Forecasting Multivariate Time Series

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 'bvarssvs'
predict(object, n_ahead, level = 0.05, ...)

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

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

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

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

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

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

# S3 method for class 'bvharsv'
predict(
  object,
  n_ahead,
  level = 0.05,
  num_thread = 1,
  use_sv = TRUE,
  sparse = 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
sparse: Apply restriction. By default, FALSE. Give CI level (e.g. .05) instead of TRUE to use credible interval across MCMC for restriction.
warn: Give warning for stability of each coefficients record. By default, FALSE.
use_sv: Use SV term
x: predbvhar 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, $$y_{n + h} \mid \Sigma_e, y \sim N( vec(y_{(n + h - 1)}^T \tilde{T}^T \Phi), \Sigma_e \otimes (1 + y_{(n + h - 1)}^T \tilde{T} \hat\Psi^{-1} \tilde{T} y_{(n + h - 1)}) )$$

n-step ahead forecasting VAR(p) with SSVS and Horseshoe

The process of the computing point estimate is the same. However, predictive interval is achieved from each Gibbs sampler sample.

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

n-step ahead forecasting VHAR with SSVS and Horseshoe

The process of the computing point estimate is the same. However, predictive interval is achieved from each Gibbs sampler sample.

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

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).

George, E. I., Sun, D., & Ni, S. (2008). Bayesian stochastic search for VAR model restrictions. Journal of Econometrics, 142(1), 553-580.

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.