = .2, daily2 = .2, daily3 = .2
  weekly = rep(.1, 3), # weekly1 = .1, weekly2 = .1, weekly3 = .1
  monthly = rep(.05, 3) # monthly1 = .05, monthly2 = .05, monthly3 = .05
)
class(bvhar_vhar_spec)
#> [1] "bvharspec"
str(bvhar_vhar_spec)
#> List of 8
#>  $ process: chr "BVHAR"
#>  $ prior  : chr "MN_VHAR"
#>  $ sigma  : num [1:3] 0.03 0.02 0.01
#>  $ lambda : num 0.2
#>  $ eps    : num 1e-04
#>  $ daily  : num [1:3] 0.2 0.2 0.2
#>  $ weekly : num [1:3] 0.1 0.1 0.1
#>  $ monthly: num [1:3] 0.05 0.05 0.05
#>  - attr(*, "class")= chr "bvharspec"