Fitting Bayesian VHAR of Minnesota Prior

This function fits BVHAR with Minnesota prior.

Usage

bvhar_minnesota(
  y,
  har = c(5, 22),
  bayes_spec = set_bvhar(),
  include_mean = TRUE
)

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

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

Arguments

y: Time series data of which columns indicate the variables
har: Numeric vector for weekly and monthly order. By default, c(5, 22).
bayes_spec: A BVHAR model specification by set_bvhar() (default) or set_weight_bvhar().
include_mean: Add constant term (Default: TRUE) or not (FALSE)
x: bvarmn object
digits: digit option to print
...: not used

Value

bvhar_minnesota() returns an object bvharmn

class. It is a list with the following components:

coefficients: Posterior Mean matrix of Matrix Normal distribution
fitted.values: Fitted values
residuals: Residuals
mn_prec: Posterior precision matrix of Matrix Normal distribution
iw_scale: Posterior scale matrix of posterior inverse-wishart distribution
iw_shape: Posterior shape of inverse-Wishart distribution ($\nu_0$ - obs + 2). $\nu_0$: nrow(Dummy observation) - k
df: Numer of Coefficients: 3m + 1 or 3m
p: 3, this element exists to run the other functions
week: Order for weekly term
month: Order for monthly term
m: Dimension of the time series
obs: Sample size used when training = totobs - 22
totobs: Total number of the observation
call: Matched call
process: Process string in the bayes_spec: "BVHAR_MN_VAR" (BVHAR-S) or "BVHAR_MN_VHAR" (BVHAR-L)
spec: Model specification (bvharspec)
type: include constant term ("const") or not ("none")
prior_mean: Prior mean matrix of Matrix Normal distribution: $M_0$
prior_precision: Prior precision matrix of Matrix Normal distribution: $\Omega_0^{-1}$
prior_scale: Prior scale matrix of inverse-Wishart distribution: $\Psi_0$
prior_shape: Prior shape of inverse-Wishart distribution: $\nu_0$
HARtrans: VHAR linear transformation matrix: $C_{HAR}$
y0: $Y_0$
design: $X_0$
y: Raw input (matrix)

It is also normaliw and bvharmod class.

Details

Apply Minnesota prior to Vector HAR: $\Phi$ (VHAR matrices) and $\Sigma_e$ (residual covariance).

$$\Phi \mid \Sigma_e \sim MN(M_0, \Omega_0, \Sigma_e)$$ $$\Sigma_e \sim IW(\Psi_0, \nu_0)$$ (MN: matrix normal, IW: inverse-wishart)

There are two types of Minnesota priors for BVHAR:

VAR-type Minnesota prior specified by set_bvhar(), so-called BVHAR-S model.
VHAR-type Minnesota prior specified by set_weight_bvhar(), so-called BVHAR-L model.

Two types of Minnesota priors builds different dummy variables for Y0. See var_design_formulation.

References

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

Examples

# Perform the function using etf_vix dataset
fit <- bvhar_minnesota(y = etf_vix[,1:3])
class(fit)
#> [1] "bvharmn"  "normaliw" "bvharmod"

# Extract coef, fitted values, and residuals
coef(fit)
#>                       GVZCLS       OVXCLS      VXFXICLS
#> GVZCLS_day      0.7239591235  0.030141906  0.0334224930
#> OVXCLS_day      0.0021163564  0.718324835  0.0094789827
#> VXFXICLS_day    0.0294772930  0.035320583  0.7377164885
#> GVZCLS_week     0.1251766305 -0.002704810 -0.0024006869
#> OVXCLS_week    -0.0030211395  0.144436818 -0.0010429118
#> VXFXICLS_week   0.0028666589  0.009321449  0.1206325735
#> GVZCLS_month    0.0482747164  0.005220543  0.0008392191
#> OVXCLS_month   -0.0022599110  0.058987040  0.0007463012
#> VXFXICLS_month  0.0007727524  0.005768089  0.0298913283
#> const           1.1216988711  0.382866382  1.9279431742
head(residuals(fit))
#>            [,1]        [,2]       [,3]
#> [1,]  0.2011463 -0.03569196  0.2373614
#> [2,] -0.5979335  0.15122959 -0.2782346
#> [3,] -0.3466428  0.50804209  3.3657096
#> [4,] -0.8431968  0.52911348 -0.2492212
#> [5,] -0.1001249  0.54094522  0.3884998
#> [6,] -0.1407253  1.48033862  0.8978522
head(fitted(fit))
#>        GVZCLS   OVXCLS VXFXICLS
#> [1,] 20.48885 32.35569 29.22264
#> [2,] 20.62793 32.27877 28.94823
#> [3,] 20.07664 32.32196 28.36429
#> [4,] 19.91320 32.77089 30.71922
#> [5,] 19.34012 33.09905 29.80150
#> [6,] 19.42073 33.36966 29.61215