predict: Forecasting Multivariate Time Series

In bvhar: Bayesian Vector Heterogeneous Autoregressive Modeling

Forecasting Multivariate Time Series

Description

Forecasts multivariate time series using given model.

Usage

## S3 method for class 'varlse'
predict(object, n_ahead, level = 0.05, ...)

## S3 method for class 'vharlse'
predict(object, n_ahead, level = 0.05, ...)

## S3 method for class 'bvarmn'
predict(object, n_ahead, n_iter = 100L, level = 0.05, num_thread = 1, ...)

## S3 method for class 'bvharmn'
predict(object, n_ahead, n_iter = 100L, level = 0.05, num_thread = 1, ...)

## S3 method for class 'bvarflat'
predict(object, n_ahead, n_iter = 100L, level = 0.05, num_thread = 1, ...)

## S3 method for class 'bvarldlt'
predict(
  object,
  n_ahead,
  level = 0.05,
  stable = FALSE,
  num_thread = 1,
  sparse = FALSE,
  med = FALSE,
  warn = FALSE,
  ...
)

## S3 method for class 'bvharldlt'
predict(
  object,
  n_ahead,
  level = 0.05,
  stable = FALSE,
  num_thread = 1,
  sparse = FALSE,
  med = FALSE,
  warn = FALSE,
  ...
)

## S3 method for class 'bvarsv'
predict(
  object,
  n_ahead,
  level = 0.05,
  stable = FALSE,
  num_thread = 1,
  use_sv = TRUE,
  sparse = FALSE,
  med = FALSE,
  warn = FALSE,
  ...
)

## S3 method for class 'bvharsv'
predict(
  object,
  n_ahead,
  level = 0.05,
  stable = FALSE,
  num_thread = 1,
  use_sv = TRUE,
  sparse = FALSE,
  med = FALSE,
  warn = FALSE,
  ...
)

## S3 method for class 'predbvhar'
print(x, digits = max(3L, getOption("digits") - 3L), ...)

is.predbvhar(x)

## S3 method for class 'predbvhar'
knit_print(x, ...)

Arguments

object	Model object
n_ahead	step to forecast
level	Specify alpha of confidence interval level 100(1 - alpha) percentage. By default, .05.
...	not used
n_iter	Number to sample residual matrix from inverse-wishart distribution. By default, 100.
num_thread	Number of threads
stable	Filter only stable coefficient draws in MCMC records.
sparse	Apply restriction. By default, FALSE. Give CI level (e.g. .05) instead of TRUE to use credible interval across MCMC for restriction.
med	If TRUE, use median of forecast draws instead of mean (default).
warn	Give warning for stability of each coefficients record. By default, FALSE.
use_sv	Use SV term
x	Any object
digits	digit option to print

Value

predbvhar class with the following components:

process

object$process

forecast

forecast matrix

se

standard error matrix

lower

lower confidence interval

upper

upper confidence interval

lower_joint

lower CI adjusted (Bonferroni)

upper_joint

upper CI adjusted (Bonferroni)

y

object$y

n-step ahead forecasting VAR(p)

See pp35 of Lütkepohl (2007). Consider h-step ahead forecasting (e.g. n + 1, ... n + h).

Let y_{(n)}^T = (y_n^T, ..., y_{n - p + 1}^T, 1). Then one-step ahead (point) forecasting:

\hat{y}_{n + 1}^T = y_{(n)}^T \hat{B}

Recursively, let \hat{y}_{(n + 1)}^T = (\hat{y}_{n + 1}^T, y_n^T, ..., y_{n - p + 2}^T, 1). Then two-step ahead (point) forecasting:

\hat{y}_{n + 2}^T = \hat{y}_{(n + 1)}^T \hat{B}

Similarly, h-step ahead (point) forecasting:

\hat{y}_{n + h}^T = \hat{y}_{(n + h - 1)}^T \hat{B}

How about confident region? Confidence interval at h-period is

y_{k,t}(h) \pm z_(\alpha / 2) \sigma_k (h)

Joint forecast region of 100(1-\alpha)% can be computed by

\{ (y_{k, 1}, y_{k, h}) \mid y_{k, n}(i) - z_{(\alpha / 2h)} \sigma_n(i) \le y_{n, i} \le y_{k, n}(i) + z_{(\alpha / 2h)} \sigma_k(i), i = 1, \ldots, h \}

See the pp41 of Lütkepohl (2007).

To compute covariance matrix, it needs VMA representation:

Y_{t}(h) = c + \sum_{i = h}^{\infty} W_{i} \epsilon_{t + h - i} = c + \sum_{i = 0}^{\infty} W_{h + i} \epsilon_{t - i}

Then

\Sigma_y(h) = MSE [ y_t(h) ] = \sum_{i = 0}^{h - 1} W_i \Sigma_{\epsilon} W_i^T = \Sigma_y(h - 1) + W_{h - 1} \Sigma_{\epsilon} W_{h - 1}^T

n-step ahead forecasting VHAR

Let T_{HAR} is VHAR linear transformation matrix. Since VHAR is the linearly transformed VAR(22), let y_{(n)}^T = (y_n^T, y_{n - 1}^T, ..., y_{n - 21}^T, 1).

Then one-step ahead (point) forecasting:

\hat{y}_{n + 1}^T = y_{(n)}^T T_{HAR} \hat{\Phi}

Recursively, let \hat{y}_{(n + 1)}^T = (\hat{y}_{n + 1}^T, y_n^T, ..., y_{n - 20}^T, 1). Then two-step ahead (point) forecasting:

\hat{y}_{n + 2}^T = \hat{y}_{(n + 1)}^T T_{HAR} \hat{\Phi}

and h-step ahead (point) forecasting:

\hat{y}_{n + h}^T = \hat{y}_{(n + h - 1)}^T T_{HAR} \hat{\Phi}

n-step ahead forecasting BVAR(p) with minnesota prior

Point forecasts are computed by posterior mean of the parameters. See Section 3 of Bańbura et al. (2010).

Let \hat{B} be the posterior MN mean and let \hat{V} be the posterior MN precision.

Then predictive posterior for each step

y_{n + 1} \mid \Sigma_e, y \sim N( vec(y_{(n)}^T A), \Sigma_e \otimes (1 + y_{(n)}^T \hat{V}^{-1} y_{(n)}) )

y_{n + 2} \mid \Sigma_e, y \sim N( vec(\hat{y}_{(n + 1)}^T A), \Sigma_e \otimes (1 + \hat{y}_{(n + 1)}^T \hat{V}^{-1} \hat{y}_{(n + 1)}) )

and recursively,

y_{n + h} \mid \Sigma_e, y \sim N( vec(\hat{y}_{(n + h - 1)}^T A), \Sigma_e \otimes (1 + \hat{y}_{(n + h - 1)}^T \hat{V}^{-1} \hat{y}_{(n + h - 1)}) )

n-step ahead forecasting BVHAR

Let \hat\Phi be the posterior MN mean and let \hat\Psi be the posterior MN precision.

Then predictive posterior for each step

y_{n + 1} \mid \Sigma_e, y \sim N( vec(y_{(n)}^T \tilde{T}^T \Phi), \Sigma_e \otimes (1 + y_{(n)}^T \tilde{T} \hat\Psi^{-1} \tilde{T} y_{(n)}) )

y_{n + 2} \mid \Sigma_e, y \sim N( vec(y_{(n + 1)}^T \tilde{T}^T \Phi), \Sigma_e \otimes (1 + y_{(n + 1)}^T \tilde{T} \hat\Psi^{-1} \tilde{T} y_{(n + 1)}) )

and recursively,

y_{n + h} \mid \Sigma_e, y \sim N( vec(y_{(n + h - 1)}^T \tilde{T}^T \Phi), \Sigma_e \otimes (1 + y_{(n + h - 1)}^T \tilde{T} \hat\Psi^{-1} \tilde{T} y_{(n + h - 1)}) )

References

Lütkepohl, H. (2007). New Introduction to Multiple Time Series Analysis. Springer Publishing.

Corsi, F. (2008). A Simple Approximate Long-Memory Model of Realized Volatility. Journal of Financial Econometrics, 7(2), 174-196.

Baek, C. and Park, M. (2021). Sparse vector heterogeneous autoregressive modeling for realized volatility. J. Korean Stat. Soc. 50, 495-510.

Bańbura, M., Giannone, D., & Reichlin, L. (2010). Large Bayesian vector auto regressions. Journal of Applied Econometrics, 25(1).

Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B. (2013). Bayesian data analysis. Chapman and Hall/CRC.

Karlsson, S. (2013). Chapter 15 Forecasting with Bayesian Vector Autoregression. Handbook of Economic Forecasting, 2, 791-897.

Litterman, R. B. (1986). Forecasting with Bayesian Vector Autoregressions: Five Years of Experience. Journal of Business & Economic Statistics, 4(1), 25.

Ghosh, S., Khare, K., & Michailidis, G. (2018). High-Dimensional Posterior Consistency in Bayesian Vector Autoregressive Models. Journal of the American Statistical Association, 114(526).

Korobilis, D. (2013). VAR FORECASTING USING BAYESIAN VARIABLE SELECTION. Journal of Applied Econometrics, 28(2).

Korobilis, D. (2013). VAR FORECASTING USING BAYESIAN VARIABLE SELECTION. Journal of Applied Econometrics, 28(2).

Huber, F., Koop, G., & Onorante, L. (2021). Inducing Sparsity and Shrinkage in Time-Varying Parameter Models. Journal of Business & Economic Statistics, 39(3), 669-683.

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