Looking at VAR and VHAR, you can learn how the models work and how to perform this package.
This package includes some datasets. Among them, we try CBOE ETF
volatility index (etf_vix). Since this is just an example,
we arbitrarily extract a small number of variables: Gold, crude oil,
euro currency, and china ETF.
var_idx <- c("GVZCLS", "OVXCLS", "EVZCLS", "VXFXICLS")
etf <-
etf_vix %>%
dplyr::select(dplyr::all_of(var_idx))
etf
#> # A tibble: 905 × 4
#> GVZCLS OVXCLS EVZCLS VXFXICLS
#> <dbl> <dbl> <dbl> <dbl>
#> 1 21.5 36.5 13.2 30.2
#> 2 21.5 35.4 12.6 28.9
#> 3 22.3 35.5 13.1 29.1
#> 4 21.6 36.6 12.8 28.5
#> 5 21.2 35.6 13.3 29.5
#> 6 21.4 34.8 13.2 29.1
#> 7 21.6 34.0 13.2 28.7
#> 8 21.1 32.6 12.8 28.0
#> 9 20.3 33.5 12.7 28.9
#> 10 19.6 33.4 12.4 28.0
#> # ℹ 895 more rowsFor evaluation, split the data. The last 19 observations
will be test set. divide_ts() function splits the time
series into train-test set.
In the other vignette, we provide how to perform out-of-sample forecasting.
h <- 19
etf_eval <- divide_ts(etf, h) # Try ?divide_ts
etf_train <- etf_eval$train # train
etf_test <- etf_eval$test # test
# dimension---------
m <- ncol(etf)This package indentifies VAR(p) model by
Yt = c + β1Yt − 1 + … + βp + Yt − p + ϵt
where ϵt ∼ N(0k, Σe)
The package perform VAR(p = 5) based on
Y0 = X0A + Z
where
$$ Y_0 = \begin{bmatrix} \mathbf{y}_{p + 1}^T \\ \mathbf{y}_{p + 2}^T \\ \vdots \\ \mathbf{y}_n^T \end{bmatrix}_{s \times m} \equiv Y_{p + 1} \in \mathbb{R}^{s \times m} $$
by build_y0()
and
$$ X_0 = \left[\begin{array}{c|c|c|c} \mathbf{y}_p^T & \cdots & \mathbf{y}_1^T & 1 \\ \mathbf{y}_{p + 1}^T & \cdots & \mathbf{y}_2^T & 1 \\ \vdots & \vdots & \cdots & \vdots \\ \mathbf{y}_{T - 1}^T & \cdots & \mathbf{y}_{T - p}^T & 1 \end{array}\right]_{s \times k} = \begin{bmatrix} Y_p & Y_{p - 1} & \cdots & \mathbf{1}_{T - p} \end{bmatrix} \in \mathbb{R}^{s \times k} $$
by build_design(). Coefficient matrix is the form of
$$ A = \begin{bmatrix} A_1^T \\ \vdots \\ A_p^T \\ \mathbf{c}^T \end{bmatrix} \in \mathbb{R}^{k \times m} $$
This form also corresponds to the other model. Use
var_lm(y, p) to model VAR(p). You can specify
type = "none" to get model without constant term.
(fit_var <- var_lm(etf_train, var_lag))
#> Call:
#> var_lm(y = etf_train, p = var_lag)
#>
#> VAR(5) Estimation using least squares
#> ====================================================
#>
#> LSE for A1:
#> GVZCLS_1 OVXCLS_1 EVZCLS_1 VXFXICLS_1
#> GVZCLS 0.93290 0.0545 0.0659 -0.0346
#> OVXCLS -0.02367 1.0047 -0.1447 0.0324
#> EVZCLS -0.00789 0.0102 0.9810 0.0199
#> VXFXICLS -0.03868 0.0109 0.0754 0.9328
#>
#>
#> LSE for A2:
#> GVZCLS_2 OVXCLS_2 EVZCLS_2 VXFXICLS_2
#> GVZCLS -0.0781 -0.04865 0.0829 0.0561
#> OVXCLS 0.0880 0.01207 0.2729 -0.1173
#> EVZCLS 0.0195 0.00255 -0.1071 -0.0383
#> VXFXICLS 0.0896 0.04278 -0.0691 0.0419
#>
#>
#> LSE for A3:
#> GVZCLS_3 OVXCLS_3 EVZCLS_3 VXFXICLS_3
#> GVZCLS 0.0424 -0.00452 -0.03245 -0.05967
#> OVXCLS -0.0272 -0.09144 -0.05764 -0.06255
#> EVZCLS -0.0123 0.00864 0.08693 0.00252
#> VXFXICLS -0.0266 -0.04810 0.00851 -0.02137
#>
#>
#> LSE for A4:
#> GVZCLS_4 OVXCLS_4 EVZCLS_4 VXFXICLS_4
#> GVZCLS -0.00793 0.01072 -0.01513 0.0616
#> OVXCLS -0.04343 -0.00377 -0.00694 0.1445
#> EVZCLS 0.00614 -0.02278 -0.01007 0.0200
#> VXFXICLS -0.00755 -0.05555 0.08783 -0.1025
#>
#>
#> LSE for A5:
#> GVZCLS_5 OVXCLS_5 EVZCLS_5 VXFXICLS_5
#> GVZCLS 0.0728 -0.01745 -0.0886 -0.017273
#> OVXCLS 0.0104 0.07151 -0.0637 0.002018
#> EVZCLS -0.0113 0.00581 0.0202 0.000498
#> VXFXICLS -0.0155 0.04192 -0.0254 0.093984
#>
#>
#> LSE for constant:
#> GVZCLS OVXCLS EVZCLS VXFXICLS
#> 0.571 0.145 0.129 0.875
#>
#>
#> --------------------------------------------------
#> *_j of the Coefficient matrix: corresponding to the j-th VAR lagThe package provide S3 object.
# class---------------
class(fit_var)
#> [1] "varlse" "olsmod" "bvharmod"
# inheritance---------
is.varlse(fit_var)
#> [1] TRUE
# names---------------
names(fit_var)
#> [1] "coefficients" "fitted.values" "residuals" "covmat"
#> [5] "df" "m" "obs" "y0"
#> [9] "p" "totobs" "process" "type"
#> [13] "design" "y" "method" "call"Consider Vector HAR (VHAR) model.
Yt = c + Φ(d) + Yt − 1 + Φ(w)Yt − 1(w) + Φ(m)Yt − 1(m) + ϵt
where Yt is daily RV and
$$\mathbf{Y}_t^{(w)} = \frac{1}{5} \left( \mathbf{Y}_t + \cdots + \mathbf{Y}_{t - 4} \right)$$
is weekly RV
and
$$\mathbf{Y}_t^{(m)} = \frac{1}{22} \left( \mathbf{Y}_t + \cdots + \mathbf{Y}_{t - 21} \right)$$
is monthly RV. This model can be expressed by
Y0 = X1Φ + Z
where
$$ \Phi = \begin{bmatrix} \Phi^{(d)T} \\ \Phi^{(w)T} \\ \Phi^{(m)T} \\ \mathbf{c}^T \end{bmatrix} \in \mathbb{R}^{(3m + 1) \times m} $$
Let T be
$$ \mathbb{C}_0 \mathpunct{:}=\begin{bmatrix} 1 & 0 & \cdots & 0 & 0 & \cdots & 0 \\ 1 / 5 & 1 / 5 & \cdots & 1 / 5 & 0 & \cdots & 0 \\ 1 / 22 & 1 / 22 & \cdots & 1 / 22 & 1 / 22 & \cdots & 1 / 22 \end{bmatrix} \otimes I_m \in \mathbb{R}^{3m \times 22m} $$
and let ℂHAR be
$$ \mathbb{C}_{HAR} \mathpunct{:}=\left[\begin{array}{c|c} T & \mathbf{0}_{3m} \\ \hline \mathbf{0}_{3m}^T & 1 \end{array}\right] \in \mathbb{R}^{(3m + 1) \times (22m + 1)} $$
Then for X0 in VAR(p),
$$ X_1 = X_0 \mathbb{C}_{HAR}^T = \begin{bmatrix} \mathbf{y}_{22}^T & \mathbf{y}_{22}^{(w)T} & \mathbf{y}_{22}^{(m)T} & 1 \\ \mathbf{y}_{23}^T & \mathbf{y}_{23}^{(w)T} & \mathbf{y}_{23}^{(m)T} & 1 \\ \vdots & \vdots & \vdots & \vdots \\ \mathbf{y}_{T - 1}^T & \mathbf{y}_{T - 1}^{(w)T} & \mathbf{y}_{T - 1}^{(m)T} & 1 \end{bmatrix} \in \mathbb{R}^{s \times (3m + 1)} $$
This package fits VHAR by scaling VAR(p) using ℂHAR
(scale_har(m, week = 5, month = 22)). Use
vhar_lm(y) to fit VHAR. You can specify
type = "none" to get model without constant term.
(fit_har <- vhar_lm(etf_train))
#> Call:
#> vhar_lm(y = etf_train)
#>
#> VHAR Estimation====================================================
#>
#> LSE for day:
#> GVZCLS_day OVXCLS_day EVZCLS_day VXFXICLS_day
#> GVZCLS 0.87561 0.0447 0.1623 -0.03772
#> OVXCLS 0.04147 0.9942 -0.0605 -0.09361
#> EVZCLS 0.00305 0.0281 0.9206 -0.00748
#> VXFXICLS 0.01021 0.0569 0.0440 0.91713
#>
#>
#> LSE for week:
#> GVZCLS_week OVXCLS_week EVZCLS_week VXFXICLS_week
#> GVZCLS 0.01622 -0.0554 -0.1608 0.0637
#> OVXCLS -0.07093 -0.0373 0.2000 0.1034
#> EVZCLS -0.00334 -0.0414 -0.0101 0.0239
#> VXFXICLS -0.03756 -0.0787 -0.0135 0.0480
#>
#>
#> LSE for month:
#> GVZCLS_month OVXCLS_month EVZCLS_month VXFXICLS_month
#> GVZCLS 0.084981 0.00359 0.0228 -0.0299
#> OVXCLS 0.045986 0.03825 -0.1564 -0.0157
#> EVZCLS -0.000597 0.02030 0.0501 -0.0138
#> VXFXICLS 0.041648 0.01263 0.0639 -0.0371
#>
#>
#> LSE for constant:
#> GVZCLS OVXCLS EVZCLS VXFXICLS
#> 0.491 0.135 0.105 0.926
#>
#>
#> --------------------------------------------------
#> *_day, *_week, *_month of the Coefficient matrix: daily, weekly, and monthly term in the VHAR model# class----------------
class(fit_har)
#> [1] "vharlse" "olsmod" "bvharmod"
# inheritance----------
is.varlse(fit_har)
#> [1] FALSE
is.vharlse(fit_har)
#> [1] TRUE
# complements----------
names(fit_har)
#> [1] "coefficients" "fitted.values" "residuals" "covmat"
#> [5] "df" "m" "obs" "y0"
#> [9] "p" "week" "month" "totobs"
#> [13] "process" "type" "HARtrans" "design"
#> [17] "y" "method" "call"This page provides deprecated two functions examples. Both
bvar_minnesota() and bvar_flat() will be
integrated into var_bayes() and removed in the next
version.
First specify the prior using
set_bvar(sigma, lambda, delta, eps = 1e-04).
bvar_lag <- 5
sig <- apply(etf_train, 2, sd) # sigma vector
lam <- .2 # lambda
delta <- rep(0, m) # delta vector (0 vector since RV stationary)
eps <- 1e-04 # very small number
(bvar_spec <- set_bvar(sig, lam, delta, eps))
#> Model Specification for BVAR
#>
#> Parameters: Coefficent matrice and Covariance matrix
#> Prior: Minnesota
#> ========================================================
#>
#> Setting for 'sigma':
#> GVZCLS OVXCLS EVZCLS VXFXICLS
#> 3.77 10.63 2.27 3.81
#>
#> Setting for 'lambda':
#> [1] 0.2
#>
#> Setting for 'delta':
#> [1] 0 0 0 0
#>
#> Setting for 'eps':
#> [1] 1e-04
#>
#> Setting for 'hierarchical':
#> [1] FALSEIn turn,
bvar_minnesota(y, p, bayes_spec, include_mean = TRUE) fits
BVAR(p).
y: Multivariate time series data. It should be data
frame or matrix, which means that every column is numeric. Each column
indicates variable, i.e. it sould be wide format.p: Order of BVARbayes_spec: Output of set_bvar()include_mean = TRUE: By default, you include the
constant term in the model.(fit_bvar <- bvar_minnesota(etf_train, bvar_lag, num_iter = 10, bayes_spec = bvar_spec))
#> Call:
#> bvar_minnesota(y = etf_train, p = bvar_lag, num_iter = 10, bayes_spec = bvar_spec)
#>
#> BVAR(5) with Minnesota Prior
#> ====================================================
#>
#> A ~ Matrix Normal (Mean, Precision, Scale = Sigma)
#> ====================================================
#> Matrix Normal Mean for A1 part:
#> GVZCLS_1 OVXCLS_1 EVZCLS_1 VXFXICLS_1
#> GVZCLS 0.7771 0.00915 0.0628 0.0193
#> OVXCLS 0.0445 0.70884 0.1111 0.0167
#> EVZCLS 0.0104 0.01068 0.7036 0.0266
#> VXFXICLS 0.0214 0.00673 0.1044 0.7674
#>
#>
#> Matrix Normal Mean for A2 part:
#> GVZCLS_2 OVXCLS_2 EVZCLS_2 VXFXICLS_2
#> GVZCLS 0.082726 -0.006736 -0.01387 -0.0020
#> OVXCLS 0.000568 0.140781 0.02419 -0.0436
#> EVZCLS -0.004778 0.000769 0.11756 -0.0114
#> VXFXICLS 0.013424 -0.003779 0.00369 0.1063
#>
#>
#> Matrix Normal Mean for A3 part:
#> GVZCLS_3 OVXCLS_3 EVZCLS_3 VXFXICLS_3
#> GVZCLS 0.03574 -0.003825 -0.01504 -0.00581
#> OVXCLS -0.01733 0.054037 0.00196 -0.01874
#> EVZCLS -0.00391 0.000223 0.05065 -0.00168
#> VXFXICLS -0.00891 -0.006167 -0.00435 0.02176
#>
#>
#> Matrix Normal Mean for A4 part:
#> GVZCLS_4 OVXCLS_4 EVZCLS_4 VXFXICLS_4
#> GVZCLS 0.02474 -0.001932 -0.011546 0.00263
#> OVXCLS -0.00987 0.030763 0.003771 0.00845
#> EVZCLS -0.00318 -0.000206 0.027289 0.00161
#> VXFXICLS -0.00898 -0.004378 0.000232 0.00582
#>
#>
#> Matrix Normal Mean for A5 part:
#> GVZCLS_5 OVXCLS_5 EVZCLS_5 VXFXICLS_5
#> GVZCLS 0.01986 -1.47e-03 -0.00970 0.00127
#> OVXCLS -0.00357 2.10e-02 0.00339 0.01017
#> EVZCLS -0.00284 1.59e-05 0.01729 0.00159
#> VXFXICLS -0.00431 -1.70e-03 0.00263 0.01207
#>
#>
#> Matrix Normal Mean for constant part:
#> GVZCLS OVXCLS EVZCLS VXFXICLS
#> 0.7271 0.3713 0.0971 1.2139
#>
#>
#> dim(Matrix Normal precision matrix):
#> [1] 21 21
#>
#>
#> Sigma ~ Inverse-Wishart
#> ====================================================
#> IW scale matrix:
#> GVZCLS OVXCLS EVZCLS VXFXICLS
#> GVZCLS 1285 375 115 287
#> OVXCLS 375 3638 131 397
#> EVZCLS 115 131 220 126
#> VXFXICLS 287 397 126 1186
#>
#> IW degrees of freedom:
#> [1] 887
#>
#>
#> --------------------------------------------------
#> *_j of the Coefficient matrix: corresponding to the j-th BVAR lagIt is bvarmn class. For Bayes computation, it also has
other class such as normaliw and bvharmod.
# class---------------
class(fit_bvar)
#> [1] "bvarmn" "bvharmod" "normaliw"
# inheritance---------
is.bvarmn(fit_bvar)
#> [1] TRUE
# names---------------
names(fit_bvar)
#> [1] "coefficients" "fitted.values" "residuals" "mn_prec"
#> [5] "covmat" "iw_shape" "df" "m"
#> [9] "obs" "prior_mean" "prior_precision" "prior_scale"
#> [13] "prior_shape" "y0" "design" "p"
#> [17] "totobs" "type" "y" "chain"
#> [21] "iter" "burn" "thin" "call"
#> [25] "process" "spec"Ghosh et al. (2018) provides flat prior for covariance matrix,
i.e. non-informative. Use set_bvar_flat(U).
(flat_spec <- set_bvar_flat(U = 5000 * diag(m * bvar_lag + 1))) # c * I
#> Model Specification for BVAR
#>
#> Parameters: Coefficent matrice and Covariance matrix
#> Prior: Flat
#> ========================================================
#>
#> Setting for 'U':
#> # A matrix: 21 x 21
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 5000 0 0 0 0
#> [2,] 0 5000 0 0 0
#> [3,] 0 0 5000 0 0
#> [4,] 0 0 0 5000 0
#> [5,] 0 0 0 0 5000
#> [6,] 0 0 0 0 0
#> [7,] 0 0 0 0 0
#> [8,] 0 0 0 0 0
#> [9,] 0 0 0 0 0
#> [10,] 0 0 0 0 0
#> # ... with 11 more rowsThen
bvar_flat(y, p, bayes_spec, include_mean = TRUE):
(fit_ghosh <- bvar_flat(etf_train, bvar_lag, num_iter = 10, bayes_spec = flat_spec))
#> Call:
#> bvar_flat(y = etf_train, p = bvar_lag, num_iter = 10, bayes_spec = flat_spec)
#>
#> BVAR(5) with Flat Prior
#> ====================================================
#>
#> A ~ Matrix Normal (Mean, U^{-1}, Scale 2 = Sigma)
#> ====================================================
#> Matrix Normal Mean for A1 part:
#> GVZCLS_1 OVXCLS_1 EVZCLS_1 VXFXICLS_1
#> GVZCLS 0.3128 0.0460 0.0205 0.0457
#> OVXCLS 0.0440 0.4128 0.0186 0.0314
#> EVZCLS 0.0174 0.0230 0.1334 0.0410
#> VXFXICLS 0.0477 0.0464 0.0481 0.3197
#>
#>
#> Matrix Normal Mean for A2 part:
#> GVZCLS_2 OVXCLS_2 EVZCLS_2 VXFXICLS_2
#> GVZCLS 0.19811 0.00287 0.00771 0.0194
#> OVXCLS 0.01595 0.23709 0.00978 -0.0111
#> EVZCLS 0.00425 0.01205 0.11715 0.0201
#> VXFXICLS 0.02725 0.01516 0.03513 0.2162
#>
#>
#> Matrix Normal Mean for A3 part:
#> GVZCLS_3 OVXCLS_3 EVZCLS_3 VXFXICLS_3
#> GVZCLS 0.13884 -0.0153 -0.00102 0.00591
#> OVXCLS -0.00741 0.1304 0.00361 -0.02452
#> EVZCLS -0.00353 0.0079 0.10724 0.01150
#> VXFXICLS 0.00797 -0.0146 0.02633 0.14628
#>
#>
#> Matrix Normal Mean for A4 part:
#> GVZCLS_4 OVXCLS_4 EVZCLS_4 VXFXICLS_4
#> GVZCLS 0.11392 -0.01775 -0.00653 0.00884
#> OVXCLS -0.01626 0.09421 0.00195 -0.00572
#> EVZCLS -0.00847 0.00565 0.10046 0.01098
#> VXFXICLS -0.00356 -0.03067 0.02292 0.10552
#>
#>
#> Matrix Normal Mean for A5 part:
#> GVZCLS_5 OVXCLS_5 EVZCLS_5 VXFXICLS_5
#> GVZCLS 0.11282 -0.0208 -0.01028 0.01136
#> OVXCLS -0.01507 0.1004 0.00155 0.00973
#> EVZCLS -0.01252 0.0104 0.09667 0.01361
#> VXFXICLS -0.00492 -0.0215 0.02342 0.10353
#>
#>
#> Matrix Normal Mean for constant part:
#> GVZCLS OVXCLS EVZCLS VXFXICLS
#> 5.49e-03 7.82e-04 8.83e-05 9.82e-03
#>
#>
#> dim(Matrix Normal precision matrix):
#> [1] 21 21
#>
#>
#> Sigma ~ Inverse-Wishart
#> ====================================================
#> IW scale matrix:
#> GVZCLS OVXCLS EVZCLS VXFXICLS
#> GVZCLS 2483 595 209 535
#> OVXCLS 595 3580 216 578
#> EVZCLS 209 216 771 354
#> VXFXICLS 535 578 354 2472
#>
#>
#> --------------------------------------------------
#> *_j of the Coefficient matrix: corresponding to the j-th BVAR lag# class---------------
class(fit_ghosh)
#> [1] "bvarflat" "normaliw" "bvharmod"
# inheritance---------
is.bvarflat(fit_ghosh)
#> [1] TRUE
# names---------------
names(fit_ghosh)
#> [1] "coefficients" "fitted.values" "residuals" "mn_prec"
#> [5] "covmat" "iw_shape" "df" "m"
#> [9] "obs" "prior_mean" "prior_precision" "y0"
#> [13] "design" "y" "p" "type"
#> [17] "chain" "iter" "burn" "thin"
#> [21] "call" "process" "spec"Consider the VAR(22) form of VHAR.
$$ \begin{aligned} \mathbf{Y}_t = \mathbf{c}& + \left( \Phi^{(d)} + \frac{1}{5} \Phi^{(w)} + \frac{1}{22} \Phi^{(m)} \right) \mathbf{Y}_{t - 1} \\ & + \left( \frac{1}{5} \Phi^{(w)} + \frac{1}{22} \Phi^{(m)} \right) \mathbf{Y}_{t - 2} + \cdots \left( \frac{1}{5} \Phi^{(w)} + \frac{1}{22} \Phi^{(m)} \right) \mathbf{Y}_{t - 5} \\ & + \frac{1}{22} \Phi^{(m)} \mathbf{Y}_{t - 6} + \cdots + \frac{1}{22} \Phi^{(m)} \mathbf{Y}_{t - 22} \end{aligned} $$
What does Minnesota prior mean in VHAR model?
For more simplicity, write coefficient matrices by Φ(1), Φ(2), Φ(3). If we apply the prior in the same way, Minnesota moment becomes
$$ E \left[ (\Phi^{(l)})_{ij} \right] = \begin{cases} \delta_i & j = i, \; l = 1 \\ 0 & o/w \end{cases} \quad \mathrm{Var}\left[ (\Phi^{(l)})_{ij} \right] = \begin{cases} \frac{\lambda^2}{l^2} & j = i \\ \nu \frac{\lambda^2}{l^2} \frac{\sigma_i^2}{\sigma_j^2} & o/w \end{cases} $$
We call this VAR-type Minnesota prior or BVHAR-S.
set_bvhar(sigma, lambda, delta, eps = 1e-04) specifies
VAR-type Minnesota prior.
(bvhar_spec_v1 <- set_bvhar(sig, lam, delta, eps))
#> Model Specification for BVHAR
#>
#> Parameters: Coefficent matrice and Covariance matrix
#> Prior: MN_VAR
#> ========================================================
#>
#> Setting for 'sigma':
#> GVZCLS OVXCLS EVZCLS VXFXICLS
#> 3.77 10.63 2.27 3.81
#>
#> Setting for 'lambda':
#> [1] 0.2
#>
#> Setting for 'delta':
#> [1] 0 0 0 0
#>
#> Setting for 'eps':
#> [1] 1e-04
#>
#> Setting for 'hierarchical':
#> [1] FALSEbvhar_minnesota(y, har = c(5, 22), bayes_spec, include_mean = TRUE)
can fit BVHAR with this prior. This is the default prior setting.
Similar to above functions, this function will be also integrated into
vhar_bayes() and removed in the next version.
(fit_bvhar_v1 <- bvhar_minnesota(etf_train, num_iter = 10, bayes_spec = bvhar_spec_v1))
#> Call:
#> bvhar_minnesota(y = etf_train, num_iter = 10, bayes_spec = bvhar_spec_v1)
#>
#> BVHAR with Minnesota Prior
#> ====================================================
#>
#> Phi ~ Matrix Normal (Mean, Scale 1, Scale 2 = Sigma)
#> ====================================================
#> Matrix Normal Mean for day:
#> GVZCLS_day OVXCLS_day EVZCLS_day VXFXICLS_day
#> GVZCLS 0.7808 0.00502 0.0595 0.0181
#> OVXCLS 0.0419 0.75268 0.1293 -0.0129
#> EVZCLS 0.0109 0.00957 0.7248 0.0213
#> VXFXICLS 0.0252 0.00347 0.1003 0.8042
#>
#>
#> Matrix Normal Mean for week:
#> GVZCLS_week OVXCLS_week EVZCLS_week VXFXICLS_week
#> GVZCLS 0.1160 -0.007669 -0.03216 -0.00181
#> OVXCLS -0.0211 0.152822 0.02794 -0.00136
#> EVZCLS -0.0103 -0.000372 0.13214 -0.00158
#> VXFXICLS -0.0177 -0.010393 0.00229 0.10274
#>
#>
#> Matrix Normal Mean for month:
#> GVZCLS_month OVXCLS_month EVZCLS_month VXFXICLS_month
#> GVZCLS 0.05269 -0.00345 -0.0110 -0.00302
#> OVXCLS 0.00556 0.05169 -0.0225 -0.01051
#> EVZCLS -0.00441 0.00410 0.0502 -0.00138
#> VXFXICLS 0.00733 -0.00321 0.0109 0.00573
#>
#>
#> Matrix Normal Mean for constant part:
#> GVZCLS OVXCLS EVZCLS VXFXICLS
#> 0.6107 0.1410 0.0741 1.1544
#>
#>
#> dim(Matrix Normal precision matrix):
#> [1] 13 13
#>
#>
#> Sigma ~ Inverse-Wishart
#> ====================================================
#> IW scale matrix:
#> GVZCLS OVXCLS EVZCLS VXFXICLS
#> GVZCLS 1268 366 114 286
#> OVXCLS 366 3742 131 378
#> EVZCLS 114 131 219 121
#> VXFXICLS 286 378 121 1189This model is bvharmn class.
# class---------------
class(fit_bvhar_v1)
#> [1] "bvharmn" "bvharmod" "normaliw"
# inheritance---------
is.bvharmn(fit_bvhar_v1)
#> [1] TRUE
# names---------------
names(fit_bvhar_v1)
#> [1] "coefficients" "fitted.values" "residuals" "mn_prec"
#> [5] "covmat" "iw_shape" "df" "m"
#> [9] "obs" "prior_mean" "prior_precision" "prior_scale"
#> [13] "prior_shape" "y0" "design" "p"
#> [17] "week" "month" "totobs" "type"
#> [21] "HARtrans" "y" "chain" "iter"
#> [25] "burn" "thin" "call" "process"
#> [29] "spec"Set δi for weekly and monthly coefficient matrices in above Minnesota moments:
$$ E \left[ (\Phi^{(l)})_{ij} \right] = \begin{cases} d_i & j = i, \; l = 1 \\ w_i & j = i, \; l = 2 \\ m_i & j = i, \; l = 3 \end{cases} $$
i.e. instead of one delta vector, set three vector
dailyweeklymonthlyThis is called VHAR-type Minnesota prior or BVHAR-L.
set_weight_bvhar(sigma, lambda, eps, daily, weekly, monthly)
defines BVHAR-L.
daily <- rep(.1, m)
weekly <- rep(.1, m)
monthly <- rep(.1, m)
(bvhar_spec_v2 <- set_weight_bvhar(sig, lam, eps, daily, weekly, monthly))
#> Model Specification for BVHAR
#>
#> Parameters: Coefficent matrice and Covariance matrix
#> Prior: MN_VHAR
#> ========================================================
#>
#> Setting for 'sigma':
#> GVZCLS OVXCLS EVZCLS VXFXICLS
#> 3.77 10.63 2.27 3.81
#>
#> Setting for 'lambda':
#> [1] 0.2
#>
#> Setting for 'eps':
#> [1] 1e-04
#>
#> Setting for 'daily':
#> [1] 0.1 0.1 0.1 0.1
#>
#> Setting for 'weekly':
#> [1] 0.1 0.1 0.1 0.1
#>
#> Setting for 'monthly':
#> [1] 0.1 0.1 0.1 0.1
#>
#> Setting for 'hierarchical':
#> [1] FALSEbayes_spec option of bvhar_minnesota() gets
this value, so you can use this prior intuitively.
fit_bvhar_v2 <- bvhar_minnesota(
etf_train,
num_iter = 10,
bayes_spec = bvhar_spec_v2
)
fit_bvhar_v2
#> Call:
#> bvhar_minnesota(y = etf_train, num_iter = 10, bayes_spec = bvhar_spec_v2)
#>
#> BVHAR with Minnesota Prior
#> ====================================================
#>
#> Phi ~ Matrix Normal (Mean, Scale 1, Scale 2 = Sigma)
#> ====================================================
#> Matrix Normal Mean for day:
#> GVZCLS_day OVXCLS_day EVZCLS_day VXFXICLS_day
#> GVZCLS 0.7670 0.00504 0.0644 0.01884
#> OVXCLS 0.0497 0.73098 0.0999 -0.00317
#> EVZCLS 0.0122 0.00907 0.7094 0.02036
#> VXFXICLS 0.0259 0.00429 0.0976 0.79368
#>
#>
#> Matrix Normal Mean for week:
#> GVZCLS_week OVXCLS_week EVZCLS_week VXFXICLS_week
#> GVZCLS 0.1259 -0.008055 -0.03361 -0.00353
#> OVXCLS -0.0222 0.168704 0.01523 -0.00218
#> EVZCLS -0.0108 -0.000896 0.14770 -0.00294
#> VXFXICLS -0.0199 -0.010546 -0.00238 0.11358
#>
#>
#> Matrix Normal Mean for month:
#> GVZCLS_month OVXCLS_month EVZCLS_month VXFXICLS_month
#> GVZCLS 0.06476 -0.00349 -0.01286 -0.00584
#> OVXCLS 0.00142 0.06915 -0.03315 -0.01117
#> EVZCLS -0.00492 0.00329 0.06611 -0.00337
#> VXFXICLS 0.00474 -0.00312 0.00592 0.01821
#>
#>
#> Matrix Normal Mean for constant part:
#> GVZCLS OVXCLS EVZCLS VXFXICLS
#> 0.5517 0.0335 0.0834 1.0025
#>
#>
#> dim(Matrix Normal precision matrix):
#> [1] 13 13
#>
#>
#> Sigma ~ Inverse-Wishart
#> ====================================================
#> IW scale matrix:
#> GVZCLS OVXCLS EVZCLS VXFXICLS
#> GVZCLS 1194 372 115 286
#> OVXCLS 372 3124 125 387
#> EVZCLS 115 125 193 119
#> VXFXICLS 286 387 119 1123