predict: Forecasting Multivariate Time Series

Description

Forecasts multivariate time series using given model.

Usage

# S3 method for varlse
predict(object, n_ahead, level = 0.05, ...)
# S3 method for vharlse
predict(object, n_ahead, level = 0.05, ...)
# S3 method for bvarmn
predict(object, n_ahead, n_iter = 100L, level = 0.05, num_thread = 1, ...)
# S3 method for bvharmn
predict(object, n_ahead, n_iter = 100L, level = 0.05, num_thread = 1, ...)
# S3 method for bvarflat
predict(object, n_ahead, n_iter = 100L, level = 0.05, num_thread = 1, ...)
# S3 method for bvarldlt
predict(
  object,
  n_ahead,
  level = 0.05,
  stable = FALSE,
  num_thread = 1,
  sparse = FALSE,
  med = FALSE,
  warn = FALSE,
  ...
)
# S3 method for bvharldlt
predict(
  object,
  n_ahead,
  level = 0.05,
  stable = FALSE,
  num_thread = 1,
  sparse = FALSE,
  med = FALSE,
  warn = FALSE,
  ...
)
# S3 method for 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 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 predbvhar
print(x, digits = max(3L, getOption("digits") - 3L), ...)
is.predbvhar(x)
# S3 method for predbvhar
knit_print(x, ...)

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

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

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).

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.