Introduction to bvhar

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)

library(bvhar)

Data

Looking at VAR and VHAR, you can learn how the models work and how to perform this package.

ETF Dataset

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

h-step ahead forecasting

For 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)

• T: Total number of observation
• p: VAR lag
• m: Dimension of variable
• s = T - p
• k = m * p + 1 if constant term, k = m * p without constant term

Models

VAR

This package indentifies VAR(p) model by

$$\Y_t = \bc + \B_1 \Y_{t - 1} + \ldots + \B_p +\Y_{t - p} + \E_t$$

where $\E_t \sim N(\mathbf{0}_k, \Sigma_e)$

var_lag <- 5

The package perform VAR(p = r var_lag) based on

$$Y_0 = X_0 A + Z$$

where

$$ Y_0 = \begin{bmatrix} \by_{p + 1}^T \ \by_{p + 2}^T \ \vdots \ \by_n^T \end{bmatrix}{s \times m} \equiv Y{p + 1} \in \R^{s \times m} $$

by build_y0()

and

$$ X_0 = \left[\begin{array}{c|c|c|c} \by_p^T & \cdots & \by_1^T & 1 \ \by_{p + 1}^T & \cdots & \by_2^T & 1 \ \vdots & \vdots & \cdots & \vdots \ \by_{T - 1}^T & \cdots & \by_{T - p}^T & 1 \end{array}\right]{s \times k} = \begin{bmatrix} Y_p & Y{p - 1} & \cdots & \mathbf{1}_{T - p} \end{bmatrix} \in \R^{s \times k} $$

by build_design(). Coefficient matrix is the form of

$$ A = \begin{bmatrix} A_1^T \ \vdots \ A_p^T \ \bc^T \end{bmatrix} \in \R^{k \times m} $$

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))

The package provide S3 object.

# class---------------
class(fit_var)
# inheritance---------
is.varlse(fit_var)
# names---------------
names(fit_var)

VHAR

Consider Vector HAR (VHAR) model.

$$\Y_t = \bc + \Phi^{(d)} + \Y_{t - 1} + \Phi^{(w)} \Y_{t - 1}^{(w)} + \Phi^{(m)} \Y_{t - 1}^{(m)} + \E_t$$

where $\Y_t$ is daily RV and

$$\Y_t^{(w)} = \frac{1}{5} \left( \Y_t + \cdots + \Y_{t - 4} \right)$$

is weekly RV

and

$$\Y_t^{(m)} = \frac{1}{22} \left( \Y_t + \cdots + \Y_{t - 21} \right)$$

is monthly RV. This model can be expressed by

$$Y_0 = X_1 \Phi + Z$$

where

$$ \Phi = \begin{bmatrix} \Phi^{(d)T} \ \Phi^{(w)T} \ \Phi^{(m)T} \ \bc^T \end{bmatrix} \in \R^{(3m + 1) \times m} $$

Let $T$ be

$$ \mathbb{C}_0 \defn \begin{bmatrix} 1 & 0 & \cdots & 0 & 0 & \cdots & 0 \ 1 / 5 & 1 / 5 & \cdots & 1 / 5 & 0 & \cdots & 0 \ 1 / 22 & 1 / 22 & \cdots & 1 / 22 & 1 / 22 & \cdots & 1 / 22 \end{bmatrix} \otimes I_m \in \R^{3m \times 22m} $$

and let $\mathbb{C}_{HAR}$ be

$$ \mathbb{C}{HAR} \defn \left[\begin{array}{c|c} T & \mathbf{0}{3m} \ \hline \mathbf{0}_{3m}^T & 1 \end{array}\right] \in \R^{(3m + 1) \times (22m + 1)} $$

Then for $X_0$ in VAR(p),

$$ X_1 = X_0 \mathbb{C}{HAR}^T = \begin{bmatrix} \by{22}^T & \by_{22}^{(w)T} & \by_{22}^{(m)T} & 1 \ \by_{23}^T & \by_{23}^{(w)T} & \by_{23}^{(m)T} & 1 \ \vdots & \vdots & \vdots & \vdots \ \by_{T - 1}^T & \by_{T - 1}^{(w)T} & \by_{T - 1}^{(m)T} & 1 \end{bmatrix} \in \R^{s \times (3m + 1)} $$

This package fits VHAR by scaling VAR(p) using $\mathbb{C}_{HAR}$ (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))

# class----------------
class(fit_har)
# inheritance----------
is.varlse(fit_har)
is.vharlse(fit_har)
# complements----------
names(fit_har)

BVAR

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.

Minnesota prior

• Litterman (1986) and Bańbura et al. (2010)
• All the equations are centered around the random walk with drift.
• Prior mean: Recent lags provide more reliable information than the more distant ones.
• Prior variance: Own lags explain more of the variation of a given variable than the lags of other variables in the equation.

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))

In 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 BVAR
• bayes_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))

It is bvarmn class. For Bayes computation, it also has other class such as normaliw and bvharmod.

# class---------------
class(fit_bvar)
# inheritance---------
is.bvarmn(fit_bvar)
# names---------------
names(fit_bvar)

Flat prior

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

Then bvar_flat(y, p, bayes_spec, include_mean = TRUE):

(fit_ghosh <- bvar_flat(etf_train, bvar_lag, num_iter = 10, bayes_spec = flat_spec))

# class---------------
class(fit_ghosh)
# inheritance---------
is.bvarflat(fit_ghosh)
# names---------------
names(fit_ghosh)

BVHAR

Consider the VAR(22) form of VHAR.

$$ \begin{aligned} \Y_t = \bc & + \left( \Phi^{(d)} + \frac{1}{5} \Phi^{(w)} + \frac{1}{22} \Phi^{(m)} \right) \Y_{t - 1} \ & + \left( \frac{1}{5} \Phi^{(w)} + \frac{1}{22} \Phi^{(m)} \right) \Y_{t - 2} + \cdots \left( \frac{1}{5} \Phi^{(w)} + \frac{1}{22} \Phi^{(m)} \right) \Y_{t - 5} \ & + \frac{1}{22} \Phi^{(m)} \Y_{t - 6} + \cdots + \frac{1}{22} \Phi^{(m)} \Y_{t - 22} \end{aligned} $$

What does Minnesota prior mean in VHAR model?

• All the equations are centered around $\Y_t + \bc + \Phi^{(d)} \Y_{t - 1} + \E_t$
• RW form: shrink diagonal elements of $\Phi^{(d)}$ toward one
  • $\Phi^{(w)}$ and $\Phi^{(m)}$ to zero
• WN form: $\delta_i = 0$

For more simplicity, write coefficient matrices by $\Phi^{(1)}, \Phi^{(2)}, \Phi^{(3)}$. If we apply the prior in the same way, Minnesota moment becomes

$$ E \left[ (\Phi^{(l)}){ij} \right] = \begin{cases} \delta_i & j = i, \; l = 1 \ 0 & o/w \end{cases} \quad \Var \left[ (\Phi^{(l)}){ij} \right] = \begin{cases} \frac{\lambda^2}{l^2} & j = i \ \nu \frac{\lambda^2}{l^2} \frac{\sigma_i^2}{\sigma_j^2} & o/w \end{cases} $$

We call this VAR-type Minnesota prior or BVHAR-S.

BVHAR-S

set_bvhar(sigma, lambda, delta, eps = 1e-04) specifies VAR-type Minnesota prior.

(bvhar_spec_v1 <- set_bvhar(sig, lam, delta, eps))

bvhar_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))

This model is bvharmn class.

# class---------------
class(fit_bvhar_v1)
# inheritance---------
is.bvharmn(fit_bvhar_v1)
# names---------------
names(fit_bvhar_v1)

BVHAR-L

Set $\delta_i$ 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

• daily
• weekly
• monthly

This 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))

bayes_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

options(orig_opts)

bvhar documentation built on April 4, 2025, 5:22 a.m.