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))
#> [,1] [,2]
#> [1,] -0.626 0.184
#> [2,] -0.836 1.595
#> [3,] 0.330 -0.820
The output of sim_matgaussian()
is a matrix.
sim_matgaussian(matrix(1:20, nrow = 4), diag(4), diag(5), FALSE)
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 1.49 5.74 9.58 12.7 18.5
#> [2,] 2.39 5.38 7.79 15.1 18.0
#> [3,] 2.98 7.94 11.82 15.6 19.9
#> [4,] 4.78 8.07 10.01 16.6 19.9
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)
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.1827 0.0894 -0.0411 -0.0924 -0.1305
#> [2,] 0.0894 0.6110 0.0860 -0.3754 -0.1015
#> [3,] -0.0411 0.0860 0.2189 -0.1649 -0.0988
#> [4,] -0.0924 -0.3754 -0.1649 0.4577 0.2136
#> [5,] -0.1305 -0.1015 -0.0988 0.2136 0.7444
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)
#> $mn
#> $mn[[1]]
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 1.19 4.96 8.03 12.7 14.6
#> [2,] 2.13 5.17 10.41 14.1 19.3
#> [3,] 3.01 7.15 10.29 14.9 18.1
#> [4,] 4.59 7.99 12.12 15.0 18.5
#>
#> $mn[[2]]
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.323 5.07 8.24 13.1 17.4
#> [2,] 2.022 5.56 9.96 13.9 19.4
#> [3,] 3.380 7.24 11.34 15.5 17.9
#> [4,] 3.755 8.79 11.84 15.6 19.6
#>
#>
#> $iw
#> $iw[[1]]
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.2887 -0.04393 0.11892 -0.2959 0.110
#> [2,] -0.0439 0.33357 0.00197 0.0106 -0.167
#> [3,] 0.1189 0.00197 1.30074 -0.0291 2.210
#> [4,] -0.2959 0.01061 -0.02913 0.5072 0.250
#> [5,] 0.1104 -0.16736 2.20957 0.2504 4.623
#>
#> $iw[[2]]
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.28118 -0.1397 0.00529 0.0283 0.038
#> [2,] -0.13965 0.2876 0.06156 -0.1575 -0.186
#> [3,] 0.00529 0.0616 0.31282 -0.1372 -0.299
#> [4,] 0.02831 -0.1575 -0.13724 0.3441 0.122
#> [5,] 0.03803 -0.1856 -0.29860 0.1217 0.738
This function has been defined for the next simulation functions.
Consider BVAR Minnesota prior setting,
\[A \sim MN(A_0, \Omega_0, \Sigma_e)\]
\[\Sigma_e \sim IW(S_0, \alpha_0)\]
build_xdummy()
build_ydummy()
sigma
: Vector \(\sigma_1,
\ldots, \sigma_m\)
lambda
m
increases, \(\lambda\) should be smaller to avoid
overfitting (De Mol et al. (2008))delta
: Persistence
eps
: Very small number to make matrix invertiblebvar_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
))
#> Model Specification for BVAR
#>
#> Parameters: Coefficent matrice and Covariance matrix
#> Prior: Minnesota
#> ========================================================
#>
#> Setting for 'sigma':
#> [1] 3.25 11.10 2.20 6.80
#>
#> Setting for 'lambda':
#> [1] 0.2
#>
#> Setting for 'delta':
#> [1] 1 1 1 1
#>
#> Setting for 'eps':
#> [1] 1e-04
#>
#> Setting for 'hierarchical':
#> [1] FALSE
sim_mncoef(p, bayes_spec, full = TRUE)
can generate
both \(A\) and \(\Sigma\) matrices.bayes_spec
, only set_bvar()
works.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))
#> $coefficients
#> [,1] [,2] [,3] [,4]
#> [1,] 0.996255 -0.19084 -0.026363 0.07600
#> [2,] -0.017158 1.07873 -0.033404 -0.00816
#> [3,] -0.225869 0.31165 0.927705 -0.48148
#> [4,] -0.002706 -0.26068 0.038566 0.84105
#> [5,] -0.023064 -0.11717 -0.030881 0.35496
#> [6,] 0.000253 -0.05523 -0.018351 0.03580
#> [7,] -0.001364 -0.06563 -0.051615 -0.33330
#> [8,] -0.025569 0.11938 -0.011720 -0.15248
#> [9,] 0.062006 -0.07985 -0.001830 0.15440
#> [10,] 0.008609 -0.03437 0.016242 0.00866
#> [11,] -0.069804 0.11559 0.003275 0.30043
#> [12,] 0.001677 0.01789 -0.000582 -0.02224
#> [13,] -0.000864 0.05893 0.038960 0.06788
#> [14,] 0.008799 -0.04337 0.005557 0.02273
#> [15,] -0.053924 0.16120 -0.006870 0.10085
#> [16,] -0.009109 0.00284 -0.008880 -0.01168
#> [17,] 0.008001 -0.05816 0.012293 -0.05256
#> [18,] -0.006107 0.03645 -0.004491 -0.00687
#> [19,] -0.011204 0.05703 0.036651 0.13185
#> [20,] -0.003148 -0.05735 0.016794 0.04086
#>
#> $covmat
#> [,1] [,2] [,3] [,4]
#> [1,] 2.615 -5.1044 0.1522 2.44
#> [2,] -5.104 32.2768 -0.0549 -12.04
#> [3,] 0.152 -0.0549 1.4573 0.31
#> [4,] 2.440 -12.0446 0.3101 30.83
sim_mnvhar_coef(bayes_spec, full = TRUE)
generates BVHAR
model setting:
\[\Phi \mid \Sigma_e \sim MN(M_0, \Omega_0, \Sigma_e)\]
\[\Sigma_e \sim IW(\Psi_0, \nu_0)\]
bayes_spec
option wants
bvharspec
. But
set_bvhar()
set_weight_bvhar()
full = TRUE
, too.(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
))
#> Model Specification for BVHAR
#>
#> Parameters: Coefficent matrice and Covariance matrix
#> Prior: MN_VAR
#> ========================================================
#>
#> Setting for 'sigma':
#> [1] 1.2 2.3
#>
#> Setting for 'lambda':
#> [1] 0.2
#>
#> Setting for 'delta':
#> [1] 0.3 1.0
#>
#> Setting for 'eps':
#> [1] 1e-04
#>
#> Setting for 'hierarchical':
#> [1] FALSE
(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
))
#> Model Specification for BVHAR
#>
#> Parameters: Coefficent matrice and Covariance matrix
#> Prior: MN_VHAR
#> ========================================================
#>
#> Setting for 'sigma':
#> [1] 1.2 2.3
#>
#> Setting for 'lambda':
#> [1] 0.2
#>
#> Setting for 'eps':
#> [1] 1e-04
#>
#> Setting for 'daily':
#> [1] 0.5 1.0
#>
#> Setting for 'weekly':
#> [1] 0.2 0.3
#>
#> Setting for 'monthly':
#> [1] 0.1 0.1
#>
#> Setting for 'hierarchical':
#> [1] FALSE