Normal-inverse-Wishart Matrix

We provide functions to generate matrix-variate Normal andinverse-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 givesnum_sim x dimmatrix. 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 ofsim_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 meanexistence) 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 ofsim_mniw(), it returns list withmn (stacked MN matrices) andiw (stacked IWmatrices). Eachmn andiw 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.

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 theRW or WN
- Governs the relative importance of the prior beliefs w.r.t. theinformation contained in the data
  - If\(\lambda = 0\), then posterior= prior and the data do not influence the estimates.
  - If\(\lambda = \infty\), thenposterior expectations = OLS.
- Choose in relation to the size of the system (Bańbura et al. (2010))
  - Asm increases,\(\lambda\) should be smaller to avoidoverfitting (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 vectorlambda = .2,# lambdadelta =rep(1,4),# 4-dim delta vectoreps =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 generateboth\(A\) and\(\Sigma\) matrices.
Inbayes_spec, onlyset_bvar() works.
Iffull = 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,]  0.996255 -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

BVHAR

sim_mnvhar_coef(bayes_spec, full = TRUE) generates BVHARmodel 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
- set_bvhar()
- set_weight_bvhar()
There isfull = TRUE, too.

BVHAR-S

(bvhar_var_spec<-set_bvhar(sigma =c(1.2,2.3),# sigma vectorlambda = .2,# lambdadelta =c(.3,1),# 2-dim delta vectoreps =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

(sim_mnvhar_coef(bvhar_var_spec))#> $coefficients#>          [,1]    [,2]#> [1,]  0.30349 -0.2954#> [2,] -0.01177  1.2277#> [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 vectorlambda = .2,# lambdaeps =1e-04,# very small numberdaily =c(.5,1),# 2-dim daily weight vectorweekly =c(.2, .3),# 2-dim weekly weight vectormonthly =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

(sim_mnvhar_coef(bvhar_vhar_spec))#> $coefficients#>          [,1]    [,2]#> [1,]  0.38047  0.2058#> [2,]  0.14585  0.5599#> [3,]  0.09613 -0.0491#> [4,]  0.05043  0.0088#> [5,]  0.05409  0.3732#> [6,] -0.00569 -0.0482#>#> $covmat#>       [,1]   [,2]#> [1,] 0.566  0.263#> [2,] 0.263 14.995

Movatterモバイル変換

Minnesota Prior

Normal-inverse-Wishart Matrix

Minnesota Prior

BVAR

BVHAR

BVHAR-S

BVHAR-L