Minnesota Prior
In bvhar: Bayesian Vector Heterogeneous Autoregressive Modeling

knitr::opts_chunk$set(
  comment = "#>",
  collapse = TRUE,
  out.width = "70%",
  fig.align = "center",
  fig.width = 6,
  fig.asp = .618
)
orig_opts <- options("digits")
options(digits = 3)
set.seed(1)

library(bvhar)

Normal-inverse-Wishart Matrix

We provide functions to generate matrix-variate Normal and inverse-Wishart.

sim_mnormal(num_sim, mu, sig): num_sim of $\mathbf{X}_i \stackrel{iid}{\sim} N(\boldsymbol{\mu}, \Sigma)$.
sim_matgaussian(mat_mean, mat_scale_u, mat_scale_v): One $X_{m \times n} \sim MN(M_{m \times n}, U_{m \times m}, V_{n \times n})$ which means that $vec(X) \sim N(vec(M), V \otimes U)$.
sim_iw(mat_scale, shape): One $\Sigma \sim IW(\Psi, \nu)$.
sim_mniw(num_sim, mat_mean, mat_scale_u, mat_scale, shape): num_sim of $(X_i, \Sigma_i) \stackrel{iid}{\sim} MNIW(M, U, V, \nu)$.

Multivariate Normal generation gives num_sim x dim matrix. For example, generating 3 vector from Normal($\boldsymbol{\mu} = \mathbf{0}_2$, $\Sigma = diag(\mathbf{1}_2)$):

sim_mnormal(3, rep(0, 2), diag(2))

The output of sim_matgaussian() is a matrix.

sim_matgaussian(matrix(1:20, nrow = 4), diag(4), diag(5), FALSE)

When generating IW, violating $\nu > dim - 1$ gives error. But we ignore $\nu > dim + 1$ (condition for mean existence) in this function. Nonetheless, we recommend you to keep $\nu > dim + 1$ condition. As mentioned, it guarantees the existence of the mean.

sim_iw(diag(5), 7)

In case of sim_mniw(), it returns list with mn (stacked MN matrices) and iw (stacked IW matrices). Each mn and iw has draw lists.

sim_mniw(2, matrix(1:20, nrow = 4), diag(4), diag(5), 7, FALSE)

This function has been defined for the next simulation functions.

Minnesota Prior

BVAR

Consider BVAR Minnesota prior setting,

$$A \sim MN(A_0, \Omega_0, \Sigma_e)$$

$$\Sigma_e \sim IW(S_0, \alpha_0)$$

From Litterman (1986) and Bańbura et al. (2010)
Each $A_0, \Omega_0, S_0, \alpha_0$ is defined by adding dummy observations
- build_xdummy()
- build_ydummy()
sigma: Vector $\sigma_1, \ldots, \sigma_m$
- $\Sigma_e = diag(\sigma_1^2, \ldots, \sigma_m^2)$
- $\sigma_i^2 / \sigma_j^2$: different scale and variability of the data
lambda
- Controls the overall tightness of the prior distribution around the RW or WN
- Governs the relative importance of the prior beliefs w.r.t. the information contained in the data
  - If $\lambda = 0$, then posterior = prior and the data do not influence the estimates.
  - If $\lambda = \infty$, then posterior expectations = OLS.
- Choose in relation to the size of the system (Bańbura et al. (2010))
  - As m increases, $\lambda$ should be smaller to avoid overfitting (De Mol et al. (2008))
delta: Persistence
- Litterman (1986) originally sets high persistence $\delta_i = 1$
- For Non-stationary variables: random walk prior $\delta_i = 1$
- For stationary variables: white noise prior $\delta_i = 0$
eps: Very small number to make matrix invertible

bvar_lag <- 5
(spec_to_sim <- set_bvar(
  sigma = c(3.25, 11.1, 2.2, 6.8), # sigma vector
  lambda = .2, # lambda
  delta = rep(1, 4), # 4-dim delta vector
  eps = 1e-04 # very small number
))

sim_mncoef(p, bayes_spec, full = TRUE) can generate both $A$ and $\Sigma$ matrices.
In bayes_spec, only set_bvar() works.
If full = FALSE, $\Sigma$ is not random. It is same as diag(sigma) from the bayes_spec.
full = TRUE is the default.

(sim_mncoef(bvar_lag, spec_to_sim))

BVHAR

sim_mnvhar_coef(bayes_spec, full = TRUE) generates BVHAR model setting:

$$\Phi \mid \Sigma_e \sim MN(M_0, \Omega_0, \Sigma_e)$$

$$\Sigma_e \sim IW(\Psi_0, \nu_0)$$

similar to BVAR, bayes_spec option wants bvharspec. But
- set_bvhar()
- set_weight_bvhar()
There is full = TRUE, too.

BVHAR-S

(bvhar_var_spec <- set_bvhar(
  sigma = c(1.2, 2.3), # sigma vector
  lambda = .2, # lambda
  delta = c(.3, 1), # 2-dim delta vector
  eps = 1e-04 # very small number
))

(sim_mnvhar_coef(bvhar_var_spec))

BVHAR-L

(bvhar_vhar_spec <- set_weight_bvhar(
  sigma = c(1.2, 2.3), # sigma vector
  lambda = .2, # lambda
  eps = 1e-04, # very small number
  daily = c(.5, 1), # 2-dim daily weight vector
  weekly = c(.2, .3), # 2-dim weekly weight vector
  monthly = c(.1, .1) # 2-dim monthly weight vector
))