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))
#> [,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))
#> [,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):
sim_mniw(2, matrix(1:20, nrow = 4), diag(4), diag(5), 7)
#> $mn
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
#> [1,] 1.19 4.96 8.03 12.7 14.6 0.323 5.07 8.24 13.1 17.4
#> [2,] 2.13 5.17 10.41 14.1 19.3 2.022 5.56 9.96 13.9 19.4
#> [3,] 3.01 7.15 10.29 14.9 18.1 3.380 7.24 11.34 15.5 17.9
#> [4,] 4.59 7.99 12.12 15.0 18.5 3.755 8.79 11.84 15.6 19.6
#>
#> $iw
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
#> [1,] 0.2887 -0.04393 0.11892 -0.2959 0.110 0.28118 -0.1397 0.00529 0.0283
#> [2,] -0.0439 0.33357 0.00197 0.0106 -0.167 -0.13965 0.2876 0.06156 -0.1575
#> [3,] 0.1189 0.00197 1.30074 -0.0291 2.210 0.00529 0.0616 0.31282 -0.1372
#> [4,] -0.2959 0.01061 -0.02913 0.5072 0.250 0.02831 -0.1575 -0.13724 0.3441
#> [5,] 0.1104 -0.16736 2.20957 0.2504 4.623 0.03803 -0.1856 -0.29860 0.1217
#> [,10]
#> [1,] 0.038
#> [2,] -0.186
#> [3,] -0.299
#> [4,] 0.122
#> [5,] 0.738
Since each matrix is stacked rowwise, it is actually not that readable and practical. 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))
- As
-
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
))
#> Model Specification for BVAR
#>
#> Parameters: Coefficent matrice and Covariance matrix
#> Prior: Minnesota
#> # Type '?bvar_minnesota' in the console for some help.
#> ========================================================
#>
#> 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
-
sim_mncoef(p, bayes_spec, full = TRUE)
can generate both \(A\) and \(\Sigma\) matrices. - In
bayes_spec
, onlyset_bvar()
works. - If
full = FALSE
, \(\Sigma\) is not random. It is same asdiag(sigma)
from thebayes_spec
. -
full = TRUE
is the default.
(sim_mncoef(bvar_lag, spec_to_sim))
#> $coefficients
#> [,1] [,2] [,3] [,4]
#> [1,] 0.196255 -0.19084 -0.026363 0.07600
#> [2,] -0.017158 0.27873 -0.033404 -0.00816
#> [3,] -0.225869 0.31165 0.127705 -0.48148
#> [4,] -0.002706 -0.26068 0.038566 0.04105
#> [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
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 wantsbvharspec
. But - 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
))
#> Model Specification for BVHAR
#>
#> Parameters: Coefficent matrice and Covariance matrix
#> Prior: MN_VAR
#> # Type '?bvhar_minnesota' in the console for some help.
#> ========================================================
#>
#> 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
(sim_mnvhar_coef(bvhar_var_spec))
#> $coefficients
#> [,1] [,2]
#> [1,] 0.06349 -0.2954
#> [2,] -0.01177 0.4277
#> [3,] 0.05543 -0.2689
#> [4,] 0.00235 0.0574
#> [5,] -0.05121 0.1928
#> [6,] -0.00891 0.0245
#>
#> $covmat
#> [,1] [,2]
#> [1,] 0.322 -0.407
#> [2,] -0.407 3.195
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
))
#> Model Specification for BVHAR
#>
#> Parameters: Coefficent matrice and Covariance matrix
#> Prior: MN_VHAR
#> # Type '?bvhar_minnesota' in the console for some help.
#> ========================================================
#>
#> 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
(sim_mnvhar_coef(bvhar_vhar_spec))
#> $coefficients
#> [,1] [,2]
#> [1,] -0.01953 0.2058
#> [2,] 0.14585 -0.2401
#> [3,] 0.01613 -0.0491
#> [4,] 0.05043 -0.1112
#> [5,] 0.02742 0.3732
#> [6,] -0.00569 -0.0749
#>
#> $covmat
#> [,1] [,2]
#> [1,] 0.566 0.263
#> [2,] 0.263 14.995